S and i^ are parallel to the
axes of co-ordinates, and the moments of inertia about those in the
plane of xy are respectively A and B + M . 0H~\^ = A, and these
being equal, any straight line through S or H in the plane of xy
is a principal axis at that point, and the moment of inertia about
it is equal to A.
If P be any point in the plane of xy, then one of the principal
axes at P will be perpendicular to the plane xy. For if ]j, q be
the co-ordinates of P, the conditions that this line is a principal
axis are
2w {x —p) z= Q
2m {y-q)z =
which are obviously satisfied because the centre of gravity is the
origin, and the principal axes the axes of co-ordinates.
The other two principal axes may be found thus. If two
straight lines meeting at a point P be such that the moments of
inertia about them are equal, then provided they are in ajrincr^
__paJ_plana the principal axes at P bisect the angles between these
two straight lines. For if with centre P we describe the momental
ellipse, then the axes of this ellipse bisect the angles between any
two equal radii vectores.
Join 8P and HP; the moments of inertia about SP, HP are
each equal to A. Hence, if PG and PT are the internal and
3—2
86 MOMENTS OF INERTIA.
external bisectors of the angle SFH; PG, FT are the principal
axes at F. If therefore with S and H as foci we describe any
ellipse or hyperbola, the tangent and normal at any point are the
Ijrincipal axes at that point.
53. Take any straight line MN through the origin, making
an angle 6 with the axis of x. Draw 8M, HN perpendiculars on
MN. The moment of inertia about it is
= Aco^^e + B sin' 9
= A-{A-B) sin'
= A-AI.(08smey
==A-M.8M\
Through P draw FT parallel to MN, and let 8Y and HZ be the
perpendiculars from 8 and // on it. The moment of inertia about
FT is then
= moment about MN+ M. MY^
= A-{-M{MY-8M){MY+8M)
= A + M.8Y.HZ.
In the same way it may be proved that the moment of inertia
about a line FG passing between H and 8 is less than A by the
mass into the product of the perpendiculars from 8 and H on FG.
If therefore tuith S a7id H as foci we describe any ellipse or
hyperbola, the moments of inertia,^ about any tc^ngent to. either of_
these curves is constant. •^•^^•^«^- ^^-i , J'^'^' ^-r.v: -tj^.^o^'- 'jn^/m^rr
It follows from this th^ the moments of inertia about the
principal axes at F are equal to B + M('
8F + HFY
\ 2
For if a and b be the axes of the ellipse we have a^ — h^ = 08"
A-B
M
and hence
A+M.8Y. HZ^A + Mb' = B + Ma' = B + j|^/ ^^ + -^^ V
and the hyperbola may be treated in a similar manner.
54. This reasoning may be extended to points lying in any
given plane passing through the centre of gravity of the body.
Let Ox, Oy be the axes in the given plane such that the product
of inertia about them is zero (Art. 33). Construct the points >S^
and H as before, so that 08'^ and OH^ are each equal to the
difference of the moments of inertia about Ox and Oy divided by
the mass. Draw 8y' a parallel through 8 to the axis of y, the
PRINCIPAL AXES. ' 87
product of inertia about Sx, Sy' is equal to that about Ox, Oy
together with the product of inertia of the whole mass collected
at 0. Both these are zero, hence the section of the momenta]
ellipsoid at >S' is a circle, and the moment of inertia about every
straight line through 8 in the plane xOy is the same and equal
to that about Ox. We can then show that the moments of
inertia about PH and PS are equal ; so that PG, PT, the internal
and external bisectors of the angle SPH are the principal dia-
meters of the section of the momental ellipsoid at P by the given
plane. And it also follows that the moments of inertia about the
tangents to a conic whose foci are >S^ and H are the same.
55. Ex. 1. To find the foci of inertia of an elliptic area. The moments of
inertia ahout the major and minor axes are M -r and ill -7 . Hence the minor axis
4 4
is the axis of greatest moment. The foci of inertia therefore lie in the minor axis
at a distance from the centre =- sja'^ - b'^, i.e. half the distance of the geometrical
foci from the centre.
Ex. 2. Two particles each of mass m are placed at the extremities of the minor
axis of an elliptic area of mass M. Prove that the principal axes at any point of
the circumference of the ellipse will be the tangent and normal to the ellipse, pro-
., T »i 5 e^
Tided in- = 7^ 1 — T-, •
31 8 1 - 2e^
Ex. 3. At the points which have been called foci of inertia two of the principal
moments are equal. Show that it is not in general true that a point exists such
that the moments of inertia about all axes through it are the same, and find the con-
ditions that there may be such a point.
Eefer the body to the principal axes at the centre of gravity. Let P be the point
required, (x, y, z) its co-ordinates. Since the momental ellipsoid at P is to be a
sphere, the products of inertia about all rectangular axes meeting at P are zero.
Hence, by Art. 13, xi/ — 0, yz = 0, zx=0. It follows that two of the three x, y, z
must be zero, so that the point must be on one of the principal axes at the centre
of gravity. Let this be called the axis of z. Since the moments of inertia about
three axes at P parallel to the co-ordinate axes are A + Mz^, B + Mz^ and C, we see
that these cannot be equal unless A — B and each is less than C There are then
two points on the axis of unequal moment which are equimomental for all axes.
[Poisson and Binet.]
J 56. Given the positions of the iwincipal axes at the centre of
\^/ gravity and the inoments of inertia about them, to find the
positions of the principal axes^', and the principal moments at any
other point P.
Let the body be referred to its principal axes at the centre of
gravity 0, let A, B, C be its principal moments, the mass of the
* Some of the following theorems were given by Sir William Thomson and
Mr Townscnd, in two articles which appeared at the same time in the Mdthcmallcal
Journal, 1840. Tlicir demonstrations arc different from those given in this treatise.
88 MOMENTS OF INERTIA.
bod}'' being taken as unity. Construct a quadric confocal with
the ellipsoid of gyration, and let the squares of its semi-axes be
a^= A +\, 1/'= B + \ c^= C + X. Let us find the moment of
inertia with regard to any tangent plane.
Let (a, /S, 7) be the direction angles of the perpendicular to
any tangent plane. The moment of inertia, with regard to a
parallel plane through 0, is
^ {A cos^ a + B cos'/3 + Ccos^y).
The moment of inertia, with regard to the tangent plane, is
formed by adding the square of the perpendicular distance be-
tween the planes, viz.
{A + X) cos" 0L + [B + \) cos'/3 + (C + X) cos'^y,
we get
moment of inertia with re-] A + B -\- C
- + A
+ d\
gard to a tangent plane] 2
B+ G-A
Thus the moments of inertia with regard to all tangent planes to
any one quadric confocal with the ellipsoid of gyration are the
same.
These planes are all principal planes at the point of contact.
For draw any plane through the point of contact F, then in the
case in which the confocal is an ellipsoid, the tangent plane
parallel to this plane is more remote from the origin than this
plane. Therefore, the moment of inertia with regard to any plane
through P is less than the moment of inertia with regard to a
tangent plane to the confocal ellipsoid through P. That is, the
tangent plane to the ellipsoid is the principal plane of greatest
moment. In the same way the tangent plane to the confocal
hyperboloid of two sheets through P is the principal plane of
least moment. It follows that the tangent plane to the confocal
hyperboloid of one sheet is the principal plane of mean moment.
Through a given point P, three confocals can be drawn, the
normals to these confocals are, by Art. 16, the principal axes at P.
By Art. 5, Ex. 3, the principal axis of least moment is normal
to the confocal ellipsoid and of greatest moment normal to the
confocal hyperboloid of two sheets.
57. The moment of inertia with regard to the point P is, by
A -\- B -\- G
Art. 14, ^ h OP^. Hence, by Art. 5, Ex. 3, the moments
•k
1/W
•i
[it
PRINCIPAL AXES. "* 89
of inertia about the normals to the three confocals through P
whose parameters are \, \, X^ are respectively
0P'-\, 0F'-\, 0P'-\.
58. If we describe any other confocal and draw a tangent
cone to it whose vertex is P, the axes of this cone are known to
be the normals to the three confocals through P. This gives
another construction for the principal axes at P.
If this confocal diminish without limit, until it becomes a
focal conic, then the principal diameters of the system at P are
the principal diameters of a cone whose vertex is P and base
a focal conic of the ellipsoid of gyration at the centre of gravity.
59. If we wish to use only one quadric, we may consider the
confocal ellipsoid through P. We know* that the normals to the
* These propositions are to be foiind in books on Solid Geometry, they may also
be proved as follows.
Let the confocal ellipsoid pass near P and approach it indefinitely. The base
of the enveloping cone is iiltimately the Indicatrix ; and as the cone becomes ulti-
mately a tangent plane, one of its axes is ultimately a perpendicular to the plane of
the Indicatrix. Now in any cone two of its axes are parallel to the principal diame-
ters of any section perpendicular to the third axis. Hence the axes of the envelop-
ing cone are the normal to the surface and parallels to the principal diameters of
the Indicatrix. But all parallel sections of an ellipsoid are similar and similarly
situated, hence the principal diameters of the Indicatrix are parallel to the princi-
pal diameters of the diametral section parallel to the tangent plane at P.
To find the principal moments, we may reason as follows. Let a tangent plane
to the ellipsoid be drawn perpendicular to any radius vector OQ of the diametral
section of OP, then the point of contact T, OQ, and OP will lie in one plane when
40 MOMENTS OF INERTIA.
other two confocals are tangents to the lines of curvature on the
ellipsoid, and are also parallel to the principal diameters of the
diametral section made by a plane parallel to the tangent plane
at P. And if D^D^ be these principal semi-diameters, we know
that
Hence, if through any point P we describe the quadric
+ -77^+^rT-r=l>
A+\ B+X C+X
the axes- of co-ordinates being the principal axes at the centre of
gravity, then the principal axes at P are the normal to this
quadric, and parallels to the axes of the diametral section made
by a plane parallel to the tangent plane at P, And if these axes
be 2Z)j and 2,D^, the principal moments at P are
0P'-\ OP'-\ + D^', OP'-\ + P,\
Ex. If two bodies have the same centre of gravity, the same principal axes at
the centre of gravity and the differences of their principal moments equal, each to
each, then these bodies have the same principal axes at all points.
60. The axes of co-ordinates being the principal axes at the
centre of gravity it is required to express the condition that any
given straight line may he a principal axis at some point in its
length and to find that point.
Let the equations to the given straight line be
^-f^y-9^^^-h _ _ _ ,^^
I m n °
OQ is an axis of the section. For draw through T a section parallel to the diame-
tral section, and let 0' be its centre, and let O'Y' be a perpendicular from 0' on the
tangent plane, which touches at T. Then OQ, O'Y' and OP are in one plane.
Now consider the section whose centre is 0' ; O'Y' is the perpendicular on the tan-
gent to an elhpse whose point of contact is T. Hence O'Y', O'T' do not coincide
unless O'Y be the direction of the axis of the ellipse. But this section is similar
to the diametral section to which it was drawn pai'allel. Hence OQ is an axis of
the diametral section.
Let PR be a straight line drawn through P parallel to OQ' to meet in P the
tangent plane which touches in T. Then RP, RT are two tangents at right angles
to the ellipse PQT. Hence
OR^ = simi of the squares of th-e semi-axes of the ellipse
because OP, OQ are conjugate diameters.
The moment of inertia about PR, a perpendicular to a tangent plane, has been
proved above to be OR"^ - X, hence the moment of inertia about a parallel through P
to the axis OQ is OP^ + qq-i _ ^_
PRINCIPAL AXES. ^ 41
then it must be a normal to the quadric
at the point at which the straight line is a principal axis.
Hence comparing the equation to the normal to (2) with (1),
we have
these six equations must be satisfied by the same values of x, y, z,
X and //..
Substituting for x, y, z from (3) in (1), we get
'^ I m n
eliminating /x from these last equations we have
f_ff l-^ ^1-i
I m _ m n _ n I _ ^ , >.
JT^^'B^^~U^~A~^ ^^'
This clearly amounts to only one equation, and is the required
condition that the straight line should be a principal axis at some
point in its length.
Substituting for x, y, z from (3) in (2), we have
which gives one value only to \. The values of X and /i having
been found, equations (3) will determine x, y, z, the co-ordinates
of the point at which the straight line is a principal axis.
The geometrical meaning of this condition may be found by
the following considerations, which were given by Mr Townsend
in the Mathematical Journal. The normal and tangent plane at
every point of a quadric will meet any principal plane in a point
and a straight line, which are pole and polar with regard to the
focal conic in that plane. Hence to find whether any assumed
straight line is a principal axis or not, draw any plane perpen-
dicular to the straight line and produce both the straight line
and the plane to meet any principal plane at the centre of gravity.
If the line of intersection of the plane be parallel to the polar
line of the point of intersection of the straight line with respect
to the focal conic, the axis will be a principal axis, if otherwise it
will not be so. And the point at which the assumed straight line
is a principal axis may be found by drawing a plane through the
42 MOMENTS OF INERTIA.
polar line perpendicular to tHe straight line. The point of inter-
section is the required point.
The analytical condition (4) exactly expresses the fact that the
polar line is parallel to the intersection of the plane.
61. Ex. 1. Given a plane - + - + --1 = 0, there is always some point in it
J 9 h
at wliicli it is a principal plane. Also this point is its intersection with the straight
line fx-A=gy — B=hz - C.
Ex. 2. Let two points P, Q be so situated that a principal axis at P intersects a
principal axis at Q. Then if two planes be drawn at P and Q perpendicular to
these principal axes, their intersection will be a principal axis at the point where
it is cut by the plane containing the principal axes at P and Q. [Mr Town send.]
For let the principal axes at P, Q meet any principal plane at the centre of
gravity in p, q, and let the perpendicular planes cut the same principal plane in
LN, MN. Also let the perpendicular planes intersect each other in RN. Then
RN is perpendicular to the plane containing the points P, Q, p, q. Also since the
polars of p and q are LN, MN, it follows that pq is the polar of the point N. Hence
the straight line RN satisfies the criterion of the last Article.
Ex. 3. If P be any point in a principal plane at the centre of gravity, then
every axis which passes through P, and is a principal axis at some pomt, lies in one
of two perpendicular planes. One of these planes is the principal plane at the
centre of gravity, and the other is a plane perpendicular to the polar line of P with
regard to the focal conic. Also the locus of all the points Q at which QP is a prin-
cipal axis is a circle passing through P and having its centre in the principal plane.
[Mr Townsend.]
Ex. 4. The edge of regression of the developable surface which is the envelope
of the normal planes of any line of curvature drawn on a confocal quadric is a
curve such that all its tangents are principal axes at some point in each.
62. To find the locus of the points at which two principal
moments of inertia are equal to each other.
The principal moments at any point P are
I^=OF'-\, I^=OP'-\ + D^^ I^= OP'-\ + D^\
If we equate I^ and I^ we have D^ = 0, and the point F must
lie on the elhptic focal conic of the ellipsoid of gyration.
If we equate Z^ and I^ we have D^ = D^, so that P is an um-
bilicus of any ellipsoid confocal with the ellipsoid of gyration. The
locus of these umbilici is the hyperbolic focal conic.
In the first of these cases we have X = — C, and I)^ is the semi-
diameter of the focal conic conjugate to OF. Hence -0./+ OF'^ =
sum of squares of semi-axes = ^1 — C -f P — C. The three prin-
cipal moments are therefore I^ = I^= OF"^ + G, 1^ = A + B—C, and
the axis of unequal moment is a tangent to the focal conic.
The second case may be treated in the same way by using
a confocal hyperboloid, we therefore have I^—I^—OF^ + B,
PRINCIPAL AXES. 43
I^ = A + C — B, and the axis of unequal moment is a tangent
to the focal conic.
63. To find the curves on any confocal quadric at which a
principal moment of inertia is equal to a given quantity I.
Firstly. The moment of inertia about a normal to a confocal
quadric is OP^ — X. If this be constant, we have OP constant,
and therefore the required curve is the intersection of that
quadric with any concentric sphere. Such a curve is a sphero-conic.
Secondly. Let us consider those points at which the moment
of inertia about a tangent is constant.
Construct any two confocals whose semi-major axes are a and
a'. Draw any two tangent planes to these which cut each other
at right angles. The moment of inertia about their intersection
is the sum of the moments of inertia with regard to the two
planes, and is therefore
= B+ C-A + a'+a".
Thus the moments of inertia about the intersections of perpendicular
tangent planes to the same confocals are the same.
Let a, a, a" be the semi-major axes of the three confocals
which meet at any point P, then since confocals cut at right
angles, the moment of inertia about the intersection of the con-
focals a , a' is
I^ = B-\-C-A + a:'^a"\
The intersection of these two confocals is a line of curvature
on either. Hence the moments of inertia about the tangents to any
line of curvature are equal to one another; and these tangents are
principal axes at the point of contact.
On the quadric a draw a tangent PT making any angles ^
and ^ — ^ with the tangents to the lines of curvature at the
point of contact P. If J^, I^ be the moments about the tangents
to these lines of curvature, the moment of inertia about the
tangent PT
= /g cos^ <^ + -^3 ^^^^ ^
= B -\- C-A+ {a" + a') cos' cf>+{a' + a") sin' cf).
But along a geodesic on the quadric a, a"sin' ^ H- a'^cos'^ is
constant. Hence the moments of inertia about the tangents to any
geodesic on the quadric arc equal to each other.
G4. Ex. 1. If a straight lino touch any two confocals whose SGini-major as.03
are a, a, the momcut of inertia about itis B + C- A + a^ + a"-.
44 MOMENTS OF INERTIA.
Ex. 2, When a body is referred to its principal axes at the centre of gravity,
show how to find the coordinates of the point P at which the three principal
moments are equal to three given quantities Iil^Jy [JuUien's Problem.]
The elhptic co-ordinates of P are evidently a^=^ (la + Ig-Zj-i?- (7+^) &c. ;
and the co-ordinates [x, y, z) may then be found by Dr Salmon's formula,
&C.
{A-B){A-C)
Ex. 3. Let two planes at right angles touch two confocals whose semi-major
axes are a, a'; and let a, a' be the values of a, a', when the confocals touch the intersec-
tion of the planes; then a^ + a"^ — a,^ + B,''^, and the product of inertia v?ith regard to
the two planes is a^a''^ - a^a'^.
65. The locus of all those points at which one of the prin-
cipal moments of inertia of the body is constant is called an equi-
momental surface.
To find the equation to such a surface we have only to put 7^
constant, this gives \ = r^ — I. Substituting in the equation to
the subsidiary quadric, the equation to the surface becomes
^^ _L '1 + - =1
x' + ^-^ + z' + A-r x' + 2/ + z'+B-I^x'+i/+z'+a-I
Throuo-h any point P on an equi-momental surface describe
the confocal quadric such that the principal axis is a tangent
to a line of curvature on the quadric. By Art. 63 one of the
intersections of the equi-momental surface and this quadric is the
line of curvature. Hence the principal axis at P about which
the moment of inertia is J is a tangent to the equi-momental
surface.
Ao-ain, construct the confocal quadric through P such that
the principal axis is a normal at P, then one of the intersections
of the momental surface and this quadric is the sphero-conic
through P. The normal to the quadric, being the principal axis,
has just been shown to be a tangent to the surface. Hence the
tangent plane to the equi-momental surface, is the plane which
contams the normal to the quadric and the tangent to the sphero-
conic.
To draw a perpendicular from the centre on this tangent
plane, we may follow Euclid's rule. Take PP' a tangent to the
sphero-conic, drop a perpendicnlar from on PP', this is the
radius vector OP, because PP is a tangent to the sphere. At P
in the tangent plane draw a perpendicular to PF, this is the
normal PQ to the quadric. From drop a perpendicidar OQ on
this normal, then OQ is a normal to the tangent plane. Hence
this construction,
If P be any ijoint on .an equi-momental surface luhose para-
meter is I and OQ a perioendicular from the centre on the tangent
PRINCIPAL AXES. 45
plane, then PQ is the j^rincipal axis at P about luhich the moment
of inertia is the constant quantity I.
The equi-momental becomes Fresnel's wave surface when
/ is greater than the greatest principal moment of inertia at the
centre of gravity. The general form of the surface is too well
known to need a minute discussion here. It consists of two
sheets, which become a concentric sphere and a spheroid when two
of the principal moments at the centre of gravity are equal.
When the principal moments are unequal, there are two singu-
larities in the surface.
(1) The two sheets meet at a point P in the plane of the
greatest and least moments. At P there is a tangent cone to
the surface. Draw any tangent plane to this cone, and let OQ
be a perpendicular from the centre of gravity on this tangent
plane. Then PQ is a principal axis at P. Thus there are an
infinite number of principal axes at P because an infinite number
of tangent planes can be drawn to the cone. But at any given
point there cannot be more than three principal axes unless two
of the principal axes be equal, and then the locus of the principal
axes is a plane. Hence the point P is situated on a focal conic,
and the locus of all the lines FQ is a normal plane to the conic.
The point Q lies on a sphere whose diameter is OP, hence the
locus of () is a circle.
(2) The two sheets have a common tangent plane which
touches the surface along the curve. This curve is a circle whose
plane is perpendicular to the plane of greatest and least moments.
Let OP' be a perpendicular from on the plane of the circle,
then P' is a point on the circle. If P be any other point on the
circle the principal axis at P is BP'. Thus there is a circular
ring of points at each of which the principal axis passes through
the same point and the moments of inertia about these principal
axes are all equal.
The equation to the equi-momental surface may also be used
for the purpose of finding the three principal moments at any
point whose co-ordinates {x, y, z) are given. If we clear the equation
of fractions, we have a cubic to determine / whose roots are the
three principal moments.
Thus let it be required to find the locus of all those points
in a body at which any symmetrical function of the three prin-
cipal moments is equal to a given quantity. We may express
this symmetrical function in terms of the coefficients by the usual
rules, and the equation to the locus is found.
Ex. 1. If an equi-momental surface cut a quadric confocal with the ellipsoid
of gyration at the centre of gravity, then the intersections are a sphero conic and a
line of curvature. But if the quadric ho an ellipsoid, hoth these cannot be I'oal.
46 MOMENTS OF INERTIA.
For if the surface cut the ellipsoid in both, let P be a point on the line of
curvature, and P' a point on the sphero-conic, then by Art. 59, 0P^ + D^ = OP'^,
which is less than A+\. But OF^ + D-^^ + I>^ = A ^ B + C + 3X, therefore I)^>B^-
C+2X, which is >J. + 2\. Hence Z>2>than the greatest radius vector of the ellip-
soid, which is impossible.
Ex. 2. Find the locus of all those points in a body at which
(1) the sum of the principal moments is equal to a given quantity I.
(2) the sum of the products of the principal moments taken two and two
together, is equal to I^.
(3) the product of the principal moments is equal to P*.
The results are
(1) a sphere whose radius is sj „ ^ , Art. 13.
(2) the surface
+ Ax' + Bf--vCz^ + AB + BG+CA i ' ' *
(3) the surface A'B'C - A 'y\^ - B'z^x^ - C'x'y^ - 2x^y''z'^ ^P,' '
where A' — A+y^ + z^, with similar expressions for B', C.
CHAPTER II.
d'alembekt's principle, &C.
66. The principles, by wliicli the motion of a single particle
under the action of given forces can be determined, will be found
discussed in any treatise on Dynamics of a Particle. These prin-
ciples are called the three laws of motion. It is shown that if
(x, y, z) be the co-ordinates of the particle at any time t referred
to three rectangular axes fixed in space, m its mass ; X^ Y, Z the
forces resolved parallel to the axes, the motion may be found by
solving the simultaneous equations,
CL QO -r-w- (Jj If "YT* Oj ^ f^
m -7- = X, on — ,f- = 1, VI -j- = Z.
dt ' dt dt
If we regard a rigid body as a collection of material particles
connected by invariable relations, we might write down the equa-
tions of the several particles in accordance with the principles just
stated. The forces on each particle are however no longer known,
some of them being due to the mutual actions of the particles, , 0^^,^'^
We assume (1) that the action between two particles is along
the line which joins them, (2) that the action and reaction be-
tween any two are equal and opposite. Suppose there are n
particles, then there will be ^n equations, and, as shown in any
treatise on Statics, 3w — 6 unknown reactions. To find the
motion it will be necessary to eliminate these unknown quanti-
ties. We may expect to find six resulting equations, and these
will be shown, a little further on, to be sufficient to determine the
motion of the body.
When there are several rigid bodies which mutually act and
re-act on each other the problem becomes still more complicated.
But it is unnecessary for us to consider in detail, either this or the
preceding case, for D'Alembert has proposed a method by which
all the necessary equations may be obtained without writing down
the equations of motion of the several particles, and without
making any assumption as to the nature of the mutual actions
except the following, which may be regarded as a natural conse-
quence of the laws of motion.
The internal actions and reactions of any system of rigid bodies
in motion are in equilibrium amongst themselves.
\y
48 d'alembert's principle.
67. To explain D'' Alembert'' s Principle.
In the application of this principle it will be convenient to
use the term effective force, which may be defined as follows.
When a particle is moving as part of a ^ Iwffld body, it is acted
on by the external impressed forces and also by the molecular
reactions of the other particles. If we considered this particle to
be separated from the rest of the body, and all these forces re-
moved, there is some one force which, under the same initial
conditions, would make it move in the same way as before. This
force is called the effective force on the particle. It is evidently
the resultant of the impressed and molecular forces on the par-
ticle.
Let r}i be the mass of the particle, {x, y, z) its co-ordinates
referred to any fixed rectangular axes at the time t. The accele-
dj^x d^v cPz
rations of the particle are -^, -T^-and -r^ . Let / be the resul-
tant of these, then, as explained in Dynamics of a Particle, the
effective force is measured by mf.
Let F be the resultant of the impressed forces, R the resultant
of the molecular forces on the particle. Then mf is the resultant
of F and R. Hence if mf be reversed, the three F, R, and mf are
in equilibrium.
We may apply the same reasoning to every particle of each
body of the system. We thus have a group of forces similar to R,
a group similar to F and a group similar to mf, these three groups
will form a system of forces in equilibrium. Now by D'Alembert's
principle the group R will itself form a system of forces in equili-
brium. Whence it follows that the group F will be in equilibrium
with the group 7nf. Hence
If forces equal to the effective forces but acting in exactly oppo-
site directions were applied at each point of the system these woidd
be in equilibrium with the impressed forces.
68. By this principle the solution of a dynamical problem is
reduced to a problem in Statics. The process would be as fol-
lows. We first choose some quantities by means of which the
position of the system in space may be fixed. We then express
the effective forces on each element in terms of these quantities.
These reversed will be in equilibrium with the given impressed
forces. Lastly, the equations of motion for each body may be
formed, as is usually done in Statics, by resolving in three direc-
tions and taking moments about three straight lines.
69. Before the publication of D'Alembert's principle a vast number of Dynami-
cal problems had been solved. These may be found scattered through the early
volumes of the Memoirs of St Petersburg, Berlin and Paris, in the works of John
d'alembert's principle. 49
Bernoulli and the Opuscules of Euler. They require for the most part the determi-
nation of the motions of several bodies with or without weight which push or pull
each other by means of threads or levers to which they are fastened or along which
they can gUde, and which having a certain impulse given them at first are then left
to themselves or are compelled to move in given lines or surfaces.
The postixlate of Huyghens, "that if any weights are put in motion by the force
of gravity they cannot move so that the centre of gravity of them all shall rise
higher than the place from which it descended," was generally one of the priuciples
of the solution : but other principles were always needed in addition to these, and
it reqitired the exercise of ingenuity and skill to detect the most suitable in each
case. Such problems were for some time a sort of trial of strength among mathe-
maticians. The Traite de Dynamique published by D'Alembert in 1743, put an end
to this kind of challenge by supplying a direct and general method of resolving or
at least throwing into equations any imaginable problem. The mechanical diffi-
culties were in this way reduced to difficulties of Pure Mathematics. See Montucla,
Vol. III. page 615, or Whewell's version of the same in his History of the Inductive
Sciences.
D'Alembert uses the following woids :— " Soient A, B, C,.&c. les corps qui com-
posent le systeme, et supposons qii'on leur ait imprime les mouvemens a, b, c, Sec.
qu'ils soient forces, k cause de leur action mutuelle, de changer dans les mouvemens
a, b, c, &c. II est clair qu'on pent regarder le mouvement a imprime au corps A
comme compost du mouvement a, qu'il a pris, et d'un autre mouvement a ; qu'on
peut de meme regarder les mouvemens h, c, &c. comme composes des mouvemens
h, p; c, 7; &c., d'ou il s'ensuit que le mouvement des corps A, B, C, &c. entr'eux
auroit ete le meme, si au lieu de leur donner les impulsions a, b, c, on leur eut
donne k-la-fois les doubles impulsions a, a; b, ,8; &c. Or par la supposition les
corps A, B, C, &c. ont pris d'eux-m^mes les mouvemens a, b, c, &c- done les mou-
vemens a, j3, 7, &c. doivent etre tels qu'ils ne derangent rien dans les mouvemens
a, b, c, &c. c'est-a-dire que si les corps n'avoient re^u que les mouvemens a, /3, 7,
&c. ces mouvemens aureient du se detruire mutuellement, et le systeme demeurer
en repos. De la resulte le principe suivant pour trouver Je mouvement de plusieurs
corps qui agissent les uns sur les autres. Decomposez ,les mouvemens a, b, c &c. im-
primes a chaque corps, chacun en deux autres a, a; b, /3; c, 7; etc. qui soient tels
que si Ton n'eut imprime aux corps que les mouvemens a, b, c, &c. ils eussent pu
conserver les mouvemens sans se nuire reciporoquement ; et que si on ne leur eut
imprimfi que les mouvemens a, /3, 7, &c. le systeme fut demeur^ en repos ; il est
clair que a, b, c, &c. seront les mouvemens que ces corps prendront en vertu de leur
action. Ce qu'il falloit trouver."
70. As an example of D'Alembert's principle let us consider
the following problem.
A heavy body is capable of motion by two hinges about a hori-
zontal axis, luhich axis is made to rotate with a uniform angular
velocity co about a vertical axis intersecting it in a point O. It is
required to find the conditions that the body may be inclined at a
constant angle to the vertical.
Let the horizontal axis which is fixed in the body be taken as
axis of y, and let two other axes also fixed in the body be taken
as a set of rectangular axes with origin 0. Let 6 be the angle
R. D. 4
50
D ALEMBERT S PRINCIPLE.
the plane of yz makes with a vertical plane through Oy. Our
object is to find the relation between 6 and a.
By h3^pothesis each particle P describes a horizontal circle
whose centre C is in the vertical through 0. If r be the radius
CP of this circle, and m the mass, the effective force on the
particle is mw^r and is directed along the radius. When reversed
this will act in the direction CP.
The impressed forces on the body are its weight which may be
supposed to act at the centre of gravity and the actions at the
hinges. To avoid these last, we shall take moments about the
axis Oy. Then the moment of the reversed effective forces toge-
ther with the moment of the weight will be zero.
Let M be the mass of the body, x, y, z the co-ordinates of the
centre of gravity, ^ its distance from the vertical plane through Oy.
The moment of the weight is Mg^. The resolved part of the
effective force parallel to Oy has no moment about Oy. The
moment of the resolved part perpendicular to the vertical plane
through Oy is mw^p if p be the distance of the particle from that
plane, The equation of moments gives if CO = u
Mg^ + tmw'pu = 0.
By projecting the co-ordinates on CO and CPwe have
u = — X sin 6 + 2 cos 0,
p = X COS ^ 4- ^ sin 0,
^ = X cos 6 + z sin 0.
Substituting we get
Mg {x COS0 + z sin 0) = a^ [^ sin 26Xm (a;^ — ^^) — cos 26't')nxz].
When the shape and structure of the body are known, the
integrals 2m (x^ — z^) and Sm xz can be found by the methods of
DALEMBERTS PRINCIPLE. 51
the preceding chapter or by direct integration. This equation''
will then give the required relation between 6 and o).
It may be noticed that the only manner in which the form of
the body enters into the equation is through its moments and
products of inertia. If we change the body into any equi-mo-
mental one, the equation connecting 6 and w will be unaltered.
So far as this problem is concerned, a body may be said to be
given Di/naniicalli/ when its mass, centre of gravity, principal
axes, and principal moments at the centre of gravity are given.
This remark will be found to be of general application.
Ex. 1. If the body be pushed along the axis of y and made to rotate about the
vertical with the same angular velocity as before, show that no effect is produced on
the inclination of the body to the vertical.
Ex. 2. If the body be a heavy disc capable of turning about a horizontal axis
Ou in its own plane, show that the plane of the disc will be vertical unless w^ > ~
where h is the distance of the centre of gravity of the disc from Oy and k the radius
of gjTation about Oy.
Ex. 3. If the body be a circular disc capable of turning about a horizontal axis
perpendicular to its plane and intersecting the disc in its circumference, show that
if the tangent to the disc at the hinge make an angle 6 with the vertical, the angular
velocity w must be a / — ^
sin '
Ex. 4. Two equal balls A and B are attached to the extremities of two equal
thin rods A a, Bb. The ends a and b are attached by hinges to a fixed point O and
the whole is set in rotation about a vertical through as in the Governor of the
Steam Engine. If the mass of the rods be neglected show that the time of rotation
is equal to the time of oscillation of a pendulum whose length is the vertical distance
of either sphere below the hinges at 0.
Ex. 5. If in the last example m be the mass of either thin rod and M that of
either sphere, i! the length of a rod, r the radius of a sphere, h the depth of either
centre below the hinge, then the length of the pendulum is , ^)-- — '— — ", — , .
° ^ l + r M{l + r)-\-lml
71. To cvpply D'Alemberfs principle to obtain the equations
of motion of a system of rigid bodies.
Let {x, y, z) be the co-ordinates of the particle m at the time
t referred to any set of rectangular axes fixed in space. Then
-T^, -jTi, and -pr, will be the accelerations of the particle. Let
X, Y, Z be the impressed accelerating forces on the same particle
resolved parallel to the axes. By D'Alembert's principle the
farces
/ ,. d^x\
4—2
52
DALEMBERTS PRINCIPLE.
^'
together with similar forces on every particle will be in equi-
librium. Hence by the principles of Statics we have the equation
and two similar equations for y and z; these are obtained by
resolving parallel to the axes. Also we have
and two similar equations for zx and xy ; these are obtained by
taking moments about the axes.
These equations may be written in the more convenient forms
-^, Zni—j- = Zml
at at
at at
(A),
a -^ I ax az \ _, , _^ _. .
j^tm[z -^^-x^yXm{zX-xZ) -
d
,(B).
In a precisely similar manner by taking the expressions for
the accelerations in polar co-ordinates we should have obtained
another but equivalent set of equations of motion.
72. Let us consider the meaning of these equations without
reference to axes of co-ordinates. The effective forces are to be
equivalent to the impressed forces. But as shown in Statics any
system of forces and therefore each of these is equivalent to a
single force and a single couple at some point taken as origin.
These resultant forces and couples must therefore be equivalent,
each to each.
If we multiply the mass m of any particle P by its velocity 'o
we have the momentum rav of the particle. Let us represent this
in direction and magnitude by a straight line PP' . Then, just as
in Statics, this momentum is eqviivalent to an equal and parallel
linear momentum at which we may represent by OM, and a
couple whose moment is mvp, where p is the perpendicular dis-
tance between OM and PP'. The plane of this couple is the
d'alembert's principle. 53
plane containing OM and PF , and it may as usual be represented
in direction and magnitude by an axis ON perpendicular to its
plane. This couple is sometimes called an angular momentum.
Let OM', ON' be the positions of these two lines after an
interval of time dt. Then MM' , NN' will represent in direction
and magnitude the linear momentum and the angular or couple
momentum added on in the time dt. Hence the effective force
on any particle m is equivalent to a single linear effective force
MM'
acting at represented by , and a single effective couple
NX'
represented by —7— .
Let V, OH be two straight lines drawn through the origin
to represent in direction and magnitude the resultant linear
momentum and resultant couple momentum of the whole system
at any time t. Let OV, OH' be the positions of these lines at
the time t + dt. Then OF is the resultant of the group OM cor-
responding to all the particles of the system, and V the resultant
VV
of the group OM'. Hence — ^ — represents the whole linear ef-
Ojb
fective force of the system at the time t. By similar reasoning
—-J- represents the resultant effective couple of the system. Thus
it appears that the points Fand i? trace out two curves in space
■whose properties are analogous to those of the hodograph in
Dynamics of a particle. From this reasoning it follows, that if
Vx be the resolved part of the momentum of a system in the
direction of any straight line Ox, and H^ the moment of the
dV d H
momentum about that straight line, then -^ and — r-^ are re-
dt dt
spectively the resolved part along, and the moment about that
straight line, of the effective force of the whole system.
Let us now refer the whole system to Cartesian co-ordinates
'as in Art. 71. "We see that m -y- , m -^ , m^ are the resolved
dt dt dt
parts of the momentum of the particle m. Hence OF is the
resultant of Sm-^-, 2)m-~, and 2m ^7. Also mioc-^ — y —
dt dt dt \ dt ^ dt
is the moment of the momentum of the particle m about the
axis of z. Hence OH is the resultant of
^ / dy dx\ ^ / dz d>/\ ^ ( dx dz\
^A'Tt-yiit)- ^'H^s-^ij' -™N-'^sJ-
Now DAlembert's principle asserts that the whole effective
forces of a system are together equivalent to the impressed forces.
54 d'alembert's principle.
Hence whatever co-ordinates may be used, if X and L be the
resolved parts and moment of the impressed moving forces re-
spectively along and about any fixed straight line which we shall
call the axis of x, the equations of motion are
dt dt
The first of these corresponds to equations (A), the second to
equations (B) of Art. 71.
We may notice the following cases.
(1) If no impressed forces act on the system, the two lines
OF, OH are absolutely fixed in direction and magnitude through-
out the motion.
(2) If all the impressed forces pass through a fixed point,
let this point be chosen as the origin, then though OF may be
variable, OH is fixed in position and magnitude.
(3) If all the impressed forces be equivalent to a system
of couples, then though OH may be variable, F is fixed in
position and magnitude*.
73. The equations of motion of Art. 71 are the general equa-
tions of motion of any dynamical system. They are, however,
extremely inconvenient in their present form. When the system
considered is a rigid body and not merely a finite number of
separate particles, the S's are all definite integrals. There are
also an infinite number of a;'s, ?/'s and s's all connected together
by an infinite number of geometrical equations. It will be neces-^
sary, as suggested in Art. 68, to find some quantities which may { ,
determine the position of the body in space and express the \1|
effective forces in terms of these quantities. These are called the i 1\
co-ordinates of the body]'. It is most important in theoretical—^ i
dynamics to choose these co-ordinates properly. They should be
(1) such that a knowledge of them in terms of the time determines
the motion of the body in a convenient manner, and (2) such that
the dynamical equations when expressed in terms of them may
be as little complicated as possible,
74. Let us first enquire how many co-ordinates are necessary
to fix the position of a body.
The position of a body in space is given when we know the
co-ordinates of some point in it and the angles which two straight
lines, fixed in the body make with the axes of co-ordinates. There
-■■■' * In a memoir on the differential coefficients and determinants of lines, Mr Cohen
has discussed some of the properties of these resultant lines. Phil. Trans. 1862.
t Sir W. Hamilton uses the phrase "marks of position," hut subsequent -writers
have adopted the term co-ordinates. Sec Cayley's Report to the Brit. Assoc, 1857.
d'alembert's principle. 55
are three geometrical relations existing between these six angles,
so that the position of a body may be made to depend on six
independent variables, viz. three co-ordinates and three angles.
These might be taken as the co-ordinates of the body. By the
term "co-ordinates of a body" is meant any quantities which de-
termine the position of the body in space.
It is evident that we may express the co-ordinates [x, y, z) of
any particle m of a body in terms of the co-ordinates of that body
and quantities which are known and remain constant during the
motion. First, let us suppose the system to consist only of a
- single body, then if we substitute these expressions for x, y, z in
the equations (A) and (B) of Art. 71, we shall have six equations
to determine the six co-ordinates of the body in terms of the
time. Thus the motion will be found. If the system consist of
several bodies, we shall, by considering each separately, have six
equations for each body. If there be any unknown reactions be-
tween the bodies, these will be included in X, Y, Z. For each
reaction there will be a corresponding geometrical relation con-
hecting the motion of those bodies. Thus on the whole we shall
have sufficient equations to determine the motion of the system.
When the motion is in two dimensions these six co-ordinates
become three. These are the two co-ordinates of the fixed point
in the body, and the angle some straight line fixed in the body
makes with a straight line fixed in space.
^^ ' 75. Let us next consider how the equations of motion formed
by resolution can be simplified by a proper choice of co-ordinates.
We must find the resolved part of the momentum and the re-
solved part of the effective forces of a system in any direction.
Let the given direction be taken as the axis of x. Let {x, y, z)
be the co-ordinates of any particle whose mass is m. The re-
cLcG
solved part of its momentum in the given direction is 'ni -rr .
. Hence the resolved part of the momentum of the whole system is
2w -y- . Let {x, y, z) be the co-ordinates of the centre of gravity
of the system and M the whole mass. Then Mx = Xmx ;
dt dt '
Hence the resolved jjart of the momentum of a system in any
direction is equal to the whole mass midtijplied into the resolved part
of the velocity of the centre of gravity.
That is, the linear momentum of a system is the same as if the
vjhole mass were collected into its centre of gravity.
56 d'alembert's principle.
In the sarm way, the resolved part of the effective forces of a
system in any direction is equal tO' the whole mass midtiplied into
the resolved part of the acceleration of the centre of gravity.
It appears from this proposition that it will be convenient to
take the co-ordinates of the centre of gravity of each rigid body in
the system as three of the co-ordinates of that body. We can then
express in a simple fos-m the resolved part of the effective forces
in any direction.
76. Lastly, let us consider how the equations of motion formed
by taking moments can be simplified by a proper choice of the
three remai^ning co-ordinates. We must find the moment of the
momentum, and the moment of the effective forces- about any
straight line.
Let the given straight line be taken as the axis of x, then
following the same notation as before; the moment of the mo^
mentum about the axis of x is
- / dz^ dy\
Now this is^ an expi:ession of the second degree. If,, then, we
substitute y = y + y', z = z + z, we get by Art. 14<
where M is the mass of the system or body under consideration.
The second' term of this expression is the moment about the
axis of a? of the momentum of a mass If moving with the centre
of gravity.
The first term' is the moment about a straight line parallel to
the axis of x, not of the actual momenta of all the several parti-
cles but of their momenta relatively to that of the centre of gravity.
In the case of any particular body it therefore depends only on the
motion of the body relatively to its centre of gravity. In finding
its value we shall suppose the centre of gravity reduced, to rest by
applying to every particle of the system a velocity equal and oppo-
site to that of the centre of gravity. Hence- we infer that
The moment of the momentum of a system about any straight
line is equal to the moment of the' momentum of the whole mass
supposed collected at its centre of gravity and moving with it,
together with the moment of the momentum of the system relative to
its centre of gravity about a straight line drawn parallel to the given
straight line through the centre of gravity.
In the same way, this proposition will be also true if for the
"momentum" of the system we substitute " effective force."
d'alembert's principle. 57
By taking the axis Ox through the centre of gravity, we see
that the moment of the relative momenta about any straight line
through the centre of gravity is equal to that of the actual
momenta.
77. It appears from the preceding article that it will be con-
venient to refer the angular motion of a body to a system of
co-ordinate axes meeting at the centre of gravity. A general
expression for the moment of the effective forces about any straight
line through the centre of gravity cannot be conveniently investi-
gated at this stage. Different expressions will be found advanta-
geous under different circumstances. There are thr^e cases to
which attention should be particularly directed : (1) when the
body is turning about an axis fixed in the body and fixed in
space ; (2) when the motion is in two dimensions, and (3) Euler's
expression when the body is turning about a fixed point. These
will be found at the beginnings of the third and fourth chapters
and in the fifth chapter respectively.
/ (J 7/ ft'lr\
78. The quantity %m [^■~±f~y-j:) expresses the moment of
the momentum about the axis of z. It is then called the angular
momentum of the system about the axis of z. There is another
interpretation which can be given to it. If we transform to polar
co-ordinates, we have
dy dx 2 dO
dt ^ dt dt
Wow \r'^d9 is the elementary area described round the origin
in the time dt by the projection of the particle on the plane of xy.
If twice this polar area be multiplied by the- mass of the particle,
it is called the area conserved by the particle in the time dt round
the axis of z. Hence
.^ f dy d'X\
is called the area conserved by the system^ in a unit of time, or
more simply the area conserved.
79. We may now enunciate two important propositions, which
follow at once from' the preceding results. It will, however, be
more useful to deduce them- from first principles.
(1) The motion of the centre of gravity of a system acted on by
any forces is the same as if all the mass were collected at the centre
of gravity and all the forces were applied at that point parallel to
their former directions.
(2) The motion of a body, acted on by any forces, about its
centre of gravity is the same as if the centre of gravity were fixed
and the same forces acted on the body.
58 d'alevibert's principle.
Taking any one of the equations (A) we have
dt\
If X, y, ~z be the co-ordinates of the centre of gravity, then
xtm = Xmx ;
.'. —77 Xm = %inX,
at
and the other equations may be treated in a similar manner.
But these are the equations which give the motion of a mass
%m acted on by forces 2mX, &c. Hence the first principle is
proved.
Taking any one of equations (B) we have
2m [x-^-y^'j=tm {xY - yX).
Let x = x + x', y =^y j^y'^ z=z-\-z', then by Art. 14 this equa-
tion becomes
Now the axes of co-ordinates are quite arbitrary, let them be
so chosen that the centre of gravity is passing through the origin
at the moment under consideration. Then x = 0, y — 0, but
-^ , -— are not necessarily zero. The equation then becomes
2m U ^-y^)= 2m {xY- y'X).
This equation does not contain the co-ordinates of the centre
of gravity and holds at every separate instant of the motion and
therefore is always true. But this and the two similar equations
obtained from the other two equations of (B) are exactly the equa-
tions of moments we should have had if we had regarded the
centre of gravity as a fixed point and taken it as the origin of
moments.
80. These two important propositions are called respectively
the principles of the conservation of the motions of translatioja and
rotation. The first was given b}'" Newton in the fourth corollary
to the third law of motion, and was afterwards generalized by
D'Alembert and Montucla. The second is more recent and seems
to have been discovered about the same time by Euler, Bernoulli
and the Chevalier d'Arcy.
81. By the first principle the problem of finding the motion
of the centre of gravity of a system, however complex the system
d'alembert's principle. 59
may be, is reduced to the problem of finding the motion of a
single particle. By the second the problem of finding the angular
motion of a free body in space is reduced to that of determining
the motion of that body about a fixed point.
In using the first principle it should be noticed that the im-
pressed forces are to be applied at the centre of gravity parallel
to their former directions. Thus, if a rigid body be moving
under the influence of a central force, the motion of the centre of
gravity is not generally the same as if the whole mass were col-
lected at the centre of gravity and it were then acted on by the
same central force. What the principle asserts is, that, if the
attraction of the central force on each element of the body be
found, the motion of the centre of gravity is the same as if these
forces were applied at the centre of gravity parallel to their
original directions.
If the impressed forces act always parallel to a fixed straight
line, or if they tend to fixed centres and vary as the distance from
those centres, the magnitude and direction of their resultant are
the same whether we suppose the body collected into its centre of
gravity or not. But in most cases care must be taken to find the
resultant of the impressed forces as they really act on the body
before it has been collected into its centre of gravity.
82, From this proposition we may infer the independence of
the motions of translation and rotation. The motion of the centre
of gravity is the same as if the whole mass were collected at that
point, and is therefore quite independent of the rotation. The
motion round the centre of gravity is the same as if that point
were fixed, and is therefore independent of the motion of that
point.
83. We may now collect together for reference the results of
the preceding articles.
Let u, V, w be the velocities of the centre of gravity of any
rigid body of mass M resolved parallel to any three fixed rect-
angular axes, let h^, h^, \ be the three moments of the momentum
relative to the centre of gravity about three rectangular axes
fixed in direction and meeting at the centre of gravity. Then the
effective forces of the body are equivalent to the three effective
forces M -j- , ^ -ji , ^^~r acting at the centre of gravity parallel
to the directions into which the velocities have been resolved, and
to the three effective couples ~ , ~ , ~ about the axes meet-
ing at the centre of gravity about which the moments were taken.
The effective forces of all the other bodies of the system may be
expressed in a similar manner.
60 d'alembert's principle.
Then all these effective forces and couples, being reversed, will
be in equilibrium with the impressed forces. The equations of
equilibrium may then be found by resolving in such directions and
taking moments about such straight lines as may be most con-
venient. Instead of reversing the effective forces it is usually
found more convenient to write the impressed and effective forces
on opposite sides of the equations.
Application of D'Alemhert's Principle to impulsive forces.
84. If a force F act on a particle of mass m always in the
same direction, the equation of motion is
dv „
where v is the velocity of the particle at the time t. Let T be the
interval during which the force acts, and let v, v be the velocities
at the beginning and end of that interval. Then
rT
m [v —v)=\ Fdt.
Jo
Now suppose the force F to increase without limit while the
interval T decreases without limit. Then the integral may have
a finite limit. Let this limit be P. Then the equation becomes
m {v —v) = P.
The velocity in the interval T has increased or decreased from
V to v. Supposing the velocity to have remained finite, let V be
its greatest value during this interval. Then the space described
is less than VT. But in the limit this vanishes. Hence the
particle has not moved during the action of the force F. It has
not had time to move but its velocity is suddenly changed from
V to v.
We may consider that a proper measure has been found for a
force when from that measure we can deduce all the effects of the
force. In the case of finite forces we have to determine both the
change of place and the change in the velocity of the particle. It
is therefore necessary to divide the whole time of action into
elementary times and determine the effect of the force during
each of these. But in the case of infinite forces which act for an
indefinitely short time, the change of place is zero, and the change
of velocity is the only element to be determined. It is therefore
more convenient to collect the whole force expended into one
measure. Such a force is called an impulse. It may be defined
as the limit of a force which is infinitely great, but acts only
during an infinitely short time. There are of course no such
d'alembert's principle. 61
forces in nature, but there are forces which are very great, and
act only during a very short time. The blow of a hammer is
a force of this kind. They may be treated as if they were im-
pulses, and the results will be more or less correct according to
the magnitude of the force and the shortness of the time of
action. They may also be treated as if they were finite forces,
and the displacement of the body during the time of action of the
force may be found.
The quantity P may be taken as the measure of the force.
An impulsive force is measured by the whole momentum gener-
ated by the impulse.
85. In determining the effect of an impulse on a body, the
effect of all finite forces which act on the body at the same time may
he omitted.
For let a finite force / act on a body at the same time as an
impulsive force F. Then as before we have
m m m m '
Bat in the limit fT vanishes. Similarly the force / may be
omitted in the equation of moments.
86. To obtain the general equations of motion of a system
acted on by any number of impulses at once.
Let u, V, lu, u', V, w be the velocities of a particle of mass m
parallel to the axes just before and just after the action of the
impulses. Let X', Y', Z' be the resolved parts of the impulse on
TO parallel to the axes.
Taking the same notation as before, we have the equation
or integrating
tm{u:~u) = Xm [ Xdt = XX' (1).
Jo
Similarly we have the equations
2to {v - v) = Sr (2),
tm{u}'-w)=^tZ' (3).
Again the equation
Sm [x -^1 - y-^j = tm (oc Y- yX)
62 d'alembert's principle.
becomes on integration
or taken between limits,
Xmlx {v' — v) —y {li —u)] = S {xY' — yX') (4),
and the other two equations become
'^mly (w'—w) — z (w— v)} = ^ [yZ' — zY') (5),
27?i \z {ii — u) — x{iu—w)] = S {zX' — xZ') (6).
In all the following investic^ations it will be found convenient
to use accented letters to denote the states of motion after impact
which correspond to those denoted by the same letters unaccented
before the action of the impulse. Since the changes in direction
and magnitude of the velocities of the several particles of the
bodies are the only objects of investigation, it will be more conve-
nient to express the equations of motion in terms of these veloci-
doG f/?/ uX
ties, and to avoid the introduction of such symbols as -^ , -^ , -j- .
87. In applying D'Alembert's Principle to impulsive forces the
only change which must be made is in the mode of measuring the
effective forces. If {u, v, w), (u, v, w) be the resolved parts of the
velocity of any particle, just before and just after the impulse, and
if mbe its mass, the effective forces will be measured by m{it—u),
ni{v' — v), and m {w' — w). The quantity mf in Art. 67 is to be
regarded as the measure of the impulsive force which, if the parti-
cle were separated from the rest of the body, would produce these
changes of momentum.
In this case, if we follow the notation of Arts. 75 and 76, the
resolved part of the effective force in the direction of the axis of ^
is the difference of the values of %m -j just before and just after
the action of the impulses, and this is the same as the difference
dz
of the values of M -y- at the same instants. In the same way the
moment of the effective forces about the axis of z will be the
difference of the values of
^ f dii dx
just before and just after the action of the impulses.
We may therefore extend the general proposition of Art. 83 to
impulsive forces in the following manner.
Let (u, V, w), {ii, v', tu) be the velocities of the centre of gravity
of any rigid body of mass J/ just before and just after the action
d'alembert's peinciple. 63
of the impulses resolved parallel to any three fixed rectangular
axes. Let [h^, \, A3), (A/, A/, 1\) be the three moments of the
momentum relative to the centre of gravity about three rect-
angular axes fixed in direction and meeting at the centre of
gravity, the moments being taken just before and just after the
impulses. Then the effective forces of the body are equivalent to
the three effective forces M{u' — u), M{v'-v), M{iv —w) acting
at the centre of gravity parallel to the rectangular axes together
with the three effective couples (7^/ — A J, []\ — AJ , {h^ — h^) about
those axes.
These effective forces and couples being revei-sed will be in
equilibrium with the impressed forces. The equations of equili-
brium may then be formed according to the rules of Statics.
Ex. 1. Two particles moving in the same plane are projected in parallel but
opposite directions with velocities inversely proportional to their masses. Find the
motion of their centre oi gravity.
v/Ex. 2. A person is placed on a perfectly smooth table, show how he may get
off.
Ex. 3. Explain how a person sitting on a chair, is able to move the chair
across the room by a series of jerks, without touchmg the ground with his feet.
V Ex. 4. A person is placed at one end of a perfectly rough board which rests on
a smooth table. Supposing he walks to the other end of the board, determine how
much the boai-d has moved. If he stejiped off the board, show how to determine its
subsequent motion.
Ex. 5. The motion of the centre of gravity of a shell shot from a gun in vacuo
is a iDarabola, and its motion is unaffected by the bursting of the shell.
Ex. 6. A rod revolving uniformly in a horizontal i^lane round a pivot at its ex-
tremity suddenly snaps in two : determine the motion of each part.
Ex. 7. A cube slides down a perfectly smooth inclined plane with four of its
edges horizontal. The middle point of the lowest edge comes in contact with
a small fixed obstacle and is reduced to rest. Determine if the cube is also reduced
to rest, and show that the resultant impulsive action along the edge will not in
general act along the inclined plane.
V Ex. 8. Two persons A and B are situated on a perfectly smooth horizontal
plane at a distance a from each other. A throws a baU to B which reaches B after
a time t. Show that A will begin to slide along the plane with a velocity ^ where
M is his own mass and m that of the ball. If the plane were perfectly rough
explain in general terms the nature of the pressures between ^'s feet and the
plane which would prevent him from sliding. Would these pressures have a single
resultant ?
Ex. 9. A cannon rests on an imperfectly rough horizontal plane and is fired
with such a charge that the relative velocity of the ball and cannon at the moment
when the ball leaves the cannon is V. If M be the mass of the cannon, m that of
64 d'alembert's principle.
the ball and [j. the coefficient of friction, show that the cannon will recoil a distance
^> I „ — on the plane.
\M+mJ 2fjLg
Ex. 10. A spherical cavity of radius a is cut out of a cubical mass so that the
centre of gravity of the remaining mass is in the vertical thi'ough the centre of the
cavity. The cubical mass rests on a perfectly smooth horizontal plane, but the
interior of the cavity is perfectly rough, A sphere of mass in, and radius b, rolls
down the side of the cavity starting from rest with its centre on a level with the
centre of the cavity. Show that when the sphere nest comes to rest, the cubical
mass has moved through a space —j^ where M is the mass of the remaiuing
portion of the cube. Will the result be the same if the cavity were imperfectly
rough or smooth ?
\ Ex. 11. Two railway engines di-awing the same train are connected by a loose
chain and come several times in succession into collision with each other; the
leading .engine being a little top-heavy and the buffers of both rather low. The
fore-wheels of the first engine are observed to jump up and down. What dynamical
explanation can be given of this rocking motion ? At what level should the buffers
be placed that it may not occur? Camb, Transac, Vol. vii.
"* Ex. 12. Sir C. Lyell in his account of the earthquake ia Calabria in 1783,
mentions two obelisks each of which was constructed of three great stones laid on
top of each other. After the earthquake, the pedestal of each obelisk was found to
be in its original place, but the separate stones above were tiu-ned partially round
and removed several inches from their position without falling. The shock which
agitated the building was therefore described as having been horizontal and vorti-
cose. Show that such a displacement would be produced by a simple rectilinear
shock, if the resultant blow on each stone did not pass through its centre of gravity.
See Mallet's Dynamics of Earthqtuikes,
y^^ CHAPTER III.
MOTION ABOUT A FIXED AXIS.
^ 88. A rigid body can turn freely ahoid an axis fixed in the
body and in space, to find the moment of the effective forces about
the axis of rotation.
Let any plane passing tliroiigli the axis and fixed in space be
taken as a plane of reference, and let 6 be the angle which any
other plane through the axis and fixed in the body makes with
the first plane. Let m be the mass of any element of the body,
r its distance from the axis, let (/> be the angle a plane through the
axis and the element m makes with the plane of reference.
The velocity of the particle m is r -^ in a direction perpendi-
cular to the plane containing the axis and the particle. The
moment of the momentum of this particle about the axis is
clearly mr"^ -j- . Hence the moment of the momenta of all the
particles is 2 ( mr^ ,^ j . Since the particles of the body are rigidly
connected with each other, it is obvious that —- is the same for
dt
in
every particle, and equal to -^ . Hence the moment of the mo-
d6
menta of all the particles of the body, about the axis is 2wr^ -j: ,
i.e. the moment of inertia of the body about the axis multiplied
into the angular velocity.
The accelerations of the particle m are r ~^ and —^(-7^)
perpendicular to, and along the directions in which r is measured)
the moment of the moving forces of m about the axis is mr^ —^ ,
® _ df
hence the moment of the moving forces of all the particles of the
body about the axis is 2 [wr'^ 7^)- ^J ^^® ''^^™6 reasoning as
R. D. 5
GQ MOTION ABOUT A FIXED AXIS.
before this is equal to 'Zmr' -jy > *-^- tl^e moment of inertia of the
body about the axis into the angular acceleration.
89. To determine the motion of a body about a fixed axis
under the action of any forces.
By D'Alembert's principle the effective forces when reversed
will be in equilibrium with the impressed forces. To avoid intro-
ducing the unknown reactions at the axis, let us take moments
about the axis.
First, let the forces be impulsive. Let w, w be the angular
velocities of the body just before and just after the action of the
forces. Then, following the notation of the last article,
co' . '%mr^ — CO . ^mr"^ — L,
where L is the moment of the impressed forces about the axis ;
, moment of forces about axis
moment of inertia about axis '
This equation will determine the change in the angular velo-
city produced by the action of the forces.
Secondly, let the forces be finite. Then taking moments about
the axis, we have
d'e ^ , ^
-^TT, . Zmr = h ;
dt' '
d^9 moment of forces about axis
df moment of inertia about axis *
This equation when integrated will give the values of 6 and
-J- at any time. Two undetermined constants will make their
appearance in the course of the solution. These are to be deter-
dO
mined from the given initial values of 6 and -r . Thus the whole
dt
motion can be found.
90. It appears from this proposition that the motion of a
rigid body about a fixed axis depends on ( 1) the moment of the
forces about that axis and (2) the moment of inertia of the body
about the axis. Let JfF be this moment of inertia, so that k is
the radius of gyration of the body. Then if the whole mass of
the body were collected into a particle and attached to the fixed
axis by a rod without inertia, whose length is the radius of gyra-
tion k, and if this system be acted on by forces having the same
moment as before, and be set in motion with the same initial
GENERAL PEINCIPLES. 67
d9
values of 6 and -^ , then the whole subsequent angular or gyra-
tory motion of the rod will be the same as that of the body. "We
may say briefly, that a body turning about a fixed axis is dyna-
mically given, when we know its mass and radius of gyration.
I /
,,,1/ 91. Ex. A "perfectly rough circular horizontal hoard is capahle of revolving
H^/ freely round a vertical axis through its centre. A man whose weight is equal to that
of the board walks on and round it at the edge: lohen he has completed the circuit
tvhat ivill be his position in space?
Let a be the radius of the board, Mk^ its moment of inertia about the vertical
axis. Let w be the angular velocity of the board, w' that of the man about the
vertical axis at any time. And let F be the action between the feet of the man and
the board.
The equation of motion of the board is by Ai't. 89,
M'k''^= -Fa (1).
dt ^
The equation of motion of the man is by Ai-t. 79,
,^ du' „
^^=^ (2)"
Eliminating F and integrating, we get
fc^w + a^w' = 0,
the constant being zero, because the man and the board start from rest. Let 0,
6' be the angles described by the board and man round the vertical axis. Then
fc)=— , w'=-r , and k-d + a"d'=0. Hence, when 8' -6 — 2ir, we have 6' = -^^- — .,27r.
dt dt' ' k^ + «2
a^ 2
This gives the angle ia space described by the man. If fc^= — we have d'=-ir.
Let V be the mean relative velocity with which the man walks along the board.
V Va 2 V
Then w'-w = — ; .*. a— - ,-^ :,= -5 -• This gives the mean angular velocity
a /c + a" o a
of the board.
On the Pendulum.
92. A body moves about a fixed horizontal axis acted on by
gravity only, to determine the motion.
Take the vertical plane through the axis as the plane of refer-
ence, and the plane through the axis and the centre of gravity as
the plane fixed in the body. Then the equation of motion is
d~d _ moment of forces , ,
df moment of inertia "
_ Mgh sin
5—2
68 MOTION ABOUT A FIXED AXIS.
where li is the distance of the centre of gravity from the axis and
il/F is the moment of inertia of the body about an axis through
the centre of gravity parallel to the fixed axis. Hence
The equation (2) cannot be integrated in finite terms, but if
the oscillations be small, we may reject the cubes and higher
powers of 6 and the equation will become
cW k' + h'
Hence the time of a complete oscillation is 27r a / r — • If
h and k be measured in feet and g = 32 18, this formula gives the
time in seconds.
The equation of motion of a particle of any mass suspended
by a string I is
which may be deduced from equation (2) by putting Zj = and
h = l. Hence the angular motions of the string and the body
under the same initial conditions will be identical if
z = '^ W.
This length is called the length of the simple equivalent
pendulum.
Through Q, the centre of gravity of the body, draw a perpen-
dicular to the axis of revolution cutting it in C. Then C is called
the centre of suspension. Produce CQ to so that CO = l. Then
is called the centre of oscillation. If the whole mass of the
body were collected at the centre of oscillation and suspended by
a thread to the centre of suspension, its angular motion and time
of oscillation would be the same as that of the body under the
same initial circumstances.
The equation (4) may be put under another form. Since
CG — h and OG = l — h, we have
GC. G0= (rad.)^ of gyration about G,
CG . CO = (rad.)" of gyration about C,
OG. OC = (rad.)^ of gyration about 0.
Any of these equations show that if be made the centre of
suspension, the axis being parallel to the axis about which k was
taken, then C will be the centre of oscillation. Thus the centres
THE PENDULUM, 69
of oscillation and suspension are convertible and the time of oscilla-
tion about each is the same.
If the time of oscillation be given, I is given and the equa-
tion (4) will give two values of h. Let these values be h , /?,.
Let two cylinders be described with that straight line as axis
about which the radius of gyration k was taken, and let the
radii of these cylinders be h^, h^. Then the times of oscillation of
the body about any generating lines of these cylinders are the
/I*
same, and are approximately equal to Stta / - .
With the same axis describe a third cylinder whose radius
is /c. Then l=2.k + - — p^, hence I is always greater than 2k,
and decreases continually as h decreases and approaches the value
h. Thus the length of the equivalent pendulum continually de-
creases as the axis of suspension approaches from without to the
circumference of this third cylinder. When the axis of suspension
is a generating line of the cylinder the length of the equivalent
pendulum is 2h. When the axis of suspension is within the
cylinder and approaching the centre of gravity the length of the
equivalent pendulum continually increases and becomes infinite
"when the axis passes through the centre of gravity.
The time of oscillation is therefore least when the axis is a
generating line of the circular C3dinder whose radius is k. But the
time about the axis thus found is not an absolute minimum. It
is a minimum for all axes drawn parallel to a given straight line
in the body. To find the a^xis about which the time is absolutely
a minimum we must find the axis about which A; is a minimum.
Now it is proved in Art. 23, that of all axes through O the
* The position of tlie centre of oscillation of a body was first correctly deter-
mined by Huyghens in his Horologium Oscillatorium published at Paris in 1673.
The most important of the theorems given in the text were discovered by him. As
D'Alembert's principle was not kno^vn at that time, Huyghens had to discover some
principle for himself. The hj^pothesis was, that when several weiglits are put in
motion by the force of gravity, in whatever manner they act on each other their-
centre of gravity cannot be made to mount to a height greater than that from which
it had descended. Huyghens considers that he assumes here only that a heavy body
cannot of itself move upwards. The next step in the argument was, that at any
instant the velocities of the particles are such that, if they were separated from
each other and properly guided, the centre of gravity could be made to mount to a
second position as high as its first position. For if not, consider the particles to
start from their last positions, to describe the same paths reversed, and then again
to be joined together into a pendulum ; the centre of gravity would rise to its first
position ; but if this be higher than the second position, the hypothesis would be
contradicted. This principle gives the same equation which the modern principle
of Vi.; ^■iva would give, and the rest of the solution is not of much interest.
70 MOTION ABOUT A FIXED AXIS.
axis about whicli the moment of inertia is least or greatest is one
of the principal axes. Hence the axis about which the time of
oscillation is a minimum is parallel to that principal axis through
G about which the moment of inertia is least. And if Mk^ be the
moment of inertia about that axis, the axis of suspension is at a
distance k measured in any direction from the principal axis.
93. Ex. 1. Eiud tlie time of the small oscillations of a cube (1) when one
side is fixed, (2) when a diagonal of one of its faces is fixed; the axis in both
cases being horizontal.
Result. If 2a be a side of the cube, the length of the simple equivalent pendu-
lum is in the first case -—^ a, and in the second case - a.
6 6
'■ Ex. 2. An elliptic lamina is such that when it swings about one latus rectum
as a horizontal axis, the other latus rectum passes through the centre of oscillation,
prove that the eccentricity is h.
Ex. 3. A ch'cular arc oscillates about an axis through its middle point perpen-
dicular to the plane of the arc. Prove that the length of the simple equivalent
pendulum is independent of the length of the arc, and is equal to twice the radius.
y- Ex, 4. The density of a rod varies as the distance from one end, find the axis
perpendicular to it about which the time of oscillation is a minimum.
Result. The axis passes through either of the two points whose distance from the
centre of gravity is — p a, where a is the length of the rod.
^ Ex. 5. Find what axis in the area of an ellipse must be fixed that the time of
a small oscillation may be a minimum.
Result. The axis must be parallel to the major axis, and bisect the semi-minor
axis.
Ex. 6. A uniform stick hangs freely by one end, the other end being close to the
ground. An angular velocity in a vertical plane is then communicated to the stick,
and when it has risen through an angle of 90", the end by which it was hangtug is
loosed. What must be the initial angular velocity so that on falling to the ground
it may pitch in an upright position ?
Result. The required angular velocity wis given by
(2?i+l)-+l
where n is any integer, and 2a is the length of the rod.
Ex. 7. Two bodies can move freely and independently under the action of
gravity about the same horizontal axis ; their masses are m, m', and the distances of
their centres of gravity from the axis are h, h'. If the lengths of their simple equi-
valent pendulums be L, L', prove that when fastened together the length of the
equivalent pendiilum will be ^^^^^ + ^''^'L\
mk + nvh'
THE PENDULUM. 71
Ex. 8. When it is required to regulate a clock withoiit stopping the pendulum,
it is usual to add or subtract some small weight from a platform attached to the
pendulum. Show that in order to make a given alteration in the going of the clock
by the addition of the least possible weight, the platform must be placed at a dis-
tance from the point of suspension equal to half the simple equivalent pendulum.
Show also that a slight error in the position of the platform will not affect the
weight required to be added.
Ex. 9. A circular table centre is supported by three legs AA\ BB', CC" which
rest on a perfectly rough horizontal floor, and a heavy particle P is placed on the
table. Suddenly one leg CO' gives way, show that the table and the particle wUl
immediately separate if jpc be greater than k^ ; where p and c are the distances of P
and respectively from the line AB joining the tops of the legs, and k is the radius
of gyration of the table and legs about the line A'B' joining the points where the
legs rest on the floor.
The condition of separation is that the initial normal acceleration of the point
of the table at P should be greater than the normal acceleration of the particle
itself.
Ex. 10. A string without weight is placed round a fixed ellipse whose plane is
vertical, and the two ends are fastened together. The length of the string is greater
than the perimeter of the ellipse. A heavy particle can slide freely on the string
and performs small oscillations under the action of gravity. Prove that the simple
equivalent pendulum is the radius of curvatui'e of the confocal ellipse jjassing
through the position of equilibrium of the particle.
>.
yV^ 94. In a clock which is regulated by a pendulum, it is neces-
sary that the time of oscillation should be invariable. As all
substances expand and contract with every alteration of tempera-
ture, it is clear that the distance of the centre of gravity of the
pendulum from the axis and the moment of inertia about that
axis will be continually altering. The length of the simple equi-
valent pendulum does not however depend on either of these
elements simply, but on their ratio. If then we can construct a
pendulum such that the expansion or contraction of its different
parts does not alter this ratio, the time of oscillation will be un-
affected by any changes of temperature. For an account of the
various methods of accomplishing this which have been suggested,
we refer the reader to any treatise* on clocks. We shall here only
notice for the sake of illustration one simple construction, which
has been generally used. It was invented by George Graham
about the year 1715.
Some heavy fluid, such as mercury, is enclosed in a cast-iron cylindrical jar
into the top of which an u'on rod is screwed. This rod is then suspended in the
usual manner from a fixed point. The downward expansion of the u'on on any
increase of temperature tends to lower the centre of oscillation, but the upward ex-
pansion of the merciuy tends on the contrary to raise it. It is requu-ed to detcr-
* Reid on Clocks; Denison's treatise on Clocks and Clockmuk'utg in Wcale's
Series, 1807; Captain Kater's treatise on Mechanics in Lardner's Cyclop(cdia, 1830.
72 MOTION ABOUT A FIXED AXIS.
mine tlie condition that the position of the centre of oscillation may on the whole
be unaltered.
Let Mk'^ be the moment of inertia of the non jar and rod about the axis of sus-
pension, c the distance of their common centre of gravity from that axis. Let I be
the length of the pendulum from the point of suspension to the bottom of the jar,
a the internal radius of the jar. Let nM be the mass of the mercury, h the
height it occupies in the jar.
The moment of inertia of the cylinder of mercury about a straight line through
its centre of gravity perpendicular to its axis is by Art. 18, Ex, 8, nM \Tn + -r] •
Hence the moment of inertia of the whole body about the axis of suspension is
^Ml2+4 + (^-2)r^^'
and the moment of the whole mass collected at its centre of gravity is
Mnil-^+Mc.
The length L of the simple equivalent pendulum is the ratio of these two, and on
reduction we have
L^^±——lI— (1).
an
Let the linear expansion of the substance which forms the rod and jar be denoted
by a and that of mercury by j8 for each degree of the thermometer. If the thermo-
meter used be Fahrenheit's, we have a = -0000065668, /3 = -00003336, according to
some experiments of Dulong and Petit. Thus we see that a and /3 are so small that
their squares may be neglected. In calculating the height of the mercury it must
be remembered that the jar expands laterally, and thus the relative vertical expan-
sion of the mercm-y is 3/3 - 2a, which we shall represent by y.
If then the temperature of every part be increased t^, we have a, I, Jc, c, all
increased in the ratio 1 + at : 1, while h is increased in the ratio 1 + yt : 1. Since L
is to be unaltered, we have
clL dL , dL , dL \ dL
da dl dk dc J dh
But 2/ is a homogeneous function of one dimension, hence
dL dL , dL ^ dL dL , ^
da dl dk dc an.
The condition becomes therefore by substitution
w _h dL
a - 7 L dh'
Let j4 , 5 be the numerator and denominator of the expression for L given by
equation (1). Then taking the logarithmic differential
/2 \ 1 ,2
\dL ^3^^-^) 2" n{l^-^ 1
L dh A ^ B B^ L '^ 2
THE PENDULUM. 73
Hence the required condition is
3(8 -a) , h c ^ i/ 2
This calculation is of more theoretical than practical importance, for the nume-
rical values of a and /3 depend a good deal on the purity of the metals and on the
mode in which they have been worked. The adjustment must therefore he finally
made by experiment.
In the investigation we have supposed a and /3 to he absolutely constant, but
this is only a very near approximation. Thus a change of 80" Fah. would alter (3
by less than a fiftieth of its value.
When the adjustment is made the compensation is not strictly correct, for the
iron jar and mercury have been supposed to be of uniform temperature. Now the
different materials of wliich the pendulum is composed absorb heat at different
rates and therefore while the temperature is changing there will be some slight
error in the clock.
95. Another cause of error in a clock pendulum is the buoy-
ancy of the air. This produces an upward force acting at the
centre of gravity of the volume of the pendulum equal to the
weight of the air displaced. A very slight modification of the
fundamental investigation in Art. 92 will enable us to take this
into account. Let V be the volume of the pendulum, D the
density of the air ; h^, h^, the distances of the centres of gravity of
the mass and volume respectively from the axis of suspension,
Mk^ the moment of inertia of the mass about the axis of suspen-
sion. Let us also suppose the pendulum to be symmetrical about
a plane through the axis and either centre of gravity.
The equation of motion is then
ilf 7,2 ~=- Mgli^ sin 6 + VDgh.^ sin ^ (1).
By the same reasoning as before we infer that if I be the
length of the equivalent pendulum
l = ^'^-^^-lI (2)-
The density of the air is continually changing, the changes being
indicated by variations in the height of the barometer. Let h be
VB
the value of h^ — h.^ -j^ for any standard density D. Suppose the
actual density to he D + 8D and let 1+ SI be the corresponding
length of the seconds pendulum, then we have by differentiation
~]i~—K~lir~' ^'^^ therefore
I " h M D '
74 MOTION ABOUT A FIXED AXIS.
If T be the time of oscillation, we have
r=27r^-, and .'. -^^^I*
96. Ex. 1. If the centres of gravity of the mass and vohime -were very nearly
coincident and the weight of the air displaced were y^ifs ^^ ^^^ weight of the
pendidiun, show that a rise of one inch in the barometer would cause an error in
the seconds pendulmn of nearly 2 sec. per day.
Ex. 2. If we affix to the pendulum rod produced upwards a body of the same
volume as the pendulum bob but of very small weight, so that the centre of gravity
of the volume lies in the axis of suspension, show that the correction for buoyancy
vanishes. This method was suggested in 1871 by the Astronomer Eoyal, but he
remarks that this construction would probably be inconvenient in practice.
Ex. 3. If a barometer be attached to the pendulum show that the rise or fall of
the mercury as the density of the ah changed could be so arranged as to keep the
time of vibration unaltered. This method was suggested first by Dr Eobinson of
Ai-magh in 1831 in the fifth volume of the memoirs of the Astronomical Society,
and afterwards by Mr Denison in the Astronomical Notices for Jan. 1873. In the
Armagh Places of Stars published in 1859, Dr Eobinson describes the difficulties
he found in practice before he was satisfied vnth the working of the clock.
The theory of this constriiction is that in differentiating equation (2) we are to
suppose P and \ variable and I constant. This gives ^^ " > — 5 [Mli^ - S {l^VD).
Let r be the rise of the barometer in the glass tube, r' the fall in the cistern, then
r = mr, where m is a known fraction depending on the dimensions of the barometer.
Let a and h be the depths of the mercmy in the tube and cistern below the axis of
suspension, 2c the diameter of the tube, p the density of the mercmy. Since ■Kc'^pr
is the quantity of mercmy added to the top of the merciury in the tube and taken
away from the cistern, we have
2
These are accurate if the barometer be merely a bent tube so that the cylinders
■fcansferred are similar as well as equal; in this case m = l. If the area of the
gistern be greater than that of the tube we have here neglected the difference of the
moments of inertia of the two cyhnders about axes through their centre of gravity.
As r is seldom more than one inch, we may write these
S [M]fi)^irc'^pT [a^ -¥),
d{3fhj) = Trc'^pr{a-h).
Since D is very small, we may neglect the variations of Vh„ when multiplied
by D. Thus we have
dD _Trc"ffp a + b-l ^
1) "^ vdL I '*'
THE PENDULUM. 75
where H— 6 - a is the height of the barometer. If the temperature of the air he
D~ H
?iT) 7\ FT
unaltered we have — - = -jj and r (1 + m) = SH. The required condition is therefore
■n-C'Hp H a + h-l ^
VD /ij I
It is clearly necessary that a + h>l. The jar of mercury in Graham's mercra-ial
pendulum might he used as the cistern of the barometer, as Mr Deuison remarks.
The height of the barometer being 30 inches this would hardly be effective unless
the pendulum was longer than the seconds pendulum, which is about 39 inches.
Prof. Eankine read a jDaper to the British Association in 1853 in which he
proposed to use a clock with a centrifugal or revolving pendulum, jDart of which
should consist of a siphon barometer. The rising and falling of the barometer wordd
affect the rate of going of the clock and thence the mean height of the mercurial
column during any long period would register itself.
Ex. 4. If the pendulum be supposed to drag a quantity of an* with it which
bears a constant ratio to the density D of the smTounding ah and adds yD to the
moment of inertia of the pendulum without increasing the moving power, show that
the change produced in the simple equivalent pendulum by a change of density §D
is given by U=y r~r- . Show that this might be included in Dr Eobinson's mode
of correcting for buoyancy.
97- In many experimental investigations it is necessary to
determine the moment of inertia of the body experimentecl on
about some axis. If the body be of regular shape and be so far
homogeneous that the errors thus produced are of the order to be-
neglected, we can determine the moment of inertia by calculation.
But sometimes this cannot be done. If we can make the body
oscillate under gravity about any axis parallel to the given axis-
placed in a horizontal position, we can determine by equation (4)
of Art. 92 the radius of gyration about a parallel axis through the
centre of gravity. This requires however that the distances of the
centre of gravity from the axes should be very accurately found.
Sometimes it is more convenient to attach the body to a pendulum-
of known mass whose radius of gyration about a fixed horizontal
axis has been previously found by observing the time of oscilla-
tion. Then by a new determination of the time of oscillation, the-
moment of inertia of the compound body, and therefore of the
given body, may be found, the masses being known.
If the body be a lamina, we may thus find the radii of gyra-
tion about three axes passing through the centre of gravity. By
measuring three lengths along these axes inversely proportional to
these radii of gyration, we have three points on a momental ellipse
at the centre of gravity. The ellipse may then be easily con-
structed. The directions of its principal diameters are the princi-
pal axes, and the reciprocals of their lengths represent on the same
scale as before the principal radii of gyration.
76 MOTION ABOUT A FIXED AXIS.
If the body be a solid, six observed radii of gyration will deter-
mine the principal axes and moments at the centre of gravity.
But in most cases some of the other circumstances of the par-
ticular problem under consideration will simplify the process.
;\ On the length of the Seconds Pendulum.
\f 98. The oscillations of a rigid body may be used to determine
the numerical value of the accelerating force of gravity. Let t be
the half time of a small oscillation of a body made in vacuo about
a horizontal axis, h the distance of the centre of gravity from the
axis, k the radius of gyration about a parallel axis through the
centre of gravity. Then we have by Art. 92,
F + A2 = 7v./,T^ (1),
where A, = ^ so that \ is the length of the simple pendulum
TT
whose complete time of oscillation is two seconds.
We might apply this formula to any regular body for which
P and h could be found by ca^lculation. Experiments have thus
been made with a rectangular bar, drawn as a wire and suspended
Jc' + h" . .
from one end. In this case — , — which is the length of the
simple equivalent pendulum is easily seen to be two-thirds of the
leno-th of the rod. The preceding formula then gives A. or g as
soon as the time of oscillation has been observed. By inverting
the rod and taking the mean of the results in each position any
error arising from want of uniformity in density or figure may
be partially obviated. It has, however, been found impracticable
to obtain a rod sufficiently uniform to give results in accordance
with each other.
99. If we make a body oscillate about two parallel axes in
succession not at the same distance from the centre of gravity, we
get two equations similar to (1), viz.
k' + h'=UT'
k' + h" = Xh'T"^ ^^^•
Between these two we may now eliminate k'^, thus
h'^ — h'^
^L^ = hr'-h'T" (8).
This equation gives A,. Since k^ has disappeared, the form and
structure of the body is now a matter of no importance. Let a
body be constructed with two apertures into which knife edges
r'
LENGTH OF THE SECONDS PENDULUM. 7/
can be fixed. By means of these resting either on a horizontal
plane or in two triangular apertures to prevent slipping, the body
can be made to oscillate through small arcs. The perpendicular
distances h, h' of the centre of gravity from the axes must then be
measured with great care. The formula will then give X.
100. In Capt. Kater's method the body has a sliding weight
in the form of a ring which can be moved up and down by means
of a screw. The body itself has the form of a bar and the
apertures are so placed that the centre of gravity lies between
them. The ring weight is then moved until the two times of
oscillation are exactly equal. The equation (3) then becomes
^-=^ ^ ^^)'
which determines A,. The advantage of this construction is that
the position of the centre of gravity, which is very difficult to find
by experiment, is not required. All we Avant is h + li , the exact
distance between the knife edges. The disadvantage is that the
ring weight has to be moved until two times of oscillation, each of
which it is difficult to observe, are made equal.
101. The equation (3) can be written in the form
We now see that if the body be so constructed that the times of
oscillation about the two axes of suspension are very nearly equal
T^ — T' will be small, and therefore it will be sufficient in the last
term to substitute for h and ]i their approximate values. The
position of the centre of gravity is of course to be found as accu-
rately as possible, but any small error in its position is of no very
great consequence, for these errors are multiplied by the small
quantity r'^ — t"^ The advantage of this construction over Kater's
is that the ring weight may be dispensed with and yet the only
element which must be measured with extreme accuracy is h + h ,
the distance between the knife edges.
102. In order to measure the distance between the knife
edges. Captain Kater first compared the different standards of
length then in use, in terras of each of which he expressed the
length of his pendulum. Since then a much more complete com-
parison of these and other standards has been made under the
direction of the Commission appointed for that purpose in 1843.
rjiil. Trans. 1857.
Having settled his unit of length, Captain Kater proceeded to
measure the distance between the knife edges by means of micro-
78 MOTION ABOUT A FIXED AXIS.
scopes. Two different methods were used, which however cannot
be described here. As an illustration of the extreme care neces-
sary in these measurements, the following fact may be mentioned.
Though the images of the knife edges were always perfectly sharp
and well defined, their distance when seen on a black ground was
'000572 of an inch less than when seen on a v^^hite ground. This
difference appeared to be the same whatever the relative illumi-
nation of the object and ground might be so long as the difference
of character was preserved. Three sets of measurements were
taken, two at the beginning of the experiments, and the third after
some time. The object of these last was to ascertain if the knife
edges had suffered from use. The mean results of these three dif-
fered by less than a ten-thousandth of an inch from each other,
, the distance to be measured being o9'44085 inches.
/^^ 103. The time of a single vibration cannot be observed di-
rectly, because this would require the fraction of a second of time
as shown by the clock to be estimated either by the eye or ear.
The difficulty may be overcome by observing the time, say of a
thousand vibrations, and thus the error of the time of a single vi-
bration is divided by a thousand. The labour of so much counting
may however be avoided by the use of "the method of coinci-
dences." The pendulum is placed in front of a clock pendulum
Avhose time of vibration is slightly different. Certain marks made
on the two pendulums are observed by a telescope at the lowest
point of their arcs of vibration. The field of view is limited by
a diaphragm to a narrow aperture across which the marks are
seen to pass. At each succeeding vibration one pendulum follows
the other more closely, and at last its mark is completely covered
by the other during their passage across the field of view of the
telescope. After a few vibrations it appears again preceding the
other. In the interval from one disappearance to the next, one
pendulum has made, as nearly as possible, one complete oscillation
more than the other. In this manner 530 half- vibrations of a
clock pendulum, each equal to a second, were found to correspond
to 532 of Captain Kater's pendulum. The advantage. of this
method of observation is such, that an error of one second in noting
the interval between two coincidences would occasion an error of
only 0*63 in the number of vibrations in 24 hours. The ratio of
the times of vibration of the pendulum and the clock pendulum
may thus be calculated with extreme accuracy. The rate of going
of the clock must then be found by astronomical means.
104. The time of vibration thus obtained will require several
coiTections which are called "reductions." For instance, if the
oscillation be not so small that we can put sin ^ = ^ in Art. 92, we
must make a reduction to infinitely small arcs. The general
method of effecting this will be considered in the chapter on Sma'l
p
XENGTH OF THE SECONDS PENDULUM. 79
Oscillations. Another reduction is necessary if we wish to reduce
the result to what it would have been at the level of the sea.
The attraction of the intervening land may be allowed for by
Dr Young's rule [Phil. Trans. 1819). We may thus obtain the
force of gravity at the level of the sea, supposing all the land
above this level were cut off and the sea constrained to keep its
present level. As the level of the sea is altered by the attraction
of the land, further corrections are still necessary if we wish to re-
duce the result to the surface of that spheroid which most nearly
represents the earth. See Gamh. Phil. Titans. Vol. x.
M. Baily gives as the length of the pendulum vibrating in half
time a mean solar second in the open air in this latitude 39"lo3
inches, and the length of a similar pendulum vibrating sidereal
SQconds 38'919.im;hes.
/ '■ 105. The observations must be made in tbe air. To correct for tbis we have to
make a reduction to a vacuum. This reduction consists of three parts : (1) The
correction for buoyancy, (2) Du Buat's correction for the air dragged along by the
pendulum, (3) The resistance of the air.
Let V be the volume of the pendulum vrhich may be found by measuring the
dimensions of the body. As the "reduction to a vacuum " is only a correction, any
small unavoidable errors in calculating the dimensions will produce an effect only
of the second order on the value of X. Let p be the density of -the air when the
body is oscillating about one knife edge, p' the density when oscillating about the
other. If the observation be made within an hotu" or two hours, we may put p = p' .
The effect of buoyancy is allowed for by supposing a force Vpg to act upwards at the
.centre of gravity of the volume of the body. If the body be made as nearly as pos-
sible symmetrical about the two knife edges this centre of gravity will be half way
between the knife edges.
Du Buat discovered by experiment that a pendulum di'ags with it to and fro a
certain mass of aii" which increases the inertia of the body without adding to the
moving force of gravity. This result has been confirmed by theory. The mass
dragged bears to the mass of air displaced by the body a ratio which depends on the
external shape of the body. Let us represent it by p, Vp. If the body be symmetri-
cal about the knife edges, so that the external shape is the same whichever edge is
made the axis of suspension, p -will be the same for each oscillation. Since this
mass is to be collected at the centre of gravity of the volume, we must add to the
fe2 of equation (1) in Ai-t. 92, and therefore also in Art. 98, the termpVp f — -— j .
Taking these two con-ections the equation (1) of Art. 98 will now become
m V 2 y \ VI 2 J
where m is the mass of the pendiUum. Similarly for the oscillation about the other
knife edge,
m V 2 / \ ml)
We must eliminate /c- as before. If the observations about the two knife
80 MOTION ABOUT A FIXED AXIS.
edges succeed each other at a short interval we may put p=p', and then Du Buat's
correction will disappear. This is of com-se a very great advantage. We then have*
the last term being very small because r and t' are nearly equal.
. The resistance of the air will be some function of the angular velocity — of the
,7/1
pendulum. Since -^ is very small we may expand this function and take only the
first power. Supposing Maclaurin's theorem not to fail, and that no coefficient of a
higher power than the first is very great, this gives a resistance proportional to
- - . The equation of motion will therefore take the form
9
Tjvliere — is the time of a complete oscillation in a vacuum and the term on the
n
right-hand side is that due to the resistance of the air. The discussion of this
equation will be found in the chapter on Small Oscillations.
106. In constructing a reversible pendulum to measure the
force of gravity, the following are points of importance.
1. The axes of suspension, or knife edges, must not be at the
same distance from the centre of gravity of the mass. They
should be parallel to each other.
2. The times of oscillation about the two knife edges should
be nearly equal.
3. The external form of the body must be symmetrical, and
the same about the two axes of suspension.
4. The pendulum must be of such a regular shape that the
dimensions of all the parts can be readily calculated.
These conditions are satisfied if the pendulum be of a rect-
angular shape with two cjdinders placed one at each end. The
external forms of these cylinders are to be equal and similar, but
one is to be solid and the other hollow, and such that by calcula-
tion of moments of inertia the distance between the knife edges is
to be as nearly as possible equal to the length of the simple equi-
valent pendulum.
5. The pendulum should be made, as far as possible, of one
metal, so that as the temperature changes it may be always similar
to itself. In this case since the times of oscillations of similar
bodies vary as the square root of their linear dimensions, it is
easy to reduce the observed time of oscillation to a standard tem-
* This formula was mentioned to the author as the one used in the late experi-
ments by Capt. Heaviside to determine the length of the seconds pendulum.
LENGTH OF THE SECONDS PENDULUM. 81
perature. The knife edges however must be made of some strong-
substance not likely to be easily injured.
107. Ex. 1. If the knife edges be not perfectly sharp, let r be the diffe-ence of
their radii of curvature, show that
— — /IT- - h r^
A
very nearly when the pendulum -sdbrates in vacuo. It appears that the correction
vanishes if the knife edges be only equally sharp. By interchangmg the knife edges
we have the same equation with the sign of r changed. By making a few observa-
tions we may thus determine r. A proiDosition similar to this has been ascribed to
Laplace by Dr Young,.
Ex. 2. A heavy spherical ball is suspended successively by a very fine wire
from two points of support A and B whose vertical distance h has been carefully
measured, thus for mi ng two pendulums- The lowest point of the ball is, on each
suspension, made to be as exactly as possible on the same level, which level is
approximately at depths a and a! below A and B respectively. If r be the radius of
the ball, wliich is small compared with a or a', and I, V the lengths of the simple
l-V 2 r^
equivalent pendulum, prove that — — = 1 - - — ^ — vray nearly. By count-
ing the number of oscillations performed in a given time by each pendulum, show
how to find ratio -, . Thence show how to find g and point out wliich lengths must
be most carefully measm'ed and which need only be approximately found, so as to
render this method effective. TMs method is mentioned in Grant's History of
Physical Astronomy, page 155, as having been used by Bessel.
108. The length of the seconds pendulum has been used as a
national standard of length. By an Act of Parliament passed in
1824, it was declared that the distance between the centres of the
two points in the gold studs in the straight brass rod then in the
custody of the clerk of the House of Commons, whereon the words
and figures " standard yard, 1760 " were engraved, shall be the
original and genuine standard of length called a yard, the brass
being at the temperature of 62" Fah. And as it was expedient that
the said standard yard if injured should be restored of the same
length by reference to some invariable natural standard, it was
enacted, that the new standard yard should be of such length that
the pendulum, vibrating seconds of mean time in the latitude of
London in a vacuum at the level of the sea, should be 89*1893
inches.
On Oct. 16, 1834, occurred the fire at the Houses of Parlia-
ment, in which the standards were destroyed. The bar of 1760
was recovered, but one of its gold pins bearing a point was melted
out and the bar was otherwise injured.
In 1838 a commission was appointed to report to the govern-
ment on the course best to be pursued under the peculiar circum-
stances of the case.
K. D. 6
82 MOTION ABOUT A FIXED AXIS.
In 1841 the commission reported that they were of opinion
that the definition by which the standard yard is declared to be
a certain brass rod is the best which it is possible to adopt. With
respect to the provision for restoration they did not recommend
a reference to the length of the seconds pendulum. " Since the
passing of the act of 1824 it has been ascertained that several
elements of reduction of the pendulum experiments therein re-
ferred to are doubtful or erroneous: thus it was shown by Dr
Young, PJdl. Trans. 1819, that the reduction to the level of the
sea was doubtful ; by Bessel, Astron. Nadir. No. 128, and by
Sabine, Phil. Trans. 1829, that the reduction for the weight of air
was erroneous ; by Baily, Phil. Trans. 1832, that the specific
gravity of the pendulum was erroneously estimated and that the
faults of the agate planes introduced some elements of doubt ; by
Kater, Phil. Trans. 1830, and by Baily, Astron. Soc. Memoirs,
Vol. IX., that very sensible errors were introduced in the operation
of comparing the length of the pendulum with Shuckburgh's scale
used as a representative of the legal standard. It is evident,
therefore, thp,t the course prescribed by the act would not neces-
sarily reproduce the length of the original yard."
The commission stated that there were several measures
which had been formerly accurately compared with the original
standard yard, and by the use of these the length of the original
yard could be determined without sensible error.
In 1843 another commission was appointed to compare all the
existing measures and construct from them a new Parliamentary
standard. Unexpected difficulties occurred in the course of the
comparison, which cannot be described here. A full account of
the proceedings of the commission will be found in a paper
contributed by Sir G. Airy to the Royal Society in 1857.
Oscillation of a Watch Balance.
109. A rod B'CB can turn freely about its centre of gravity
C which is fixed, and is acted on by a very fine spiral spring CPB.
The spring has one end C fixed in position in such a manner that
the tangent at C is also fixed, and has the other end B attached
to the rod so that the tangent at B makes a constant angle with
the rod. The rod being turned through any angle, it is required
to find the time of oscillation. This is the construction used
in watches, just as the pendulum is used in clocks, to regnilate
the motion.
Let Cx be the position of the rod when in equilibrium, and
let be the angle the rod makes with Cx at any time t, Mk^ the
moment of inertia of the rod about C. Let p be the radius of
OSCILLATION OF A WATCH BALANCE. 83
curvature at any poiut P of the spring, p^ the value of p when in
equihbrium. Let {x, y) be the co-ordinates of P referred to C as
origin and Cx as axis of x. Let us consider the forces which act
on the rod and the portion BP of the spring. The forces on the
rod are X, Y the resolved parts of the reaction at C parallel to the
axes of co-ordinates, and the reversed effective forces which are
equivalent to a couple MF' -r^ . The forces on the spring are, the
reversed effective forces which are so small that they may be
neglected, and the resultant action across the section of the spring
at P. This resultant action is produced by the tensions of the
innumerable fibres which make up the spring, and these are
equivalent to a force at P and a couple. When an elastic spring-
is bent so that its curvature is changed, it is proved both by
experiment and theory that this couple is proportional to the
change of curvature at P. We may therefore represent it by
E[ j , where E depends only on the material of which the
spring is made and on the form of its section.
Taking moments about P to avoid introducing the unknown
force at P, we have
This equation is true whatever point P may be chosen. Con-
sidering the left side constant at any moment and {x, y) variable,
this becomes the intrinsic equation to the form of the spring.
Let BP=s, multiply this equation by ds and integrate along
the whole length I of the spiral spring, we have
ds
Now — is the angle between two consecutive normals, hence
ds . '
— is the angle between the extreme normals. Now at A
P
the normal to the spring is fixed throughout the motion, therefore
6—2
/
84; MOTION ABOUT A FIXED AXIS.
f/ds ds\
\{— is the angle between the normals at B in the two
positions in which 6 = 6 and ^ = 0. But since the normal at B
makes a constant angle with the rod, this angle is the angle 6
which the rod makes with its position of equilibrium. Also if
X, y be the co-ordinates of the centre of gravity of the spring at
the time t, we have \xds = xl, lyds = yl. Hence the equation of
motion becomes
72/3 7^
Mk' ^=-j6+Yx- Xy.
Let us suppose that in the position of equilibrium there is no
pressure on the axis C, then X and Y will, throughout the motion,
be small quantities of the order 6. Let us also suppose that the
fulcrum G is placed over the centre of gravity of the spring when
at rest. Then if the number of spiral turns of the spring be
numerous and if each turn be nearly circular, the centre of gravity
will never deviate far from C. So that the terms Yx and X^ are
each the product of two small quantities, and are therefore at least
of the second order. Neglecting these terms we have
M¥^l=-^6.
df I
Hence the time of oscillation is 27r
MkH
E '
It appears that to a first approximation the time of oscillation
is independent of the form of the spring in equilibrium, and
depends only on its length and on the form of its section.
This brief discussion of the motion of a watch balance is taken
from a memoir presented to the Academy of Sciences. The
reader is referred to an article in Liouville's Journal, 1860, for a
further investigation of the conditions necessary for isochronism
and for a determination of the best forms for the spring.
Pressures on the fixed axis.
110. A body moves about a fixed axis under the action of any
forces, to find the pressures on the axis.
First. Suppose the body and the forces to be symmetrical
about the plane through the centre of gravity perpendicular to
the axis. Then it is evident that the pressures on the axis are
reducible to a single force at C the centre of suspension.
Let F, G be the actions of the point of support on the body
resolved along and perpendicular to GO, where is the centre
PRESSURES ON THE FIXED AXIS.
85
of gravity. Let X, Y be the sum of the resolved parts of
the impressed forces in the same directions, and L their moment
round C.
Let CO = h and 6 = angle which CO makes with any straight
line fixed in space.
Taking moments about C, we have
d'e ^ L
de M{¥ + K') •"" ^ ^'
The motion of the centre of gravity is the same as if all the
forces acted at that point. Now it describes a circle round C;
hence, taking the tangential and normal resolutions, we have
, d'^e Y+G ,„,
, (dff\' X+F
Equation (1) gives the values of -y^ and j- , and then the
pressures may be found by equations (2) and (3).
If the only force acting on the body be that of gravity, let 6
be measured from the vertical. If the body start from rest in
that position which makes CO horizontal, we have
X = Mg cos e, Y=- Mg sin d, L = - Mgh sin 6 ;
d^9 cjh . .
df t + h^
integrating, we have
'de\^ ^ 2qh „
JQ
but when ^ = ^, -j, vanishes, therefore (7=0; substituting these
values (2) and (3), wc get
86
MOTION ABOUT A FIXED AXIS.
G = Mff Bine. j^-^,
where 6 is the angle which CO makes with the vertical.
Let -ylr be the angle the direction of the pressure at makes
with the line CO, the angle being measured from CO downwards
to the left, then
cotA/r= f l + 3pJcot^,
which is a convenient formula to determine the direction of the
pressure*.
111. Secondly. Suppose either the body or the forces not to
be symmetrical.
Let the fixed axis be taken as the axis of z with any origin
and plane of xz. These we shall afterwards so choose as to sim-
plify our process as much as possible. Let x, y, i be the co-ordi-
nates of the centre of gravity at the time t.
Let lo be the angular velocity of the body, f the angular
acceleration, so that f = —- ,
' -^ dt
Now every element m of the body describes a circle about the
axis, hence its accelerations along and perpendicular to the radius
vector r from the axis are — wV and fr. Let Q be the angle
* Let M.i2 be the resultant of F and G, and let a'=q—- and h = (i
then-^ + -J=.-,
to a and 6 measured along and perpendicular to CO. Then the resultant pressure
varies as the diameter along which it acts. And the direction may be found thus;
let the auxiliary circle cut the vertical in F, and let the perpendicular from V on
CO cut the ellipse in li. Then CR is the direction of the pressure.
Construct an ellipse with C for centre and axes equal
PEESSURES ON THE FIXED AXIS. 87
which r makes with the plane of xz at any time, then from the
resokition of forces it is clear that
_— = - &)V cos —fr sin 6 = — co'^x —/>/,
CLb
similarly -j^ = — ^V +/»•
These equations may also be obtained by differentiating the
equations a; = r cos ^, y =r ^vaO twice, remembering that r is
constant.
Conceive the body to be fixed to the axis at two points, distant
a and a from the origin, and let the reactions of the points on
the body resolved parallel to the axes be respectively F, G, H \
F', a', H'.
The equations of motion of Art. 71 then give
df
%mX+F+F' = tm ^ = 2m (- o>'x -fy)
^-(o'M^-fMy (1),
tmY-V G+G' = tm,^^,=Xm (- co'y +fx)
= -03'My+fMx (2),
%mZ-\-H+H' = Xm^,=0 (3).
Taking moments about the axes, we have
d^z d"^
tm{yZ-zY) - Ga- G'a=Xm {y^^-z-^}j
= ur'Zmyz —f%mxz (4) :
by merely introducing z into the results in (2),
%m{zX-xZ)-{-Fa-^F'a=%m(z--^-x~^
= — (o^Xmxz —fZinyz (5),
^m{xY-yX) =tm{x^^-y^)
^ o d(ii
= 2^mr . -j-
dt
= Mk\f (6).
88 MOTION ABOUT A FIXED AXIS.
Equation (6) serves to determine/ and w, and equations (1),
(2), (4), (5) then determine F, O, F', G'; ^and H' are indeter-
minate, but their sum is given by equation (3).
Looking at these equations, we see that they would be greatly
simplified in two cases.
First, if the axis of ^ be a principal axis at the origin,
Xmxz = 0, %myz = 0,
and the calculation of the right-hand sides of equations (4) and
(5) would only be so much superfluous labour. Hence, in at-
tempting a problem of this kind, we should, when possible, so
choose the origin that the axis of revolution is a principal axis
of the body at that point.
Secondly, except the determination of f and w by integrating
equation (6), the whole process is merely an algebraic substitution
of/ and (o in the remaining equations. Hence our results will
still be correct if we choose the plane of xz to contain the centre
of gravity at the moment under consideration ; this will make
^ = 0, and thus equations (1) and (2) will be simplified.
112. If the forces which act on the body be impulsive, the
equations will require some alterations.
Let ft), ft)' be the angular velocities of the body just before and
just after the action of the impulses. In the case in which the
body and forces are symmetrical, the equations (1), (2), (3) of
Art. 110 become respectively
"'""' = 71/ (/.^ + /i'0 ^^^'
A(ft)'-ft)) = i^ • (2),
= ^ (3),
where all the letters have, the same meaning as before, except
that F, G, X, Y are now impulsive instead of finite forces.
Let us next consider the case in which the forces on the body
are not symmetrical. Let u, v, lu, u, v , w be the velocities
resolved parallel to the axes of any element m whose co-ordinates
are x, y, z. Then m = — yw, u' = — yw, v = xw, v = xw, and
w, w are both zero.
The several equations of Art. Ill will then be replaced by the
following :
^X + F^F = Sm (li -u) = - tmy (ft)' - ft))
= -Jlf^(ft)'-ft)) (1),
PEESSUEES ON THE FIXED AXIS. 89
^Y+Q+G' = tm [v'-v) = %mx («' - «)
= Mx{(o' -(o) (2),
XZ+H+H' = (3),
2 {yZ— zY) — Ga— G'd = 2m [y (m —w) — z {v — ?;)}
= — "Zinxz . (o)' — ft)) (4),
t {zX-xZ) +Fa + Fa' = tm [z {v! -u)-x{w'-w)]
— — tmyz . (ft)' — ft)) (5),
t(xY-yX) =%mix' + f).{ai'-co) (6).
These six equations are sufficient to determine &)', F, F\
G, G' and the sum H+ H' oi the two pressures along the axis.
These equations admit of simplification when the origin can
be so chosen that the axis of rotation is a principal axis at that
point. In this case the right-hand sides of equations (4) and (5)
vanish. Also if the plane of xz be chosen to pass through the
centre of gravity of the body, we have ^ = 0, and the right-hand
side of equation (1) vanishes.
113. Ex. A door is suspended by two hinges from a fixed axis making an angle
a with the vertical. Find the motion and pressures on the hinges.
Since the fixed axis is evidently a principal axis at the middle point, we shall
take this point for origin. Also we shall take the plane of xz so that it contains the
centre of gravity of the door at the moment nnder consideration.
The only force acting on the door is gravity, which may be supposed to act at
the centre of gi-avity. We must first resolve this parallel to the axes. Let —^, w = 0; hence
k'^w^ = 2gx sin a (cos (p - cos j3) )
and k'^f =- g sin a sin (p . X )'
By substitution in the first four equations F, F', G, G', may be found.
^, 114. It should be noticed that these equations do not depend
/ on the form of the body, but only on its moments and products
of inertia. We may therefore replace the body by any equi-
momental body that may be convenient for our purpose.
This consideration will often enable us to reduce the compli-
cated forms of Art. Ill to the simpler ones given in Art. 110.
For though the body may not be symmetrical about a plane
through its centre of gravity perpendicular to the axis of sus-
pension, yet if the momental ellipsoid at the centre of gravity be
symmetrical about this plane we may treat the body as if it were
really symmetrical. Such a body may be said to be Dynamically
Symmetrical. If at the same time the forces be symmetrical
about the same plane, and this will always be the case if the axis
of suspension be horizontal and gravity be the only force
acting, we know that the pressures on the axis must certainly
reduce to a single pressure, which may be found by Art. 110.
115. Ex. 1. A uniform heavy lamina in the form of a sector of a circle is
^x, suspended by a horizontal axis parallel to the radius which bisects the arc, and
6\ / oscillates under the action of gravity. Show that the pressures on the axis are
equivalent to a single force, and find its magnitude.
Ex. 2. An equilateral triangle oscillates about any horizontal axis situated in
its own plane, show that the pressures are equivalent to a single force and find its
magnitude.
mP; PRESSURES ON THE FIXED AXIS. 91
116. If a body be set in rotation about any axis which is
a principal axis at some point in its length, and if there be
no impressed forces acting on the body, it follows at once from
these conditions that the pressures on the axis are equivalent
to a single resultant force acting at 0. ^ Hence if be fixed in
space, the body will continue to rotate about that axis as if it also
were fixed in space. Such an axis is called a permanent axis of
rotation at the point 0.
If the body be entirely free and yet turning about an axis
of rotation which does not alter its position in space, we may
suppose any point we please in the axis to be fixed. In this case
the axis must be a principal axis at every point of its length.
It must therefore by Art. 49 pass through the centre of gravity.
The existence of principal axes was first established by Segner
in the work Specimen Theories Turhinum. His course of in-
vestigation is the opposite of that pursued in this treatise. He
defines a principal axis to be such that when a body revolves
round it the forces arising from the rotation have no tendency
to alter the position of the axis. From this dynamical definition
he deduces the geometrical properties of these axes. The reader
may consult Prof. Cayley's report to the British Association on the
special problems of Dynamics, 1862, and Bossut, Histoire de
MatMmatique, Tome ii.
^j 117. Suppose the body to start from rest and to be acted on
^"7 by a couple, let us discover the necessary conditions that the
pressures on the fixed axis may be reduced to a single resultant
pressure. Supposing such a single resultant pressure to exist, we
can take as origin that point of the axis at which it is intersected
by the single resultant. Then the moments of the two pressures
on the axis of rotation about the co-ordinate axes will vanish.
Hence since co = the equations (4), (5), and (6) of Art. 11^ become
L^-f%mxz, M=-ftmyz, N = Mhy,
where we' have written L, M, Niov the three moments 1' be the momentum thus generated we have
If Vq and Jfl be the values of v and b when the gun is fired without a ball, we have
2m VI cf "^
Since Vq is greater than v, this equation would show that, for considerable charges,
Dr Hutton's formula will give too small a value for v. The value of v^ is however
very imperfectly known.
122. A gun is placed in front of a heavy 'pendulmn, which
can turn freely about a horizontal axis. The hall strikes the pen-
dtdum horizontally at a distance ifroni the axis of suspension. It
p>enetrates into the wood a short distance and communicates a
momentum to the pendulum. The chord of the arc being measured
as before by a piece of tape, find the velocity of the bullet.
The time, wliicti the bullet takes to penetrate, is so short that
we may suppose it completed before the pendulum has sensibly-
moved from its initial position. If we follow the same notation
as before, the moment of inertia of the pendulum and ball about
Pv. D. 7
98 MOTION ON A FIXED AXIS.
the axis of suspension will be Mk'^ + mi^, and the distance of the
centre of gravity will be —^ . Following the same reasoning,
we find
_h^Jg (MIc" + mi')^ {Mh + mi)^
ci VI
If the gun be placed as nearly as possible opposite the centre
of gravity of the pendulum, we may put h = i in the small terms,
and since M is large compared with in the formula takes the
simple form
M + m hh ,-
where I is the distance of the centre of oscillation of the pen-
dulum and ball from the axis of suspension.
The inconvenience of this construction as compared with the
former is that the balls remain in the pendulum during the time
of making one whole set of experiments. The weight, and the
positions of the centres of gravity and oscillation, will be changed
by the addition of each ball which is lodged in the wood. Even
then the changes produced in the pendulum itself by each blow
are omitted. A great improvement was made by the French in
conducting their experiments at Metz in 1839, and at L'Orient
in 1842. Instead of a mass of wood, requiring frequent renewals,
as in the English pendulum, a permanent recepteur was substi-
tuted. This receiver is shaped within as a truncated cone, which
is sufficiently long to prevent the shot from passing entirely
through the sand with which it is filled. The front is covered
with a thin sheet of lead to prevent the sand from being shaken
out. This sheet is marked by a horizontal and by a vertical
line, the intersection corresponding to the axial line of the cone,
so that the actual position of the shot when entering the re-
ceiver can be readily determined by these lines.
Ex. 1. Show that after each bullet has been fired into a ballistic pendulum
constructed on the English plan, h must be increased by ^ (i - h) and I by -=j.(t - 1)
nearly in order to prepare the formula for the next shot.
Ex. 2. Dr Haughton found that, for rifles fired with a constant charge, the
initial velocity of the bullet varies as the square root of the mass of the bullet in-
versely and as the square root of the length of the gun directly. Show from this,
that the force developed by the explosion of the powder diminished by the friction
of the barrel is constant as the ball traverses the riflie.
Dr Hutton found that in smooth bores the velocity increases in a ratio some-
what less than the square root of the length of the gun, biit greater than the cube
root of the length. Show that this might be expected from the decreased friction in
a smooth bore as compared with a rifle.
THE BALLISTIC PENDULUM. 99
Ex. 3. If the velocity of a bullet issuing from the mouth of a gun 30 inches
long be 1000 feet per second, show that the time the bullet took to traverse the gun
was about -^^ of a second.
Ex. 4. It has been found by experiment that if a bullet be fired into a large
fixed block of wood, the penetration of the bullet into the wood varies nearly as the
square of the velocity, though as the velocity is very much increased the depth of
penetration falls short of that given by this rule. Assuming this rule, show that
the resistance to penetration is constant and that the time of penetration is the
ratio of twice the sj^ace to the initial velocity of the bullet. In an experiment of
Dr Hutton's a ball fired with a velocity of 1500 feet per second was found to pene-
trate about 14 inches into a block of sound dry elm : show that the time of penetra-
tion was -jri-j. of a second.
CHAPTER IV.
MOTION IN TWO DIMENSIONS.
On the Equations of Motion.
123. The position of a body in space of two dimensions
may be determined by the co-ordinates of its centre of gravity,
and the angle some straight line fixed in the body makes with
some straight line fixed in space. These three have been called the
co-ordinates of the body, and it is our object to determine them
in terms of the time.
It will be necessary to express the effective forces of the body
in terms of these co-ordinates. The resolved parts of these
effective forces parallel to the axes have been already found in
Art. 79, all that is now necessary is to find their moment about
the centre of gravity. If {x, y) be the co-ordinates of any
particle of mass m referred to rectangular axes meeting at the
centre of gravity and parallel to axes fixed in space, this moment
has been shown in Art. 72 to be equal to -r: , where
, dy , dx^
^ ( ,dy ,dx\
Let 6 be the "angular co-ordinate" of the body, i.e. the angle
some straight line fixed in the body makes with some straight line
fixed in space. Let {r, ^') be the polar co-ordinates of any par-
ticle m referred to the centre of gravity of the body as origin.
Then r is constant throughout the motion, and -— is the same
dB
for every particle of the body and equal to ~j- . Thus the an-
gular momentum h, exactly as in Art. 88, is
^^^(^'i-^'J)=^^'(^
dt J
at
THE EQUATIONS OF MOTION. 101
where Mh^ is the moment of inertia of the body about its centre
of gravity.
The angle 6 is the angle some straight line fixed in the body
makes with a straight line fixed in space. "Whatever straight
lines are chosen -j is the same. If this be not obvious, it may
be shown thus. Let OA, O'A' be any two straight lines fixed in
the body inclined at an angle a to each other. Let OB, O'B' be
two straight lines fixed in space inclined at an angle ^ to each
other. Let AOB=e, A' O'B' =6', then ^' + yS = 6' + a. Since
a and /3 are independent of the time, ~j1 = ~37 - ^J this propo-
sition we learn that the angular velocities of a body in two di-
mensions are the same about all points.
The general method of proceeding will be as follows.
Let {x, y) be the co-ordinates of the centre of gravity of
any body of the system referred to rectangular axes fixed in space,
M the mass of the body. Then the effective forces of the body
are together equivalent to two forces measured by M-^ , M-~
acting at the centre of gravity and parallel to the axes of co-
ordinates, together with a couple measured by il/F -7^ tending to
turn the body about its centre of gravity in the direction in which
6 is measured. By D'Alembert's principle the effective forces of all
the bodies, if reversed, will be in equilibrium with the impressed
forces. The dynamical equations may then be formed according
to the ordinary rules of Statics.
For example, if we took moments about a point whose co-
ordinates are (j9, q) we should have an equation of the form
where L is the moment of the impressed forces and the other
letters have the same meaning as before. In this equation {p, q)
may be the co-ordinates of any point whatever, whether fixed
or moving. Just as in a statical problem, the solution of the
equations may frequently be much simplified by a proper choice
of the point about which to take moments. Thus if we wished
to avoid the introduction into our equations of some unknown
reaction, we might take moments about the point of application
or use the principle of virtual velocities. So again in resolving
102 MOTION IN TWO DIMENSIONS.
our forces we might replace the Cartesian expressions M -^ ,
M -^ by the polar forms
^^--..ff)V--i|(4t)
. {d'r (c
for the resolved parts parallel and perpendicular to the radius
vector. If V be the velocity of the centre of gravity, p the radius
of curvature of its path, we may sometimes also use with advantage
the forms J/^- and M — for the resolved parts of the effective
at p
forces along the tangent and radius of curvature of the path of the
centre of gravity.
V 124. As we shall have so frequently to use the equation
formed by taking moments, it is important to consider other forms
into which it may be put. Let the point about which we are
to take moments be fixed in space, so that it may be chosen as
the origin of co-ordinates. Then the moment of the effective
forces on the body M is
d { nr f dy dx\ ^^-.odO
The attention of the reader is directed to the meaning of the
several parts of this expression. We see that, as explained in
Art. 72, the moment of the effective forces is the differential
coefficient of the moment of the momentum about the same point.
The moment of the momentum by Art. 76 is the same as the moment
about the centre of gravity together with the moment of the whole
mass collected at the centre of gravity, and moving with the velocity
of the centre of gravity. The moment round the centre of gravity
is by the first Article either of Chap. iii. or Chap. iv. equal to
Mh^ -T- and the moment of the collected mass is, M [x -~- — y -=r],
dt \ dt "^ dtj
where {x, y) are the co-ordinates of the centre of gravity. Hence
in space of two dimensions we have for any body of mass M
angular momentum round) _ ,,r / dy dx\ ifi^d^
the origin \~ ^^^ V dt'^Tt)'^ ^^^ dt '
If we prefer to use polar co-ordinates, we can put this into
another form. Let (r, 0) be the polar co-ordinates of the centre of
gravity, then,
angular momentum round] _ -.j. ^d^ ^„^d9
the origin | dt dt'
If V be the velocity of the centre of gravity, and j^ the per-
pendicular from the origin on the tangent to its direction of
THE EQUATIONS OF MOTION. 103
motion, the moment of momentum of the mass collected at the
centre of gravity is Mvp, so that we also have
angular momentum round] ,^ -nnodQ
the origin J -^ at
It is clear from Art. 76 that this is the instantaneous angular
momentum of the body about the origin, whether it is fixed or
moveable, though in the latter case its differential coefficient with
regard to t is not the moment of the effective forces.
Since the instantaneous centre of rotation may be regarded as
a fixed point, when we have to deal only with the coordinates and
with their first differential coefficients with regard to the time, we
' have
angular momentum round thel ,^ , 2 7.2% ^^
instantaneous centre J ^ dt'
If Mlc'^ be the moment of inertia about the instantaneous
centre, this last moment may be written MW^ -j, .
In taking moments about any point whether it be the centre
of gravity or not, it should be noticed that the 3Ik^ in all these
formulse is the moment of inertia with regard to the centre of
gravity, and not with regard to the point about which we are
taking moments. It is only when we are taking moments about
the instantaneous centre or about a fixed point that we can use
the moment of inertia about that point instead of the moment
of inertia about the centre of gravity, and in that case our expres-
sion for the angular momentum includes the angular momentum
of the mass collected at the centre of gravity.
V
J f' 125. Suppose we form the equations of motion of each
Y body by resolving parallel to the axes of co-ordinates and by
taking moments about the centre of gravity. We shall get
three equations for each body of the form
M -j-^ = Fcos (fi + R cos yjr + ...
I
J/-^ = F sin (ji + Ji sin ^}r + ... j" (1),
Mk'^= Fp +Mq +...I
where F is any one of the impressed forces acting, on the body,
whose resolved parts are i^cos (f>, Fsin (p, and whose moment
about the centre of gravity is Fp, and R is any one of the re-
actions. Tliesc wc shall call the Dynamical equations of the body.
104 MOTION IN TWO DIMENSIONS.
Besides these there will be certain geometrical equations
expressing the connections of the system. As every such forced
connection is accompanied by a reaction and every reaction by
some forced connection, the number of geometrical equations will
be the same as the number of unknown reactions in the system.
Having obtained the proper number of equations of motion
we proceed to their solution. Two general methods have been
proposed.
First Method. Differentiate the geometrical equations twice
with respect to t, and substitute for -^ , -^ , -j-^ , from the
dynamical equations. We shall then have a sufficient number
of equations to determine the reactions. This method will be of
great advantage whenever the geometrical equations are of the
form
Aa; + Btj+ CO = D (2),
where A, B, C, D are constants. Suppose also that the dynamical
equations are such that when written in the form (1) they contain
only the reactions and constants on the right-hand side without
any x, y, or 6. Then, when we substitute in the equation
. dJ^x -r,dh/ ^ d^O
obtained by differentiating (1), we have an equation containing
only the reactions and constants. This being true for all the
geometrical relations, it is evident that all the reactions will be
constant throughout the motion and their values may be found.
Hence when these values are substituted in the dynamical equa-
tions (1), their right-hand members will all be constants and the
values of x, y, and 6 may be found by an easy integration.
If however the geometrical equations are not of the form (2),
this method of solution will usually fail. For suppose any geo-
metrical equation took the form
2 1 2 2
X -^y = c ,
containing squares instead of first powers, then its second dif-
ferential equation will be
d^x d^y fdxV /di/V ^
cL o(* ft "1/
and though we can substitute for -j-^ , -~ , we cannot, in general,
Ctu Ctu
eliminate the terms ( -t- j and i fr] ■
t
THE EQUATIONS OF MOTION.
105
126. The reactions in a dynamical problem are in many
cases produced by the pressures of, some smooth fixed obstacles
which are touched by the moving bodies. Such obstacles can only
push, and therefore if the equation showed that such a reaction
changes sign at any instant, it is clear that the body will leave the
obstacle at that instant. This will occasionally introduce discon-
tinuity into our equations. At first the system moves under
certain constraints, and our equations are found on that suppo-
sition. At some instant which may be determined by the vanish-
ing of some reaction, one of the bodies leaves its constraints and
the equations of motion have to be changed by the omission of
this reaction. Similar remarks apply if the reactions be produced
by the pressure of one body against another.
It is important to notice that when this first method of solu-
tion applies, the reactions are constant throughout the motion, so
Q that this kind of discontinuity can never occur. If a moving
^ body be in contact with another, they will either separate at the
beginning of the motion or will always continue in contact.
\ ^' 127. Suppose that in a dynamical system we have two bodies
''' which press on each other with a reaction _R; let us consider
how we should form the corresponding geometrical equation.
We have clearly to express the fact that the velocities of the
points of contact of the two bodies resolved along the direc-
tion of R are equal. The following proposition will be often
useful. Let a body be turning about a point with an angular
clB
velocity "tT = « in a direction opposite to the hands of a watch,
and let G be moving in the direction GA with a velocity V. It
is required to find the velocity of any point P resolved in any
^"
direction PQ, making an angle (^ with GA. In the time dt the
wliole body, and therefore also the point P, is moved through a
space Vdt parallel to GA, and during the same time P is moved
perpendicular to GP through a space w.GP . dt. Resolving
parallel to PQ, the whole displacement of P
= ( Fcos - w . GP sin GPN) dt.
106 MOTION IN TWO DIMENSIONS.
If GN=p be the perpendicular from G on PQ, we see that the
velocity of P parallel to PQ is = V cos cfi — cojj.
It should be noticed that this is independent of the position of
P on the straight line PQ. It follows that the velocities of all
points in any straight line PQ resolved along PQ are the same.
In practice, therefore, we only use that point in the direction
of PQ which is most convenient, and this is generally the foot of
the perpendicular from the centre of gravity.
If (x, y, 6), [x , y , 9) be the co-ordinates of the two bodies,
q, q the perpendiculars from the points [x, y), {w, y') on the direc-
tion of any reaction R, yjr the angle the direction of B makes with
the axis of x, the required geometrical equation will be
dx . dy . , dd dx , dy' . , dO' ,
^cost + Jsmf+^^=^cosf + -^sm^/. + ^^.
If the bodies be perfectly rough and roll on each other
without sliding, there will be tivo reactions at the point of contact,
one normal and the other tangential to the common surface of the
touching bodies. For each of these we shall have an equation
similar to that just found. But if there be any sliding friction
this reasoning will not apply. This case will be considered a little
further on.
128. Second Method of Solution. Suppose in a dynamical
system two bodies of masses M, M' are pressing on each other
with a reaction R. Let the equations of motion of M be those
marked (1) in Art. 125, and let those of M' be obtained from
these by accenting all the letters except R, yjr and t, and writing
— R for R, yjr and t being of course unaltered. Let us multiply
(X.-C (X'?/ clO
the equation of if by 2-7-, ^ -f. , ^ -r. respectively, and those of
M' by corresponding quantities. Adding all these six equations,
we get
fdx d'x dy d'y „ dO d'6\
_, „ / , dx . , dti d6\ . „
^ „ / , dx . , dii dO
+ 2i2(^cos^/.^ + smt^ + ^^
^ „ / , dx . , dy' , dd'
_2Jj(cost^+smt^+5^
The coefficient of R will vanish by virtue of the geometrical
equation obtained in the last Article. And this reasoning will
apply to all the reactions between each two of the moving bodies.
THE EQUATIONS OF MOTION. 107
Suppose the body M to press against some external fixed
obstacle, then in this case R acts only on the body M, and it's
coefficient will be restricted to the part included in the first
bracket. But the velocity of the point of contact resolved along
the direction of R must vanish, and therefore the coefficient of
R is again zero.
Let A be the point of application of the impressed force F,
df
and let -j be the velocity of ^ resolved along the direction of action
df
of F. Then we see that the coefficient of 2F is -~- . It also
dt
. . df
follows from the definition of -~ that Fdf is what is called in
Statics the virtual moment of the force F. "^
We have thus a general method of obtaining an equation free
from the unknown reactions of perfectly smooth or perfectly
rough bodies. The rule is, Multiply the equations having
M^r^, M^r^, Mh^ —rr, , &c. on their left-hand sides by ~ ,
df df df ^ dt'
-y , -J- , &c., and add together all the resulting equations for all
Civ Cut
the bodies. The coefficients of all the unknown reactions will be
found to be zero by virtue of the geometrical equations.
The left-hand side of the equation thus obtained is clearly
a perfect differential. Integrating we get
where C is the constant of integration.
In practice it is usual to omit all the intermediate steps and
write down the resulting equation in the following manner:
where ?7is the integral of the virtual moment of the forces. -1^>
This is called the equation of Vis Viva. Another proof will
be given in the chapter under that heading.
129. The left-hand side of this equation is called the vis viva
of the whole system. Taking any one body M, we may say that
M-m-'y- 10 ,, , 10 ^, ,
(a + o) I -^ I =5rcos9i; .*. — (/(I -cos0)=5^ cos 0; .'. eosi^ = — . It may be re-
marked that this result is independent of the magnitudes of the spheres.
Ex. 1. If the spheres had been smooth the upper s^jhere would have left the
lower sphere when cos — -|.
/ Ex. 2, A rod rests with one extremity on a smooth horizontal plane and the
other on a smooth vertical wall at an inclination a to the horizon. If it then slijjs
down, show that it will leave the wall when its inclination is siu"^ (!; sin a)._
EXAMPLES.
113
^J
Ex. 3. A beam is rotating on a smooth liorizontal plane about one extremity,
■wMch is fixed, under the action of no forces except the resistance of the atmosphere.
Supposing the retarding effect of the resistance on a small element of the beam of
length a to be Aa (vel.)^, then the angular velocity at the time t is given by
[Queens' Coll.]
x/
w fi \Mie
Ex. 4. An inclined plane of mass M is capable of moving freely on a smooth
horizontal plane. A perfectly rough sphere of mass m is placed on its inclined face
and rolls down under the action of gravity. If a;' be the horizontal space advanced
by the inclined plane, x the part of the plane rolled over by the sphere, prove that
(ilf + ?») x' = mx cos a,
^x - cos ax' = \g sin at^,
where a is the inclination of the plane to the horizon.
Ex. 5. Two equal perfectly rough spheres are placed in unstable equilibrium,
one on the top of the other ; the lower sphere resting on a perfectly smooth table.
The sUghtest disturbance being given to the system, shew that the spheres will
continue to touch each other at the same points and if 6 be the inclination to the
vertical of the straight line joining the centres,
-r-\ =2ga{l-cos2e).
(F + a2 + a2sin2(
Ex. 6. Two imequal perfectly smooth spheres are placed in unstable equihbrium
one on the top of the other ; the lower sphere resting on a perfectly smooth table.
A very slight distm-bance being given to the system, shew that the spheres will
separate when the straight hne joining the centres makes an angle ^ with the verti-
cal, given by the equation — ^
lower and m of the upper sphere.
cos^ ^ - 3 cos + 2 = 0, where M is the mass of the
Ex. 7. A sphere of mass M and radius a is constrained to roll on a perfectly
rough curve of any form and initially the velocity of its centre of gravity is V. If
the initial velocity were changed to V, shew that the normal reaction would be
Y'2 _ y2
increased by M and that the friction would be unaltered, p being the
p-a r o
radius of curvature of the curve at the point of contact.
\ 135. A rod OA can turn about a liinge at 0, while the end A rests on a smooth
wedge lohich can slide along a smooth horizontal plane through O. Find the motion.
Let a = the inclination of the wedge, iHf =its mass and x— 00.
11. D.
114 MOTION IN TWO DIMENSIONS.
Let Z = the lengtli of the beam, m = its mass and d = A 00.
Let ^ = tlie reaction at A. Then we have
the dynamical equations,
cPx R sin a - ,
d^^^^r ^ ''
,g Rl . COS (a - ^) - mg - cos 6
'dt^ mB """"^ ^ ''
and the geometrical equation,
x = -. .sin (a -61) (3).
sin a
It is obvious we must apply the second method of sokition. Hence
„,,cZxd% „ ,„d9d^d -, ^dd „„( . dx ^ , „.d9)
The coefficient of R is seen to vanish by differentiating equation (3). Inte-
grating we have
This result might have been written down at once by the principle of vis viva.
For the vis viva of the wedge is clearly If ( — J and that of the rod Mh^ \]t] '
The vh-tual moment of the forces is -mgdy where y is the altitude above OC of the
centre of gravity of the rod OA, hence twice the force function is C—%ngy. Since
y=^lsm. 6, this reduces to the result already written down.
Substituting from (3) we have
( sm'' a
) f dQ\ ^
cos2{a-^) + mA;- (-jr) =C-mgl sin 9 (4).
If the beam start from rest when 9 — j3, then C=mgl sin /3.
This equation cannot be integrated any further. We cannot therefore find 9 in
terms of t. But the angular velocity of the beam, and therefore the velocity of the
wedge, is given by the above equation.
^C' 136. Tivo rods A B, BO are hinged together at B and can freely slide on a
smooth horizontal plane. The extremity A of the rod AB is attached by another
hinge to a fixed point on the table. An elastic string AC, xohose unstretched length
is equal to AB or BO, joins A to the extremity O of the rod BC. Initially the two
rods and the string form an equilateral triangle and the system is started with an
angular velocity round A. Find the greatest length of the elastic string during the
motion. Find also the angular velocities of the rods ivhen they are at right angles^
and the least value of fi that this may be possible.
Let the length of either rod be 2a, mF the moment of inertia of either about its
a^
centre of gravity, so that F = — . Let D and JE be the middle points of the rods,
and let {r, 9) be the polar co-ordinates of E referred to A as origin.
The only forces on the system are the reaction of the hinge at A and the tension
of the elastic string A 0. If we search for any direction in which the sum of the
resolved parts of these vanishes, we can find none, since the direction of the
EXAMPLES. 115
reaction is at present unknown. But since the lines of action of botli these forces
pass thi-ough A, their moments about A vanish, and therefore, by Art. 132, the
angular momentum about A is constant throughout the motion and equal to its
initial value. Let w, w' be the angular velocities oi AB, £C at any instant t. The
angular momentum of BO about A is by Art. 124 m {r^ -^ + A;^w'). The angular
momentum of AB is by the same article mfP + a^) w, since IB is turning about A
as a fixed point. The initial values of these are respectively ^^(Sa^O + Zc'^O), and
m{h^ + a'^)Q, suice w, w' and ^ are each initially equal to fi, and r is initially
equal to the perpendicular from A on the opposite side of the equilateral triangle
formed by the system. Hence
m(Jc^ + a^)co + inF-a)' + mr-^^ = m{2h^ + id')Q (1).
We may obtain another equation by the use of the principle of vis viva. The
vis viva of the rod BC is by Art. 129 m j (—X + r" (^ + Fw'^ j . The vis viva of
AB is by the same article m Qfi + oP) ofi since it is turning round 4 as a fixed point.
The initial values of these are respectively in (Sa^ + ¥) fl^ and m [h^ + a?) Q,K If T be
the tension of the string, p its length at time t, the force function of the tension is
fi-T) dp. According to the rule given in Statics to calculate virtual moments,
the minus sign is given to the tension because it acts to diminish p ; and the limits
are 2a to p because the string has stretched from its initial length 2a to p. By
o - 2a (p - 2a)^
Hooke's law T=E f—^ — , so that, by integration, the force function = - E — -^ — .
The reaction at A does not appear by Art. 128. The equation of vis viva is
therefore
m(P + a>Hmj(jy + r^(|)%7.V^|=m(2P + 4a^)J^^-i:^^^ (2).
There are only two possible independent motions of the rods. We can turn A B
about A and BC about B, aU other motions, not compounded of these, are incon-
sistent with the geometrical conditions of the question. Two dynamical equations
R — 9
116 MOTION IN TWO DIMENSIONS.
are sufficient to determine these, and these we have jiist obtained. AU the other
equations which may be wanted must be derived from geometrical considerations.
We must now express the geometrical conditions of the question. Let be the
supplement of the angle ABO, then
r^- = 5a^ + 4.a^ cos - (6),
^ . d^p de ,
and since ^ = -^ - w, we have
at dt
^^^'^\di~'^j f^cos^ + ~3 sin'-^JK-w) (7).
Also from the triangle j4 5(7
/)2 + 2a2=2r2 (8).
dr do
From these eight equations we can eliminate w, w', r, ~r- , p, \p and -j- . We shall
then have a differential equation of the first order to solve, containing and -7- .
It is required to find the greatest length of the elastic string during the motion.
At the moment when p is a maximum, -r^ = and the whole system is therefore
moving as if it were a rigid body. We therefore have for a single moment w, w' and
— all equal to each other and ^- = 0. The two first equations become, when we
dt '■ dt
have substituted for ¥■ its value — ,
(5a2 + 3r2)w = 14a2
(5a2 + 3r2) w^ = Ua"^ Q!' - M_ (^ - 2af
Eliminating w and substituting for r from (8) we have the cubic
(3p^ + 16a^)(^ - 2a) J^:!!^ . (^ + 2a),
which has one positive root greater than 2a.
It is also required to find the motion at the instant lohen the rods are at right
angles. At this moment = ^ and hence by (3) r=a^Jb, by (5) --r-= -—7= a {00' -co),
_ Ir.i 4-Ai,i\. Snb
dt 6
do 1
by (7) — — - (a)' + 4w). Substituting in equations (1) and (2) we get
17
4w+w' = -0 "I
2 ma 2
EXAMPLES. 117
From these two equations we may easily find w and w'. It is easUy seen that the
values of w, w' will not be real unless (^^>— {J'2 - 1)^.
7 ma
We may often save ourselves the trouble of some ehmination if we form the
equations derived from the principles of angular momentum and vis viva in a
shghtly different manner. The rod £C is tm-ning round B with an angular velocity
w', while at the same time B is moving perpendicularly to AB with a velocity 2aw.
The velocity of E is therefore the resultant of am' perpendicular to BC and 2aw per-
pendicular to 41?, both velocities, of course, being appHed to the point E. When
we wish our results to be ex]3ressed in terms of w, w' we may use these velocities to
express the motion of E instead of the polar co-ordinates {r, 6).
Thus in applying the principle of angular momentum, we have to take the
moment of the velocity of E about A. Since the velocity 2flw is perpendicular to
AB, the length of the perpendicular from A on its direction i&AB together with the
projection of i?JS on J. 5, which is 2a -f- a cos 0. Since the velocity aw' is perjjen-
dicular to BE, the length of the perpendicular from A on its liae of action is BE
together with the projection of AB on BE, which is a -f 2a cos (p. Hence the angu-
lar momentum of the rod BG about A is, by Ai-t. 124,
MiF w'-f 2;;iaw (2a + a cos (p) + maw' (a + 2a cos (p).
The principle of angular momentum for the two rods gives therefore
m (F + 5a^ + 2a^ cos 0) w + m {Jc^ + a^ + 2a'^ cos (p) w' = vi (2k^ + 4a^) fi.
The right-hand side of this equation, being the initial value of the angular momen-
tum, is derived from the left-hand side by putting cos 0= - i and w = w'=fi.
In applying the principle of vis viva, we require the velocity of E. Eegarding
it as the resultant of 2aca and aw' we see that, if v be this velocity,
v^ = (2au)^+ [au')^ + 2 . 2aw . aw' cos (p.
The initial value being found, as before, by putting cos (p— -h, w = w'=fi, the princi-
ple of vis viva gives, by Art. 129,
m (i^ + 5a^) or + m {¥- + a^) w'- + ima" ww' cos = m (2^^ + 4a2) 0^ _ ^ \PjZ. — I _
The force function is found in the same manner as before. If we join to this equa-
tion (4) given above, and substitute p=4a cos - , we have just three equations to find
w, w', and tp. If these quantities are all that are require:!, as in the two cases con-
sidered above, this form of solution has the advantage of brevity. When /> is a
maximum, we put w = w', when the rods are at right angles, we put cos ^ = 0. The
equations then lead to the results already given.
■*" 137. The bob of a heavy pendulum contains a spherical cavity which is filled
with loater. To determine the motion.
Let be the point of suspension, G the centre of gravity of the solid part of the
pendulum, MK^ its moment of inertia about and let 00 =h. Let C be the centre
of the sphere of water, a its radius and OC=c. Let m be the mass of the water.
If we suppose the water to be a perfect fluid, the action between it and the case
must, by the definition of a fluid, be normal to the spherical boundary. There will
therefore be no force tending to turn the fluid round its centre of gravity. As the
pendulum oscillates to and fro, the centre of the sphere Avill partake of its motion,
but there will be no rotation of the water.
118 MOTION IN TWO DIMENSIONy,
The effective forces of the water are by Art. 123 equivalent to the effective force of
the whole mass collected at its centre of gravity together with a couple ml? -j-
where w is the angular velocity of the water, and ml? its moment of inertia about a
diameter. But w has just been proved zero, hence this couple may be omitted. It
follows that in all problems of this kind where the body does not turn, or turns with
uniform angular velocity, we may collect the body into a single particle placed at
its centre of gravity.
The pendulum and the collected fluid now form a rigid body turning about a
fixed axis, hence if 6 be the angle CO a fixed line in the body makes with the
vertical, the equation of motion by Art. 88 is
,72 O
{MK^ + mc^) -T-^ + {Mh + mc) gsinO^ 0,
where in finding the moment of gravity, 0, G and C have been supposed to lie in a
straight line.
The length L' of the simple equivalent pendulum is, by Art. 92,
" Mh + mc
Let ml? be the moment of inertia of the sphere of water about a diameter.
Then if the water were to become solid and to be rigidly connected with the case,
the length L of the simple equivalent pendulum would be, by similar reasoning,
MK^ + m{c^ + ¥)
Z =
Mh+mc
It appears that L'
J 2a J 2a dt^
the limits being the same as before. This gives
,^ mq sin 6 ,^ . ,„ „ ,
r= .^ , 2a - x) 2a - 3a;),
which vanishes when the tendency to break is a maximum, and is a maximum at a
distance from the fixed end equal to two-thirds of the length of the rod.
To find the tension at P we must resolve along the rod. If the result be called
X, we have
r da ^ r du , . [dd^
If the rod start from rest at an inclination a to the vertical, we find, by integrating
fdd\^ 2,(1
the equation of motion, ( — 1 = ^ (cosa- cos 6^). Hence
Z=^2 (2a - x) { - 4a cos ^ -)- 3 (cos a - cos 6) (2a + x)].
From these equations we may deduce the following results. (1) The magnitudes
of the stress couple and of the shear are independent of the initial conditions.
(2) The magnitude of either the couple or the shear at any given point of the rod
varies as the sine of the inchnation of the rod to the vertical. (3) The ratio of the
'magnitudes of the stress couples at any two given points of the rod is always the
same, and the same proposition is also true of the shear, (4) The tension depends
on the iuitial conditions and unless the rod start from rest in the horizontal position,"
the ratio of the tensions at any two given points varies with the position of the rod.
141. A rigid hoop coinpletely cracked at one point rolls on a perfectly rough
horizontal plane and.is acted on by no forces but gravity. Prove that the lorencli
couple at the point of the hoop most remote from the crack loill be a maximum lohen-
cver, the crack being lower than the centre, the inclination of the diameter through
2
the crack to the horizon is tan~^ — , [The Math. Tripos, 1864.]
Let 0} be the angular velocity of the hoop, a its radius. The velocity of any
point P of the hoop is the resultant of a velocity aw parallel to the horizontal plane
and an equal velocity aw along a tangent to the hoop. The first is Constant in
direction and magnitude and therefore gives nothing to the acceleration of P. The
latter is constant in magnitude but variable in direction and gives au^ as the
acceleration which is directed along a radius of the hoop. Let A be the cracked
point, B the other end of the diameter, C the centre, the inclination of A CB to
ON STEESS. 123
the horizon. Let PP' be any element on the upper half of the circle, BCP = (p.
Then the wrench couple, or tendency to break, at B is proportional to
'' n
[-au^a sin ^ /— • [^o^^- Exam.]
Ex. 3. A wu'e in the form of the portion of the curve r = a (1 + cos 0) cut off by i^
the initial line rotates about the origin with angular velocity w. Prove that the
IT 12 J2
tendency to break at the point ^ = ^ is measured by m — = — w^a^. [St John's Coll.]
On Friction between Imperfectly/ Bough Bodies.
142. When one body rolls on another under pressure, the two
bodies yield slightly, and are therefore in contact along a small
area. At every point of this area there is a mutual action be-
tween the bodies. The elements just behind the geometrical
point of contact are on the point of separation and may tend to
adhere to each other, those in front may tend to resist com-
pression. The whole of the actions across all the elements are
equivalent to (1) a component R, normal to the common tan-
gent plane, and usually called the reaction ; (2) a component F
in the tangent plane usually called the friction ; (3) a couple L
about an axis lying in the tangent plane and which we shall call
the couple of rolling friction ; (4) if the bodies have any relative
angular velocity about their common normal, a couple N about
this normal as axis which may be called the couple of twisting
friction.
124 MOTION IN TWO DIMENSIONS.
143. These two couples are found by experiment to be in
most cases very small and are generally neglected. But in certain
cases where the friction forces are also small, it may be necessary
to take account of them.
144. When one body presses against another over any small
area, the force of friction acts in such a direction and with such a
magnitude that it is just sufficient to prevent sliding. Both the
magnitude and direction of friction may, therefore, be unknown
beforehand, and their determination will be part of the problem
under consideration. It is found by experiment that no more
than a certain amount of friction can be called into play, and
when more is required to keep the bodies from sliding on each
other, sliding will begin. This amount is called limiting friction.
The magnitude of this limit is found to bear a ratio to the normal
pressure which is very nearly constant for the same two bodies.
Though all experimenters have not entirely agreed with each other
as to the accuracy of this result, yet it has been found generally that,
if the relative motion of the two bodies be the same at all points of
the area of contact, this ratio is nearly independent of the extent of
the area and of the relative velocity. If, however, the bodies have
remained in contact for some time under pressure in a position
of equilibrium, it is found that, for the more compressible bodies,
the ratio is a little greater than after motion has begun. This
ratio has been called the coefficient of friction of the materials of
the two bodies. Its constancy is generally assumed by mathema-
ticians. When the friction which can be called into play is insuf-
ficient to prevent sliding, the bodies slide on each other. In this
case the magnitude of the friction is equal to its limiting value,
and the direction of the friction is opposite to that of relative
motion.
145. If the bodies be perfectly rough, the coefficient of friction
is infinite, and there is no limit to the amount of friction which
can be called into play. There can, therefore, be no sliding be-
tween the bodies.
146. Discontinuity of motion will often occur when a body
moves under the action of friction. Suppose the body rolls on a
rough surface, the friction called into play just prevents sliding,
and is possibly variable in magnitude and direction. By writing
down and solving the equations of motion we can find the ratio of
the friction F to the normal pressure B. If this ratio be always
less than the coefficient //. of friction, enough friction can always
be called into play to make the body roll on the rough surface.
In this case we have obtained the true motion. But if at any
. F
instant the ratio -p thus found should be greater than the co-
I
IMPEKFECT FRICTION. 125
efficient of friction, the point of contact will begin to slide at that
moment. In this case the equations do not represent the true
motion. To correct them we must replace the unknown friction
F by fxR, and remove the geometrical equation which expresses
the fact that there is no slipping between the bodies. The equa-
tions must now be again solved on this new supposition. It is of
course possible that another change may take place. If at any
instant the velocities of the points of contact become equal to
each other, all the possible friction may not be called into play.
At that instant the friction ceases to be equal to /xi^ and becomes
again unknown in magnitude and direction.
Discontinuity may also arise in other ways. When, for example,
one body is sliding over another, the friction is opposite to the
direction of relative motion, and numerically equal to the normal
reaction multiplied by the coefficient of friction. If then, during
the course of the motion the direction of the normal reaction
should change sign, while the direction of motion remains un-
altered ; or if the direction of motion should change sisfn while
the normal reaction should remain unaltered, the sign of the
coeflficient of friction must be changed. This may modify the
dynamical equations and alter the subsequent motion. The same
cause of discontinuity operates when a body moves in a resisting
medium, when the law of resistance is an even function of the
velocity, or any function which does not change sign when the
direction of motion is changed.
In some cases the motion may be rendered indeterminate by
the introduction of friction. Thus, we have seen in Art. Ill, that
when a body swings on two hinges, the pressures on the hinges
resolved in the direction of the straight line joining them cannot
be found. The sum of these components can be found, but not
either of them. But there was no indeterminateness in the
motion. If however these hinges were imperfectly rough, there
would be two friction couples, one at each hinge, acting on the
body. The common axis of these couples would be the straight
line joining the hinges. The magnitude of each would be equal
to the pressure resolved along its axis multiplied by a constant
depending on the roughness of the hinge. If the hinges were
unequally rough, the magnitude of the resultant couple would
depend on the distribution of the pressure on the two hinges. In
such a case the motion of the body would be indeterminate.
' 147. A homogeneous sphere is placed at rest on a rough inclined plane, the
coefficient of friction being /m, determine zvhether the sphere ivill slide or roll.
Let F be the friction required to make the sphere roll. The problem then
becomes the same as that discussed in Ai't. 133. We have, therefore, - =| tan a,
where a is the inclination of the plane to the horizon.
126 MOTION IN TWO DIMENSIONS.
If tlien f tan a be not greater tlian /x, the solution given in tlae article referred
to is the correct one. But if ^ < f tan a the sphere will begin to slide on the
inclined plane. The subsequent motion will be given by the equations
m -T-2='>ng sin a -fiR ]
0— -mg cos a + R y
d^x d'9 I
ma^r-^ + ml? -r-r, = mga sin a I
dt^ dr J
whence we have, remembering that h" = f a^,
d'^x 1
— 2 z= gr (sin a - /i cos a)
' d^o . g ]'
•772=1/^- cos a
dP a J
Since the sphere starts from rest, we have by integration
a;=|<7f" (sin a-^cos a) l
6> = f ^ ^ i2 cos a I
a J
The velocity of the point of the sphere in contact with the plane is
dx dO . , . „ ,
- — a-^=qt (sma-i ucos a).
dt dt ^ ^ -^ '
But since, by hjrpothesis, /t is less than f tan a, this velocity can never vanish.
The friction therefore wUl never change to rolling friction. The motion has thus
been completely determined.
148. A homogeneous sphere is rotating about a horizontal diameter, and is
gently placed on a rough horizontal plane, the coefficient of friction being fi. Deter-
mine the subsequent motion.
Since the velocity of the point of contact with the horizontal plane is not zero,
the sphere will evidently begin to slide, and the motion of its centre will be along a
straight line perpendicular to the initial axis of rotation. Let this straight line be
taken as the axis of x, and let 6 be the angle between the vertical and that radius of
the sphere which was initially vertical. Let a be the radius of the sphere, w^^•^ its
moment of inertia about a diameter, and Q, the initial angular velocity. Let R be
the normal reaction of the plane. Then the equations of motion are clearly
d^x „ 1
0=mg-R \ (1),
dt^ I
whence we have
--i^g
1
i-
a J
(Px
d^9_ , -^^ (2)-
dt"
dt
dR
Integrating, and remembering that the initial value of -y- is 0, we have
x = \iJ.gt'^
e=^t-iix^t'^ '
IMPERFECT FRICTION. 127
But it is evident that these equations cannot represent the whole motion, for
they -would make t- , the velocity of the centre of the sphere, increase continually.
This is quite contrary to experience. The velocity of the point of the sphere in
contact with the plane is
dx do ^ „
This vanishes at a time t.=S- — , ,^ (4). '
At this instant the friction suddenly changes its character. It now becomes
only of sufficient magnitude to keep the point of contact of the sphere at rest. Let
F be the friction required to effect this. The equations of motion will then be
Q=mg-R \ (5),
mk^ -rrz — - Fa I
dt^ J
and the geometrical equation will be a; = a9.
Differentiating this twice, and substituting from the dynamical equations, we
get F {a^ + k^) = 0, and therefore F— 0. That is, no friction is required to keep the
point of contact of the sphere at rest, and therefore none will be called into play.
The sphere will therefore move rmiformly with the velocity which it had at the
time t-^. Substituting the value of t-^^ in the expression for — obtained from equa-
tions (3) we find that this velocity is faQ. It appears therefore that the sphere
will move with a uniformly increasing velocity for a time S- — and will then move
uniformly with a velocity f aO. It may be remarked that this velocity is independ-
ent of fl.
If the plane be perfectly rough, fi is infinite, and the time i^ vanishes. The
sphere therefore immediately begins to move with a uniform velocity =f rtO.
149. In this investigation the couple of rolling friction has been neglected. Its
effect would be to diminish the angular velocity. The velocity of the lowest point
of the sphere would then tend to be no longer zero, and thus a small sliding friction
will be required to keep that point at rest. Suppose the moment of the friction-
couple to be measm'ed by fmg, where / is a constant. Introducing this into the
equations (5) the third is changed into
mF -^^= -Fa -fmg,
the others remaining unaltered. Solving these as before we find
a^ + k^'
We see from this that F is negative and retards the sphere. The effect of the
couple is to call into play a friction-force which gradually reduces the sphere to
rest.
As the sphere moves in the air we may wish to determine the effect of its resist-
ances. The chief part of this resistance may be pretty accurately represented by a
128 MOTION IN TWO DIMENSIONS.
force ?»/3 — acting at the centre in the direction opposite to motion, v being the
velocity of the sphere and /3 a constant whose magnitude depends on the density of
. the au-. Besides this there will be also a small friction between the sphere and air
whose magnitude is not known so accurately. Let us suppose it to be represented
by a couple whose moment is vn-^v'^ where 7 is a constant of small magnitude. The
equations of motion can be solved without difficulty, and we find
tan-i../to_tan- V J^^ = -^^^^ t,
where V is the velocity of the sphere at the epoch from which t is measured.
h 150. In order to determine by experiment the magnitude of
rolling friction, let a cylinder of mass M and radius r be placed on
a rough horizontal plane. Let two weights whose masses are P
and P + ^ be suspended by a fine thread passing over the cylinder
and hanging down through a slit in the horizontal plane. Let F
be the force of friction, L the couple at the point of contact A of
the cylinder with the horizontal plane. Imagine p to be at first
zero, and to be gradually increased until the cylinder just moves.
When the cylinder is on the point of motion, we have by resolving
horizontally F= aud by taking moments L =pgr. Now in the
experiments of Coulomb and Morin p was found to vary as the
normal pressure directly, and as r inversely. When p was great
enough to set the cylinder in motion, Coulomb found that the
acceleration of the cylinder was nearly constant, and thence we
may conclude that the rolling friction was independent of the
velocity. M. Morin found that it was not independent of the
length of the cylinder.
The laws which govern the couple of rolling friction are similar
to those which govern the force of friction. The magnitude is
just sufficient to prevent rolling. But no more than a certain
amount can be called into play, and this is called the limiting
rolling couple. The moment of this couple bears a constant ratio
to the magnitude of the normal pressure. This ratio is called the
coefficient of rolling friction. It depends on the materials in con-
tact, it is independent of the curvatures of the bodies, and, in
some cases, of the angular velocity.
No experiments seem to have been made on bodies which
touch at one point only and have their curvatures in all direc-
tions unequal. But since the magnitude of the couple is indepen-
dent of the curvature, it seems reasonable to assume that the
axis of the rolling couple, when there is no twisting couple, is the
instantaneous axis of rotation.
In order to test these laws of friction let us compare the
results of the following problem with experiment.
IMPERFECT FRICTION. 129
151. A carriage on n pairs of wheels is dragged on a level horizontal plane ly a
horizontal force 2P xoith uniform motion. Find the magnitude of P.
Let the radii of the wheels be respectively i\, r.^, &e., their weights lo-^, w„ &c.,
and the radii of the axles pj, p^, &c. Let 2T7 be the whole weight of the carriage,
2Q]^, 2Qo, &c. the pressures on the several axles, so that TF=2Q. Let the pressures
between the wheels and axles be Ri, R^, -^ga. [King's Coll.]
154, Four equal rods each of length 2a and mass m are freely jointed so as to
form a rhombus. The system falls from rest tvith a diagonal vertical under the action
of gravity and strikes against a fixed horizontal inelastic plane. Find the subse-
quent motion.
Let AB, BC, CD, DA be the rods and let AC be the vertical diagonal impinging
on the horizontal plane at A. Let V be the velocity of every point of the rhombus
just before impact and let a be the angle any rod makes with the vertical.
Let u, V be the horizontal and vertical velocities of the centre of gravity and w
the angular velocity of either of the upper rods just after impact. Then the
effective forces on either rod are equivalent to the force m{v- V) acting vertically
and mu horizontally at the centre of gravity and a couple mk^a tending to increase
the angle a. Let R be the impulse at C, the direction of which by the rule of
symmetry is horizontal. To avoid introducing the reaction at £ into our equations,
let us take moments for the rod BC about B and we have
mPw + m {v - F) a sin a - m^ia cos a— - R .2a cos a (1).
IMPULSIVE FORCES. 133
Each of the iower rods will begin to turn round its extremity 4 as a fixed point.
If w' be its angular velocity just after impact, the moment of the momentum about
A just after impact wiU be m(F+a-)w' and just before will be mFa sin a. The
difference of these two is the moment about A of the effective forces on either of
the lower rods. We may now take moments about A for the two rods A£, BC
together and we have
m (F + a^) w' - m Fa sin a - mly^o} + m (« - 7) a sin a + mtt , 3a cos a = i2 . 4a cos a . . . (2).
The geometrical equations may be found thus.
Since the two rods must make equal angles with the vertical during the whole
motion we have
w' = w (3).
Again, since the two rods are connected at B the velocities of the extremities of
the two rods must be the same in direction and magnitude. Resolving these hori-
zontally and vertically, we have
w + awcos a=:2aw'cosa (4),
?;-aw sin a=2aw'siua (5).
These five equations are sufficient to determine the initial motion.
Eliminating R between (1) and (2), substituting for n, v, w' in terms of w from
the geometrical equations, we find
^3 Fsing
'^ 2 ■ a(l + 3sin2a) ^''
In this problem we might have avoided the introduction of the unknown reaction
R by the use of Virtual Velocities. Suppose we give the system such a displace-
ment that the incUnation of each rod to the vertical is increased by the same
quantity 5a. Then the principle of Virtual Velocities gives
mFwStt -m{v- V)d (3a cos a) + 7nuS [a sin a) + m IJc^ + c(P) u'da + m V8 {a cos a) = 0,
which reduces to
(2^2 ^ ^2) (J _ Y(i sin a + 3 (u - V)a sin a + ua cos a = 0,
and the solution may be continued as before.
Ex. 1. Prove that the rhombus loses by the impact „ . , of its
'■ 1 -t- 3 sin2 a
momentum.
Ex. 2. Show that the direction of the impulsive action at the hinges B or D
makes with the horizon an angle whose tangent is — .
tan a
To find the subsequent motion. This may be found very easily by the method of
Vis Viva. But in order to illustrate as many modes of solution as possible, we
shall proceed in a different manner. The effective forces on either of the
iipper rods will be represented by the differential coefficients m-r, m-r, mlc^—r-,
^ '' clt dt' cW
and the moment for either of the lower rods will be m (Jc^ + a^) -^ . Let be the
dt
angle any rod makes with the vertical at the time t. Taking moments in the same
way as before, we have
mJc^ -j- + 7n-- a sin 6 -m -j-a cos 0= - R.2a cos 6 + mga sin 6 (1)',
dt dt dt
j„dii} dv . ^
—. — mk^ —r +vi~i-a am +i,^ ,
dt dt dt dt
,,., „ diji' ,„rfcj dv . „ du ^ n -n 4 „ ^ . „ ,„v,
m {k^ + a^)—i — mk^ -j +m~a am O + m-j .3a cos = R .ia cos 6 + 2mga sm 0. . . (2)'.
134 MOTION IN TWO DIMENSIONS.
The geometrical equations are the same as those given above, with 9 written
for a.
Eliminating E and substituting for u, v, we get
(2lf^ + a^) -^ +a2 |9 sin^ — (w sin6i) + cos ^ y{u cos 6)1 = iga sin d;
Clt ( CiZ Clt )
rJO
multiplying both sides by w == -r- and integrating, we get
{2 (^'3 + a^) + 8ci2 sinS 6} w^=C- 8ga cos 0.
Initially when 9 = a, w has the value given by equation (6). Hence we find that
the angular velocity w when the inclination of any rod to the vertical is 6 is
given by
/-. « • o o, « 9F^ sin^a 3o ,
(l + 3sin2^) w? = -— J . - — ., . o + — (cos a - cos d).
4a^ 1 + 6 sm^ a a '
Ex. 1. A square is moving freely about a diagonal with angular velocity w,
when one of the angular points not in, that diagonal becomes fixed; determine the
impulsive pressm'e on the fixed point, and show that the instantaneous angular
velocity wiU be — . [Christ's Coll.]
Ex. 2. Three equal rods placed in a straight line are jointed by hinges to one
another ; they move with a velocity v perpendicular to their lengths ; if the middle
point of the middle one become suddenly fixed, show that the extremities of the
other two will meet in a- time -j^— , a being the length of each rod. [Coll. Exam.]
Ex. S".. The points ABCD are the angular points of a square; AB, CD are two
equal similar rods connected by the string EC: The point A receives an impulse
in the direction A I), show that the initial velocity of A is seven times that of the
point B. [Queens' CoU.]
Ex. 4. A series of equal beams AB, BC, CD is connected by hinges; the
beams are placed on a smooth horizontal plane, each at right angles to the two
adjacent, so as to form a figure resembling a set of ^ steps, and' an impulse is given
at the end A along AB: determine the impulsive action at any hinge. [Math.
Tripos.]'
Result. If X„ be the impulsive action at the w*'' angular point, show that
^2^1 - 5X„+o - 2Z2^3^0 and X,„+a - 5Zo„+i - 2Z2„=0. Thence find X„.
155.. A' free lamina of any form is turning in its oion plane ad'out an instanta-
neous centre of rotation S and impinges on an oistaele at P, situated in the straight
line joining the centre of gravity G to S. To Jiiid the point P lohen the magnitude
of the blow is a maximum*.
Fifst, let the obstacle V'he a fixed point.
Let GP—x, and let R be the force of the blow. Let SG — Ti, and let w, w' be the
angular velocities about the centre of gravity before and after the impact. Then /tw
* Poinsot,.Sur la percussion des corps, Liouvillc's Journal, 1857; translated in
the Annals of PldlosoiJhy, 1S5S.
IMPULSIVE FORCES. 185
is the linear velocity of G just before the impact; let v' be its Hnear velocity just
after the impact. We have the eqiiations
,, "\\ «■
V - /i W = - -TV
M )
and supposing the point of impact to be reduced to rest,
v' + «w' = (2).
Substituting for w' and v' from (1) in equation (2), we get
r, !<- ,., X + A.
X' + h-
This is to be made a maximum. Equating to zero its differential coefficient
with respect to x, we get
a;2 + 2/jx-Z;3 = (3);
One of these values of x is positive and the other negative. Both these corre-
spond to maximum points of percussion, but opposite in du'ection. Thus there is a
point P with which the body strikes in front and a point P' with which it strikes
in rear of its own translation in space more forcibly than with any other point.
Ex. 1. Show that the two points P, P' are equally distant from S, and if be
the centre of oscillation with regard to /S as a centre of suspension, SP'^ = S6 . SO.
Ex. 2. If P be made a point of suspension, P' is the corresponding centre of
oscillation. Also PP' is harmonically divided in G and 0.
Ex. 3. The magnitudes of the blows at P, P' are inversely proportional to their
distances from G.
Secondly, let the obstacle be a free i^article of mass m.
Then, besides the equations (1), we have the equation of motion of the particle
r>
m. Let V be its velocity after impact, .*. F' = - .
The point of impact in the two bodies will have after impact the same velocity,
hence instead of equation (2) we have V'=v' + xia'.
Eliminating w', v', V, we get
(Al + m) k'-' + 7nx-
This is to be made a maximum. Equating to zero its differential coefficient
with respect to x, we find
..:.^A^ + ^^(l + f) (4).
This point does not coincide with that found when t^e obstacle was fixed, unless
m is infinite. To find when it coincides with the centre of oscillation, we must put
31 X -{-It
k'^ = xli. This gives — — -- — , or iil=x + li\>e the length of the simple equivalent
m 11
pendulum,— — -r- . Since F'=— , it is evident that when Jl is a maximum V
m h m
is a maximum. Hence the two points found by equation (4) might be called the
centres of greatest communicated velocity.
138 MOTION IN TWO DIMENSIONS.
There are other singular points in a moving body whose positions maybe found;
thus we might inquire at what point a body must impinge against a fixed obstacle,
that first the linear velocity of the centre of gravity might be a maximum, or
secondly, that the angular velocity might be a maximum. These points, respec-
tively, have been called by Poinsot the centres of maximum Eeflexiou and Conver-
sion. Eeferring to equations (1), we see that when v' is a maximum R is either a
maximum or a minimum, and hence it may be shown that the first point coincides
with the point of greatest impact. When w' is a maximum, we have to make
0) - ■r—rr^ = maximum. Substituting for R, this gives x^ -2-j-x-l?—0. If be the
centre of oscillation, we have GO—r- . L^t this length be represented by h'. Then
this equation becomes
x^-2h'x-k^=-0 (5).
The roots of this equation are the same functions of h' and h that those of
equation (3) are of h and k, except that the signs are opposite. Now S and are
on opposite sides of 0, hence the positions of the two centres of maximum Con-
version bear to and G the same relation that the positions of^ the two centres of
maximum Eeflexion do to S and G. If the point of suspension be changed from S
to 0, the positions of the centres of maximum Eeflexion and Conversion are inter-
changed.
Ex. A free lamina of any form is tm-ning in its own plane about an instanta-
neous centre of rotation S and impinges on a fixed obstacle P, situated in the
straight line joinmg the centre of gravity G to S. Find the position, of P, firsts that
the centre of gravity may be reduced to rest, secondly, that its velocity after impact
may be the same as before but reversed in direction.
Result. In the first case, P coincides either with G or with the centre ol oscil-
lation. In the second case the points x— GP are found from the equation
where SG — h. [Poinsot.]
L 156. TwQ bodies impinge on each other ^to explain tlie natm^e
of the action which takes place between them.
When two spheres of any hard material impinge on each
other, they appear to separate almost immediately, and a finite
change of velocity is generated in each by their mutual action.
This sudden change of velocity is the characteristic of an im-
pulsive force. Let the centres of gravity of the spheres be
moving before impact in the same straight line with velocities
u and V. Then after impact they will continue to move in the
same straight line, and let u, v be the velocities. Let m, m' be
the masses of the spheres, R the action between them, then we
have by Article 152,
Ry
u —u =
711
R ^
ffi. j
[ (!)■
IMPULSIVE FORCES. 137
These equations are not sufficient to determine the three quan-
tities u', v, R. To obtain a third equation we must consider what
takes place during the impact.
Each of the balls will be slightly compressed by the other, so
that they will na longer be perfect spheres. Each also will, in
general, tend to return to its original shape, so that there will be
a rebound. The period of impact may therefore be divided into
two parts. First, the period of compression, while the distance
between the centres of gravity of the two bodies is diminishing,
and secondly the period of restitution, while the distance between
the centres of gravity is increasing. At the termination of this
second period the bodies separate.
The arrangement of the particles of a body being disturbed by
impact, we ougKt to determine the relative motions of the several
parts of the body. Thus we might regard each body as a collec-
tion of free particles connected by their mutual actions. These
particles being thus set in motion might continue always in motion
oscillating about some mean positions.
It is however usual' to assume that the changes of shape and
structure are so small that the effect in. altering the position of the
centre of gravity and the moments of inertia of the body may be
neglected, and that the whole time of impact is so short that the
motion of the body in that time may be neglected. If for any
bodies these assumptions are not true, the effects of their impact
must be deduced from the equations of the second order. We
may therefore assert that at the moment of greatest compression
the centres of gravity of the two spheres are moving with equal
velocity.
The ratio of the magnitude of the action between the bodies
during the period of restitution to that during compression is
found to be different for bodies of different materials. In some
cases this, ratio is so small that the force during the period of re-
stitution may be neglected. The bodies are then said to be inelastic.
In this case we have just after the impact u' = v . This gives
M= jiu — v), whence u = .-7-.
m -{• m m+ m
If the force of restitution cannot be neglected, let R be the
whole action between the balls, i^^ the action up to the moment
of greatest compression. The magnitude of M must be found by
experiment. This may be done by determining the values of u'
and v, and thus determining R by means of equations (1). These
experiments were made in the first instance by Newton, and the
result is that -^ is a constant ratio depending on the material of
the balls. Let this constant ratio be called 1 + c. The quantity
138 MOTION IN TWO DIMENSIONS.
e is always less than unity, in tlie limiting case when e = 1 the
bodies are said to be perfectly elastic.
The value of e being supposed known the velocities after
impact may be easily found. The action R^ must be first calcu-
lated as if the bodies were inelastic, then the whole value of R
may be found by multiplying this result by 1 + e. This gives
-^ mm . . _ ,
R = — \ 7 (u -v)(l + e),
whence u and v may be found by equations (1).
P'
157. As an example, let us consider how the motion of the reel discussed in
Art. 153 would be affected if the string were elastic.
Since the point of the reel in contact with the string has no velocity at the
moment of greatest compression, the impulsive tension found in the article referred
to, measures the whole momentum communicated to the reel from the beginning of
the impact up to the moment of greatest comi^ression. By what has been said in
the last article, the whole momentum communicated from the beginning to the
termination of the period of restitution will be found by multiplying the tension
found in Art. 153 by 1 + e, if e be the measme of the elasticity of the string. This
gives
T = |mt;(l+e).
The motion of a reel acted on by this known impulsive force is easily found.
Resolving vertically we find
m (v' -v)= -^ mv (1 + e).
Taking moments about the centre of gi'avity
imW-di — \ mva (1 + e) ,
whence v' and w' may be found.
0'' Ex. A uniform beam is balanced about a horizontal axis through its centre
of gravity, and a perfectly elastic baU is let fall from a height h on one extremity ;
determine the motion of the beam and ball.
Result. Let 31, m be the masses of beam and ball, 2a = length of beam, V, V
the velocities of ball at the moments just before and after impact, w' the angular
velocity of the beam. Then
6mF v'-V ^"^-^^
ip
{WI+Sm)a' ' Sm + M'
158. Hitherto we have only considered the impulsive action
nbrmal to the common surface of the two bodies. If the bodies
are rough there will clearly be an impulsive friction called into
play. Since an impulse is only the integral of a very great force
acting for a very short time, we might suppose that impulsive
friction obeys the laws of ordinary friction. But these laws are
founded on experiment, and we cannot be sure that they are
correct in the extreme case in which the forces are very great.
This point M. Morin undertook to determine by experiment at
the express request of Poisson. He found that the frictional
IMPULSIVE FORCES. 139
impulse between two bodies whicli strike and slide bears to the
normal impulse the same ratio as in ordinary friction, and that
this ratio is independent of the relative velocity of the striking
bodies. M, Morin's experiment is described in the following
example.
159. A box AB -which can he loaded with shot so as to be of any proposed
weight has two vertical beams AC, BD erected on its lid ; CD is joined by a cross
piece and supports a weight eqnal to mcj attached to it by a string. The weight of
the loaded box is Mg. A string AEF passes horizontally from the box over a
smooth pulley E and supports a weight at F equal to [M+m)giJL. The box can
shde on a horizontal plane whose coeificient of friction is fx, and therefore having
been once set in motion, it moves in a straight line with a uniform velocity which
we will call V. Suddenly the string supporting m^r is cut, and this weight falls into
the box and immediately becomes fixed to the box. Show that an impulsive fric-
tion is called into play between the box and the horizontal plane. Prove that if
the velocity of the box immediately after the impulse is again equal to Y, the coeffi-
cient of impulsive friction is equal to that of finite friction. Find also the whole
space passed over by the box in any time which includes the impact.
160. When two inelastic bodies impinge on each other at
some point A, the points in contact at the beginning of the im-
pact have a relative velocity both aloDg the common tangent
plane at A and also along the normal. Thus two reactions will be
called into play, a normal force and a friction, the ratio of these
two being fx, the coefficient of friction. As the impact proceeds
the relative normal velocity gets destroyed, and is zero at the
moment of greatest compression. Let R be the whole momentum
transferred normally from one body to the other in this very
short time. This force R is an unknown reaction, to determine it
we have the geometrical condition that just after impact the
normal velocities of the points in contact are equal. This condi-
tion must be expressed in the manner explained in Art. 127.
The relative sliding velocity at A is also diminished. If it
vanishes before the moment of greatest compression, then during
the rest of the impact, only so much friction is called into play,
and in such a direction, as is necessary (if any be necessary) to
prevent the points in contact at A from sliding, provided that
this amount is less than the limiting friction. Let F be the
whole momentum transferred tangentially from one body to the
other. This reaction F is to be determined by the condition
that just after impact the tangential . velocities of the points
in contact are equal. If, however, the sliding motion does not
vanish before the moment of greatest compression, then the whole
of the friction is called into play in the direction opposite to that
of relative sliding, and we have F = fiR. Generally we may dis-
tinguish these two cases in the following manner. In the first
case it is necessary that the values of F and R found by solving
140 MOTION IN TWO DIMENSIONS.
the equations of motion should be such that Ftx, the sphere will separate slightly from the wall before sufficient
friction has been called into play to reduce the tangential velocity of the point of
contact to zero. In this case we must replace F by fj-R in the equations. At the
moment of gi-eatest compression we have as before v' — O. This gives R = niv cos, a.
By substituting in the equations the motion of the sphere may be found. The
initial velocity of the point of contact is easily seen to be u' -aui' = v (sin a - ;« | cosa).
If this were negative, the friction at the £nd of the impact would be acting in the
direction of relative motion, which is impossible. This solution is therefore correct
only if f tan a > yw.
If the sphere be imperfectly elastic, a normal force of restitution is called into
play equal to emv cos a. If then f mv sin a be < ^a (1 + e) mv cos a, the friction neces-
sary to destroy the tangential velocity of the point of contact is less than the
limiting friction. In this case by writing F=^mvB,viia, R={\ + e)mv cos, a in the
equations of motion, we can find u', v' and w. If ^mv sin a be > /* (1 + e) mv cos a,
we must put i2 = (1 + e) mv cos a, F = fj. [1 -{■ e) mv cos a, and the same equations will
now give u', v' and w'.
164. Two rough bodies of any form ivipinge on each other in
a given manner. It is required to find the motion just after
impact.
Let G, 0' be the centres of gravity of tiie two bodies, A the
142 MOTION IN TWO DIMENSIONS.
point of contact. Let U, V be the resolved velocities of G just
before impact, parallel to the tangent and normal respectively
at A; It, V the resolved velocities at any time t after the com-
mencement of the impact, but before its termination. Then t is
indefinitely small. Let O be the angular velocity of the body,
whose centre of gravity is G, just before impact, co the angular
velocity after the interval t. Let M be the mass of the body,
k its radius of gyration about G. Let GN be a perpendicular
from G on the tangent at A, and let AJV = x, NG=y. Let
accented letters denote corresponding quantities for the mother
body.
Let R be the whole momentum communicated to the body M
in the time t of the impact by the normal pressure, and let i^be
the momentum communicated by the frictional pressure. We
shall suppose these to act on the body whose mass is M in the
directions NG, NA respectively. Then they must be supposed to
act in the opposite directions on the body whose mass is M'.
Since R represents the whole momentum communicated to
the body Mm. the direction of the normal, the momentum com-
municated in the time dt is dR. As the bodies can only push
against each other, dR must be positive, and, by Art. 126, when
dR vanishes, the bodies separate. Thus the magnitude of R may
be taken to measure the progress of the impact. It is zero at the
beginning, gradually increases throughout, and is a maximum at
the termination of the impact. It will be found more convenient
to choose R rather than the time t as the independent variable.
The dynamical equations are by Art. 152
M{u-U)=-F ^
M{v-V)=R [ (1),
M¥ {w - n) = Fy + Rx J
M' {u - TT) = F ^
M' {v - r)=-R [ (2).
M'k"{a,'-n'):=Fy'-Ra;' J
The relative velocity of sliding of the points in contact is by
Art. 127
S —■ u — yco — u' — y'co' (.3),
and the relative velocity of compression is by the same article
C = v + x'(o' — V — XO) (4).
Substituting in these equations from the dynamical equations
we find
S=S,-aF-bR (5), ■
IMPULSIVE FORCES. 143
C=C,-hF-aR (6),
where S,= U - y^- U' -y'^' (7),
G^=V'+xn.' -V-xD. (8),
-A JL ^ y" fQ^
^~ M'^ M''^ Mk'"^ M'k" ^ ^'
/ /
These may be called the constants of the nnpact. The first
two S^, Cq represent the initial velocities of sliding and com-
pression. These we shall consider to be positive ; so that the
body if is sliding over the body Jf at the beginning of the com-
pi'ession. The other three constants a, a, h are independent of
the initial motion of the striking bodies. The constants a and a
are essentially positive, while b may have either sign. It will be
found useful to notice that aa > If.
165. When 6 = 0, the discussion of these equations, . as in
Art. 168, does not present any difficulty, but in the general case
it is more easy to follow the changes in the forces, if we adopt a
graphical method. Let us draw two lengths AB, AF along the
normal and tangent at A in the directions NQ, ^iV respectively,
to represent the magnitudes of R and F at any moment of the
imj)act. Then if we consider AR and AF to be the co-ordinates
of a point P, referred to AR, AF SiB axes of R and F, the changes
in the position of P will indicate to the eye the changes that
take place in the forces during the progress of the impact. It will
be convenient to trace the two loci determined hj 8=0, (7 = 0.
By reference to (5) and (6) we see that they are both straight
lines. These we shall call the straight lines of 7io sliding and of
greatest compression. To trace these, we must find their inter-
cepts on the axes of F and R. Take
AG= ^, AS=^, AC'=^j~\ AS' = ^,
a a
then SS', CG' will be these straight lines. Since a and a are
necessarily positive, while b has any sign, we see that their inter-
cepts on the axes of F and R respectively are positive, while their
intercepts on the axes of R and F must have the same sign.
Since aa' > 6^ the acute angle made by the line of no sliding with
the axis of F is greater than that made by the line of greatest
compression, i.e. the former line is steeper to the axis of Pthan
the latter. It easily follows that the two straight lines cannot
144
MOTION IN TWO DIMENSIONS.
intersect in the quadrant contained by RA produced and FA
produced.
166. In the beginning of the impact the bodies slide over
each other, hence, as explained in Art. 144, the whole limiting
friction is called into play. The point P therefore moves along a
straight line AL, defined by the equation F= jmR, where /a is the
coefficient of friction. The friction will continue to be limiting
until P reaches the straight line 88'. If R^ be the abscissa of
S
this point we find Rq = -, . This gives the whole normal
blow, from the beginning of the impact, until friction can change
from sliding to rolling. If R^ is negative, the straight lines AL
and 88' will not intersect ou the positive side of the axis of i'^.
In this case the friction will be limiting throughout the impact.
If R^ is positive the representative point P will reach 88 '. After
this only so much friction is called into play as will suffice to
prevent sliding, provided this amount is less than the limiting
friction. If the acute angle which 88' makes with the axis of ^
be less than tan~^ /i, the friction dF necessary to prevent sliding
will be less than the limiting friction fjudU. Hence P must
travel along 88' in such a direction that the abscissa R con-
tinues to increase positively. In this case the friction will not
again become limiting during the ianpact.
But if the acute angle which 88' makes with the axis of R be
greater than tan~^;tt, the ratio of dF to dR will be numerically
greater than fx, and more friction is necessary to prevent sliding
than can be called into play. The friction will therefore continue
to be limiting, and P, after reaching 88', must travel along a
straight line, making the same angle with the axis of R that AL
does. But this angle must be measured on the opposite side of
the axis of R, for when the point P has crossed 88' the direction
IMPULSIVE FORCES. 145
of relative slidiug and therefore the direction of friction is
changed. In this case it is clear that the friction will continue
limiting throughout the impact.
When P passes the straight line CG\ compression ceases and
restitution begins. But the passage is marked by no peculiarity
except this. If R^ be the abscissa of the point at which P crosses
CC , the whole impact, for experimental reasons, is supposed to
terminate when the abscissa of P is R,^ = R^ (1+e), e being the
measure of the elasticity of the two bodies.
It is obvious that a great variety of cases may occur according
to the relative positions of the three straight lines AL, SS' and
CC. But in all cases the progress of the impact may be traced
by the method just explained, which may be briefly stated thus.
The representative point P travels along AL, until it meets 8S'.
It then proceeds either along SS', or along a straight line
making the same angle with the axis of R as AL does, but on the
opposite side. The one along which it proceeds is the steeper to
the axis of F. It travels along this line in such a direction as to
make the abscissa R, increase. The complete value of R for the
whole impact is found b}^ multiplying the abscissa of the point at
which P crosses CC by 1 + e. The complete value of F is the
corresponding ordinate of P. Substituting these in the dyna-
mical equations (1) and (2), the motion just after impact may be
easily found.
If the bodies be smooth, the straight line AL coincides with
the axis of R. The representative point P must travel along the
axis of R and the complete value of R for the whole impact is
found by multiplying the abscissa of (7 by 1 + e.
167. It is not necessary that the friction should keep the
same direction during the impact. The friction must keep one
sign when P travels along AL. But when P reaches SS', its
direction of motion changes, and the friction dF called into play in
the time dt may have the same sign as before or the opposite.
But it is clear that the friction can change sign only once during
the impact.
It is possible that the friction may continue limiting through-
out the impact, so that the bodies slide on each other throughout.
The necessary conditions are that either the straight line SS'
must be less steep to the axis of F than AL, or the point P
must not reach the straight line SS' until its abscissa lias be-
come greater than P,^. The condition for the first case is, that
h must be greater than fxa. The abscissae of the intersections
S
of AIj with SS' and CC arc respectively R^ = — and
R. D. 10
146 MOTION IN TWO DIMENSIONS.
c
B, = x ^ . The condition for tlie second case is necessary,
that B^ must be positive, and B^ either negative or positively
greater than B^ (1 + e).
168. Ex. 1. Show that the representative point P as it travels in the manner
directed in the text must cross the line of greatest compression, and that the
abscissa H of the point at which it crosses this straight hne must he positive.
Ex. 2. Show that the conic whose equation referred to the axes of E and F is
aF^ + 2hFR + a'E^ = e, where e is some constant, is an ellipse, and that the straight
lines of no sliding and greatest compression are parallel to the conjugates of the
axes of F and F, respectively. Show also that the intersection of the straight lines
of no sliding and greatest compression must lie in that angle formed by the conju-
gate diameters which contains or is contained by the first quadi-ant.
Ex. 3. Two bodies, each turning about a fixed point, impinge on each other,
find the motion just after impact.
Let G, G', in the figm-e of Art. 164, be taken as the fixed points. Taking
moments about the fixed points, the results will be nearly the same as those given
in the case considered in the text.
Initial Motions.
169. Suppose a system of bodies to be in equilibrium and
that one of the supports suddenly gives way. _ It is required to
find the initial motion of the bodies and the initial values of the
reactions which exist between the several bodies.
The problem of finding the initial motion of a dynamical
system is the same as that of expanding the co-ordinates of the
moving particles in powers of the time t. Let {x, y, 6) be the
co-ordinates of any body of the system. For the sake of brevity
let us denote by accents differential coefficients with regard to the
time, and let the suffix zero denote initial values. Thus x^'
denotes the initial value oi -r^ . By Taylor's theorem we have
f f
x = a+a;^" r^ + x^" --+ (1) :
the term x^' is omitted because we shall suppose the system to
start from rest.
First, let only the initial values of the reactions he required.
The dynamical equations will contain the co-ordinates, their second
differential coefficients- with regard to t, and the unknown re-
actions. There will be. as many geometrical equations as re-
actions. From these we have to eliminate the second differential
\
INITIAL MOTIONS. 147
coefficients and find the reactions. The process will be as follows,,
which is really the same as the first method of solution described
m Art. 125.
Write down the geometrical equations, differentiate each twice
and then simplify the results by substituting for the co-ordinates
their initial values. Thus, if we use Cartesian co-ordinates, let
(f) (x, tj, 6) =0 be any geometrical relation, we have since x^' = 0,
The process of differentiating the equations may sometimes
be much simplified when the origin has been so chosen that the
initial values of some at least of the co-ordinates are zero. We
may then simplify the equations by neglecting the squares and
products of all such co-ordinates. For if we have a term x?, its
second differential coefficient is 2 {xx" + x'^), and if the initial
value of a; is zero, this vanishes.
The geometrical equations must be obtained by supposing the
bodies to have their displaced position, because we require to
differentiate them. But this is not the case with the dynamical
equations. These we may write down on the supposition that
each body is in its initial position. These equations may be
obtained according to the rules given in Art. 125. The forms
there given for the effective forces admit in this problem of some
simplifications. Thus since r^ = 0, 0„' = 0, the accelerations along
and perpendicular to the radius vector take the simple forms r''
and r^o". So again the acceleration — along the normal vanishes.
If, for example, we know the initial direction of motion of the
centre of gravity of any one of the bodies, we might conveniently
resolve along the normal to the path. This will supply an equa-
tion which contains only the impressed forces and such tensions
or reactions as may act on that body. If there be only one re-
action, this equation will suffice to determine its initial value.
We may also deduce from the equations the values of x^',
?/o"> ^o"' ^^^ ^^^'^ ^y substituting in equation (1) we have found
the initial motion up to terms depending on f.
170. Secondly, let the initial motion he required. How many
terms of the series (1) it may be necessary to retain will depend
on the nature of the problem. Suppose the radius of curvature
of the path described by the centre of gravity of one of the bodies
to be required. We have
x'y" — y'x'
10—2
148 MOTION IN TWO DIMENSIONS,
and by differentiating equation (1)
X — Xq -r Xq I -t Xq -r^ + ...
&c. =&c.;
r II > n / II in ur n\ " , / 'i \v iv "\ " i
xy -yx ={x,y^ -x, yo) ^+ {x, y^' -x, y, ) ^^- "-
results which may also be obtained by a direct use of Taylor's
theorem.
If then the body start from rest, the radius of curvature is
zero. But if x^'y^" — x^"yl' = 0, we have
P " iv iv '' •
To find these differential coefficients we may proceed thus.
Differentiate each dynamical equation twice and then reduce
it to its initial form by writing for x, y, 0, &c. their initial values,
and for x, y , & zero. Differentiate each geometrical equation
four times and then reduce each to its initial form. We shall
thus have sufficient equations to determine x'J , x^" , x^", &c., R^,
Rg, Rq", &e., where R is any one of the unknown reactions. It
will often be an advantage to eliminate the unknown reactions
from the equations before differentiation. We shall then have
only the unknown coefficients x^", x^" , &c, entering into the equa-
tions.
If we know the direction of motion of one of the centres
of gravity under consideration, we can take the axis of ?/ a tangent
to its path. Then we have p = |- , where x is of the second order,
y of the first order, of small quantities. We may therefore neg-
lect the squares of x and the cubes of y. This will greatly sim-
plify the equations. If the body start from rest we have x^ = 0,
and if x^' = 0, we may then use the formula
K
171. Ex. A circular disc is hung up by three equal strings attached to three
points at equal distances in its circumference, and fastened to a peg vertically over
the centre of the disc. One of these strings is suddenly cut. Determine the initial
circumstances of motion.
INITIAL MOTIONS. 149
Let be the peg, AB the circle seen by an eye in its plane. Let OA be the
string which is cut and G be the middle point of the chord joining the points of the
circle to which the two other strings are attached. Then the two tensions, each
equal to T, are thi-oughout the motion equivalent to a resultant tension R along CO.
If 2a be the angle between the two strings, we have
^=2rcosa.
Let I be the length of OC, ^ be the angle GOC, a be the radius of the disc. Let
{x, y) be the co-ordinates of the displaced position of the centre of gravity with
reference to the origin 0, x being measured horizontally to the left and y vertically
downwards. Let d be the angle the displaced position of the disc makes with AB.
By drawing the disc in its displaced position it will be seen that the co-ordinates
of the displaced position of G are a; - Z sin /3 cos ^ and ?/ - Z sin j8 sin 6.^ Hence since
the length OG remains constant and equal to I we have
a;2 + 2/2 _ 2Z sin /3 (as oo&d + y sin 6) — P cos^ |3.
Suppose the initial tensions only to be required. It will be sufficieni to differ-
entiate this twice. Since we may neglect the squares of small quantities, we
may omit x^, put cos ^=1, sin d — 6. The process of differentiation will not then be
veiy long, for it is easy to see beforehand what terms will disappear when we equate
the differential coefficients {x', y', ff) to zero, and put for (a;, y, 6) their initial values
(0, I cos jS, 0). We get
yd' cos /3 = sin /3 (xq" + 1 cos /S^o'O-
This equation may also be obtained by an artifice which is often useful. The
motion of G is made up of the motion of C and the motion of relatively to G.
Since G begins to describe a cii'cle from rest, its acceleration along CO is zero.
d'^9
Again, the acceleration of relatively to C when resolved along CO is GG -rr^ cos /3.
The resolved acceleration of G is the sum of these two, but it is also
2/o" cos /3 - Xq" sin /3.
Hence the equation follows at once.
In this case we require the differential equations only in their initial form.
These are
mxQ"=E()Sinp
my^" = m^ - i^o cos /3
mh-di^'= Mq I sin p cos /3j
where m is the mass of the body. Substituting in the geometrical equation we find
^ cosS
l + r-2sin2/3cos2/3
150 MOTION IN TWO DIMENSIONS.
The tension of any string, before the string OA was cut, may he found hy the
mg_
'< cos 7 '
rules of Statics, and is clearly T, = - — ^ , where 7 is the angle A OG. Hence the
' 6 cos y
change of tension can be found.
172, Ex. 1. Two strings of equal length have each an extremity tied to a
weight C and their other extremities tied to two points A, B hx the same horizontal
line. If one be cut the tension of the other is instantaneously altered in the ratio
l:2cos2^. [St Pet. Coll.]
Ex. 2. An elliptic lamina is supported with its plane vertical and transverse
axis horizontal by two weightless pegs passing through the foci. If one pin be
released show that if the eccentricity of the ellipse be \/ >- , the pressure on the
other pin will be initially unaltered. [Coll. Exam.] . _
Ex. 3. Three equal particles A, B, Q repelling each other with any forces, are
tied together by three strings of rmequal length, so as to form a triangle right-
angled at .4. If the string Joining B and C be cut, prove that the instantaneous
changes of tension of the strings joining BA, CA are | Tcos B and | Tcos C respec-
tively, where B and C are the angles opposite the strings joining CA , AB respec-
tively, and T is the repulsive force between B and C.
Ex. 4. Two uniform equal rods, each of mass m, are placed in the form of the
letter X on a smooth horizontal plane, the upper and lower extremities being con-
nected by equal strings ; show that whichever string be cut, the tension of the other
is the same function of the inclination of the rods, and initially is | mg sin a, where
a is the initial inclination of the rods. [St Pet. Coll.]
Ex. 5. A horizontal rod of mass m and length 2a hangs by two parallel strings
of length 2a attached to its ends : an angular velocity w being suddenly communi-
cated to it about a vertical axis through its centre, show that the initial increase
of tension of either string equals —— — , and that the rod will rise through a space
^ . [Coll. Exam.]
Ex. 6. A particle is suspended by three equal strings of length a from three
points forming an equilateral triangle of side 2b in a horizontal plane. If one
string be cut the tension of each of the others is instantaneously changed in the
ratio ^iJ^ty [Coll. Exam.]
Ex. 7. A sphere resting on a rough horizontal plane is divided into an infinite
number of solid liaes and tied together again with a string ; the axis through which
the plane faces of the lines pass being vertical. Show that if the string be cut
the pressure on the plane is diminished instantaneously in the ratio iSir^ : 2048,
[Emm. Coll.]
EELATIVE MOTION. 151
On Relative Motion or Moving Axes.
173. In many djmamical problems the relative motion of
the different bodies of the system is frequently all that is required.
In these cases it will be an advantage if we can determine this
without finding the absolute motion of each body in space. Let
us suppose that the motion relative to some one body (A) is
required. There are then two cases to be considered, (1) when
the body (A) has a motion of translation only, and (2) when it
has a motion of rotation only. The case in which the body (A)
has a motion both of translation and rotation may be regarded
as a combination of these two cases. Let us consider these in
order.
174. Let it be required to find the motion of any dynamical
system relative to some moving point C. We may clearly reduce
C to rest by applying to every element of the system an accelera-
tion equal and opposite to that of G. It will also be necessary to
suppose that an initial velocity equal and opposite to that of C
has been applied to each element.
Let /be the acceleration of C at any time t If every particle
m of a body be acted on by the same accelerating force f parallel
to any given direction, it is clear that these are together equi-
valent to a force fXtn acting at the centre of gravity. Hence to
reduce any point (7 of a system to rest, it will be sufficient to
apply to the centre of gravity of each body in a direction opposite
to that of the acceleration of (7 a force measured by Mf, where
M is the mass of the body and y the acceleration of C.
The point G may now be taken as the origin of co-ordinates.
We may also take moments about it as if it were a point fixed in
space.
Let us consider the equation of moments a little more minutely.
Let (r, 6) be the polar co-ordinates of any element of a body
whose mass is m referred to G as origin. The accelerations of
the particle are -j-^ ~ '" ( 77; ) ^^^ ~ 77> ( ^77 J ' ^^^^S ^^^ perpen-
dicular to the radius vector r. Taking moments about C, we get
moment round G of the impressed forces
plus the moment round G of the reversed
effective forces of G supposed to act at the
^centre of gravity.
dO
If the point G be fixed in the body and move with it, -j-
will be the same for every element of the body, and, as in Art. 88,
we have %n -j- ( ^'^ -rr ) = MJc' -^-r, .
at \ at J at
^ d_f.,d6\_
dt V dt,
152 MOTION IN TWO DIMENSIONS.
175. From the general equation of moments about a moving
point G we learn that we may use the equation
cl(o _ moment of forces about G
dt moment of inertia about G
in the following cases.
First. If the point G be fixed both in the body and in space ;
or, if the point G being fixed in the body move in space with
uniform velocity ; for the acceleration of G is zero.
Secondly. If the point G be the centre of gravity ; for in that
case, though the acceleration of G is not zero, yet the moment
vanishes.
Thirdly. If the point G be the instantaneous centre of rota-
tion*, and the motion be a small oscillation or an initial motion
which starts from rest. At the time t the body is turning about C,
and the velocity of G is therefore zero. At the time t + dt, the
body is turning about some point G' very near to G. Let GG'= da-,
then the velocity of G is (oda. Hence in the time dt the velocity
of (7 has increased from zero to wdcr, therefore its acceleration is
ft) -77 . To obtain the accurate equation of moments about G we
must apply the effective force S«i '(^ -f in the reversed direction
at the centre of gravity. But in small oscillations w and ^- are
"= ^ dt
both small quantities whose squares and products are to be
neglected, and in an initial motion (o is zero. Hence the moment
of this force must be neglected, and the equation of motion will
be the same as if G had been a fixed point.
It is to be observed that we may take moments about any
point very near to the instantaneous centre of rotation, but it will
usually be most convenient to take moments about the centre in
its disturbed position. If there be any unknown reactions at the
centre of rotation, their moments will then be zero.
176. If the accurate equation of moments' about the instan-
taneous centre be required, we may proceed thus. Let L be the
moment of the impressed forces about the instantaneous centre,
* If a body be in motion in one plane it is known that the actual displacement
of every particle in the time dt is the same as if the body had been turned through
some angle udt about some fixed point G. This may be proved in the same way as
the corresponding proposition in Three Dimensions is proved in the next Chapter.
See Art. 183. The point C is call-ed the instantaneous centre of rotation, and w is
called the instantaneous angular velocity. See also Salmon's Higher Plane Curves,
1852, Arts. 216 and 264.
EELATIVE MOTION. 153
the centre of gravity, r the distance between the centre of
gravity and the instantaneous centre C, M the mass of the body ;
then the moment of the impressed forces and the reversed
effective forces about G is
L-Moy^^.rco^GC'G:
at
If h be the radius of gyration about the centre of gravity, the
equation of motion becomes
dv
writinsf for cos GG'C its value -^ .
111. Ex. 1. Tim heavy particles whose masses are m and m' are connected by
an inextensihle string, lohich is laid over the vertex of a double inclined plane lohose
mass is M, and ivhich is capable of moving freely on a smooth horizontal plane.
Find the force which must act on the wedge that the system may be in a state of
relative equilibrium.
Here it vidll be convenient to reduce the wedge to rest by applying to every
particle an acceleration / equal and opposite to that of the wedge. Supposing this
done the whole system is in equilibrium. If F be the required force, we have by
resolving horizontally {M+m + m')f=F.
Let a, a be the inclinations of the sides of the wedge to the horizontal. The
particle m is acted on by mg vertically and m/ horizontally. Hence the tension of
the string is m(£^sina+/cosa). By considering the particle m', we find the
tension to be also m' (g sin a -/cos a'). Equating these two we have
„ m sm a -m sm a
1 = —, ; <)•
m cosa +m cos a
Hence F is found.
178. Ex. 2. A cylindrical cavity whose section is any oval curve and whose
generating lines are horizontal is made in a cubical mass ivhich can slide freely on a
smooth horizontal plane. The surface of the cavity is perfectly rough and a sphere is
placed in it at rest so that the vertical plane through the centres of gravity of the
mass and the sphere is perpendicular to the generating lines of the cylinder. A
momentum B is communicated to the cube by a blow in this vertical plane. Find the
motion of the sphere relatively to the cube and the least value of the blow that the
sphere may not leave the surface of the cavity.
Simultaneously with the blow B there will be an impulsive friction between the
cube and the sphere. Let M, m be the masses of the cube and sphere, a the radius
of the sphere, h its radius of gyration about a diameter. Let Fq be the initial
velocity of the cube, Vq that of the centre of the sphere relatively to the cube, w^ the
initial angular velocity. Then by resolving horizontally for the whole system, and
taking moments for the sphere alone about the point of contact, we have
m{v,+ V,)^MV, = B)
a[v, + V,;) + k-o:, = Q' ^^^'
154 MOTION IN TWO DIMENSIONS.
and since there is no sliding
V^) - «Wo = .*. (2).
To find the subsequent motion, let (x, y) be the co-ordinates of the centre of the
sphere referred to rectangular axes attached to the cubical mass, x being horizontal
and y vertical, then the equation to the cylindrical cavity being given, y is a known
function of x. Let \p be the angle the tangent to the cavity at the point of contact
of the sphere makes with the horizon, then tan r^ = ~ . Let Y be the velocity of
the cubical mass, then, by Art. Kl,
m(^+Y\ + MV=B (3).
If Tq be the initial vis viva and y^ the initial value of y, we have by the equation
of vis viva
m
l(J+7y+(|y+Fa,2J+i)fr=^ = To-2mr7(2/-2/o) W,
where w is the angular velocity of the sphere at the time t. If v be the velocity of
the centre of the sphere relatively to the cube, we have since there is no sliding
v^aij]. Eliminating V and w from these equations, we have
(I)'. j(l+tan=«(l + |')-^j = C,-2„ (5),
where Cy = p- + 2^yo"
(31+ in) M + (i)/ + m) —
( CI?)
This eqtiation gives the motion of the sphere relatively to the cube.
To find the pressure on the cube, let us reduce the cube to rest. Let R be the
pressui'e of the sphere on the cube, then the whole effective force on the cube is
R sin \p parallel to the axis of x. By Art. 174 we must therefore apply to every
particle an acceleration — ^j-^- opposite to this effective force. The sphere will
then be acted on by ^ i2 sin \j/ in a horizontal du'ection ui addition to the reaction
R, the friction and its own weight. Resolving the forces on its centre along a
normal to its path we have
dt) +U) i- = ^ + j^^«-^^— ^cos^ (6).
where p is the radius of curvature of the path of the centre of the sphere. Elimi-
dx
nating — by the help of the equation of vis viva, we have
c-%+pc„.,(i.|:-«;^)=i,^. (,,
where mgF-p f 1 + -^ — ^ ) ( "*■ "^ M ^^^^ ^ )' ^^^ '^^ P ^'^^^ ^°^ change sign, is
essentially a positive quantity.
At the point where the sphere leaves the surface of the cavity R vanishes.
Putting R — 0, we have an equation to determine \p at this point, C being a known
function of the initial conditions. If the sphere is to go all round the cylindrical
cavity, the values of cos \p given by this equation must be all imaginary or numeri-
RELATIVE MOTION.
155
cally greater ttau unity. If the spliere is just to go all round, then R must be
positive throughout and must vanish at the point where it is least. In this case we
have S, and -j- simultaneously zero.
dlogp , f^ F
Differentiating we have
d^p
M+m
»'^)K^
+ 3-
3m
M + m'
cos^i/' ) sin;/' (8).
This equation, since p is given as a function of ip from the equation to the cyhnder,
determines xp; C is then known from (7) when R is put equal to zero, and thence
the requii'ed value of B.
We may notice that the position of the point at which R is to be put zero is
independent of the initial conditions, and depends on the form of the cavity and the
ratio of the masses of the cube and sphere. This point cannot be at the highest
point of the cavity unless the radius of curvature of the cavity is at that point a
maximum or minimum. If the section of the cavity be a circle or an ellipse
having its major axis horizontal, then the equation to find yp is satisfied only when
ip — TT. In this case we find as the least value of the blow B to be given to the cube
that the sphere may go all round
— = jj/+ {M+m) Ki . U {M+m)§+ (m+ {M+m) ^
where .a and j3 are the semi-axes of the ellipse.
179. Next, let us consider tlie.case in whicli we wish to refer
tlae motion to two straight lines 0^, Or}, turning round a fixed
origin with angular velocity on.
Let Ox, Oy be any fixed axes and let the angle xO^= 6. Let
^ = OM, 7} = FM be the co-ordinates of any point P.
It is evident that the motion of P is made up of the motions
of the two points M, N by simple addition. The resolved parts of
the velocity of M are -^ and ^co along and perpendicular to OM.
(XfTJ
The resolved parts of the velocity of N are in the same way -77
and 7](o along and perpendicular to ON. By adding these with
their proper signs we have
156 MOTION IN TWO DIMENSIONS.
velocity of P]_d^_
parallel to 0^\~di~ '^^'
velocity oi P]_dr}
parallel to O'tj] dt
In the same way by adding the accelerations of M and N we
have
acceleration of P 1 _ d"^^ ^ ^ 1 d . ^ ,
parallel to 0| j ~ "df ~ ^"^ ~ ^ dt ^"^ '^^'
acceleration of P I _ c?^?7 2 1 d .^^ .
parallel to On) ~ d? ~ "^"^ '^ ^~dt ^^"'"^-
d^x d^v
By using these formnlse instead of -^r ^^^ ~J^ '^^^ ^^J refer
the motion to the moving axes 0^, Otj.
In a similar manner we may use polar co-ordinates. In this
case if (r, (J)) be the polar co-ordinates of P, we have
acceleration of P] d^r fd(f)
— T — — H &)
along rad. vect. J df \dt
acceleration of P] _ 1 d ( ^ /d4>
perp. to rad. vect.J r dt \ \dt
180. Ex. 1. Lfit the axes 0^, Or] be oblique and make an angle a with each
other, prove that if the velocity be represented by the two components u, v parallel
to the axes,
u = -j — w| cot a - W57 cosec a,
v= -7- + W17 cot a + w^ cosec a.
In this case PM is parallel to Or). The velocities of M and N are the same as
before. Their resultant is, by the question, the same as the resultant of m and v.
By resolving in any two directions and equating the components we get two equa-
tions to find u and v. The best directions to resolve along are those perpendicular
to 0^ and Orj, for then u is absent from one of the equations and v from the other.
Thus u ox V may be found separately when the other is not wanted.
Ex. 2. If the acceleration be represented by the components X and F, prove
A=^ — wit cot a - WW cosec a,
at
r= -rr+(j}VCOia + caU cosec a,
dt
These may be obtained in the same way by resolving velocities and accelerations
perpendicular to Of and Or],
RELATIVE MOTION.
157
181. Ex, A particle under the aetion of any forces moves on a sviooth curve
which is co7istrained to turn with angular velocity w about a fixed axis. Find the
motion relative to the curve.
Let us suppose the motion to be in three dimensions. Take the axis of Z as
the fixed axis, and let the axes of ^, r] be fixed relatively to the curve. Then the
equations of motion are
dt^
d^
dt
■(1),
=Z + Rn
where X, T, Z are the resolved parts of the impressed accelerating forces resolved
parallel to the axes, R is the pressm'e on the cm-ve, and (Z, m, n) the dhection-
cosines of the direction of R. Then since R acts perpendicularly to the curve
,(?f dtj di; „
ds ds as
Suppose the moving curve to be projected orthogonally on the plane of |, rj, let
a be the arc of the projection, and z/=— be the resolved part of the velocity parallel
to the plane of projection. Then the equations may be written in the form
d^^ ^ „. (?w „ .d-n ^,
d^
df^
^T + (a^7]
db} ^ „ ,d^ _
dt da
The two terms 2wi;' — and - 2uyv' ~ may be regarded as the resolved parts of a
force 2uv' acting in a direction whose direction-cosines are
drj ^ — d^
l' =
da'
m' = -
da
w':=0.
These satisfy the equation V ~ + m' ^ + n' ~r— 0.
ds ds ds
Hence the force is perpendicular to the tangent to the curve, and also perpen-
dicular to the axis of rotation. Let R' be the resultant of the reaction R and of the
force 2wv'. Then R' also acts perpendicularly to the tangent, let {I", m", n") be the
direction-cosines of its direction.
The equations of motion therefore become
d^P _ „. du _,„,
d'^ri ,_ „ do} ^ _, „
d^
dt'^
= Z-\-R'n"
.(2).
158 MOTION IN TWO DIMENSIONS.
These are the equations of motion of a jDarticle moving on a fixed curve, and
acted on in addition to the impressed forces by two extra forces, viz. (1) a force wV
tending directly from the axis, where r is the distance of the particle from the axis,
and (2) a force -=- r perpendicular to the plane containing the particle and the axis,
and tending opposite to the direction of rotation of the curve.
In any particular problem we may therefore treat the curve as fixed. Thus
suppose the curve to be tm-ning round the axis with uniform angular velocity.
Then resolving along the tangent we have
dv ^d% ,,% „f73 „ dr
as as as as as
where r is the distance of the particle from the axis. Let V be the initial value of
V, Tfj that of r. Then
v"-- V^=^2f{Xdx+ Tdy + Zdz) + cj"{r^-ro^).
Let ■I'o be the velocity the particle would have had under the action of the same
forces if the curve had been fixed. Then
v^^ -V- = 2f{Xdx+Ydy + Z dz).
Hence v" - Vq^ = w^ (r^ - j'o^) .
The pressure on the moving curve is not equal to the pressm'e on the fixed
curve. The pressure E on the moving cm-ve is clearly the resultant of the pressure
R' on the fixed eru've, and a pressure 2ww' acting perpendicular both to the curve
and to the axis in the direction of motion of the curve.
Thus suppose the curve to be plane and revolving uniformly about an axis per-
pendicular to its plane, and that there are no impressed forces. We have, resolving
along the normal,
— = - wV SUKp + R,
P
where (p is the angle r makes with the tangent.
If p be the perpendicular dravm from the axis on the tangent, we have, there-
fore,
R= — + u-p + 2wv.
P
This example might also have been advantageously solved by cylindrical co-ordi-
nates. The fixed axis might be taken as axis of z and the projection on the i^lane
of xy referred to polar co-ordinates. This method of treating the question is left to
the student as an exercise.
Ex. If w be variable, we have in a similar manner
1) civ I ^— —
iv = - -f w% + 2u}V + -r Jr'^ -j»^.
p dt
■ RELATIVE MOTION, 159
EXAMPLES*.
1. A circtilar hoop, which is free to move on a smooth horizontal plane, carries
on it a small ring -th of its weight, the coefficient of friction between the two being
At. Initially the hoop is at rest and the ring has an angnlar velocity u about the
centre of the hoop. Show that the ring will be at rest on the hoop after a time
1 + n
fJ.0}
/ 2. A heavy chciilar wire has its plane vertical and. its lowest point at a height
h above a horizontal plane, A small ring is projected along the whe from its
highest point with an angiilar velocity about its centre equal to ttu */ y at the
instant that the wire is let go. Show that when the wire reaches the horizontal
plane, the particle will just have described n revolutions.
3. A heavy uniform sphere rolls on a rough plane and is acted on by a fixed
centre of force in the plane varying inversely as the square of the distance ; if the
sphere be projected along the plane from a given point in it, in a direction opposite
to that of the centre of force, find the roughness of the plane at any point, suppos-
ing the whole of it to be reqmred.
4. Two equal uniform rods of length 2a, loosely jointed at one extremity, are
placed symmetrically upon a fixed smooth sphere of radius -— — , and raised into a
horizontal position so that the hinge is in contact with the sphere. If they be
allowed to descend imder the action of gravity, show that, when they are first at
rest, they are inchned at an angle cos~^ ^ to the horizon, that the points of contact
with the sphere are the centres of oscillation of the rods relatively to the hinge,
that the pressure on the sphere at each point of contact equals one-fourth the
weight of either rod, and that there is no strain on the hinge.
5. Two circular discs are on a smooth horizontal plane; one, whose radius is n
times that of the other, is fixed: an elastic string wi'aps round them so that those
portions of it not in contact with the discs are common interior tangents, the
natmal length of the string being the sum of the circumferences. The moveable
disc is drawn from the other till the tension of the string is T, prove that if it be
now let go, the velocity acquired when it comes in contact with the fixed disc
will be — A / — , where m is the mass of the moving disc, X the modu-
lus of elasticity, a the radius of the moving disc.
6. Two straight equal and uniform rods are connected at their ends by two
strings of equal length a, so as to form a parallelogram. One rod is supijorted at
its centre by a fixed axis about which it can tiu'u freely, this axis being perpendicu-
lar to the plane of motion which is vertical. Show that the middle point of the
lower rod wiU oscillate in the same way as a simple pendulum of length a, and that
the angular motion of the rods is independent of this oscillation.
* These examples arc taken from the Examination Papers which have been set
in the University and in the Colleges.
160 . MOTION IN TWO DIMENSIONS.
Ji^ 7. A fine string is attached to two points A, B in tlae same horizontal plane,
and carries a weight W at its middle point. A rod whose length is AB and weight
W, has a ring at either end, through which the string passes, and is let fall from
the position AB. Show that the string must be at least ^ AB, in. order that the
weight may ever reach the rod.
Also if the system be in equilibrium, and the weight be sUghtly and vertically
I AT}
displaced, the time of its small oscillations is 27r . / .
V 3^V3
8. A fine thread is enclosed in a smooth circular tube which rotates freely
about a vertical diameter ; prove that, in the position of relativ-e equilibrium, the
inclination [6) to the vertical, of the diameter through the centre of gravity of the
thread, wUl be given by the equation cos 6 = — ^ — - , where u is the angular
aw^cosjS °
velocity of the tube, a its radius, and 2a^ the length of the thread. Explain the
case in which the value of aw^ cos j3 lies between g and - g.
9. A smooth wire without inertia is bent into the form of a helix which is
capable of revolving about a vertical axis coinciding with a generating line of the
cylinder on which it is traced. A small heavy ring slides down the helix, starting
from a point in which this vertical axis meets the helix: prove that the angular
velocity of the helix will be a maximum when it has turned through an angle
given by the equation cos^ 9 + ta.n'' a+ 6 sin 29 = 0, a being the inclination of the
helix to the horizon.
10. A spherical hollow of radius a is made in a cube of glass of mass M, and a
particle of mass m is placed within. The cube is then set in motion on a smooth
horizontal plane so tliat the particle just gets round the sphere, remaining in con-
tact with it. If the velocity of projection be Y, prove that V^=5ag + 4:ag — .
M
11. A perfectly rough bail is placed within a hollow cylindrical garden-roller at
its lowest point, and the roUer is then drawn along a level walk with a uniform
velocity V. Show that the ball wiU roll quite round the interior of the roller, if
F^ be> y g{b- a), a being the radius of the ball, and h of the roller.
12. AB, BC are two equal uniform rods loosely jointed at B, and moving with
the same velocity in a direction perpendicular to their length ; if the end A be sud-
denly fixed, show that the initial angular velocity of ^5 is three times that of BC.
Also show that in the subsequent motion of the rods, the greatest angle between
them equals cos~^ f , and that when they are next in a straight line, the angular
velocity of BC is nine times that of AB.
13. Three equal heavy uniform beams jointed together are laid in the same
right line on a smooth table, and a given horizontal impulse is applied at the
middle point of the centre beam in a direction perpendicular to its length ; show
that the instantaneous impulse on each of the other beams is one-sixth of the given
impulse.
14. Three beams of like substance, joined together so as to form one beam,
are laid on a smooth horizontal table. The two extreme beams are equal in length,
and one of them receives a blow at its free extremity in a direction perpendicular to
its length. Determine the length of the middle- beam in order that the greatest
possible angular velocity may be given to the third.
EXAMPLES. 161
Result. If m be tlie mass of either of the outer rods, ^m that of the inner rod,
P the momentum of the blow, w the angular velocity communicated to the third
/I 4 4S\ /—
rod, then mau ( o + q + "^ ) —■^- Hence when w is a maximum ^ = \^'d.
15. Two rough rods A, B are placed parallel to each other and in the same
horizontal plane. Another rough rod G is laid across them at right angles, its
centre of gravity being half way between them. If C be raised through any angle a
and let fall, determine the conditions that it may oscillate, and show that if its
length be equal to twice the distance between A and B, the angle 6 through which
it will rise in the n^^ oscillation is given by the equation sin d=[-\ . sin a,
16. A rod moveable in a vertical plane about a hinge at its upper end has a
given uniform rod attached to its lower end by a hinge about which it can tm-u
freely in the same vertical plane as the upper rod ; at what point must the lower
rod be struck horizontally in that same vertical plane that the upper rod may
initially be unaffected by the blow ?
17. A ball spinning about a vertical axis moves on a smooth table and impinges
directly on a perfectly rough vertical cushion; show that the vis viva of the ball is
4-0
diminished in the ratio 10 4--14 tan^ 9 : — + 49 tan^ 6, where e is the elasticity of the
ball and 6 the angle of reflexion,
18. A rhombus is formed of four rigid uniform rods, each of length 2a, freely
jointed at their extremities. If the rhombus be laid on a smooth horizontal table
and a blow be applied at right angles to any one of the rods, the rhombus will begin
to move as a rigid body if the blow be applied at a point distant a (1 - cos a) from
an acute angle, where a is the acute angle.
19. A rectangle is formed of fom- uniform rods of lengths 2a and 2h respectively,
which are connected by hinges at their ends. The rectangle is revolving about its
centre on a smooth horizontal plane with an angular velocity n, when a point in
one of the sides of length 2a suddenly becomes fixed. Show that the angular
Telocity of the sides of length 26 immediately becomes — n. Find also the
ba + ib
change in the angular velocity of the other sides and the impulsive action at the
poiat which becomes fixed.
20. Three equal uniform inelastic rods loosely jointed together are laid in a
straight line on a smooth horizontal table, and the two outer ones are set in
motion about the ends of the middle one with equal angular velocities (1) in the
same direction and (2) in opposite directions. Prove that in the first case, when
the outer rods make the greatest angle with the direction of the middle one pro-
duced on each side the common angular velocity of the three is — , and in the
second case after the impact of the two outer rods the triangle formed by them will
move with uniform velocity — =- , 2a lacing the length of each rod.
21. An equilateral triangle formed of three equal heavy uniform rods of length
a hinged at their extremities is held in a vertical i^lane with one side horizontal and
the vertex do^vnwards. If after falling through any height, the middle point of the
R. D. 11
162 MOTION IN TWO DIMENSIONS.
Tipper rod be suddenly stopped, the impulsive strains on tlie upper and lower hinges
will be in the ratio of sjl'd to 1. If the lower hinge would just break if the system
fell through a height -— , prove that if the system fell through a height -^ the
lower rods would just swing through two right angles.
22. A perfectly rough and rigid hoop rolling down an inclined plane comes in
contact with an obstacle in the shape of a spike. Show that if the radius of the
T
hoop=r, height of spike above the plane - and F= velocity just before impact, then
the condition that the hoop will surmount the spike is F^> V 5"' J 1 - sin ( «+ ^ ) | ,
a being the inclination of the plane to the horizon.
Show that unless 72<\f gr. sin ( a+ ,^ ) , the hoop will not remain in contact
with the spike at aU.
If this inequahty be satisfied the hoop will leave the spike when the diameter
through the point of contact makes an angle with the horizon
( 9 72 ( TV
23. A flat circular disc of radius a is projected on a rough horizontal table,
which is such that the friction upon an element a is cV^ma where Fis the velocity
of the element, m the mass of a unit of area : find the path of the centre of the disc.
If the initial velocity of the centre of gravity and the angular velocity of the
disc be UqOSq, prove that the velocity u and angular velocity w at any subsequent
time satisfy the relation ( - — ^ ;, — . = —r, — .
24. A heavy circular lamina of radius a and mass M rolls on the inside of a
rough circular arc of twice its radius fixed in a vertical plane. Find the motion.
If the lamina be placed at rest in contact with the lowest point, the impulse which
must be applied horizontally that it may rise as high as possible (not going all
round), without falling off, is Mj'dag.
25. A string without weight is coiled round a rough horizontal cylinder, of
which the mass is M and radius a, and which is capable of turning round its axis.
To the free extremity of the string is attached a chain of which the mass is m and
the length I ; if the chain be gathered close up and then let go, prove that if 6 be
the angle thi-ough which the cylinder has turned after a time t before the chain is
fully stretched, Mad
=t(^'--)
26. Two equal rods AC, BC, are freely connected at C, and hooked to A and B,
two points in the same horizontal line, each rod being then inclined at an angle a to
the horizon. The hook B suddenly giving way, prove that the direction of the strain
,,.,,, , , ,/l + 6sin2a 2-3cos2a'
at C IS instantaneously shitted through an angle tan"-' I r: — ■
■ 6 cos^ a ' 3 sin a cos a
EXAMPLES. 163
27. Two particles A,B are connected by a fine string ; A rests on a rongli liori-
zontal taWe and B hangs vertically at a distance I below the edge of the table. If
A be on the point of motion and B be projected horizontally with a velocity u, show
that A will begin to move with acceleration —- , and that the initial radius of
/J. + 1 I
curvature of ^'s path will be {/x + 1) I, where /j, is the coefficient of friction.
28. Two particles {m, m') are connected by a string passing through a small
fixed ring and are held so that the string is horizontal ; their distances from the
ring being a and a', they are let go. If p, p' be the initial radii of curvatui'e of
■■ . ,-, ,1 . «i ™' T 1 1 1 1
then- paths, prove that ~ = — , and - + - = - + -.
^^ p p p p a a
29. A sphere whose centre of gravity is not in its centre is placed on a rough
table ; the coefficient of friction being fi, determine whether it will begin to slide or
to roll.
30. A circular ring is fixed in a vertical position upon a smooth horizontal
plane, and a small ring is placed on the circle, and attached to the highest point
by a string, which subtends an angle a at the centre ; prove that if the string be cut
and the circle left free, the pressures on the ring before and after the string is cut
are in the ratio M+m sin^ a : Mcos a, m and M being the masses of the ring and
circle.
31. One extremity C of a rod is made to revolve with uniform angular velocity
n in the circumference of a circle of radius a, while the rod itself is made to revolve
in the opposite direction with the same angular velocity about that extremity. The
rod initially coincides with a diameter, and a smooth ring capable of sliding freely
along the rod is placed at the centre of the circle. If r be the distance of the ring
from C at the time t, prove »*=-=- (e"* + e~"«) + - cos 2nt.
o o
32. Two equal uniform rods of length 2a are joined together by a hinge at one
•extremity, then- other extremities being connected by an inextensible string of length
21. The system rests upon two smooth pegs in the same horizontal line, distant 2c
from each other. If the string be cut prove that the initial angular acceleration of
either rod wiU be g ^^,^, ^^^'l, .
__ + ___8.-c?
33. A smooth horizontal disc revolves with angular velocity J p. about a verti-
cal axis at which is placed a material particle attracted to a certain point of the disc
by a force whose acceleration is ^ax distance; prove that the path on the disc will
be a cycloid.
11—2
CHAPTER V.
MOTION OF A RIGID BODY IN THREE DIMENSIONS.
Translation and Rotation.
182. If the particles of a body be rigidly connected, then
whatever be the nature of the motion generated by the forces,
there must be some general relations between the motions of the
particles of the body. These must be such that if the motion of
three points not in the same straight line be known, that of every
other point may be deduced. It will then in the first place be
our object to consider the general character of the motion of
a rigid body apart from the forces that produce it, and to reduce
the determination of the motion of every particle to as few in-
dependent quantities as possible : and in the second place we
shall consider how when the forces are given these independent
quantities may be found.
183. One point of a moving rigid body being fixed, it is re-
quired to deduce the general relations between the motions of the
other 2^0 hits of the body.
Let be the fixed point and let it be taken as the centre of
a moveable sphere which we shall suppose fixed in the body.
Let the radius vector to any point Q of the body cut the sphere
in P, then the motion of every point Q of the body will be repre-
sented by that of P.
If the displacements of two points A, B, on the sphere in any
time be given as AA' , BB' , then clearly the displacement of any
other point P on the sphere may be found by constructing on
A'B' as base a triangle A'P' B similar and equal to APB. Then
PP' will represent the displacement of P. It may be assumed as
evident, or it may be proved as in Euclid, that on the same base
and on the same side of it there cannot be two triangles on the
same sphere, which have their sides terminated in one extremity
of the base equal to one another, and likewise those terminated in
the other extremity.
Let D and E be the middle points of the arcs AA', BB', and
let DC, EC ho. arcs of great circles drawn perpendicular to A A',
TRANSLATION AND ROTATION.
165
BB' respectively. Then clearly GA= GA' and CB= CB', and
therefore since the bases AB, A'B' are equal, the two triangles
liE'W'
A CB, A' GB' are equal and similar. Hence the displacement of
G is zero. Also it is evident since the displacements of and G
are zero, that the displacement of every point in the straight line
0(7 is also zero.
Hence a body may he brought from any position, tvhich we may
call AB, into another A'B' by a rotation about OC as an axis
through an angle POP' such that any one point P is brought into
coincidence ivith its neio position P'. Then every point of the body
will be brought from its first to its final position.
184. A body is referred to rectangular axes x, y, z, and the
origin remauiing the same the axes are changed to x , y', z', accord- _
ing to the scheme in the margin. Show that this is equivalent ^
to turning the body round an axis whose equations are any two V
of the following three :
y>
through an angle 6, where
ijX+ (&2- 1) y + b^z=0,
c^x + C2y + ic^-1)z = 0,
3 - 4sin^~ = ai + 62 + C3.
What is the condition that these thi-ee equations are consistent ?
Take two points one on each of the axes of z and z' at a distance h from the
origin. Their co-ordinates are (0, 0, h) {a^h, h^li, cji) therefore their distance is
J2 (1 - C3) h. But it is also 2A sin 7 sin - ; .-. 2sin2- sin2 7=l -C3. We have by
similar reasoning 2 sin^^ sin" a = l -a^ and 2 sin2-sin^|3=l -&2J whence the equa-
tion to find 6 follows at once.
185. When a body is in motion we have to consider not
merely its first and last positions, but also the intermediate posi-
166 MOTION IN THREE DIMENSIONS.
tions. Let ns then suppose AB, A'B' to be two positions at any
indefinitely small interval of time dt. We see that when a body
moves about a fixed point 0, there is, at every instant of the
motion, a straight line OG, such that the displacement of every
point in it during an indefinitely short time dt is zero. This
straight line is called the instantaneous axis.
Let dO be the angle through which the body must be turned
round the instantaneous axis to bring any point P from its posi-
tion at the time t to its position at the time 1 4 dt, then the
ultimate ratio of dO to dt is called the angular velocity of the
body about the instantaneous axis. The angular velocity may
also be defined as the angle through which the body would turn
in a unit of time if it continued to turn uniformly about the
same axis throughout that unit with the angular velocity it had
at the proposed instant.
186. Let us now remove the restriction that the body is
moving with some one point fixed. We may establish the fol-
lowing proposition.
Every displacement of a rigid body may be represented hy a
combination of the two foUotuing motions, (1) a motion of trans-
lation whereby every particle is moved parallel to the direction of
motion of any assumed point P rigidly connected with the body
and through the same space. (2) A motion of rotation of the whole
body about some axis through this assumed point P.
It is evident that the change of position may be effected by
moving P from its old to its new position P' by a motion of trans-
lation and then retaining P' as a fixed point by moving any two
points of the body not in one straight line with P into their
final positions. This last motion has been proved to be equivalent
to a rotation about some axis through P'.
Since these motions are quite independent, it is evident that
their order may be reversed, i.e. we may rotate the body first and
then translate it. We may even suppose them to take place
simultaneously.
It is clear that any point P of the body may be chosen as the
base point of the double operation. Hence the given displace-
ment may be constructed in an infinite variety of ways.
187. To find the relations betiueen the axes and angles of rota-
tion when different points P, Q are chosen as bases.
Let the displacement of the body be represented by a rotation
6 about an axis PP and a translation PP'. Let the same dis-
placement be also represented by a rotation 6' about an axis Q8
and a translation QQ'. It is clear that any point has two dis-
TRANSLATION AND EOTATION. - 167
placements, (1) a translation equal and parallel to PP', and (2) a
rotation through an arc in a plane perpendicular to the axis of
rotation PR. This second displacement is zero only when the
point is on the axis PR. Hence the only points whose displace-
ments are the same as the base point lie on the axis of rotation
corresponding to that base point. Through the second base point
Q draw a parallel to PR. Then for all points in this parallel, the
displacements due to the translation PP', and the rotation 6
round PR, are the same as the corresponding displacements for
the point Q. Hence this parallel must be the axis of rotation
corresponding to the base point Q. We infer that the axes of
rotation corresponding to all base points are parallel.
188. The axes of rotation at P and Q having been proved
parallel, let a be the distance between them. The rotation 6
about PR will cause Q to describe an arc of a circle of radius a
and angle 6, the chord Qci of this arc is 2a sin ^ and is the dis-
placement due to rotation. The whole displacement of Q is the
resultant of Qq and the displacement of P. In the same way the
rotation 6' about QS will cause P to describe an arc, whose chord
B'
Pp is equal to 2a sin — . The whole displacement of P is the
resultant of Pp and the displacement of Q. But if the displace-
ment of Q is equal to that of P together with Qq, and the dis-
placement of P is equal to that of Q together with Pp, we must
have Pp and Qq equal and opposite. This requires that the two
rotations 6, 6' about PR and QS should be equal and in the same
direction. We infer that the angles of rotation corresponding to
all base points are equal.
189. Since the translation QQ' is the resultant of PP' and
Qq, we may by this theorem find both the translation and rotation
corresponding to any proposed base point Q when those for P are
given.
Since Qq, the displacement due to rotation round PR, is per-
pendicular to PR, the projection of QQ' on the axis of rotation is
the same as that of PP'. Hence the projections on the axis of rota-
tion of the displacements of all points of the body are equal.
190. An important case is that in which the displacement is
a simple rotation 6 about an axis PR without any translation. If
any pointy distant a from PR be chosen as the base, the same
displacement is represented by a translation of Q through a chord
a n
Qq = 2a sin - in a direction making an angle — ^ — with the plane
QPR and a rotation which must be cfjual to 6 about an axis which
168
MOTION IN THREE DIMENSIONS.
must be parallel to PR. Hence a rotation about any axis may he
replaced hy an equal rotation about any parallel axis together with
a motion of translation.
191. When the rotation is indefinitely small, the proposition
can be enunciated thus, a motion of rotation oodt about an axis
PR is equivalent to an equal motion of rotation about any parallel
axis QS, distant a from PR, together with a motion of translation
awdt perpendicular to the plane containing the axes and in the
direction in which QS moves.
192. It is often important to choose the base point so that
the direction of translation may coincide with the axis of rotation.
Let us consider how this may be done.
Let the given displacement of the body be represented by a
rotation 6 about PR, and a translation, PP'. Draw P'N perpendi-
cular to PR. If possible let this same displacement be represented
by a rotation about an axis QS, and a translation QQ' along Q8.
By Arts. 187 and 1^^ QS must be parallel to PR and the rotation
about it must be 6. This translation will move P a length QQ'
along PR, and the rotation about QS will move P along an arc
perpendicular to PR. Hence QQ' must equal Pi\^ and NP' must
be the chord of the arc. It follows that QS must lie on a plane
bisecting NP' at right angles and at a distance a from PR where
a
NP' = 2a sin- , or, which is more convenient, at a distance y from
a
the plane NPP' where NP' = 2y tan - . The rotation 6 round QS
is to bring iV^to P' and is in the same direction as the rotation Q
round PR. Hence the distance y must be measured from the
TRANSLATION AND ROTATION. 169
middle point of NP' in tlie direction in whicli that middle point
is moved by its rotation round PR.
Having found tlie only possible position of Q8, it remains to
sliow that the displacement of Q is really along QS. The rotation
6 round PP, will cause Q to describe an arc whose chord Qq is
n
parallel to P'N and equal to 2a sin ^ . The chord Qq is therefore
equal to NP', and the translation NP' brings q back to its position
at Q. Hence Q is only moved by the translation PN, i.e. Q is
moved along QS.
193. It follows from this reasoning that any displacement of
a body can be represented by a rotation about some straight line
and a translation parallel to that straight line. This mode of con-
structing the displacement is called a screw. The straight line is
sometimes called the central axis and sometimes the axis of the
screw. The ratio, of the translation- to the angle of rotation is
called the pitch of the screw.
194. The same displacement of a body cannot be constructed
by two different screws. For if possible let there be two central
axes AB, CD. Then AB and CD by Art. 187 are parallel. The
displacement of any point Q on CD is found by turning the body
round AB and moving it parallel to AB, hence Q has a displace-
ment perpendicular to the plane AB Q and therefore cannot move
only along CD.
195. "When the rotations are indefinitely small, the construc-
tion to find the central axis- may be simply stated thus. Let the
displacement be represented by a rotation wdt about an axis PR
and a translation Vdt in the direction PP'. Measure a distance
F sin P'P-R
y = from P perpendicular to the plane P'PR on that
side of the plane towards which P' is moving. A parallel to PR
through the extremity of y is the central axis.
196. Ex. 1. Given the displacements A A', BB', CC of three points of a body
in direction and magnitude, but not necessarily in position, find the direction of
the axis of rotation corresponding to any base point P.
Tkrough any assumed point draw Oa, 0/3, O7 parallel and equal to A A', BB',
CC. If Op be the direction of the axis of rotation, the projections of Oa, 0/3, Oy
on Op are all equal. Hence Op is the perpendicular drawn from on the plane
0/37. This also shows that the direction of the axis of rotation is the same for all
base points.
Ex, 2. If in the last example the motion bo referred to the central axis, find
the translation along it.
It is clearly equal to Op.
17Q MOTION IN THREE DIMENSIONS.
Ex. 3. Given the displacements AA', BB' of two points A, B of the body and
the direction of the central axis, find the position of the central axis. Draw planes
through J.4', BB' parallel to the central axis. Bisect AA\ BB'h^ planes perpen-
dicular to these planes respectively and parallel to the direction of the central axis.
These two last planes intersect hi the central axis.
Composition of Rotations.
197. It is often necessary to compoimd rotations about axes
OA, OB which meet at a point 0. But as the only case which
occurs in Rigid Dynamics is that in which these rotations are
indefinitely small we shall first consider this case with some par-
ticularity, and then indicate generally the mode of proceeding
when the rotations are of finite magnitude.
198. To explain what is meant hy a body having angular
velocities about more than one axis at the same time.
A body in motion is said to have an angular velocity w about
a straight line, when, the body being turned round this straight
line through an angle a)dt, every point of the body is brought
from its position at the time t to its position at the time t + dt.
Suppose that during three successive intervals each of time dt,
the body is turned successively round three different straight lines
OA, OB, OG meeting at a point through angles (o^dt, co./It,
co^dt. Then we shall first prove that the final position is the same
in whatever order these rotations are effected. Let P be any
point in the body, and let its distances from OA, OB, G, respect-
ively be Tj, r^, r^. First let the body be turned round OA, then
P receives a displacement co^r^dt. By this motion let r^ be in-
creased to r^ + dr^, then the displacement caused by the rotation
about OB will be in magnitude co^ (r^ + dr^ dt. But according to
the principles of the Differential Calculus we may in the limit
neglect the quantities of the second order, and the displacement
becomes co^r^dt. So also the displacement due to the remaining
rotation will be (o^r^dt. And these three results will be the same
in whatever order the rotations take place. In a similar manner
we can prove that the directions of these displacements will be
independent of the order. The final displacement is the diagonal
of the parallelepiped described on these three lines as sides, and
is therefore independent of the order of the rotations. Since then
the three rotations are quite independent, they may be said to
take place simultaneously.
When a body is said to have angular velocities about three
different axes it is only meant that the motion may be determined
as follows. Divide the whole time into a number of small in-
tervals each equal to dt. During each of these, turn the body
COMPOSITION OF EOTATIONS. 171
round the three axes successively, through angles w^dt, w^dt, w^dt.
Then when dt dimmishes without limit the motion during the
whole time will be accurately represented.
199. It is clear that a rotation about an axis OA may be
represented in magnitude by a length measured along the axis.
This length will also represent its direction if we follow the same
rule as in Statics, viz. the rotation shall appear to be in some
standard direction to a spectator placed along the axis so that
OA is measured from his feet at towards his head. This di-
rection of OA is called the positive direction of the axis.
200. If tivo angidar velocities about two axes OA, OB he
represented in magnitude and direction by the two lengths OA, OB ;
then the diagonal OC of the parallelogram constructed on OA, OB
as sides will he the residtant axis of rotation, and its length will
represent the magnitude of the resultant angular velocity. This
Prop, is usually called " The parallelogram of angular velocities."
Let P be any point in G, and let PM, PN be drawn per-
pendicular to OA, OB. Since OA represents the angular ve-
locity about OA and PM is the perpendicular distance of P
from OA, the product OA . PM will represent the velocity of P
due to the angular velocity about OA. Similarly OB.PX will
represent the velocity of P due to the angular velocity about
OB. Since P is on the left-hand side of OA and on the right-
hand side of OB, as we respectively look along these directions,
it is evident that these velocities are in opposite directions.
Hence the velocity of any point P is represented by
OA.PM-OB.PN
= OP[OA. sin CO A - OB . sin COB]
= 0.
Therefore the point P is at rest and (9(7 is the resultant axis
of rotation.
Let 03 be the angular velocity about OC, then the velocity
of any point A in OA is perpendicular to the plane AOB and is
represented by the product of w into the perj)endicular distance
of A from OC =^ w . OA sin CO A. But since the motion is also
172 MOTION IN THREE DIMENSIONS.
determined by the two given angular velocities about OA, OB, the
motion of the point A is also represented by the product of OB
into the perpendicular distance of A from OB = OB . OA sin B OA ;
sm COA
Hence the angular velocity about OC is represented in mag-
nitude by OC.
From this proposition we may deduce as a corollary "the
parallelogram of angular accelerations." For if OA, OB repre-
sent the additional angular velocities impressed on a body at
any instant, it follows that the diagonal OC will represent the
resultant additional angular velocity in direction and magnitude.
201. This proposition shows that angular velocities and an-
gular accelerations may be compounded and resolved by the same
rules and in the same way as if they were forces. Thus an an-
gular velocity co about' any given axis may be resolved into two,
CO cos a and w sin a, about axes at right angles to each other and
TT . . .
making angles a and ^ — a with the given axis.
If a body have angular velocities w^, oy^, co^ about three axes
Ox, Oy, Oz at right angles, they are together equivalent to a
single angular velocity w, where tu = V&)^^ + w^ -1- w^, about an
axis making angles with the given axes whose cosines are re-
spectively — , — , — .. This may be proved, as in the corre-
^ "^ CO Qi Oi
sponding proposition in Statics, by compounding the three angular
velocities, taking them two at a time.
It will however be needless to recapitulate the several propo-
sitions proved for forces in Statics with special reference to an-
gular velocities. We may use " the triangle of angular velocities "
or the other rules for compounding several angular velocities
together, without any further demonstration.
202. A body has angular velocities co, co about two 'parallel
axes OA, O'B distant a fro77i each other, to find the residting
Tnotion.
Since parallel straight lines may be regarded as the limit of
two straight lines which intersect at a very great distance, it
follows from the parallelogram of angular velocities that the two
given angular velocities are equivalent to an angular velocity
about some parallel axis 0" C lying in the plane containing OA,
O'B.
COMPOSITION OF ROTATIONS. 173
Let x be the distance of this axis from OA, and suppose it
to be on the same side of OA as OB. Let II be the angular
velocity about it.
Consider any point P, distant y from OA and lying in the
plane of the three axes. The velocity of P due to the rotation
about OA is wy, the velocity due to the rotation about OB is
(ii'{y — a). But these two together must be equivalent to the
velocity due to the resultant angular velocity 12 about 0"G, and
this is 11 {y — x),
•■• «?/ + «' (y - a) = ^ (y - ^)-
This equation is true for all values of ?/, .*. n = tw + w', a^ = — .
This is the same result we should have obtained if we had
been seeking the resultant of two forces co, co' acting along OA,
OB.
If ct) = — ft)', the resultant angular velocity vanishes, but x is in-
finite. The velocity of any point P is in this case &)?/+ co' {y— a) — aco,
which is independent of the position of P,
The result is that two angular velocities, each equal to co but
tending to turn the body in opposite directions about two parallel
axes at a distance a from each other, are equivalent to a linear
velocity represented by aco. This corresponds to the proposition
in Statics that " a couple" is properly measured by its moment.
We may deduce as a corollary, that a motion of rotation co
about an axis OA is equivalent to an equal motion of rotation
about a parallel axis O'B plus a motion of translation aco perpen-
dicular to the plane containing OA, O'B, and in the direction in
which O'B moves.
203. To explain a certain analogy which exists between Statics
and Dynamics.
All propositions in Statics relating to the composition and
resolution of forces and couples are founded on these theorems :
1. The parallelogram of forces and the parallelogram of
couples.
2. A force F is equivalent to any equal and parallel force
together with a couple i^), where p is the distance between the
forces.
Corresponding to these we have in Dynamics the following
theorems on the instantaneous motion of a rigid body :
1. The parallelogram of angular velocities and the parallelo-
gram of linear velocities.
174 MOTION IN THREE DIMENSIONS.
2. An angular velocity a is equivalent to an equal angular
velocity about a parallel axis together with a linear velocity
equal to cop, where p is the distance between the parallel axes.
It follows that every proposition in Statics relating to forces
has a corresponding proposition in Dynamics relating to the
motion of a rigid body, and these two may be proved in the
same way.
To complete the analogy it may be stated (i) that an angular
velocity like a force in Statics requires, for its complete determina-
tion, five constants, and (ii) that a velocity like a couple in Statics
requires but three. Four constants are required to determine the
line of action of the force or of the axis of rotation, and one to
determine the magnitude of either. There will also be a conven-
tion in either case to determine the positive direction of the line.
Two constants and a convention are required to determine the
positive direction of the axis of the couple or of the velocity and
one the magnitude of either.
It is proved in Statics that a system of forces and couples is
generally equivalent to a single force and a single couple, and
that these may be reduced to a resultant R acting along a line
called the central axis, and a couple about that axis. Or they
may also be reduced to a resultant R of the same magnitude
as before, acting along any line parallel to the central axis at'
any chosen distance c from it, together with a couple G' about
an axis perpendicular to the line whose length is c, and in-
clined to the resultant R at an angle 6. Then we know that
Cf' = \/ G^ + R^(?, and is a minimum when c = 0, and also that
tan d — —7^ .
Cr
The same train of reasoning by which these results were
established, will establish the following proposition. The instan-
taneous motion of a bod}^ having been reduced to a motion of
translation and one of rotation, these are equivalent to a motion
of rotation tw about a line called the central axis, and a trans-
lation V along that axis. Or they may also be reduced to a
rotation w of the same magnitude as before about any line par-
allel to the central axis, and at any chosen distance c from it,
together with a translation V along a line perpendicular to the
line c, and inclined to the axis of w at an angle Q. Then we
know that F' = V K '■^ -f- cW, and is a minimum when c = 0, and
also that tan ^ = ^ . In a similar manner many other propositions
may be established.
204. Ex. 1. The locus of points in a body moving about a fixed point •wbicbi
at any proposed instant have the same actial velocity is a circular cylinder.
COMPOSITION OF ROTATIONS. 175
Ex. 2. The geometrical motion of a body is represented by angular velocities
inversely proportional to /3-7, 7 -a, a-/3 about three lines forming three edges
of a cube which do not meet nor are parallel. Prove that the body rotates about
the line
(/3 - 7) re - «a = (7 - a) ?/ - a/3 = (a - /3) 3 - ay,
2a being an edge of the cube, the centre being the origin, and the axes parallel to
the edges.
Ex. 3. A body has an angular velocity w about the axis
x-a ^ y-^ ^z-y
I m »i '
where P + m^ + 71^ = 1. The motion is equivalent to rotations Iw, mo), nui about the
co-ordinate axes, and translations (7717 - m/3) w, {na - ly) w, (?j3 - ma) w in the direc-
tions of the axes.
This follows from the analogy of forces in Statics to angular velocities in
Dynamics. See Art. 203.
Ex. 4. A body has equal angular velocities about two axes which neither meet
nor are parallel. Prove that the central axis of the motion is equally inchned to
each of the axes.
205. When the rotations to be compoimded are finite in magnitude, the rule to
find the resultant is somewhat more complicated. Let the given rotations be (1) a
rotation about an axis OA through an angle 9; (2) a subsequent rotation about an
axis OB through an angle 6', and let both these axes be fixed in space. Let lengths
measured along OA, OB represent these rotations in the manner explained in
Art. 199.
Let the directions of the axes OA, OB cut a sphere whose centre is at in ^
and B. On this sphere measure the angle BA C equal to ^ in a direction opposite
a
to the rotation round OA and also the angle ABO equal to - in the same direction
as the rotation round OB and let the arcs intersect in C. Lastly, measure the
angles BAC, ABC respectively equal to BAC, ABO, but on the other side of AB.
The rotation round OA will then carry any point P in 00 into the straight
line 00' and the subsequent rotation 9' about OB will carry the point P back into
00. Thus the points in 00 are unmoved by the double rotation and OC is therefore
the axis of the single rotation by which the given displacement of the body may be
constructed. The straight line 00 is called the resultant axis of rotation. If the
order of the rotations were reversed, so that the body is rotated first about OB and
then about OA, the resultant axis would be OC.
176 MOTION IN THREE DIMENSIONS.
If the axes OA, OB were fixed in tlie body, the rotation 9 about OA would bring
OB into a position OB'. Then the body may be brought from its first into its
last position by rotations 9, 9' about the axes OA, OB' fixed in space. Hence the
same construction will again give the position of the resultant axis and the rotation
about it.
To find the magnitude 9" of the rotation about the resultant axis OOv^e notice
that if a point P be taken in OA, it is unmoved by the rotation 9 about OA, and the
subsequent rotation 9' about OB will bring it into the position P', where PP' is
bisected at right angles by the plane OBC. But the rotation 9" about OC must
give P the same displacement, hence in the standard case 9" is twice the external
angle between the planes OCA, OCB. If the order of the rotations be reversed, the
rotation about the resultant axis 00' would be twice the external angle at C", which
is the same as that at O. So that though the position of the resultant axis of rota-
tion depends on the order of rotation the resultant angle of rotation is independent
of that order.
206. A rotation represented by twice any internal angle of the spherical
triangle ABC is equal and opposite to that represented by twice the corresponding
external angle. For since the sum of the internal and external angles is tt, these
two rotations only differ by Stt; and it is evident that a rotation through an angle
27r cannot alter the position of any point of the body. This is merely another way
of saying that when a body turns about a fixed axis it may be brought from one
given position to another by turning the body either way round the axis.
207. The rule for compounding finite rotations may be stated thus:
If ABC be a spherical triangle, a rotation round OA from C to B 'through tivice
the internal angle at A, followed by a rotation round OB from k to G through tivice
the internal angle at B is equal and opposite to a rotation round 00 from B to A
through twice the internal angle at 0.
It will be noticed that the order in which the axes are to be taken as we travel
round the triangle is opposite to that of the rotations.
As the demonstrations in Ai-t. 205 are only modifications of those of Bodrigues,
we may call this theorem after his name.
208. Ex. 1. If two rotations 9, 9' about two axes OA, OB at right angles be
compounded into a single rotation cp about an axis OC, then
9' 9 9 9' , A 9 9'
tan COA = tan — cosec -, tan COB — tan - cosec — and cos ^ = cos -- cos - .
2 2 A A Z Z ii
209. From Eodrigues' theorem we may deduce Sylvester's theorem by drawing
the polar triangle A'B'C. Since a side B C is the supplement of the angle A, a
rotation represented in direction and magnitude by 25'C" differs from that repre-
sented by 1A in the opposite direction by a rotation through an angle 27r. But a
rotation through 27r cannot alter the position of the body, hence the two rotations
IB'C and 1A are equivalent in magnitude but opposite in direction. If therefore
A'B'O' be any spherical triangle, a rotation represented by tivice B'C followed by a
rotation twice C'A' produces the same displacement of the body as a rotation twice
B'A '. By a rotation B'C is meant a rotation about an axis perpendicular to the
plane of B'C which wiU bring the point B' to C.
210. The following proof of the preceding theorem was given by Prof. Donkin
in the Phil, Mag. for 1851. Let ABC be any triangle on a sphere fixed in space,
COMPOSITION OF EOTATIONS.
177
ajSy a triangle on an equal and concentric sphere moveable about its centre. The
sides and angles of a/Sy are equal to those of ABC, but differently arranged, one
triangle being the inverse or reflection of the other. If the triangle a/37 be placed
in the position I, so that the sides containing the angle a may be in the same great
circles with those containing A, it is obvious that it may slide along AB into the
position n, and then along BO into the position III ; into which last position it
might also be brought by sliding along AC. To slide 0^37 along AB is equivalent
to moving j3 and a each through an arc twice the arc AB about an axis perpen-
dicular to the plane oi AB. A similar remark applies when the triangle slides
along BO or A C. Hence, twice the rotation AB followed by twice the rotation BC
produces the same displacement as twice the rotation AC.
211. If it be required to compound the rotations about two parallel axes, the
construction of Eodrigues requires only a slight modification. Instead of arcs
drawn on a sphere, let planes be drawn through the axes making with the plane
containing the axes the same angles as before; their intersection will be the
resultant axis. One case deserves notice. If 6= -6', the resultant axis is at
infinity. A rotation about an axis at infinity is evidently equivalent to a translation.
Hence a rotation -^ about any axis 04 followed by an equal and opposite rotation
a
about a parallel axis O'B distant a from OA is equivalent to a translation 2a sin -
perpendicular to a plane through OA making an angle - with the plane containing
the axes and in the direction of the chord of the arc described by any point in OA .
These results also follow easily fi'om Ai't. 190.
212. Any given displacement of a body may be represented by two finite rotations,
one about any given straight line and the other about some other straight line which
does not necessarily intersect the first. When a displacement is thus represented,
the axes are called conjugate axes and the rotations are called conjugate rotations.
Let OA be the given straight line and let the given displacement be represented
by a rotation about a straight line OR and a translation OT. We ^vi^h to resolve
this rotation about OR into two rotations, one about OA to be followed by a
rotation about OB, where OB is some straight line perpendicular to 02'. To do
this we follow the rule in Art. 205, we describe a sphere whose centre is and
radius unity and let it intersect OA, OR, OT in A, R and T. Make the angle ARB
R. D. 12
178
MOTION IN THREE DIMENSIONS.
equal to the supplement of ^ and produce RB to B so tliat TB — ^ and join AB.
By the triangle of rotations the rotation ^ is now represented by a rotation about
OA which we may call ^, followed by a rotation about OB which we may call & .
By Art. 211 the rotation Q' is equivalent to an equal rotation 0' about a parallel
axis CD, together with a translation, which may be made to destroy the translation
OT. This will be the case if the angle OT makes with the plane of OB, CD be
ir — 6' • ■
on the one side or the other of OT according to the direction of the rotation,
and if the distance r between AB, CD be such that 2r sin -- = OT.
The whole displacement has thus been reduced to a rotation 6 about OA followed
by a rotation 6' about CD.
213. Analytically, we might reason thus — A screw motion is given when we
know (1) its axis, (2) the rotation about it, (3) the translation along it. The axis is
known when its inchnation to two of the axes and the two co-ordinates of the point
in which it cuts the plane of xy are given. Thus six constants are required to
determine a screw.
Let a given screw be resolved into two screws. We have then twelve constants,
but since they are together equivalent to the given screw there are six relations
between the constants. We are therefore at liberty to choose any six relations we
please between these twelve constants. We might, for example, resolve a given
screw into two screws of any given pitches, the remaining four constants being
chosen to make the axis of one screw coincide with any given straight line. If the
given pitch of each screw be zero, the screws are reduced to simple rotations, and
thus any displacement can be reduced to two conjugate rotations. It has been
shown in the preceding article that the two rotations are real.
214. Ex. Show that any screw may be resolved into two real screws having
the axis of one in a given direction and the axis of the other intersecting the first
at a given angle.
215. Any two successive displacements of a body may be represented by two
successive screw motions. It is required to compound these.
Let the body be screwed first along the axis OA with linear displacement a and
COMPOSITION OF ROTATIONS. 179
angle of rotation 6, and secondly along the axis CD with displacement a' and angle
6'. Let OC be the shortest distance between OA and CD, and for the sake of the
perspective let it be called the axis of y. Let be the origin and let the axis of x
be parallel to CD, so that OA lies in the plane of xz. Let OC=r, and the angle
AOx = a. Draw a plane xOT making with the plane of xz an angle - , and let it cut
yz in OT. Draw another plane AOR making with xz an angle - , and cutting the
plane xOT in OR.
■ Produce AO io & point P, not marked in the figiure, so that PO = a, and let us
choose P as a base point to which the whole displacement of the body may be
referred. The rotation 6' is equivalent to a rotation 6' about Ox together with a
at
translation along 0T=2r^va.- by Art. 190. By Art. 205 the rotation 9 about OA
followed by 6' about Ox is equivalent to a rotation Q about OR where Q, is twice the
SMole ART, so that sin- = sin^. -. — =;-. The whole displacement is now repre-
° 2 2 svo-Rx
sented by (1) a translation of the base point P to 0, (2) the rotation fi, (3) a further
nf
linear translation which is the resultant of the translations 2r sin — along OT and
a' along Ox. By Art. 186 these displacements may be made in any order, being all
connected with the same base point. They may therefore be compoimded into a
single screw by the rule given in Art. 192. This is called the resultant screw. A
screw equal and opposite to the resultant screw will bring the body back to its
original position.
The angle of rotation of the residtant screw is Q, and its axis is parallel to OR
by Art. 187. It follows by Ai't. 206 that the sine of half the angle of rotation of
each screw is proportional to the sine of the angle between the axes of the other
two screws.
To find the linear displacement along the axis of the resultant screw, we must by
Art. 189 add together the projections on OR of the three displacements OT, a, a'. The
projection of 0T=2r sin - cos TR = 2rco^ Ty . cos TR which is twice the projection
of the shortest distance r on the axis of rotation. If T be the Unear displace-
ment, we have T—2rc.os.Ry + a(io?,RA + a'c,osRx.
216. If the component screws be simple rotations we have a = 0, a.' = 0, and it
may be shown without difficulty that T sin - = 2r sin- sin— sin a. It has been
shown in Art. 212 that any displacement may be represented by two conjugate
rotations in an infinite number of ways, but it now follows that in all these
r sin - sin ^ sin a is the same. WTien the rotations are indefinitely small, and equal
to udt, bi'dt respectively, this becomes \ ruw sin a; that is, the product of an angular
velocity into the moment of its conjugate angular velocity aboiit its axis is the same
for aU conjugates representing the same motion.
Ex. 1. If the component screws be simple finite rotations, show that the equa-
tions to the axis of the resultant screw are
/?' fi' /9' /?' /?' 6' f2
-.rtan0' + y sin- + 2Cos- = r sin-, ycos- -2sin- = rsin - cos 0' cot— ,
12—2
180 MOTION IN THREE DIMENSIONS.
where „, w, about the co-ordinate axes. We have to
dx
dz
find the resolved velocities ^ , -~ , ^ of a^ particle whose co-
dt' dt' dt ^
ordinates are x, y, z.
These angular velocities are supposed positive when they tend
the same way round the axes that positive couples tend in Statics.
Thus the positive directions of co^, co^, Wg are respectively from y
to z, from z to x, and from x to y.
182 MOTION IN THREE DIMENSIONS.
Let US determine the velocity of P parg-llel to the axis of z.
Let PN be the ordinate z, and let PM be drawn perpendicular
to Ox. The velocity of P due to the rotation about Ox is clearly
w^PM. Resolving this along ^P we get w^Pil/ sin NPM = (o^y.
Similarly that due to the rotation about Oy is — w^x ; and that due
to the rotation about Oz is zero. Hence the whole velocity of P
parallel to Oz is
and the velocities parallel to the other axes are
dx
dii
f^ = o>,x-<.,z.
220. The quantities w^, w^, co^ are called the angular veloci-
ties of the body about the axes of x, y, z respectively, but they
must be carefully distinguished from the angular velocities of any
particular particle of the body about the same axes. Let P be
any particle of the body whose co-ordinates are x, y, z, and draw
PL = r perpendicular to the axis of z. Let 6 be the angle x ON,
then the instantaneous angular velocity of P about 0^ is -j- .
But T'^-^ = x-^ — y-j- = w^^'^-xz(i,^ — yzw^, by substituting
for -^ , -J- , their values just found ;
dd xz yz
dt ^ r ^ r
Hence the angular velocity of a particle about Oz is the same
as that of the body when the particle lies in the plane of xy, or
when it lies in the plane given hy y = — x — .
If the axes be themselves moving in any manner, these equa-
tions only give the linear velocities of the particle relatively to the
axes. Thus suppose the directions of the axes to be fixed in space,
but the origin to be in motion with a velocity F whose resolved
parts parallel to the axes are respectively u, v, tu. Then the
velocities in space resolved parallel to the axes will be
u' = u + o)^z- (o^y \
v =v -\- (O^X — co^z > .
vj'=w + w,7/ — «,/; -'
FIXED AXES. 183
221. The motion being given, as before, by the linear veloci-
ties (w, Y, w) of some point O a^id the angular velocities {co^, co^,
ft)g), fi^id the equations to the central axis.
Let the same motion be also represented by tbe linear ve-
locities li , V , u) parallel to the axes, of some other point 0' and
by angular velocities w/, w^', 0)3' about axes parallel to the co-
ordinate axes and meeting in 0'. Let (f, 77, ^) be the co-ordinates
of O . We have now two representations of the same motion, both
these must give the same result for the linear velocities of any
point. Hence
u + w^z - w^y = u + o)/ [z-^)- W3' (2/ - 77) 1
V +(o^x— C0j^z = v' + 6)3' (a; — ^) — ft)/ (^ — ^) j" (l)j
tu+ co^y — co^x = w'+ 03^ {y — v) — ^2 (^ "" ^) J
must be true for all values of x, y, z.
This gives cd^ = co^, w^ = co^, (o^' = (o^, so that whatever origin
is chosen, the angular velocity is always the same in direction and
magnitude. See Art. 188.
Also (^, 7), ^) may be so chosen that the velocity of 0' is along
the axis of rotation ; in this case we have {u, v, w) proportional
to (ft)^, (o^, 0)3). The equation to the locus of 0' is therefore
u + w,^^ - 0)^7) ^ v + w^^ - coj; _ w + w^rj - ft)^g ^
«i "2 «3 '
By multiplying the numerator and denominator of each of
these fractions by ty^, 03^, co^ respectively, and adding them to-
gether, we see that each of them is
~ XP ■
The motion of the body is thus represented by a motion of
translation along the straight line whose equations are (2) and
an angular velocity equal to O about it.
This straight line has been called the central axis, and the
fraction just written down is equal to the ratio of the velocity
of translation along the central axis to the angular velocity about
it, i.e. the pitch of the screw.
If the motion be such that uco^ + vco^ + wco^ = 0, and &>,,«,
6)3 do not all vanish, each of the equalities in (2) is zero, and
hence by equation (1) u' = 0, v' — O, w = 0. The motion is there-
fore equivalent to a rotation about the central axis, without
translation. This is also evident from the analogy explained
in Art. 203.
222. When the rotations are finite the corresponding formnlffi are somewhat
more complicated. Lot the given displacement of the body be a rotation through a
184 MOTION IN THREE DIMENSIONS.
finite angle 9 about an axis passing through the origin whose direction cosines are
{I, m, n). It is required to find the changes produced in the co-ordinates [x, y, z) of
any point P.
Let PP' be the chord of the arc described by P and let Q be the middle point
of PP'. Let x + ^x, y+Sy, z+5z be the co-ordinates of P' and ^, rj, f those of Q.
Since the abscissae of Q is the arithmetic mean of those of P and P' we have
^=x-^~] similarly 97:=?/+ -^ , f =2+^. Let QM be a perpendicular from Q on
the axis, then PP' =z2 QM tan- .
Ji
Let (\, IX, v) be the direction cosines of PP', then since PP' is perpendicular to
the axis, we have IX + niix + nv — Q, and since it is also perpendicular to OQ, we have
^\ + rilx + ^v — (}, hence
\ IJ. _ V
m^ - nrj n^ -l^ l-rj-m^'
The sum of the squares of the denominators is
which is OQ^ - 0M^= QIP. Hence each of these ratios is = p— .
Now 5a; is the projection of PP' on the axis of x,
.'. Sx = 2Q3f. tan-X = 2tan - (mf-n??);
a
similarly 5?/ = 2 tan ^ («?- l^), Sz = 2 tan j^ (^7?-m^), which are the required formula.
If the origin have a linear displacement whose resolved parts parallel to the axes
are (a, h, c), we must add those displacements to the values of 5x, dy, Sz found by
solving these equations., Let the co-ordinates of the middle point of the tvhole dis-
placement of P be represented by f', t]', f. Then we have, as before, ^'=x + — &c.,
but since 5a;, dy, 8z, are increased, by a,, b, c we must write l'-s> V-91 ^"'-9
for I, 7], f. "We thus obtain
5aj=a + 2tan- \m (^'-2) "'^ ^'''2)) '
with similar expressions for 8y and 82.
223. The equations to the central axis follow from these expressions without
difficulty. The whole displacement of any point in the central axis is along the
axis, so that (^', 77', f ') the co-ordinates of the middle point of the displacement are
co-ordinates of a point in the axis, and 5a;, dy, dz are pro]3ortional to (I, m, n) the
direction cosines of the axis. Hence
,.2..n|j,„(r-|)-..(v4)j _ .....n|j. (!■-;)-, (r-l)j
I m
c + 2tangjz(/-|)-m(r-|)j
n
Each of these is evidently equal to la + mh + nc, which is the linear displacement
along the central axis. The results of this and the preceding Article are due to
Eodrigues.
FIXED AXES. 185
224. Ex. Let the restraints on a body be such that it admits of two motions
A and B each of which may be represented by a screw motion, and let m, m' be the
pitches of these screws. Then the body must admit of a screw motion compounded
of any indefinitely small rotations ix>dt, ca'dt about the axes of these screws accom-
panied of course by the translations mwdt, m'u'dt. Prove that (1) the locus of the axes
of all these screws is the surface z (x^ + y'^) = 2axy. (2) If the body be screwed along
any generator of this surface the pitch is c + a cos 26, where c is a constant which is
the same for all generators and 6 is the angle the generator makes with the axis of
X. (3) The size and position of the sm-face being chosen so that the two given
screws A and B He on the surface with their appropriate pitch, show that only one
surface can be drawn to contain two given screws. (4) If any three screws of the
surface be taken and a body be displaced by being screwed along each of these
through a small angle proportional to the sine of the angle between the other two,
the body after the last displacement wili occupy the same position that it did before
the first.
This surface has been called the cylindroid by Prof. Ball, to whom these four
theorems are due.
225. Ex. 1. If an instantaneous motion be given by the linear velocities
{u, V, w) along and the angular velocities (w^, Wg, W3) about the co-ordinate axes,
•C^/ II ~~ Q Z — Jl
show that the equations to the conjiigate of -— -=^ = - — - = are
I VI n
= III + mv + niu,-,.
= {f-oc)u + {ff-y)v + ih-z)w.
The first equation follows from the fact that the direction of motion of any
point on the conjugate is perpendicular to- the given axis, and the second from the
fact that the direction of motion is also perpendicular to the straight line joining
the point to (/, g, h).
Ex. 2. If an instantaneous motion be represented by a screw along the axis of
2, the linear and angular velocities being V and Q, prove that the equations to the
X - ■ "f 1/ CI z — ft V V
conjugate of — — — = are inx -ly + n-=0 and gx-fy - -{z-hj — O.
t 171 Tt ifl \l
Ex. 3. The locus of the conjugates of all axes of instantaneous rotation which
are parallel to a fixed straight line is a plane parallel to the central axis and to the
fixed straight line.
Ex. 4. The locus of the conjugates of all axes of instantaneous rotation
which pass through a given point is a plane. If two axes intersect, their conjugates
also intersect.
226. If the instantaneous motion of a body be represented by two conjugate
rotations about two axes alright angles, a plane can be drawn through either axis
perpendicular to the other. The axis in the plane has been called the characteris-
tic of that plane, and the axis perpendicular to the plane is said to cut the i^lane in
its focus. These names were given by M. Chasles in the Comptes Rendus for 1843.
Some of .the following examples were also given by him, though without demonstra-
tions.
X
1/
z
Wj
"2
'^■i
I
VI
n
X
y
z
Wj
'^■2
W3
/
9
hr
186 MOTION IN THREE DIMENSIONS.
Ex, 1. Show that every phxue has a characteristic and a focus.
Let the central axis cut the plane in 0. Resolve the linear and angular veloci-
ties in two directions Ox, Oz, the first in the plane and the second perpendicular to
it. The translations along Ox, Oz may he removed if we move the axes of rotation
Ox, Oz parallel to themselves, hy Art. 202. Thus the motion is represented hy a
rotation about an axis in the plane and a rotation about an axis perpendicular to
it. It also follows that the characteristic of a plane is parallel to the projection of
the central axis.
Ex. 2. If a plane he fixed in the body and move with the body, it intersects
its consecutive position in its characteristic. The velocity of any point P in the
plane when resolved perpendicular to the plane is proportional to its distance fi'om
the characteristic, and when resolved in the plane is proportional to its distance
from the focus and is perpendicular to that distance.
Ex. 3. If two conjugate axes cut a plane in F and G, then FG passes through
the focus.
If two conjugate axes be projected on a plane, they meet in the characteristic of
that plane.
Ex. 4. If two axes CM, CN meet in a point C, their conjugates lie in a plane
whose focus is C and intersect in the focus of the plane CMN.
This follows from the fact that if a straight line cut an axis the direction of
motion of every point on it is perpendicular to the straight line only when it also
cuts the conjugate.
Ex. 5. Any two axes being given and their conjugates, the four straight lines
lie on the same hyperboloid.
Ex. 6. If the instantaneous motion of a body be given by the linear and angu-
lar velocities {u, v, w) (wj, Wj, Wg), prove that the characteristic of the plane
Ax+By + Cz + I):::^0
is its intersection vrith
A (u + WgS - W3?/) +B {v+u.^x- W12) + C{w + w-^y - w.^x) = 0,
and its focus may be found from
U + (j}^ — W32/ _ '1'+ WjSC - W^2 _ W -1- Wj?/ - WgiK
A 'B ~ O •
For the characteristic is the locus of the points whose directions of motion are
perpendicular to the normal to the plane, and the focus is the point whose direction
of motion is perpendicular to the plane.
What do these equations become when the central axis is the axis of 2 ?
Ex. 7. The locus of the characteristics of planes which pass through a given
straight line is a hyperboloid of one sheet ; the shortest distance between the given
straight line and the central axis being the direction of one principal diameter, and
the other two being the internal and external bisectors of the angle between the given
straight line and the central axis. Prove also that the locus of the foci of the
planes is the conjugate of the given straight line.
Ex. 8. Let any surface A be fixed in a body and move with it, the normal
planes to the trajectories of all its points envelope a second surface B. Prove that
if the surface B be fixed in the body and move with it, the normal planes to the
euler's equations. 187
trajectories of its points will envelope the surface A : so that the surfaces A and B
have conjugate properties, each surface being the locus of the foci of the tangent
planes to the other.
Prove that if one surface is a quadric the other is also a quadric.
Ex. 9. A body is moved from any position in sj^ace to any other, and every
point of the body in the first position is joined to the same i^oint in the second
position. If all the straight lines thus found be taken ■which pass through a given
point, they mil form a cone of the second order. Also if the middle points of all
these lines be taken, they will together form a body capable of an infinitesimal
motion, each point of it along the line on which the same is situate. Cayley's
Report to the Brit. Assoc, 1862,
Euler's Equations.
227. To determine the general equations of motion of a body
about a fixed point.
Let the fixed point be taken as origin, and let x, y, z be the
co-ordinates at time t of any particle m referred to any rectangular
axes fixed in space. Let Xm, Ym, Zm be the impressed forces
acting on this element parallel to the axes of co-ordinates, and
let L, M, N be the moments of all these forces about the axes.
d^x
Then by D'Alembert's Principle, if the effective forces m -^ ,
on -j^ , m -ji ^6 applied to every particle m in a reversed direc-
tion, there will be equilibrium between these forces and the im-
pressed forces. Taking moments therefore about the axes, we have
„ / d'^y d^x\ „ ,^,
and two similar equations.
To simplify these equations, let w^, w^, &>, be the angular velo-
cities about the axes. Then -r- = coz — w„y, -/ = w^x — w^z,
dt ' ~^ dt '
dz
^ = co^y-co^x;
d^x d(i)„ d(o.,
-•To = z
df dt -^ dt
y-Tif-^^'j ^^^y - ^i^) - «« ^^*^ - ^x^),
d^y dw., d(i>^ , , , .
^ = a; -^-- - 2 -^ + ft), (w/j - «,7/) - ft), (f»,7/ - w^x).
188 MOTION IN THREE DIMENSIONS.
Substituting in equation (1) we get
Xm {x^ + rf) -^ - tmyz . -^- - %mxz . -^
\ = N.
- 'Xmxy . (ft)/ - ft)/) + Sm (a;^ - f) co^w^ - Xmyz . w^w^ i
+ "XniXZ . ft)^&)^ J
The other two equations may be treated in the same manner.
The coefficients in tliis equation are the moments and products
of inertia of the body with regard to axes fixed in space and are
therefore variable as tlie body moves about. Let us then take a
second set of rectangular axes OA, OB, OC fixed in the body, and
let ft)^, ft>2, «3 be the angular velocities about these axes. Since
the axes Ox, Oy, Oz are perfectly arbitrary, let them be so chosen
that the axes OA, OB, OG are passing through them at the
moment under consideration. Then o3^ = w^, (Oy = a}^, co^— co^. If
the principal axes at the fixed point have been chosen as the set
of axes fixed in the body, and A, B, C be the moments of inertia
about them, the equation takes the form
C^-{A-B)co,co, = N,
in which all the coefficients are constants.
228. We shall now show that — = ~ . This may appear
at first sight to follow at once from the equation co^ = ft),. But it is
not so ; ft>3 denotes the angular velocity of the body about C fixed
in the body, while w^ denotes the angular velocity about a line Oz
fixed in space and determined by the condition that at the time t
C coincides with it. At the time t + dt OG will have separated
from Oz and we cannot therefore assert a priori that the angular
velocity about OC will continue to be the same as that about Oz.
We have to prove that this is the case as far as the first order of
small quantities. Let OR, OR' be the resultant axes of rotation
at the times t and t 4- dt, i. e. let a rotation D.dt about OR bring
OG into coincidence with Oz at the time t, and let a further
rotation fl'dt about OR' bring OG into the position OG' in space
at the time t+dt. Then according to the definition of a differ-
ential coefficient
day, rt . n' cos R'G'-n cos RG
-dt^^""^ dt '
d(o^_J^ O' cos jR's — O cos jR^
dt dt
Since a rotation about OR' brings OG from the position Oz to
OG', R'G' and R'z differ by quantities of the second order, and
therefore these two differential coefficients are ultimately equal.
da
d(o
da
dl3
-~ 9 _
- (O
■ (0 —
dt
dt
'dt
'dt
Also -77 is the angular rate at which A separates from a
euler's equations. 189
229. The following demonstration of this equality has been
given by the late Professor Slesser of Queen's College, Belfast, and
is instructive as founded on a different principle. Let A, B, G be
the points in which the principal axes cut a sphere whose centre
is at the fixed point. Let OL be any other axis, and let 12 be
the angular velocity about it. Let the angles LOA, LOB, LOG
be called respectively a, yS, 7. Then by Art. 201
n = &>j cos a + fo)^ cos /3 + Wg cos 7 ;
dn d(o. dw. ^ d(j),
. dcL . d/S . dy
-a,,sma^-a,,sm/3-^-a,3sm7^.
Now let the line OL be fixed in space and coincide with OG
Sit the moment under consideration. Then a—^, /S=-, 7 = 0;
therefore
dt
. . dB
fixed point at G, this is clearly co^. Similarly -—= — &) . Hence
JO _ cZojg rpi d^x _ doy^ do3y _ doo^ dw^ doi^
dt ^ dt ' dt ~ dt ^ dt ~ dt ' dt ~ dt
230. The three equations of motion of the body referred to
the principal axes at the fixed point are therefore
A^-{B-C)<.,o., = L,
B'^'-{G-A)co,ay^ = M,
G^-^^-{A-B)co,a>, = K
These are called Euler's equations.
231. We know by D'Alembert's principle that the moment
of the effective forces about any straight line is equal to that of
the impressed forces. The equations of Euler therefore indicate
that the moment of the effective force about the principal axes
at the fixed point are expressed by the left-hand sides of the above
equations. If there is no point of the body which is fixed in
space, the motion of the body about its centre of gravity is the
same as if that point were fixed. In this case, if A, B, be the
principal moments at the centre of gravity, the left-hand sides of
Euler's equations give the moments of the effective forces about
190 ~ MOTION IN THREE DIMENSIONS.
the principal axes at the centre of gravity. If we want the
moment about any other straight hne passing through the fixed
point, we may find it by simply resolving these moments by the
rules of Statics.
232. Ex. 1. If 2T -Aw-^^ + Bo3.2^ + Cw.^ and G be tlie moment of the impressed
forces about the instantaneous axis, Q, the resultant angular velocity, prove that
Ex. 2. A body turning about a fixed point is acted on by forces which tend to
produce rotation about an axis at right angles to the instantaneous axis, show that
the angular velocity cannot be uniform unless two of the principal moments at the
fixed point are equal. The axis about which the forces tend to produce rotation is
that axis about which it would begin to turn if the body were placed at rest.
233. To determine the pressure on the fixed point.
Let X, y, z be the co-ordinates of the centre of gravity referred
to rectangular axes fixed in space meeting at the fixed point, and
let P, Q, R be the resolved parts of the pressures on the body in
these directions. Let jx be the mass of the body. Then we have
dt
d^x
and two similar equations. Substituting for -j^ its value in terms
a\., o3y, co^ we have
and two similar equations.
If we now take the axes fixed in space to coincide with the
principal axes at the fixed point at the moment under considera-
tion we may substitute for -^^ and -^ from Euler's equations.
We then have
/. |a,,(5+ C-A) (^+5) - i<+^^>} = P+2:mX-/. (f ^-f 2/) ,
with similar expressions for Q and R.
234. Ex. If G be the centre of gravity of the body, show that the terms on
the left-hand sides of the equations which give the pressures on the fixed point are
the components of two forces, one fi"^ . Gil along GH which is a perpendicular on
the instantaneous axis 01, CI being the resultant angular velocity, and the other
i2'2, GiT perpendicular to the plane OGK, where GK is a perpendicular on a straight
B - C C — A
line OJ whose direction cosines are proportional to — j— WgWg, — -g— WgW^,
WjWg, and fi'4 is the sum of the squares of these quantities.
euler's equations.
191
235. To determine the geometrical equations connecting the
motion of the body in space ivith the angidar velocities of the body
about the three moving axes, OA^ OB, 00.
Let the fixed point be taken as the centre of a sphere of
radius anity ; let X, Y, Z and A, B, C be the points in which the
sphere is cut by the fixed and moving axes respectively. Let ZG,
BA produced if necessary, meet in E. Let the angle XZG = '^,
ZG = 6, EGA = (j). It is required to determine the geometrical
relations between 6, cf), y^r, and (o^, co^, co^.
Draw GN perpendicular to OZ. Then since -yjr is the angle
the plane COZ makes with a plane XOZ fixed in space, the velo-
city of C perpendicular to the plane ZOG is GN -^ , which is the
same as sm
dt
, the radius 00 oi the sphere being unity.
dO
the velocity of along ZG is -j- .
Also
Thus the motion of is re-
presented by -J- and sin 6 -~- respectively along and perpendi-
cular to ZO. But the motion of G is also expressed by the angular
velocities w^ and w^ respectively along BO and GA. These two
representations of the same motion must therefore be equivalent.
Hence resolving along and perpendicular to ZG we have
de
dt
^df
= ft)j sin (j) + 0).^ cos (f)
f^'m 6 ~~ = — 0), cos 6 + ct)„ sin 6
dt I T i r J
192 MOTION IN THREE DIMENSIONS.
Similarly by resolving along GB and CA we have
dd . , dyjr ■}
0)^ = -J- sm (p — ■—- sm U cos (p
dd , , d^ . n ■ ,\
"^^ ^ dt ^"^^ '^^dt^'^^ ^^^^ "^ J
These two sets of equations are precisely equivalent to each
other and one may be deduced from the other by an algebraic trans-
formation.
In the same way by drawing a perpendicular from E on OZ we
may show that the velocity of E perpendicular to ZE is -~ sin ZE,
and this is the same as ~j~ cos 6. Also the velocity of A relative
to E along EA is in the same way -^ sin CA, and this is the
same as -^. Hence the whole velocity of A in space along AB
is represented by -p cos ^ + -~^ . But this motion is also ex-
pressed by Wg. As before these two representations of the same
motion must be equivalent. Hence we have
d-\lr n d(b
If in a similar manner we had expressed the 'motion of any
other point of the body as B, both in terms of w^, w^, w^ and
6, (f), 'xjr, we should have obtained other equations. But as we cannot
have more than three independent relations, we should only
arrive at equations which are algebraic transformations of those
already obtained.
236. Ex. li p, q, r be the direction cosines of OZ with regard to the axes OA,
OB, 00, show that these equations may be put into the symmetrical form
dp ^ do ^ dr ^
Any one of these may be obtained by differentiating one of the expressions
p — -sin cos (p, g = sin ^sin 0, r = cos^. The others may be inferred by the
rule of symmetry.
237. It is clear that instead of referring the motion of the body
to the principal axes at the fixed point, as Euler has done, we
may use any axes fixed in the body. But these are in general so
complicated as to be nearly useless. When, however, a body is
making small oscillations about a fixed point, so that some three
rectangular axes fixed in the body never deviate far from three
axes fixed in space, it is often convenient to refer the motion to
euler's equations.
193
these even though they are not principal axes. In this case
CO , CO , 0) are all small quantities, and we may neglect their
products and squares. The general equation of Art. 227 reduces in
this case to
dco^ ^ do)^ ^ do), ^ j^j
dt dt dt
where the coefficients have the usual meanings given to them in
Chap. I. We have thus three linear equations which may be
written thus :
F
dt
d(o,
'dt
+ B
dt
dt
dco
It
dw^
~dt
dt
dt
dt
238. It appears from Euler's Equations that the whole changes of w^, u.,, wj
are not due merelj' to the direct action of the forces, but are in part due to the
centrifugal force of the particles tending to carry them away from the axis about
which they are revolving. For consider the equation
~dt
N A-B
+
C C
Of the increase dw.^ iu the time dt, the part
N
C
-B
C
dt is due to the du-ect action of
w^Wo f^* ^^ ^^6 ^^ ^^^ centrifugal
the forces whose moment is N, and the part
force. This may be proved as follows.
If a hody he rotating about an axis 01 ivith an angular velocity w, then the
vioment of the centrifugal forces of the xvhole body about the axis Oz is (A -B) Wiw^.
Let P be the position of any particle m and let x, y, z be its co-ordinates. Then
x-OR, y=RQ, z=QP. Let PS be a perpendicular on 01, let OS=u, and PS^r.
Then the centrifugal force of the particle m is w^rm tending from 01.
11. 1).
1.*]
194 MOTION IN THREE DIMENSIONS.
The force wVjji is evidently equivalent to the four forces u"xm, uPym, urzm, and
- u^um acting at P parallel to x, y, z, and u respectively.
The moment of w"xm round Oz— - u^xym
co-ym = (i}-xym
uhm =0
these three therefore produce no effect.
The force - u^um parallel to 01 is equivalent to the tlu'ee, - uoj^ iiin, - tau^ urn,
— uca^um, acting at P parallel to the axes, and their moment round Oz is evidently
wum (w,?/-wox). Now the direction cosines of 01 being — , — ^, -3, we get by
^ i-^ ■^ ' www
projecting the broken line x, y, z on 01, u= —x+ — y+ —z; therefore sub-
stituting for u, the moment of centrifugal forces about Oz is
= (uixy + w^w^y^ + w^^WgT/g - Wj^w^a;^ - Wg^y — WgWjCcs) iii.
Writing S before every term, and supposing the axes of x, y, z, to be principal
axes, then the moment of the centrifugal forces about the principal axis Oz
= Uj^oj.^'Zmiy^ - x^) = w^w, {A - B).
Let the moments of the centrifugal forces about the principal axes of the body
be represented by L', M', N', so that
L'={B-C)ui.o3^, i)f' = (C-yl) W3W1, N' = {A-B)w-^^u^,
and let G be their resultant couple. The couple G is usually called the centrifugal
couple.
Since i'w]^+ Jf 'w^ + iV'wg = 0, it follows that the axis of the centrifugal couple is
at right angles to the instantaneous axis.
Describe the momental ellipsoid at the fixed point and let the instantaneous
axis cut its surface in I. Let OH be a perpendicular from on the tangent plane
at I. The direction cosines of OH are proportional to Aw-^, Bw^, Cwj. Since
^Wj^Z/' + -Sw2ilf' +Cw3W=0, it follows that the axis of the centrifugal couple is at
right angles to the perpendicular OH.
The plane of the centrifugal couple is therefore the plane lOH.
If [xlc^ be the moment of inertia of the body about the instantaneous axis of
rotation we have ¥■= -y^^, and T^^ai-w^ is the Vis Viva of the body. We may
then easily show that the magnitude G of the centrifugal couple is (? = T tan ([>,
■where ^ is the angle lOH.
This couple wUl generate an angular velocity of known magnitude about the
diametral line of its plane. By compounding this with the existing angular velocity,
the change in the position of the instantaneous axis might be found.
Expressions for Angular Momentum.
239. We may now investigate convenient formuloe for the
angular momentum of a body about any axis. The importance
of these has been already pointed out in Art. 77. In fact, the
general equations of motion of a rigid body as given in Art. 71,
EXPEESSIOXS FOR ANGULAR MOMENTUM. 1.95
cannot be completely expressed until these formulae have been
found.
When the body is moving in space of two dimensions about
either a fixed point, or its centre of gravity regarded as a fixed
point, the angular momentum about that point has been proved in
Art. 88 to be Mlc^a where Mk^ is the moment of inertia, and co
the angular velocity about that point. Our object is to find cor-
responding formulae when the body is moving in space of three
dimensions. Follov/ing the same order as in Euler's Equations,
we shall first find the angular momentum about any fixed straight
line in space, taken as the axis of z and passing through the
fixed point; secondly, the momentum about any fixed straight line
in the body and also passing through the fixed point, and lastly, we
shall show how the angular momenta about other axes may be
found.
240. A body is turning about a fixed point in any manner, to
determine the moments of the momentum about the axes, i.e. to find
the areas conserved round those axes. See Chap. ii. Art. 78.
Let (a?, y, z) be the co-ordinates of any particle m of the body
referred to axes fixed in space meeting at the fixed point. Let
0)^, &)^, ct)^ be the angular velocities of the body about the fixed
axes. Then the moment of the momentum about the axis of z is
7 ^ ( dy dx\
' \ dt ^ dtj
dx clt/
Substituting for -^ , -^ their values
dx ^
dii
we have li^ = Sm {x^ + y"^) &>, — {l,mxz) w^ — {%myz) w^.
241. The coefficients of w^, co^, &>, are the moments and pro-
ducts of inertia of the body about the axes, and if the axes be
fixed in space, these will generally be variable. In some cases it
will be found more convenient to take as axes of reference three
straight lines fixed in the body.
Let (Uj, ft)^, 0)3 be the angular velocities of the body about rect-
angular axes Ox, Oy , Oz fixed in the body and meeting at the
fixed point 0. Since in the expression given above for h^ the
fixed axes may be any whatever, let them be chosen so that the
moving axes coincide with them at the time t. Then, w^— w^,
13—2
196 MOTION IN THREE DIMENSIONS.
C(>j,= &)2, (1)^=0)^, and tlie moment of the momentum about the
moving axis of z will be expressed by the form
Ii^ = OcOg — E(o^ — Dco^,
where C=^m{x"^ +y"^), E = ^mx'z', D = %my'z'.
These will be constant throughout the motion, and their values
may be found by the rules given in Chapter I.
If the axes fixed in the body be principal axes, then the pro-
ducts of inertia will vanish. The expressions for the moments of
the momentum will then take the simple forms
h^ = Ao}^ j
K = ^^'2 [ >
where A, B, C are the principal moments of the body.
Let the direction-cosines of the axes fixed in space but moving
with reference to axes fixed in the body be given by the following
diagram ; where, for example, h^ is the cosine of
the angle between the axes of z and y'. It has
just been proved that the resultant of the mo-
menta of all the particles of the body is equiva-
lent to the three "couples" /i/, h^, h^ about the
axes Ox , Oy, Oz . Hence the moment of the
momentum about the axis of z which is fixed in space may be
written in the form
which will be frequently found useful.
242. It may be required to find the moment of the momen-
tum about axes neither fixed in space nor in the body, but moving
in any arbitrary manner. This will be expressed by the same
form as if the axes were fixed. If co^, Wy, &), be the angular
velocities about these axes, the moment required will be
= %m {a? +y'^) co^ — (Zmxz) co^ — i^myz) co^.
If the axis of z coincide with the instantaneous axis of rotation,
60^ = 0, G)^=0, and co^ is the resultant angular velocity. The ex-
pressions for the moments of the momentum or areas conserved
about the axes of x, y, z become respectively
— (^mxz) a\, — (Zmyz) co^ , 'tm [x^ +• y"^) w, .
The axis of the couple which is the resultant of the moments
of the momentum about the axis is sometimes called the resultant
axis of angular momentum and sometimes the resultant axis of
areas. It is to be remarked that this axis does not in general
X,
y>
z
x
a„
«2'
a.
y
K
K
h
z
C.'
C2>
^3
ON MOVING AXES AND RELATIVE MOTION. 197
coincide with the instantaneous axis of rotation. The two are
coincident only when the axis of rotation is a principal axis. If a
body be turning about a straight line, which we may call the axis
of ^, as instantaneous axis, it is a common mistake to suppose that
the angular momentum about a perpendicular axis is zero. We
see from the last remark that this is not generally true.
If it be required to find the moment of the momentum about
the axis of z of a rigid body moving in any manner in space, we
may use the principle proved in Chapter ii. Art. 76.
In the case of a system of rigid bodies, the moment of their
momenta may be found by adding up the separate moments of the
several bodies.
Ex. 1. A triangular area A CB wliose mass is M is turning round tlie side CA
with an angular velocity w. Show that the angular momentum about the side CB
is yV Mob sin^ Cw, where a and h are the sides containing the angle C.
Ex. 2. Two rods OA, AB, are hinged together at A and suspended from a
fixed point 0. The system turns with angular velocity w about a vertical straight
line through so that the two rods are ui a vertical plane. If 6, (p be the inclina-
tions of the rods to the vertical, a, h their lengths, M, M' their masses, show that
the angular momentum about the vertical axis is
w [(I M + Jf' ) a^ sin2 6 + M'ab sin ^ sin + 1 M W sin^ 0.]
Ex. 3. A right cone, whose vertex is fixed, has an angular velocity w com-
municated to it about its axis OC, while at the same time its axis is set moving in
TT
space. The semi-angle of the cone is - and its altitude is h,. If B be the inclina-
tion of the axis to a fixed straight line Oz and i/' the angle the plane zOG makes
with a fixed plane through Oz, prove that the angular momentum about Oz is
\MWia (sin^ -T- + i cos 6), where 31 is the mass of the cone.
Ex. 4, A rod AB is suspended by a string from a fixed point and is moving
in any manner. If {I, m, n) {p, q, r) be the direction cosines of the string and rod
referred to any rectangular axes Ox, Oy, Oz, show that the angular momentum
about the axis of z is
where M is the mass of the rod, and a, h the lengths of the rod and string.
On Moving Axes and Relative Motion.
243. In many cases it will be found convenient to refer the
motion of the body under consideration to axes moving in space in
some manner about a fixed origin. If we refer the motion of these
axes to other axes fixed in space we shall have the inconvenience
of two sets of axes. For this reason their motion at any instant is
sometimes defined l^y angular velocities [6^, 0^, 0.,) about them-
198 MOTION IN THREE DIMENSIONS.
selves. In this case we are to regard the axes as if they were
a material system of three straight lines at right angles whose
motion at any instant is given hy three coexistent angular velo-
cities about axes instantaneously coincident with them.
When the axes are moving we may suppose the motion of the
body to be determined by the three angular velocities (o^, co^, Wg
about the axes, in the same manner as if the axes were fixed for
an instant in space. The position of the body at the time t+ dt
may be constructed from that at the time t by turning the body
througli the angles co^dt, w^dt, co^dt successively round the instan-
taneous position of the axes. But it must be remembered that
(o^dt does not now give the angle the body has been turned
through relatively to the plane xz, but relatively to some plane
fixed in space passing through the instantaneous position of the
axis of z. The angle turned through relatively to the plane of xs
is ((Dg — ^3) dt.
244. To find the resolved lyart of the velocity of any particle
parallel to the vioving axes.
The resolved parts of the velocity of any point whose co-
die fill Ci ^
ordinates are [x, y, z) are not given by ~ii •> 'jI ^ ~Ji • These are
the resolved velocities of the particle relatively to the axes. To
find the motion in space we must add to these the resolved veloci-
ties due to the motion of the axes themselves. If w^e supposed the
particle to be rigidly connected with the axes, it is clear that its
velocities would be expressed by the forms given in Art. 219 with
^1' ^2' ^3 substituted for Wj, o).^, eo,. So that the actual resolved
velocities of the particle are
dx
" = dt -
-y^ + ^o,,,
dt
-ze^ + xe^,
dz
-x0, + 7je^.
245. To find the accelerations of any ixirticle parallel to the
axes we may proceed thus.
The velocities of the particle at the time t resolved parallel to
the axes Ox, Oy, Oz are respectively {u, v, w). At the time t-\- dt,
the axes have been turned into the position Ox', Oy', Oz' by
rotations equal to O^^dt, 6^dt, O^dt round the axes Ox, Oy, Oz
respectively, and the velocities of the particle parallel to the axes
in their new position are
dii ,, dv ,, dv> J
u + ~r dt, v + -rdt, w + 7 dt.
dt dt dt
ON MOVING AXES AND RELATIVE MOTION. 199
Describe a sphere of unit radius whose centre is at the fixed
orio-in and let all these axes cut the sphere in the points x, y, z,
x, y , z respectively. Thus we have tvfo spherical triangles xyz
and x'y'z, all whose sides are right angles. The resolved part of
the velocity of the particle at the time t-\-dt along the axis of z is
[w+ -r.dt\ co^zx -\- iv + -jdt\ cos^j/'+ Uu + --^dt\ co^ zz.
By the rotation round Oy, x has receded from z by the arc 6^dt,
and by the rotation round Ox, y has approached z by the arc d^dt.
Therefore
zx = zx + 6^ dt,
zy —zy — 0^ dt.
Also the cosine of the arc zz differs from unity by the squares
of small quantities. Substituting these we find that the compo-
nent velocity of the particle at the time t-{- dt parallel to the axis
of z is ultimately
vj + -~r- dt — uB^ dt + vQ, dt.
dt 2^1
But the acceleration is by definition, the ratio of the velocity
gained in any time dt to that time. Hence if Z be the acceleration
resolved parallel to the axis of z, we have
„ dw ^ /,
Similarly if X and Y be the accelerations parallel to the axes
of X and y, we have
246. Ex. 1, Let the motion be referred to oblique moving axes so that the
sides of the spherical triangle xyz are a, b, c and the angles A, B, C. Let the equal
quantities sin a sin & sin C, sin 6 sine sin J, sin c sin a sin B be called /i. Prove
that if the velocity be represented by the three components u, r, w parallel to these
axes, then the resultant acceleration parallel to the axis of z is
„ dw du , cfo „ ^
Z — -— + ~- cos b + -r cos a - u9m + vd,ix,
dt dt dt ' ^
with similar expressions for X and Y.
This may be done by the use of the spherical triangles xyz, x'y'z', by first proving
that zx' = b + 6„ dt sin c sin ^4, zy'=a-6^dt sin c sin B, and then substitiating as before.
Ex. 2. Prove in the same way that if x, y, z be the co-ordinates referred to
oblique axes, and u', v', w' the resultant velocities parallel to the axes,
, dz dx , dy
with similar expressions for v' and v'.
200 MOTION liS^ THREE DIMENSIONS.
Ex. 3. Prove also that the equations connecting ic, v, w with the co-ordinates
dz
- cot B - cot A
z X y
with two similar expressions f or li and v.
Since to' is the resolved velocity parallel to z of (u, v, w,) we have
u cos h + v cos a + iv — to',
with similar expressions for v,' and v'. By solving these we get the required values
of u, V, w.
Ex, 4. If the whole acceleration be represented by the three components
X, T, Z parallel to the axes, prove that the expressions for these in terms of uvw,
may be obtained from those given in the last example by changing x, y, z into u, v, w
and u, V, lo into X, Y, Z.
247. To express the geometrical conditions that a straight line
whose equations with reference to the moving axes are given is fixed
in direction in space.
Let the equation to tlie given straight line be
X —f_ y — g __z — h
p q r '
and let the equations be so prepared that {p, q, r) are the
direction cosines of the line. Let a straight line be drawn through
the origin parallel to this given straight line and let a point P be
taken on this at any given distance L from the origin 0. Then
the co-ordinates of P are pL, qL, rL respectively. Since the
straight line OP is fixed in direction in space, the resolved parts
of the velocity of P parallel to the axes are zero. Hence we have
^-Lq0,+ Lr9,= O,
and two similar equations. The required geometrical conditions
are therefore
When it is necessary to refer the motion of these moving axes
to other axes fixed in space, Ave may either use the equations of
this article or those of Art. 235. Takinsr the notation of the
ON MOVING AXES AND RELATIVE MOTION. 201
article referred to, it is obvious (the axes being treated as a body
consisting simply of three straight lines) that we shall have the
results
-^ sin 6 = — 6^ cos 6 + 6^ sin S
at 1 r 2 r 1
do /I • I /I I
-y = t/j sm 9 + c7„ cos (p >■ •
at at
These equations will determine 6, cp, -v/^ in terms of the angular
velocities 0^, 0.^, 6^.
248. To express the geometrical conditions that a point whose
. co-ordinates ivith reference to the moving axes are (x, y, z) is fixed
in space.
This may be done by equating to zero the resolved parts of the
velocity of the point as given in Art. 244. If the origin of the
moving axes be fixed, the conditions are
and two similar equations. If the origin be in motion, let u^, i\, w^
be the resolved parts of its velocity parallel to the axes, then the
required conditions are clearly
and two similar equations.
249. Ex. Let the direction cosines of a straight line OM fixed relatively to the
moving axes be (X, /i, i') and let it be required to refer the motion of OM to some
straight line OL fixed ra space whose du'ection cosines at the time t are [p, q, r).
Let the angle L03I be 6 and let ^ be the angle the plane LOM makes with any
fixed plane in space passing thi'ough OL. Then show that
cos 6 —p\ + qij. + TV, \
sin^ e''~ = 9^ (p - X cos O) + 0.2(q-iJL cos 6) + 0.^ [r - v cos d)\ '
If 0^, 0.„, be the resolved parts of the angular velocities about OL, OM respec-
tively, the last equation may be written in the form
sin2^^=^;-6'„.cos6'.
dt
If the straight line OM be not fixed relatively to the axes, then (X, /x, v) will be
variable and we must add to the right-hand side of the second equation the deter-
minant
A d,j. d\\ ( dv djjX ( d\ ^ dA
202 MOTION IN THREE DIMENSIONS.
The mode of proof may be indicated as follows. Let P be a point in OM at a
distance unity from and let P move about with OM. The moment of its velocity
about OL is siu^ 6 -- . But if [x, y, z) be the co-ordinates of P, its velocities paral-
lel to the axes are given by Art. 244, and the moments of these velocities about the
axes wlU be L—yw-zv, M=m-xw, N—xv-yu. Hence the moment about OL
wiU be
If we effect these substitutions, and since OM is itnity, replace x, y, z by X, ju, v, we
get the results in the example.
250. To explain a method of changing from fixed to moving
axes.
If a body be moving about a fixed point and we have esta-
blished any general proposition referring its motion to fixed axes
meeting at the fixed point, then we may use the following method
to infer the corresponding proposition referring the motion to axes
moving in any proposed manner about the origin. Suppose the
general equation established to be
T|r jo),, -^% &c [ = 0,
dt
where w^, oOy, w^ are the angular velocities about the fixed axes.
Let CO , co^, CO3 be the angular velocities of the body about the
moving axes and let the motion of the axes be defined by the
angular velocities 6^, 6^, 9^ about themselves.
The fixed axes being arbitrary in position, let them be so
chosen that, at the moment under consideration, the three moving
axes are passing through them, so that the two sets are for an
instant coincident. Then, by referring to Art. 243, we see that
we may write ft)^=ft>^, (Oy = (o,^, (£>^= w^, but we cannot assert
—^ = — ^ , for the movino: axes at the time t + dt are not coinci-
dt dt
dent with the fixed axes.
To determine the relation between — — * and — r^ we may proceed
dt dt
thus. Let OL be any straight line fixed in space making with the
moving axes the angles a, /S, 7. Let O be the angular velocity of
the body about this straight line. Then as in Art. 229
O = ft)^ cos a + cOg cos /3 + (U3 cos 7,
(Zn dw. dco^ „ do),
:. -^— = -^ cos a H — r." cos p -\ — ^^ cos 7
dt dt dt ^ dt '
dec . ^d/3 . dj
-a,,sma^^-a,,sm^^^-a)3sm7^y.
ox MOVING AXES AND EELATIVE MOTION. 203
Since OL is any fixed line in space, let it be so chosen that the
niovinsf axis of z coincides with it at the time t. Then a
Q
8= - , and 7 = 0, also —r will be -r^' Since a is the angle OL
2 at at
d% .
makes with the moving axis of x, —r is the rate at which the axis
dt
of X is separating from a fixed straight line coincident Avith the
dB
axis of z and this is clearly Q^. Similarly -^- = — 9^, hence
dw^
dt
dco^
dt
— ft)
A
+
ft)A-
Similarly
dco^
dt
_ doy^
dt
-«.A
+
^A
dw^
dt
_d(o^
dt
-«3^.
+
<^A
If we substitute these expressions in the given general equation
we shall have the corresponding equation referred to moving axes.
If the moving axes be fixed in the body, and move with it, we
have ^j = &)j, 6^=Qy^, 0^= o)^. In this case the relations will
, dco^ do). d(o„ dco„ dco. dw„ . . , oor.
become ~^ = -j^, —^ = ^ , -^' = ^ , as m Art. 229.
dt dt dt dt dt at
The preceding proof of the relation between — r-^ and —r^ is a
^ o ^ dt dt
simple corollary from the parallelogram of angular velocities. The
result will therefore be true for any other magnitude which obeys
the "parallelogram law." In fact the demonstration is exactly the
same. Now linear velocities and linear accelerations do obey this
law. Hence the expressions obtained in Arts. 244, 245, for the
velocities {u, v, tu) and the accelerations (X, Y, Z) may be deduced
from the one proved above.
If the general equation -v^ = should contain the velocity or
acceleration of any particle of the body, then to obtain the corre-
sponding equation referred to moving axes, Ave must substitute for
these velocities or accelerations the expressions found in Arts. 244
and 24.5.
251. If the general eqiiation should contain ——^ or any other second differen-
tial coefficients, the expressions to be substituted for them become more compli-
cated.
204 MOTION IN THREE DIMENSIONS.
Since — - , -—^ , -^ , being angular accelerations, follow the parallelogram law,
at at at
we have
d^ fdu3^ „ „ \ fdw., „ .\ ^ fdo,io . „\
'dt ^ Vdt ~ '^^ •^'^ "^ V '^^^ " ^ \'dt ~ '^^ ^ ^ '^^ V *^°^ ^ V "^ "^^ "^ '^^ V ^^^ '^'
We may repeat the same reasoning and we shall finally obtain
So we may proceed to treat third and higher differential coefficients.
252. A body is turning about a fixed point in any manner,
to determine the moments of the effective forces about the axes.
Let {x, y, z) be the co-ordinates of any particle m of the body
referred to axes fixed in space and meeting at the fixed point,
and let h^, h^, \ be the moments of the momentum about the
axes.
The moment of the effective forces about the axis of z is
^ / dSi d^x\
^'''[''df-yw)'
dh
and this may be written in the form --^ . Thus the moments of
the effective forces about axes Ox, Oy, Oz fixed in space are
respectively -^,S -jf > -jf y where h^, \, h^ have the values
found in Art. 240.
Let A/, /i/, hi be the moments of the momentum, found by
Art. 242, about axes Ox , Oy, Oz moving in space about the fixed
origin. Let 6^, 6^, 6^ be the angular velocities of these axes about
their instantaneous directions. Then since moments or couples
follow the parallelogram law, we see by the proposition of Art. 250
that the moments of the effective forces about the moving axes
are respectively
If the moving axes be fixed in the body, we have ^^ = 6)^,
0^=0)^, 6^ = (o^, and the equations admit of some simplification.
If the axes be iha principal axes we have h^ = Aw^, h^ = Bw.^,
ON MOVING AXES AND EELATIVE MOTION. 205
A3' = CcDg, and the moments of the effective forces take the simple
forms
di
COM,
where A, B, C are the principal moments. See Art. 230.
If it be required to find the moment about the axis of z of the
effective forces on a rigid body moving in any manner in space,
we may use the principle proved in Chap. II. Art, 72.
In the case of a system of rigid bodies, the moment of their
effective forces may be found by adding up the separate moments
of the several bodies.
2.53. To obtain the general equations of motion of a system of
rigid bodies.
These equations have been already obtained in Chap. ii. Art. 83,
when the system is referred to axes fixed in space. If the axes be
moveable we must replace the accelerations -jI ^ ~ii y ~n7 ^7 the
corresponding forms in Art. 24*5 and the couples ~~ , -~ , -^
at cLti at
by the expressions in Art. 252.
Thus, suppose we refer the motion to three axes moving in
space about a fixed origin 0. Let X, Y, Z be the impressed
forces on any rigid body of the system, including the unknown
reactions of the other bodies of the system. Let L, M, N be the
moments of these forces about axes drawn through the centre of
gravity of the body parallel to the co-ordinate axes. Let m be the
mass of the body. Then if we adopt the notation of Arts, 245
and 252, the equations of motion for the rigid body under con-
sideration will be
au a , a -^
dv ^ ^ I
rii
^-^^^^ + ^^^3 = -,
dw ^ n Z
at ^ ^ m
206 MOTION IN THREE DIMENSIONS.
and ^-/^;^3 + /^X = a]
at 2 3 „ .
rih ' '
dt
where ]\, IiJ, h.^ have the values given in Art. 240 *.
Similar equations will apply for each body of the system.
Besides these dynamical equations there will be the geome-
trical equations expressing the connections of the system. As
every such forced connection is accompanied by some reaction,
the number of geometrical equations will be the same as the
number of unknown reactions in the system.
Thus we have sufficient equations to determine the motion.
254. If two of the principal moments at the fixed origin
are equal, it is often convenient to choose as axis of z the axis
00 of unequal moment, and as axes of x, y two other axes OA,
OB moving in any manner round 00. Let ^ be the angle the
plane oi AOG makes with some plane fixed in the body and pass-
ing through 00. Then we have 9^= ay^, 0.^= a>^, and 6^ = cOg^- ^ .
Also by Art. 241, we have li^=Aoi^, h.^ — Aa^, h^' = Cco^. The
equations of motion of Art, 253 now become
dt
J
In this case the most convenient geometrical equations to
express the relations of these moving axes to straight lines fixed
in space will be those given in Art. 235.
Since -— is arbitrary, it may be chosen to simplify either the
dynamical or the geometrical equations.
* Tlie equations of Art. 253 were first given in this form by Prof. Slesser to
whom the equations of Art. 254 had been shown by the author. It appears however
that similar results had been previously published in Liouville's Journal in 1858.
ON MOVING AXES AND RELATIVE MOTION. 207
First, we may put -^ = — 03^. The dynamical equations then
become
at ^ ^
at
Secondly, we may so choose -Jj that ^ = 0. In this case the
plane COA always passes through a straight line OZ fixed in
space. The geometrical equations then become,
dO d-4r . ri dy dyjr ^
^ = c.„ ^sm^=-a,„ --^+ — cos^ = a,3.
'2.00. If three principal moments at the fixed origin be
equal, there are three sets of axes such that when the motion is
referred to them, the equations take a simple form.
First. We may choose axes fixed in space. Since every axis
is a principal axis in the body, the general equations of motion
become
da^ _ L dco^ _ M c?&>3 JSf
llt~A' Ht^A' ~dt^A'
The geometrical equations of Art. 235 are not required.
Secondly. We may choose one axis as that of 0(7 fixed in
space and let the other two move round it in any manner, then as
in Art. 254, the equations of motion become
dt '' dt A
dco^ , dx M
dt ^ ' dt A
d^ _^
dt ~ A
Thirdly. We can take as axes any three straight lines at right
angles moving in space in any proposed manner. The equations
of motion are then by Art. 253
d(o /) , z) ^
^^^ n , n N
208 MOTION IN THREE DIMENSIONS.
The geometrical equations will then be the same as those
given in Art. 285 or Art. 247.
256. Ex. An eUipsoid, whose centre is fixed, contracts by cooling and being
set in motion in any manner is under the action of no forces. Find the motion.
The principal diameters are principal axes at throughout the motion. Let us
take them as axes of reference. The expressions for the angular momenta about
the axes are by Art. 241 h-^^'=Aw^, h2=B(i}„, Ji^'=Cuy The equations of Art. 253
then become
|(5c.„)-(C-^)c.3c.., = [.
Multiplying these equations by A co-^, Bw^, Cw.^, adding and integrating we see
that A^u}i^ + B^oo2^ + C'^w.J^ is constant throughout the motion. To obtain another
integral, let A—AQf{t), B — BQf{i), C=C(,f{t) where f{t) expresses the law of cool-
ing which has been supposed such that the body changes its form very slowly. Let
w^/(f) = 0^, u.-^f[t) — ^.2, U3f(t)=Q^, and put — =-7T-r, then the equations become
at J (t)
and two similar equations. These may be treated as in the chapter on the motion
of a body under no forces. Liouville's Journal.
2.57. The theory of relative motion is best understood by-
viewing it in as many aspects as possible. We shall, therefore,
now consider a method of determining the motion which is more
elementary, and does not, in the result, make an exclusive use of
Cartesian co-ordinates.
Let it be required to refer the motion of a particle P to any
given system of moving axes. The motion of these axes during
a,ny interval of time dt may be constructed by a screw-motion
along and round some straight line 01. Let Uclt be the transla-
tion along and Q.dt the rotation round 01. Let P^ be the position
of P at the time t, and let P^ be attached to the given axes and
move with them during the interval dt. Let / represent the
acceleration of P^ in direction and magnitude. The particle P
will, of course, separate from P^ ; but, as is explained in Dynamics
of a Particle, the actual acceleration of P in space is the resultant
of its acceleration relative to P^ treated as a fixed point, and the
acceleration f of P^.
To find the acceleration relative to P^, we must treat P^ as a
fixed point. Draw P^z parallel to 01 and let P^y' be the projec-
tion of the direction of the relative motion of P on a plane perpen-
dicular to P^z, and let P^x be perpendicular to P^y and P^^z.
These axes are taken for the purposes of description, and but little
ON MOVING AXES AND EELATIVE MOTION. 209
use will be maxie of co-ordinates. Let these axes move during the
time dt, so as to preserve unchanged the angles they make with
the given axes of reference. Let PJP^ be the displacement of P
relative to P^, and let P^P^ make an angle 6 with P^z', so that
PqP^ sin 6 is the projection of the relative displacement on the
plane of x'y'. Since these axes, in the interval of time dt, have
turned round P^z through an angle D.dt, the x co-ordinate of P,
after that interval, is greater than what it would have been if
referred to axes fixed in space by P^P^ sin OVldt, while the y and z
co-ordinates are unaltered. We have here, according to the
rules of the Differential Calculus, retained only the lowest powers
of the small quantities which occur. Hence, if the acceleration
of P relative to these axes be compounded with an acceleration
equal and opposite to that which would produce a displacement
P^P^ sin 6£ldt, we shall have the acceleration of P relative to axes
whose directions are fixed in space, but having the moving point
Pq as origin. Let Fbe the velocity of the particle relative to the
moving axes, then- P^P^— Vdt in the limit, and therefore the
change hx in the x co-ordinate of P is hx = VVl sin 6 {dt)'\ If
f be the acceleration corresponding to this displacement, we
have hx = |/' (dty. Comparing these two expressions we see
that f = 2 Vll sin 0. This acceleration must be supposed to act
along the positive direction of the axis of x'.
The general conclusion is that the acceleration of P in space is
the resultant of the accelerations f, —f, and the acceleration
relative to the given moving axes.
The equations of motion of a particle being comprised in the
formula, "acceleration in any fixed direction equals the impressed
force divided by the mass," it is more convenient to transpose the
terms / and — /' to the other side of the equation with opposite
signs, we then have the following theorem :
In finding the motion of a imrticle of mass m ivith reference
to any moving axes, ive may treat the axes as if they tvere fixed in
space, provided we regard the jxirticle as acted on, in addition to the
impressed forces, hy two forces:
(1) a force equal and opposite to that tvhich luould constrain
the particle to remain fixed to the moving axes, and which is mea-
sured hy mf where f is the reversed acceleration of the point of
moving space occupied hy the particle,
(2) a force p)erpendicidar to both the direction of relative
motion of the particle and to the central axis or axis of rotation of
the moving axes, and which is measured hy 2mVn sin 6, where V
IS the relative velocity of the jmrticle, 12 the resultant angular velocity
of the moving axes, and 6 the angle between the direction of tiie
velocity and the axis of rotation.
R. D. 14
210 MOTION IN THREE DIMENSIONS.
To find the direction of this last force^ we notice that in the
investigation, the rotation O has been supposed to be, as usual,
from the positive direction of x to the positive direction of y , and
that the positive direction of y is a tangent to the projection of
the relative velocity of P. Since the force acta along the positive
direction of x , we have this rule : Stand with the hack along the
aocis of rotation, so that the rotation appears to he in the direction
of the hands of the tuatch ; then vietuing the particle receding from
the axis of rotation, the force acts on the left hand. We may call
these forces respectively the force of moving space, and the com-
pound centrifugal force of -the particle.
258. This method of determining the relative motion of a particle was first
given by Clairaut in 1742, and afterwards the same rule was demonstrated in a
different manner by Coriolis. The arguments of the former were criticized and
improved by M. Bertrand in a paper published in the nineteenth volume of the
Journal Poly technique. We have here followed, with but slight variations, M.
Bertrand's mode of proof, as being the most diiSerent of any from the analytical
methods given in this chapter. But it will be important to perceive the connection
between the two methods of expressing the relative motion, and this will be
explained in the next article.
259. Let us refer the motion of P to any moving axes having
a fixed origin, and let X, Y, Z be the impressed forces on the
particle resolved parallel to the axes. If we eliminate u, v, w
from the equations of Art. 244 and Art. 245 we get
with similar expressions for Y and Z. Here A, B, G are functions
of 6^, 6^, 6g and their differential coefficients with regard to t,
which it is unnecessary to write down. If x, y, z were constants,
all the terms of X would disappear except the three last. These
then with the corresponding terms in Y and Z express the acce-
leration of a point P^ rigidly attached to the axes, but occupying
the instantaneous position of P. The second and third terms of
X taken together, with the corresponding terms of_ Y and Z,
express the resolved parts of an acceleration perpendicular both
to the resultant axis of the rotations 6^, 6^, 6^, and to the direc-
ft 'Y* fill fi ^
tion of the velocity which is the resultant ^^ ^ ' 3 ' ^ • ^7
adding up the squares we easily find the magnitude of the re-
sultant acceleration to be 211 V sin Q, where 11, Y and Q have the
meaning given in Art. 257*.
'to o
* Another demonstration by the use of polar co-ordinates is given in Vol. xii. of
the Quarterly Journal of Mathematics, by the Rev. H. W. Watson.
ON MOVING AXES AND EELATIVE MOTION. 211
To determine the manner in which these forces should be
applied, we must transpose the terras which represent them to the
other sides of the equations. The first equation will then become
and the other two will take similar forms. These are the equa-
tions of motion of a particle referred to fixed axes, moving under
the same impressed forces as before, but with two additional forces.
These are, first, a force equal and opposite to that represented by
mf, where / is the acceleration of the point of moving space occu-
pied by the particle; and secondly, a force whose magnitude has
been shown to be 2??i FX2 sin 6. To determine the direction of this
force, let the axis of z be taken along the instantaneous axis of
rotation of the moving S23ace, and let the plane of yz be parallel to
the direction of motion of the particle, then ^^ = 0, ^2 = and
-p = 0. We then easily see that this force disappears from the
ci tl cl Z ftp
equations giving m -~ and m -^_j ; while in that giving m --^^ ,
dv
we have the single term 2m -- 6^. The magnitude of this force is
obviously 2m VCl sin 6, and it acts along the positive direction of
the axis of x. This is the left-hand side when the receding parti-
cle is viewed from the axis of rotation and the rule given at the
end of Art. 257 is therefore established.
When these equations have been integrated, the arbitrary con-
stants are to be determined from the initial values of x, y, z, -^ ,
-^ dt '
dij dz
-^> -ji' These differential coefficients are clearly the components
of the initial velocity of the particle, taken relatively to the mov-
ing axes.
260. Ex. If the particle be constrained to move along a curve which is itself
moving in any manner, the compound centrifugal force, being perpendicular to the
direction of the relative velocity of the particle, may be included in the reaction of
the curve. The only force which it is necessary to impress on the particle is the
force of the moving space. If the curve be turning about a fixed axis with an
angular velocity fi in the manner described in Art. 181, the components of the
accelerating force of moving space are clearly fi-r tending directly from the axis of
rotation, and - r perpendicular to the plane containing the particle and the axis.
Here r, as in the article referred to, is the distance of the particle from the axis.
261. In finding the compound centrifugal force it will be
found useful to remember, that we may resolve the angular velo-
11—2
212 MOTION IN THKEE DIMENSIONS.
city n or the linear velocity V in any manner that we please,
and find the forces due to each of the components separately.
Though we have thus more than two forces which must be applied
to the particle, yet, by making a proper resolution, some of these
may produce either no effect, and may therefore be omitted, or
may produce an effect which it may be easy to take account of.
262, When we wish to determine the motion of a rigid body
by this method, we must consider each particle to be acted on by
the two forces corresponding to the position and velocity of that
particle. This will generally require an integration to be per-
formed ; which, though not difficult, is not always convenient.
The forces of moving space for any body are the same as the
effective forces of an imaginary body occupying the instantaneous
position of the real body, and moving with the space occupied by
it. The resultant of these forces may, therefore, be found by the
method indicated in Art. 83.
The components of the compound centrifugal forces on any
particle are, by Art. 259, algebraic functions of 37, ^/f ' T/ * ^^
may, therefore, use Art. 14 to help us in finding the resultants
of the compound centrifugal forces of the whole body. If M
be the mass of the body, V the velocity of its centre of gravity,
n the angular velocity of the moving space, 6 the angle between
the direction of V and the axis of H, then the compound centri-
fugal forces of all the particles of the body are equivalent to a
force 2MVn sin 6 acting at the centre of gravity perpendicular
both to its direction of motion and the axis of II, together with
the compound centrifugal forces of the body after the centre of
gravity has been reduced to rest.
To find these latter forces, let us refer the body to the princi-
pal axes at the centre of gravity as axes of co-ordinates. Let
w , ft) , tOg be the resolved angular velocities of the body, £l^,fl^, II3
the resolved parts of ft about these axes; A, B, G the principal
moments of inertia at the centime of gravity. Then, by Art. 259,
the compound centrifugal forces on any particle of the body whose
co-ordinates are {x, y, z) and mass m, are
with similar expressions for Fand Z. The centre of gravity being
at the origin, the resultant forces of these are easily seen by inte-
gration to be all zero, while the resultant couples about the axes
are
i = a),ft„(^-F^-C')-«,ft,(^+ C-i?),
with similar expressions for M and N.
ON MOVING AXES AND RELATIVE MOTION. 213
263. Ex. 1. A disc of mass M is constrained to move in a i^lane under any
forces while the plane turns about a straight line parallel to the plane and distant
a from it with angular velocity Q,. Show that in finding the motion of the disc, we
may regard the plane as fixed, provided we impress on the disc in addition to the
given forces, (1) a force MQ,'^r~Ma ~- acting through the centre of gravity tending
directly fi'om the projection of the axis of rotation on the plane, where r is the
distance of the centre of gravity from the projection, (2) a couple i^fl^ -^vhere F is
the prodiict of inertia about two rectangular axes in the plane intersecting at the
centre of gravity, and respectively paraUel to the axis and perpendicular to it.
The constants of integration are to be determined from the initial conditions taken
relatively to the moving plane.
Ex. 2. A disc of mass M is constrained to move in a plane under any forces
while the plane turns with angular velocity fi about a straight line perpendicular to
its plane and cutting the plane in the point 0. Show that we may regard the plane
as fixed provided we impress on the disc (1) a force MQ,"r acting at the centre of
gravity and tending directly from the axis, where r is the distance of the centre of
gravity from the axis, (2) a force Mr — acting at. the centre of gravity perpendicular
to r in the direction opposite to the rotation, (3) a couple Ml? -^ , where MJc^ is the
moment of inertia of the disc about an axis tkrough its centre of gravity perpen-
dicular to its plane, (4) a force 2M FO acting at the centre of gravity perpendicu-
lar to its direction of motion, where V is the velocity of the centre of gravity.
Ex. 3. A sphere of mass M moves in space, show that the compound centri-
fugal forces of all its elements are equal to (-1) a resultant force 2MVQ, sin 6 acting
at the centre of gravity, where Fis the velocity of the centre of gravity and fl the
angular velocity of the moving space and the angle the direction of V makes with
the axis of O, (2) a couple MTfi^.ui sin 0, where w is the angiJar velocity of the
sphere, the angle its instantaneous axis makes with the axis of fi, and the plane
of the couple is parallel to the axes of 12 and w.
On Motion relative to the Earth.
264. The motion of a body on tlie surface of the earth is not
exactly the same as if the eartli were at rest. As an illustration
of the use of the equations of this chapter, we shall proceed to
determine the equations of motion of a particle referred to axes of
co-ordinates fixed in the earth and moving with it.
Let be any point on the surface of the earth whose latitude
is X. Thus X is the angle the normal to the surface of still water
at makes with the plane of the equator. Let the axis of z be
vertical at and measured positively in the direction opposite to
gravity. Let the axes of x and y be respectively a tangent to the
meridian and a perpendicular to it, their positive directions being
respectively south and west. In the figure the axis of y is dotted
214
MOTION IN THREE DIMENSIONS.
to indicate tliat it is perpendicular to the plane of the paper. Let
6) be the angular velocity of the earth, b the distance of the point
from the axis of rotation.
We may reduce the point to rest by applying to every
point under consideration an acceleration equal and opposite to
that of 0, and therefore equal to co^b and tending from the axis of
rotation. We must also apply a velocity equal and opposite to
the initial velocity of 0. This velocity is cob. The whole figure
will then be turning about an axis 01, parallel to the axis of
rotation of the earth with an angular velocity co.
When the particle has been projected from the earth it is
acted on by the attraction of the earth and the applied accelera-
tion co^b. The attraction of the earth is not what we call gravity.
Gravity is the resultant of the attraction of the earth and the
centrifugal force, and the earth is of such a form that this resultant
acts perpendicular to the surface of still water. If it were not so,
particles resting on the earth would tend to slide along the sur-
face. It appears, therefore, that the force on the particle, aftei- O
has been reduced to rest, is equal to gravity. Let this be repre-
sented by g. Besides this there may be other forces on the par-
ticle, let their resolved parts parallel to the axes be X, Y, Z.
Since the earth is turning round 01 with angular velocity «*,
the resolved part about Oz is w sin A,, since the angle lOz is the
complement of w; since the rotation is from west to east, the
resolved angular velocity is from ;/ to x, which is the negative
direction, hence 6^ = — (o sin X. The resolved angular velocity
round Ox is (o cos A, and is from y to z, which is the positive
direction, hence 9^ = co cos \. Also since 01 is perpendicular to
Qi/ 0=0. Hence, by Art. 244, the actual velocities of any
particle whose co-ordinates are (.t, y, z), are
ON MOTION KELATIVE TO THE EARTH.
215
dx ^ . ^ 1.
w = -^- +w smXy
dy ^ • > I
i; = -^ — ft) cos A, ^ — w sin\ a? !>
at
dz
w = '^ + oi cosXy
CLZ
To find the equations of motion it is only necessary to substitute
these in the equations of Art. 245.
The resulting equations may be simplified if we neglect such
small quantities as the difference between the force of gravity at dif-
ferent heights. If a be the equatorial radius of the earth and g the
force of gravity at a height z, we have g =g{l ^ j nearly. Now
(o^a is the centrifugal force at the equator, which is known to be
1 z
7-^^ g. Hence if we neglect the small term (/ - we must also
289 0!-
neglect co^z. The equations will therefore become
d'^x _ • -^ dy _-.
dt dt
d'^y ^ ^ dz ^ . ^ dx „ <
-~ — 1(0 cos A ^7 — Zft) sm A -^ = Jr r ,
dt^ dt dt
d'^z c^ -. dy „
J
where the terms (X, Y, Z) include all the accelerating forces,
except gravity, which act on the particle. These equations agree
with those given by Poisson, Journal Poly technique, 1838.
265. If we do not neglect the term containing w, the equa-
tions of motion are
Uj CC (Z7/
-j-2 + 2cosAsm/3^+2ft)CosAcos^-^ = -^---^
the origin being taken at the point of suspension.
If the oscillation be sufficiently small z will differ from I by
small quantities of the order a^ where a is the semi-angle of oscil-
lation. The last equation then shows that T differs from mg by
quantities of the order coa at least. If then we neglect terms of the
ox MOTION EELATIVE TO THE EARTH. 219
order wa^ and a^, we may put mg for T in tlie two first equations
. . dz
and neglect the terms containing w -^ . The equations of motion
thus become the same as for a pendulum attached to a fixed
point. The solutions of the equations are clearly
x = Aeo& (^1 t+Cy ^=Bsm [^^It + D^ .
The small oscillations of a pendulum on the earth referred to
axes turning round the vertical with angular velocity w sin A, are
therefore the same as those of an imaginary pendulum suspended
from an absolutely fixed point.
Let us then suppose the pendulum to be drawn aside so as to
make with the vertical a small angle a and then let go. Relatively
therefore to the axes moving round the 'vertical with angular
velocity w sin X we must suppose the particle to be projected with
a velocity ZsinacosinX, perpendicular to the initial plane of dis-
placement. We have then when t = 0, oc = la, y = 0, -,- = 0,
-- = lao) sin A,, It is then easy to see that in the above values
of 00 and 7/, C and D are both zero and that the particle de-
scribes an ellipse, the ratio of the axes being co sin I a/ ~- ^^^
effect of the rotation of the earth is to make this ellipse turn
round the vertical with uniform angular velocity « sin A, in a
direction from south to west. If the angle a be not so small
that its square may be neglected, it is known by Dynamics of a
particle that, independently of all considerations of the rotation
of the earth, there will be a progression of the apsides of the
ellipse. It is therefore necessary for the success of the experi-
ment that the length I of the pendulum should be very great.
This motion of the apsides depending on the magnitude of a is in
the opposite direction to that caused by the rotation of the earth
and cannot therefore be mistaken for it.
It also appears that the time of oscillation is unaffected by the
rotation of the earth, provided the arc of oscillation be so small
that the effects of forces whose magnitude contains the factor coo!'
may be neglected.
270. In Chapter iv. we have considered the motion of a system of bodies
constrained to remain in a fixed plane. Since no plane can be found which does
not move with the earth, it is imiDortant to determine what effect the rotation of the
earth will have on the motion of these bodies. Let lis treat this as an example of
the method of Coriolis given in Ai-t. 257.
Let the plane make an angle a with the axis of the earth. Let a point in
this plane be on the siu-face of the earth and let it be reduced to rest. Then, as
220 MOTION IN THREE DIMENSIONS.
proved in Art. 264, the moving bodies wliile in the neighbourhood of are acted on
by their weights in a direction normal to the surface of the earth. The earth is
now tin-ning ronnd an axis through parallel to the axis of figure with a constant
angular velocity w. Let this angular velocity be resolved into two, viz., w sin a
about an axis perpendicular to the plane and w cos a about an axis in the plane.
Now the square of w is to be rejected, hence by the principle of the superposition of
small motions, we may determine the whole effect of these two rotations by adding
together the effects produced by each separately.
It is a known theorem that if a particle be constrained to move in a plane which
turns round any axis in that plane with a constant angular velocity w cos a, the
motion may be found by regarding the plane as fixed and impressing an accelera-
tion uh- cos^ a on the particle, where r is the distance of the particle from the axis.
This may be deduced, as in Axt. 260, from the theorem of Coriolis. This impressed
acceleration is to be neglected because it depends on the square of w. The angular
velocity « cos a has therefore no sensible effect.
If the bodies be free to move in the plane, the effect of the rotation w sin a is to
turn the axes of reference round the normal to the plane drawn through the point
0. If then we calculate the motion without regard to the rotation of the earth,
taking the initial conditions relative to fixed space, the effect of the rotation of the
earth may be allowed for by referring this motion to axes turning round the normal
with angular velocity w sin a. For example, if the body be a heavy particle sus-
pended by a long string from a point fixed relatively to the earth, it is really
constrained to move in a horizontal plane, and the reasoning given above shows
that the plane of oscillation will appear to a sjjectator on the earth to revolve with
angular velocity w sin a round the vertical.
If the bodies be constrained to revolve with the plane, it wiU be required to find
the motion relatively to that plane. We must therefore apply to each particle the
force of moving space and the compound centrifugal force. If r be the distance of
any particle of mass m from 0, the former is mrw^ sin^ a. This is to be neglected
because it depends on the square of w. The latter is therefore the only force to be
considered. By Art. 262, the compound centiifugal forces on all the particles of a
body are equivalent to a force at the centre of gravity and three couples. In our
case these couples are easUy seen to be zero. For if the plane be taken as the plane
of xy, we have 0^ = 0, O2 = 0i (^i = ^, Wj = 0. Hence L, M, iVare all zero. If, there-
fore, m be the mass of a body, V the relative velocity of its centre of gravity, the
effect of the rotation of the earth may be found according to the rule given in Art.
257, by impressing on the body a force equal to 2mVu sin a, acting at the centre of
gravity, in the plane of motion and perpendicular to the direction of motion of the
centre of gravity.
The ratio of this force to gravity for a particle moving 32 feet per second, is at
47r
iiiost ^, „^ „^ , which is less than a five thousandth. This is so small that, except
24.60.60
nnder special cu-cumstances, its effect will be imperceptible.
271. Ex. 1. In Foucault's experiment, a long penduhun is suspended from a
point over the centre of a circular table, and the arc of oscillation is seen to pass
from one diameter to another. Show that the arc of the circular rim of the table
described by the plane of oscillation in one day is equal to the difference in length
between two parallels of latitude one through the centre and the other through the
ON MOTION RELATIVE TO THE EAETH. 221
northern or southern extremity of the rim. This theorem is due to the late Prof.
Young.
Ex. 2. A heavy particle is suspended from a fixed point of support by a string
of length a. It performs elHptic oscillations whose major and minor semi-axes are
6 and c. If 6 and c be smaU compared with a, prove that the apses will advance,
3 be
in each complete revolution of the particle, through an angle ., — 2ir nearly. If b
and c be not small compared with a but be very nearly equal, the apse will advance
through an angle
1 1^2.
where sin a = - in each complete revolution of the particle.
a
Ex. 3. A pendulum, at rest relatively to the earth, is started in any direction
with a small angular velocity, show that the oscillations will take place in a vertical
plane turning uniformly round the vertical so that the pendulum becomes vertical
once in each half oscillation.
Ex. 4. Let 6 be the angle a pendulum of length I makes with the vertical, and
the angle the vertical plane containing the pendulum makes with a vertical plane
which tm-ns round the vertical with uniform angular velocity w sin \ in a direction
from south to west. Prove that when terms depending on w^ are neglected the
ecLuations of motion become
(f)
d f . ^ „ dd>\ „ . „ , , , ^, ^ do
— sm- e ^ ] = 2 sm- 9 cos (rf) + S) w cos \ ^- ,
dt\ dtj '^ ^' dt'
where A is an arbitrary constant, and the other letters have the meanings given to
them in Art. 267. See M. Quet in Liouville's Journal, 1853.
These equations will be found convenient in treating the motion of a pendulum.
They may be easily obtained by transforming those given in Art. 239 to polar co-
ordinates.
Ex. 5. A semi-circular arch ACB is fixed with its plane vertical on a horizontal
wheel at A and B, and may thus be moved with any degree of rapidity from one
azimuth to another. A rider slides along the inner edge of the arch which is
graduated and may be fixed at any degree marked thereon. A spiral spring by
means of which a slow vibration is obtained with comparatively a short length is
attached at one end to a jpin in the axis of the semicircle so that the point of
attachment may be in the axis of rotation and at the other end it is fixed to a
similar pin in a parallel position fixed to the rider. The vertical semicircle is not
placed in a diameter of the horizontal wheel but parallel to it at such a distance as
not to interrupt the eye of the observer from the vertical plane passing through the
diameter, and in which plane the wire in all its positions remains.
If the rider be placed at an angular distance from the highest point of the
arch and the wire set in vibration in any plane, show that the plane of vibration of
the wire will make a complete revolution relatively to the arch whUe the arch turns
round sec complete revolutions. This is best observed by fixing the eye on a lino
222 MOTION IN THREE DIMENSIONS.
in the same plane with the wire while walking round with the wheel during its
rotation. This apparatus was devised by Sir C. Wheatstone to illustrate Foucault's
mechanical proof of the rotation of the earth. Proceedings of the Royal Society,
May 22, 1851.
272. Hitherto we have considered chiefly the motion of a
single particle. The effect of the rotation' of the earth on the
motion of a rigid body will be more easily understood when the
methods to be described in the following chapters have been read.
If, for example, a body be set in rotation about its centre of
gravity, it will not be difficult to determine its motion as viewed
by a spectator on the earth, when we know its motion in space.
It seems, therefore, sufficient here to consider the peculiarities
which these problems present, and to seek illustrations which do
not require any extended use of the equations of motion.
273. The effect of the rotation of the earth is in general so
small compared with that of gravity, that it is necessary to fix the
centre of gravity in order that the effects of the former may be
perceptible. Even when this is done, the friction on the points of
support and the other resistances, cannot be wholly done away
with. If, however, the apparatus be made with care that these
resistances should be small, the effects of the rotation of the earth
may be made to accumulate, and after some time to become
sufficiently great to be clearly perceptible.
If a body be placed at rest relatively to the earth and free to
turn about its centre of gravity as a fixed point, it is actually in
rotation about an axis parallel to the axis of the earth. Unless
this axis be a principal axis, the body would not continue to rotate
about it, and thus a change would take place in its state of
motion. By referring to Euler's equations, we see that the change
in the position of the axis of rotation is due to the terms
{A — B) 0)^(0^, {B — C) co^co^, (G — A) 0)^(0^. The body having
been placed apparently at rest, co^, co^, cOg are all small quan-
tities of the same order as the angular velocity of the earth ; these
terms are, therefore, all of the order of the squares of small quan-
tities. Whether they will be great enough to produce any visible
effect or not will depend on their ratio to the frictional forces
which could be called into play. But since these frictional forces
are just sufficient to prevent any relative motion, these terms will
in general be just cancelled by the frictional couples introduced
into the right-hand sides of Euler's equations. The body will,
therefore, continue at rest relatively to the earth.
In order that some visible effect may be produced, it is usual
to impress on the body a very great angular velocity about some
axis. If this be the axis of co^, the terms in Euler's equations,
which are due to the centrifugal forces, and which contain w as a
ON MOTION RELATIVE TO THE EARTH. 223
factor, become greater than when co^ had no such initial value.
The greater this initial angular velocity, the greater these terms
will be, and the more visible we may expect their effects on the
body to be.
If the angular velocity thus communicated to the body be
sufficient to turn it only once in a second, it will be still
24 X 60 X 60 times as great as the angular velocity of the earth.
In these problems, therefore, we may regard the angular velocity
of the earth as so small, compared with the existing angular
velocities of the body, that the square of the ratio may be neg-
lected.
As an example of the application of these principles, we have
selected one case of Foucault's pendulum, which seems to admit of
an elementary solution.
274. The centre of gravity of a solid of revolution is fixed,
ivliile the axis of figure is constrained to remain in a plane fixed
relatively to the earth. The solid being set in ivtation about its
axis of figure, it is required to find the motion.
Let us refer the motion to moving axes. Let the centre of
gravity be the origin, the plane of yz the plane fixed relatively to
the earth. Let the axis of figure be the axis of z, and let it make
an angle ^ with the projection of the axis of rotation of the earth
on the plane of yz. Let this projection, for the sake of brevit}^, be
called the axis of %. Let p be the angular velocity of the earth
about its axis, a the angle the normal to the plane of yz makes
with the axis of the earth. The motion of the moving axes is
given by
6^=jp COBOL -T-jj, d^=p &in a Qva.'y^, 6^= p ^ma co^x-
Let Wj, Wg, cogbe the angular velocities of the body about the
moving axes; A, A, G the principal moments of inertia at the
centre of gravity. Let R be the reaction by which the axis of
figure is constrained to remain in the fixed plane, then R acts
parallel to the axis of x. Let h be the distance of its point of
application from the origin. The angular momenta about the
axes are respectively
h^ = A(i>^, h^ = Ao)^, h^= Coi^.
Substituting in Art. 230, the equations of motion are
A~'-A(oJ^+C.6'..+ 7lrA,6' -0
224< MOTION IN THREE DIMENSIONS.
Since the axis of z is fixed in the body, we see by Art. 243,
that (x)^ = Q^, w^—B^. The last equation of motion, therefore,
shows that Wg is constant. It should however be remembered that
CO is not the apparent angular velocity of the body as viewed by
a spectator on the earth. If Hg be the angular velocity relatively
to the moving axes, we have by Art. 24.3, Hg = 0)3—^3, so that
O3 + p sin a cos y^ — constant.
Thus the body, if so small a difference could be perceived, would
appear to rotate quicker the nearer its axis approached the pro-
jection of the axis of the earth's rotation on the fixed plane.
The first equation of motion after substitution for w^, co^, ^2' ^3»
their values in terms of y^, becomes
A -jY — A'^ sin^ a sin % cos % + Cnj) sin a sin % = 0,
where n has been written for Wg.
The second term may be rejected as compared with the third,
since it depends on the square of the small quantity ^. We have,
therefore,
d\ C . .
-^ = - — wp smasm^.
By Art. 92, this is the equation of motion of a pendulum
tinder the action of a force constant in magnitude, and whose
direction is along the axis of y, i.e. the projection of the axis of
rotation of the earth on the fixed plane. The body being set in
rotation about its axis of figure, we see that that axis will imme-
diately begin to approach one extremity or the other of the axis of
y with a continually increasing angular velocity. When the axis
of figure reaches the axis of %, its angular velocity will begin to
decrease, and it will come to rest when it makes an angle on the
other side of the axis of % equal to its initial value. The oscilla-
tion will then be repeated continually.
The axis of figure will oscillate about that extremity of the
axis of %, which, when y is measured from it, makes the coefficient
on the right-hand side of the last equation negative. This extre-
mity is such, that when the axis of figure is passing through it,
the rotation n of the body is in the same direction as the resolved
rotation p of the earth.
275. If we compare bodies of different form, we see that the
G
tim^e of oscillation depends oidy on the ratio -j . It is otherwise
independent of the structure or form of the body. The greater
this ratio the quicker will the oscillation be. For a solid of
revolution, it appears from the definitions in Art. 4, that this
, ON MOTION RELATIVE TO THE EARTH. 225
ratio is greatest when 2m/ = 0. In this case the ratio is equal
to 2, and the bodj is a circular disc or ring,
276. If we compare the different planes in which the axis may
be constrained to remain, we see that the motion is the same for all
planes making the same angle with the axis of the earth. It is
therefore independent of the inclination of the plane to the horizon
at the place of observation. The time of oscillation will be least,
and the motion of the axis most perceptible when a.— -,i.e. when
the plane is parallel to the axis of rotation of the earth. If the
plane be perpendicular to the axis of the earth, the axis of figure
will not oscillate, but if the initial value of -^ is zero, it will
at
remain at rest in whatever position it may be placed.
277. Ex. 1. Show that a person furnished with the particular form of Fou-
cault's pendulum just described, could, without any Astronomical observations,
determine the latitude of the place, the direction of the rotation of the earth, and
the length of the sidereal day. This remark is due to M. Quet, who has given a
different solution of this problem in Liouville's Journal, vol. xviii.
Ex. 2. If the body be a rod, and its centre of gravity supported without friction,
prove that it could rest in relative equilibrium either parallel or perpendicular to
the projection of the earth's-axis on the plane of constraint. If it be placed in any
other position, its motion will be very slow, depending on ^^, but it will oscillate
about a mean position perpendicular to the projection of the earth's axis.
Ex. 3. If the axis of figure be acted on by a frictional force producing a
retarding couple, whose moment about the axis of x bears a constant ratio /* to the
moment of the reactional couple about the axis of y, and if the fixed plane be
parallel to the axis of the earth, find the small oscillations about the position of
equilibrium. Show that the position at any timet is given by
X=£e-''cos[(^-xft+M'],
where 2A\=iJ.{Cn-2Ap) and L and M are two constants depending on the initial
conditions.
Ex. 4. The centre of gravity of a solid of revolution is fixed, while the axis of
figure is constrained to remain in the surface of a smooth right cone fixed relatively
to the earth. Show that the axis of figure will oscillate about the projection of the
axis of rotation oi the earth on the surface of the cone, and that the time of a com-
/ A sin e ,
plete small oscillation about the mean position will be 2ir .. / -^ -. — ^, wnere e
is the semi-angle of the cone, /3 the inclination of its axis to the axis of the earth,
and the other letters have the same meaning as before. This result is due to
M. Quet.
Ex. 5. Two equal heavy rods CA, CB are connected by a hinge at C, with a
spring so that they tend to make a known angle with each other. The free ends
A and B are then tied together and the whole is suspended by a string OC attached
II. D. 15
226 MOTION OF TWO DIMENSIONS.
to the hinge. The system is left to itself until it is at rest relatively to the earth.
If the string which fastens A and B be now cut, the arms separate from each other.
Show that the system will immediately have an apparent angular velocity round
the vertical equal to p sin X, where I, I' are the moments of inertia of the
system about the vertical OC respectively before and after the string joining A and
B was cut, p is the angular velocity of the earth about its axis and \ is the latitude
of the place. In which direction wiU the system turn? This apparatus was
devised by M. Poinsot who considered that the experiment would be so effective
that the latitude of the place could be deduced from the observed angular velocity.
See Comptes Bendus, 1851, Tome xxxii. page 206.
Ex. 6. If a river is flowing due north, prove that the pressure on the eastern
bank at a depth z is increased by the change of latitude of the running water in
the ratio gz + bvw sin I : gz, where b is the breadth of the stream, v its velocity, I the
latitude and w the angular velocity of the earth about its axis. [Math. Tripos, 1875.]
CHAPTER VI.
ON MOMENTUM.
278. The term Momentum has been given as the heading of
this Chapter, though it only expresses a portion of its contents.
The object of the Chapter may be enunciated in the following
problem. The circumstances of the motion of a system at any time
Iq are given. At the time t^ the system is moving under other
circumstances. It is required to determine the relations which
may exist between these two motions. The manner in which
these changes are effected by the forces is not the subject of
enquiry. We only wish to determine what changes have been
effected in the time t^ — t^. If the time t^ — t^ be very small, and
the forces very great, this becomes the general problem of im-
pulses. This also will be considered in the Chapter.
Let us refer the system to any fixed axes Ox, Oy, Oz. Then
the six general equations of motion may, by Art. 71, be written in
the form
Sm -772 = %mZ I
at I
Integrating these from t =t^io t = t^,y^e have
^ dz
^ I dti dx
^'^'^''dt-yTt
= z,m I Zdt,
u J to
tmf'ixY-yY) dt.
Let a force P act on a moving particle m during any time
t^ — t^, and let this time be divided into intervals each equal
to dt. At the middle of each of these intervals let a line be
drawn from the position of m at that instant, to represent, at the
same instant, the value of mPdt both in direction and magnitude.
Then the resultant of these forces, found by the rules of Statics,
may be called the whole force expended in the time t^—t^. Thus
I mZdt is the whole force resolved parallel to the axis of Z.
These equations then show that
15—2
228 MOMENTUM,
(1) The change produced by any forces in the resolved part
of the momentum of any system is equal in any time to the whole
resolved force in that direction.
(2) The change produced by any forces in the moment of the
momentum of the system about any straight line is, in any time,
equal to the whole moment of these forces about that straight line.
When the interval i^ — t^ is very small, the " whole force "
expended is the usual measure of an impulsive force, and the
preceding equations are identical with those given in Art. 86.
It is not necessary to deduce these two results from the equa-
tions of motion. The following general theorem, which is really
equivalent to the two theorems enunciated above, may be easily
obtained by an application of D'Alembert's principle.
279. If the movientum of any particle of a system in motion
he compounded and resolved, as if it ivere a force acting at the
instantaneous position of the particle, according to the rules of
Statics, then Hie momenta of all the particles at any time t^ are
together equivalent to the momenta at any previous time t^ together
with the whole forces which have acted during the interval.
In the case in which no forces act on the system, except the
mutual actions of the partidles, we see that the momenta of all
the particles of a system at any two times are equivalent ; a result
which has been already enunciated in Art. 72. The two princi-
ples of the Conservation of Linear Momentum and Conservation of
Areas may be enunciated as follows.
If the forces which act on a system be such that they have no
component along a certain fixed straight line, then the motion
is- such that the linear momentum resolved along this line is
constant.
If the forces be such that they have no moment about a cer-
tain fixed straight line, then the moment of the momentum or
area conserved about this straight line is constant.
It is evident that these principles are only particular cases of
the results proved in Art. 79.
280. Ex. Suppose that a simple particle m describes an"
orbit about a centre of force 0. Let v, v be its velocities at any
two points P, P' of its course. Then mv' supposed to act along
the tangent at P' if reversed would be in equilibrium with mv
acting along the tangent at P together with the whole central
force from P to P'. If p, p be the lengths of the perpendiculars
from on the tangents at P, P', we have, by taking moments
about 0, vp = v'p, and hence vp is constant throughout the
MOMENTUM. 229
motion. Also if the tangents meet in T, the whole central force
expended must act along the line TO, and may be found in terms
of V, v by the rules for compounding velocities.
Ex. Two particles of masses m, m' move about the same centre of force. If
h, h' be the double areas described by eacli per unit of time, prove that mh + m'h'
is unaltered by an impact between the particles.
281. Ex. Suppose three particles to start from rest attracting
each other, but under the action of no external forces. Then the
momenta of the three particles at any instant are together equiva-
lent to the three initial momenta and are therefore in equilibrium.
Hence at any instant the tangents to their paths must meet in
some point 0, and if parallels to their directions of motion be
drawn so as to form a triangle, the momenta of the several parti-
cles are proportional to the sides of that triangle.
If there are n particles it may be shown in the same way that
the n forces represented by mv, m'v', &c. are in equilibrium, and if
parallels be drawn to the directions of motion and proportional to
the momenta of the particles beginning at any point, they will
form a closed polygon.
If F, F' , F" be the resultant attraction on the three particles,
the lines of action of F, F', F" also meet in a point. For let
X, Y, Z be the actions between the particles mm", mm, mm' ,
taken in order. Then F is the resultant of — y and Z; F' of — Z
and X; F" of -X and Y. Hence the three forces F, F' , F"
are in equilibrium*, and therefore their lines- of action must meet
in a point 0'. Also the magnitude of each is proportional to the
sine of the angle between the directions of the other two. This
point is not generally fixed, and does not coincide with 0.
If the law of attraction be proportional to the distance, the
two points 0, 0' coincide with the centre of gravity G, and are
fixed in space throughout the motion. For it is a known propo-
sition in Statics that with this law of attraction, the whole attrac-
tion of a system of particles on one of the particles is the same as
if the whole system were collected at its centre of gravity. Hence
0' coincides with G. Also, since each particle starts from rest,
the initial velocity of the centre of gravity is zero, and therefore,
by Art. 79, 6^ is a fixed point. Again, since each particle starts
from rest and is urged towards a fixed point G, it will move in the
straight line joining- its initial position with G. Hence coin-
cides with G. When the law of attraction is proportional to the
distance, it is proved in Dynamics of a Particle, that the time of
reaching the centre of force from a position of rest is independent
* This proof is merely an amplification of the following. The three forces
F, F', F", being the internal re-actions of a system of three bodies, are in equiU-
brium by D'Alcmbert's Principle.
230 MOMENTUM.
of the distance of that position of rest. Hence all the particles of
the system will reach G at the same time, and meet there. If Xni
be the sum of the masses, measured by their attractions in the
, . . . -, , , 1 27r
usual manner, this time is known to be 7 ,— — ,
^ vSm
282. Ex. Three pai^ticles wJiose masses are m, m', m", mutu-
ally attracting each other, are so projected that the triangle formed
hy joining their positions at any instant remains always similar to
its original form. It is required to determine the conditions of
projection.
The centre of gravity will be either at rest or will move uni-
formly in a straight line. We may therefore consider the centre
of gravity at rest and may afterwards generalise the conditions of
projection by impressing on each particle an additional velocity
parallel to the direction in which we wish the centre of gravity to
move. Let be the centre of gravity, P, P , F" the positions of
the particles at any time t. Then by the conditions of the ques-
tion the lengths OP, OP', OP" are always to be proportional, and
their angular velocities about are to be equal. Since the moment
of the momenta of the system about is always the same, we
have
- mr^ + rdr'Si + i'n!'r"'^n = constant,
where r, r, r" are the distances OP, OP', OP", and n is their
common angular velocity. Since the ratios r \ r : r" are con-
sta^nts, it follows from this equation that mr^n is constant, i.e. OP
traces out equal areas in equal times. Hence by Newton, Section 11,
the resultant force on P tends towards 0.
Let p, p, p" be the sides P'P", P'P, PP' of the triangle
formed by the particles, and let the law of attraction be ,_.. ,^ .
Then since the resultant attraction oim, m" on m passes through 0,
^ sin P'PO = ^ sin P 'PO,
P P
but since is the centre of gravity,
m'p" sin PPO = m"p' sin P"PO.
Hence either the three particles are in one straight line or
p"*+i = p''^^^ If k = —l the law of attraction is " as the distance."
If k be not = — 1, we have p = p", and the triangle must be equi-
lateral.
Conversely, suppose the particles to be projected in directions
making equal angles with their distances from the centre of
gravity with velocities proportional to those distances, and sup-
pose also the resultant attractions towards the centre of gravity to
MOMENTUM. 231
be proportional to those distances, then in all the three cases the
same conditions will hold at the end of a time dt, and so on con-
tinually. The three particles will therefore describe similar orbits
about the centre of gravity in a similar manner.
First, let us suppose that the three particles are to be in one
straight line. To fix our ideas, let m lie between m and m", and
between m and m. Then since the attraction on any particle
must be proportional to the distance of that particle from 0, the
three attractions
m m" m" m m, m'
{PP'f^ {PF'y {F'PJ {FFy {PF'J {FF'f
must be proportional to OF, OF, OF". Since XmOF=0, these
FF"
two equations amount to but one on the whole. Let z = ppv ,
1 OF _ m + m" (1 + z) OF' _ — «i + m'z
FF m + m +m FF m + m + m
Then we have
which agrees with the result given by Laplace, by whom this
problem was first considered.
In the case in which the attraction follows the law of nature
k = 2 and the equation becomes
mz' {(1 + zf - 1} - ni (1 + zY (1 - z') - m" {(1 + zf - z'] = 0.
This is an equation of the fifth degree, and it has therefore
always one real root. The left side of the equation has opposite
signs when 0=0 and ^ = go , and hence this real root is positive.
It is therefore always possible to project the three masses so that
they shall remain in a straight line. Laplace remarks that if m
be the sun, m the earth, and m" the moon, we have very nearly
3 / 7~, 77 -1
z=\ / — 7, = tkt; • If then originally the earth and moon had
V Sm 100 * "^
been placed in the same straight line with the sun at distances
from the sun proportional to 1 and 1 + rr-rr, , and if their velocities
had been initially parallel and proportional to those distances, the
moon would have always been in opposition to the sun. The
moon would have been too distant to have been in a state of
continual eclipse, and thus would have been full every night. It
has however been shown by Liouville, in the Additions a la
Conncdssance des Temps, 1845, that such a motion would be un-
stable.
232 MOMENTUM.
The paths of the particles will be similar ellipses having the
centre of gravity for a common focus.
Secondly. Let us suppose that the law of attraction is " as the
distance," In this case the attraction on each particle is the
same as if all the three particles were collected at the centre of
gravity. Each particle will describe an ellipse having this point
for centre in the same time. The necessary conditions of projec-
tion are that the velocities of projection should be proportional to
the initial distances from the centre of gravity, and the directions
of projection should make equal angles with those distances.
Thirdly. Let us suppose the particles to be at the angular
points of an equilateral triangle. The resultant force on the par-
ticle m is
^, cos FPO + ^cos P'PO.
The condition that the forces, on the particles should be pro-
portional to their distances from shows that the ratio of this
force to the distance OP is the same for all the particles. Since
mp" cos FPO + m"p'' cos P"PO ={m + m+ m") OP,
it is clear that the condition is initially satisfied when p = p = p".
Hence, by the same reasoning as- before, if the particles be pro-
jected with equal velocities in directions making equal angles with
OP, OP', OP" respectively, they will always remain at the angular
points of an equilateral triangle.
Ex, 1. Show that if the three particles attracted each other according to the
law of nature, the paths of- the particles, when at the corners of an equilateral
triangle, are equal ellipses having for a common focus. Find the periodic time.
Ex. 2. If four particles be placed at the corners of a quadrilateral whose sides
taken in order are a,, 6, c, d and diagonals p, p', then the particles could not move
under their mutual attractions so as to remain always at the corners of a similar
quadrilateral unless
(/)>'" - &"(Z»).(o" + a") + {a^C' - p^p"^) [l'^ + cZ») + (6"cZ" - a"0 (p" + p'") = 0,
where the law of attraction is the inverse [n- 1)^"°^ power of the distance.
Show also that the mass at the iatersection of 5, o- divided by the mass at
intersection of c, d is equal to the product of the area formed by a, p', d divided by
the area formed by a, I, p and the difference -^ - ^ divided by the difference
1- i
pn in •
These results may be conveniently arrived at by reducing one angular point as
A of the quadrilateral to rest. The resolved part of all the forces which act on each
particle perpendicular to the straight line joining it to A will then be zero. The
case of three particles may be treated in the same manner. The jDrocess is a little
shorter than that given in the text, but does not illustrate so well the subject of the
chapter.
MOMENTUM. 233
283. When the system under consideration consists of rigid
bodies we must use the results of Art. 75 to find the resolved part
of the momentum in any direction. The moment of the momentum
about any straight line may also be found by Art. 76 in Chap. Ii,
combined with Art. 123 in Chap, iv, if the motion be in two
dimensions, or Art, 240 in Chap. V, if the motion be in three
dimensions.
284. Ex. A disc of any form is moving in its own plane in
any manner. Suddenly any point O in the disc is fixed, find the
angidar velocity of the disc about O.
Let us suppose that just before became fixed the centre of
gravity G was moving with velocity V, and that p is the length of
the perpendicular from on the direction of motion. Also let a
be the angular velocity of the body about its centre of gravity.
Just after has become fixed, let the body be turning about
with angular velocity &)'. het MJif he the moment of inertia of
the disc about the centre of gravity, and let 0G = r.
The change in the motion of the disc is produced by impulsive
forces acting at during a short time t^ — t^. These forces have
no moment about 0. Hence the moment of the momentum about
is the same just after and just before the impact. Just before
became fixed, the moment of the momentum about G was
Mk^a3 (Art. 123), and the moment of the momentum of the whole
mass collected at G was 3IVp. Hence the whole moment of the
momentum about is the sum of these two (Art. 76). Just after
has become fixed the body is turning about 0, hence by Art. 123
the moment of the momentum about is 31 (k^ + r") co'. Equating
these we have
M{k' + r) ft)' = Mk'co + MVp;
, Fft) + Vp
.'. ft) = — r5 9^ •
k' + r^
Ex. A circular area is turning about a point A on its circumference. Suddenly
A is loosed and another point B also on the circumference is fixed. Show that if AB
is a quadi'ant, the angular velocity is reduced to one-third its value, and if .4 5 is a
thii-d of the circumference, the area will be reduced to rest.
28.5. Ex. A disc of any form is turning about an axis Ox
sit^iated in its own plane with an angidar velocity ft). Suddenly
this axis is let free and another axis Ox', also situated in the plane
of the disc, becomes fixed, it is required to find the neiu angular
velocity ft)' about Ox'.
The change in the motion of the disc is caused by the action
of the impulsive forces due to the sudden fixing of the axis Ox'.
These act at points situated in Ox and have no moment about
234
MOMENTUM.
Ox. Hence the moment of the momentum about Ox' is the same
just before and just after Ox is fixed.
Let da- be any element of the area of the disc ; y, y its dis-
tances from Ox, Ox. Then yw, y'(o' are the velocities of da just
before and just after the impact. The moments of the momentum
about Ox just before and just after are therefore yy'wda and
y^wda. Summing these for the whole area of the disc, we have
(o'Xy'^da = (o^yy'da (1).
First, let Ox, Ox be parallel, so that the point is at in-
finity. Let h be the distance between the axes, then y' =y — ]i.
Hence we have
(o'^y'da = (£> \%y'^da — h^yda].
Let A, A' be the moments of inertia of the disc about Ox,
Ox respectively, y the distance of the centre of gravity from Ox,
M the mass of the disc. Then we have
A'oi' = a>{A-Mhy)..
Secondly, let Ox, Ox' not be parallel. Let be the origin
and the angle xOx = a, then y' = y cos a — a? sin a. Let F be the
product of inertia of the disc about Ox, Oy where Oy is perpen-
dicular to Ox. Then by substitution in (1) we have
A'ca' = CO {A cos a. — F&m. a).
Ex. 1. An elliptic area of eccentricity e is turning about one latus rectum.
Suddenly this latus rectum is loosed and the other fixed. Show that the angular
velocity is
of its former value.
Ex. 2. A right-angled triangular area ACB is turning about the side AC.
Suddenly AC is loosed and BC fixed. If C be the right angle, the angular velocity
BO
is of its former value.
286. A rigid body is moving freely in space in a known
manner. Suddenly either a straight line or a point in the body
becomes fixed. To determine the initial subsequent motion.
MOMENTUM. 235
This proposition will include the last two articles as par-
ticular cases. It is obvious that all the impulsive actions on the
body pass through the fixed straight line or the fixed point.
Hence the moments of the momentum of the body about the
fixed axis in the first case or about any axis through the fixed
point in the second case are unaltered by the impulsive forces.
First. Let a straight line suddenly become fixed. Let it be
taken as the axis of z.
Let MK^ be the moment of inertia of the body about the axis
of z, and O the angular velocity after the straight line has become
fixed. Suppose that the body when moving freely was turning
with angular velocities w^., &)^, 03^ about three straight lines Gx,
Qy Gz through the centre of gravity parallel to the axes of co-
ordinates. And let the co-ordinates of the centre of gravity be
X, y, z.
Then
C'o), - (Sm.'^;') 0,, - (tm/y ) a,, -H if (^ I - ^ f ) = MK\ O,
where C is the moment of inertia of the body about Gz , and
'%mz X , '%mz'y' are calculated with reference to the axes Gx ,
Gy , Gz.
Secondly. Let a point in the moving body be suddenly
fixed in space. Take any three rectangular axes Ox, Oy, Oz,
and three parallel axes Gx, Gy\ Gz through the centre of
gravity G. Let w^, w^, 6)^ be the known angular velocities of
the body about the axes Gx, Gy', Gz before the point became
fixed, O3., O^, O^ the unknown angular velocities about Ox, Oy,
Oz after has become fixed.
Then, following the same notation as before, we have by
Art. 240,
A'oi, - {tm x'y) co^ - {^m x'z) w, ^-tml^j-^-l -J j
= A[1^— {tm xy) 11^ — (^m xz) fl,.
B'coy — (2m y'z') w, — {Xmy'x) co^ + Xm ( ^ 77 ~ ^ "^j
= BQ,y — (^myz) fl, — (2m yx) D.^.
C'ca.^ — (2m z'x) (o-,. — (2m z'y') co^ + 2m [x-r- — y -j-j
= Cn., — (2?H zx) 11^ — (2m zy) €1^.
These equations determine O^, U,^, O,, It is obvious that
they may be greatly simplified by so choosing the axes that one
236 MOMENTUM.
of the two sets Ox, Oy, Oz or Gx\ Oy, Gz may be a set of
principal axes.
287. If the body be turning about an axis GI through the
centre of gravity G just before the point is fixed, the terms
containing the co-ordinates of the centre of gravity disappear
from the equations. They now admit of an easy geometrical
interpretation. The equation to the momental ellipsoid at the
centre of gravity is
AX" ■VB'Y^+ G'Z' - ^l^my'z YZ- 2Sm z'x ZX
-^Xmx'y XY=Me\
It is therefore cfear that the left-hand sides of these equations are
proportional to the direction-cosines of the diametral plane of a
straight line whose direction-cosines are proportional to (co^, Wy, wj.
In the same way if we construct the momental ellipsoid at 0, the
right-hand sides are proportional to the direction-cosines of the
diametral plane of the axis (12^, II^,, HJ. Thus the instantaneous
axes of rotation, before and after is fixed, are so related that
their diametral planes with regard to the momental ellipsoids at
G and respectively are parallel.
We may also deduce this result, without difficulty, from
Art. 117. The motion of the body about the axis GI may be
produced by an impulsive couple in the diametral plane of GI
with regard to the momental ellipsoid at G. Let us then suppose
the body at rest and' fixed, and let it be acted on by this couple.
It follows from-^ the same article, that the body will begin to turn
about an axis 01' which is such that its diametral plane with
reo-ard to the momental ellipsoid at is parallel to the plane of
the couple.
The direction of the blow at may also be easily found. The
centre of gravity being at rest suddenly begins to move perpen-
dicular to the plane containing it and the axis OF. This is
obviously the direction^ of the blow.
288. Ex. 1. A sphere in co-latitude 6 ie hung up hy a point Oin its surface in equi-
librium under the action of gravity. Suddenly the rotation of the earthis stopped, it
is required to determine the motion of the sphere. [Math. Tripos, 1857.]
Let G be the centre of the sphere, its point of suspension, and a its radius.
Let C be the centre of the earth. Let uS' suppose the figure so drawn that the
sphere is moving away from the observer.
Let w = angular velocity of the earth, then if CG—ixa, the sphere is turning
about an axis Gp parallel to CP, the axis of the earth with angular velocity w, while
the centre of gravity is moving with velocity ixa sin 6 . w.
Let OC, Op, and the perpendicular to the plane of OC, Op be taken as the axes
of X, y, z respectively, and let O^. , Oj, , 0^ be the angular velocities about them just
after the rotation of the earth is stopped.
MOMENTUM.
237
By Art. 286, the angular momenta about Ox, just before and just after the
rotation was stopped, are equal to each other ;
.-. Ml? (^ cos e^Mi?n^,
where Mh^ is the moment of inertia of the sphere about a diameter.
Again, the angular momenta about Oy are equal to each other ;
.-. - if i-2 w sin ^ + if^i a^ w sin i9 = iff {I? + oF) Q,y.
Lastly, the angu;lar momenta about Oz are equal; .•. 0=3fX-^fi,
Solving these equations, we get
0,, = w sin 6
-Ti^ + lia^ . .-2 + 5/x
-7^5 — ^ = w sm ^ — = .
But fi^= w cos 0. Adding together the squares of 0^ , fi^ , 0^ we have
fi2 = aj2 003^61 +
-2 + 5fxy
j sin." el
where is the angular velocity of the sphere about its instantaneous. axis,
Ex. 2. A particle of mass M, without velocity, is suddenly attached to the sur-
face of the earth at the extremity of a radius vector making an angle with the
axis of the earth. If E be the mass of the earth before the addition of M, A and G
its principal moments of inertia at the centre, w the angular velocity about its axis,
prove that
w_ EMAr^ sin^
B~ "^ {E+M) AC+ EM Or' cos^ '
cot = COt ^ +
E+M
A
E ' Mr^ sin (9 cos (9 '
where Q is the initial angular velocity about an axis parallel to the axis of the earth
and is the angle the initial axis of rotation makes with the axis of the earth.
Ex. 3. A body having a point fixed is turning with angular velocity w about
an axis 01 whose direction cosines referred to the principal axes at are (l, m, n).
Suddenly, an axis OF whose direction-cosines are {V, m', n') is fixed. Show that
the angular velocity about 01' is given by
{AV^ + Bm'^JrCn'^) u' = {AW + Bmm' + C'nn') w,
where A, B, C are the principal moments at 0.
238 MOMENTUM.
Ex. 4. A regular homogeneous prism whose normal section is a regular polygon
of n sides roUs on a perfectly rough plane. Prove that, when the axis of rotation
changes from one edge to another, the angular velocity is reduced in the ratio
/ 2 +7 cos — \
8 + cos —
289. In these examples the changes produced in the motion
were sudden, but the method of proceeding is the same if the
changes are gradual.
Ex. 1. A bead of mass m slides on a circular wire of mass M and radius a,
and the wire can turn freely about a vertical diameter. Prove that, if w, fi be the
angular velocities of the wire when the bead is respectively at the extremities of a
horizontal and vertical diameter, - = 1 + 2 -^-j. .
w M
Ex. 2. If the earth gradually contracted by radiation of heat, so as to be always
similar to itself as regards its physical constitution and form, prove that when every
radius vector has contracted an ji*'' part of its length, where n is small, the angular
velocity has increased a 2?i*'' part of its former value.
Ex. 3. If two railway trains each of mass M were to travel in opposite direc-
tions from the pole along a meridian and to arrive at the equator at the same time,
2ilfa^
prove that the angular velocity of the earth would be decreased by -=p , where a is
the equatorial radius of the earth and EB its moment of inertia about its axis of
figure.
What would be the effect if one train only were io' travel from the pole to the
equator ?
Ex. 4. A fly alights perpendicularly on a sheet of paper lying on a smooth
horizontal plane and proceeds to describe the curve r=f{d) traced on the sheet of
paper, the equation to the curve being referred to the centre of gravity of the paper
as origin. Supposing the fly to be able to prevent himself from slipping on the
paper, show that his angular velocity in space about the common centre of gravity
of the paper and fly is equal to -^ —^4- 5 -r- , where M and ??i are the masses
of the paper and the fly and k is the radius of gyration of the paper about its centre
of gravity. Hence find the path of the fly in space.
Ex. 5. Suppose the ice to melt from the polar regions twenty degrees round
each pole to the extent of something more than a foot thick, enough to give 1^^ feet
over those areas or -066 of a foot of water spread over the whole globe, which would
in reahty raise the sea-level by only some such undiscoverable difference as fth of
an inch or an inch, then this would slacken the earth's rate as a time-keeper by one-
tenth of a second per year. This and the next example are taken from the Phil.
Mag. They are both due to Sir W. Thomson.
If E be the mass of the earth, a its radius, Jc its radius of gyration about the
polar axis, w its angular velocity before the melting, then we have by the principle
THE INVARIABLE PLANE. 239
of angular momentum — = - 5^;7-o cos d (1 + eos 6), where M is the mass of the ice
melted and 6 is twenty degi-ees. Substituting for the letters their known numerical
values, the value of 5w is easily found.
Ex. 6. A layer of dust is formed on the earth A feet thick, where h is small, by
the fall of meteors reaching the earth from all directions. Show that the change in
the length of the day is nearly — 7) <^^ ^ *^^y '^here a is the radius of the earth
ui feet, p and D the densities of the dust and earth respectively. If the density of
the dust be twice that of water and h = -^-^ express this in numbers.
The Invariable Plane.
290. It is shown in Art. 72 of Chap, il, that all the momenta
of the several particles of a system in motion, are together equi-
valent to a single resultant linear momentum at any assumed
origin 0, represented in direction and magnitude by a line OV,
together with an angular momentum about some line passing
through 0, represented in direction and magnitude by a line OH.
Let h^, h^, /ig be the moments of the momenta of the particles
about any rectangular axes Ox, Oy, Oz meeting in 0, so that
h=t
m
V dt ^ dt) '
with similar expressions for h^, J\, and let
Then the direction-cosines of OH are -y , ~ , -^ and the an-
li h II
gular momentum itself is represented by h.
If no external forces act on the system then by Art. 72 or Art.
279 \, 1\, h^ are constant throughout the motion, hence OH is
fixed in direction and magnitude. It is therefore called the in-
variable line at 0, and a plane perpendicular to OH is called the
invariable plane at 0.
If any straight line OL be drawn through making an angle 6
with the invariable line OH at 0, the angular momentum about
OL is h cos 6. For the axis of the resultant momentum-couple
is OH, and the resolved part about OL is therefore OH cos 0.
Hence the invariable line at may also be defined as that axis
through about which the moment of the momentum is greatest.
At different points of the system the position of the invariable
lino is different. But the rules by which they are connected are
the same as those which connect the axes of the resultant couple
of a system of forces when the origin of reference is varied. These
240 MOMENTUM.
have been already stated in Art. 203 of Chap. V, and it is un-
necessary here to do more than generally to refer to them.
291. The position of the invariable plane at the centre of
gravity of the solar system may be found in the following manner.
Let the system be referred to any rectangular axes meeting in the
centre of gravity. Let &> be the angular velocity of any body
about its axis of rotation. Let MF' be its moment of inertia
about that axis and (a, ^, j) the direction-angles of that axis.
The axis of revolution and two perpendicular axes form a system
of principal axes at the centre of gravity. The angular momentum
about the axis of revolution is Mk^co, and hence the angular mo-
mentum about an axis parallel to the axis of z is Mk^co cos 7. The
moment of the momentum of the whole mass collected at the
centre of gravity about the axis of s is Mix -~ — y-^\ , hence we
have
/.3 = S-¥F«cos7 + 2:if(a.^-2/§).
The values of 1\, h^, may be found in a similar manner. The posi-
tion of the invariable plane is then known.
292. The Invariable Plane may be used in Astronomy as a
standard of reference. We may observe the positions of the
heavenly bodies with the greatest, care, determining the co-ordi-
nates of each with regard to any axes we please. It is, however,
clear, that unless these axes are fixed in space, or if in motion
unless their motion is known, we have no means of transmitting
our knowledge to posterity. The planes of the ecliptic and the
equator have been generally made the chief planes of reference.
Both these are in motion and their motions are known to a near
degree of approximation, and will hereafter probably be known more
accurately. It might, therefore, be possible to calculate at some
future time, what their positions in space were when any set of
valuable observations were made. But in a very long time some
error may accumulate from year to year and finally become con-
siderable. The present positions of these planes in space may also be
transmitted to posterity by making observations on the fixed stars.
These bodies, however, are not absolutely fixed, and as time goes
on, the positions of the planes of reference would be determined
from these observations with less and less accuracy. A third
method, which has been suggested by Laplace, is to make use of
the Invariable Plane. If we suppose the bodies forming our
system, viz. the sun, planets, satellites, comets, &c., to be subject
only to their mutual attractions, it follows from the preceding
articles that the direction in space of the Invariable Plane at the
centre of gravity is absolutely fixed. It also follows from Art. 79
THE INVARIABLE PLANE. 241
that the centre of gravity is either at rest or moves uniformly in
a straight Hne. We have here neglected the attractions of the
stars. These, however, are too small to be taken account of in
the present state of our astronomical knowledge. We may, there-
fore, determine to some extent the positions of our co-ordinate
planes in space, by referring them to the Invariable Plane as being
a plane which is more nearly fixed than any other known plane in
the solar system. The position of this plane may be calculated at
the present time from the present state of the solar system, and at
any future time a similar calculation .may be m-ade founded on the
then state of the system. Thus a knowledge of its position cannot
be lost. A knowledge of the co-ordinates of the Invariable Plane
is not, however, sufficient to determine conversely the position of
our planes of reference. We must also .know the co-ordinates of
some straight line in the Invariable Plane whose direction is also
fixed in space. This, as Poisson has suggested, is supplied by the
projection on the Invariable Plane of the direction of motion
of the centre of gravity of the system. If the centre of gravity
of the solar system were at rest or moved perpendicularly to
the Invariable Plane, this would fail. In any case our knowledge
of the motion of the centre of gravity is not at present sufficient
to enable us to make much use of this fixed direction in space.
293. If the planets and bodies forming the solar system can
be regarded as spheres whose strata of equahdensity are concen-
tric spheres, their mutual attractions act along the straight lines
joining their centres. In this case the motions of their centres
will be the same as if each mass were collected into its centre of
gravity, while the motion of each about its centre of gravity
would continue unchanged for ever. Thus we may obtain another
fixed plane by omitting these latter motions altogether. This is
what Laplace has done, and in his formulae the terms depending
on the rotations of the bodies in the preceding values of h^, \, h^
are omitted. This pkne might be called the Astronomical Invari-
able Plane to distinguish it from the true Dynamical Invariable
Plane, The former is perpendicular to the axis of the momentum
couple due to the motions of translation of the several bodies,
the latter is perpendicular to the axis of the momentum couple
due to the motions of translation and rotation.
The Astronomical Invariable Plane is not strictly fixed in
space, because the mutual attractions of the bodies do not strictly
act along the straight lines joining their centres of gravity, so that
the tei'ms omitted in the expressions for h^, \, h„ are not abso-
lutely constant. The effect of precession is to make the axis of
rotation of each body describe a cone in space, so that, even though
the angular velocity is unaltered, the position in space of the Astro-
nomical Invariable Plane must be slightly altered. A collision
between tAvo bodies of the system, if such a thing wore possible,
R, D. 16
242 MOMENTUM.
or an explosion of a planet similar to that by which Olbers sup-
posed the planets Pallas, Ceres, Juno and Vesta, &c., to have been
produced, might make a considerable change in the sum of the
terms omitted. In this case there would be a change in the
position of the Astronomical Invariable Plane, but the Dynamical
Invariable Plane would be altogether unaffected. It might be
supposed that it would be preferable to use in Astronomy the
true Invariable Plane. But this is not necessarily the case, for
the angular velocities and moments of inertia of the bodies form-
ing our system are not all known, so that the position of the
Dynamical Invariable Plane cannot be calculated to any near
degree of approximation, while we do know that the terms into
which these unknown quantities enter are all very small or nearly
constant. All the terms rejected being small compared with
those retained, the Astronomical Invariable Plane must make
only a small angle with the Dynamical Invariable Plane. Al-
though the plane is very nearly fixed in space, yet its intersection
with the Dynamical Invariable Plane, owing to the smallness of
the inclination, ' may undergo considerable changes in course of
time.
In the Mecanique Celeste, Laplace calculated the position of
the Astronomical Invariable Plane at the two epochs, 1750 and
1950, assuming the correctness for this period of his formulae for
the variations of the eccentricities, inclinations and nodes of the
planetary orbits. At the first epoch the inclination of this plane
to the ecliptic was 1°.7689, and longitude of the ascending node
114°.3979; at the second epoch the inclination will be the same as
before, and the longitude of the node 114''.3934.
294. Ex. 1. Show that the invariable plane at any p9int of space in the
straight line described by the centre of gravity of the solar system is parallel to
that at the centre of gravity.
Ex. 2. If the invariable planes at all points in a certain straight line are
parallel, then that straight line is parallel to the straight line described by the
centre of gravity.
Impulsive Forces in Three Dimensions.
295. To determine the general equations of motion of a body
about afxed point under the action of given impulses.
Let the fixed point be taken as the origin, and let the axes
of co-ordinates be rectangular. Let (O^., Xl^, XIJ, [co^, Wy, &)J be
the angular velocities of the body just before and just after the
impulse, and let the differences eo^ — O^., (Oy — Hy, co^ — O^ be
called ft) ', ft),/, ft) ' Then w' w', (oj are the angular velocities
generated by the impidse. By D'Alembert's Principle, see Art. 87,
IMPULSIVE FORCES. 243
the difference between the moments of the momenta of the par-
ticles of the system just before and just after the action of the
impulses is equal to the moment of the impulses. Hence by-
Art. 240,
AcoJ — (Zmxy) Wy — (tmxz) &)/ = L \
Ba)J - {Xmyz) &>; - {tmijx) &>; = ill i (1),
Cw^ — i^mzx) Q)J — (%mz}f) wj = N J
where L, M, N are the moments of the impulsive forces about the
axes.
These three equations will suffice to determine the values of
w^, (oj, ft)/. These being added to the angular velocities before
the impulse, the initial motion of the body after the impulse is
found.
296. Ex. 1. Show tliat these equations are independent of each other.
This follows from Art. 20 where it is shown that the eliminaut of the equations
cannot vanish.
Ex. 2. Deduce these equations from the general equations of motion referred to
moving axes given in Art. 253.
Ex. 3. Show that if the body be acted on by a finite number of given impulses
following each other at infinitely short intervals, the final motion is independent of
their order.
297. It is to be observed that these equations leave the axes
of reference undetermined. They should be so chosen that the
values of A, 'S^mxy, &c. may be most easily found. If the posi-
tions of the principal axes at the fixed point are known they will
in general be found the most suitable.
In that case the equations reduce to the simple form
Aco:=L^
■ Bcp,J=mI (2).
The values of coj, (oj, coj being known, we can find the pres-
sures on the fixed point. For by D'Alembert's Principle the
change in the linear momentum of the body in any direction is
equal to the resolved part of the impulsive forces. Hence if
F, G, H, be the pressures of the fixed point on the body
2X + F= M . ~ by Art. 86 |
= M{foJz-co:Tj)hyAi±219[. (3).
tY+G = M{co:x-coJz) I
298. Ex. A uniform disc bounded by an arc OP of a parabola, the axis ON,
and the ordinate PN, has its vertex fixed. A blow B is given to it perpendicular
16—2
244
MOMENTUM.
to its plane at tJie other extremity P of the curved boundary. Supposing the disc to
be at rest before the application of the bloiv, find the initial motion.
Let the equation to the parabola be ?/^=4aa; and let the axis of z be perpendicular
to its plane. Then 71mxz — 0, '2,myz — 0. Let /* be the mass of a unit of area and
let ON=c. Also
'2mxy—iJL\\xijdxdy = iJ.\ x'-~ dx — 2fi I ax'^dx = -fji.ac^,
'^—^lJ-\y^dx=—rij.a^c', £=fi\ x^ydx=~iJi.a^c'- and C=A+£'hj Axt. 7.
^ Jo i-^ jo 7
The moments of the blow B about the axes are L = Bsf^e, M=- Be, N-Q. The
equations of Art. 295 will become after substitution of these values
^fx-a c-Uy--ixac*w^ = -Bc \
-. "7 9 / dC
From these w^, osy may "be found. By eliminating B we have -^ = — ^^ — . Hence
7
-if NQ be taken equal to ^iVP, the disc will begin to rotate about OQ. The re-
75 B
sultant angular velocity will be — ^ OQ.
A\} flClC
299. When a body free to turn about a fixed point is acted on by any number
of impulses, each impulse is equivalent to an equal and parallel impulse acting at
the fixed point together with an impulsive couple. The impulse at the fixed point
can have no effect on the motion of the body, and may therefore be left out of con-
sideration if only the motion is wanted. Compounding all the couples, we see that
the general problem may be stated thus : — A body moving about a fixed point is
acted on by a given impulsive couple, find the change produced in the motion. The
analytical solution is comprised in the equations which have been written down in
Art. 295. The following examples express the result in a geometrical form.
Ex. 1. Show from these equations that the resultant axis of the angular
velocity generated by the couple is the diametral line of the plane of the couple
with regard to the momental ellipsoid. See also Art. 117.
IMPULSIVE FOECES. 245
Ex. 2, Let G be the magnitude of the couple, p the perpendicular from the
fixed poiat on the tangent plane to the momental ellipsoid parallel to the j)lane
of the couple G. Let O he the angular velocity generated, r the radius vector of
the elhpsoid which is the axis of Q,. Let Me"^ be the parameter of the ellipsoid.
Prove that — = — .
12 fr
Ex. 3. If fi^i f^i/i ^z be angular velocities about three conjugate diameters of the
momental ellipsoid at the fixed point, such that their resultant is the angular
velocity generated by an impulsive couple G, A', B\ 0' the moments of inertia
about these conjugate diameters, prove that
^'fl;c=Gcosa, i?'Oy = (? cos /3, C"02 = (?cos7,
where a, /S, 7 are the angles the axis of G makes with the conjugate diameters,
Ex. 4. If a body free to turn about a fixed point be acted on by an impulsive
couple G, whose axis is the radius vector r of the ellipsoid of gyration at 0, and if -p
be the perpendicular from on the tangent plane at the extremity of r, then the
axis of the angular velocity generated by the blow will be the perpendicular p and
the magnitude fi is given by 6^ = MprQ,.
Ex, 5. Show that if a body at rest be acted on by any impulses, we may take
moments about the hiitial axis of rotation, according to the rule given in Art. 89, as
if it were a fixed axis.
300.' Ex, 1, When a body turns about a fixed point the product of the moment
of inertia about the instantaneous axis into the square of the angular velocity is
called the Vis Viva. Let the vis viva generated from rest by any impulse be IT
and let the vis viva generated by the same impulse when the body is constrained to
turn about a fixed axis passing through the fixed- point be 2r'. Then prove that
T' = T cos- d, where 6 is the angle between the eccentric lines of the two axes of
rotation with regard to the momental elhpsoid at the fixed point.
Ex. 2. Hence deduce Lagrange's theorem, that the vis viva generated from
rest by an impulse is greater when the body is free to tiu-n about the fixed point,
than when constrained to turn about any axis through the fixed point.
Ex. 3. If a body be moving in any manner about a fixed point and an axis
through the fixed point be suddenly fixed, show that if the vis viva 2r be changed
into 2T', we have T'=Tgos^6, where ^ has the same meaning as before.
301. To determine the motion of a free body acted on by any
given impulse.
Since the body is free, the motion round the centre of gravity-
is the same as if that point were fixed. Hence the axes being any
three straight lines at right angles meeting at the centre of
gravity, the angular velocities of the body may still be found by
e({uations (1) and (2) of Art. 295.
To find tlie motion of the centre of gravity, let {U, V, W),
(//, V, m) bo the resolved velocities of the centre of gravity just
246 ' MOMENTUM.
before and just after tlie impulse. Let X, Y, Z be the com-
ponents of the blow, and let M be the whole mass. Then by
resolving parallel to the axes we have
M{u-U)=X, Miv- V)=Y, 3I{w-JV) = Z.
If we follow the same notation as in Art. 295, the differences
u— U, V —V, w — IF' may be called w', v, iil .
302. Ex. 1. A body at rest is acted on by an impulse whose components parallel
to the principal axes at the centre of gravity are (X, Y, Z) and the co-ordinates of
whose point of application referred to these axes are {j}, q, r). Prove that if the
resulting motion be one of rotation only about some axis,
A {B-OpYZ+B {0-A) qZX+ C{A-B) rZF^O.
Is this condition sufficient as well as necessary ? See Art. 221.
Ex. 2. A homogeneous cricket-ball is set rotating about a horizontal axis in
the vertical plane of projection with an angular velocity 0. Wlien it strikes the
grornid, supposed perfectly rough and inelastic, the centre is moving with velocity
F in a dhection making an angle a with the horizon, prove that the du'ection of the
motion of the ball after impact wUl make with the plane of projection an angle
tan~i ;; --= , where a is the radius of the baU.
5 Fcos a
803. The equations of Art. 301 completely determine the
motion of a free body acted on by a given impulse, and from these
by Art. 219 we may determine the initial motion of any point of
the body. Let {p, q, r) be the co-ordinates of the point of appli-
cation of the blow, then the moments of the blow round the axes
are respectively qZ — rY, rX—pZ, pY— qX. These must be
written on the right-hand sides of the equations of Art. 295. Let
{p, q', r) be the co-ordinates of the point whose initial velocities
parallel to the axes are required. Let (w^, v^, w^, {u^, v^, w^ be
its velocities just before and just after the impulse. Let the rest of
the notation be the same as that used in Art. 295. Then
u^ — u^ — II + WyV — wlq,
with similar equations for v^ — v^ , w^ — w^. Substituting in these
equations the value of w', v , w , w^, tw^', &)/ given by Art. 301 we
see that u^ — u^, '^2~'^i' '^^si~'^^i ^^® ^^^ linear functions of X, Y,
Z of the first degree of the form
u^-u^ = FX-vGY^BZ,
where F, G, U sue functions of the structure of the body and the
co-ordinates of the two points.
804. When the point whose initial motion is required is the
point of application of the blow, and the axes of reference the
principal axes at the centre of gravity, these expressions take the
simple forms
IMPACT OF ROUGH ELASTIC BODIES. 247
The right-hand sides of these equations are the differential
coefficients of a quadratic function of X, Y, Z, which we may call
E. It follows that for all blows at the same point P of the same
body the resultant change in the velocity of the point P of appli-
cation is perpendicular to the diametral plane of the direction of
the blow with regard to a certain ellipsoid whose centre is at P,
and whose equation is -E'= constant.
The expression for E may be written in either of the equiva-
lent forms
2^= ^!±Z!+^ + _1_ [(^Ap' + Bq'+ Cr') [AX' -\-BY'+ CZ')
-{ApX+BqY+CrZy}
= ^^^^^^^ + ^{qZ-rYy + l{rX-pZ) + lipY-qXy.
In this latter form we see that it is
= M{u" + v" + w") + AcoJ' + BcoJ' + Ceo;',
which is the vis viva of the motion generated by the impulse.
ImjKict of Bough Elastic Bodies.
805. The problem of determining the motion of any two bodies
after a collision involves some rather long analysis and yet there
are some points in which it differs essentially from the same
problem considered in two dimensions. We shall, therefore, first
consider a special problem which admits of being treated briefly,
and will then apply the same princijDles to the general problem
in three dimensions,
806. Two rough ellipsoids moving in any manner imjyinge on
each other so that the extremity of a principal diameter of one
strikes the extremity of a principal diameter of the other, and at
that instant the tltree principal diameters of one are 2^ci'>''^il&l to
those of the other. Find the motion just after imqmct.
Let us refer the motion to co-ordinate axes parallel to the prin-
cipal diameters of either ellipsoid at the beginning of the impact.
Then since the duration of the impact is indefinitely small and
the velocities are finite, the bodies will not have time to clian"e
248 MOMENTUM.
their position, and therefore the principal diameters will be par-
allel to the co-ordinate axes throughout the impact.
Let U, V, W be the resolved velocities of the centre of gravity
of one body just before impact ; u, v, w the resolved velocities at
any time t after the beginning of the impact, but before its termi-
nation. Let n^, n^, n^ be the angular velocities of the body just
before impact about its principal diameters at the centre of gravity ;
0)^, CO, J, &)j, the angular velocities at the time t. Let a, b, g be the
semiaxes of the ellipsoid, and A, B, G the moments of inertia at
the centre of gravity about these axes respectively. Let M be the
mass of the- body. Let accented letters denote the same quan-
tities for the other body. Let the bodies impinge at the extremi-
ties of the axes of c, c .
Let P, Q, R be the resolved parts parallial to the axes of the
momentum generated in the body i/ by the blow during the time
t. Then —P, — Q, — R are the resolved parts of the momentum
generated in the other body in the same time.
The equations of motion of the body M are
A{co,-^,) = Qc
i?(a),-nj = -Pcl (1), -
G {oi, - ft,) = )
M {u-U) = P^
■M{v-V) = q\ (2).
3I{w-W)^Rr
There will be six corresponding equations for the other body
which may be derived from these by accenting all the letters on
the left-hand sides and writing — P, — Q, — R and — c for P, Q, R
and c on the right-hand sides of these equations. Let us call these
new equations respectively (8) and (4).
Let S be the velocity with which one ellipsoid slides along the
other, and 6 the angle the direction of sliding makes with the
axis of X, then
>S' cos 6 = u -\- cw'j — u-\- coOy (5),
/S'sin = v' — c'coj. —v-\- cco^ (6).
Let (7 be the relative velocity of compression, then
G-w -w. (7).
Substituting in these equations from- the dynamical equations
we have
S cos ^ S.coii 9, -pP. (8),
/S^sin^=>S;sin^„-^f9 (9),
C=G,-rR (10),
IMPACT OF ROUGH ELASTIC BODIES. 249
where
s^ cos e^^u' + c'fi; -u+cQ]
s,Bm0^=v'-caj - v+caA (11),
c^=.w'-w )
^ M "^ i/' '^ B^ B' \
^ = M+M' + A+A'\ (^'^•
These are the constants of the impact. S^, G^ are the initial
velocities of sliding, and ^^ the angle the direction of initial sliding
makes with the axis of x. Let us take as the standard case that
in which the body M' is sliding along and compressing the body M,
so that 8q and G^ are both positive. The other three constants
J), q, r are independent of the initial motion and are essentially
positive quantities.
307. Exactly as in two dimensions we shall adopt a graphical
method of tracing the changes which occur in- the frictions. Let
us measure along the axes of x, y, z three lengths OP, OQ, OB to
represent the three reactions P, Q, B. Then if these be regarded
as the co-ordinates of a point T, the motion of T will represent
the changes in the forces. It will be convenient to trace the loci
given hj S=-0, G=0. The locus given by 8 = is a straight
line parallel to the axis of B, which we may call the line of no
sliding. The locus given by C— 0"is a plane parallel to the plane
P, Q, which we may call the plane of greatest compr-ession. At
the beginning of the impact one ellipsoid is sliding along the other,
so that according to Art. 144 the friction called into play is limit-
ing. Since P, Q, B are the whole resolved momenta generated in
the time t; dP, dQ, dB will be the resolved momenta generated
in the time dt, the two former being due to the frictional, and the
latter to the normal blow. Then the direction of the resultant of
dP, dQ must be opposite to the direction in which one point of
contact slides over the other, and the magnitude of the resultant
must be equal to ^idB, vihere /x is the coefficient of friction. We
have therefore
f ,,g^ Y"y^"l (13).
dQ S^smd^-qQ
{dpy + {dQy = f.'{dBY (14).
The solution of these equations will indicate the manner in
which the representative pohit T approaches the line of no sliding.
250 MOMENTUM.
The equation (13) can be solved by separating the variables.
We get
1 1
{S, cos e^ -pPy^ = a {S, sin 6, - qQ) ^,
where a is an arbitrary constant. At the beginning of the motion
P and Q are zero, hence we have
f S.cos e^ -pP \j,^ ( S, sin 0, - gQ \l , .
\ S,cos6, J [ S,sm0^ J ^ ^^'
which may also be written
^cos6'\- fSsm9\l
S=sJ'^)^k('^)^^ (17).
This equation gives the relation between the direction and the
velocity of sliding.
808. If the direction of sliding does not change during the
impact must be constant and equal to 0^. We see from (16)
that if p = q, then = 0^; and conversely if 0—0^, /S would be
constant unless p = q. Also if sin 0^ or cos 0^ be zero, 8 would
be zero or infinite unless 0=0^. The necessary and sufficient
condition that the direction of friction should not change during
the impact is therefore p = q or sin 20^ = 0. The former of these
two conditions by (12) leads to
c'i^-l]+c\(^,-^)^0 (18).
A BJ^ \A' B'j
If this condition holds, we have by (13) P= Q cot ^^ and
therefore by (14)
F = ^n cos 0\_
Q=fiRsm0j " ^ ^^•
It follows from these equations that when the friction is 'limit-
ing, the representative point T nwves along a straight line making
an angle tan~^ /jl with the axis of B, in such a direction as to meet
the straight line of no sliding.
309. If the condition 2^ = ^ ^^^^s not hold, we may, by dif-
ferentiating (8) and (9) and eliminating P, Q, and S, reduce the
determination of B in terms of to an integral.
By substituting for S from (17) in (8) and (9), we then have
P, Q, B expressed as functions of 0. Thus we have the equations
to the curve along which the representative point T travels.
The curve along which T travels may more convenienth^ be
IMPACT OF ROUGH ELASTIC BODIES. 251
defined by the property that its tangent by (14) makes a constant
angle tan"^/Awith the axis of R and its projection on the plane
of PQ is given by (15). And it follows that this curve must
meet the straight line of no sliding, for the equation (15) is satis-
fied by pP= Sq cos 6q, qQ = Sq sin 6^.
810. The whole progress of the impact may now be traced
exactly as in the corresponding problem in two dimensions. The
representative point T travels along a certain known curve, until
it reaches the line of no sliding. It then proceeds along the line
of no sliding, in such a direction that the abscissa JR increases.
The complete value B^ of B for the whole impact is found by
multiplying the abscissa B^ of the point at which T crosses the
plane of greatest compression by 1 + e so that R.^ = B^{l + e), if e
be the measure of the elasticity of the two bodies. The complete
values of the frictions called into play are the ordinates of the
position of T corresponding to the abscissa B — B^. Substi-
tuting these in the dynamical equations (1), (2), (3), (4), the
motion of the two bodies just after impact may be found.
311. Let us consider an example. Since the line of no
sliding is perpendicular to the plane of PQ, P and Q are constant
when T travels along this line. So that when once the sliding
friction has ceased, no more friction is called into pla5^ If there-
fore sliding ceases at any instant before the termination of the
impact as when the bodies are either very rough or perfectly rough,
the whole frictional impulses are given by
p^ S.co&e ^ ^ _ 8^ sin 0^
P ' <1 '
If cr be the arc of the curve whose equation is (15) from the
origin to the point where it meets the line of no sliding, then the
representative point T cuts the line of no sliding at a point whose
abscissa is> B—-. If the bodies be so rough that -< — -, the
^ . ^ '!
jooint T will not cross the plane of greatest compression until after
it has reached the line of no sliding. The whole normal imjDulse
G
in this case is therefore given by i2 = — " (1 + e). Substituting
these values of P, Q, R in the dynamical equations, the motion
just after impact may be found.
312. Ex. 1. If 9 be the augle the direction of sliding of one ellipsoid over the
other makes with the axis of x, prove that 6 continually increases or continually
decreases throughout the impact. And if the initial value of 9 lie between and ,
then 9 approaches - or zero according as ^^ is > or < q. Show also that the
representative point reaches the line of no sliding when 9 has either of these values.
252 MOMENTUM.
Ex. 2. If tlie bodies be such that the dii'ection of sliding continues unchanged
during the impact and the sliding ceases before the termination of the impact,
S r
the roughness must be such that u. > -pr — -^ .
Ex. 3. If two rough spheres impinge on each other, prove that the direction of
sliding is the same throughout the impact. This jDroposition was first given by
Coriolis. Jeu de billard, 1835.
Ex. 4. If two inelastic solids of revolution impinge on each other, the vertex
of each being the point of contact, prove that the dii-ection of sliding is the same
throughout the impact. This and the next proposition have been given by
M. PhilliiDS in the fourteenth volume of Liouville's Journal.
Ex. 5. If two bodies having their principal axes at their centres of gravity
parallel impinge so that these centres of gravity are in the common normal at the
point of contact and if the initial direction of sliding be parallel to a principal axis
at either centre of gravity, then the direction of sliding vfill be the same throughout
the impact.
Ex. 6. If two ellipsoids of equal masses impinge on each other at the extremi-
ties of their axes of c, c', and if aa' = W and ca =hc', prove that the direction of
friction is constant throughout the impact.
313. Two rough bodies moving in any manner impiiige on each other. Find the
motion just after impact.
Let us refer the motion to co-ordinate axes, the axes of x, y being in the tangent
plane at the point of impact and the axis of z along the normal. Let U, V, W be
the resolved velocities of the centre of gravity of one body just before impact,
te, V, IV the resolved velocities at any time i after the beginning, but before the
termination of the impact. Let 0^, O^,, fl^ be the angular velocities of the same
body just before impact about axes parallel to the co-ordinate axes, meeting at the
centre of gravity; w^, Wy, w^. the angular velocities at the time t. Let A, B, C, D,
E, F be the moments and products of inertia about axes parallel to the co-ordi-
nate axes meeting at the centre of gravity. Let M be the mass of the body. Let
accented letters denote the same quuntities for the other body.
Let P, Q, R be the resolved parts parallel to the axes of the momentum
generated in the body M from the beginning of the impact, up to the time t. Then
_ P, -Q, -B are the resolved parts of the momentum generated in the other body
in the same time.
Let [x, y, z) {x', y', z') be the co-ordinates of the centres of gravity of the two
bodies referred to the point of contact as origin. The equations of motion are
therefore
A {u,^-n^)-F(u:y-Q,) -B{w^-Q^)= -yR + zQ.
-F(c^^-^^)+Bio^y-Q,j)-I>{w,-U^)=~zP + xR[ (1).
M(v- V)^q[ (2).
3f{w-W) = B}
We have six similar equations for the other body, which differ from these in
having all the letters, except P, Q, R, accented, and in having the signs of P, Q, R
changed. These we shall call equations (3) and (4).
IMPACT OF ROUGH ELASTIC BODIES. 253
Let S be the velocity witli wliich one body slides along the other and let 6 be the
angle the direction of sliding makes with the axis of x. Also let G be the relative
velocity of compression, then
jS'cos^ = m'- b}yz' + iojy' -u + u,jZ - u^ ■>
(S sin 9 = v' - Ug'x' + co^'z' -v + w^ — w^z \ (5) .
C=w' — w^y' + My'od - |io + w^ - UyX]
If we substitute from (1) (2) (3) (4) in (5) we find
/S'o cos 6*0 - -S cos = aP +/(3 + ei? N
S(,smeQ~Sshie = fP + hQ + dR{ (6),
Co-C^eP + clQ + cR]
where 8^, 6q, Cq are the initial values of S, 9, O and are found from (5) by writing
for the letters their initial values. The expressions for a, b, c, d, e, f are rather
comjDlicated, but it is unnecessary to calculate them.
314. We may now trace the whole progress of the impact by the use of a
graphical method. Let us measure from the point of contact 0, along the axes of
co-ordinates, three lengths OP, OQ, OR to represent the three reactions P, Q, R.
Then if, as before, these be regarded as the co-ordinates of a point T, the motion of
T will represent the changes in the forces. The equations to the line of no sliding
are found by putting S = in the first tvv'o of equations (6). We see that it is a
straight line.
The equation to the plane of greatest compression is found by putting (7=0 in
the third of equations (6).
At the beginning of the impact one body is sliding along the other, so that the
friction called into play is limiting. The path of the representative point as it
travels from is given, as before, by
dP do ,„ ,_,
— ~^-u.dR (7).
cos 6* sm^ ^ ^ '
When the representative point T reaches the line of no shding, the sliding of
one body along the other ceases for the instant. After this, only so much friction is
called into play as will suffice to prevent sliding, provided this amoimt is less than
the limiting friction. If therefore the angle the line of no shding makes with the
axis of R be less than tan~^/i, the point T will travel along it. But if the angle be
greater than tan^^^u., more friction is necessary to prevent shding than can be called
into play. Accordingly the friction will continue to be hmiting, but its dh-ectiou
will be changed if S changes sign. The point T will then travel along a curve given
by equations (7) with 6 increased by tt.
The complete value R., of E for the whole impact is found by multiplying the ab-
scissa R of the point at which T crosses the plane of greatest compression by 1 -f- e,
where e is the measure of elasticity, so that R.2 = ^i (! + '')• The complete values
of P and Q are represented by the ordinates corresponding to the abscissa E^. Sub-
stituting in the dynamical equations, the motion just after impact may be found.
315. The path of the representative point before it reaches the line of no
sliding must be found by integrating (7). By differentiating (6) we have
d (iS'cos e) _ adP+fdQ + edR _ a/x cos 6 +fiJ. sin ^ + e
d {S sin e) ~ JdP^ hdQ + ddR ~ fy- cos d + fifi sin O-bd'
254 MOMENTUM.
w'liich reduces to
^ + ^^cos25+/sin25+-cos^+-sm5
1 dS_ 2 2 A^' /"
S dd~ a-b . „„ ^ na , '^ rt <^ ■ /I '
sm 2^ +/COS 2^ + - cos ^ - - sm ^
2 jj, /J.
From this equation we may find 5^ as a function of 6 in tlie form f^, bo d^, ca > e^.
Show that by turning the axes of reference round the axis of'R through the
proper angle we can make / zero.
EXAMPLES. 255
Ex. 2, Prove that the line of no sKdiug is parallel to the conjugate diameter of
the plane containing the frictions P, Q. And the plane of greatest compression is
the diametral plane of the reaction R.
Ex. 3. The line of no sliding is the intersection of the polar planes of two
points situated on the axes of F and Q and distant respectively from the origin
— and 7- — -. — - . The plane of greatest compression is the polar plane of a
Sq cos Oq Sf)SlQ.do
2E
point on the axis of R distant -zp from the origin.
Ex. 4. The plane of PQ cuts the elhpsoid of Ex. 1 in an ellipse, whose axes
divide the plane into four quadrants ; the line of no sliding cuts the plane of PQ in
that quadrant in which the initial sliding Sq occurs.
Ex. 5. A parallel to the line of no sliding through the origin cuts the plane of
greatest compression, in a point whose abscissa R has the same sign as Cq. Hence
show, from geometrical considerations, that the representative point T must cross
the plane of greatest compression.
EXAMPLES*.
1. A cone revolves round its axis with a known angular velocity. The altitude
begins to diminish and the angle to increase, the volume being constant. Show
that the angular velocity is proportional to the altitude.
2. A ckcular disc is revolving in its own plane about its centre ; if a point in
the circumference become fixed, find the new angular velocity.
3. A uniform rod of length 2a lying on a smooth horizontal plane passes
through a ring which permits the rod to rotate freely in the horizontal plane. The
middle point of the rod being indefinitely near the ring any angular velocity is
impressed on it, show that when it leaves the ring the radius vector of the middle
point will have swept out an area equal to -^ .
D
4. An elliptic lamina is rotating about its centre on a smooth horizontal table.
If Wj, Wj, W3 be its angular velocities respectively when the extremities of its major
axis, its focus, and the extremity of the minor axis become fixed, prove
5. A rigid body moveable about a fixed point at which the principal moments
are A, B, (7 is struck by a blow of given magnitude at a given point. If the angular
velocity thus impressed on the body be the greatest possible, prove that [a, b, c)
being the co-ordinates of the given point referred to the principal axes at 0, and
{/, m, n) the direction cosines of the blow, then
al + bm + en = 0,
* These examples are taken from the Examination Papers which have been set
in the University and in the College?!.
256 MOMENTUM.
6. Any triangular lamina ABC has the angular point C fixed and is capable of
free motion about it. A blow is struck at B perpendicular to the plane of the
triangle. Show that the initial axis of rotation is that trisector of the side AB
which is furthest from B.
7. A cone of mass vi and vertical angle .2a can move freely about its axis, and
has a fine smooth groove cut along its surface so as to make a constant angle /3 with
the generating lines of the cone. A heavy particle of mass P moves along the
groove under the action of gravity, the system being initially at rest with the
particle at a distance c from the vertex. Show that if be the' angle through
which the cone has turned when the particle is at any. distance r .from the vertex,
then
m¥' + P.r^sin.^a_ 29 sin a. cot /3
mk^ + Pc^ sin^ a
k being the radius of gyration oi the cone about its axis.
8. A body is turning about an axis through its centre of gravity, a point in the
body becomes suddenly fixed. If the new instantaneous axis be a principal axis
with respect to the point, show that the locus of the point is a rectangular
hyperbola.
9. A cube is rotating with angular velocity w about a diagonal, when one of its
edges which does not meet the diagonal suddenly becomes fixed. Show that the
w
angular velocity about this edge as axis =:^ p.
10. Two masses m, m' are connected by a fine smooth string which passes
round a right circular cylinder of radius a. The two particles are in motion in one
plane under no impressed forces, show that if A be the sum of the absolute areas
swept out in a time t by the two unwrapped portions of the string,
d^-A 1 /I 1 \ ^
■cW 2 \m m J
T being the tension of the string at any time.
11. A piece of wire in the form of a circle lies at rest with its plane in contact
with a smooth horizontal table, when an insect on it suddenly starts walking along
the arc with uniform relative velocity. Show that the wire revolves round its
centre with uniform angular velocity while tha,t centre describes a circle in space
with uniform angular velocity.
I 12. A uniform cu'cular wire of radius a, moveable about a fixed point in its
circumference, lies on a smooth horizontal plane. An insect of mass equal to that
of the wire crawls along it, starting from the extremity of the diameter opposite to
the fixed point, its velocity relative to the wire being uniform and equal to V.
Prove that after a time t the wire will have turned through an angle
— ^ tan 1 (—7- tan -r- ) .
\J3 2a;
Vt 1 , _ / 1 , Vt
^ p tan 1 ( — -
13. A smaU insect moves along a uniform bar of mass equal to itself, and
length 2a, the extremities of which are constrained to remain on the circumference
2a
of a fixed circle, whose radius is --- . Supposing the insect to start from the middle
V3
EXAMPLES. 257
poiut of- the bar, auci its velocity relatively totlie bar to be uniform and equal to V;
1 Vt
prove that the bar in time t will turn through an angle — t= tan~^ — .
v/3 «
14. A rough circular disc can revolve freely in a horizontal plane about a vertical
axis through its centre. An equiangular spiral is traced on the disc having the
centre for pole. An insect whose mass is an n*^ that of the disc crawls along the
curve starting from the point at which it cuts the edge. Show that when the insect
reaches the centre the disc will have revolved through an angle log ( 1 +
where a is the angle between the tangent and radius vector at any poiut of the
sphal.
15. A uniform circular disc moveable about its centre in its own plane (which
is horizontal) has a fine groove in it cut along a radius, and is set rotating with
an angular velocity w. A small rocket whose weight is an nth of the weight of the
disc is placed at the inner extremity of the groove and discharged ; and when it has
left the groove, the same is done with another equal rocket, and so on. Find the
angular velocity after 7i of these operations, and if 7i be indefinitely increased, show
that the Umiting value of the same is we~^,
16. A rigid body is rotating about an axis through its centre of gravity, when a
certain point of the body becomes suddenly fixed, the axis being simultaneously set
free; find the equations of the new instantaneous axis; and prove that, if it be
parallel to the originally fixed axis, the point must lie in the line represented by
the equations «-/« + 6-m?/ + e%2 = 0, {h^-c^)j+(c'^-a^)~ + (a^-b^)- = 0; the prin-
cipal axes through the centre of gravity bemg taken as axes of co-ordinates, a, b, c
the radii of gyration about these lines, and I, m, n the drreetion-cosines of the
originally fixed axis referred to them.
17. A solid body rotating with uniform velocity w about a fixed axis contains
a closed tubular channel of small uniform section filled with an incompressible fluid
in relative equihbrium ; if the rotation of the solid body were suddenly destroyed
the fluid would move in the tube with a velocity — — - , where A is the area of the
projection of the axis of the tube on a plane perpendicular to the axis of rotation
and I is the length of the tube,
18. A gate without a latch in the form of a rectangular lamina is fitted with a
universal joint at the upper corner and at the lower corner there is a short bar
normal to the plane of the gate and projecting equally on both sides of it. As the
gate swings to either side from its stable position of rest, one or other end of the
bar becomes a fixed point. If h be the height of the gate, h tan a its length and 2^8
the angle which the bar subtends at the ui^per corner, show that the angular
velocity of the gate as it passes through the position of rest is impulsively dimin-
ished in the ratio -. — ; ^- — —-, and the time between successive impacts when the
sm-a + tan''|3
oscillations become small decreases in the same ratio, the weights of the bar and
joint being neglected.
K D. 17
CHAPTER VII.
VIS VIVA.
The Force-function and Work.
318. If a particle of mass m be projected along the axis of x
with an initial velocity V and be acted on by a force F in the
same direction, the motion is given by the equation m -7^ = F.
Integrating this with regard to t, if v be the velocity after a
time t, we have,
m{v- V)=\ Fdt.
•^
If we multiply both sides of the differential equation of the
second order by -77 and integrate, we get*
^m{v'-V')=rFdcc.
* It is seldom that Mathematicians can be found engaged in a controversy
such, as that which raged for forty years in the last century. The object of the
dispute was to determine how the force of a body in motion was to be measured.
Up to the year 1686, the measure taken was the product of the mass of the body
into its velocity. Leibnitz, however, thought he perceived an error in the common
opiaion, and undertook to show that the proper measure should be, the product of
the mass into the square of the velocity. Shortly aU Europe was divided between
the rival theories. Germany took part with Leibnitz and BernoiiUi ; while Eng-
land, true to the old measure, combated then- arguments with great success,
France was divided, an illustrious lady, the Marquise du Chatelet, being first a
warm supporter and then an opponent of Leibnitzian opinions. Holland and Italy
were in general favourable to the German philosopher. But what was most strange
in this great dispute was, that the same problem, solved by geometers of opposite
opinions, had the same solution. However the force was measured, whether by
the first or the second power of the velocity, the result was the same. The argu-
ments and replies advanced on both sides are briefly given in Montucla's History,
and are most interesting. For this however we have no space. The controversy
was at last closed by D'Alembert, who showed in his treatise on Dynamics that the
whole dispute was a mere question of words. When we speak, he says, of the force
of a moving body, we either attach no clear meaning to the word or we understand
only the property that certain resistances can be overcome by the moving body. It
FORCE-FUNCTION AND WORK. 259
The first of these integrals shows that the change of the mo-
mentum is equal to the time-integral of the force. B}'" applying
similar reasoning to the motion of a dynamical system we have
been led in the last chapter to the general principle enunciated in
Art. 279, and afterwards to its application to determine the changes
produced by very great forces acting for a very short time. The
second integral shows that half the change of the vis viva is equal
to the space-integral of the force. It is our object in this chapter
to extend this result also, and to apply it to the general motion of
a system of bodies.
319. For the purposes of description it will be convenient to
give names to the two sides of this equation. Twice the left-hand
side is usually called the vis viva of the particle, a term introduced
by Leibnitz about the year 1695. Half the vis viva is also called
the kinetic energy of the particle. Many names have been given
to the right-hand side at various times. It is now commonly
called the work of the force F. When the force does not act
in the direction of the motion of its point of application the term
"work" will require a more extended definition. This we shall
discuss in the next article.
320. Let a force F act at a point ^ of a body in the direction
AB, and let us suppose the point A to move into any other po-
sition A very near A. If be the angle the direction AB of
the force makes with the direction AA' of the displacement of
the point of application, then the product i^.^^'.cos^ is called
the work done by the force. If for ^ we write the angle the
direction AB of the force makes with the direction AA opposite
to the displacement, the product is called the work done against
the force. If we drop a perpendicular A'AI on AB, the work done
by the force is also equal to the product F.AM, where AM is to be
estimated as positive when in the direction of the force. If F' be
the resolved part of F in the direction of the displacement, the
work is also equal to F. AA'. If several forces act, we can in the
same way find the work done by each. The sum of all these is
the work done by the whole system of forces.
is not then by any simple considerations of merely the mass and the velocity of the
body that we must estimate this force, but by the nature of the obstacles overcome.
The greater the resistance overcome, the greater we may say is the force ; provided
we do not understand by this word a pretended existence inherent in the body, but
simply use it as an alnridged mode of expressing a fact. D'Alembert then points
out that there are different kinds of obstacles and examines how their different
kinds of resistances may be used as measures. It will perhaps be sufficient to
observe, that the resistance may in some cases be more conveniently measured
by a space-integral and in others by a time-integral. See Montucla's Histoi-y,
Vol. III. and "Whewell's History, Vol. ii.
17—2
260 VIS VIVA.
Thus defined, the work done by a force, corresponding to any
indefinitely small displacement, is the same as the virtual moment
of the force. In Statics, we are only concerned with the small
hypothetical displacements, we give the system in applying the
principle of Virtual Velocities, and this definition is therefore
sufficient. But in Dynamics the bodies are in motion, and we
must extend our definition of work to include the case of a dis-
placement of any magnitude. When the points of application of
the forces receive finite displacements we must divide the path
of each into elements. The work done in each element may be
found by the definition given above. The sum of all these is the
whole work.
It should be noticed that the work done by given forces as the
body moves from one given position to another, is independent
of the time of transit. As stated in Art. 318, the work is a space-
integral and not a time-integral.
321. If two systems of forces he equivalent, the work done hy
one in any small dis2}lacement is equal to that done by the other.
This follows at once from the principle of Virtual Velocities in
Statics. For if every force in one system be reversed in di-
rection without altering its point of application or its magnitude,
the two systems will be in equilibrium, and the sum of their
virtual moments will therefore be zero. Restoring the system of
forces to its original state, we see that the virtual moments of the
two systems are equal. If the displacements are finite the same
remark applies to each successive element of the displacement,
and therefore to the whole displacement.
322. We may now find an analytical expression for the work
done by a system of forces. Let {x, y, z) be the rectangular
co-ordinates of a particle of the system and let the mass of this
particle be m. Let (X, Y, Z) be the accelerating forces acting
on the particle resolved parallel to the axes of co-ordinates. Then
mX, niY, mZ are the dynamical measures of the acting forces.
Let us suppose the particle to move into the position x + dx,
y -t- dy, z-\- dz] then according to the definition the work done by
the forces will be
[m
Xdx + mYdy + mZdz) (1),
the summation extending to all the forces of the system. If the
bodies receive any finite displacements, the whole work will be
•^ni
j{Xdx+ Ydy + Zdz). (2),
the limits of the integral being determined by the extreme
positions of the system.
FORCE-FUNCTION AND WORK. 261
823. When the forces are such as generally occur in nature,
it will be proved that the summation (1) of the last Article is a
complete differential, i.e. it can be integrated independently of any
relation between the co-ordinates x, y, z. The summation (2) can
therefore be expressed as a function of the co-ordinates of the
system. When this is the case the indefinite integral of the
summation (2) is called the force-function. This name was given
to the function by Sir W. R. Hamilton and Jacobi independently
of each other.
If the force- function be called U, the work done by the forces
when the bodies move from one given position to another is the
definite integral U^— IT^, where U^ and U^ are the values of U,
corresponding to the two given positions of the bodies. It follows
that the work is independent of the mode in which the system
moves from the first given position to the second. In other words,
the work depends on the co-ordinates of the two given extreme
positions, and not on the co-ordinates of any intermediate posi-
tion. Wh«n the forces are such as to possess this property, i.e.
when they possess a force-function, they have been called a con-
servative system of forces. This name was given to the system
by Sir W. Thomson.
324. There will he a force-function, first, luhen the external forces
tend to fixed centres at finite distances and are functions of the
distances from those centres ; and secondly, when the force due to
the mutual attractions or repulsions of the particles of the system
are functions of the distances between the attracting or repelling
particles.
Let m(^ [r) be the action of any fixed centre of force on a
particle m distant r, estimated positive in the direction in which r
is measured, i.e. from the centre of force. Then the summation
(1) in Art. 322 is clearly %m(^ {r) dr. This is a complete differ-
ential. Thus the force-function exists and is equal to 2m j(j)(7-)dr.
Let mm'cf) (r) be the action between two particles m, m whose
distance apart is r, and as before let this force be considered
positive when repulsive. Then the summation (1) becomes
Xmm' (f)(r) dr. The force-function therefore exists, and is equal
to Xmm J (f> (r) dr.
If the law of attraction be the inverse square of the distance,
(f) (r) = 2 and the integral is - . Thus the force-function differs
from the Potential by a constant quantity.
325. It is clear that there is nothing in the definition of the
force -function to compel us to use Cartesian Co-ordinates. If
262 VIS VIVA.
P, Q, &c. be forces acting on a particle, dp, dq, &c. their virtual
velocities, m the mass of the particle, then the force-function is
U= tm j{Pdp + Qdq + &c.),
the summation extending to all the forces of the system.
Ex. 1. If (p, (p, z) be the cylindrical or semi-polar co-ordinates of the particle
m; P, Q, Z the resolved parts of the forces respectively along and perpendicular to
p and along z, prove that dU'='2m{Pdp + Qpd((> + Zdz],
Ex. 2. If (r, 6, cp) be the polar co-ordinates of the particle m; P, Q, iZ the
resolved parts of the forces respectively along the radius vector, perpendicular to it
in the plane of d and perpendicular to that plane, prove that
dt7=Sm {Pdr+Qrd9 +Jlr sm ed(f>).
Ex. 3, If {x, y,z) be the oblique Cartesian co-ordinates of m; X, Y, Z the
components along the axes, prove that
dV'= 2m {X {dx + vdy + fidz) + Y {vdx + dy + \dz) + Z {/idx + \dy + dz) },
where (X, p., v) are the cosines of the angles between the axes yz, zx, xy respectively.
This example is due to Poinsot.
Ex. 4. If the system be refei-red to rectangular axes moving about a fixed
origin, show that the force-fmiction may be found by writing for dx, dy, dz, in
Art. 322 the values of ndt, vdt, ivdt given in Art. 244.^
326. If a system receive any small displacement ds parallel to
a given straight line and an angular displacement dd round that
line, then the partial differential coefficients -,- and -^ represent
respectively the resolved part of all the forces along the line and the
moment of the forces about it.
Since cZ Z7 is the sum of the virtual moments of all the forces
due to any displacement, it is independent of any particular co-
ordinate axes. Let the straight line along which ds is measured
be taken as the axis of z. Taking the same notation as before,
dU= 2m [Xdx + Ydy + Zdz).
But dx =0, dy = 0> and dz = ds, hence we have
dU
dU=ds. XmZ; .-. -^ = XmZ.
ds
Here d U means the change produced in U by the single dis-
placement of the system, taken as one body, parallel to the given
straight line, through a space ds.
Again, the moment of all the forces about the axis of z is
tm {xY— yX), but dx = - ydO, dy = xdO, and dz = 0. Hence the
above moment
^ Ydy + Xdx + Zdz dJJ
FORCE-FUNCTION AND WORK. 263
Here dU is the change produced in U by the single rotation
of the system, taken as one body, round the given axis through
an angle dO.
327. As considerable use will be made of the force -function, the student will
find it advantageous to acquire a facility in writing down its form. The following
examples have therefore been given.
Ex. An elastic string whose unstretched length is I is stretched, find the work
done by the tension when the string is stretched from a length r to a length /.
Let p be any length intermediate between r and r' and let E be the coefficient of
elasticity. The tension is T=E~- and acts opposite to the direction in which p
is measured. The work done while p becomes p + dp is therefore equal to - Tdp.
The force-function is therefore - JTdp. If this be integrated and taken between
E
the limits r to r', we find the required work equal to -^, {('' -l)"^- {r-iy}.
It follows from this that the work required to stretch an elastic string from one
length to another is the product of the arithmetic mean of the initial and final
tensions into the extension of the string.
328. Ex. 1. A system of bodies falls under the action of gravity. If M be
the whole mass, Ji the space descended by the centre of gravity of the whole system,
the work done by gravity is 3Igh,
Let the axis of z be vertical and let the positive dii-ection be downwards. Then
in the summation (1) of Art. 322, X=0, T=0 and Z=g. Hence dU='2mgdz. If i
be the depth of the centre of gravity below the plane of xij, and C be any constant,
we find U=Mgz + G. Taking this between limits we easily obtain the result given.
The theoretical unit of work is the work done by a dynamical unit of force
acting through a unit of space. We may use the result of this example to supply a
practical unit. The work required to raise the centre of gravity of a given mass a
given height at a given place may be taken as the unit of work. English engineers
use a pound for the mass and a foot for the height, and the unit is then called a, foot-
pound. The term Horse-power is used to express the work done per unit of time.
The unit of horse-power is usually taken to be 33000 foot-pounds per minute. The
duty of a steam-engine is the actual work done by the consumption of a imit quan-
tity, usually a bushel, of coal.
Ex, 2. A force communicates to a particle whose mass is equal to that of a
cubic foot of water a velocity of one foot per minute. Find the work done in foot-
pounds.
Ex. 3. Prove that the amount of work required to raise to the surface of the earth
the homogeneous contents of a very small conical cavity whose vertex is at the
centre of the earth, is equal to that which would be expended in raising the whole
mass of the contents, through a space equal to one-fifth of the earth's radius from
the surface, supposing the force of gravity to remain constant. [Coll. Exam.]
329. Ex. 1. If m, m' be the masses of two particles attracting each other with
a force — 7 where r is the distance between them, show that the work done when
they have moved from an infinite distance apart to a distance r is .
This follows from Art. 321.
2G4 VIS VIVA.
Ex. 2. If the particles composing any mass were separated from each other,
work might be obtained from their mutual attractions by allowing the particles to
approach each other. The work thus obtained is greatest when the particles are
collected together from infinite distances. If dv be an element of volume of a solid
mass attracting according to the law of nature, p the density of the element, V the
potential of the solid mass at the element dv, prove that the work performed iu
. 1 r
collecting the particles composing the mass from infinite distances is - / V/jdv.
Let m-^, m.y, mg, &c. be the masses of any particles, 1\^, i\^, &c. the distances
between the masses m^, m^, m^, ni^, &c. in any arrangement. Then as before
• ^ . T • rr 'ni-iinn m „vi., „
the work done in collecting them from mfimte distances is t/ = — *— H — ^—2+ (fee,
'•12 ^S3
which may be written U— 2 — . Now if F, be the potential at the particle m^ of
aU the particles except m-^ in the given arrangement, V^= ~ -\ — - + ... If Fg, F3, &c.
*'i2 ''13
have similar meanings we may write the work in the form
In finding the potential of any solid mass at any point P we may omit the
matter within any indefinitely small element enclosmg P if its density be finite.
For, since "potential is mass divided by distance," and the mass varies as the cube
of the linear dimensions, it follows that the potential of similar figiu-es at points
similarly situated must vary as the square of the linear dimensions and must vanish
when the mass becomes elementary and the distance indefinitely small. In
applying, therefore, the form Z7=^ SFni to a sohd body we may write pdv for m and
take F to be the potential of the whole ma^s at the element dv,.
The problem of determining how much work can be obtained from the bodies
forming the solar system by allowing them to consolidate into a solid mass has
been considered by several philosophers. Sir W. Thomson has calculated that the
potential energy or the work which can be obtained from the existing solar system
is 380,000 X 1033 foot-pounds. Edin. Trans. 1854.
Ex. 3. The particles composing a homogeneous sphere of mass M and radius
r were originally at infinite distances from each other. Prove that the work done
by then* mutual attractions is - — ..
-' 5 r
Ex. 4. The particles of a homogeneous ellipsoid whose mass is il/and semiaxes
a, b, c are collected from infinite distances, show that the work done is
^-M^ r- ^' - .
10 Jo ^(a2+X)(&2 + x)(c2 + Ay
330. Ex. 1. An envelope of any shape and whose volume is v, contains gas
at a uniform pressure p. Assuming that the pressure of the gas per unit of area
is some function of the volume occupied by it, prove that the work done by the
pressures when the volume increases from v = a to v — b is I pdv.
Ja
FORCE-FUNCTION AND WOEK. 26o
Divide the surface into elementary ai*eas each equal to da-, then pda- is the
j)ressiu:e on da: When the volume has increased to t? + dv, let any element da take
the position da' and let dn be the length of the perpendicular drawn from the
central point of da' on the plane of da, then pda dn is the work done by the pressure
on da and p) jdadn is the work done over the whole area. But dadn is the volume
of the oblique cylinder whose base is da and opposite face da' ; so that jdadn is the
whole increment of volume. The whole work done when the volume increases by
dv is therefore pdv.
Ex. 2. A spherical envelope of radius a contains gas at pressure P, assuming
that the pressure of the gas per unit of area is inversely proportional to the volume
occupied by it, prove that the work required to compress the envelope into a sphere
of radius h is 4,ira^P log y .
Ex. 3. An envelope of any shape contains gas and the shape is altered without
altering the volume. Show that the work done over the whole surface is zero.
331. Ex. 1. An impulsive force acts on a body in a fixed direction in space. Show
that if F be the whole momentum communicated by tke force ; w^, Mj the velocities
of the point of application resolved in the direction of the force, just before
and just after the impulse, then the work done by the impulse is — * — ^ F. This
proposition is given in Thomson and Tait's Natural Philosophy .
Let us regard the impulse as the limit of a finite force acting in the fixed direc-
tion for a very short time T. Let the direction of the axis of x be taken jjarallel to
the fixed direction and let X be the whole momentum communicated dming a time
t measured from the commencement of the impulse. Here t is any time less than
T and X vaiieo from zero to i^ as i varies from to T. Also, since X is the whole
momentum up to the time t, -^ is the moving force on the body at the time t. Let
u be the resolved velocity of the point of application at the time t, then j/^ and w^
are the values of u when ^ = and t = 2\ Since udt is the space described in the
dX
time dt by the point of application of the force -y- , the work done in the time T is
I — - udt. This is the same as I udX. Now, when the time t is indefinitely small,
the velocity u is known by Art. 3o3 to be a hnear function of X, so that we may write
u=Uf) + LX vfheve JL is a constant depending on the natm-e of the body. Substi-
tuting this value of u, we have the work equal to j (z
where Z^, F^, ^1, E-^ are the values of X, T, Z, E when t = T.
We may eliminate the form of the body and express the work in terms of the
resolved velocities of the point of application just after the termination of the im-
pulse. Since E-^ is a homogeneous quadratic function of Z^, T^, Z^ we have
Substituting we find
work=^-^+ "^ Z, + ^«-+3 Y, + "^' Z,.
332. A spherical membrane is stretched into a sphere whose radius is r. Let
Tds be the tension across any elementary arc ds when the membrane is stretched,
where T is a known function of r depending on the nature of the material. Then
the work done by the tensions, when the membrane is stretched into a sphere of
Cb
radius 6 is Stt I Trdr.
JO,
Let the centre of the sphere be taken as origin and let us refer any point on the
sphere to polar co-ordinates (r, 6, ). The adjacent sides of an elementary area
are rd9, r sin dd(p. The tensions across rdO and the opposite side are each equal to
Trdd. When the radius r increases by dr, the distance between these sides is
increased by dr sin ddcp, this being the differential of an adjacent side. Hence the
work done by these tensions is Tr dd . dr sin 6 d(p. Let us now consider the remain-
ing two sides of the element. The tensions across rsin^c?^ and the opposite side
are each equal to Tr sin ddrfo. When the radius r increases by dr, the distance
between these sides is increased by drdd. Hence the work done by these tensions
is Tr sin 6d{x,y,z,t) = (1)
be any geometrical relation connecting the co-ordinates of the
particle w. This may be regarded as the equation to a moving
surface on which the particle is constrained to rest. The quanti-
ties Bx, By, Bz are the projections on th^ axes of any arbitrary
displacement of the particle m consistent with the geometrical
relations which hold at the time t. They must therefore satisfy
the equation
dx dy "^ dz
» cl K* flit Cm 2i •
The quantities -j. Bt, -^ Bt, -p Bt are the projections on the
axes of the displacement of the particle due to its motion in the
time Bt. They must therefore satisfy the equation
dx dt dy dt dz dt dt
VIS VIVA AND ENERGY. 269
7 1
Hence unless ~- is zero througlioiit the whole motion we can-
djc dii clz
not take 8x, Si/, Sz to be respectively equal to -^ St, -4- St, -j, St.
The equation -j^ — expresses the condition that the geometrical
equation (1) should not contain the time explicitly.
336. If a system be under the action of no external forces, we liaveX=0, Y=0,
Z=0, and hence the vis viva of the system is constant.
If, however, the mutual reactions between the particles of the system are such
as would appear in the equation of virtual moments, then the vis viva of the system
will not be constant. Thus, even if the solar system were not acted on by any
external forces, yet its vis viva would not be constant. For the mutual attractions
between the several planets are reactions between particles whose distance does not
remain the same, and hence the sum of the virtual moments will not be zero.
Again, if the earth be regarded as a body rotating about an axis and slowly con-
tracting from loss of heat in course of time, the vis viva will not be constant, for
the same reason as before. The increase of angular velocity produced by this
contraction can be easily foiuid by the conservation of areas.
337. Let gravity be the only force acting on the system. Let the axis of z be
vertical, then we have Z = 0, F=0, Z— -g. Hence the equation of vis viva becomes
Tlmv'^ - 2,niv^ = — 2Mg [z' - z).
Thus the vis viva of the system depends only on the altitiide of the centre of
gravity. If any horizontal plane be drawn, the vis viva of the system is the same
whenever the centre of gravity passes through the plane.
338. The equation of Virtual Velocities in Statics is known
to contain in one formula all the conditions of equilibrium. In
the same way the general equation
may be made to give all the equations of motion by properly
choosing the arbitrary displacements Sx, Sy, Sz. In Article 334
we made one choice of these displacements and thus obtained an
equation in an integrable form.
If we give the whole system a displacement parallel to the
axis of z we have Sx = 0, Sy = 0, and Sz is arbitrary. The equa-
(Pz
tion then becomes 2!m ~ — %mZ, which represents any one of the
three first general equations of motion in Art. 71.
If we give the wholo system a displacement round the axis of
of z througli an angle Sd, we have Sx = — yS9, Sy = xSd, Sz = 0.
270 VIS VIVA.
The equation then becomes 2wt [x -t~^ - y -tj^] == Sm {x Y— yX),
whicli represents any one of the three last general equations of
motion in Art. 71.
339. The principle of Vis Viva was first used by Huyghens
in bis determination of the centre of oscillation of a body, but in
a form different from that now used. See the note to page 69.
The principle was extended by John Bernoulli and applied by
his son, Daniel Bernoulli, to the solution of a great variety of
problems, such as the motion of fluids in vases, and the motion of
rigid bodies under certain given conditions. See Montucla, Histoire
de Mathematique, Tome ill.
The great advantage of this principle is that it gives at once a
relation between the velocities of the bodies considered and the
variables or co-ordinates which determine their positions in space,
so that when, from the nature of the problem, the position of all
the bodies may be made to depend on one variable, the equation
of vis viva is sufficient to determine the motion. In general the
prin civile of vis viva will give a first integral of the equations of
motion of the second order. If, at the same time, some of the
other principles enunciated in Art. 278 may be applied to the
bodies under consideration, so that the whole number of equa-
tions thus obtained is equal to the number of independent co-
ordinates of the system, it becomes unnecessary to write down
any equations of motion of the second order.
340. Ex. If a system in motion pass through a position of equilibrium, i. e. a
position in which it would remain in equilibrium under the action of the forces if
j)laced at rest, prove that the vis viva of the system is either a maximum or a
minimum. Courtivron's Theorem, Mem, de I'Acacl. 17-i8 and 1749.
341. Suppose a weight mg to be placed at any height h
above the surface of the earth. As it falls through a height z,
the force of gravity does work which is measured by mgz. The
weight has acquired a velocity v, half of its vis viva is ^mv^ which is
known to be equal to mgz. If the weight fall through the re-
mainder of the height h, gravity may be made to do more work
measured by mg{h—z). When the weight has reached the
ground, it has fallen as far as the circumstances of the case
permit, and no more work can be done by gravity until the weight
has been lifted up again. Throughout the motion we see that
when the weight has descended any space z, half its vis viva,
together with the work that can be done during the rest of the
descent is constant and equal to the work done by gravity during
the whole descent h.
If we complicate the motion by making the weight work
some machine during its descent, the same theorem is still true.
VIS VIVA AND ENERGY. 271
By the principle of vis viva, proved in Art. 834, half the vis viva
of the particle, when it has descended any space z, is equal to the
work '}ngz which has been done by gravity during this descent,
diminished by the work done on the machine. Hence, as before,
half the vis viva together with the difference between the work
done by gravity and that done on the machine during the re-
mainder of the descent is constant and equal to the excess of the
work done by gravity over that done on the machine during the
whole descent.
Let us now extend this principle to the general case of a
system of bodies acted on by any conservative system of forces.
342. Let us select some position of a moving system of bodies
as a position of reference. This may be an actual final position
passed through by the system in its motion, or any position
which it may be convenient to choose, into which the system
could be moved. Suppose the system to start from some position
which we may call A, and at the time t, to occupy some position P.
Then at the time t, half the vis viva generated is equal to the
work done from A to P. Hence half the vis viva at P together
with the work which can be done from P to the position of refer-
ence is constant for all positions of P.
To express this, the word energy has been used. Half the vis
viva is called the kinetic energy of the system. The work which
the forces can do as the system is moved from its existing position
to the position of reference is called the 'potential energy of the
system. The sum of the kinetic and potential energies is called
the energy of the system. The principle of the conservation of
energy may be thus enunciated : —
Wlien a system moves under any conservative forces, the sum of
the kinetic and potential energies is constant throughout the motion.
843. The distinction between work done and potential energy
may be analytically stated thus. The force-function has been
defined in Art. 323 to be the indefinite integral of the virtual
moment of the forces. As the system moves the work done is
the definite integral taken with its lower limit fixed and its upper
limit determined by the instantaneous position of the system.
The potential energy is the definite integral taken with its upper
limit fixed and its lower limit determined by the instantaneous
Coriolis, Helmlioltz and others have suggested that it would be more con-
venient if the Vis Viva were defined to be half the sum of the products of the
masses into the squares of the velocities. See Phil. Trans. 185 1, p. 89. But this
change in the meaning of a term so widely established in Em-ope would be very
likely to cause some confusion. It seems better for the present to iise another
name, such as kinetic energy.
272 VIS VIVA.
position of the system. The terms potential energy and actual
energy are due to Prof. Rankine.
344. Ex. 1. A particle describes an ellipse freely about a centre of force in its
centre. Find the whole energy of its motion.
Let m be tlie mass of the particle, r its distance at any time from tlie centre,
fxr tlie accelerating force on the particle. If coincidence of the particle with the
centre of force be taken as the position of reference, the potential energy by Art. 343
( - mfxr) dr = ^ jn/tr^. If »•' be the semi-conjugate of /■, the velocity of the
~-i:
particle is sf/mr' and the kinetic energy is therefore -m/xr'^. As the particle de-
scribes its ellipse round the centre of force, the sum of the potential and kinetic
energies is equal to -m/x {a^+ b^) where a and h are the semi-axes of the ellipse.
Ex. 2. A particle describes an ellipse freely about a centre of force in the
centre. Show that the mean kinetic energy during a complete revolution is equal
to the mean jDotential energy; the means being taken with regard to time.
Ex. B. If in the last example the means be taken with regard to the angle
described round the centre, the difference of the means is ^ m/j. (a - 6)^.
Ex. 4. A mass M of fluid is running round a circular channel of radius a with
velocity u, another equal mass of fluid is running round a channel of radius b with
velocity v, the radius of one channel is made to increase and the other to decrease
until each has the original value of the other, show that the work required to pro-
duce the change is - T^ - p j {P - o?) M. [Math. Tripos, 1866.]
345. In applying the principle of vis viva to any actual cases, it will be im-
portant to know beforehand what forces and internal reactions may be disregarded
in forming the equation. The general rule is that all forces may be neglected
which do not appear in the equation of Viitual Velocities. These forces may be
enumerated as follows :
A. Those reactions whose virtual velocities are zero.
1. Those whose line of action passes through an instantaneous axis ; as rolling
friction, but not shding friction nor the resistance of any medium.
2. Those whose line of action is perpendicular to the direction of motion of
the point of application ; as the reactions of smooth fixed surfaces, but not those of
moving surfaces.
B. Those reactions whose virtual velocities are not zero and which therefore
would enter into the equation, but which disappear when joined to other re-
actions.
1. The reactions between particles whose distance apart remains the same ; as
the tensions of inextensible strings, but not those of elastic strings.
2. The reaction between two rigid bodies, parts of the same system, which roll
on each other. It is necessary however to include both these bodies in the same
equation of vis viva.
VIS VIVA AND ENERGY. 273
C. All tensions which act along inextensible strings, even though the strings
are bent by passing through smooth fixed riags.
For let a string whose tension is T connect the particles m, vi', and pass through
a ring distant respectively r, r' from the particles. The virtual velocity is clearly
T5r + TSr', because the tension acts along the string. But since the string is
inextensible 5r+ 5?'' = ; therefore the virtual velocity is zero.
346. To determine the vis viva of a rigid body in motion.
If a body move in any manner its vis viva at any instant is
equal to the vis viva of the whole mass collected at its centre of
gravity, together with the vis viva round the centre of gravity con-
sidered as a fixed point : or
The vis viva of a body = vis viva due to translation
+ vis viva dice to rotation.
Let X, y, z be the co-ordinates of a particle whose mass is m
and velocity v, and let x, y, ^ be the eo-ordinates of the centre of
gravity G of the body. Let x = x + ^, y = y + r]> z = z+^. Then
by a property of the centre of gravity 2m^ = 0, Xmrj = 0, ^m^= 0.
Hence Xm -^ = 0, Sm ~ ='0, %m -,- = 0. Now the vis viva of a
■dt at at
body is
sit) \dtj \dtj
Substituting for x, y, z, this becomes
dt "" dt dt"^ dt dt dt '
All the terms in the last line vanish as they should, by
Art. 14. The first term in the first line is the vis viva of the
whole mass 2m, collected at the centre of gravity. The second
term is the vis viva due to rotation round the centre of gravity.
This expression for the vis viva may be put into a more con-
venient shape.
347. First. Let the motion be in two dimensions. Let v be
the velocity of the centre of gravity, r, 6 its polar co-ordinates
referred to any origin in the plane of motion. Let r^ be the
distance of any particle whose mass is m from the centre of gravity,
and let v^ be its velocity relatively to the centre of gravity. Let
ft) be the angular velocity of the whole body about the centre of
gravity, and MIc' its moment of inertia about the same point.
R. D. 18
274 VIS VIVA.
The vis viva of the whole mass collected at G is Mv"^, which
may by the Differential Calculus be put into either of the forms
The vis viva about G is ^mv^. But since the body is turning
about G, we have v^ = r^co. Hence Xmv^ = w^ . ^mr^ = co^ . Mk\
The whole vis viva of the body is therefore
^oyiv^ = MW + MkW.
If the body be turning about an instantaneous axis, whose
distance from the centre of gravity is r, we have v = r(o. Hence
Xmv' = Mco^ {r' + k') = Mk'W,
where Mk'^ is the moment of inertia about the instantaneous axis.
348. Secondly. Let the body he in motion in space of three
dimensions.
Let V be the velocity of G',r, 6, (f> its polar co-ordinates re-
ferred to any origin. Let co^, w^, a^ be the angular velocities
of the body about any three axes at right angles meeting in G,
and let A, B, G be the moments of inertia of the body about
the axes. Let f , 77, ^ be the co-ordinates of a particle m referred
to these axes.
The vis viva of the whole mass collected at G is Mlf, which
may be put equal to
according as we wish to use cartesian or polar co-ordinates.
The vis viva due to the motion about G is
^--^»{(l)^(IT-(f)}-
^""^ dt'^''^^"'^'^'dt = '''^~''^^''dt'^''^'^~''^^-
Substituting these values, we get, since A = Sm [if -|- ^),
5 = 2m(r + r), C=Sm(f + 77''),
- 2 {tm^ri) w^(o,j - 2 {tmrjl;) w^w, - 2 (Sm^^) w,&)^.
If the axes of co-ordinates be the principal axes at G, this re-
duces to
VIS VIVA AND ENERGY. 275
If the body be turning about a point 0, whose position is
fixed for the moment, the vis viva may be proved in the same
way to be
where A', B', C are the principal moments of inertia at the
point 0, and co^, cOy, oi^ are the angular velocities of the body
about the principal axes at 0.
349. Ex. 1, A rigid body of mass M is moving in space in any manner and
its position is determined by the co-ordinates of its centre of gravity and the angles
d, be the angle that radius of the sphere which was initially
perpendicular to the plane makes with the axis of y. Let [x, y) be the co-ordinates
of P the centre of the sphere, and Mk"^ the moment of inertia of the sphere about a
diameter.
If the sphere were fixed relatively to the plane its effective forces would be Mn^x
and Mnhj parallel to the axes, and Mh^ T~^ round the centre of gravity. Also the
* This theorem is due to Coriolis, see the Journal Polyteck. 1831,
VIS VIYA AND ENEEGY. 279
impressed force, gravity, is equivalent to g sin nt and -gcosnt parallel to the
moving axes. Hence the equation of Vis Viva for relative motion becomes
^di/dxy fdyy fd4>y) „ dx dy . dx dy
dt'
Here -7, = a — - and -^ = 0, We have therefore
dt (it dt
h^\ d^x „
J7S =n^x + g smnt.
04:)
This equation might also have been derived from the formulae for moving axes
2
r
dt^
2
given in Art. 179. If Jc^ — ~a^, this equation leads to
a:= _ A4 sin nf +^eVfi+ i?e-"\/f «,
12 n-'
where A, B are two constants which depend on the initial conditions of the
question,
353. To determine the change in the vis viva of a movifig
system produced by any collisions between the bodies or by any
explosions. (Carnot's Theorem.)
Let v^, v,j, v^, vj, vj, vj be the resolved parts of the velocities
of any particle m of the system before and after the impulse.
Then the momenta m {vJ — v^), m {vJ — v^), m (vJ — v^), being
reversed and taken throughout the whole system, are by
D'Alembert's Principle in equilibrium with the impulses. But
these last are themselves in equilibrium. Hence the former
set are also in equilibrium. Therefore by Virtual Velocities,
Xm {{vJ - vJ Bx + (vJ - vJ By + « - V,) Bz] = 0,
where Bx, By, Bz are any small arbitrary displacements of the
jDarticles impinging on each other, which are consistent with the
geometrical conditions of the system during the time of action of
the impulse.
During the impact, it is one geometrical condition that the
particles impinging on each other have no velocity of separa-
tion normal to the common surface of the bodies of which they
form a part.
First. Let the bodies be devoid of elasticity. Then the
above geometrical conditon will hold just after the moment of
greatest compression as well as during the impact. Hence we
can put Sx = vJBt, By = vJBt, Bz = vJBt. The equation now be-
comes
tm [{vJ - V,) v^ + (v; - v^) v^ + {vi - V,) V,'} = ;
and
280 VIS VIVA.
This may be put into the form
= - sm [{v: - v:i' + « - V.;,' + {v: - v:f].
Therefore in the impact of inelastic bodies vis viva is always
lost.
Secondly. Let an explosion take place in any body of the
system. Then the geometrical equation above spoken of will
hold just before the impulse begins as well as during the ex-
plosion, but it will not hold after the particles of the body have
separated. Hence we must now put hx = v^St, hy = v^ht, hz = v^U.
As before, we have
Xm {v,vj + vX + V^l) = Sm {vj" + v/ + v,'),
= + tm {« - v:)^ + « - tg^ + (v! - vy].
Therefore in cases of explosion vis viva is always gained.
Thirdly. Let the particles of the system be perfectly elastic.
Then the whole action consists of two parts, a force of compres-
sion as if the particles were inelastic, and a force of restitution of
the nature of an explosion. The circumstances of these two forces
are exactly equal and opposite to each other. By examining
these two expressions it is easy to see that the vis viva lost in
the compression is exactly balanced by the vis viva gained in the
restitution.
354. It should be noticed that Carnot's demonstration^ does
not exclusively apply to collisions, but to all impulses which are
such as do not appear in the equation of Virtual Velocities.
Let a system be moving in any way, and let us suddenly intro-
duce some new restraints, by which some of the particles are
compelled to take new courses. The impulses which produce this
change of motion are of the nature of reactions, and are such
that in the subsequent path their virtual moments are zero. It
follows from Carnot's first theorem, that vis viva will be lost, and
the amount of vis viva lost is equal to the vis viva of the relative
motion.
Let there be two systems at rest, in all respects the same
except that one is subject to some restraints from which the
other is free. Let both these be set in motion by equal im-
pulses, and let X, Y, Z be the components of these. Then, if
YIS VIVA AND EXEEGY. 281
accented letters refer to the more free system and twice accented
letters to the other, we have
Sw (vJSx + &c.) = t {XSx + &c.))
tm {vJ'Sx + &c.) = t IxSx + &c.) j '
where Sx, Sy, Sz are any arbitrary displacements consistent with
the geometrical conditions. Since both systems may be displaced
in the manner in which the less free system actually begins to
move, we may put Sx = vJ'St, &c. "We therefore have
Xm {vjvj' + &c.) = Sm {vj" + &c.).
It again follows from Carnot's first demonstration that the
vis viva of the constrained system is less than that of the free.
Generally, the greater the constraints impressed on a system at rest,
the less will he the vis viva generated by any given impidses. This
theorem is in part due to Lagrange, it has been generalized by
Bertraud in his notes to the Mecanique Analytique.
355. Let two systems be in all respects the same and moving in the same
manner. Let us suppose that suddenly some of the constraints are removed from
one system and at the same instant let both be acted on by equal impulses. Then
following the same notation as before, we have
HmKvx -Vx) 5a; + &c.} = S (A'5a; + &c.),
2m {(?;/ - Vx) 5a; + &c.} = S {Xdx + &c.).
If we make Sx — Vx"St, &o. we obtain
Sm K- V + &c.) = "Em {v/" + &c.),
and we may deduce from this eauation theorems similar to those of the last article.
Let us now give these two systems any other displacement which is permitted
by the geometrical relations common to both. Let this displacement be represented
by bx, = vJ"U, &c. Then as before we have
2m (i'>/' + &c.) =: S?ft (i-Zv^'" + &c-.).
From this and the last equation we easily find
2m {« - Vx"? + &c.} = 2wi {{vj - vj'f + &c.} + 2m {(v/ - vJ"Y + &c.}.
Let a^, a.2, &c. be the positions of the particles m^, m^, &c. just before the action of
the impulses ; a{, aj, &c. , ft/', a.2", &c. their jDOsitions just after, in the more free
and constrained systems respectively, a^'", a.2", &c. their positions after any hypo-
thetical displacement. Then
27ft [a'a'")^ = 29)1 [a' a"? + Urn {a"a"'f.
Hence we infer that the motion of the more constrained system is such that
2m {a'a"Y is less than if the particles took any other courses, consistent with, all
the geometrical relations.
If we suppose the systems to be acted on by a series of indefinitely small im-
pulses, these impulses may be regarded as finite forces. We may therefore infer
the following theorem, which is called Gauss' principle of least constraint.
The motion of a system of material points connected by any geometrical rela-
tions is always as nearly as possible in accordance with free motion; i.e. if the
282 VIS VIVA.
constraint during any time dt is measured by tlie sum of tiie products of the mass
of each particle into the square of its distance at the end of that time from the
position it would have taken if it had been free, then the actual motion during the
time dt is such that the constraint is less than if the particles had taken any other
positions.
M. Gauss remarks that the free motions of the particles -when they are incom-
patible with the geometrical conditions of the system are modified in exactly the
same way as geometers modify results, which have been obtained by observation,
by applying the method of least squares so as to render them compatible with the
geometrical conditions of the question.
356. To determine the mean vis viva of a system of material points in stationary
motion. Clausius' Theorem*.
By stationary motion is meant any motion in which the points do not continually
remove further and fm-ther from their original position, and the velocities do not
alter continuously in the same direction, but the points move within a limited
space and the velocities only fluctuate within certain limits. Of this nature are all
periodic motions, such as those of the planets about the sun and the vibrations of
elastic bodies, and further, such irregular motions as are attributed to the atoms
and molecules of a body in order to explain its heat.
Let X, y, z be the co-ordinates of any particle in the system and let its mass
be m. Let X, Y, Z be the components of the forces on this particle. Then
m — = Z. We have by simple differentiation,
dt^ dt\ dtj \dtj dt
and therefore
2\dt) —2 "^4 df' '
Let this equation be integrated with regard to the time from to t and let the
integral be divided by t, we thereby obtain
in which the application of the suffix zero to any quantity implies that the initial
value of that quantity is to be taken.
The left-hand side of this equation and the first term on the right-hand side are
evidently the mean values of ^ f^Y and -^^^ during the time t. For a periodic
motion the duration of a period may be taken for the time t ; but for irregular
motions (and if we please for periodic ones also) we have only to consider that the
time t, in proportion to the times during which the point moves in the same durec-
tion in respect of any one of the directions of co-ordinates is very great, so that in
the course of the time t many changes of motion have taken place, and the above
expressions of the mean values have become sufficiently constant. The last term
of the equation, which has its factor included in square brackets, becomes, when
the time is periodic, equal to zero at the end of each period. When the motion is
* This and the next article are an abridgement of Clausius' paper in the Phil.
Mag., August, 1870.
VIS VIVA AND ENEEGY. 283
not periodic, but irregularly varying, the factor in brackets does not so regularly
become zero, yet its value cannot continually increase witli the time, but can only
fluctuate within certain limits ; and the divisor t, by which the term is affected,
must accordingly cause the term to become vanishingly small with very great values
of t. The same reasoning will apply to the motions parallel to the other co-ordi-
nates. Hence adding together our results for each particle, we have, if v be the
velocity of the particle m,
mean - 'Emv'^= -~ mean - S {Xx + Yy + Zz).
The mean value of the expression - ^ S (Xx -f Fy -t- Zz) has been called by Clausius
the virial of the system. His theorem may therefore be stated thus, the mean
semi vis viva of the system is equal to its virial.
357. In order to apply this theorem to heat, let us consider a body as a system
of material particles in motion. The forces which act on the system will in general
consist of two parts. In the first place, the elements of the body exert on each
other attractive or repulsive forces, and secondly, forces may act on the system from
without. The viiial will therefore consist of two parts, which are called the
internal and external virial.
If {r) a;' = (r) ^ — — - . And since for the
two other co-ordinates corresponding equations may be formed, we have for the
internal vhial - « S {Xx+ Yy + Zz) = - Zr(p (?•).
As to the external forces, the case most frequently to be considered is where the
body is acted on by a imiform pressure normal to the surface. If p be this pres-
sure, da- an element of the sm-face, I the cosine of the angle the normal makes with
the axis of x, - - SA'ic -- xp lda= ~ j jxdydz. If V be the volume of the body this
1 3
is ^pV, and therefore the whole external virial is ^pV.
Ex. Show that the virial of a system of forces is independent of the origin
and the directions of the axes supposed rectangular.
The first result is clear, since in stationary motion 2A = 0, &c. The second
follows from the equality Xx +Yy + Zz= Rp, where jR is the resultant of X, F, Z, and
p is the projection of the radius vector on the direction oiR.
Newton s Principle of Similitude.
858. What are the conditions necessary that two systems of
particles which are initially geometrically similar should also be
mechanically similar, i.e. the relative positions of the particles in
one system at time t should also be similar to the rekxtive posi-
tions in the other system at time t', where t' bears to ^ a constant
ratio ?
284 VIS VIVA.
In other words, a model is made of a machine, and is found to
work satisfactorily, what are the conditions that a machine made
according to the model should work as satisfactorily ?
Since all the equations of motion may be deduced from the
principle of Virtual Velocities, that principle seems to afford the
simplest method of investigating any general theorem in Dyna-
mics. It has also the advantage of not requiring us to consider
the unknown reactions, if there be any in the system. This mode
of proof is given by M. Bertrand in Galiier xxxii. of the Journal
de Vecole Poly technique.
359. Let [x, y, z) be the co-ordinates of any particle of mass
m in one system referred to any rectangular axes fixed in space,
and let (X, Y, Z) be the resolved part of the impressed moving
forces on that particle. Let accented letters refer to correspond-
ing quantities in the other system.
Then the principle of Virtual Velocities supplies the two
following equations :
UX—m -jj ] 8x 4- &c. > — 0,
iX'—m —j-r^- j hx + &c. (- = 0.
It is evident that one of these equations will be changed into
the other if we put X' = FX, T' = FY, &c., x = Ix, y = ly, &c.,
m' = fxin, &c., t' = it, &c., where F, I, fi, r are a.11 constants,, pro-
vided iJbl— Ft^. In two geometrically similar systems, we have
but one ratio of similarity, viz. that of the linear dimensions, but
in two mechanically similar systems we have three other ratios,
viz. that of the masses of the particles, that of the forces which
act on them, and that of the times at which the systems are to be
compared. It is clear that if the relation just established hold
between these four ratios of similitude, the motion of the two
systems will be similar.
Suppose then the two systems to be initially geometrically
similar, and that the masses of corresponding particles are pro-
portional each to each, and that they begin to move in parallel
directions with like motions and in proportional times, then they
will continue to move with like motions and in proportional times
provided the external moving forces in either system are propor-
, mass X linear dimensions ... ,, , , i •,•
tional to T-- ^9 . omce the resolved velocities
(time)
of any particle are -j- , &c., it is clear that in two similar systems
the velocities of corresponding points at corresponding times are
PRIXCIPLE OF SIMILITUDE. 285
, , linear dimensions -r^ v • i_ l^
proportional to -. . it we elmimate the time
time
between these two relations, we may state, briefly, that the con-
dition of similitude between two systems is that the moving
„ , T , • 1 , mass X (velocity)^
lorces must be proportional to ,-; ^^ -■ •
^ ^ linear dimensions
360. M. Bertrand remarks, that in comparing the working of
a model with that of a large machine, we must take care that all
the forces bear their proper ratios. Supposing the model to be
made of the same material as the machine, the weights of the
several parts will vary as their masses, and therefore as the
cubes of the linear dimensions. Hence we infer that the velocity
of working the model must be made to be proportional to the
square root of its linear dimensions. The times of describing
corresponding arcs will also be in the same ratio.
If there be any forces besides gravity which act on the model,
these must bear the same ratio to the corresponding forces in the
machine, if the model is to be similar to the machine. Hence the
impressed forces must be made to vary as the cubes of the linear
dimensions. For example, in the case of a model of a steam-
engine, the pressure of the steam on the piston varies as the
product of the area of the piston into the elastic force. Hence,
the elastic force of the steam used must be proportional to the
linear dimensions of the model.
Supposing the impressed forces in the tAVO systems to have,
each to each, the proper ratio, the mutual reactions between the
parts of the systems will, of themselves, assume the same ratio. For
if, by giving proper displacements according to the principle of
Virtual Velocities, we form equations of motion to find these reac-
tions, it is easy to see that they will be, each to each, in the same
ratio as the forces. Since sliding friction varies as the normal
pressure, and is independent of the areas in contact, these frictions
will bear their proper ratio in the model and machine. This, how-
ever, is not the case with rolling friction. Recurring to Art. 150,
we see that the rolling friction varies inversely as the diameter of
the wheel, and will, therefore, bear a greater ratio to the other forces
in the model than in the machine. If the resistance of the air be
proportional to the product of the area exposed into the square of
the velocity, this resistance will bear the proper ratio in the
model and the machine.
361. As an example, let us apply the principle to the ease of a rigid body
oscillating about a fixed axis under the action of gravity. That the motions of two
pendulums may be similar they must describe equal angles, corresponding times
are therefore proportional to their times of oscillation. Since the forces vary as the
mass into gravity, we see that when a pendulum oscillates through a given angle,
286 VIS VIVA.
the square of the time of oscillation must vary as the ratio of the linear dimensions
to gravity.
As a second example consider the case of a particle describing an orbit round
the centre of attraction whose force is equal to the product of the inverse square
of the distance into some constant n. The principle at once shows that the square
of the periodic time must vary as the cube of the distance directly and as fx in-
versely. This is Kepler's third law.
362. In the twenty-ninth volume of the Annales de Chimie (Paris, 1825) Savart
describes numerous experiments which he made on the notes sounded by similar
vessels centaining air. He says that if we construct cubical boxes and set the air
in motion as is ordinarily done in organ pipes we find that the number of vibrations
in a given time is proportional to the reciprocals of the linear dimensions of the
masses of air. This law was verified between extreme limits, and its truth tested
with a great many notes. He says he frequently used the law during his researches,
and never once found it led him wrong. This result having been obtained for
cubes, it was natural to examine if the same law held for prismatic tubes on
square bases. After a great many experiments he found the same law to be true.
He then tested the law with conical pipes in which the opening was always
of the same solid angle, then with cylindrical pipes, then with pipes whose
bases were equilateral triangles. These he made to sound in different ways, put-
ting the mouth-piece for instance at different points of the lengtli of the tube. In
all cases the same law was found to hold, for tubes whose diameters were very
small compared with their lengths as well as for those whose diameters were very
great. This law he again found applicable to masses of air set in motion by communi-
cation from other vibrating bodies. Hence he infers this general law which he
enunciates as an experimental fact.
When masses of air are contained in two similar vessels, the number of vibra-
tions in a given time [i. e. the pitch of the note sounded] is proportional inversely
to the linear dimensions of the vessel.
This theorem of Savart's follows at once from the principle of Similarity. Divide
the similar vessels into corresponding elements, then the motion of these elements
will be similar each to each if the forces vary as '- ' . But by Mar-
(time)''*
riotte's law the force between two elements varies as the product of the area of
contact into the density. Hence the times of oscillation of corresponding particles
of air must vary as the hnear dimensions of the vessel.
363. The first person who gave a theoretical explanation of Savart's law was
Cauchy, who showed, in a Memoire presented to the Academy of Sciences in 1829,
that it followed from the linearity of the equations of motion. He refers to the
general equations of motion of an elastic body v/hose particles are but shghtly dis-
placed even though the elasticity is different in different directions. These equa-
tions which serve to determine the displacements (f , yj, f) of a particle in terms of
the time t and the co-ordinates {x, y, z) are of two kinds. One applies to all points
of the interior of the elastic body and the other to all points on its surface. These
fi,re to be found in all treatises on elasticity. An inspection of these equations
shows that they will continue to exist if we replace f , 17, f, x, y, 2, t by k'^, kt], k^, kx,
ny, KZ, Kt, where k is any constant provided we alter the accelerating forces in the
ratio K to 1. Hence if these accelerating forces are zero, it wiU be sufficient to
PRINCIPLE OF SIMILITUDE. 287
increase the dimensions of the elastic body and the initial values of the displace-
ments in the ratio 1 to /f, in order that the general yahies of ^, rj, f and the dura-
tions of the vibrations should vary in the same ratio. Hence we deduce Cauchy's
extension of Savart's law, viz. if we measure the pitch of the note given by a body,
by a plate or an elastic rod, by the number of vibrations produced in a unit of time ;
the pitch will vary inversely as the linear dimensions of the body, plate or rod, sup-
posing all its dimensions altered in a given ratio.
864. These results may be also deduced from the theory of
dimensions. Following the notation of Art. 818, a force F is
measured by m -^ . We may then state the general principle,
that all dynamical equations must be such that the dimensions of
terms added together are the same in space, time and mass, the
-■. . n p T* ,-1 . 1 mass * space
dimensions oi lorce being taken to be — y-. — —^ — .
^ (time)^
Let us apply this to the case of a single pendulum of length I,
oscillating through a given angle a, under the action of gravity.
Let m be the mass of the particle, F the moving force of gravity,
then the time t of oscillation can be a function only of F, I, m
and a. Let this function be expanded in a series of powers of
F, I and in. Thus
T^^AFH'm'',
where A being a function of a only is a number. Since t is of no
dimension in space, we have p+ q = 0. Also t is of one dimen-
sion in time ; .'. —2p = 1. Finally r is of no dimensions in mass;
.'. p +r = 0. Hence p = — ^, q — r = \, and since p, q, r have
each only one value, there is but one term in the series. We
infer that ia any simple pendulum r =A a / ^ where A is an
undetermined number.
365. Ex. 1. A particle moves from rest towards a centre of force whose attrac-
tion varies as the distance in a medium resisting as the velocity, show by the
theory of dimensions that the time of reaching the centre of force is independent of
the initial position of the particle.
Ex. 2. A particle moves from rest in vacuo towards a centre of force whose
attraction varies inversely as the n'^'^ power of the distance, show that the time of
w-f- 1
reaching the centre of force varies as the — „— th power of the initial distance of the
particle.
288 VIS VIVA.
Lagi^ange's Eqitations.
866. Our object in this section is to form the general equation
of motion of a dynamical system freed from all the unknown
reactions and expressed, as far as is possible, in terms of any kind
of co-ordinates which may be convenient in the problem under
consideration.
In order to eliminate the reactions we shall use the principle
of Virtual Velocities. This principle has already been applied to
obtain the equation of Vis Viva by giving the system that par-
ticular displacement which it would have taken if i't had been left
to itself. But since every dynamical problem can, by D'Alembert's
principle, be reduced to one in statics, it is clear that by giving
the system proper displacements, we must be able to deduce, as
in Art. 338, not Vis Viva only, but all the equations of motion.
367. Let {x, y, z) be the co-ordinates of any particle m oi the
system referred to any fixed rectangular axes. These are not
independent of each other, being connected by the geometrical
relations of the system. But they may be expressed in terms of
a certain number of independent variables whose values will de-
termine the position of the system at any time. Extending the
definition given in Art. 73, we shall call these the co-ordinates
of the system. Let these be called 6, (f>, 'yjr, &c. Then x^ y, s, &c.
are functions of 6, cf), &c. Let
x:=f{t,e,cf>,&c.) ...(1),
with similar equations for y and z. It should be noticed that these
equations are not to contain -j- , -^ , &c. The independent
variables in terms of which the motion is to be found may be any
we please, with this restriction, that the co-ordinates of every
particle of the body could, if required, be expressed in terms of
them by means of equations which do not contain any differ-
ential coefficients with regard to the time.
The number of independent co-ordinates to which the position
of a system is reduced by its geometrical relations, is sometimes
spoken of as the number of the degrees of freedom of that body.
Sometimes it is referred to as being the number of independevt
motions which the system admits of
In the following investigations total differential coefficients
with regard to t will be denoted by accents. Thus -r: and --^
will be written x' and x".
LAGRANGE'S EQUATIONS, 289
If 2 T be the vis viva of the system, we have
2T=Xm{a^^-Yy'^ + z") (2);
we also have, since the geometrical equations do not contain
6 , (f) , &c.,
-'=t+l^'4^'+*^ (^)'
with similar equations fory' and z'. In these the differential co-
efficients ^r-,i7\> *^c. are all partial. Substituting these in the
at do
expression for 2 T, we find
2T=F(t,0,cf>,&c.e',(fi',&c.).
When the system of bodies is given, the form of F will be
known. It will appear presently that it is only through the form
of F that the effective forces depend on the nature of the bodies
considered ; so that two dynamical systems which have the same
jp'are dynamically equivalent.
It should be noticed that no powers of 6', ', &c. above the
second enter into this function, and when the geometrical equa-
tions do not contain the time explicitly, it is a homogeneous
function of 6', <^', &c. of the second order,
868. To find the virtual moments of the momenta of a system,
and also of Hie effective forces cory^esponding to a displacement pro-
duced by varying one co-ordinate only.
Let this co-ordinate be 6, and let us follow the notation al-
ready explained. Let all di'fferential coefficients be partial, unless
it be otherwise stated, excepting those denoted by accents. Since
x, y , z' are the components of the velocity, the virtual moment of
the momenta will be 2m {x'hx-\-y'hy -^ z'hz), where hx, Sy, Sz are
the small changes produced in the co-ordinates of the particle m
by a variation 80 of 6. This is the same a-s
^ ( ,dx ,dy idz\^^
If 2T be the vis viva given by (2) of the last article
dT ^ f ,dx'
dO'
But differentiating (.3) partially with regard to 6', we see
that ~, = -j^. . Hence the virtual moment of the momenta is
dT
equal to -r^ W.
ad
-R. D. 19
290 VIS VIVA.
The virtual moment of the effective forces will be
,,dx ,, dy ,, dz"
^ / ,,clx ,, dy ,1 ciz\ ^^
This may be written in the form
d ^ f , dx ^ p_\ ^ _ . / , d dx
dt
Sm(a.'^ + &c.)-2m(..'|^J + &c.),
where the -^ represents a total differential coefficient with regard
to t. We have already proved that the first of these terms is
-T- -T?r, . It remains to express the second term also as a differ-
dt do ^
ential coefficient of T. Differentiating the expression for 2T
partially with regard to 9
dT ^ [ , dx \
But differentiating the expression for x with regard to
CtOu (Jb JO Ct Ju /\f (Jj JU ,/ n
dO dddt d&' ddd(j)
and this is the same as -r; ^7i • Hence the second term may be
dt dO "^
dT
written -^7, , and the virtual moment* of the effective forces is
da
^, . fddTdT\ .^
therefore (^^^-^jS^.
* The following explanation ■will make the argument clearer. The virtual
moment of the effective forces is clearly the ratio to dt of the difference between
the virtual moments of the momenta of the particles of the system at the times
t + dt and t, the displacements being the same at each time. The virtual moment
of the momenta at the time t is first shown to be -— -, 5^. Hence I — -; + -T,-rT,dt ) S^
dd \dd' dt do' J
is the virtual moment of the momenta at the time t + dt corresponding to a dis-
placement W consistent with the positions of the particles at that time. To
make the displacements the same, we must subtract from this the virtual moment
of the momenta for a displacement which is the difference between the two displace-
ments at the times t and t-\-dt. Since 5a; = ^~oi9, this difference for an abscissa is
dO
' 1 dt 59. We therefore subtract on the whole 2m \x' — i ■—]dt + &c. 50, and
fdx\
\de)
dt \de) ^" ""■ "^ --— -™ -- -- "— -- (- at \do) """""'^•i
dT
this is shown to be - „ dt 59.
do
LAGRANGE'S EQUATIONS. 291
869. To deduce the general equations of motion referred to any
co-ordinates.
Let U be the force-function, then C/" is a function of 6, ^, &c.
and t. The virtual moment of the impressed forces corresponding to
a displacement produced by varying 6 only is -ttt^^- ^^^ ^J
D'Alembert's principle this must be the same as the virtual
moment of the effective forces. Hence
ddT_dT_dU
dt dB' dd~ dd '
^. .' , , d dT dT dU
similarly we nave — -ytt — ^t = tt >
•^ dt d(}i d(f) d(f>
&c. = &c.
It may be remarked that if V be the potential energy we
must Avrite — Ffor U. We then have
d^dT_dT dV^
dt dd' dd^ dB '
with similar equations for (^, -v/r, &c.
In using these equations, it should be remembered that all the
differential coefficients are partial except that with regard to t.
These are called Lagrange's general equations of motion. Lagi'ange only con-
siders the case in wliicli the geometrical equations do not contain the time ex-
plicitly, but it has been shown by Vieille, in Liouville's Journal, 1849, that the
equations are still true when this restriction is removed. In the proof given above
we have included Vieille's extension, and adopted in part Sir W. Hamilton's mode
of proof, Phil. Trans., 1834. It differs from Lagrange's in these respects ; firstly, he
makes the arbitrary displacement such that only one co-ordinate varies at a time,
and secondly, he operates directly on T instead of 2mx'^,
870. To deduce the general equations of motion for Im-
pulsive forces.
Let ZU^ be the virtual moment of the impulsive forces pro-
duced by any displacement of the system. Then from the geo-
metry of the system, we can express S U^ in the form
hU^ = PW-V QB', &c. Then as in
fdT dT \
Art. 368 the virtual moment of the momenta is = {-rah ~~;i~a'''] ^^*
The Lagrangian equations of impulses may therefore be written
do; do; '
with similar equations for j), and i/r, &c.
871. If we compare this equation with the general principle
of Art. 295, viz. that the momenta of the particles just after an
impulse compounded with the reversed momenta just before are
equivalent to the impulse, we see that it will be convenient to
dT
call -Tff the component of the momenta with regard to 6, a name
only slightly altered from that suggested in Thomson and Tait's
Natural Philosophy. More briefly we may say that the ^-com-
dT
ponent of the momentum is -rw, . In the same way we may
d d T d T
define the 6 component of the effective forces to be j j^ — -^ .
372. These equations for impulsive forces are not given by Lagrange. They
seem to have been first deduced by Prof. Niven from the Lagrangian equation
^AT _dT_dU
dt dd' de" dd '
We may regard an impulse as the limit of a very large force acting for a very
short time. Let Iq, t-^ be the times a,t which the force begins and ceases to act. Let
us integrate this equation between the limits t = tQ to t-t-^^. The integral of the first
term is I — 1 ' which is the difference between the initial and final values of --:, .
\_dd'At, dd
dT
The integral of the second term is zero. For ;j^ is a function of 0, rjy, &c. 9', (p', &c.
which though variable remains finite dming the time t-^ - 1^. If A be its greatest
value during thi^ time, then the integral is less than A (f-^ - 1^) which ultimately
.. 1 rdT~\ti dU^ „
vanishes. Hence the Lagrangian equation becomes 7^/ ^ ~dd ' ^ paper
in the Mathematical Messenger for May, 1867.
373. Other expressions for the virtual moments of the momenta and of the
effective forces may be found when T is expressed in terms of the angular velocities
of the bodies of the system instead of the co-ordinates. Thus taking any one body,
if [x,y, z) be the co-ordinates of its centre of gravity, w^;, Wy, oi^ the angular velocities
about rectangular axes meeting at the centre of gravity, M its mass, A, B, C, &c.
its moments and products of inertia, v the velocity of its centre of gravity, then by
Art. 348,
2 r-= Mv' + A w^- -I- i? w/ 4- Cw,2 - 2 2) wy w, - 2£'w,w^ - 2/'w^w^.
Lagrange's equations. 293
The virtual moment of the momenta will then be by Ex. 3. of Art. 349
dT , dT ^ dT^ dT ,„ dT ,^ dT ^ ,
-^, 5x + -^,dy + —,Sz + -^ 59 + ~~ Sep + ,- 5\!/,
dx dy dz aw^ duy doi^
and by Ex. 4 the virtual moment of the effective forces will be if the directions of
the axes axe fixed in s])ace
d dT ^ , d dT ^^ ^
-3- , , 5a3 + &c. + - V- 5i9 + &c.,
dt dx dt awj.
where 5a;, dy, 5z are the hnear displacements of the centre of gravity and 5^, Z(p, d\f/
the angular displacements of the body about the axes of w^., cjy, w^,. If the axes be
inoving we have merely to substitute for the coefficients of 5x,, &c. the corresponding
expressions given in the example just referred to,
374. Before proceeding to discuss some properties of Lagrange's equations, let
us illustrate their use by the foUowing problems.
A body, two of ivhose principal moments at the centre of gravity are equal, turns
about a fixed point situated in the axis of unequal moment under the action of
gravity. To determine the conditions that there may be a simple equivalent
pendulum.
Def. If a body be suspended from a fixed point under the action of gravity,
and if the angular motion of the straight line joining to the centre of gravity be
the same as that of a string of length I to the extremity of which a heavy particle is
attached, then I is called the length of the simple equivalent pendulum. This is an
extension of the definition in Art. 92.
Let OC be the axis of unequal moment, A, A, C the principal moments at the
fixed point, and let the rest of the notation be the same as in Ai't. 349, Ex. 1. Then
2T=A{e'-^ + sin^d^'') + C{(p' + fcose.f,
V— Mgh cos 6 + constant,
where h, is the distance of the centre of gravity from the fixed point, and gravity is
supposed to act in the positive direction of the axis of s. Lagrange's equations will
be found to become
- {AO')-A sin e cos Oxp"^ + Cf {(p' + ^' cos 6) sin ^ = - Mgh sin 0,
|^{a(0'+^'cos^)} = O,
d
7Jt7 (0' + '/'' cos 6) cose + A siu^ Of] = 0.
Integrating the second of Lagrange's equations we have
(j) + i^-'cos 6=n,
where n is some constant expressing the angular velocity about the axis of unequal
moment. Integrating the third we have
d^p
Cn cos 6 + A sin^ 0— =a,
dt
where a is another constant expressing the moment of the momentum about the
vertical through 0.
294 VIS VIVA.
There is an error, sometimes made in using Lagrange's equations, which Ave
should here guard against. If W3 be the angular velocity about 00, we know by
Euler's equations, Art. 230, that w, is constant. If n be this constant, the Vis Viva
of the body might have been correctly written in the form
2r= A (6i'2 + sin2 d^p''^) + Cn'^.
But if this value of The substituted in Lagrange's equations, we should obtain
results altogether erroneous. The reason is, that, in Lagrange's equations, all the
differential coefficients except those with regard to t are partial. Though W3 is
constant, and therefore its total differential coefficient with regard to t is zero, yet
its partial differential coefficients with regard to 6, ^" + £u>/ + CwgS.
We cannot take (w;^, cj.^, Wg) as the independent variables because the co-ordinates of
every particle of the body cannot be expressed in terms of them without introducing
differential coefficients into the geometrical equations. Let us therefore express
^1, ws, wj in terms of 6, '
^^ dT _ duy „ , dT . dw. „ doja , „
be seen by differentiating the expressions for Uj, w^. Also by Art. 326, if N be
! moment of the forces
Substituting we have
the moment of the forces about the axis of C, rr- = N.
d', &c., -v. = -ttt ^'+ j^-, ^" + &c.,
at cLu cLu
subtracting this from the last expression we have
dT dU ^ dU . ,
Integrating, we have the equation of Vis Viva
T-U=li,
where h is an arbitrary constant, sometimes called the constant of Vis Viva.
377. Ex. As an illustration of the application of Lagrange's equations to
impulsive forces, let us consider the example already discussed in Art. 154.
Let X be the altitude of the centre of gravity of the rhombus at any time, then x
and a may be taken as the independent variables.
We have
Let P be the impulsive action between the rhombus and the plane, then the
virtual moment of the impulsive forces is
5 Z7= P5 (a; - 2a cos a) = P5;c + 2a sin a P5a,
The Lagrangian equations are therefore
4(a;i'-a5o') = -P |
4 (fe'-^ + a?) (a-^ - ftoO = 2a P sin a ) °
Now the initial and final values of x' are Xq — — V, x-[^ - 2a sin aw ; those of a'
are o.^ — '^, o/= w. Hence eliminating P we have
,_3 F sina
~ 2 a 1 + 3 sin^ a '
the same result as before.
378. Sir W. R. Hamilton has put the general equations of
Lagrange into another form, which is found to be more con-
venient for the investigation of the general properties of a dyna-
mical system. This transformation may be made to depend on
the following lemma.
Let jTj he a function of 6, 0, &c., 6' , =M, ---^ = v, -r-l = w. Hence show geometricaUy that, if T^-h be the
ad dip df " ' i
, . . dT2 ^, dT^ . dT^ ,,
reciprocal quadnc, -—- =; 6 , —~ = , -^^ = ^ .
du dv dw
380. To express the Lagrangian equations in the Hamiltonian
form.
If a system be acted on by any impulses, the Lagrangian
(7 /T7\
where the bracket implies that 6^ — 0^', (/>,' — <^/, &c. are to be
written for 6', where the bracket
on the right-hand side implies that (P, Q, &c.) are to be written
for {ti, V, &c.) after differentiation.
298 VIS VIVA.
881. If a system be acted on by any finite forces, the La-
grangian equations of motion may be written in tlie typical form
d dL dL _
It dd'~dd~'
•where L = T+ U, so that L is the difference between the kinetic
and potential energies. Since U does not contain {9', ^', &c.) the
equations of transformation may be written in the form
_dL _dT _i^_^£.
'^~dd'~dd'* ^~dcf>'~d<}i"
Also Lagrange's equations may be written in the form
, dL , dL „
Let H be the reciprocal function of L, then these equations
change into
du dv
, dH , dH
which are called the Hamiltonian equations.
When the geometrical equations do not contain the time ex-
plicitly, jT is a homogeneous quadratic function of {& , 0', &c.), and
therefore
ud' + vj> + &c. = 2r.
Hence H=- L + uO' + v({>' +&c. = T-U.
Thus H is the su7n of the kinetic and potential energies, and
is therefore the whole energy of the system.
382. Ex. To deduce the equation of Vis Viva from the Hamiltonian . equa-
tions.
Since H is s. function of (9, 0, &c.), {u, v, &c.) we have, if accents denote total
differential coefficients with regard to the time,
„, dH dH ^, dH , „ dH
dt dd du dt
so that the total differential coefficient of H with regard to t is always equal to the
partial differential coefficient. If the geometrical equations do not contain the
time explicitly, this latter vanishes and therefore we have H—h, where ^ is a con-
stant.
383. Ex. 1. To deduce Euler's equations of motion from the Hamiltonian
equations.
Lagrange's equations. 299
Taking the same notation as in the corresponding proposition for Lagrange's
equations, Art. 375, we have
dT clT
u=^rzr, — Au}. sin (h+Bu.-, cos d>, v = -—,— Cuo,
dd' 1 •^ - ^' (^^' i'
dT
w = — — , = {- Ao}-^ cos (p + Bw.2 sin (p) sin 6 + Cwj cos d.
To express T in terms of [u, v, iv) we must find (wj^, w^, Wg). "We have
, „ . cos
AcJ}^ = M sm o!) + (u cos - iv) —. — - ,
^ / ^ > sin A
Bw., =M cos - (u cos ^ - w^
Also H= I (A w^- + B w.s + awgS) _ ?7.
/777" r777"
As we only require one of Euler's equations, let us use -r- = -v'
The former of these gives Aw-. — -^ + iJw, -j-^ - -r- — - ^ -^ ^
^ '- d(p -^ dip dip dt
..,.,, . i?w, -r, Aoj^ dU r^dw.,
which IS the same as ^ w, — — - - Boo.j —— - T^ = - C' -^^ ,
^ A B d(p dt
and this leads at once to the third Euler's equation in Art. 230. The latter of the
two Hamiltonian equations leads to one of the geometrical equations of Art. 235.
Thus the sis Hamiltonian equations ar-e equivalent to all the thi'ee dynamical and
the three geometrical Euleriau equations.
384 Ex. 1. The position in space of a hody, of mass M, is given by {x, y, z) the
rectangular co-ordinates of its centre of gravity, and (0, cp, ^) the angular co-ordi-
nates of its principal axes at the centre of gravity, as used in Art. 235. If two of its
principal moments are equal and if (f, -q, f, u, v, w} be the (x, y, z, 6, ^x ^, + &c.j ^z(f.,x ^ + &c.J ,
by Ai't. 3G8. Hence
— 'Zi{fx-^x'dx + &c.).
In this case, therefore, if U be the force-function of the conservative forces, F the
function just defined, 05^, ^S^, &c. the virtual moments of the remaining forces,
Lagrange's equations may be written
d_dT _dT_dU dP
diW dd~ de '^dd''^'
with similar equations for (p, ip, &c. The use of this function was suggested by
Lord Eayleigh in the Proceedings of the London Mathematical Society, June, 1873.
The function F was called by him the Dissipation Function.
387. Ex. 1. If any two particles of a dynamical system act and react on each
other with a force whose resolved parts in three fixed directions at right angles are
proportional to the relative velocities of the particles in those directions, show that
these may be included in the dissipation function F. If V^, Vy, V^ be the com-
ponents of the velocities, /^jFj., Ma^y; H^z ^^^^ components of the force of repulsion,
the part of F due to these is ^ S (fj^V^^ + fi^ V,/ +fj.^ F/). This example is taken from
the paper just referred to.
Ex. 2. A solid body moves in a medium which acts on every element of the
sm-face with a resisting force partly frictional and partly normal to the sm-face.
LAGRANGE'S EQUATIONS. 301
Each of these when referred to a unit of area is eqnal to the velocity resolved in its
own direction multiplied by the same constant k. Show that these resistances may
be included in a dissipation function F,
jP= - 1 {o- (w2 + 1)2 + w^) + Au^^ + Bw/ + Cw/ - 2DiOyio^ - 2Eo)^uy^ - 2i^w^Wy},
where a is the area; A, B, &c. the moments and products of inertia of the surface
of the body and (w, v, w) the resolved velocities of the centre of gravity of a:
888. To explain how Lagrange's equations can he used in
some cases when the geometrical equations contain differential
coefficients with regard to the time.
It has been pointed out in Art. 367, that the independent
variables 6, 4>, &c. used in Lagrange's equations must be so
chosen that all the co-ordinates of the bodies in the system can
be expressed in terms of them without introducing 6', cf)', &c.
But when we have to discuss a motion like that of a body rolling
on a perfectly rough surface, the condition that the relative
velocity of the points in contact is zero may sometimes be ex-
pressed by an equation which, like that given in Art. 127, may
necessarily involve differential coefficients of the co-ordinates.
In some cases the equation expressing this condition is integrable.
For example ; when a sphere rolls on a rough plane, as in
Art. 133, the condition is x—a6'=0, which by integration
becomes x— ad = b where b is some constant. In such cases we
may use the condition as one of the geometrical relations of the
motion, thus reducing by one the number of independent vari-
ables.
But when the conditions cannot easily be cleared of differ-
ential coefficients, it will be often convenient to introduce the
reactions and frictions into the equations among the non-con-
servative forces in the manner explained in Art. 886. Each
reaction will have an accompanying equation of condition, and
thus we shall always have sufficient equations to eliminate the
reactions and determine the co-ordinates of the system.
The elimination of the reactions may generally be most easily
effected by recurring to the genei'al equation of Virtual Velocities,
and giving only such displacements to the system as may make
the virtual moments of these forces disappear. Suppose, to fix
our ideas, a body is rolling on a perfectly rough surface. Let
6, , &c. be the six co-ordinates of the body, then by Art. 127,
there will be three equations of the form
L^ = A^e' + B^^'+...^0 (1),
the other two being derived from this by writing 2 and 3 for the
suffix. These three equations express the fact that the resolved
802 VIS vivi.
velocities in three directions of the point of contact are zero. The
equation of virtual velocities may be written
■where TJ is the force-function of the impressed forces. Since the
virtual moments of the reactions at the point of contact have been
omitted, this equation is not true for all variations of 6, (f), &c.,
but only for such as make the body roll on the rough surface.
But the geometrical equations L^, L^, L^ express the fact that
the body rolls in some manner, hence hd, 8(}), &c. are connected
by three equations of the form
A^8e + B^B+...=0 (3).
If we use the method of indeterminate multipliers*, the equa-
tions of virtual velocities will be transformed in the usual manner
into
d dT dT_dU dX dL dL, .
TtdO'~dd~ dO'^^dd'^'^ dd''^'^ d9 ^ ^'
with similar equations for the other co-ordinates ^, t/^, &c. These
joined to the three equations L^, L^, L^ are sufficient to determine
the co-ordinates of the body and X, ix, v.
This process will be very much simplified, if we prepare the
geometrical equations L^, L^, L^ by elimination, so that one dif-
ferential coefficient, as &, is absent from all but the first equation,
another, as 0', absent from all but the second, and so on. When
this has been done, the equation for 6 becomes
dt d9' dd ~ dd ^ d6' ^''^'
Thus \ is found at once. The values of /a and v may be found
from the corresponding equations for ^, -v/r. We may then sub-
stitute their values in the remaining equations.
fS89. The method of indeterminate multipliers is really an
introduction of the unknown reactions into Lagrange's equations.
* If we multiply the geometrical equations (3) by X, ix, v respectively and sub-
tract tliem from (2) we get
yf-^- — -^-x'^-^- — »- —O 561 =
l_dtdd' do dd dd' ^ dd' " dO' J
Now there will be as many indeterminate multiples X, yu., v as there are geome-
trical equations (3) connecting the quantities 5^, 50, &c., i.e. there are as many
multipliers as there are dependent variations. By properly choosing X, /x, v the
coefficients of these variations may be made to vanish, and then the coefficients of
the independent variations must vanish of themselves. Hence the coefficient of
each variation in this summation will be separately zero.
lageange's equations. 303
Thus let i?j, R^, -Rg be the resolved parts of the reaction at the
point of contact in the_ directions of the three straight lines used
in forming the equations L^, L^, L^. Then L^, L^, L^ are propor-
tional to the resolved relative velocities of the points of contact.
Let these velocities be kJj^, kJL^, /CgL^. Then if 6 only be varied
the virtual velocity of B is k^A^SO which may be written
dL
1^1— iff 80- Similarly the virtual velocities of R^ and R^ are
'^2 'Iff ^^ ^^^ '^3 ~iiv ^^' Hence, by Art. 385, Lagrange's equa-
tions are
d dT dT _dU -n dLx „ dZ^ „ dJU,
dtd&~Td~dd'^ "'^^ d9' "^ "'^^ d& + "'^^^ dd' ■
Comparing this with the equations obtained by the method of
indeterminate multipliers we see that A,, jjb, v are proportional to
the resolved parts of the reactions. The advantage of using the
method of indeterminate multipliers is that the reactions are
introduced with the least amount of algebraic calculation, and in
just that manner which is most convenient for the solution of the
problem.
The method of indeterminate multipliers may sometimes be
used with advantage when the geometrical equations do not
contain & , ^' , &c., but are too complicated to be conveniently
solved. Thus if
f{t,d,4>, ...) = ()
be a geometrical equation, connecting 6, ^, &c., we have, as in
Art. 335,
|8^ + |8^ + ...=0.
This may be treated in the same manner as the equations
L^, L.^, X3 in the preceding theory. We thus obtain the equation
d^ dT_dT^dU df
dt dd' de~ d9^ dd'^ '"
with similar equations for ^, ^jr, &c.
390. Ex. Form by Lagrange's method the equations of motion of a homoge-
neous sphere rolling on an inclined plane under the action of gravity.
Let the axis of x be taken down the plane along the line of greatest slope and
let the axis of y be horizontal and that of 2 normal to the plane. . Let [x, y, a) bo
the co-ordinates of the centre of graYity of the sphere, 6, , ^ the angular co-ordi-
nates of throe diameters at right angles fixed in the sphere in the manner explained
in Art. 235. Then, if the mass be taken as unity, the Vis Viva is by Art. 319
2r = x'2 + T/'2 -)- ^2 {(0' + f COS oy- + e"^-y sin" 0^p'%
304 VIS VIVA,
The resolved velocities parallel to the axes of x and y of the point of the sphere
in contact with the plane are to be zero. These conditions will he found to lead
to the equations L^ = x' - ad' cos - a\p' sin 6 sin = 0,
L^=y' + a6' sin <}>-a\p' sin ^cos0=:O.
Also if g he the resolved part of gravity along the plane and C any constant
U=gx+ (7.
The general equation of motion is
dtdq' dq dq dq' dq' '
where q stands for any one of the five co-ordinates x, y, 0, f, I
Jc"^ — ((p'+f cos 6) =
The last equation shows that (p'+ \p'cos0 is constant. From this we infer that
the angular velocity of the sphere about a normal to the plane is constant through-
out the motion. EUminating /* from the two preceding equations and substituting
for xj/" from the last, we find
_ ^ = ^" cos (p + ^" sin ^ sin 9f> - e'(j>' sin (j> f (p'^' sin ^ cos ^ + 6' 4/' cos 9 sin 0.
But this is — . In the same way we find -^ = ^ . Substituting these values
a "" k^ a
of X and ix in the first two of Lagrange's equations, we have
."(l4:)=,,and,"(l+^-:)=0.
These are the equations of motion of a projectile. Hence the centre of gravity
describes a parabola as if it were imder a constant acceleration equal to .,-.,,
tending along the line of greatest slope.
If we had used some of the other expressions for the virtual moments given in
Art. 373, the solution of this problem would have been much simphfied. Thus let
w^, u3y, 0}^ be the angular velocities of the sphere about axes meeting at the centre
of gravity parallel to the co-ordinate axes. Then
2 T^ a;'2 + y'^ + P (w^2 + ^^s + ^^^S)^
and the equations of condition are
x'-ao)y = 0, y'+aux — ^-
Displace the sphere by rolling it along a small arc parallel to the axis of x
through an angle S^. Then we have
d dT ,„ d dT ^„ dU ,„ ,, ^^dwy
- —,add + -j -—d0=-r a80, .'. ax" + F' -rj' = ga.
dt dx dt duy dx dt
Similarly rolling the sphere parallel to the axis of y and twisting it round the
axis of Wj, we have
- ay" + Z;* ^" = 0, and B '-^' = 0.
" dt dt
These, by elimination of Ux, wy, w^, lead to the same result as before.
LEAST ACTION AND VARYINO ACTION.
305
Frinciples of Least Action and Varying Action.
891. Let [q^, q^, q^, &c.) be the co-ordinates of a system of
bodies, and let q stand for any one of these. Let 2 Z" be the vis
viva of the whole system and U the force-function, and let
L = T + U. As before let accents denote differential coefficients
with regard to the time.
Let us imagine the system to be moving in some manner,
which we will call the actual motion. Then q^, q^, &c. are all
functions of t, and it is generally our object to find the form of these
functions. Let us suppose the system to move in some slightly
different manner, i.e. let q^, q^, &c. be functions of t slightly
different from their actual forms. Let us call the motion thus
represented a neighbouring motion. We may pass, in our minds,
from the actual motion to any neighbouring motion by the process
called variation in the calculus of that name. By the fundamental
theorem in that calculus
+
S ^' (^^ - ^'^^)
where the letter % implies summation for all the co-ordinates
q^, q^, &c. and, as implied by the square brackets, the terms
outside the integral sign are to be taken between limits.
The co-ordinates being independent of each other, each sepa-
rate term under the integral sign vanishes by Lagrange's equa-
tions, and we have therefore
J tit
Ldt =
rJT flT
dT '
- im + ^J^rBq
where H is the reciprocal function of L, by Art. 378.
The integral L dt has been called by Sir W. R. Hamilton
J u
the iirincipal function, and is usually represented by the letter S.
If the geometrical equations do not contain the time explicitly,
we have //= T — U. In this case the equation of vis viva will
hold, and if h be the constant of vis viva we have
S f ' L dt = -h {St^ - St,)
+
R. D.
20
306 '- VIS VIVA.
392. Other functions may be used instead of >S'. Let us put
v=s+[mi,
The function Fis called the characteristic function.
If the geometrical equations do not contain the time explicitly,
we have H=h, where h is a constant which may be used to repre-
sent the whole energy of the system. In this case
V=S+h{t,-t,)
= r{T+U)dt+f\T- U)dt
J to -It,
= 2 r Tdt.
J to
The function V therefore expresses the whole accumulation of the
vis viva, i.e. the action of the system in passing from its position
at the time t^ to its position at the time ty
393. In the proof of tliese theorems we have supposed that all the forces are
conservative. If in addition to the impressed forces there are any reactions, such
as rolling friction, which cannot be taken account of by reducing the number of
independent co-ordinates, we must use Lagrange's equation in the form
d dL dL _
dt dq' dq
where, as explained in Art. 385, PSq is the virtual moment of these reactions coiTe-
sponding to a displacement dq. In this case the quantity under the integral sign
will not vanish unless the variations are such that
:^P[Bq-q'5t)=0.
Now q being the value of any co-ordinate in the actual motion at the time t,
q+Sq is its value in a neighbouring motion at the time t + St. But q'8t is the
change of q in the time 5t, hence q + dq- q'St is the value of the co-ordinate in the
neighbouring motion at the time t. The neighbouring motions must therefore be
such that the virtual moments of the reactions corresponding to a displacement of
the system from any position in the actual motion into its position in a neighbour-
ing motion at the same time is zero. With this restriction on the variations, the
two equations which express the variations of S and V wUl still be true.
394. The two fundamental equations, giving the values of
BS and SV, will be found to lead to many important theorems
which we shall now proceed to consider.
Let us call the positions of the system at the times ^o and t^ the
initial and terminal positions, and let us suppose these fixed, so
LEAST ACTION AND VARYING ACTION. 307
that the actual motion and all its neighbouring motions are to
have the same initial and terminal positions. In this case hq
vanishes at each limit, and the two fundamental equations take
the form*
J t-o
28 [*' Tdt = {t- %) Bh,
* We may easily establish these theorems without the use of Lagrange's
theorems. Let {x, y, z) be the rectangular co-ordinates of any particle and let m be
the mass of this particle. Let X, Y, Z be the components of the impressed accele-
rating forces on it. Then
L = ^2m (ic'S + y'2 + 2^) + U,
and by the fundamental theorem in the Calculus of Variations
'f>^' (£ - 1 ,f ) <»-^-") ^'- [^ g (»' -'»•)];; •
If we substitute for L and remember that T is a homogeneous function of
x', y', z', this becomes
5 f 'Ldt^[{U-T)5t + I.mx'8x]' + f 'i:m(X-x") {Sx-x'dt)dt.
"^to to "^ to
If we consider the positions of the system at the times tfy and fj to be given, Sx
is zero in the part taken between limits.
If the time of transit be given it is unnecessary to vary the time. Putting dt = 0,
the part under the integral sign vanishes by the principle of virtual velocities. The
rti
part outside the integral sign as also zero and therefore 5 / Ldt = 0.
If the time be varied, Sx - x'5t is the projection on the axis of x of the displace-
ment of the particle m from its position in the actual motion at time t to its position
in a neighbouring motion at the same time. Hence the part under the integral
sign vanishes as before by the principle of virtual velocities. Let us suppose that
the geometrical conditions do not contain the time explicitly, then T - U= h and
L — 2T-h. The equation then becomes
25 [ 'Tdt- [h{ht)-f' = [ - hStf .
*' ^0 io to
'to
From the general value of the variation in Cartesian co-ordinates we can also
deduce the values of 56' and oV given in the text. For the term l,mx' is clearly the
dT
virtual moment of the momenta, and this by Art. 368 is -r-, 57. The method
dq
followed in the text seems however to be preferable.
Lagrange has given a general view of his transformation from Cartesian co-
ordinates which seems worthy of notice. Let L be any function of x, x', &c.
t/, »/', &c. and of (, and let the variables x, y, fee. be transformed into others
20—2
rti
If li be given we have 5 / Tdt = 0.
308 VIS VIVA.
where it has been supposed that the geometrical equations do not
contain the time explicitly.
If the time of transit of the system from its initial to its
terminal position be also given, we have St^= St^, and therefore
Hence Z cZi is either a maximum or a minimum. It cannot
be the former, since by causing the bodies to take circuitous
paths we may make it as large as we please. It is therefore a
minimum.
If the constant Ti be given, or which is the same thing (since
the terminal position is given) if the energy of the system be
given, we have Bh = 0, and therefore S Tdt = 0. We may
J ti
now infer the two following theorems.
Let any two positions of a dynamical system be given, the
actual motion is such that i Tdt is less than if the system were
constrained, without violating any geometrical conditions, to
move in some other manner from the one position to the other
with the same energy; these other motions being such that,
throughout, T is the same function of the co-ordinates and their
differential coefficients.
This is called the principle of Least Action.
3i, q^, &c. by writing for x, y, &c. any functions of g-^, q^, SiC and of t. The func-
tion L is tlius expressed in two ways, and by comparing tbe two values of 5 / ^Ldt
given by the Calculus of Variations, we see that the integral of
(S-*°-)^'-('4-*"-)M
may be completely found. Hence this expression must be a perfect differential
with regard to t, quite independently of the operation 5. But this cannot be unless
it vanishes, because it contains only the variations 5a;, 5g, &c. and not the
differential coefficients of these variations. We have therefore the general equa-
tion of transformation
\dx dtdx' ') ~ \dq dt dq' 'J
where the S implies summation for all the variables x, y,,.&(i. or q^^, q„, &c.
lix, y, &c. be Cartesian co-ordinates the left-hand side of this equality vanishes
by virtual velocities. Hence S ( -^ — &c. J S^ must also vanish. The ^'s being all
independent, we are led to Lagrange's equations.
LEAST ACTION AND VARYING ACTION. 309
In the same way if the system moves in the varied motion,
not with the same energy, but in the same time, from the one
given position to the other, then j Ldt is a, minimum.
395. Maupertuis conceived that lie could establisli a priori by theological argu-
ments, that all mechanical changes must take place in the world so as to occasion
the least possible quantity of action. In asserting this, it was proposed to measure
the action by the prodiict of velocity and space ; and this measure being adopted,
mathematicians, though they did not generally assent to Maupertuis' reasonings,
found that his principle expressed a remarkable and useful truth, which might be
established on known mechanical grounds. Whewell's History of the Inductive
Sciences, Vol. ii. p. 119.
396. Conversely, from either of these theorems we may deduce
the motion of any system, by making Ldt or \ Tat a minimum
according to the rules of the Calculus of Variations*. That this
* Lagrange's equations are the ordinary equations supplied by the Calculus of
Variations when we make / Ldt a minimum under known conditions. Sh" W.
Hamilton put these equations under a form (see Art. 381) which is very useful in
Dynamics. It is an interesting question to determine what is the corresponding
transformation when Z is a function of differential coefficients higher than the first.
This was considered by Ostrogradsky in a Memoire sur les equations differentielles
relative ati probleme des Isoperimetres, published in the Memoirs of the Academy of
Sciences at St Petersburgh in 1850. The Memoir is rather difficult on account of
the immense length of the algebraical transformations. The following short ac-
count may therefore prove useful.
Let I; be a function of t and of m variables, of which q is any one, and let it be
a function of the first n differential coefficients of q with regard to t.
d^q
Let Qk stand for the partial differential coefficient of L with regard to — ^ , and
let 4 = Q*.-QVi + Q'V2-
where, as usual, accents denote differential coefficients with regard to t, and let h
accents be denoted by {h). The relations between these variables are, therefore,
and so on up to Q„_i = -^^^ - Q'^
— dL
and the last is ^w~j~u)
By the principles of the Calculus of Variations, the minimum is given by the
typical equation Qo = 0. [When
\ m-
310 VIS VIVA,
process will really lead to the equations of motion may be seen by
simply reversing our steps. Thus granting that S j Ldt= under
the known conditions, we have
When L contains no differential coefficient above the first, Sir W. Hamilton
eliminated the m first differential coefficients typified by q' by introducing m new
variables typified by Q,-^ = -—. Let us in the same way eliminate the highest differ-
ential coefficients typified by g'"' and introduce instead the m new variables typified
by^. Let
H==L-:E{Q,q'+Q,q"+ ...+ Q,r').
where the S refers to summation for all the ^'s. Let 2^"' be found from the equa-
tion Q„ = : and let its value be substituted in this expression for H so that H is
" dqW
now a function of t, q, q'...q^^~'^\ Qd Q^-'-Qn- Since L was originally a function of
t, q, 5'... 2'"' it is now a function of i, q, q'.-.q^'^"^^ and Q^.
We have by differentiation
^3. -3(^+1)=- I oW (2),
provided ^ + 1 is not n. In that case
clH dL dq'-''^ _ dq^"^
but the first and third of these terms destroy each other, so that the theorem (2) is
also true when /c + 1 = ?i. Also
rig»> " dq^^> "*" d2("> dq^^ ^^ ^" dq^"^ '
Here the second and fourth terms destroy each other. Tlie first and third, by
d -
dt*
(1), become Q'^+j or -j Q^+i' Thus all the equations may be written in the typical
HamUtonian form
dQ,+, '^^
dH _ d~
which are true for all values of k from h = to Jc=n-1. Thus there are 2n equa-
tions corresponding to each q.
We may show in the same way as in Ai-t. 382, that the total differential coeffi-
cient of H with regard to t is equal to its partial differential coefficient. So that
when L, and therefore H, are not explicit functions of t, we have as one integral
if =/i, where A is a constant. Writing this at length it becomes
L = -E(Q,q'+Q.2q"+-)+h
which is the integral continually used in the Calculus of Variations. We see that
this integral corresponds to the equation of Vis Viva in Dynamics.
LEAST ACTION AND VARYING ACTION. 811
for all variations. The Bc/s being all arbitrary and independent,
each coefficient under the integral sign must vanish separately,
and this leads to the typical Lagrange's equation.
Ex. 1. There is another method of deducing Lagrange's equations from the
principle of Least Action which is worthy of notice. We are to make / ' Tdt a
minimum, subject to the condition T- U=h. By Lagi'ange's rule in the Calculus
of Variations we are to make
sf{T+\iT-U-h)}dt = 0,
without regard to the given condition, and afterwards make X such a function of t
that the given condition is satisfied. This will be found an excellent exercise in
the Calculus of Variations.
The solution maybe indicated as follows. Putting W=T+\{T-U)we have
with the same notation as before
a/w.=[,«]+./(f -If ) ft-,*,.,, [zf (,,-,*,] ,
and this must be equal to hd / \di. The integrals are to be taken between the
limits, which are omitted for the sake of brevity.
First, let us consider the part outside the integral sign. The initial and final
positions being given Sq = 0, and we have
, dW
W6t - S -5-. q'ot = A5 f\dt = h\5t.
do' ^ J
dq
This equation is satisfied by 5( = 0, but since the time of transit is not to be the
same in the actual and varied motions, this factor must be rejected. Also Tia a,
dT
homogeneous function of the g's, hence "L ——,q' = 2T. Substituting for W its
value and using this equation we find (1 + X) T+\U+h\ — 0. But X is such that
T- 11=11, hence (1 + 2X) 7=0 and .-. X = - -.
Next, let us consider the part under the integral sign. By the rule in the
Calculus of Variations this gives at once the typical equation
dq dt dq'
Substituting for W we have the typical Lagrange's equation.
Ex. 2. If we add to the conditions given in the principle of Least Action, the
condition that the time of transit is to be always the same, show that the minimum
does not in general lead to Lagrange's equations. Following the notation of the
1 A
last Article, show that the minimum for a given time is determined by X= -;; + ^,
SI 2 VIS VIVA.
where A is an arbitrary constant to be chosen so that the constant of vis viva has
1
2*
its given value, whUe the absolute minimum is determined by \ = - jr
397. When the geometrical equations do not contain the time explicitly the
symbol H. or li may be vised to express the energy of the system. If we represent
the energy by JS, Sir W. K. Hamilton's fundamental equation may be written
25/ Tdt^t5E+ ^j-,5q
This equation has been applied to the motion of a system of bodies oscillating
in such a manner that the motion repeats itself in all respects at some constant
interval. Let this interval be i. Suppose that some disturbance is given to the
system by the addition of a quantity of energy 5^. Let the system be such that ,
the motion still recurs after a constant interval, and let this interval be now
i + 5i. The symbols of variation in Hamilton's equation may be used to imply a
change from one kind of motion to the other. If the time t be taken equal to the
period i of complete recurrence, the initial and terminal states of motion are the
same and therefore the last term vanishes when taken between the limits. The
equation reduces to 25 / Tdt — idE. Let T^ be the mean vis viva of the system
during a period of complete recurrence of the motion, then / Tdt — iT^. We there-
fore have 1^=2 ^^^>.
This equation may be put into another form. Let P„j be the mean potential
energy of the system dm'ing a period of complete recurrence ; then we have
5F^ + dT^=5E,
5P„i-5r„j = 2?'„^-T ,
which serve to determine the change in the mean potential and kinetic energies
when any additional energy 5E is added to the system.
These or equivalent equations have been applied by Bolzman, Clausius and
Szily to the Dynamical Theory of Heat. The papers of the two latter are in
various numbers of the Fhilosophical Magazine extending from 1870 to the present
time. The second of the equations above written may be called Clausius' equation.
398. Ex. 1, If the period of complete recurrence of a dynamical system be not
altered by the addition of energy, prove that this additional energy is equally dis-
tributed into potential and kinetic energy.
Ex. 2. A quantity of energy dE is communicated to a system whose mean
semi-vis-viva during a period of complete recurrence is T^. This is repeated
continually, so that at last the mean vis viva and the period of complete recurrence
are the same as at first. Prove that ~-=0.
This example is due to M. Szily, and is important in the Dynamical Theory of
Heat.
GENERAL EQUATIONS OF MOTION. 31 J
On the Solution of the General Equations of Motion.
899. Sir W. R. Hamilton has applied liis fundamental theo-
rem expressing the variation of the Principal and Characteristic
functions to obtain a new method of solving dynamical problems.
Let (a^, a/, a^, a/, &c.) be the values of [q^, q[, q^, q^, &c.)
when t = t^, and let T^ be the same function of (a^, a,', &c.) that
T is of {q^, ql, &c.). We have then when t is written for the
upper limit
It is clear that both S and V may be regarded as functions of
the time and the initial conditions of the system of bodies, i.e. we
may regard either of these quantities as a function of t, a^, a^, &c.,
a/, a/, &c. Also the co-ordinates q^, q^, &c. are functions of t and
the same initial conditions. Though these functions are in general
unknown, yet we can conceive the initial velocities a/, a^', &c.
eliminated, so that 8 and Fare now functions of t, and a^, a^, &c.,
^v 9i> ^'^' ^^® co-ordinates of the system at the times t^ and t.
Let S be thus expressed, then, by the equation for BS, we have
the typical equations
dS^dT dS^_dJ\
dq dq ' da da
Since T is not a function of €[' , the first of these equations
contains no differential coefficient of a co-ordinate higher than the
first. This equation, therefore, represents typically all the first
integrals of the equations of motion.
Since T^ contains only the initial co-ordinates and the initial
velocities, the second equation has no differential coefficient of
any co-ordinate in it. This equation, therefore, represents typically
all the second integrals of the motion.
Besides these we have the two equations
dry jj. do _ jj
Tt^~ ' dt,^ "
where, if the geometrical equations do not contain the time ex-
plicitly, we may put h for //, h being a constant. In this case
314 VIS VIVA.
the integrals may be used to connect the constant of vis viva with
the constants {a, a , &c.).
Comparing Art. 394 with these results we see that 8 is such
a function, that all the equations of motion and their integrals are
included in the statement that hS is a known function of the
variation of the limits. If we keep the limits fixed, we get
Lagrange's equations; if we vary the limits we get the integrals,
400. In just the same way, if we regard q^, q^, &c. as
functions of t, the initial co-ordinates and their initial velocities,
we may eliminate t also by means of the equation
dT
H=-U-T+X^q',
aq -^
which reduces to 11= T— U when the geometrical equations do
not contain the time explicitly.
Let us suppose V to be expressed in this manner as a function
of the initial co-ordinates, the co-ordinates at the time t, and of
H. Then, by the equation for hV,
dV_dT dV__dT\ dV_
dq dq ' da da ' dH
Supposing V to be known, the first of these equations gives in
a typical form all the first integrals of the equations of motion.
The second supplies as many equations as there are co-ordinates
($'i' 9.iy ^c-)- When the geometrical equations do not contain the
time explicitly these do not contain t, but they all contain Ji.
One of them, therefore, reduces to the relation between this
dV
constant and the constants {a, a, &c.). The equation -jr = t will
give another second integral of the equations of motion containing
the time.
401. Ex. liQ^ f (Zqp' -^ ff) dt, where p = Y" pi'ove that SQ = lH5t + 'SqSp]* .
Thence show that if Q be expressed as a function of the initial and terminal
components of momentum, viz. (Jj, b^, &c.) and (p^, jj^, &c.) and of the time, then
^ = (7, ^=-a, ^ = H. This result is due to Sir W. E. Hamilton.
dp ^ db dt
402. Ex. 1. A homogeneous sphere of unit mass rolls down a perfectly rough
fixed inclined plane. If the position of the sphere is defined by the distance q of
the point of contact from a fixed point on the inclined plane, show that
where g is the resolved part of gravity down the plane and =I>2> ^^- ^^^ eliminate q^', q^, &c. Let the
reciprocal function of H thus found be
JI=F{q^,p^, q^,p„ &c.).
F[q,,
316 VIS VIVA.
But Pj = -v— , p^ = -J— , &c. and H= — -^. Hence S must
satisfy the equation
dS r^f dS dS „ \ ^
in J ust the same way, jo^ = -j— , p^ = -^ , &c. and the equa-
tion of vis viva gives H=h. Hence Fmust satisfy the equation
dV dV . \ ,
^,9„^^,&c.J=/.
If we consider the initial value of T, we shall have another
equation of a similar form with a^, a^, &c. written for q^, q^, &c.,
and t^ for t. It is necessary that the functions should satisfy both
these equations.
Ex. Taking tlie same circumstance of motion as in Ex. 1 of Art. 402, show
that the differential equation to find ^ is -j ( -^ J - gq — h. Integrate this equa-
tion and thence find the motion.
404. When there are several independent variables, the equation to find V is
of the form
where (B^-^, B^^, &c.) are functions of q-^, q.^, &c. only. The left-hand side of this
equation, by Ex. 2 of Art. 384, may be written in the form of a determinant. We
dV dV
have only to replace u, v, &c. by their values ■— - , - — , &c.
We thus have, in general, a partial differential equation to find V, and
Sir W. Hamilton gave no rule to determine which integral is to be taken. This
rule has been supphed by Jacobi in the following proposition.
Suppose a solution to have been found containing n-1 constants^ besides h, and
the constant lohich may be introduced by simple addition to the function V. These
need not be the initial values of q-^, qa ■ • • ^ut may he any constants whatever. Let
them be denoted by a^, ar,...a^_^, so that
V = f (qj, q2...qn, «!, "a • ■ • an-i) + an (2).
Then the integrals of the dynamical equations ivill be
rlr^'-'^'-str^"-' <"•
aT='+' w-
* An integral of a partial differential equation has been called by Lagrange
"complete," when it contains as many arbitrary constants as there are independent
variables. It is implied that the constants enter in such a manner into the inte-
gral that they cannot by any algebraic process be reduced to a smaller number.
For instance, if two of the constants enter in the form aj + Cj, they amount on the
whole to only one.
GENERAL EQUATIONS OF MOTION.
317
where p^, ^2---^n-\ <^nd e are n new arbitrary constants. A7id the first integrals of
the equations maij he xvritten in the form
df dT df dT ^
5 — — i — /. T — = 5 — n O-'C. =a''C
dq^ dq^' dqg dq^
Let the expression for the semi- vis-viva he
5).
^=2^1l'?l'^ + ^12?lV + &C.
•(6),
where the coefficients A-^-^, A-^^, &c. are functions of 5^, q^, &c. only.
Let Q-^, Q2...Q„he siich functions of g,, q^.-.^n ^^^ ^^^ constants, that they
may satisfy identically the n equations
dq^
df
dq,
&C.
=A.,^Q^+A.^,Q.,+ ...
= &o.
,(7).
Then from the mode in which the differential equation to find V has heeu
formed, in Art. 403, we know Q^, Q.^ will also satisfy identically the equation
V + h = ~A,,Q^^ + A,,Q,Q^ +
(8).
Firstly, we shall prove that Q^ = g/, Q^ — q^, &c., it will then follow that tha
equations (5) are satisfied. Differentiating equations (3) and (4), we have
d^f dq^ ^ d'-f dq^ ^ ^ ^^-^
' " , dt
da^ dq-j^ dt da^ dq^
&c.
d^f dq, , dj dq., ^
dh dq^ dt dh dq^ dt
(9).
These are the equations to find -4^ , -—-, &c.
dt dt
But differentiating (7) with regard to a^, we have
d'f
■-A.
1 — ■"-"12 "7 +
da^ da^
.(10),
da^ dq^
dai dq^ '^ da^ da^
&c. = &c.
because iji,^i2) ^^- ^^^ ^<^* functions of the constants. Multiplying these equations
by Q^, Qj..., and adding, we get
Since the equation (8) is an identical equation the quantity in hractets on the
right-hand side does not contain a^, being equal to U+h. Hence the expression on
the left-hand side vanishes. Thus we have an equation connecting Q^, $„... ex-
actly similar to the first of equations (9). Similarly by differentiating equations (7)
with respect to Oj...^ successively, we shall have equations similar to the second,
(fee. and last of equations (9). We have therefore exactly the same equations to
find <5i, Q,... and 7/, q^'.... Hence Q.y = q{, Q^ = q.I, &c.
318 VIS VIVA.
Secondly, we shall prove that (3) and (4) satisfy the equations of motion. Let
us consider the equation of Lagrange *,
cl dT clT _dU
dt dq^ dq^ dq^ '
When q^', (//...g',/ have been expressed in terms of q^, q^.. q^ aiid the constant
by means of equations (5), we have identically
C/ + A = 2 4 n?i'2 + ^is^iV + • • •
Therefore, differentiating partially,
But differentiating (5) written at length, with regard to g^, we have
dql dql _^_ ,^i_ >dA^._
'^^^dq,^^'dq,^-'dq,'^ ^'dq^ ^'dq, -
, da' , dq^' d^f ,dA^-. ,dJ„
2' dq-^ ^^ dq^ dq^ dq^ ^^ dq-^ dq^
&C. = &C.
Hence, substituting.
dU_ dY ,,J^., ^dA,.,^_dA^
* We may also show that the Jacobian integrals satisfy the Hamiltonian form
of the equations of motion. The peculiar relation of the differential equation to
the Hamiltonian function H adapts it to this process. If we substitute the value
of F given by (2) in the differential equation (l),the result is an identical equation.
dV
Differentiating this identity with r-egard to each of the n constants and replacing —
^ .-1 .; d'H dH dH dH r. ^ ^ n
by p, we get n equations of the form -, — V + t ^ — V + . . . = to find
dp^ dq^ da dp^ dq^ da.
~,r- . -r- J &c. These are the same as the equations (9) in the text, hence
— — = fl'. Again, differentiating H partially with regard to q^, we have
dp
dH dH dH dH dj ^
1 — H — ^ + ...=0.
dq^ dp-^dq^'^ dp,dq^dq^
But all the terms of this equation except the first are together equal to the total
differential coefficient -^ . Hence -r- — — 4^ . The investigations of Hamilton and
dt dqi dt
Jacobi apply to a system of free particles mutually attracting each other referred to
jCartesian co-ordinates. In the text the reasoning has been applied to a system of
bodies referred to any co-ordinates.
GENERAL EQUATIONS OF MOTION, 319
Next let lis consider the expression for T: we see that the partial differential
coefficient
dT 1 dA.. ,„ dA-,^ , , ,
j.—^-dis^^:'^''-^^-
is the same as the latter part of the expression for -=— .
Also — ; = V^ , therefore taking the total differential coefficient, we have
dq-^ dq^
d_dT__^ , dj ,
dt d^' ~ dq;^ ^1 "^ dq^dq.^ ii +■■■■>
which is the same as the first part of the expression for -— . Hence the differen-
tial equation of motion is satisfied.
We have also, since T is homogeneous,
„^ dT , dT , df , df , df
dq-^' ^^ dq^' " <^
But when Cy, c^, ... are considered as variables, the equations (1) are the integrals
of the differential equations (3). Hence repeating the same process, we have
dc^ _ dc-^ dH dCy dH dc^
dt dp dq dq dp "' dt
dc-y^ dK dcy dK
dp dq dq dq
— H^ = &C.
dt
where the differential coefficients on the left-hand side are total, and those on the
right-hand side partial.
Hence, using the identities (4), we get
cZcj dc^ dK (Zcj dK ,-■.
dt dp dq dq dp
with similar expressions for -~ , &o.
dc
If K be given as a function of p, q, &c. and t, we have -— i , &a. expressed as
functions of p, q, &c. and t. Joining these equations to those marked (1) we find
Cj, Cg ... as functions of t. If K be given as a function of Cj, Oj, ... and t we may
continue thus,
dK__^d(^ (IKdc^ dK _dKdc^ dK dc^
dp dc^ dp dc^ dp "' dq ~ dc^ dq dc.-, dq
dc
Substituting in the expression for — 1 , we get
dt l_dq dp dp dq J dc^ Ldq dp dp dq J dc^ '"'
where the S means summation for all values of ^, q, viz. p-^, q-^, p.^, q^, &c.
Since by hypothesis Cj, Cg, ... are supposed expressed as functions oi p.^, q^, &c.
and t, these coefficients may be found by simple differentiation. It will, of course,
be more convenient to express them in terms of c^, Cj, &c. and t by substituting
for p^, q^, &c. their values given by the integrals (1).
407. On effecting this substitution it will bo found that ( disappears from the
expressions. This may be proved as follows. Let A be any coefficient, so that
GENERAL EQUATIONS OF MOTION. 321
4 = 2!-^-^ — r^-r^ , we have to prove that A being regarded as a function
L dq^ dp dp dciJ
d . A
otp-^, q^, &c. and t, the total differential coefficient — ^ is zero. Now
d.A dA dA , dA .
The letters p^, q^, &c. enter into the expression for A only through c^ and c^.
d A
Let us consider only the part of — ^ due to the variation of c^, then the part due
to the variation of Cg may be found by interchanging c^ and c^, and changing the
d A
sign of the whole. The complete value of -^ is the sum of these two parts.
d.A
The part of — ^ — due to the variation of Cj ig
^ rrfCg f d dc^ d\ dH d^c^dH ) _ dc^ {d^de-^ d^c-^ dA d'^c^ dll
^ \_dp \dq dt dpdq dq dq^ dp '") dq (dp dt dp'^ dq dpdq dp
dc
If we substitute for -^ its value given by the identity (4), we get
^ rde^ \ dc^ d^H dc^ d^^H ) dc^ [dc-^ d'^E dc^d^H\~\
\_dp (dp dq^ dq dpdq) dq' [dp dpdq dq dp^ ) A'
If we now iaterchange c^ and Cg we get the same result. Hence when the two
d.A
parts of ~ — are added together, tke signs being opposite, the sum is zero.
[etc dc dc dc I
-— -r^ — T^ -T^ I , where the S means summa-
dq dp dp dq J
tion for aU the values of p, q, be represented shortly by (Cj, c^). Then in any
dynamical problem if K be the disturbing function, the variations of the parameters
c^, c^, ... are given hy -^ — (c^, c^ ^— + (Cx > C3) -^ + . . ., where all the coefficients are
fimctions of the parameters only and not of t.
This equation may be greatly simplified by a proper choice of the constants
Cj, Cj, ... In the Mecanique Analytique of Lagrange, it is shown that if the con-
stants chosen be the initial values of 2>i. P^f- ^^^ ^v ?2>-"> ^^' "' ft 7)"- ^^^
\, fM, V, ... respectively, then the equations become
da_ _ dK d^_ dK . -^
di~~d\' 'di~~~dij.' *l
dX_ dK dfj,_ ^ ^ i
dt~' da' dt~ rf^ ' J
It is assumed in the demonstration that K is a function of q^, q„,... only. This
simplification has been extended by Sir W. HamUtou and Jacobi to other cases, but
for this we must refer the reader to books which treat on theoretical dynamics.
409. It follows from the investigation in Art. 407, that if two integrals of a
dynamical problem be found, viz. Cj—a, c^=§, where Cj and Cj stand for some
functions of p^, q^, p„, q^, ... and t, and a and ^ are constants, then (Cj, c„) is also
constant. So that (cj, c^) = y, where 7 is a constant, is either a third integral of
the equations of motion or an identity. If it is an integral it may be either a
new integral or one derivable from the two c^ and c^ already found.
R. D. 21
322 VIS VIVA.
EXAMPLES*.
1. A screw of Archimedes is capable of turning freely about its axis, which is
fixed in a yertical position : a heavy particle is placed at the top of the tube and
runs down through it ; determine the whole angular velocity communicated to the
screw.
Result. Let n be the ratio of the mass of the screw to that of the particle,
a=the angle the tangent to the screw makes with the horizon, h the height
descended by the particle. Then the angular velocity generated is
=x/:
2(//i cos^ a
a^ {n + 1) (tc 4- sin^ a) '
2. A fine circular tube, carrying within it a heavy particle, is set revohdng
about a vertical diameter. Show that the difference of the squares of the absolute
velocities of the particle at any two given points of the tube equidistant from the
axis is the same for all initial velocities of the particle and tube.
3. A circular wire ring, carrying a small bead, hes on a smooth horizontal
table ; an elastic thread the natural length of which is less than the diameter of
the ring, has one end attached to the bead and the other to a point in the wire ;
the bead is placed initially so that the thread coincides very nearly with a diameter
of the ring ; find the vis viva of the system when the string has contracted to its
original length.
4. A straight tube of given length is capable of turning freely about one ex-
tremity in a horizontal plane, two equal particles are placed at different points
within it at rest, an angular velocity is given to the system, determine the velocity
of each particle on leaving the tube.
5. A smooth cncular tube of mass M has placed within it two equal particles
of mass m, which are connected by an elastic string whose natural length is | of
the circumference. The string is stretched until the particles are in contact and
the tube is placed flat on a smooth horizontal table and left to itself. Show that
when the string arrives at its natural length, the actual energy of the two particles
is to the work done in stretching the string as
2 (M2 + Mm + m2) : {3f+ 2m) (2if + m).
6. An endless flexible and inextensible chain in which the mass for unit length
is ju. through one continuous half and p.' through the other half is stretched over
two equal perfectly rough uniform circular discs (radius a, mass M) which can turn
freely about their centres at a distance h in the same vertical line. Prove that the
time of a small oscillation of the chain under the action of gravity is
=V-
4- (Tra + h) {fj, + //)
7. Two particles of masses m, m' are connected by an elastic string of length a.
The former is placed in a smooth straight groove and the latter is projected in a
* These examples are taken from the Examination Papers which have been set
in the University and in the Colleges.
EXAMPLES. • 323
direction perpendicular to the groove with a velocity V. Prove that the particle m
will oscillate through a space , , and if m he large compared with m' the time
m + m
of oscillation is nearly -=- ( 1 - j- J .
8. A rough plane rotates with uniform angular velocity n about a horizontal
axis which is parallel to it but not in it. A heavy sphere of radius a being placed
on the plane when in a horizontal position, rolls down it under the action of
gravity. If the centre of the sphere be originally in the plane containing the
moving axis and perpendicular to the moving plane, and if x be its distance from
this plane at a subsequent time t before the sphere leaves the plane, then
x= 7=: -^ - 84a - 60c (e^ ' -e ^^ ) - r- — , sm 7i«,
24^35 V "' J 12 71-
c being the distance from the axis to the plane measured upwards.
9. The extremities of a uniform heavy beam of length 2a slide on a smooth
wire in the form of the curve whose equation is r = a (l-cos(9) the prime radius
being vertical and the vertex of the curve downwards. Prove that if the beam
be placed in a vertical position and displaced with a velocity just sufficient to
1 ( \/^ f iJ^f )
bring it into a horizontal position tan ^ = - |e^ si - e~^ za l, where d is the angle
throiTgh which the rod has turned after a time t.
10. A rigid body whose radius of gyration about G the centre of gravity is h, is
attached to a fixed point C by a string fastened to a point A on its surface. CA{=h)
and AG {—a) are initially in one line, and to G is given a velocity F at right angles
to that line. No impressed forces are supposed to act, and the string is attached
so as always to remain in one right line. If 6 be the angle between AG and AO
n
f(Ld\ 2 Yi^^~ ^"* ^^^' 2
at time t, show that (^-) = v^ „ ■ . " m ,., and if the amplitude of d, i.e.
2 sin~i — — =: be very small, the period is
2Va6 ' V^a{a + l)
11, A fine weightless string having a particle at one extremity is partially
coiled round a hoop which is placed on a smooth horizontal plane, and is capable
of motion about a fixed vertical axis through its centre. If the hoop be initially at
rest and the particle be projected in a direction perpendicular to the length of the
string, and if s be the portion of the string unwound at any time t, then
m + /x
where b is the initial value of s, m and y« the masses of the hoop and particle, a the
radius of the hoop and V the velocity of projection.
12. A square formed of four similar uniform rods jointed freely at their ex-
tremities is laid upon a smooth horizontal table, one of its angular points being
fixed : if angular velocities w, w' in the plane of the table be communicated to the
two sides containing this angle, show that the greatest value of the angle (2a)
between them is given by the equation cos 2a = -7: „—,!,- .
w + w -
21—2
324 ' VIS VIVA.
13. Two particles of masses m, m' lying on a smooth horizontal table are con-
nected by an inelastic string extended to its full length and passing through a small
ring on the table. The particles are at distances a, a' from the ring and are pro-
jected with velocities u, v' at right angles to the string. Prove that if rnv'-a^ — mlv'^a'^
their second apsidal distances from the ring wiU be a', a respectively.
14. If a uniform thin rod PQ, move in consequence of a primitive impulse
between two smooth curves in the same plane, prove that the square of the angular
velocity varies inversely as the difference between the sum of the squares of the
normals OP, OQ to the curves at the extremities of the rods, and y\ of the square
of the whole length of the rod.
15. A small bead can slide freely along an equiangular spiral of equal mass
and angle a which can turn freely about its pole as a fixed point. A centre of
repulsive force F is situated in the pole and acts on the particle. If the system
start from rest when the particle is at a distance a, show that the angular velocity
Fdr
of the spiral when the particle is at a distance Tc from the pole is
7wA;2(i + 2cot2a)
where mh^ is the moment of inertia of the spiral about its pole.
16. The extremities of a uniform beam of length 2a, slide on two slender rods
without inertia, the plane of the rods being vertical, their point of intersection
fixed and the rods inchned at angles j and - j to the horizon. The system is set
rotating about the vertical line through the point of intersection of the rods with
an angular velocity w, prove that if d be the inclination of the beam to the vertical
at the time t and a the initial value of d
,fdey (3 cos2 a + sin2 a)2 , ,^ , • n . ^ ^9 , ■ • a\
4—1 -Y— ., ^ ■ ., 1 to°= (3cos«a4-sm2a)w3 + _!^/suict_giii^),
\dtj 3cos2^ + sin^^ '■ ' a^
17. A perfectly rough sphere of radius a is placed close to the intersection of
the highest generating lines of two fixed equal horizontal cylinders of radius c the
axes being inclined at an angle 2a to each other, and is allowed to roll down be-
tween them. Prove that the vertical velocity of its centre in any position will be
sin a cos rf) 1 -^-^ — ^v ^^-^ I . where ^2 - ^ »
jCg - - also form a geometrical progression. In this way we find - which is the
value of X corresponding to the position of equilibrium.
The position of equilibrium being known, the interval between two successive
passages of the system through it is also a known function of a and b, and thus a
third equation may be formed.
Ex. 3. A body performs rectilinear vibrations in a medium whose resistance is
proportional to the velocity, under the action of an attractive force tending towards
a fixed centre and proportional to the distance therefrom. If the observed period
of vibration is T and the co-ordinates of the extremities of three consecutive semi-
vibrations arep, q, r; prove that the co-ordinate of the position of equilibrium and
the time of vibration if there were no resistance are respectively
^"-5' andl'll + i''— ''~«Vi-'
M^<
p + r~2q ( TT^ \ r-q^
[Math. Tripos, 1870.]
ONE DEGREE OF FREEDOM. 829
414, Wlien the coefficients are functions of tlie time, tlie equation can be
integrated only by some artifice suited to the particular case under consideration.
Let the equation be
d'^x dx
then a few useful methods of solution will be indicated in the following examples.
7)^ 1 dii7
Ex. 1, If Q.-^ ~ 9~ii^^ ^ positive constant, viz. n^, prove that the successive
oscillations of the system wiU be performed in the same time, though the extent of
the oscUlations may follow any law.
This may be proved by clearing the equation of the second term in the usual
way, i.e. put x—^e''^-''"'^'
Ex. 2. If r=0 and — p - — —— = a, where a ia a constant, prove that
x^e'^-^o'^^'U sin j ^1 - jf'^Jqdt + i?| .
Thence show that if / Jqdi does not become infinite, the time of oscillation is
independent of the arc of oscillation but the successive oscillations are not per-
formed in the same time.
This may be proved by writing t = ip{X), and then so choosing the form of \{/
that the coefficient of x in the differential equation becomes unity or some constant.
Ex. 3. A system oscillates about a position of equilibrium and its motion is
determined by the equation -y^ + qx = 0, where g is a known function of t, which
during the time under consideration always lies between j3^ and ^"^, the latter being
the greater. If the system be started with an initial co-ordinate x^, and an initial
velocity Xq in a direction away from the position of equiUbrium, show that the
system wUl begin to return before x becomes so great as */ XQ^-i- -~^. If ±m, ^m'
be two successive maximum values of x, prove that vi' cannot be so great as ^ m,
P
and that the time from one maximum to the next lies between -^ and — , .
415. When the arc of oscillation is not small, the equation cannot always be
reduced to the linear form, and no general rule can be given for its solution. In
many cases it is important to ascertain if the motion of the system is tautochro-
nous. Various methods of determiaing this wUl be shown in the following
examples.
Ex. 1. Show that if the equation of motion be
dh; f , , ,. .da;
= ( a homogeneous function of — and x of the first degi-ee ) ,
then, in whatever position the system is placed at rest, the time of arriving at the
position determined by ac = is the same.
330 SMALL OSCILLATIONS.
/l dx\
Let the homogeneous function be written ^f[--rf) • I^e* ^ ^^^ ^ ^^ t^^ co-
ordinates of two systems starting from rest in two different positions, and let x = a,
l — Ka initially. It is easy to see that the differential equation of one system is
changed into that of the other by writing ^ = kx. If therefore the motion of one
system is given by x-(p{t, A, B), that of the other is given by ^=^K(p(t, A', £').
To determine the arbitrary constants, A, B and A', B', we have exactly the same
conditions, viz. when t — O, = a and ;^=0. Since only one motion can follow
from the same initial conditions we have A'=A, and B' = B. Hence throughout
the motion ^=kx and therefore x and ^ vanish together. It follows that the
motions of the two systems are perfectly similar.
This result may also be obtained by integrating the differential equation. If we
put-— =», we ^ndi x = A(p it + B). When i=0, ~ = 0, and therefore 0'(B)=:O.
X at at
Thus £ is known and x vanishes when (y) and if possible so choose the form of (p, that —^ becomes a homogeneous
function of y and -=^ of the first degree. If this can be done, the motion is, by Ex. 1,
tautochronous.
Ex. 3. If the motion of any system is tautochronous according to Lagrange's
formula in vacuo, it will also be tautochronous in a resisting medium, if the effect
of the resistance is to add on to the differential equation of motion a term propor-
tional to the velocity. This theorem is due to Lagrange.
Ex. 4. A particle, acted on by a repulsive force varying as the distance and
tending from a fixed point, is constrained to move along a rough curve in a medium
resisting as the velocity, find the curve that the motion may be tautochronous by
Lagrange's rule.
Let V be the velocity, s the arc to be described, r the radius vector of the
particle, p the perpendicular on the tangent, p the radius of curvature. Let ar be
FIRST METHOD OF FORMING THE EQUATIONS OF MOTION. 831
the repulsive force, 5 the coefficient of friction. Then omitting the resistance by
Ex. 3, the equations of motion are
— -- ap + B
P
Eliminating the pressure E, we have
By Lagrange's rule, the motion is tautochronous if, when / (s) = a&^5 -ajr^-p'', we
h f is)
find -= - *^— —-' . This wiU be found to give /) = (! + h'^)p, which is an epicycloid.
P J {v
First Method of forming the Equations of Motion.
416. When the system under consideration is a single body,
there is a simple method of forming the equation of motion which
is sometimes of great use.
First, let the motion be in two dimensions.
It has been shown in Art. 175, that if we neglect the squares of
small quantities we may take moments about the instantaneous
centre as a fixed centre. Usually the unknown reactions will be
such that their lines of action will pass through this point, their
moments will then be zero, and thus we shall have an equation
containing only known quantities.
Since the body is supposed to be turning about the instan-
taneous centre as a point fixed for the moment, the direction of
motion of any point of the body is perpendicular to the straight
line joining it to the centre. Conversely when the directions of
motion of two points of the body are known, the position of the
instantaneous centre can be found. For if we draw perpendiculars
at these points to their directions of motion, these perpendiculars
must meet in the instantaneous centre of rotation.
The equation will, in general, reduce to the form
, ,, 2 cP9 _ /moment of impressed forces aboutN
df \ the instantaneous centre / '
where 6 is the angle some straight line fixed in the body
makes with a fixed line in space. In this formula Mh"^ is the
moment of inertia of the body about the instantaneous centre,
and since the left-hand side of the equation contains the small
factor -. .^ we may here suppose the instantaneous centre to have
382
SMALL OSCILLATIONS.
its mean or undisturbed position. On the right-hand side there is
no small factor, and we must therefore be careful either to take
the moment of the forces about the instantaneous centre in its
disturbed position, or to include the moment of any unknown
reaction which passes through the instantaneous centre.
Ex. If a body with only one independent motion can be in equilibrium in
the same position under two different systems of forces, and if L^, L^ are the
lengths of the simple equiYalent pendulums for these systems acting separately,
then the length L of the equivalent pendulum when they act together is given by
417. Ex. A homogeneous hemisphere performs small oscillations on a perfectly
rough horizontal plane : find the motion.
Let C be the centre, .G the centre of gravity of the hemisphere, N the point of
contact vnth the rough plane. Let the radius = a, CG = c, O—^NCG.
Here the point 1 + ^' , where GA' meets OAE in D. Also
since one body rolls on the other, the arc AP=qxcA'P, .-. pcf)=p'(j),
^ P + P
Again, in order to take moments about P, we require the
horizontal distance of G from P; this may be found by projecting
the broken line PA' +A'G on the horizontal. The projection of
PA' = PA' cos {a+ 6) = p then the expression for L takes the form
AG'~AG ' AF
Jc' + r'
L
= G'N.
The equilibrium is therefore stable or unstable according as G'
lies within or without the circle of stability.
421. Two points A, B of a body are constrained to describe given curves, and
the body is in equilibrium under the action of gravity. A small disturbance being
given, find the time of an oscillation.
Let C, D be the centres of curvature of tlie given curves at the two points A , B.
Let AG, BD meet in 0. Let G be the centre of gravity of the body, GE a perpen-
dicular on AB. Then in the position of equilibrium OG is vertical. Let i, j be
the angles CA, BD make with the vertical, and let a be the angle AOB. Let
A', i?'.. .denote the positions into which A, i?...have been moved when the body has
been turned through an angle 6. Let ACA' = (p, BI)B' = 4)'. Since the body may
be brought from the position AB into the position A'B' by turning it about
through an angle 6, we have — a4^ = OR ^^^^ ^^' ^^ ^'^^^^^^l perpen-
dicular to 00, and we have GG'^OG . 9. Also let x, y be the projections of 00' on
the horizontal and vertical through 0. Then by projections
X cos j + y sin J — distance of 0' from OD — OD . 0',
cos i- 2/ sin { = distance of 0' from 0C==0O. (f>;
0D.smi.4>'+ 00. sin j.(l>
FIRST METHOD OF FORMING THE EQUATIONS OF MOTION. 837
Now taking moments about 0' as the centre of instantaneous rotation, we have
,f ^^ OD . OB sin i OC.OA sin/ \
^ v. BD sm a CA sin a J
where h is the radius of gyration about the centre of gravity.
Hence if L be the length of the simple equivalent pendulum, we have
l? + OG' _ O B. OB sint OC.OA sinj
Z BD ' sin a AG ' sin a *
CoR. If the given curves, on which the points A, B are constrained to move, be
straight hues, the centres of curvature G and D are at infinity. In this case, we
may put -=-t, — - 1, -r^— - 1> aiid the expression becomes
^^±^^OG-OB.'^-OA.'^.
L sin a sm a
If OA and OB be at right angles, this takes the simple form
where F is the projection on OG of the middle point of AB.
422. A body oscillates about a position of equilibrium under the action of
gravity, the radius of curvature of the path of the centre of gravity being hiow7i,
find the time of oscillation.
Let A be the position of the centre of gravity of the body when it is in its
position of equilibrium, G the position of the centre of gravity at the time (. Then
since in equilibrium the altitude of the centre of gravity is a maximum or mini-
mum, the tangent at A to the curve AG is horizontal. Let the normal GO to the
curve at G meet the normal at A in C. Then when the oscillation becomes indefi-
nitely small Cis the centre of curvature of the curve at A. Let AG = s, the angle
ACG=\p, and let i? be the radius of curvature of the curve at A.
Let d be the angle turned round by the body in moving from the position of
equilibrium into the position in which the centre of gravity is at G' ; then -,- is the
at
angular velocity of the body. Since G is moving along the tangent at G, the
R. D. 22
S38 SMALL OSCILLATIONS.
centre of instantaneous rotation Kes in the normal GO, at siicli a point 0, that
dO ds ^^ ds
Oe^=vel.of^=-,.-.(?0=^.
Let MJo^ he the moment of inertia of the body about its centre of gravity, tlien
taking moments about 0, we have
{B + 0G^) — = -ff.Oa sin yp.
Now ultimately when the angle 6 is indefinitely small ^ = jT = p '> •'• *1^®
equation of motion becomes
Hence if L be the length of the simple equivalent pendulum we have
423. "When the system of bodies in motion admits of only one independent
motion, the time of a small oscUlation may frequently be deduced from the equa-
tion of Vis Viva. This equation will be one of the second order of small quantities,
and in forming the equation it will be necessary to take into account small quanti-
ties of that order. This \sill sometimes involve rather troublesome considerations.
On the other hand the equation will be free from all the unknown reactions, and
we may thus frequently save much elimination.
The method of proceeding wUl be made clear by the following example, by
which a comparison may be made with the method of the last article.
2'he motion of a body in space of two dimensions is given by the co-ordinates x, y
of its centre of gravity, and the angle 6 lohich any fixed line in the body mahes with
a line fixed in space. The body being in equilibrium under the action of gravity it
is required to find the time of a small oscillation.
Since the body is capable of only one independent motion, we may express {x, y)
as functions of 6, thus
x=F{d), y=f{e).
Let MF be the moment of inertia of the body about its centre of gravity, then the
equation of Vis Viva becomes
(syH-dT-^K")'-^-^-
where C is an arbitrary constant.
Let a be the value of d when the body is m the position of equilibrium, and
suppose that at the time t, e = a + -7^ there arc two positions of equilibrium of the rod. It
will be found by substitution that the position in which the rod is
inclined to the vertical is stable, and the other position unstable.
344 SMALL OSCILLATIONS.
If <^ <-ri tbe only position in which the rod can rest is vertical,
and this position is stable.
If 71 = 0, the body is in a position of neutral equilibrium. To
determine the small oscillations we must retain terms of an order
higher than the first. By a known transformation we have
Hence the left-hand side of equation (3) becomes (p + k^) ^ .
de y df ~ dt V dt)
and side of equation {I
The right-hand side becomes by Taylor's Theorem
-j-^ f ^i cos a — ^ CO r sm 2a 1 j— ^ + occ.
When 71 = 0, we have a = ^ and w'' = -^ . Making the neces-
sary substitutions the equation of motion becomes
Since the lowest power of & on the right-hand side is odd
and its coefficient negative, the equilibrium is stable for a displace-
ment on either side of the position of equilibrium. Let a be the
initial value of & , then the -time T of reaching the position of
equilibrium is
put & = a<}>, then
V gl 'Kjl^^'o.'
Hence the time of reaching the position of equilibrium varies
inversely as the arc. When the initial displacement is indefi-
nitely small, the time becomes infinite.
This definite integral may be otherwise expressed in terms of the Gamma
function. It may be easily shown that j
Jo
429. This problem might have been easily solved by the
first method. For if the two perpendiculars to Ox, Oy at
A and B meet in N, N is the instantaneous axis. Taking mo-
ments about N, we have the equation
OSCILLATIONS WITH TWO OR MORE DEGREES OF FREEDOM. 345
dr
{P-h F) ^=gl cose- r^(o' {I + ry sin cos 6
= gl cos 9 — ^ 0)"^ sin 6 cos
=fie).
Then the position of equilibrium can be found from the equa-
tion /(a) = Oand the time of oscillation from the equation
430. Ex. 1. If the mass of the rod AB is M show that the magnitude of the
couple which constraias the system to turn round Oy with uniform angular velocity is
3/ -:r- w -r sm 29.
A at
Would the magnitude of this couple be altered if Ox or Oy had any mass ?
Ex. 2. The upper extremity of a uniform beam of length 21 is constrained to
slide on a smooth horizontal rod without inertia, and the lower along a smooth
vertical rod through the upper extremity of which the horizontal rod passes : the
system rotates freely about the vertical rod, prove that if a be the inclination of the
beam to the vertical when in a position of relative equilibrium, the angular velocity
of the system will be ( rr-^ — \ , and if the beam be slightly displaced from this
■' \u cos a J
position show that it will make a small oscillation in the time
— — ~ . [Coll. Exam.].
|-^ (sec a+ 3 cos a) I*
In the example in the text the system is constrained to turn round the vertical
with uniform angular velocity, but in this example the system rotates freely. The
angular velocity about the vertical is therefore not constant, and its smaU variations
must be found by the principle of angular momentum.
Oscillations with two or more Degrees of Freedom.
431. When the position of a system of bodies depends on
several independent co-ordinates, the equations to determine the
motion become rather complicated. In order to separate the
difficulties of analysis from those of dynamics, we shall consider
the case in which the system depends on two independent co-
ordinates, though the remarks about to be made will be for the
most part quite general, and will apply, no matter how many
co-ordinates the system may have. In the sequel we shall con-
sider Lagrange's general method of forming the equations when
the system has n co-ordinates.
346 SMALL OSCILLATIONS.
432. The equations of motion of a dynamical system per-
forming small oscillations with two independent motions are of
the form
To solve these, we eliminate either x or y; \i D stand for -j- ,
we have
AD'+BD + G, FD^+GD+H x = 0,
A'B' + B'D + C, FD^ + G'B + H'
with a similar equation for y. If 'AB stand for the determiDant
A B\ • • 1
.,' this biquadratic becomes, when x is omitted,
AJB'HAG+BF)B'+iAS+Ba+CF)D'+(m+CG) B+CH=0.
If the roots of this biquadratic be m^, m^, m^, m^, we have by
the theory of Linear Differential Equations
where M , M^, M^, M^ are arbitrary constants. Similarly we have
The Jf's are not independent of the if' s, for by substituting in
either differential equation and taking any M and M' as typical
of all,
(^m' + Bm + C) M= - (Fni' + Gm + H) M'.
There are therefore just four arbitrary constants, and these are to
be determined by the initial values ^^ ^' I/' ~^' ~^-
433. If the position of the system depends on three indepen-
dent co-ordinates x, y, z, we shall have three equations of motion
similar to the two at the beginning of this article. These may be
solved in the same way. In this case we obtain a subsidiary equa-
tion of the sixth degree to determine the exponentials which
occur in the variables. The relations between the _ coefficients of
corresponding exponentials can be^ found by substitution in any
two of the equations of motion.
In certain cases it may be more convenient to choose x or y
to be itself a differential coefficient of a co-ordinate. In this case
the biquadratic or sextic equation will reduce to a cubic or
quintic.
OSCILLATIONS WITH TWO OR MOEE DEGREES OF FREEDOM. 847
434<. It appears from this summary that the character of the
motion depends on the forms of the roots* of this biquadratic.
* If the general character of the motion is required it will be necessary to
analyse the biquadratic. Rules by which this is made to depend on a cubic
equation are given in most of the books on the theory of equations, but as the final
results are not stated, it will be useful to give here a short analysis for reference.
Let the biquadratic be
so that the invariants are 7=ae-45d + 3c^ and J—ace + 2bcd-ad?-eh'^-c^. This
last may also be written in the form of a determinant. It will generally be found
convenient to clear the equation of the second term. Let the equation so trans-
formed be
where H-lP-ac and G = 'ib^ -iale+a-d. By using the invariants or by actual
transformation, it is easy to see that Iw^^F+BH^ and a?J=4.R^~G^-a^IH.
Let A be the discriminant, i. e. A=i3-27J'^ then it is proved in all books on
the theory of equations that if A is negative and not zero the biquadratic has two
real and two imaginary roots. If A is positive and not zero, the roots are either all
real or all imaginary.
Usually we can distinguish whether the roots are all real or all imaginary by
ascertaining if the biquadratic has or has not a real root, thus if a and e have opposite
signs one root is, and therefore all the roots are, real. In any case we may use the
following criterion. Let Ka'^=9H^- - F^Vm"^ - la^. Then if a, j3, y, 5 be the roots
of the transformed equation it is easy to prove
3R_ a^ + ^^ + y^+d'^ ^
If all the roots are real H must be finite and positive. Since the arithmetic
mean of four positive quantities is greater than their geometrical mean, it is clear
that K is also positive, and can vanish only when aU the squares of the roots are
equal. If all the roots are imaginary, let them he p :i:jp' ^J - 1, ~p^q' ^f-l. We
then have
Sir _2£-{p2±f} ■]
i^ 2 I
If ZT is positive or zero, it is easy to see that K must be negative. If therefore
H and K are both positive, the four roots are real, if either is negative or zero, the
four roots are imaginary.
If the discriminant A is zero, but 1 and / not zero, it is known that the
biquadratic has two roots equal. If two of the roots are real and equal and the other
348 SMALL OSCILLATIONS.
If any one of the roots is real and positive, x and y will ultimately-
become large, unless the initial conditions are such that the term
depending on this root disappears from the values of x and y. If
the roots are all real and negative, the motion will gradually
disappear and the system will come to rest at the end of an
infinite time.
If two of the roots are imaginary, we have a pair of imaginary
exponentials with imaginary coefficients, which can be rationa-
lized into a sine and a cosine. This rationalization will be however
unnecessary if, as usually happens, only the character of the oscil-
lations is required. Suppose the roots to be a + ^ V(~ 1)> we have
X = e"' (iVj cos pt + N^ sin 'pi) -1- &c.,
where iV^, N^ are arbitrary constants. There will be a similar
expression for y with W written for N. Thus the period of the
oscillation is — . The oscillation will ultimately become very
large or vanish away, according as a is positive or negative. If
a= 0, the oscillations will continue throughout of the same mag-
nitude.
If it be required to find not merely the character of the motion,
but also the motion resulting from given initial conditions, it will
be necessary to determine the relations between the arbitrary
constants which enter into the expression for x and y. This may
be effected very easily in the following manner. Let TJ' -\-fD + g
be the factor which equated to zero gives the imaginary roots,
then / and g are known in terms of a and p. Let us now substi-
tute —fD — g for D^ in the two first equations of Art. 432. They
reduce to equations of the form
two imaginary, we see by putting g' zero that if 77" is positive or zero, K must be
negative. Hence if H and K are both positive all the roots are real, ii H or K is
negative or zero, two roots are real and two imaginary. If G is zero, there are then
two pairs of equal roots. In this case K is zero, and these roots are all real if //
is positive, all imaginary if H is negative.
Lastly if A is zero and also both I and J zero. The biquadratic has three roots
equal, and therefore all the roots are real. If il = also, the four roots are all equal
and real.
OSCILLATIONS WITH TWO OR MORE DEGREES OF FREEDOM, 349
dv
where B^, G^, &c. are some constant coefficients. Eliminating -^
from these equations, we have an equation of the form
where K and L are constants, so that when the two terais of x,
which depend on this factor, are known, the corresponding terms
of 2/ can be found immediately. If there be another pair of
imaginary roots, we obtain by a similar process a similar equation
with d^ifferent constants for K and i, to find the corresponding
terms in y.
If two of the roots are equal, say 77^^ = m^, then, by the theory
of Linear Equations, we know that
where N^ and iV^, &c. are arbitrary constants. If three roots
are equal, there will be a term with f and so on. The expres-
sion for y will of course contain similar terms. Let it be
The terms containing ^ as a factor will at first increase with t,
and if m^ is positive or zero will become very great, but if m^ is
negative, they will ultimately vanish. The motion will, in the
latter case, be stable if the initial increase of the terms is not such
that the values of x and y become large, i. e. if the system is not
at first so much disturbed that the motion cannot be considered
as a small oscillation.
In some cases the relations between the constants in the ex-
pressions for X and y are such that the coefficients of both the
terms containing the factor t vanish*. When this occurs the four
* To prove this let us find the relations between the constants. Substituting
the vahies of x and y in the two first equations of Art 432, we find
{Avij^ + Bm^ + C)N^^- {Fm^^ + Gm^ + H) N.{,
{Am^^ + Bm^ + C)N^ + {2Am^ + B) N„ = - [Fm-^^ + 6m^+H) iV/ - {2Fmi + G) N^,
with two similar equations which may be obtained from these by accenting the
letters A, B, C, F, G, H. If then
Amj^+Bm^+C =0^ Fm^~ + Gm^ + H =0)
A'm^^ + B'm-^ + C" = ) ' F'm^^ + G'm-^ + H'= \ *
while the two expressions
{2Amj^ + B) {2F'm-^ + G') and {2A'm^ + B) {2Fm^ + G)
are unequal, we have N^, N„' both zero, and N^, N-^' both arbitrary. If the two
expressions just written down were equal also, it may be shown that the biquadratic
to find D would have three equal roots.
350 SMALL OSCILLATIONS.
arbitrary constants will be N^, F^', M^ and M^. In such cases the
motion is stable for all initial conditions.
435, The most important case is that in which there are no
real exponentials in the values of x and y. li AG -\- BF and
BH-\- CG both vanish, there will be no odd powers in the sub-
sidiary biquadratic. The biquadratic may now be regarded as
quadratic in I)''. If its roots are real and negative, let them be
—p^ and — q^. The expression for oc will then take the form
x = N^ sin (pt + v^ + N^ sin [qt + v^,
where N^, N^, v^, v^ are arbitrary constants. The corresponding
terms in y may be found by the rule just given. Eliminating
^ between the two given equations of motion, let the result be
€1%
Then writing — p^ for -^ , we have
G'-A'p' B' dx
^~ R'-F'f R' -Ff dt
C — A'v^ B'p
^~ S'-Fy ^' ^'"^ (i^«+ ^i) - jj' _^y ^1 COS ipt + v,)
- ^'ZfY ^' '""' ^^^ "^' "'^^ " H^-W '^^ ""' ^^^ "^ '^^'
436. In many cases it will be found impracticable to solve
the biquadratic on which the character of the motion depends.
If however we only wish to ascertain whether the position of
equilibrium, or the steady motion about which the system is^ in
oscillation, is stable or unstable, we may proceed without solving
the biquadratic.
With the reservations in the case of equal roots mentioned in
Art. 434, the necessary and sufficient conditions for stability are,
that the real roots and the real parts of the imaginary roots should
be all negative. It is proposed here to investigate a method of
easy application to decide whether the roots are of this character.
Let the biquadratic be written in the form
^ (D) = aD' + hD' + cB'+dD + e = 0.
Let us form that symmetrical function of the roots which is the
OSCILLATIONS WITH TWO OR MORE DEGREES OF FREEDOM. 351
product of the sums of the roots taken two and two. If this be
called ~^, we find*
a'
X = hcd — acr — el/
— 1
~ 2
2a
h
c
I
d
c
d 2e
It will be convenient to consider first the case in which X is
finite.
Suppose we know the roots to be imaginary, say a ± p^— 1,
and ^ ±q V^-
Then I = 4.;S {(a + ^f + {p + qf] ((a + ^Y + (p - qT].
Thus, c(/3 always takes the sign of — , and a + /3 always takes the
sign of — . Thus the signs of both a and yS can be determined ;
and if a, h, X have the same sign, the real parts of the roots are
all negative.
Suppose, next, that two of the roots are real and two imagi-
nary. Writing q V— 1 for q, so that the roots are a +|) V— 1 and
/3 + q', we find
§= 4a^ {[(« + ^y +/ - q'J + 4>pY].
* This value of X may be found in seTeral ways more or less elementary. If
we substitute D=E±Z in the given biquadratic and equate to zero the even and
odd powers of Z, we have
aZ'i + {6aE^ + dhE + c)Z'- + AE* + bE'' + cE^ + dE+e = 0)
{'i(tE + b)Z^ + {iaE^ + 3bE"'-i-2cE + d)Z = o\'
Rejecting the root Z — and eliminating Z we have
6ia^E'^+ + hcd-ad'^-eh'^ = 0,
where only the first and last terms of the equation are retained, the others not
being required for our present pvirpose. Since x—E± Z it is clear that each value
of E is the arithmetic mean of two values of x. We have an equation of the sixth
degree to find E because there are six ways of combining the four roots of the
biquadratic two and two. The product of the roots of the equation in E may be
deduced in the usual manner from the first and last terms, and thence the value
of X is seen to be that given in the text.
If we eUminated E we should obtain an equation in Z whose roots are the
nrithmetic means of the dififerencea of tlic roots of the given equation taken two
and two.
352 SMALL OSCILLATIONS.
X
Just as before, a/3 takes the sign of — , and a+ ^ takes tlie sign
b e
of . Also, S^ — q^ takes the sign of the last term - of the bi-
quadratic. This determines whether ^ is numerically greater or
less than 4ae.
437. If tlie equation on wliicli the motion of tlie system depends is of the fifth
degree, we may proceed in the same way. Let the equation be
and let us suppose the coefficient a to be positive.
Form, as before, the product of the sums of the roots taken two and two. If this
, X „ , -^=1 he -ad he-af I
be -7 , we find , , , . .
a* \ be - af ae - cf \
Let us consider first the case in which X is finite.
Suppose that there are four imaginary roots ai^psJ-1, ^±qj-l, and one
real root 7. Then 7 has the sign opposite to /, and a^ takes the sign of X, while
2(a + /3) +7= --. If then / be positive, 7 is negative. If 6 be positive and
(p ( J negative, the root 7 is numerically less than - , so that a + /3 is negative.
If therefore a, h, /, X, and -(pi | be all positive, a, /3, 7 will be all negative.
Suppose that there are two imaginary roots a^pj-l, and three real roots
(3, 7, 5. Then, if all the coefficients are positive, j3, 7, 5 are negative, and X takes
the sign opposite to a; so that, if X be also positive, a, /3, 7, 3 will be aU negative.
Suppose aU the roots to be real; then, if all the coefficients be positive, the
roots win be all negative, and not otherwise; and it is also clear that X, being the
product of ten negative quantities, will be positive.
In both these cases, if the real roots and the real parts of the imaginary roots
be negative, it is clear that (pi J must have the sign opposite to a.
Conversely, if all the real roots and the real parts of the imaginary roots be
negative, the factors of the equation, and therefore the equation itself, must have
all the coefficients of the same sign.
R. D. 23
354 SMALL OSCILLATIONS.
We therefore conclude that it is necessary and sufficient for stahility of equili-
brium that every coefficient of the equation, -(pi ) , and also X, should be posi-
tive.
The case in which X is zero may be treated in the same manner as in the
biquadratic.
As it is very seldom that equations beyond the fourth or fifth degrees present
themselves in dynamics, it is unnecessary to consider any other cases in detail.
A more general method of proceeding will be indicated in a note.
438. It will be often found advantageous to trace the more
complicated cases of motion by the help of a figure. There are
various methods of effecting this, some being more suited to illus-
trate one kind of motion, others to illustrate another. We might,
for instance, follow the method indicated in Art. 412. Let the
abscissa of a point P represent on any scale the time elapsed since
some epoch, and let the ordinate represent the value of x. In the
same way the curve traced out by another point Q will represent
the changes of y. Suppose, for example, we wished to trace the
motion represented by
X = N ^m.pt + N sin ^pt,
the coefficients being equal in magnitude. There will be no
difficulty in tracing the two curves x^=N^\n'pt and x^ = N^m 2pt.
Let these be the two dotted lines. We obtain the required curve
by adding the ordinates corresponding to each abscissa. Let this
be the continuous line.
In the figure the axis of the abscissfe is not drawn. It clearly
joins the two extreme points on the right and left hand sides.
We see from a simple inspection of the figure, that the motion
consists of a violent oscillation to each side of the mean position
OSCILLATIONS WITH TWO OR MORE DEGREES OF FREEDOM. 355
followed by a very slight one, and so on alternately. This figure
resembles that used in Astronomy to trace the changes of magni-
tude of the equation of time throughout the year.
439. Ex. 1. Show that the motion represented by x = N sin pt + N sin Spt
consists of two large oscillations to one side of the mean position followed by two
equally lai'ge ones to the other side and so on continually.
Ex. 2. Trace the motion represented by a!!=iVsin2^« + iVsin 3pt, and point out
the difference between the two parts of the large oscillation.
440. Let us trace the motion represented by x—N^ sin {pt + v^ + N^ sin {qt + v.^,
where iVj and N^ are both positive, firstly when p and g are nearly equal, and
secondly when j) is small compared with q.
In the first case, consider any time at which pt + v-^^ and qt+v.^ differ from each
other by an even multiple of tt. At this instant the two trigonometrical terms
have the same sign, and, since p and q are nearly equal, they will increase and
decrease together for several oscillations, how many will depend on the nearness of
j3 and q to each other. The value of x will therefore vary between the Umits
±(iV^ + iV2). Next consider any time at which pt + v-^^ and qt + v.^ differ by an odd
multiple of tt. The two trigonometrical terms have opposite signs and will continue
to have opposite signs for several oscillations. The value of x will therefore vary be-
tween the limits ± [N^ - N.^. We see that the motion of that part of the dynamical
system which depends on the co-ordinate x undergoes a periodic change of character.
At one time, this part of the system is oscillating with an arc N-^-{-N^, after an
interval equal to , the arc of oscillation is N-^^ - N.^. If iV^ and N^ are nearly
equal, this last arc may be so small, that the motion is invisible to the eye.
Thus there wiU be alternate periods of comparative activity and rest. This alter-
nation is sometimes called beats. Usually the two co-ordinates x and y will be so
related that the period of comparative rest in one will coincide with the period of
comparative activity in the other. When this is the case there will he an alternate
transference of energy from one part of the system to another and back again,
441. Ex. Show that, if ^ and q be unequal, x may be written in the form
flj^iVsin- {pt + v^ + qt + Vq + d),
where N'^N-^^ + N.^^ + 'i.N-^N.^cosipt + v-^^-qt-v.^,
. d N^-N,. 1. ^ , ,
2 " n\n. 2 ^^ + "i - !^* - "a)-
Thence show that when p and q are nearly equal, the oscillation will appear to the
eye to be harmonic, but the arc of oscillation will slowly vary between the limits
442. Next, let p be small compared with q. In this case qt + v„ increases by
27r whUe ^f + j/j alters only ^by~ 27r, so that the second trigonometrical term goes
through all its changes while the first is only very slightly altered. The system
will therefore appear to oscillate about a mean position determined by the iustan-
23—2
356 SMALL OSCILLATIONS.
taneous value of the first trigonometrical term. Thus tlie oscillations will appear
to be harmonic to the eye, while the apparent mean position will travel first to one
side and then to the other of the real mean.
443. Ex. Investigate the following geometrical construction to represent the
motion
±x = Ni sinpt + N^ sin qt.
Let q be greater than p in the standard case and let x have a sign such that iV^ is
. . a — 7)
positive. Describe a circle with centre and radius equal to - — - iV,. Let another
q 1
circle with centre and radius equal to - N-^ touch the first circle externally at a
point A. Measure CP equal to A^„ in the direction OC, so that if N^ is negative
CP must be measured in the opposite direction. If the second circle be now made
to roll on the first, the point P traces out an epitrochoid. If C and P' be cor-
responding positions of the centre of the moving circle and the generating point,
then the distance of P' from the fixed straight line OA is the value of x, while
the angle C'OA is equal to pt.
Apply this to trace the motion when p and q are nearly equal.
The third or Lagrange's method of forming the equations of motion.
444. Let a system of bodies be in equilibrium under any con-
servative forces. When disturbed into any other position let U be
the force function, 2T the vis viva. Let the position of the systembe
defined by n co-ordinates 6, <^, &c., which are such that they vanish
in the position of equilibrium. Then if the system oscillate about
the position of equilibrium, 6, <^, &c. will be small throughout the
whole motion. As before, let accents denote differential coeffi-
cients with regard to t.
Let us suppose that the geometrical equations do not contain
the time explicitly, then by Art. 367 T may be expressed as a
homogeneous function of 6', " + &c (1).
Here the coefficients A^^, &c. are all functions of 6, 4>, &c., and
we may suppose them to be expanded in a series of some powers
of these co-ordinates. If the oscillations of the system are so small
that we may reject all powers of the small quantities 6, (f), &c.
except the lowest which occur, we may reject all except the con-
stant terms of these series. We shall therefore regard the coeffi-
cients J-jj, &c. as constants.
In the same way we may expand ?7 in a series of powers of
6, (f), &c. In this series the terms containing the first powers will
vanish, because by the principle of virtual velocities
dU== 2m (Xdx + Ydy + Zdz)
LAGRANGE'S METHOD. 857
vanishes in the position of equilibrium. Hence we may put
2U=2U, + a^,e' + 2a^,e-{-&c (2),
where ZT^ is a constant, which is easily seen to be the value of IT
in the position of equilibrium. It is necessary for the success of
Lagrange's method that both these expansions should be possible.
We have now to substitute these values of T and U in the 7i
Lagrange's equations
dtdd' de dd ^ ''
with similar equations for (^, '^. Since the expression for T does
not contain 6, 0, &c., we have
The n equations (3) therefore iSecome
A,J'' + A^,^'' + ... = a,J + a,,^ + ...\.:' (^)-
&c. = &c. J
These are Lagrange's equations to determine the small oscillations
of any system about a position of equilibrium, under any conserva-
tive forces, provided the geometrical equations do not contain the
time explicitly, and are not functions of the differential coeffi-
cients of the co-ordinates.
These equations may be obtained in a variety of ways. In
many cases it is more convenient to use the process of taking
moments and resolving. The advantage of using Lagrange's method
is that the whole motion is made to depend on one function only,
viz. T^-U.
445. We shall now proceed to the solution of the equations.
We notice that these equations do not contain any differential
coefficients of the first order. This will be the case when a dyna-
mical system oscillates about a position of equiUbrium under con-
servative forces. This peculiarity greatly simplifies the solution.
Instead of using exponentials, as in Art. 432, which (when we want
anything more than the periods) have afterwards to be ration-
alized, we may now conveniently introduce the trigonometrical
expressions at once. Let us then put
= L^ sin {j)^t -t- G(j) + jL, sin (^;./ + a^ + &c. 1
(^ = i¥, sin {p^t + a,) + M^ sin (/>,/ + ot^) + c^c. ^ (5),
&c. = &c. j
358 SMALL OSCILLATIONS.
which may be written in the typical form
6 = Lsm ipt + a), <^ = M sin {pt + a), &c.
If we substitute in equations (4) we have
{A,,f + a,,)L + {A^,f + aJM-\-ko. = 0\ (6).
&c. &c. =0]
Eliminating L, M, &c., we have a determinantal equation
&c. &c. &c.
= (7),
which, it will be observed, is symmetrical about the leading
diagonal. This equation is of the w**^ degree to find p^. It will be
presently shown that its roots are real. Taking any root positive
or negative, the equations (6) determine the ratios of if, N, &c. to L ;
and we notice that these ratios will also be all real. If all the
roots are positive, the equations (5) will give the whole motion,
with 2?i arbitrary constants, viz. L^, L^...Ln\ oi^, a^....a^. These
have to be determined by the initial values of 6, (f>, &c., & , ^', &c.
If any root be negative, the corresponding sine will resume its
exponential form, the coefficient being rationalized by giving the
coefficient L an imaginary form.
That the determinant should contain no odd powers of p is
just what we should have expected a priori. In our preliminary
assumption (5) each sine is really the sum of two exponentials
with indices of opposite signs. The equation therefore of Art. 432
to determine p should here give pairs of equal roots of opposite signs.
The equation (7) may be written down without difficulty as
soon as the values of T and U have been expanded in powers of
6' , &c., 6, &c., respectively. In finding the times of oscillation of a
system about a position of equilibrium, it is not necessary to go
through all the intermediate steps; we may, if we please, write
down at once the determinantal equation. The rule will be as
follows. Omitting the accents in the expression for T, and the
constant term in U, equate to zero the discriminant of p'^T + XJ.
The roots of the equation thus formed are the values of p. If we
require the motion as well as the periods, we shall require equa-
tions (6). But these may be also very simply found in the follow-
ino" manner. Omitting accents as before and taking any of the
values of ^ just found, form the equations*"
* These equations are given by Lagrange.
lageange's method. 359
^^{fT+U) = 0,^{p^T+U) = 0,&c (8).
The 9, (f), Sc. in these equations may he replaced by the coefficients
required in the equations (5).
If we solve these equations we see that the ratios of L, M, &c.
are equal to the ratios of the minors of the constituents of any one
line in the determinant (7).
Ex. 1. A rod AB whose lengtli is 2a and mass m is suspended from a fixed
point by a string OA the length of which is I. The rod oscillates under gravity
in a vertical plane, find the periods of the small oscillations.
Let d, (p he the angles the string and the rod make with the vertical. Proceed-
ing as in Ai't. 136, we find that when powers of 6 and (p higher than the second are
neglected,
Forming the discriminant oi 2}^T+ U and dividing out the common factor m, we
have
IpH^-gl alp^ 1=0.
I alp^ p^{F' + a^)~ag \
This quadratic gives two values of p^. If these he ^^i^ and p^^, we have
e =L-^ sin {p-^t + a^)-\-L.2 sin (pzt + a^), '\
'l'--^^''-^^'^(lht+a,)-L,P^^Binip.,t + a,)\'
Ex. 2. Show that when the determinant (7) is zero, the ratios of the minors of
the constituents of any one hne are equal to the ratios of the corresponding minors
of the constituents of any other line.
Ex. 3. If (T^, Z7i), (Tj, U^), &c. he the values of T and U- Uq when (L^,
M-^, &c.), (ig, M2, &c.) are substituted for {6', (j>', &c.) or {0, ^, &c.), prove that
2'i2'i'+ U^ = Q, T^-p^^ U^ = 0, &c.
This follows from equations (8) by Euler's theorem on homogeneous functions.
446. In order to determine the values of p^, it will often be necessary to expand
the determinant. This may be done by the use of Taylor's theorem. Let A be
the discrimiaant of T and let 11 represent the operation
then the determinant when expanded becomes
ApSn + n ( A) jp2n-2 + lP{A)p^'^-*+...=0.
If A' be the discriminant of U and 11' the operation 11 when the great and small
letters are interchanged, we may write the equation in the form
A' + n' (A') p2 + n™ (A')p* +...=0.
When there arc only tlu-ec independent co-ordinates, we may adopt the notation
iised in the chapter on Invariants in Dr Salmon's Conies,
360 SMALL OSCILLATIONS.
Ex. 1. If a system be in a position of equilibrium, show that the equi-
librium will be stable if -n(A), IT- (A), -113(A), &c. be all positive.
Firstly, we may show that A is necessarily positive, and secondly that these
are then the conditions that the roots of the equation (7) are all real.
Ex. 2. If ,9 J. be the sum of the products of each Mh minor of the discriminant
A' into the conjugate minor of A, prove that Sk is the coefficient of jr'^.
Ex. 3. The same dynamical system can oscillate about the same position of
equilibrium under two diiferent sets of forces. If p-^,p2... and o-j , o-g . . . be the
IDeriods of the oscillations when the two sets act separately, E^, R2... the periods
when they act together, prove that S -„ +S — = S — .
p^ a- R^
This follows from the fact that IT (A) contains A^^ &c. only in their first powers.
Ex. 4. Two different systems of bodies when acted on by the same set of forces
oscillate in periods p^, p^... and o-^, ffi ... If R^, J?^... be the periods when they are
both set in oscillation by the same set of forces, prove that S/)^ + So-^ = 2-K^.
Ex. 5. Prove that the equation giving the periods of the oscillations may be
expressed as a determinant of 2n rows and columns by using Sir W. Hamilton's
equations given in Art. 381.
447. If we refer the motion of the system to any other co-ordinates f, 77, f, &c.
which vanish in the position of equilibrium, it is clear that when d, (p, \p, &c. are
expressed in terms of ^, &c. and the squares of small quantities neglected, we shall
have equations of the form
= [i-i^ + IJii-n + &c. \ (9).
&c. =&c. J
Now 6, (f), &c. being expanded in a series of sines as in equations (5) it is clear
that ^, 77, (fee. will be expanded in a series of the same sines but with different co-
efficients. Hence the values of j)^ found from the determiuantal equation will be
the same whatever co-ordinates the system is referred to. The ratio of the
coefficients of the several powers of y are therefore invariable.
If /x be the determinant of transformation, we know that A becomes t/?\. Hence
all the other coefficients will be altered in the same ratio. The quantities A, 11 (A),
n^(A), &c. are therefore called the invariants of the dynamical system.
448. To show that the values ofp^ are all real*.
Since T is essentially a positive quantity for all values of 6',
+ -4f^ir + ...-,
and since A^^ is positive, this transformation is real. In the same
way B^ must be positive, and we may repeat the process. We
thus have
2r=p + 77'^ + &c.,
where ^ = V^c/) + -^ ^/r + ...,
and it is clear that this process may be repeated continually.
We may take ^, rj, &c. as co-ordinates of the system because
they are independent of each other and vanish in the position of
equilibrium. We thus have
where /jj, f^^, &c. are all real constants. The determinantal
equation now takes the form
P' +fni /i2. &c.
A> /+/22> &C.
&c, &c. &c.
= 0.
When there are only three co-ordinates, this is the discrimi-
nating cubic used in Solid Geometry, and we know that its roots
are all real. When there are more than three co-ordinates, it is
proved in Dr Salmon's Higher Algebra, Lesson VI., that the roots
are all real.
449. To explain what is meant by the principal co-ordinates
of a dynamical system.
When we have two homogeneous quadratic functions of any
number of variables, one of which is essentially positive for all
values of the variables, it is known that by a real linear trans-
formation of the variables we may clear both expressions of the
terms containing the products of the variables, and also make the
coefficients of the squares in the positive function each equal to
unity. If the co-ordinates 0, cf), &c. be changed into f, rj, &c. by
the equations (9) of Art. 447, we observe that 6', , yjr,
&c,, which are all variable, the ratios of these co-ordinates to each
other are constant throughout the motion*. For referring to
the equations (9) of Art. 447, we see that when 7], ^ are all zero,
and only ^ is variable,
\ f^i '"
* This property is mentioned by Lagrange, who on several occasions uses
principal co-ordinates though not the name.
Lagrange's method. 863
452, When some of the roots of the equation to find p^ are
equal, we know by the theory of linear differential equations
that either terms of the form [At -\- B) sin pt enter into the
values of 6, (f), &c., or else there must be an indeterminateness
in the coefficients L, M, &c. given by equations (8). Referring
the system to principal co-ordinates we see that the first alter-
native is in general excluded. If two values of p^ were equal,
say 5^j = 622' ^^® trigonometrical expressions for ^ and t] have
equal periods, but terms which contain i as a factor do not make
their appearance. The physical peculiarity of this case is that
the system has more than one set of principal, or harmonic
oscillations. For it is clear that, without introducing any terms
containing the products of the co-ordinates into the expressions
for T or tl, we may change ^, 77 into any other co-ordinates f^, t]^,
which make ^ + 77^ = ^^ + v^i the other co-ordinates ^, &c. re-
maining Unchanged. For example we may put ^ = ^^ cos a — rj^ sin a
and »7 = 1^1 sin a + ?;^ cos a, where a has any value we please.
These new quantities ^^, t]^, ^, &c. Avill evidently be principal
co-ordinates, according to the definition of Art. 449.
One important exception must however be noticed, viz. when
one or more of the values of p are zero. If, for example, 5,^ =
we have ^ = At + B, where A and B are two undetermined con-
stants. The physical peculiarity of this case is that the position
of equilibrium from which the system is disturbed is not solitary.
To show this, we remark that the equations giving the position
of equilibrium are -7^- = 0, -7- = 0, &c., where U has the value
given at the end of Art. 449. These in general require that ^,
7), &c. should all vanish, but if 6^^ = they are satisfied whatever
f may be, provided rj, ^, &c. are zero. These values of f must
however be very small, because the cubes of ^, 7], &c. have been
rejected. It follows therefore that there are other positions of
equilibrium in the immediate neighbourhood of the given position.
Unless the initial conditions of disturbance are such as to make
the terms of the form At + B zero, it may be necessary to examine
the terms of the higher order to obtain an approximation to the
motion.
453. The motion being referred to any co-ordinates 0, (p...
it may be required to find the principal oscillation. This may be
done by finding X, fju, &c. in equations (9) Art. 447, by the analy-
tical process of clearing the two quadratic expressions of the terms
containing the products, in the manner explained in Art. 449.
We may also proceed thus, Let the system be performing the
2-77
principal oscillation whose period is — . Then in the equations
P\
(5), L^, il/g, &c., Xg, il/3, &c. are all zero, hence 6, , tp as the Cartesian
co-ordinates of a point P, it is clear that the position of P at any instant will give
the position of the system. Omitting the accents in T and the constant term in U,
the equations T=a, U—-^, where a and j3 are any constants, represent two
quadric sm'faces which have their centres at the origin. These have a common set
of conjugate diameters which may be found by the following process. Let 6, 4>, yp
be the co-ordinates «f any point on one of the three conjugates. Then, since the
diametral planes of this point in the two quadrics are parallel, we have
dT _dU dT_dU dT_dU
'^dd~dd' '^d(p~d, yp contain only a single trigonometrical term (Art. 450) when the
system is performing a principal oscillation, we see that the' representative point P
moves with an acceleration tending to the origin and varying as the distance there-
from.
455. As an example of this geometrical analogy let us consider the following
problem. A rigid body, free to move about a fixed jyoint 0, is under the action of
any forces and makes small oscillations about a position of equilibrium; find the
principal oscillations.
Let OA, OB, OC be the positions of the principal axes in the position of
equilibrium, OA', OB', OC their positions at the time t. The position of the body
maybe defined by the angles between (1) the planes AOO, AOC, (2) the planes
BOO, BOC, (3) the planes GOA, COA'. Let these be called 6, , \p be, as
in Art. 444,
U = ^ctu^^ + ai20(p + &c.
Then, following the analogy, as P moves along a radius vector OD' of the quadric
/OP \2
U= - 13, the work is - I y-y-, \ /3. Hence this quadric possesses the property that
the work done by the forces when the body is twisted through a given angle round
any radius vector varies inversely as the square of that radius vector. If the equi-
librium is stable, the work due to a rotation about every diameter miist be negative,
the quadric must therefore be an ellipsoid.
It now follows from the general theorem that the body will perform a principal
oscLllation if it is set in rotation about any one of the three conjugate diameters of
the momental ellipsoid and the ellipsoid U, and wiU therefore continue to oscillate
as if that diameter were fixed in space.
The quadric U has been called the ellipsoid of the potential. This name was
given to it by Prof. Ball, who arrived at the theorem just proved by a different
com-se of reasoning. See his Theory of Scretvs, Art. 126. The following application
is also due to him.
456. When the only force acting on the body is gravity, the ellipsoid of the
potential is a surface of revolution about a vertical axis. For the inverse square of
any radius vector measures the work done in turning the body through a given
small angle about that radius vector. But the work is also proportional to the
vertical distance through which the centre of gravity has been elevated from its
position in equilibrium vertically under the point of support. Hence all radii
vectores which make the same angle with the vertical are equal. Fiu'ther the
vertical radius vector is infinite, for the work done in rotating the body about
a vertical axis is zero. The ellipsoid of the potential is therefore a right circular
cylinder with its axis vertical.
The common conjugate diameters of these two quadrics are obviously the
vertical and the two common conjugate diameters of the two ellipses in which the
diametral plane of the vertical with regard to the momental ellipsoid intersects the
momental elUpsoid and the cylinder.
The principal oscillation about the vertical conjugate is performed in an infinite
time and would therefore cause the body to depart far from the position of equi-
librium. Bat this is contrary to supposition. The initial axis of rotation must
therefore be in the plane of the other two conjugates, i. e. must be in the diametral
plane of the vertical with regard to the momental ellipsoid, and it will remain in
this plane throughout tlie whole of the subsequent motion.
Since these conjugate diameters project into the conjugate diameters of tlio
366 SMALL OSCILLATIONS.
horizontal section of the cylinder, it is clear that two vertical planes each contain-
ing one of the principal or harmonic axes are at right angles to each other.
457. Ex. Show that the mean kinetic energy of a dynamical system oscillating
about a position of equilibrium is equal to the mean potential energy, the mean
being taken for any long period, and the position of equilibrium being the position
of reference.
Eefer the motion to principal co-ordinates and let
2r=^'2 + ,,'2 + &c., 2{U- Uo)= -i^i^f -252V-&C.
Then we find ^ =^ E sin {p-^t + a^), r}=FsiQ.{p.2t + a). Substituting these in T and
Uq-U we have the instantaneous kinetic and potential energies. The means of
these are obviously the same, and equal to - {E^pi^ + F'^p^^ + &c.).
If the system remain in the position of equilibrium the Hamiltonian character-
istic function S= TJ^t. If the system be disturbed and after any time t again pass
through the position of equilibrium, the value of S for these two neighboming
modes of passing from one position to another in the same time must be equal.
Hence / {T-^V)d,t— U^t, i.e. the mean values of the kinetic and potential energies
•'0
are equal.
458. Ex. Find the energy of a dynamical system oscillating about a position
of equilibrium referred to any co-ordinates.
By referring the system to its principal co-ordinates, we can easily show that
the energy is the sum of the energies of its principal oscillations. Let the system
be referred to any co-ordinates 6, (p, &c. and let it perform the principal oscillation
whose type is, by equation (5),
T^M =&<'. = sin (:pii+ai).
Substituting in the expression for T, we have T — T-^pj^cos^{pjt + a^). Kepeating
this for all the principal oscillations, we have
kinetic energy = I\^i^ cos^ {pit + a^) + T^p^ cos^ {p>} + Oo) + &c.
where T^, Tg' ^^- ^^^ ^^^^ values of Twhen Zj, ili^, &c., L^, M„, &c. are substituted
for B', -^ + T^p.^ + ...
459. Ex. 1. A new constraint is introduced into a dynamical system, so that the
general co-ordinates 6, , &o. are constrained to vary in the ratio I, m, &c. If we
jnit d = lsin{p't + a.), (p = m?,m(p't-^a), &c., and if T', V be the values of T and
U— JJ^ when I, m, &c. are substituted for 6', , &c. Since tbe system starts
from rest, 6', (/>', &c. will all be very small quantities in the be-
ginning of the motion. If we reject all powers of 0', (f)', &c.
except the lowest which occur, we may regard A^^, &c. as con-
stants whose values are found by substituting for 0, ^, &c. their
initial values. Further, since the initial position of the system
is not a position of equilibrium, the first differential coefficients
of U with regard to 0, <^, &c. will not be zero. Let the initial
values of these differential coefficients be respectively a^, a^, &c.
The equations of motion are now
AJ"+A,," + ...=a, I
AJ"+A,,cj>"+...=a, [.
&c. = &c.]
From these equations we may determine the initial values
of 0", +... = Q\
H0' + K(l>'+...=o\ (10),
&c. = o)
where E, H, &c. are in general functions of 0, j>, &c., each of
which may be expanded in the form
E=E^ + E^0 + E^4> + ...
The equations of motion of Art. 388 will be
A J" + &c. = ftj + a J + &c. + \E+ ijlH+&c.\
AJ" ^ k(i. = a, + aj + &c. + \F ^- {xK-v &c\ ........ (II).
&c. = &c. j
Since the system has been disturbed from a position of equi-
librium, these equations are satisfied by = 0, , &c.
can be so chosen that the first powers in the expansion of C/'are
absent. In this case E^, E,^, &c. disappear from the equations,
so that it is unnecessary to calculate the geometrical equations
(10) beyond terms of the first order. The coefficients will then
be constant, and the equations can be integrated. As explained
in Art. 388, we may now reduce the number of variables 9, (j), &c.
to the proper number of independent co-ordinates. We may
therefore proceed as in Art. 444, without introducing \, fi, &c.
into the equations.
If, however, we prefer to retain the quantities X^, /u-^, &c., we
Bee by equations (10) and (13) that we may obtain the periods
exactly as in Art. 4i5, by equating the discriminant oi p^T+ U'
to zero, where
U' = U+\(E^e + F^<}> + ...) + Ix^iE^d + K, F^, the
equation (1) shows that T k< T^+ V^— V^. Thus throughout the
subsequent motion the vis viva is restricted between zero and a
small positive quantity, and therefore the motion of the system
can never be great.
Also, since T is necessarily positive, the system ca,n never
deviate so far from the position of equilibrium that F should
become greater than T^ + F^. These tvv'o results may be stated
thus.
R. D. 24
370 SMALL OSCILLATIONS,
If a system he in equilibrium in a position in ivhich the potential
energy of the forces is a minimum or the work a maximum for all
displacements, then the system if slightly disjjlaced will never acquire
any large amount of vis viva, and will never deviate far from the
•position of equilibrium. The equilibrium is then said to he stable.
463. If the potential energy be an absolute maximum in the
position of equilibrium, V is less than V^ for all neighbouring
positions. By the same reasoning we see that T is always greater
than T^+ V^— V^, and the system cannot approach so near the
position of equilibrium that V should become greater than T^ 4- V^.
So far therefore as the equation of vis viva is concerned there is
nothing to prevent the system from departing widely from the
position of equilibrium. To determine this point we must examine
the other equations of motion*.
If any principal oscillation could exist, let the system be placed
at rest in an extreme position of that oscillation, then the sys-
tem will describe that principal oscillation and will therefore pass
through the position of equilibrium. But if T^ be zero, V can
never exceed V^, and can therefore never become equal to V^.
Hence the system cannot pass through the position of equilibrium.
It is unnecessary to pursue this line of reasoning further, for
the argument will be made clearer in the next proposition.
46 k We may also deduce the test of stability from the equa-
tions which determine the small oscillations of a system about a
position of equilibrium. Let the system be referred to its prin-
cipal co-ordinates, and let these be 6, 0, &c. Then we have
2T=d'^ + 4>'^+
2{U-U,) = h^e' + b^c^^ + ...:
where h^, h^, &c. are all constants, and C^ is the value of U in the
position of equilibrium. Taking as a type any one of Lagrange's
equations
ddT_clT^c7U
dt dd' d6 ~ de '
we have
0"- h^e = 0,
* This demonstration is twice given by Lagrange in his Mecanique Analytique.
In the form in which it appears in the first part of that work, V is expanded in
powers of the co-ordinates, which are supposed very small ; but in Section vi. of
the second part, this expansion is no longer used, and the proof appears almost
exactly as it is given in this treatise up to the asterisk. The demonstration in the
next proposition is simplified from that of Lagrange by the use of principal
co-ordinates.
ENERGY TEST OF STABILITY. 371
with similar equations for (f), yjr, &c. If b^ is positive, this equation
will give 6 in terms of real exponentials, and the equilibrium will
be unstable for all disturbances which affect 0, except such as
make the coefficient of the term containing the positive exponent
zero. If 5j is negative, will be expressed by a trigonometrical
term, and the equilibrium will be stable for all disturbances which
affect 9 only. In this demonstration the values of 6^, b^, &c. are
supposed not to be zero.
If in the position of equilibrium Z7 is a maximum for all
possible displacements of the system, we must have b^, b^, &c. all
negative. Whatever disturbance is given to the system, it will
oscillate about the position of equilibrium, and that position is
then stable. If Z7 is a maximum for some displacements and a
minimum for others, some of the coefficients b^, b^, &c. will be
negative and some positive. In this case if the system be dis-
turbed in some directions, it will oscillate about the position of
equilibrium; if disturbed in other directions, it may deviate more
and more from the position of equilibrium. The equilibrium is
therefore stable for all disturbances in certain directions, and un-
stable for disturbances in other directions. If Z7 is a minimum
in the position of equilibrium for all displacements, the coefficients
b^, b^, &c. are all positive, the equilibrium will then be unstable
for displacements in all directions. Briefly, we may sum up the
results thus,
The system will oscillate about the position of equilibrium for
all disturbances if the potential energy is a minimum for all dis-
placements. It will oscillate for some disturbances and not for others
if the potential energy is neither a maximum nor a minimum. It
will not oscillate for any disturbance if the potential energy is a
maximum for all displacements.
It appears from this theorem that the stability or instability of
a position of equilibrium does not depend on the inertia of the
system but only on the force function. The rule is, give the
system a sufficient number of small arbitrary displacements, so
that all possible displacements may be compounded of these. By
examining the work done by the forces in these displacements we
can determine whether the potential energy is a maximum or
minimum or neither.
Ex. 1. A perfectly free particle is in equilibrium under the attraction of any
number of fixed bodies. Show that if the law of attraction be the inverse square,
the equilibrium is unstable. [Earnshaw's Theorem.]
Let be the position of equilibrium, Ox, Oij, Oz any three rectangular axes,
^ly ^iy ^y
thou if V bo the potential of the bodies, 6^ = -— , &2 = -— ^, 6j, = _^. But since
uX ^l) (m^
the sum of these is zero, 6^, i^, 63 cannot all have the same sign.
Ex. 2. Hence show that if any number of particles, mutually repelling each
24— -2
372 SMALL OSCILLATIONS.
other, be contained in a vessel, and be in equilibrium, the equilibrium will be
unstable unless they all lie on the containing surface. [Sir W. Thomson, Camb.
Math. Journal, 1845.]
465. We may in certain cases apply the energy criterion to determine when
a given motion is stable. Let a dynamical system be in motion in any
manner under a conservative system of forces, and let E be its energy. Then
jE" is a known function of the co-ordinates d, <}>, &c. and their first difierential co-
efficients 6', ', &c. ; this is constant and equal to h for the given motion. Sup-
pose that either some or all of the other first integrals of the equations of motion
are also known, let these be
i?i {6, e', &c.) = Ci, Fg {e, 6', &c.) = Cg, &c. =&c.
For the purposes of this proposition, let us regard 6 and 6', and 0', &c. as inde-
pendent variables, except so far as they are connected by the equations just written
down. Then if E be an absolute maximum, or an absolute minimum, for all
variations of B, ff, &c. (those corresponding to the given motion making E con-
stant), the motion is stable for all disturbances which do not alter the constants
Ci, Cg, &c.
This result follows from the same reasoning as in Art. 462, which we may
briefly recapitulate thus. Let as many of the letters as is possible be found
from the first integrals in terms of the rest, and substituted in the expression for
E. Let ^, ^', &c. be these remaining letters, then we have
E ^f {yp, f, &c., Ci, Cg, &c.) = h.
Let the system be started in some manner slightly different from that given, then
the constant h is altered into h + 8h. First let ^ be a minimum along the given
motion, then any change whatever of the letters ^, ^', &c. increases E, and it
follows that the disturbed motion cannot deviate so far from the given motion
that the change in E becomes greater than 5h. Similarly, if E be an absolute
maximum, the same result will follow.
The same argument will apply to any first integral of the equations of motion,
besides the energy integral. If any one of the functions F^, F^, &o., which con-
tains all the letters, be an absolute maximum or minimum, then the motion is
stable for all displacements which do not alter the constants of the other integrals
used.
When the system is disturbed from a position of equilibrium which is defined,
as in Art. 444, by the vanishing of the co-ordinates 6,
where A-^^, A-^^, &c. are all constants, and U is independent of 6', 4>', &c. Here
the terms which constitute the kinetic energy, being necessarily positive and
vanishing with 6', cp' &c., are evidently a minimum for all variations of 6', ', &c.
We see, without the use of any other integrals, that if - Z7 be a minimum for
all variations of d, 4>, &c., E will be an absolute minimum, and that therefore
the equilibrium is stable.
466. It often happens that the expression for the energy is not a function of
some of the co-ordinates, though it is a function of the differential coefficients of
all the co-ordinates with regard to the time. When this is the case, the system
admits of what we shall caU a steady motion. Let x, y, &c. be the co-ordinates
which are absent from the expression for the energy E, and let ^, o?, &c. be the
ENERGY TEST OF STABILITY. 373
remaining co-ordinates, then E is a function of f, rj, &c., ^', V, &c,, x', y', &g. If
we form tlie equations of motion by Lagrange's rule (Art. 369), these equations
■will contain |, rj, |', -t]', ^", rj", x', y', x", ?/", &c. , &c. Since these equations do
not contain t explicitly, they may be satisfied by putting x'—a, y' = h, &c., ^ = a,
9j=/3, &c., where a, b, &c., a, j8, &c. are constants to be determined by substituting
in the equations. If 6 stand for any one of the co-ordinates, it is evident that
dT dT
-~- and 3—, will both be constants after the substitution is made. The constants
da dd
must therefore satisfy the typical equation — =0 (Art. 369). Since x, y, &c.
do
are absent from the expressions for T and U, this is an identity if we write any of
these co-ordinates for 6. Hence we have as many equations, viz.
dJT+TP, d{T+U) _
d| ~' dv ^''
as there are co-ordinates ^, tj, &c. present in the expressions for T and U. The
quantities a, b, &c. are therefore undetermined except by the initial conditions,
while a, ;8, &c. may be found in terms of a, b, &c. by these equations. These
equations may be conveniently remembered by the following rule. In the Lagran-
gian function, tvhich is the difference between the kinetic and potential energies,
write for the differential coefficients, their assumed constant values in the steady
motion, viz. x'~a, &g., §' = 0, &c. Differentiating the result partially loith regard
to each of the remaining co-ordinates, loe obtain the equations of steady motion.
. 467. To determine if this motion is stable, we must by Art. 465 use the integrals
3— 7=M, -T— =i;, &c., where u, v, &c. are constants. Let
dx dy
T=^^{xx)x'^ + {xi)x'^'-^&(i (2),
where the coefficients of the accented letters, viz. the quantities in brackets, are
all known functions of ^, 17, &c., but not of x, y, &c. The integrals may then be
written in the form
[xx) x' + [xy) y' -{■ ... = m- {x^) ^' - (xtj) yf -&C.
{xy)x'+{yy)y'+...= v-{yi)^'-{^j-r,)'n'-&c.y (3).
&c. =&c.
For the sake of brevity, let us call the right hand sides of these equations u- X,
v-Y, &c. Since T is a quadratic function of the accented letters, we may write
it in the form
T = I m r- + m rv +&G.+'^x'{u+x)+^^y'{v+Y)+&c.
If we substitute in the terms after the first &c. the values of x', y' given by (3)
we obtain the determinant
2A
0, u + X, v+Y, &c.
u-X, (ocx) (xy), &c.
v-Y, (xy), {yy), &c.
&c.
where A is the discriminant of T, when ^', 97', &c. have been put zero. If we change
the signs of A', Y, &c., this determinant is unaltered, hence when expanded such
terms as uX, vX, &c. cannot occur. If therefore, we put
374 SMALL OSCILLATIONS.
2A
« V ..
u (xx) (xy) ..
.(4),
and expand the first determinant, we have
T^F+lB^^r- + J^i,i'v'+ (5),
■where the terms after F express some homogeneous quadratic function of f, t)', &c.
When ^', 7)', &c. are put zero, the process of finding F is exactly that described
in Art. 378, as the Hamiltonian method of forming the reciprocal function.
Following the same proof* as in that Article, we may show that if | be any letter
contained in T, we have -tl = - •:tc ■ Hence the equations of steady motion (1) maij
also he written in the form
d{F-v) ^^ d{F-^J) ^^^
where F — U xs the energy expressed as a functioji of n, v, &c. instead ofx', y', &c.,
the other accented letters, viz. |', rj', &c. being put equal to zero either before or
after differentiation.
Further T is essentially positive for all values of a;', y', &c. and therefore for
such as make u, v, &c. all zero. Hence the quadratic expression Bj^f'2 + &c. is a
minimum when |', t)', &c. are zero. If then the function F — U is a minimum for all
variations of f , 17, &c. , the steady motion given by (6) is stable for all disturbances
which do not alter the momenta u, v, &c.
468. If the energy be a function of one only of the co-ordinates, though it is a
function of the differential coefficients of all of them, ive may show conversely that
the steady motion ivill not be stable unless F -JJ is a minimum.
Let ^ be this single co-ordinate, then following the same notation as before, we
have by Vis Viva
\B^^^^-^F-U=h.
Differentiating with regard to t, and treating B-^^ as constant because we shall
neglect the square of |', we obtain
* Taking the notation of Ai-t. 378, the proof is as follows. The total differential
of T^ when aU the letters vary is
^T.= -'lide-f^di^[-'^^^^)de'^e'du^&o.^,
as before, the quantity in brackets vanishes, and hence when T^ is expressed as a
dT dT
fimctiou of d, = F{t), &c Then exactly as in Art. 428, we substi-
tute 6 =f[t) + X, (j) = F{t) + y, &c., in the equations of motion.
The squares of x, y, &c. being neglected, we have certain linear
equations to find a?, y, &c. These equations can, however, seldom
be solved unless we can make t disappear explicitly from them.
When this can be done the linear equations can be solved by the
usual known methods, and the required oscillations are then found.
In what follows we shall first illustrate the method just de-
scribed by forming the equations in a few interesting cases from
the beginning. We shall then generalize the process and obtain a
determinantal equation analogous to that given by Lagrange for
oscillations about a position of equilibrium. This equation will be
adapted to all cases which lead to differential equations with
constant coefficients.
471. Ex. 1. To find the motion of the balls in Watfs Governor of the steam
engine.
The mode in wMch this works to moderate the fluctuations of the engine is well
known. A somewhat similar apparatus has been used to regulate the motion of
clocks, and in other cases where uniformity of motion is required. If there be any
increase in the driving power of the engine, or any diminution of the load, so that
the engine begins to move too fast, the balls, by their increased centrifugal force,
open outwards, and by means of a lever either cut off the driving power or increase
the load by a quantity proportional to the angle opened out. If on the other hand
the engine goes too slow, the baUs fall inward, and more driving power is called
into action. In the case of the steam engine the lever is attached to the throttle-
valve, and thus regulates the supply of steam. It is clear that a complete adapta-
tion of the driving power to the load cannot take place instantaneously, but the
machine will make a series of small oscillations about a mean state of steady
motion. The problem to be considered may therefore be stated thus : —
Two equal rods OA, OA', each of length I, are connected with a vertical spindle
by means of a hinge at which permits free motion in the vertical plane A OA'. At
A and A' are attached two baUs, each of mass m. To represent the inertia of the
other parts of the engine we shall suppose a horizontal fly-wheel attached to the
spindle, v/hose moment of inertia about the spindle is I. When the machine is in
uniform motion, the rods are inclined at some angle a to the vertical, and turn
round it with uniform angular velocity n. If, owing to any disturbance of the
motion, the rods have opened out to an angle with the vertical, a force is called
into play whose moment about the spindle is - j8 (d-a). It is required to find the
oscillations about the state of steady motion.
Let 4> be the angle the plane AOA' makes with some vertical plane fixed in
space. The equation of angular momentum about the spindle is
clt
[(I+2mFsin2^)^j=:-^(^-a) (1),
THE GOVERNOR. 877
where ink^ is the moment of inertia of a rod and ball about a perpendicular to the
rod through 0, the balls being regarded as indefinitely small heavy particles. The
semi Vis Viva of the system is
and the moment of the impressed forces on either rod and ball about a horizontal
through perpendicular to the plane AOA' is o'Ja= -mgh sin 0, where h is the dis-
tance of the centre of gravity of a rod and ball from 0. Hence by Lagrange's
d (IT dT dU
^'-sin^cos^^y^-^sin^ (2),
where a has been written for - . This equation might also have been obtained by
II
taking the acceleration of either ball, treated as a particle, in a direction perpen-
dicular to the rod in the plane in which 6 is measured.
To find the steady motion we put d^a, -^=«, the second equation then gives
n^cos a =- . To find the oscillations, we put ^ = a + i(;, -^=« + 2/. The two equa-
a "^
tions then become
(^■11 (foe 1
(I + 2m¥- ski2 a)^ + ^mhH sin 2a ^ = - jSa; j
—^ — n sin 2ay — i n^ cos 2a - - cos a j a;
To solve these equations, we must write them in the form
( sin2aD +
(1)2 -I- n^ sin- a) a; - ?i sin 2a?/ = 0-*
where the symbol D stands for the operation — . Ehminating y by cross multipli-
cation we have
[(2-i^+^^''^)^'+"'^'"'"('+'''''''^+2i)''''27&'^^^"'"]^=^-
The real root of this cubic equation is necessarily negative because the last term
is positive. The other two roots are imaginary because the term D^ has dis-
appeared between two terms of like signs. Also the sum of the three roots being
zero, the real parts of the two imaginary roots must be positive. Let these roots
therefore be - 2p and p±q^J -1. Then
X = He-^J"^ + KeP* sin ((jft + L),
where II, K, L are three undetermined constants depending on the nature of the
initial disturbance. Thus it appears that the oscillation is unstable. The balls
will alternately approach and recede from the vertical spindle with increasing
violence.
378 SMALL OSCILLATIONS.
472. A common defect of governors is that they act too quickly, and thus
produce considerable oscillation of speed in the engine. If the engine is -worldug
too violently, the governor cuts off the steam, but owing to the inertia of the parts
of the machinery, the engine does not immediately take up the proper speed.
The consequence is that the balls continue to separate after they have reduced
the supply of steam to the proper amount, and thus too much steam is cut off.
Similar remarks apply when the balls are approaching each other, and a con-
siderable oscillation is thereby produced. This fault may be very much modified
by applying some resistance to the motion of the governor.
In the same way when the motion of clock-work is regulated by centrifugal
balls, it is found as a matter of observation that there is a strong tendency to
irregularity. If the balls once receive in the slightest degree an elliiitic motion,
the resistance ^ {9 -a) by which the motion of the balls is regulated may tend to
render the ellipse more and more elliptical. To correct this some other resistance
must be called into play. This resistance should be of such a character that it
does not affect the circular motion and is only produced by the ellii^ticity of the
movement.
One method of effecting this has been suggested by Sir G. Airy. The elliptic
motion of the balls may be made to cause a sHder on the vertical spindle to rise
and fall. If this be connected with a horizontal circular plate in a vertical
cylinder of slightly greater radius, and filled with water, the shder may be made
to move the plate up and down by its oscillations. Thus the shder may be
subjected to a very great resistance, tending to diminish its oscillations, while its
place of rest, as depending on statical, or slowly altering forces, is totally un-
affected. Memoirs of the Astronomical Society of London, Vol. xx., 1851.
The general effect of the water will be to produce a resistance varying as the
velocity, and may therefore be represented by a term - 7 -r- on the right hand of
equation (2). The solution being continued as before, the cubic will now take the
form
If the roots of this cubic are real, they are all negative, and the value of x takes the
form
x = Ae-''^ + Be-i^t+Ce-''*,
where - X, - /x, -v are the roots, and A, B, C are three undetermined constants.
If one root only is real, that root is negative, and if the other two he p^qsl -1 the
value of X takes the form
X - He-''^ + Kei"- sin [qt + L),
where H, K, L as before are undetermined constants.
In order that the motion may be stable it is necessary that i) should be negative.
The analytical condition* of this is
* If the roots of the cubic ax^ + 5x- + ca; + d = be a; = a i ^ ^/( - 1) and 7, we have
— r= 2a-l-7, - =27a + a" + /3^, — —{a^ + B")y, whence we easily deduce 5 —
a a ci a-'
= -2a{(a-f-7)"-|-/32}; hence he- ad and a have always opposite signs. See Art. 436.
THE GOVERNOR. 379
If 7 be sufficiently great this condition may be satisfied. The uniformity of
motion of the rods round the yertical will then be disturbed by an oscillation whose
magnitude is continually decreasing and whose period is — . By properly choosing
the magnitude of I when constructing the instrument, the period may sometimes
be so arranged as to produce the least possible ill effect. If the period be made
very long the instrument will work smoothly. If it can be made very short there
will be less deviation from chcular motion.
In this investigation no notice has been taken of the frictions at the hinge and
at the mechanical appliances of the Governor, which may not be inconsiderable.
These in many cases tend to reduce the oscillation and keej) it within bounds.
473, In the case of "Watt's Governor if any permanent change be made in the
relation between the driving power and the load, the state of uniform motion which
the engine will finally assume is different from that which it had before the change.
Thus, when the engine is driving a given number of looms, let the rods OA, OA' of
the Governor be incUned to each other at an angle 2a and be revolving about the
vertical with an angular velocity n. If some large number of the looms is sud-
denly disconnected from the engine, the balls will separate from each other, and the
rods will become inclined at some other angle 2a'. In this case, if n' be the angular
velocity about the vertical, n'^ cos a' = «^cos a. The rate of the engine is therefore
altered, it works quicker with a less load than with a greater. This is a great
defect of Watt's Governor. For this reason it has been suggested that the term
Governor is inappropriate, the instrument being in fact only a moderator of the
fluctuations of the engine.
This defect may be considerably decreased by the use of Huyghens' parabolic
pendulum. In this instnrment the centres of gravity ^ , ^' of the balls are made to
move along the arc of a parabola whose axis is the axis of revolution. Let AN be
an ordinate of the parabola, AG the normal, then NG is constant and equal to L,
where 2L is the latus rectum. Regarding the balls as particles, and neglecting the
inertia of the rods which connect them with the throttle valve, we see by the
triangle of forces that the balls wUl rest in any positions on the parabola, if
n^L=g, where n is the angular velocity of the balls about the vertical through 0.
It is also clear that when the angular velocity is not that given by this formula, the
balls (unless placed at the vertex) must slide along the arc. Let us now consider
how this modification of the governor affects the working of the engine. When the
load is diminished the engine begins to quicken ; the balls separate and the steam is
cut off. It is clear that equilibrium will not be established until the quantity of
steam admitted is just such as to cause the engine to move at exactly the same rate
as before.
Ex. Show that when the inertia of the rod and balls are taken account of,
the centre of gravity of either ball and rod must be constrained to describe a
parabola whose latus rectum is independent of the radius of the ball, if the
Governor is to cause the engine always to move at a given rate.
474. The reader who may be interested in the subject of Governors may refer
to an article by Sir G. Airy, Vol. XI. of the Memoirs of the Astronomical Society,
1840, where four different coiistructious are considered. Ho may also consult an
380 SMALL OSCILLATIONS.
article by Mr Siemens in the Phil. Trans, for 1866, and a brief sketch of several
kinds of governors by Prof. Maxivell in the Phil. Mag. for 1868. An account of
some experiments by Mr Ellery, on Huyghens' parabolic pendulum, may be found
in the Astronomical Notices for December, 1875.
475. Ex. 2. It has been shown in Art. 282 that if three particles be placed at the
corners of an equiangular triangle and properly projected, they will move under
their mutual attractions so as always to remain at the angular points of an equi-
lateral triangle. These we may call Laplace's three particles. It is our present
object to determine if this motion is stable or unstable*.
Let the mass M of the particle to be reduced to rest be taken as unity, and let
m, m' be the masses of the other two. Let r, r', R be the distances between the
particles Mm, Mm', mm'; and let 0', (p, \p be the angles opposite to these distances.
If ^, ^' be the angles r, r' make with a straight line fixed in space, and if the law of
attraction be the inverse /cth power of the distance, the equations of motion are
d'^r ^d9\^ 1 + m m'cos^ m'cosA „
V' ^'\dtj
dt'^ \dtj ^K J.'" jfj«
1 d / „ d9\ m' sin \J/ m' sin X-\2ahnD+-^ m{K + l^^^^
2hnDx+ a5i)V+ J2&wD-^(K+l)m|z+ ja62)2_^„i(K + l)a 7=0.
* In a brief note in Jullien's Problems, Vol. n. p. 29, it is mentioned that this
question has been discussed by M. Gascheau in a These de M^canique, the particles
being supposed to attract each other according to the law of nature. The result
arrived at is that the motion is stable when the square of the sum of the masses is
greater than 27 times the sum of the products of the masses taken two and two.
No reference is given to where M. Gascheau's work can be found, and the author is
therefore unable to give a description of the process employed.
Laplace's three particles. 381
476. To solve these we i)ut x = Ae^\ y^Be^*, X=Ge^\ Y=He^K Substituting
and eliminating the ratios of A, B, G and H we obtain a determinantal equation
whose constituents are the coefficients of x, y, X and Y with X written for D, This
equation will give six values of X. We see at once that one factor is X. This might
have been expected, because we know that a variation of y with x, X and Y all zero,
is a possible motion. Again, some variation of x and y with X and Y both zero is
also a possible motion, hence some factor of the determinant can be found by ex-
amining the first two columns. By subtracting from the first 2n times the second
column we find that this factor is b\^ - {/c - 3) (1 + ?» + m') = 0.
To find the other factors we divide the determinant by the factors ah-eady
found. Then subtracting the first row from the third and the second from the
fourth we have three zeros in the first column and two in the second. The
expansion is then easy. We see that there is another factor X, also
3
¥\^ + &X2(3 - k) (1 + m + m') + . (1 + k)^("i + »*' + mm') = 0.
The two zero roots give x = A^ + A.2t with similar expressions y, X and Y. But
K+1 A
by substitution in the equations of motion we see that x—A-^^, y=B^ ^ nt,
X—Q> and F=0. These roots therefore indicate merely a permanent change in the
size of the triangle. On examining the other values of X^, we find (1) The motion
cannot be stable unless k is less than 3. (2) The motion is stable whatever the
masses may be, if the law of force be expressed by any positive power of the dis-
tance or any negative power less than unity. (8) The motion is stable to a first
approximation if
{M+m + m'f /l+K-y
Mm + Mim! -(- mm \3 - /c/ '
where M, m, m' are the masses. To express the co-ordinates in terms of the time,
we must return to the differential equations of the second order. The results are
rather long, and it may be sufficient to state that when, as in the solar system, two
of the masses are much smaller than the third, the inequahties in their angular
distances, as seen from the large body, have much greater coefficients than their
linear distances from the same body.
477. To form the general equations of oscillation of a dynami-
cal system about a state of steady motion.
Let the system he referred to any co-ordinates 6, (J), yjr, &c.
Let the state of motion about which the system is oscillating be
determined by 6 =f (t), ^=F [t), &c., then as explained in Art. 470
we shall put 6=f{t) + x, ^=-F{t)+y, &c. Let the Lagrangian
function L (see Art. 381) be expanded in powers of ic, y, &c., as
follows :
i = i„ + ^j.x -1- J^?/ -f &c. -f B^x + B,^y' -f &c.
-f ^ {A^^x^ + 2A^^xy + &c.) + | (D^^af' + 2BJy' \ &c.)
882 SMALL OSCILLATIONS.
We shall now define a steady motion to be one in which all the
coefficients in this expansion are independent of the time. The
physical characteristic of such a motion is that when referred to
proper co-ordinates the same oscillations follow from the same dis-
turbance of the same co-ordinate at whatever instant it may be
applied to the motion. If the coefficients are not constant for the
co-ordinates chosen it may be possible to make them constant by
a change of co-ordinates. There are obviously many systems of
co-ordinates which may be chosen, and a set may generally be
found by a simple examination of the steady motion. If there are
any quantities which are constant during the steady motion, such
as those called f, t), &c. in Art. 466, these may serve for some of
the co-ordinates, others may be found by considering what quanti-
ties appear only as differential coefficients or velocities, for example
those called x, y, &c. in the same Article. If none of these are
obvious, we may sometimes obtain them by combining the existing
co-ordinates. Practically these will be the most convenient
methods of discovering the proper co-ordinates.
478. To obtain the equations of motion we must now substi-
tute the value of L in the Lagrangian equations
d_clL_ clL _ A ^ ^Q
cltdx dx '
and reject the squares of small quantities. The steady motion
being given by x, y, &c. all zero, each of these must be satisfied
when we omit the terms containing x, y, &c. We thus obtain the
equations of steady motion, viz.
^^ = 0, J^, = 0, &c. = 0,
which by Taylor's theorem are the same as the equations (1) of
steady motion given in Art. 466.
Omitting these terms and retaining the first powers of all the
small quantities we obtain the equations of small oscillations, of
which the following is a specimen :
To solve these we write x = Le^*, y = Me^^, &c. Substituting and
eliminating the ratios of L, M, &c. we obtain the following deter-
minantal equation
ABOUT STEADY MOTION.
383
A^^--iu
&C.
&c.
K?^'-A.
A3^^-^23
-(^32-^23)^
As^^-^sa
&c.
&c.
&c.
&C.
&c.
= 0.
If in this equation we write — X for X, the rows of the new
determinant are the same as the columns of the old, so that the
determinant is unaltered. When expanded the equation contains
only even powers of A-.
479. Regarding this as an equation to find X^ we notice that
if the roots are all real and negative, each of the co-ordinates x, y,
&c. can be expressed in a series of trigonometrical terms having
different periods; the motion will therefore be stable. If any one
of the roots is imaginary or if any one is real and positive, there
will be both positive and negative real exponentials entering into
the expressions for x, y, &c. and therefore the motion will be un-
stable. The condition of dynamical stability is therefore tha,t the
roots of this equation must all be of the form \= ± jms/ — 1, where
IM is some real quantity.
480. It follows also that when a system, under the action of
forces which have a potential, oscillates about a stable state of
steady motion, the oscillations of the co-ordinates are represented
by trigonometrical terms of the form A sin (\t + a) which are not
accompanied by any real exponential factors such as those which
occurred in the problem of the Governor.
We see further that there will in general be as many finite
values of A,^ and therefore as many trigonometrical terms of differ-
ent periods as there are co-ordinates. It often happens, as ex-
plained in Art. 477, that some of the co-ordinates are absent from
the expression for L, appearing only as differential coefficients.
Suppose for example 6 to be absent; then A^^, A^^, &c. are all
zero, and we may divide A, both out of the first line and the first
column of the fundamental determinant. We therefore have two
zero values of X, while at the same time the number of finite
values of X^ is diminished by unity. Hence the number of trigo-
nometrical terms of different periods cannot exceed the number of
384 SMALL OSCILLATIONS.
co-ordinates whicli explicitly enter into the Lagrangian function.
For example in Art. 374, the function T— t/'has only the co-ordi-
nate 6 explicitly expressed, the others <^' and \|r' appearing only as
differential coefficients. It follows that if a top is disturbed from
a state of steady motion, 'there will be but one period in the
oscillation.
481. The relations between the coefficients L, M, &c. in the
exponential values of x, y, &c. may be obtained without difficulty
if we remember that the several lines of the fundamental determi-
nant are really the equations of motion. Taking any one line ;
multiply the first constituent by L, the second by M, &c. and
equate the sum to zero. We thus obtain as many equations as
there are co-ordinates. On the whole we shall have, exactly as in
Art. 445, twice as many arbitrary constants as there are co-ordi-
nates, all the other constants being determined by the equations
just found. The arbitrary constants are determined by the initial
values of the co-ordinates and their differential coefficients.
But, unlike Art. 445, the quantity X occurs in the first power
in each of these equations, so that the ratios of L, M, &c. thus
found may be imaginary. The expressions for the co-ordinates
when rationalized may therefore take the form
x = A^ sin {\t + a J + A^ sin (\i +a^ + ...
y=B^ sin (X^t + /3 J + i?^ sin {\t + ^,) + ...
z = &c.
where a^ is not necessarily equal to yS^^, nor ci^ to yS^, &c., though
they are connected together.
482. When the initial conditions are such that every co-
ordinate is expressed by a trigonometrical term of one and the
same period, the system is said to be performing a principal or
harmonic oscillation. Thus each trigonometrical term corresponds
to a principal oscillation, and any oscillation of the system is
therefore said to be compounded of its principal oscillations. The
physical characteristic of a principal oscillation is that the motion
of every part of the system is repeated at a constant interval.
483. The stability of the motion depends on the nature of the
roots of the fundamental determinant. If we expand the determi-
nant we may use the methods given in the theory of equations to
discover if the roots are all of the proper form. This however is
often tedious and we may sometimes settle the point by a simple
examination of the determinant as it stands.
J
ABOUT STEADY MOTION. 385
In practice it frequently happens that the determinant is
reduced to two rows. If the invariants be written
the conditions of stability are
(1) A is positive.
(2) (C21 ~ ^12)^ - ® is positive and greater than 2 'jAB,
These conditions may also be expressed thus. Let a and /3 be
the roots of the quadratic formed by omitting the terms containing
Cj2 and Cgj. Then by Art. 448, a and /3 are real. If a and yS are
both negative the motion is stable. If both are positive, the
C ~ (7 .
motion is stable or unstable according as --^— - ^^ is numerically
y B
greater or less than fja + tj^, the roots being taken positively. If
a and /3 have opposite signs, the motion is unstable.
Whatever may be the number of co-ordinates, it may be shown
that the motion caanot be stable unless the discriminant of
A^^x^ + 'i.A^^xy + &;c. is positive or negative according as the
number of rows is even or odd^
The following theorem is also useful. Beginning with the
fundamental determinant we may form a series of determinants,
each being obtained from the preceding by erasing the first line
and the first column. As we may supplement the fundamental
determinant with a row and a column of zeros added on at the
bottom and right-hand side with unity at the right-hand bottom
corner, we may suppose the series of determinants to terminate
with unity. Let us substitute in the series any negative value of
X^ and count the number of Variations of sign in the series. Then
as X.^ changes from — 00 to 0, there cannot be fewer negative roots
between any two given values of X,^ than there are losses in the
number of variations of sign corresponding to the two values of X^.
If there be more negative roots than losses the excess must be an
even number.
484. Ex. A homogeneous sphere of unit mass and radius a is suspended from
a fixed point by a string of length h, and is set in rotation about tlie vertical diame-
ter. When the sphere is slightly disturbed, let hx, by and b be the co-ordinates of
the point on the surface to which the string is attached ; &a; + a^, by -f ai], and b + a
the co-ordinates of the centre, the fixed point being the origin and the axis of 2
being vertical and downwards. Also let x — 'P + f where 1, the ratio of a^ to a.^ i^
real and different from unity. Hence we must have |-<1. Let
then ;(- = sin ^ ; and therefore a = cos 26 + sin 2^ V— 1.
2c ~
Hence, by what we proved before,
(cos 26 + sin 26 V^)"^' = (cos 26 - sin 26 V^)"^' ;
.-. sin 2 (w + 1) ^ = 0, or ^ = sin
«7r
2c 2 (w + 1) '
27r
and the period of any term = — .
If m and I be indefinitely small and n indefinitely large, the
loaded string may be regarded as a uniform string of length
{n + 1) 1 = L and mass nm = Hi stretched between two fixed points
25—2
388 S^ULL OSCILLATIONS.
witli a tension T. In this case the expression just found reduces
to^ = 7rzY^.
487. If we substitute tliese values of ^ in the expressions for a^ and a^, we
easily find
_-, . Mir . (^ . . i'f )
2,, = 2C,sin^-^ .sm j 2ci sm ^^^^^^ + a.j ,
where Cj has been written for 2A ^ -1, a^ for a, and the symbol 2 implies summa-
tion for all integer values of i from i = l to i=n. This expression has n terms,
and thus we have 2re arbitrary constants, viz. Cj, Cg ••. C'„ and o^, 02 ... a„. These
are to be determined by the known initial values of yi,y^, &c. and -p , -^ , &c.
To find these it will be more convenient to write the expression in the form
.^„ . hiir . in ^ . itr ) _„ . Tcitr ir> . • i'"' )
T/k = z,Ei sm sm j 2ct sm ;r- -, } + ^F, sm — :, cos \ 2ct sm
71+1 2(71+1) ' 71 + 1 (
2(7l+l)i
Putting t — 0, we have the two tyioical equations
Mo=2JPiSm^^,
— \-^\ =2EiSm z sm^r^ :-.
2c\_dtjQ ' 71 + 1 2(71 + 1)
It is a theorem in Trigonometry that if i, i' be any integers between and
71+1, the sum of the series Ssin sin — -^ taken from Z;=l to Z;=7i is zero
71 + 1 71 + 1
71 + 1
when i is different from i' and the sum is equal to — -— when i = i'. This may be
proved by expressing the general term of the series as the difference of two cosines,
thus separating the given series into two series, each consisting of cosines of angles
in arithmetical progression. Summing these from ^ = to /i;=7i when i and i' are
both even or both odd, and from k = l to k^n when i is even and i' odd, we easily
find the whole sum to be zero when i and i' are unequal. This change in the limits
of the summation only adds a term which is zero to one end of the original series
and therefore does not affect the sum. When i and i' are equal the value of the
series may be found in a similar manner.
This theorem will at once enable us to find the general values of Ei and Fi.
Let us multiply both sides of the first typical equation by the coefficient of J^^ and
Bum all the series of which it is the type. We have
^ (- . . Uv ) 71 + 1
%\[y,]oBin^=-^F,.
where 2 implies summation for all values of k from lc = l to k = n. Treating the
second equation in the same way, we have
2c sin
^^^•2![tl«'^^J="-f'^.•
2(71+1)
488. Lagrange in his Mecanique Analytique has applied his general equations
of motion to the solution of the preceding problem. He has also determined the
THE CAVENDISH EXPERIMENT. 389
osciUationg of an inextensible string charged with any number of weights, and
suspended by both ends or by one only. Though several solutions of these pro-
blems had been given before his time, he considers that they were aU more or less
incomplete.
489. Ex. 1. A light elastic string of length nl and coefficient of elasticity E is
loaded with n particles each of mass m, ranged at intervals I along it beginning at
one extremity. If it be suspended by the other extremity, prove that the periods of
its vertical oscillations will be given by the formula tt * / -=- cosec - , where
i = 0, 1, 2 ... Ji- 1 successively. Hence show that the periods of vertical oscillation
of a heavy elastic string will be given by the formula ^. — ^ a / —^ , where L is the
length of the string, M its mass, and i is zero or any positive integer. [Math.
Tripos, 1871.]
Ex. 2. An infinite number of equal particles, each of mass m, are placed in a
row at distances each equal to I and mutually repel each other so that the force
between any two is 7n^f'(D), where D is the distance between those two. A disturb-
ance is given to the system such that each particle makes oscillations in the direc-
tion of the row whose extent is very small compared with I. Show that the
disturbance of the k^^ particle, counting from any one particle, is given by the series
o—
Sa cos — {vtiikl), where S implies summation for all values of X, and
.=ijrn\v-m ('47+2V'(2.) C-^)%^c.j*,
and 9= — . Thence show that all very long waves travel with the same velocity.
A
lf/(z)=/tz~", show that v is infinite unless n is greater than 3, [Phil. Mag.]
The Cavendish Experiment.
490. As an example of the mode iu which the theory of small
oscillations may be used as a means of discovery we have selected
the Cavendish Experiment. The object of this experiment is to
compare the mass of the earth with that of some given body. The
plan of effecting this by means of a torsion-rod was first suggested
by the Rev. John Michell. As he died before he had time to
enter on the experiments, his plan was taken up by Mr Cavendish,
who published the result of his labours in the Fhil. Trans, for
1798. His experiments being few in number, it was thought
proper to have a new determination. Accordingly in 1837, a
grant of £500 was obtained from the Government to defray the
expenses of the experiments. The theory and the analytical
formulae were supplied by Sir G. Airy, while the arrangement
of the plan of operation and the task of making the experiments
were undertaken by Mr Baily. Mr Baily made upwards of two
thousand experiments with balls of different weights and sizes,
and suspended in a variety of ways, a full account of which is
390
SMALL OSCILLATIONS.
given in the Memoirs of the Astronomical Society, Yol. xiv.
The experiments were, in general, conducted in the following
manner.
491. Two small equal balls were attached to the extremities
of a fine rod called the torsion-rod, and the rod itself was sus-
pended by a string fixed to its middle point G. Two large
spherical masses A, B were fastened on the ends of a plank
which could turn freely about its middle point 0. The point
was vertically under G and so placed that the four centres of
gravity of the four balls were in one horizontal plane.
First, suppose the plank to be placed at right angles to the
torsion- rod, then the rod will take up some position of equilibrium
called the neutral position, in which the string has no torsion.
Let this be represented in the figure by Gol. Now let the masses
A and B be moved round into some position B^A^, making a
not very large angle with the neutral position of the torsion-rod.
The attractions of the masses A and B on the balls will draw the
torsion-rod out of its neutral position into a new position of equi-
librium, in which the attraction is balanced by the torsion of the
string. Let this be represented in the figure by GE^. The angle
of deviation Efix and the time of oscillation of the rod about this
position of equilibrium might be observed.
Secondly, replace the plank AB Sit right angles to the neu-
tral position of the rod, and move it in the opposite direction until
the masses A and B come into some position A^B^ near the rod
but on the side opposite to B^A^. Then the torsion-rod will
perform oscillations about another position of equilibrium GE^
under the influence of the attraction of the masses and the torsion
of the string. As before, the time of oscillation and the deviation
E^Ca might be observed.
In order to eliminate the errors of observation, this process
was repeated over and over again, and the mean results taken.
I
THE CAVENDISH EXPERIMENT; 391
The positions B^A^ and A^B^, into which the masses were alter-
nately put, were as nearly as possible the same throughout all the
experiments. The neutral position Ca of the rod very nearly
bisected the angle between B^A^ and A^B^, but as this neutral
position, possibly owing to changes in the torsion of the string,
was found to undergo slight changes of position, it is not to be
considered in any one experiment coincident with the bisector
of the angle AfiB^.
Let Cx be any line fixed in space from which the angles may
be measured. Let h be the angle xGa, which the neutral position
of the rod makes with Cx ; A and B the angles which the al-
ternate positions, B A^ and A^B^, of the straight line joining the
A + B
centres of the masses, make with Cx ; and let a = — ^ — . Also
let X be the angle which the torsion-rod makes with Cx at the
time t.
Supposing the masses to be in the position A^B^, the moment
about CO of their attractions on the two balls and on the rod will
be a function only of the angle between the rod and the line A^B^;
let this moment be represented by (f) (A — x). The whole appa-
ratus was enclosed in a wooden casing to protect it from any
currents of air. The attraction of this casing cannot be neglected.
As it may be different in different positions of the rod, let the
moment of its attraction about CO be '\fr{x). Also the torsion of
the string will be very nearly proportional to the angle through
which it has been twisted. Let its moment about CO be E{x—h).
If then / be the moment of inertia of the balls and rod about
the axis CO, the equation of motion will be
I^^<^{A-x) + -E(,a-b)
In''
Then ic = e + i sin {nt + L'),
where L and L' are two arbitrary constants. We sec therefore
that in the position of equilibrium the angle the torsion-rod
392 SMALL OSCILLATIONS.
makes with the axis of x is e, and the time of oscillation about
the position of equilibrium is — .
Let us now suppose the masses to be moved into their alternate
position A^B^ ; the moment of their attraction on the balls and
rod will now he — (}> {x — B). The equation of motion is therefore
Let a = x — ^, then substituting for B its value 2a — A, we
find by the same reasoning as before
£c = e' + iVsin (nt + N'),
where n has the same value as before and
4> {A- a) + yjr (a) - E (a - b)
e =a +
In'
In these expressions, the attraction yjr (a) of the casing, the
coefficient of torsion E and the angle b are all unknown. But
they all disappear together, if we take the difference between
e and e. We then find
(f) (A — a) _e — e
-■(^ (A),
where T is the time of a complete oscillation of the torsion-rod
about either of the disturbed positions of equilibrium. Thus the
attraction (}){A — a) can be found if the angle e — e between the
two positions of equilibrium and also the time of oscillation about
either can be observed.
492. The function <}) [A —a) is the moment of the attraction
of the masses and the plank on the balls and rod, when the rod
has been placed in a position Cf, bisecting the angle A^CB^ be-
tween the alternate positions of the masses. Let M be the mass
of either of the masses A and B, m that of one of the small balls,
m that of the rod. Let the attraction of M on m be represented
by jM -Yfi' > where D is the distance between their centres. If
(p, q) be the co-ordinates of the centre of A^^ referred to (7/" as the
axis of X, the moment about C of the attraction of both the masses
on both the balls is
where c is the distance of the centre of either ball a, b from the
centre C of motion. Let this be represented by /nMmP. The
moments of the attraction of the masses on the rod may by inte-
THE CAVENDISH EXPEEIMENT. 393
gration be ^onnd =fiMmQ, where ^ is a known function of the
linear dimensions of the apparatus. The attraction of the plank
might also be taken account of. Thus we find
^o
<}>{A-a)= ixMirtiP + m'Q),
If r be the radius of either ball, we have
1= 2m
which may be represented by 1= mP'+m'Q', where P' and Q' are
known functions of the linear dimensions of the rod and balls.
Hence we find by substituting in equation (A)
mP+m'Q _ e-e f27ry
^ 'mP' +771 Q'~ 2 '\tJ'
Let E be the mass of the earth, R its radius and g the force
E *
of gravity, then.^ = A* "eg • Substituting for fM, we find
M_e-l f27ry _1^ 7n"^ ^^
E 2 \T)-gB^'n^p^ '
Til
The ratio — , was taken equal to the ratio of the weights of
the ball and rod weighed in vacuo, but it would clearly have been
more accurate to have taken it equal to the ratio weighed in air.
For since the masses attrabt the air as well as the balls, the pres-
sure of the air on the side of a ball nearest the attracting mass is
greater than that on the furthest side. The difference of these
pressures is equal to the attraction of the mass on the air displaced
by the ball.
493. By this theory the discovery of the mass of the earth
has been reduced to the determination of two elements, (1) the
time of oscillation of the torsion-rod, and (2) the angle e — e'
between its two positions of equilibrium when under the influence
of the masses in their alternate positions. To observe these, a
small mirror was attached to the rod at G with its plane nearly
perpendicular to the rod. A scale was engraved on a vertical
plate at a distance of 108 inches from the mirror, and the image
of the scale formed by reflection on the mirror was viewed in
a telescope placed just over the scale. The telescope was fur-
* In Baily's experiment, a more accurate value of g was used. If e be the ellip-
ticity of the earth, m the ratio of centrifugal force at the equator to equatoreal
gravity, and X the latitude of the place, we have fi' = Mn5|l-2£ + (-??i-e) cos'xL
394 \ SJIALL OSCILLATIONS.
nished with three vertical wires in its focus. As the torsion-rod
turned on its axis, the image of the scale was seen in the telescope
to move horizontally across the wires and at any instant the
number of the scale coincident with the middle wire constituted
the reading. The scale was divided by vertical lines one-thirteenth
of an inch apart and numbered from 20 to 180 to avoid negative
readings. The angle turned through by the rod when the image
of the scale moved through a space corresponding to the interval
111
of two divisions was thereiove rr^ . rr-r^ . ■^ = 7S"' 4:6. But the
LO lOo A
division lines were cut diagonally and subdivided decimally by
horizontal lines ; so that not only could the tenth of a division
be clearly distinguished, but, after some little practice, the frac-
tional parts of these tenths. The arc of oscillation of the torsion-
rod was so small that the square of its circular measure could be
neglected ; but as it extended over several divisions it is clear
that it could be observed with accuracy. A minute description
of the mode in which the observations were made would not find
a fit place in a treatise on Dynamics, we must therefore refer the
reader to Baily's Memoir.
In this investigation no notice has been taken of the effect
of the resistance of the air on the arc of vibration. This was,
to some extent at least, eliminated by a peculiar mode of taking
the means of the observations. In this way also some allowance
was made for the motion of the neutral position of the torsion-rod.
494. The density of water in which the weight of a cubic
inch is 252"725 grains (7000 grains being equal to one pound
avoirdupois) was taken as the unit of density. The final result
of all the experiments was that the mean density of the earth
is 5-6747.
495. Two other methods of finding the mean density have
been employed. In 1772 Dr Maskelyne, then Astronomer Royal,
suggested that the mass of the earth might be compared with
that of a mountain by observing the deviation produced in a
plumb-line by the attraction of the latter. The mountain chosen
was Schehallien, and the density of the earth was found to be
a little less than five times that of water. See Phil. Trans.
1778 and 1811. From some observations near Arthur's Seat, the
mean density of the earth is given by Lieut.-Col. James, of the
Ordnance Survey, as 5"316. See Fhil. Trans. 1856.
The other method, used by Sir G. Airy, is to compare the
force of gravity at the bottom, of a mine with that at the surface,
by observing the times of vibration of a pendulum. In this way
the mean density of the earth was found to be 6'566. See Phil.
Trans. 1856.
OSCILLATIONS OF THE SECOND ORDER. 395
Oscillations of the Second Order.
496. The equations of small oscillations are formed on the following principle.
Some small quantities are selected as the co-ordinates of the system, and all powers
of these above the first are neglected. The assumption is tacitly made that the
order of magnitude of the terms is not materially altered by the process of solving
the equations ; so that a small term, which should by the rule be neglected in
forming the differential equations, cannot become of importance in the final
integrals. This assumption, however, is not strictly correct. In the Lunar and
Planetary theories, where something more is wanted than the mere periods of
oscillations, there are many instances of small terms in the differential equations,
which become of great magnitude in the result. We require some rule to dis-
tinguish the small terms which become of importance from those which remain
insignificant. For the sake of simplicity we shall consider the case in which the
system depends on two independent co-ordinates, though the remarks are for the
most part quite general.
497. Referring to Ai-t. 432, let PsinXt be some small periodic term which
occurs on the right-hand side of the first of the two differential equations of
motion. To simplify the solution, let us write for the trigonometrical term ita
exponential value, and fix our attention on the part — ■==. "^ "^^^ or, as we shall
2 \/ — 1
write it, Q/'^. Let/(Z>) stand for the determinant which is the operator on a; in
the third equation of Art. 432. Also let F{D} be the minor of the leading con-
stituent ; the value of x is then known to be
^ = l^Qei^i + M^e'^^*+
The term Qe^ in the differential equation is the analytical representation of
some small periodical force which acts on the system. The first term of the
expression for x is the direct effect of the force, and is sometimes called the
■forced vibration in the co-ordinate x. The quantities m^, m^, &c. being generally
imaginary, the remaining terms are also trigonometrical and are sometimes called
the free or natural vibrations in the co-ordinate. In the analytical theory of linear
differential equations, the forced vibration is called the particular integral and the
free vibration the complementary function.
498. If we examine the coefficient of the forced vibration in x we shall see that
it is large only if /(/*) is very small or zero. Since the roots of the equation
f(fji.) = are m^, m^, &c. the rule may be simply stated thus : amj small periodical
term whose coefficient in the differential equation is less than the standard of quantities
to be neglected may rise into importance if its period is nearly equal to one of the
free vibrations of the system.
Suppose the dynamical system to have two of its free periods equal and let it
be acted on by a small force whose period is nearly equal to this free period. The
divisor/ (fjt.) of the forced vibration will be a small quantity of the second order and
the magnitude of the term may be much greater than if the free periods were
unequal. When such a case occurs in the Lunar theory, the term is said to rise
two orders.
396 SMALL OSCILLATIONS.
499. This principle admits of an elementary explanation in some cases. Let a
system oscillating with one degree of freedom be acted on by a smaU periodical
force at some point A. The force will act sometimes to accelerate the motion of A
and sometimes to retard it, and thus the magnitude of the vibration will not become
very great. But if the period of the force be equal to that of the point A, the force
may continually act to increase the motion of A in whatever direction A is moving.
Thus the extent of the vibration will be continually increasing. For example,
every one knows how a heavy swing can be set in violent oscillation by a series
of small pushes and pulls applied at the proper times.
If the period of the force be only nearly equal to that of the point A, a time
will come when the force acts continually to decrease the motion of A. Thus the
oscillation will not increase indefinitely, but will alternately slowly increase and as
slowly decrease.
500. A remarkable use of this principle was made by Capt. Kater in his
experiments to determine the length of the seconds' pendulum. It was important
to determine if the support of his pendulum was perfectly firm. He had recourse
to a delicate and simple instrument invented by Mr Hardy a clockmaker, the
sensibility of which is such that had the slightest motion taken place in the support
it must have been instantly detected. The instrument consists of a steel wire,
the lower part of which is inserted in the piece of brass which forms its support,
and is flattened so as to form a delicate spring. On the wire a small weight slides
by means of which it may be made to vibrate in the same time as the pendulum
to which it is to be applied as a test. When thus adjusted it is placed on the
material to which the pendulum is attached, and should this not be perfectly firm,
the motion will be communicated to the wire, which in a little time will accompany
the pendulum on its vibrations. This ingenious contrivance appeared fully adequate
to the purpose for which it was employed, and afforded a satisfactory proof of the
stability of the point of suspension. See Pkil. Trans. 1818.
501. It generally happens that the small terms rejected in the equations of
motion are functions of the co-ordinates and their differential coefficients. To
take account of these terms we proceed by successive approximation. Suppose the
co-ordinates x, y to determine the oscillation about some state of steady motion, and
to be zero for that motion. As a first approximation we obtain (Art. 432)
sc = M-^e + Ifge ' +
with a corresponding expression for y, where mj, wis, &c. give the free periods, and
i/j, M^, &c. are aU small quantities of the first order. If we now substitute these
values of x and y in any small term of a high order which occurs in the differential
equation, it becomes a series of exponentials of the form
Pglpnii+gOTj + ...){
where p, q, &c. are positive integers whose sum is equal to the order of the term.
By the principle explained in Art. 498, the corresponding forced vibration cannot
be important unless ^m^ + gwij + . . . is very nearly equal to one of the quantities
mi, mi, &c. In the same way, in any approximation, if the periods of the terms
are not such that an equality of this nature can be very nearly true, the next
approximation to the motion will not produce any important terms. Even if such
a relation does approximately hold, yet, if the order of the term to be examined is
great, the term will probably remain insignificant.
OSCILLATIONS OF THE SECOND ORDER. S97
502. As an example let us consider the case of a planet describing a circle
about the sun, considered as fixed in the centre. If slightly disturbed the changes
in the radius vector and longitude will be very small and will correspond to what
we have called x and y. From the theory of elliptic motion, we know that these
will be approximately
a;=a-aeco3 (ni + a), j/ = 6< + c + 2e sin (n< + a),
27r
where a, 5, c and e are all small quantities, and — is the period of the planet. Com-
paring these with the expressions for x and y given in Art. 432, we see that the
free periods for x are given by m = 0, 711= ^nj -1, and for y, by jw=0, m=0,
m=iznj-l, one period being absent from x. We infer that any small periodical
force may produce a considerable disturbance both in the radius vector and in the
longitude of the planet, if its period is nearly equal to that of the planet or is very
long. Since there are two equal free periods in the longitude corresponding to
m = 0, those small forces whose periods are very long may be expected to rise two
orders ia the longitude. If any such forces act on the planet it will be necessary to
examine into their effects. Small forces, whose periods are different from these,
and whose magnitude is beneath the standard of quantities to be retained, may be
disregarded.
503. If the period of the small disturbing force Qe'^^ be equal to one of the free
periods, the solution changes its character. The forced vibration now takes the
form —^ Qt^*. This may indicate that the motion of the system will, after a
time, become very different from that which we took as a first approximation. We
may have therefore to amend our first approximation by including in it the effect
of this force. We may then enquire how far tliis modified first approximation
indicates that the undisturbed motion is stable or unstable. When this force is
included in the equations, the equations wQl probably be no longer linear, and it
may be impossible to solve them or to find a solution sufficiently accurate to serve
as a first approximation throughout the whole motion,
504. In many cases however the effects of some of these forces may be included
in the first approximation by slightly altering the free periods. Referring to Art.
432, let us suppose that on substituting our first approximation in the small terms,
we have on the right-hand side of the two first equations
R^e^^^R^e^^U...)
These are supposed to have arisen from some relations of the form
pmi + qvi2+ ...=mi (2).
Let us take as our amended first approximation
x = N^e'%* + N^e''^*+...] ,3^
where iV^, (fee. N/, Sec. are, as before, small quantities of the first order, and
ni=mi + dm^, 11^ = 7112 + 8m^, &c. where Srw^, Sm^, &c. are qiiantities of the order
Qi, &c. i?i, &c. If we substitute the amended values of x, y in the small terms,
they will become
398 - SMALL OSCILLATIONS.
instead of (1), provided the relations represented by (2) apply also to the indices
Kj, 7?2, &c. Here Q/, &c. ^/, &c. differ from Q^, &c. E^, &c. by quantities of the
order Q^2. Substituting the values (3) in the differential equations of Art. 432,
and rejecting the squares of Q^, &c, J?,, &e. , we obtain
{A'v? + B'n + C')N+{F'n^^G'n-\-E')N'=R\ ^ ''
where the suifixes have been dropped for the sake of generality. These two equa-
tions determine n and A"', leaving N to be determined by the initial conditions.
The test of the success of the amended first approximation is that the values of n
thus found satisfy the relation (2).
505. The condition may also be stated thus. Consider the determinant given
in Art. 432, which when expanded is equal to /(D). After substitution of the first
approximation in the small terms of the higher orders in the equations, perform on
these equations the operations indicated by the minors of the constituents in the
first column, and add the results together. We have an equation of the form
f{D)x^^^e^^*■\■\e'^^*■\■...
where the coefficients A^, Ag, &c. are aU functions of Mi, M^, &c., to^, Wg, &c.
Following the same reasoning as in the last Article, and amending our first approxi--
mation, we find
5m,~ - .}, — r, 5?n2=-r--r7^— -, &c.
1 Ml/' (mi)' 2 MJ'iMi)'
If these satisfy the relations typified by
pdm^ + qSm^-b ... =S?Mi,
the effect of the disturbing cause is to modify the free periods of the system without
affecting the stabihty of the undisturbed motion.
506. Having in this way amended the first approximation, we may proceed to
the second by substitution in the small term, and so on. If the several stages can
be so arranged that no term makes its appearance which can become greater than
our previous approximation, we may consider that we have obtained a correct repre-
sentation of the motion.
507. Ex. 1. A pendulum sivings in a very rare medium, resisting partly as the
velocity and partly as the square of the velocity, to find the motion.
Let 6 be the angle the straight line joining the point of support to the centre
of gravity of the pendulum makes with the vertical. Then the equation of
motion is
dW g . „ ^ de
-(sy (^)'
where I is the length of the simple equivalent pendulum, 2k and ii the coefficients
of the resistance divided by the moment of inertia of the pendulum about the
axis of suspension. Let g^hv'. Since d is small we may write the equation in the
form
d^9 ,^ „ d9 fdey ,e-^
Since k and 6 are very small, we might at first suppose that it would be
sufficient as a first approximation to reject all the terms on the right-hand side.
OSCILLATIONS OF THE SECOND ORDER. 399
This gives 0=aeinnt, the origin of measurement of t being so chosen that t and 6
vanish together. If we substitute this in the small terms we get
which gives
-— - + n^Q = - 2k71 . a cos nt + - n^a^ sin nt + &c. ,
^ = a sin nf - Ktt . i sin ni + — J na?t cos nt + &e.
lb
These additional terms contain t as a factor, and show that our first approximation
was not sufficiently near the truth to represent the motion except for a short
time. To obtain a sufficiently near first approximation we must include in it the
small term 2k 3- , we have therefore
at
dt^ dt
This gives 6 = ae""^ • sin mt, where for the sake of brevity we have put n^ - k^ = wi'.
In our second approximation we shall reject all terms of the order a^ or o'k
unless they are such that after integration they rise in importance in the manner
explained in Ai't. 498. We thus get
5-5 + 2/c ^ + n^e = - ~—^ ^^{l + cos 2mt) + - a^ -7- 3 sm mt - sm 3m<)
dt^ ai 2 '64'
Ha?Ke ^"M -- +^cos2mi+msin2mn',
where all the terms on the right-hand side after the first are of the third order, and
are to be rejected unless they rise in importance. To solve this, let us first consider
the general case
-j-^ + 2K-i- + n'^d=e~^'^^ .{Ami rmt + B cos rmt).
at cit
Put 6 = e "^"^ (L sin rmt + ill cos rmt). Substituting we get
L {(p - l)V + mP (1 -r^)} +2 (_p- 1) KrmM=A
M{{p - l)V + m2 (1 - r^)] _ 2 (i> - 1) KrmL=B
A
Now K is very small. If then r be not equal to unity, we have L-
IV? (1 - r-) '
^=~rR v\ nearly; but if r = l, wehave I/= — — — . M=— — — nearly.
ni2 (1 - r2) -^ ' 2 (2? - 1) Kin 2 (^ - 1) kvi
The case of ^j = 1 does not occur in our problem. It appears that those terms only
in the differential equation which have r = l give rise to terms in the value of x
which have the small quantity k in the denominator. Hence in the differential
equation the only term of the third order which should be retained is the first.
We thus find, putting successively r=0, r=2, r=l,
= ae-"^ Bin mf - '^'e - 2«^ + ^% - 2«« cos 2mf + ^ e - ^""^ cos mi.
2 fa olKin
This equation determines the motion only during any one swing of the pendu-
lum ; when the pendulum turns to go back yn changes sign. Let us suppose the
pendulum to be moving from left to right, and lot us find the lengths of the arcs of
descent and ascent. To do this, we must put -y- ~ 0. Let the equation be written
400 SMALL OSCILLATIONS.
in the form 0=f{t), then if we neglect all the small terms, -3- vanishes when
mt= dz~. Put then mt= -~+x where a; is a small quantity, we have
/'(""•^'C- !;)+/"(- 24)£=»-
Now
/'(«) = ae -*i5 (m cos »n«-K sin m«)-^e" 2"* ( - 2K + -^cos2mi+ ■5-sin2m< j
+ „^ e "3"^ ( _ jtt sin mf - 3ic cos jw«).
32k?w
A sufficiently near approximation to the value of /" {t) may be found by differ-
K 4 iXCLK 71 CL^
entiating the first term of the value of /' {t). We thus find x= 5 ^ — ;
the second of these terms being smaller than the other two might be neglected.
We also find as the arc of descent
Kir KIT Kir Sktt
Similarly to find the arc of ascent we put int=^ + y. This gives y= -— — ,
and the arc of ascent ia
Kir Kir Kjr Sktt
e = ae -^u.o?e -yj.a. +^^e
In these expressions for the arcs of descent and ascent the terms containing x
and y are very small, and assuming k not to be extremely small, these terms will be
neglected*.
Now a is different for every swing of the pendulum, we must therefore eliminate
KTT
a. Let M„ and m„+i be two successive arcs of descent and ascent, and let X = e 2m ,
so that \ is a little less than unity. Then we have
eliminating a we have very nearly
* If these terms are not neglected the equation connecting the successive arcs of
descent and ascent becomes
1 X^ 2 .,^.5.^ n'^x 1-X4
Now 1-X*=-^ nearly, so that this additional term is very small compared with
m
that retained.
SMALL OSCILLATIONS. 401
The successive arcs are, therefore, such that — + - ia the general term of a
«„ c
Kir
geometrical series whose ratio is e"*^ . The ratio of any arc u„ to the following arc
K7T Kit
which continually decreases with the arc. In any series of oscillations the ratio is
at first greater and afterwards less than its mean value. This result seems to agree
with experiment.
To find the time of oscillation. Let f^, t.^ he the times at which the pendulum is
at the extreme left and right of- its arc of oscillation. Then
The time of oscillation from one extreme position to the other is t^ - t^ which is
equal to — . This result is independent of the arc, so that the time of oscillation
remains constant throughout the motion. The time is however not exactly the
same as in vacuo, but is a little longer ; the difference depending on the square of
the small quantity k.
Ex. 2. If in Art. 418 a first approximation to the motion is ^ = ^ sin [at + 3),
show that a second will be
= ^ sin [at + 5) + 1 (6 + c) 42 + 1 (35 + c) A^- cos 2 (at + B)
, , rs sin a 1 s^ ( cos a ds sin 2a sin a )
where b = — — -. , c = ^ j y- + r ( >
^2 4. 7-2 ' 2 s cos a - »• ( s dff r p 1
and ff is the length of the arc of either cylinder,
A general method of solving problems of this kind, both for two and three
dimensions, is given in the Proceedings of the London Mathematical Society, Vol. v.
page 101, 1874.
Ex. 3. A rigid body is suspended by two equal and parallel threads attached
to it at two points symmetrically situated with respect to a principal axis through
the centre of gravity which is vertical, and being turned round that axis through a
small angle is left to perform small finite oscillations. Investigate the reduction to
infinitely small oscillations, [Smith's Prize.]
EXAMPLES*.
1. A uniform rod of length 2c rests in stable equilibrium with its lower end
at the vertex of a cycloid whose plane is vertical and vertex downwards, and passes
through a small smooth fixed ring situated in the axis at a distance 6 from the
vertex. Show that if the equihbrium be slightly disturbed, the rod will perform
* These examples are taken from the Examination Papers which have been set
in the University and in the Colleges.
R. D. 26
402 • SMALL OSCILLATIONS.
small oscillationg with its lower end on the arc of the cycloid in the time
4ir a/ ^n /,o V. — r-^f where 2a is the length of the axis of the cycloid.
V 3g (b^ - 4:ac)
2. A small smooth ring slides on a circular wire of radius a which is con-
strained to revolve aboiit a vertical axis in its own plane, at a distance c from the
centre of the wire, with a uniform angular velocity a/ x — ; show that the ring
^ csj2 + a
will he in a position of stable relative equilibrium when the radius of the circular
wire passing through it is inclined at an angle 45" to the horizon ; and that if the
ring be slightly displaced, it wiU perform a small oscillation in the time
3. A uniform bar of length 2a suspended by two equal parallel strings each of
length b from two points in the same horizontal line is turned through a small
angle about the vertical line thi'ough the middle point, show that the time of a
fbj^
small oscillation is 27r . / -
V 9
4. Two equal heavy rods connected by a hinge which allows them to move
in a vertical plane rotate about a vertical axis through the hinge, and a string
whose length is twice that of either rod is fastened to their extremities and
bears a weight at its middle point. If M, M' be the masses of a rod and the
particle, and 2a the length of a rod, prove that the angular velocity about the
, . „ . /""% M + 2M'~ ,
vertical axis when the rods and string form a square is \ / . ^ — , ana
^ 2a^/2 ^
if the weight be slightly depressed in a vertical direction the time of a small
... ^. . „ /4av/2 M + 3if'
oscillation is 2it '
^/^
15^ M^2M'
5. A ring of weight W which slides on a rod inclined to the vertical at an angle
a is attached by means of an elastic string to a point ia the plane of the rod so
situated that its least distance from the rod is equal to the natural length of the
string. Prove that if Q be the inclination of the string to the rod when in
equilibrium, cot 6 - cos Q = — cos a, where w is the modulus of elasticity of the
string. And if the ring be slightly displaced the time of a small oscillation will be
27r A. / r-TT , where I is the natural length of the string.
V wg X - ^\u^Q
6. A circular tube of radius a contains an elastic string fastened at its highest
point equal in length to - of its circumference, and having attached to its other
8
extremity a heavy particle which hanging vertically would double its length. The
system revolves about the vertical diameter with an angular velocity */„• ^'^"^^
the position of relative equilibrium and prove that if the particle be slightly dis-
2ir^ + Bco^-^Ca>,^=T. (1),
where T is an arbitrary constant.
Again, multiplying the equations respectively by Aw^, E'){\-iO^){\-co^) (6).
The integration of equation (6) * can be reduced without diffi-
culty to depend on an elliptic integral. The integration can be
effected in finite terms in two cases ; when A=B, and when
G^ = TB, where B is neither the greatest nor the least of the three
quantities A, B, G. Bot^i. these cases will be discussed further on.
Ex. If right lines are measured along the three principal axes of the body from
the fixed point, and inversely proportional to the radii of gyration round those axes,
the sum of the squares of the velocities of their extremities is constant throughout
the motion.
509. It will generally be supposed that A, B, C are in order of magnitude, so
that A is greater than B, and B than C. The axis of B will be called the axis of
mean moment. If we eliminate w^ from the equations (1) and (2), we have
AT- G^=B [A - B) w^^ + C [A - C) ui^,
which is essentially positive. In the same way we can show that CT- 0^ is nega-
G^
tive. Thus the quantity — may have any value lying between the greatest and
least moments of inertia.
The three quantities \, \> \ i^i -A^t. 508 are all positive quantities ; for since
B + 0- A is positive, and — or < B.
It foUows from equations (5) that throughout the motion bp must lie between X^
and the greater of the quantities X^ and X3.
* Euler's solution of these equations is given in the ninth volume of the Quarterly
Joxirnal, p. 361, by Prof. Cayley. Kirchhoff's and Jacobi's integrations by elliptic
functions are given in an improved form by Prof. Greenhill in the fourteenth
volume, pages 182 and 265. 1876.
406 MOTION UNDER NO FORCES.
510. The solution in terms of elKptic integrals has been effected in the follow-
ing manner by Kirchhoff. If we put
then yfc is called the modulus of F, and must be less than unity if F is to be real for
all values of 0. The upper limit is called the amplitude of the elliptic integral
F and is usually written am F. In the same way sin ,. ,.
~dW^ = cos -^= cos ) fc2 sin d> cos d> dd) , „ .
.(1).
These equations may be made identical with Euler's equations if we put
F=\(t~T)a,ndi
Wi = aAam\(i-T)
W2=:6sinamX {t — r) I (2),
W3= c cos am X (i - r) j
A-B _ c\ A-C _ b\ B-C ^ aX
C ~ ah' B " ca' A be : ^''
We have introduced here six new constants, viz. a, h, c, \, h and t. With these
we may satisfy the three last equations and also any initial values of w^, Wg, Wj.
The solution if real wUl also be complete.
When <=T we have from (2) w^ = a, w^ = 0, and o}^ = c. Hence by Art. 608
Aa^ + Cc^=T, ^V + CV = G3;
^_ G'^-CT 2_ AT-Q""
*''°' ~A{A-C)' '^'CiA-C)'
Dividing the second of equations (3) by the first, we have
62 A-CC ^„ AT-G^
c2 A-BB' ' B{A-B)'
Multiplying the first and second of equations (3), we obtain
{A-B)(G^~CT)
^''~ ABC
The ratios of the right-hand sides of (3) are as c^ : b^ : Jc"a^, and these have just
been found. Hence if the signs of a, 6, c, \ be chosen to satisfy any one of the
three equalities, the signs of aU will be satisfied.
Dividing the last of equations (3) by either of the other two, we find
B-C AT-G^ . 72_ A-0 G^-BT
A-BG^~CT'' •*■ iA-B)((;f^-VT)'
poinsot's and mac cullagh's constructions. 407
If 0^ > BT and A, B, C are in descending order of magnitude, tlie values of
a', h^, c^ and X^ are all positive. Also ^^ is positive and less than unity. Tlie
solution is therefore real and complete.
If G^ < BT we must suppose 4 , B, C to be in ascending order of magnitude to
ohtain a real solution. If we may anticipate a phrase used by Poinsot, and which
wiU be explained a little further on, we may say that the expression for w^ in this
solution is to be taken for the angular velocity about that principal axis which is
enclosed by the polhode.
If CP^ BT we have F = 1 and
r^ d(j> 1 1 + sin ^
Jo cos <{)
sin am F=
2^"^l-sin^'
fi-F
Substituting in equations (2) the elliptic functions become exponential.
If B = C vre have P^O and in this case jF=^, so that am F=F. If we again
substitute in equations (2) the elliptic functions become trigonometrical.
The geometrical meaning of this solution will be given a little fm-ther on.
Poinsot's and MaoCullagKs constructions for the motion.
511. The fundamental equations of motion of a body about a
fixed point are
^V + ^V+C'V=^* (1).
Aoi^ + Ba>.^-\-Coy^'=^T (2).
These have been already obtained by integrating Euler's
equations, but they also follow very easily from the principles of
Angular Momentum, and Yis Viva.
Let the body be set in motion by an impulsive couple whose
moment is G. Then we know by Art. 279, that throughout the
whole of the subsequent motion, the moment of the momentum
about every straight line which is fixed in space, and passes
through the fixed point 0, is constant, and is equal to the mo-
ment of the couple G about that line. Now by Art. 241, the
moments of the momentum about the principal axes at any
instant are A(o^, Bf^^, G(o^. Let a, /3, 7 be the direction angles
of the normal to the plane of the couple G referred to these
principal axes as co-ordinate axes. Then we have
Ao)^ = G cos a 1
B(o^= Gco^^l (.S),
Cwg = G cos 7 J
adding the squares of these we get equation (1).
'408 . MOTION UNDER NO FORCES.
Throughout the subsequent motion the whole momentum of
the body is equivalent to the couple O, It is therefore clear
that if at any instant the body were acted on by an impulsive
couple equal and opposite to the couple O, the body would be
reduced to rest,
512. It follows from Art. 290, that the plane of this couple
is the Invariable plane and the normal to it the Invariable line.
This line is absolutely fixed in space, and the equations (3) give
the direction cosines of this line* referred to axes moving in the
body.
It appears from these equations, that if the body be set in
rotation about an axis tvhose direction cosines are [l, m, n) when
referred to the principal axes at the fixed point, then the direction
cosines of the invariable line are proportional to Al, Bm, Cn. If
the axes of reference are not the principal axes of the body at the
fixed point, the direction cosines of the invariable line will, by
Art. 240, be proportional to Al — Fm — En, Bm — Dn — Fl, and
On — El — Dm, where the letters have the meaning given to them
in Art. 15.
513. Since the body moves under the action of no impressed
forces, we know that the Vis Viva will be constant throughout the
motion. Hence by Art. 348, we have
where T\ is a constant to be determined from the initial values
of ft)j, cwg, 0)3.
The equations (1), (2), (3) will suffice to determine the path
in space described by every particle of the body, but not the posi-
tion at any given time.
* That the straight line whose equations referred to the moving principal axes
are - — = -~— = 7^— is absolutely fixed in space may be also proved thus, if we
A(i}^ -0W2 ^^3
assume the truth of equation (1) in the text. Let x, y, z be the co-ordinates of
any point P in the straight line at a given distance r from the origin, then each of
the equalities in the equation to the straight line is equal to -^ and is therefore con-
stant. The actual velocity of P in space resolved parallel to the instantaneous
position of the axis of x is
dx 1' [ , du)^ , „ „,
But this is zero, by Euler's equation. Similarly the velocities parallel to the other
axes are zero.
t It should be observed that in this Chapter T represents the whole vis viva of
the body. In treating of Lagrange's equations in Chapter vii. it was convenient to
let T represent lialj the vis viva of the system.
POINSOTS CONSTRUCTION. 409
514. To explain Poinsofs representation of the motion by
means of the momental ellipsoid.
Let the momental ellipsoid at the fixed point be constructed,
and let its equation be
Ax" + By"" + Cz^ =^ Me\
Let r be the radius vector of this ellipsoid coinciding with the
instantaneous axis, and p the perpendicular from the centre on
the tangent plane at the extremity of r. Also let co be the an-
gular velocity about the instantaneous axis.
The equations to the instantaneous axis are — =^ =— , and
&>! (Wg 6)3
if {x, y, z) be the co-ordinates of the extremity of the length r,
each of these fractions is equal to — .
Substituting in the equation to the ellipsoid, we have
/J_ r
~ V 3Ie' 6 '
Again the expression for the perpendicular on the tangent
plane at [x, y, z) is known to be —^ = —^ , substi-
2) jyj. €
tuting as before we get
r.p = -^.e.
The equation to the tangent plane at the point (x, y, z) is
Ax^+By7j-]rCz^=M6\
substituting for {x, y, z) we see that the equations to the perpen-
dicular from the origin are
Aw. Bco„ Ci
<*>„
but these are the equations to the invariable line. Hence this
perpendicular is fixed in space.
From these equations we infer
(1) The angular velocity about the radius vector round which
the body is turniny varies as that radius vector.
410 MOTION UNDER NO FORCES,
(2) The resolved part of the angular velocity about the 'per-
pendicular on the tangent plane at the extremity of the instan-
taneous axis is constant. This theorem is due to Lasransfe.
For the cosine of the angle between the perpendicular and
the radius vector = - . Hence the resolved angular velocity is
p T . .
= ft) - = -7:3, which is constant.
r G
(3) The perpendicular on the tangent plane at the extremity
of the instantaneous axis is fixed in direction, viz. normal to the
invariable plane, and constant in length.
The motion of the momental ellipsoid is therefore such that,
its centre being fixed, it always touches a fixed plane, and the
point of contact, being in. the instantaneous axis, has no velocity.
Hence the motion may be represented by supposing the momental
ellipsoid to roll on the fixed plane with its centre fixed.
515. Ex, 1, If the body while in motion be acted on by any impnlsive couple
whose plane is perpendicular to the invariable line, show that the momental ellipsoid
will continue to roll on the same plane as before, but the rate of motion will be
altered.
Ex. 2, If a plane be drawn through the fixed point parallel to the invariable
plane, prove that the area of the section of the momental ellipsoid cut off by this
plane is constant throughout the motion.
Ex, 3, The sum of the squares of the distances of the extremities of the princi-
pal diameters of the momental ellipsoid from the invariable line is constant through-
out the motion. This result is due to Poinsot.
Ex. 4. A body moves about a fixed point under the action of no forces. Show
that if the surface Ax"^ + By^ + Cz^ ^ M{x^ + y^ + 2^)^ be traced in the body, the principal
axes at being the axes of co-ordinates, this surface throughout the motion will
roll on a fixed sphere.
516. To assist our conception of the motion of the body, let
us suppose it so placed, that the plane of the couple G, which
would set it in motion, is horizontal. Let a tangent plane to the
momental ellipsoid be drawn parallel to the plane of the couple G,
and let this plane be fixed in space. Let the ellipsoid roll on this
fixed plane, its centre remaining fixed, with an angular velocity
which varies as the radius vector to the point of contact, and let
it carry the given body with it. We shall then have constructed
the motion which the body would have assumed if it had been
left to itself after the initial action of the impulsive couple G^.
* Prof. Sylvester has pointed out a dynamical relation between the free rotating
body and the ellipsoidal top, as he calls Poinsot's central ellipsoid. If a material
poinsot's construction. .411
The point of contact of the ellipsoid with the plane on which
it rolls traces out two curves, one on the surface of the ellipsoid,
and one on the plane. The first of these is fixed in the body and
is called the polhode, the second is fixed in space and is called the
herpolhode. The equations to any polhode referred to the prin-
cipal axes of the body may be found from the consideration that
the length of the perpendicular on the tangent plane to the ellip-
soid at any point of the polhode is constant. Hence its equations
are
Eliminating y, we have
A{A-B)x'' + C{G-B)z'=(^-B\Me\
Hence if B be the axis of greatest or least moment of inertia,
the signs of the coefiicients of x^ and z^ will be the same, and the
projection of the polhode will be an ellipse. But if B be the
axis of mean moment of inertia, the projection is an hyperbola.
A polhode is therefore, a closed curve drawn round the axis of
greatest or least moment, and the concavity is turned towards the axis
of greatest or least moment according as -™- is greater or less than
the mean moment of inertia. The boundary line which separates
the two sets of polhodes is that polhode whose projection on the
plane perpendicular to the axis of mean moment is an hyperbola
whose concavity is turned neither to the axis of greatest, nor to
the axis of least moment. In this case G^ = BT, and the projec-
tion consists of two straight lines whose equation is
A{A-B)x^-G{B- C)z''=0.
This polhode consists of two ellipses passing through the axis
of mean moment, and corresponds to the case in which the per-
pendicular on the tangent plane is equal to the mean axis of the
ellipsoid. This polhode is called the separating polhode.
Since the projection of the polhode on one of the principal
planes is always an ellipse, the polhode must be a re-entering
curve.
ellipsoidal top be constructed of uniform density, similar to Poinsot's central ellip-
soid, and if with its centre fixed it be set rolling on a perfectly rough horizontal
plane, it will represent the motion of the free rotating body not in space only, but
also in time : the body and the top may be conceived as continually mo\dng round
the same axis, and at the same rate, at each moment of time. The reader is referred
to the memoir in the rhilosopliical Transactions for 18G6.
412 MOTION UNDER NO FOECES.
517. To find the motion of the extremity of the instantaneous
axis along the polhode which it describes we have merely to sub-
stitute from the equations
_ ^2 _ ^ _ ^ _ /^ 2.
X y
in any of the equations of Art. 508. For example we thus obtain
dt \ M A e" ' '
BG
^' = [A-C)[A-B) (- ^^' + "')' ^'-^ ^''
Ex. 1. A point P moves along a polhode traced on an ellipsoid, show that the
length of the normal between P and any one of the principal planes at the centre
is constant. Show also that the normal traces out on a principal plane a conic
similar to the focal conic in that plane. Also the measure of curvature of an
ellipsoid along any polhode is constant.
Ex. 2. Show that the line OJ used in Art. 234 to find the pressure on the
fixed point is at right angles to the invariable line, and parallel to the tangent
plane to the momental ellipsoid at the point where the invariable line cuts it.
Showalso that ^-= - .* + .^^-^^Z^ _ I>^T^^iP.P.^V.)G^T+^.fi^ ^^^^^
are the sum of the products A, B, taken respectively one, two and three together.
518. Since the herpolhode is traced out by the points of
contact of an ellipsoid rolling about its centre on a fixed plane,
it is clear that the herpolhode must always lie between two circles
which it alternately touches. The common centre of these circles
will be the foot of the perpendicular from the fixed centre on
the fixed plane. To find the radii let OL be this perpendicular,
and /be the point of contact. Let LI= p. Then we have
maccullagh's construction. 413
The radii will therefore be found by substituting for w' its
greatest and least values. But by Art. 509, these limits are \,
and the greater of the two quantities \, \.
The herpolhode is not in general a re-entering curve ; but if
the angular distance of the two points in which it successively
touches the same circle be commensurable with 27r, it will be re-
entering, i.e. the same path will be traced out repeatedly on the
fixed plane by the point of contact.
519. To explain MacCullaglis representation of the motion
hy means of the ellipsoid of gyration.
This ellipsoid is the reciprocal of the momental ellipsoid, and
the motion of the one ellipsoid may be deduced from that of the
other by reciprocating the properties proved in the preceding
Articles. We find,
(1) The equation to the ellipsoid referred to its principal
axes is
A^ B'^ a~ M-
(2) This ellipsoid moves so that its superficies always passes
through a point fixed in space. This point lies in the invariable
C
line at a distance -; . from the fixed point. By Art. 509 we
\/MT
know that this distance is less than the greatest, and greater than
the least semi-diameter of the ellipsoid.
(3) The perpendicular on the tangent plane at the fixed point
is the instantaneous axis of rotation, and the angular velocity of
the body varies inversely as the length of this perpendicular.
1 /T
If p be the length of this perpendicular, then ^ ~~ \/ iTf-
(4) The angular velocity about the invariable line is constant
and = 7^ .
The corresponding curve to a polhode is the path described on
the moving surface of the ellipsoid by the point fixed in space.
This curve is clearly a sphero-conic. The equations to the sphero-
conic described under any given initial conditions are easily seen
to be
2u. 2^ 2_ o' ^\y\l^L
These sphero-conics may be shown to be closed curves round
the axes of greatest and least moment. But in one case, viz.
414
MOTION UNDER NO FORCES.
r<2
when -™=^, where B is neither the greatest nor least mo-
ment of inertia, the sphero-conic becomes the two central circular
sections of the ellipsoid of gyration.
The motion of the body may thus be constructed by means of
either of these ellipsoids. The momental ellipsoid resembles the
general shape of the body more nearly than the ellipsoid of gy-
ration. It is protuberant where the body is protuberant, and
compressed where the body is compressed. The exact reverse of
this is the case in the ellipsoid of gyration.
520. MacCullagh has used the elHpsoid of gyration to obtain a geometrical
interpretation of the solution of Euler's equations in terms of elliptic integrals.
The ellipsoid of gyration moves so as always to touch a point L fixed in space.
Let us now project the point ii on a plane passing through the axis of mean
moment and making an angle a with the axis of greatest moment. This projection
may be effected by drawing a straight line parallel to either the axis of greatest
moment or least moment. We thus obtain two projections which we will call
P and Q. These points will be in a plane PQL which is always perpendicular to
the axis of mean moment. As the body moves about the point L describes on
the surface of the elHpsoid of gyration a sphero-conic KK', and the points P, Q
describe two curves pp', qq' on the plane of projection OBD. If the sphero-conic
as in the figure enclose the extremity A of the axis of greatest moment, the curve
inside the ellipsoid is formed by the projection parallel to the axis of greatest
moment, but if the sphero-conic enclose the axis of least moment, the inner curve
is formed by the projection parallel to that axis. The point P which describes the
inner curve will obviously travel round its projection, whUe the point Q which
describes the outer curve will oscillate between two limits obtained by drawing
tangents to the inner projection at the points where it cuts the axis of mean
moment.
maccullagh's construction. 415
Since the direction cosines of OL are proportional to ^Wj, Bo3.t,, Cwg it is easy to
see that, if x, y, zare the co-ordinates of L,
— =-^ = — = -=—1=- (1)
Aw^ Bw^ Cwg G JmT
Let OP=p, OQ=p', and let the angles these radii vectores make with the plane
containing the axes of greatest and least moment be ^ and 4>' measured in the
direction £D so that DOP= - 4>, DOQ= -4>': we then have
-pshi=:y = Bw.^{3IT}-i
pcos^sina = ^ = (7w3(J/rrM""
p'cos(p'cosa = x — Au}^{MT)~i)
-p'sm' ^y=£w.,{MT)-h) ^'^^•
It is proved in treatises on solid geometry that, if the plane on which the
projection is made is one of the circular sections of the ellipsoid, the projections
will he circles. This result may be verified by finding p or p' from these equations.
Remembering that p and p' are constants, let us substitute in Euler's equation
from (2) and the first of equations (3). We have
P -yT = "TTT sJ^T pp' sm a cos a cos (jt.
Since p' cos 4>' is the ordinate of Q, we see that the velocity of P varies as the
ordinate of Q, and in the savie way the velocity of Q varies as the ordinate of P.
To find the constants p, p' we notice that p is the value of y obtained from
the equations to the sphero-conic when 3=0. We thus have
,^ {AT-G^)B ,2^ (G^-CT)B
^ ~ MT(A-£)' ^ MT{B-C)'
the latter being obtained from the former by interchanging the letters A and C,
Hence
/velocity\ ^JA - B i-^, — ^^ /ordinateN
ofQ r^jim'^^'~^^ \ oip )'
V ofQ ; ]JaBO
521, Since p' sin ^' = p sin 0, we have by substitution
P
where X^ has the same value as in Art. 510. Let us suppose expressed in terms
of t by the elliptic integral
p/- d4>
M«-r)= P
Jo
^^4-
sm^<6
P'
SO that 0=amX(<-T). Substituting this value of in equations (2) or (3), we
obtain the values of w^, Wj, W3 expressed in terms of the time.
Ex. Investigate the corresponding theorem for the momental ellipsoid.
41 G MOTION UNDER NO FORCES.
522. If a body be set in rotation about any principal axis at
a fixed point, it will continue to rotate about that axis as a per-
manent axis. But the three principal axes at the fixed point do
not possess equal degrees of stability. If any small disturbing
cause act on the body, the axis of rotation will be moved into a
neighbouring polhode. If this polhode be a small nearly circular
curve enclosing the original axis of rotation, the instantaneous
axis will never deviate far in the body from the principal axis
which was its original position. The herpolhode also will be a
curve of small dimensions, so that the principal axis will never
deviate far from a straight line fixed in space. In this case the
rotation is said to be stable. But if the neighbouring polhode be
not nearly circular, the instantaneous axis will deviate far from
its original position in the body. In this case a very small dis-
turbance may produce a very great change in the subsequent
motion, and the rotation is said to be unstable.
If the initial axis of rotation be the axis OB of mean mo-
ment, the neighbouring polhodes all have their convexities turned
towards B. Unless, therefore, the cause of disturbance be such
that the axis of rotation is displaced along the separating polhode,
the rotation must be unstable. If the displacement be along the
separating polhode, the axis may have a tendency to return to its
original position. This case will be considered a little further on,
and for this particular displacement the rotation may be said to
be stable.
If the initial axis of rotation be the axis of greatest or least
moment, the neighbouring polhodes are ellipses of greater or less
eccentricity. If they be nearly circular, the rotation will certainly
be stable ; if very elliptical, the axis will recede far from its initial
position, and the rotation may be called unstable. If OC be the
axis of initial rotation, the ratio of the squares of the axes of the
neighbouring polhode is ultimately -tttd — pi • I^ i-^ therefore
necessary for the stability of the rotation that this ratio should not
differ much from unity.
It is well known that the steadiness or stability of a moving
body is much increased by a rapid rotation about a principal axis.
The reason of this is evident from what precedes. If the body
be set rotating about an axis very near the principal axis of
greatest or least moment, both the polhode and herpolhode will
generally be very small curves, and the direction of that principal
axis of the body will be very nearly fixed in space. If now a
small impulse/ act on the body, the effect will be to alter slightly *
the position of the instantaneous axis. It will be moved from one
polhode to another very near the former, and thus the angular
position of the axis in space will not be much affected. Let CI
be the angular velocity of the body, co that generated by the im-
THE INVARIABLE AND INSTANTANEOUS CONES. 417
pulse, then, by the parallelogram of angular velocities, the change
in the position of the instantaneous axis cannot be greater than
sin"^ jr- . If therefore O be great, co must also be great, to produce
any considerable change in the axis of rotation. But if the body
has no initial rotation H, the impulse may generate an angular
velocity (o about an axis not nearly coincident with a principal
axis. Both the polhode and the herpolhode may then be large
curves, and the instantaneous axis of rotation will move about
both in the body and in space. The motion will then appear
very unsteady. In this manner, for example, we may explain
why in the game of cup and ball, spinning the ball about a ver-
tical axis makes it more easy to catch on the spike. Any motion
caused by a wrong pull of the string or by gravity will not produce
so great a change of motion as it would have done if the ball had
been initially at rest. The fixed direction of the earth's axis in
space is also due to its rotation about its axis of figure. In rifles,
a rapid rotation is communicated to the bullet about an axis in
the direction in which the bullet is moving. It follows, from
what precedes, that the axis of rotation will be nearly unchanged
throughout the motion. One consequence is that the resistance
of the air acts in a known manner on the bullet, the amount of
which may therefore be calculated and allowed for.
On the Cones described by the Invariable and Instantaneous Axes.
523. It is clear from what precedes that there are two im-
portant straight lines whose motions we should consider. These
are the invariable line and the instantaneous axis. The first of
these is fixed in space, but as the body moves the invariable line
describes a cone in the body, which by Art. 519 intersects the
ellipsoid of gyration in a sphero-conic. This cone is usually called
the Invariable Gone. The instantaneous axis describes both a
cone in the body and a cone in space. By Art. 514, the cone de-
scribed in the body intersects the momental ellipsoid in a polhode,
and the cone described in space intersects the fixed plane on
which the momental ellipsoid rolls in a herpolhode. These two
cones may be called respectively the instantaneous cone and the
cone of the herpolhode.
524. There are various ways in which we may study the
properties of these cones. We may have recourse to solid geo-
metry. We may examine their curves of intersection with the
momental ellipsoid or the ellipsoid of gyration, as Poiusot and
MacCullagh have done. Wc may also examine by the help of
spherical trigonometry their curves of intersection with a sphere
R. D. 27
cosines of the instantaneous axis are — , — ^ , — . From the
CO 0) (O
418 • MOTION UNDER NO FORCES.
whose centre is at the fixed pointy and which is either fixed in the
body or fixed in space at our pleasure. This will be found con-
venient when we wish to use a diagram.
525. Let the principal axes at the fixed point be taken as the
axes of co-ordinates. The axes of reference are therefore fixed in
the body but moving in space. By Art. 512, the direction-cosines
of the invariable line are -~ , -~ , —-^ ; and the direction-
Cr Cr 6r
3 axis are -^ ,
w
equations (1) and (2) of Art. 511, we easily find
If we take the co-ordinates x, y, z to be proportional to the
direction-cosines of either of these straight lines and eliminate w^,
<«2) c^a by the help of this equation, we obtain the equation to the
corresponding cone described by that straight line. In this way
we find that the cones described in the body by the invariable
line and the instantaneous axis are respectively
AT-G"" „ BT-G' „ CT-G' „ ^
A {A T- G') x' + B {BT- G') f+G {CT- G') z' = 0.
These cones become two planes when the initial conditions are
such that G' = BT.
Ex. 1. Show that the circular sections of the invariable cone are parallel to
those of the ellipsoid of gyration and perpendicular to the asymptotes of the focal
conic of the momental eUipsoid.
526. There is a third straight line whose motion it is sometimes convenient to
consider, though it is not nearly so important as either the invariable hne or the
instantaneous axis. If x, y, z be the co-ordinates of the extremity of a radius vector
of an ellipsoid referred to its principal diameters as axes and if a, h, c be the semi-
axes, the straight line whose direction-cosines are - , ^ , -is called the eccentric line
a b c
of that radius vector. Taking this definition, it is easy to see that the direction-
cosines of the eccentric line of the instantaneous axis with regard to the momental
ellipsoid are Wj. /-, Wg . /^, f^sx/r' '^^^^^ ^^^ ^Iso the direction-cosines
of the eccentric line of the invariable line with regard to the ellipsoid of gyration.
This straight line may therefore ^be called simply the eccentric line and the cone
described by it in the body may be called the eccentric cone.
Ex. 1. The equation to the eccentric cone referred to the principal axes at the
fixed point is
{AT- G"-) X' + (BT- G") ?/2 + (CT- G~) z'-=.0.
THE INVAEIABLE AND INSTANTANEOUS CONES.
419
This cone has the same circular sections as the momental ellipsoid and cuts that
ellipsoid in a sphero-conic.
Ex, 2. The polar plane of the instantaneous axis with regard to the eccentric
cone touches the invariable cone along the corresponding position of the invariable
line. Thus the invariable and instantaneous cones are reciprocals of each other
with regard to the eccentric cone.
527. Let a sphere of radius unity be described with, its centre
at the fixed point about which the body is free to turn. Let
this sphere be fixed in the body, and therefore move with it in
space. Let the invariable line, the instantaneous axis, and the
eccentric line cut this sphere in the points L, I, and ^respectively.
Also let the principal axes cut the sphere in A, B, G. It is clear
that the intersections of the invariable, instantaneous, and eccen-
tric cones with this sphere will be three sphero-conics which are
represented in the figure by the lines KK', JJ', DD', respectively.
The eye is supposed to be situated on the axis OA* viewing the
sphere from a considerable distance. All great circles on the
sphere are represented by straight lines. Since the cones are co-
axial with the momental ellipsoid, these sphero-conics are sym-
metrical about the principal planes of the body. The intersections
of these principal planes with the sphere will be three arcs of
great circles, and the portions of these arcs cut off by any sphero-
conic are called axes of that sphero-conic. If we put s = in the
equations to any one of the three cones, the value of - is the
tangent of that semi-axis of the sphero-conic which lies in the
plane of xy. Similarly, putting 2/ = 0, we find the axis in the
plane of xz. If (a, h), [a, h'), (a, IB) be the semi-axes of the
invariable, instantaneous, and eccentric sphero-conics respectively,
we thus find
27—2
4)20 MOTION UNDER NO FORCES.
tana tana' tana NAT--W 1
B A ^AB ^/G'-BTs/AB'
tan b _ tan b' _ tan^g _ *JAT- G^ 1
G ~ A ~7jd~ ^/G^-CT TTd'
The first of these two sets gives the axes in the plane AOB,
the second those in the plane AOG. The former will he imagi-
nary if G^ < BT. In this case the sphero- conies do not cut the
plane AOB. The sphero-conics will therefore have their con-
cavities turned towards the extremities of the axes OA or G, i. e.
towards the extremities of the axes of greatest or least moment
according as G^ is > or < BT,
Ex. 1. If we put l-e^= . „ we may define e to be the eccentricity of the
sphero-conic whose semi-axes are a and b. If e and e' be the eccentricities of the
invariable and eccentric sphero-conics respectively, prove that <^^ = js ^ — 79 ^^^
B—C
c'^=j— p, so that both these eccentricities are independent of the initial conditions.
f QS.\l
Ex. 2. If the radius of the sphere had been taken equal to ( -^^ \ instead of
unity, show that it would have intersected the ellipsoid of gyration along the invari-
(MT^\ ^
— TTj-)", it would have intersected the
momental ellipsoid along the eccentric ellipse.
Ex. 3. A body is set rotating with an initial angular velocity n about an axis
which very nearly coincides with a principal axis OC at a fixed point O. The
motion of the instantaneous axis in the body may be found by the following
formulae. Let a sphere be described whose centre is O, and let / be the extremity
of the radius vector which is the instantaneous axis at the time t. If [x, y) be the
co-ordinates of the projection of I on the plane A OB referred to the principal axes
OA, OB, then
X = Jb (B - G) L sin {pitt + M),
j/= sJa (A~C)L cos (pnt + M),
where p^=- — ~ j^ ~ — ' , and L, M are two arbitrary constants depending on the
initial values of x, y.
Ex. 4. If in the last question L be the point in which the sphere cuts the
invariable line, if (p, 6) be the spherical polar co-ordinates of C with regard to
L as origin, and a the radius of the sphere, then
p2 =,i2 ^^ 12 1 2AB -C(A-i-B) + {A-B)CcoB2 (pnt+M)},
^ T CT-G"' raHt
THE INVAEIABLE AND INSTANTANEOUS CONES. 421
528. To find the motion of the invariable line and the instan-
taneous axis in the body.
Since the invariable line OL is fixed in space and the body-
is turning about 01 as instantaneous axis, it is evident that the
direction of motion of OL in the body is perpendicular to the
plane lOL. Hence on a sphere whose centre is at the arc IL
is normal to the sphero-conic described by the invariable line. This
simple relation will serve to connect the motions of the invariable
line and the instantaneous axis along their respective sphere- .
conies.
529. Let V be the velocity of the invariable line along its
sphero-conic, then since the body is turning about 01 with an-
gular velocity to, and OL is unity, we have « = w sin LOT. But
T ... T
by Art. 514 ^ = w cos i 07. Eliminating w we have v = j^ tan LOT.
530. Produce the arc IL to cut the axis AK in N, so that
ZZV is a normal to the sphero-conic described by the invariable
line. Taking the principal axes at the fixed point as axes of
reference, the direction-cosines of OL and 01 are respectively
proportional to Aco^, Bco^, Cco^, and oo^, w^, co^. The equation to
the plane L 01 is
(B - C) w^w^x + {G-A) w^w^y ^{A-B) w^w^z = 0.
This plane intersects the 'plane of xy in the straight line ON^
hence putting 2; = 0, we find the direction-cosines of ON to be
proportional to {A — C)ci)^, {B— C) co^, and 0. Hence
CO. L0N=i^4^£M±ms^^i^,
Gs/{A-Cyoy; + {B-Cf(o^^
The numerator of this expression is easily seen to be G^ — GT.
Expanding the quantity under the root we have
A',' - 2(7 {Ao>l + Bco^) + C (o),^ + <),
which is clearly the same as
g^ - C'co^^ -2G{T- Gco^) + G' (o)' - w/).
Substituting we find
G''-GT
con L ON =
G^G''-2CT+GW
C \fGW - T'
tan L ON ^ q2 _ qj,
422 MOTION UNDER NO FOECES.
But 5 = w COS LOI, .'. tan LOI= V^'^'-^ . Hence the
G T
.. tanLOI G'-GT . . . . , , n . ,
ratio fY)]\r = — TTni — > ^^c^ '^^ tkerejove constant throughout
the motion.
Treating the other principal planes in the same way, we see
that this proposition supplies us with a geometrical meaning for
the three expressions -j^ — 1, -^—1, and 'pm~ ^'
Combining this result with that given in the last Article, we
see that the
velocity of i ] G' - CT ^^^ ^
along its conic] CG '
where n is the angle LON. If we adopt the conventions of
spherical trigonometry, n is also the length of the arc normal to
the sphero-conic intercepted between the curve and the principal
plane AB oi the body,
Ex. 1. If tlie focal lines of the invariable cone cut the sphere in 8 and S', these
points are called the foci of the sphero-conic. Prove that the velocity of L
resolved perpendicular to the arc SL is constant throughout the motion and equal
to ;=; I' — 775 -[ ' If LMhe an arc of a great circle perpendicular to the
( AB )
axis containing the foci, and p he the arc SL, prove ako that
dt C ( AB
Ex, 2. Prove that the velocity of L resolved perpenxiicular to the central radius
, ,, . AT-G^ . , r
vector AL is — r-?; — cot AL,
AQ
Ex. 8. If r, r', r" be the lengths of the arcs joining the extremity^ of a princi-
pal axis to the extremities L, I, E of the invariable line, instantaneous axis, and
eccentric Hne respectively; d, ff, 6" the angles these arcs make with any principal
plane A OB, prove that
cos r cos r' _ cos r"
'at^G^^^^' gJTt'
tan d _ t ang' _ tan 6"
~C~ -~B~-'JW^'
where f=arci/. This theorem wiU enable us to discover in what manner the
motions of the three points L, I, E are related to each other.
Ex. 4. Show that the velocity of the instantaneous axis along its sphero-conic
is tan n' cos f, where n' is the length of the normal to the instantaneous
T AB
sphero-conic intercepted between the curve and the arc AB, and f=arc LI,
THE CONE OF THE HERPOLHODE. 423
Comparing this result with the corresponding formula for the motion of L given
in Art. 530, we see that for every theorem relating to the motion of L in its sphero-
conic there is a corresponding theorem for the motion of I. For example, if *S" he a
focus of the instantaneous sphero-conic, we see that the velocity of / resolved per-
pendicular to the focal radius vector S'l bears a constant ratio to cos LI, This
^ ^ ,. . G i(AT-G^){G'--BT))h
constant ratio is -^-^ ] ^ f^r [ .
CI ( AH I
Ex. 5. Show that the velocity of the eccentric line along its sphero-conic is
, tan n", where n" is the length of the are normal to the sphero-conic inter-
sJaect
cepted between the curve and the principal arc A B.
Ex. 6. Prove that (velocity of E)'^ - (velocity of Z)2 = constant. Show also that
[AT- (?2) [BT- G^) {CT- G2)
this constant =
ABCG'^T
Ex. 7. The motion of L along its sphero-conic is the same as that of a particle
acted on by two forces whose directions are the tangents at L to the arcs LS, LS'
joining L to the foci of the sphero-conic and whose magnitudes are respectively
proportional to sin LS cos LS' and sin LS' cos LS.
531. The instantaneous axis describes a cone in space, which
has been called the cone of the herpolhode. The equation of
this cone cannot generally be found, but when it can be determined
we have another geometrical representation of the motion. For
suppose the two cones described by the instantaneous axis in
space and in the body to be constructed. Since each of these
cones will contain two consecutive positions of their common
generator, they will touch each other along the instantaneous
axis. Then the points of contact having no velocity the motion
will be represented by making the cone fixed in the body roll on
the cone fixed in space.
532. To find the motion of the instantaneous axis in space.
Since the invariable line OL is fixed in space, it will be con-
venient to refer the motion to OL as one axis of co-ordinates.
Let the angle the instantaneous .axis 01 makes with OL be called
f, and let (^ be the angle the plane lOL makes with any plane
passing through OL and fixed in space.
During the motion the cone described by 01 in the body rolls
on the cone described by 01 in space. It is therefore clear that
the angular velocity of the instantaneous axis in space is the
same as its angular velocity in the body. Describe a sphere
whose centre is at and radius unity, and let this sphere be
fixed in the body. Let L, /be the intersections of the invariable
line and instantaneous axis with the sphere at the time t, L\ T
their intersections at the time t-Vdt. Then /L, TL' are con-
secutive normals to the sphero-conic KK' traced out by the in-
variable line and therefore intersect each other in some point P
424
MOTION UNDER NO FORCES.
which may be regarded as a centre of curvature of the sphero-
conic. Let p = PL. Then clearly
velocity of / resolved] _ /velocity\ sin (p + ^)
perpendicularly to IL] \ of Zr / *
T
= ^ tan ^ (cos X, 4- cot p sin ^) ;
sin
^ _ T / tang
dt G \ tanp
tan^n
But it may be proved that in any sphero-conic tanp= .^ ,
tan L
where n is the length of the normal intercepted between the
curve and that axis which contains the foci, and 21 is the length
of the ordinate through either focus, and is usually called the
latus rectum. Substituting for tan p, and remembering that
tan g
tanw
/tan^ b^
, by Art. 530, and tan I = , we get
GT
d^_T T ( G^-CT \
G^ G\
tan a
cot^ g.
dt G G\ CT J ' Vtan a,
If we substitute for tan a and tan h their values, we get
d^_T [AT- G') {BT- G') {CT- G') ,
dt~ G^ ABCGr ^'
This result was first discovered by Poinsot.
533. Since the resolved angular velocity about the invariable
T
line is constant, vv^e easily find « = -^ sec g. Substituting this
value of ft) in equation (6) of Art. 508, we find a relation between
^ and -y- , which however is too complicated to be of much use.
THE ROLLING AND SLIDING CONE. 425
The values of -^ and -^ in terms of t have now both been
at at
found ; from these the motion of the instantaneous axis in space
can be deduced.
Ex, 1. Show that the angular velocity v' of the instantaneous axis in space or
in the body is given by
where w is the resultant angular velocity of the body and \, "K.^, \ 3 have the mean-
ings given to them in Art. 508, This result is due to Poinsot,
Ex, 2. The length of the spiral between two of its successive apsides, described
in absolute space, on the surface of a fixed concentric sphere, by the instantaneous
axis of rotation, is equal to a quadrant of the spherical ellipse described by the same
axis on an equal sphere moving with the body. This is Booth's Theorem.
Ex. 3. If the eccentric line intersect in the point E the unit sphere which is
fixed in the body and has its centre at the fixed point, prove that
/velocityY_Td0
534. Let be the fixed point, 01 the instantaneous axis.
Let the angular velocity co about 01 be resolved into two, viz.
T
a uniform angular velocity —, about the invariable line OL, and
an angular velocity w sin lOL about a line OH lying in a plane
fixed in space perpendicular to the invariable line, and passing
through the fixed point 0. Let this fixed plane be called the
invariable plane at 0. As the body moves, OH will describe a
cone in the body which will always touch this fixed plane. The
velocity of any point of the body lying for a moment in OH is
unaffected by the rotation about OH, and the point has therefore
only the motion due to the uniform angular velocity about OL.
We have thus a new representation of the motion of the body.
Let the cone described by OH in the body be constructed, and
let it roll on the invariable plane at with the proper angular
velocity, while at the same time this plane turns round the in-
T
variable line with a uniform angular velocity -^ . The cone de-
scribed by OH in the body has been called by Poinsot the Boiling
and Sliding Cone.
535. To find a construction for the sliding cone. Its generator
OH is at right angles to OL, and lies in the plane LOL. Now
OL is fixed in space ; let OL be the line in the body which, after
an interval of time dt, will come into the position OL. Since the
body is turning about OL, the plane LOL' is perpendicular to the
plane L OL, and hence OH is perpendicular to both OL and OL'.
That is, OH is perpendicular to the tangent plane to the cone
420 MOTION UNDER NO FORCES.
described by OL in the body. The cone described by OH in the
body is therefore the reciprocal cone of that described by OL.
The equation to the cone described by OL has been shown to be
AT- a' , , BT-G' 2 GT-G' ^ ^
— ^-^ +— 5^2/^ + — 0— ^ =^-
Hence the equation to the cone described by OH is
A , . B . . G
' W ri'^y "I" /Trm ri2,^ — ^*
AT-G'' BT-G'^ ^ GT-G
The focal lines of the cone described by OH are perpendicular
to the circular sections of the reciprocal cone, that is the cone
described by OL. And these circular sections are the same as
the circular sections of the ellipsoid of gyration. Hence the focal
lines lie in the plane containing the axes of greatest and least
moment, and are independent of the initial conditions.
This cone becomes a straight line in the case in which the
cone described by OL becomes a plane, viz. when the initial con-
ditions are such that G^ = BT.
53G. To find the motion o/ OH in space and in the hody.
Since OL, OH and 01 are always in the same plane the
motion of OH in space round the fixed straight line OL is the
same as that of 01, and is given by the expression for —- in
Art. 532.
To find the motion of OH in the body it will be convenient
to refer to the figure of Art. 532. Produce the arcs PL, PL'
to H and H' so that LH and LH' are each quadrants. Then
H and H' are the points in which the axis OH intersects the
unit sphere at the times t and t + dt.
We have therefore
/velocity\ /velocityN ^^^ l,^ "*" 2 j T , ^ ,
\ oiH j = l ofL j- sinp =^tanrcotp.
Substituting for tan p as before we may express the result in
terms of ^ or w at our pleasure.
Since the cone described by OH in the body rolls on a plane
which also turns round a normal to itself at 0, it is clear that the
angular velocity of OH in the body is less than the angular
velocity of OH in space by the angular velocity of the plane, i. e.
/velocityN _d4_T
\ oiH )'"dt G'
MOTION OF THE PRINCIPAL AXES.
427
Motion of the Principal Axes.
537. To find the angular motions in space of the principal
axes.
Since the invariable line OL is fixed in space it will be con-
venient to refer the motion to this straight line as axis of z.
Let OA, OB, 00 he the principal axes at the fixed point 0, and
let, as before, a, yS, 7 be their inclinations to the axis OL or OZ.
Let X, //-, V be the angles the planes LOA, LOB, LOG make
with some fixed plane LOX passing through OL. Our object is
to find -7- and -j- with similar expressions for the other axes.
This problem is really the same as that discussed in Art. 285, but
it will be found advantageous to make a slight variation on the
demonstration.
Describe a sphere whose centre is at the fixed point, and
whose radius is unity. Let the invariable line, the instantaneous
axis and the principal axes cut this sphere in the points L, I,
A, B, G respectively. The velocity of A resolved perpendicular
to LA will then be sin a -j- . But since the body is turning round
01 as instantaneous axis, the point A is moving perpendicularly
to the arc LA, and its velocity is co sin lA. Resolving this per-
pendicular to the arc LA, we have
dX
sin a -77 = ^^^ ^ cos LA is the resolved part of the angular
428 MOTION UNDER NO FolSf §:
velocity about OA, which is w^. We have therefore
. . d\ T
sm a -?: = 7^ — «i cos a,
at ix
a result which follows immediately from Art. 249. Since
6^ cos a = Ao)^, we have
. ^ dX T Gcos^a ,,,
^^^"«-jr^ — A- (^)-
This result may also be written in the form
d\ T AT-G' ,2 /ON
-dra^-AG-'''^ (')•
da
538. To find -r we may proceed in the following manner.
We have cos a= -^, cos (3 = -^^r, cos 7 = -jf.
^^_(5-O)a,,a>3=0,
Substituting in Euler's equation
dco^
W
we have sin a -3- = f -^ — --^ j 6^ cos /3 cos 7 (3),
But by Art. 608 cos ot, cosyS, cos 7 are connected by the equations
cos^a cos^/3 cos^7_ T "j
~A"^~B~'^~G~~a'[ (4).
cos'^a + cos^/8 + cos^7 = 1 j
If we solve these equations so as to express cos^S, cos 7
in terms of cos a, we easily find
. , fday G^ (G'-CT A-C „ \fG^-BT A-B „ \ ...
539. Since the left-hand side of equation (5) is necessarily real, we see that the
values of cos^a are restricted to lie between certain limits. If the axis whose
motion we are considering is the axis of greatest or least moment let B be the axis
of mean moment. In this case cos^ a must lie between the limits — —= — ■ - — - and
G'^ A-C
G^~BT A
if both be positive. By Art. 509 the former of these two is positive
and less than unity ; this is easily shown by dividing the numerator and the de-
nominator by ACG'^. If the latter is positive the spiral described by the principal
axes on the surface of a sphere whose centre is at the fixed point lies between two
concentric circles which it alternately touches. If the latter limit is negative cos a
has no inferior limit. In this case the spiral always lies between two small circles
on the sphere, one of which is exactly opposite the other.
ix., ^'OF THE PRINCIPAL AXES. 429
If the axis considered is the axis of mean moment, cos^ a must lie outside the
same two limits as before. Both these are positive, but one is gi-eater and the
other less than unity. The sphal therefore lies between two small circles opposite
each other.
dt
tion makes -5- imaginary. Thus -^ always keeps one sign. It is easy to see that
if the initial conditions are such that -=• is less than the moment of inertia about
the axis which describes the spiral we are considering, the angular velocity will be
greatest when the axis is nearest the invariable Une and least when the axis is
furthest. The reverse is the case if -=• is greater than the moment of inertia.
540. Ex. 1. Let Oil! be any straight line fixed in the body and passing
through and let it cut the ellipsoid of gyration at in the point M. Let OM' be
the perpendicular from O on the tangent plane at M. If OM=r, 01^'=^, and if
i, i' be the angles OM, OM' make with the invariable line OL, prove that
. ..dj T G
BUX^ l-T = yi cos I cos I,
dt Q pr
where j is the angle the plane LOM makes with some plane fixed in space passing
through OL. This follows from Art. 249 or from Art. 537.
Ex. 2. If KLK' be the spiral traced out by the invariable line in the manner
described in Art. 527, show that
T T ^ TT /vectorial area\
\ = ^t + LAK-i^ L^^ j'
where \ is the angle described by the plane containing the invariable line and the
principal axis OA.
Ex. 3. If \p be the angle described in space by the plane containing the invari-
able line and any straight line OM, fixed in the body, passing through and
cutting the sphere in M, prove that
, T^ rAnT /vectorial area \
where MN is any spherical arc fixed in the body and cutting in N the sphero-conic
described by the invariable line.
Ex. 4. If we draw three straight lines OA , OB, 00 along the principal axes at
the fixed point of equal lengths, the sum of the areas conserved by these lines on
the invariable plane is proportional to the time. [Poinsot.]
Ex. 5. If the lengths OA, OB, 00 be proportional to the radii of gyration
about the axes respectively, the sum of the areas conserved by these lines on the
invariable plane will also be proportional to the time. [Poinsot.]
430 MOTION UNDER NO FOUQES.
Motion of the hody when two principal axes are equal.
541. Let the body be rotating with an angular velocity w
about an instantaneous axis 01. Let OL be the perpendicular
on the invariable line. The momental ellipsoid is in this case a
spheroid, the axis of which is the axis of unequal moment in the
body. Let the equal moments of inertia be A and B. From
the symmetry of the figure it is evident that as the spheroid rolls
on the invariable planes, the angles iyOC, XO/ are cow5ia?i^, and
the three axes 01, OL, G are always in one plane. Let the angles
LOC = %IOG=i.
Following the same notation as in Art. 508, we have
Wg = &) cos i, ct)/ + oi^ = (o^ sin^ {,
G'= (A' sm'i+C cos" i)(o\
T = {AsmH+Gcos^'i)co\
We therefore have
C(o„ C cos i
COS 'V =^ ^—
' jA'sinH+C'cos'i'
This result may also be obtained as follows. In any conic if
i and 7 be the angles a central radius vector and the perpendicular
on the tangent at its extremity make with the minor axis, and if
a, h be the semi-axes, then tan 7 = — 2 tan i. Applying this to the
a
momental spheroid, we have
A
tan y= y^ tan i.
The angle ^ being known from the initial conditions, the angle 7
can be found from either of these expressions. The peculiarities
of the motion will then be as follows.
The invariable line describes a right cone in the body whose
axis is the axis of unequal moment, and whose semi-angle is 7.
The instantaneous axis describes a right cone in the body
whose axis is the axis of unequal moment, and whose semi-angle
is ^.
The instantaneous axis describes a right cone in space, whose
axis is the invariable line, and whose semi-angle is i ~ 7.
The axis of unequal moment describes a right cone in space
whose axis is the invariable line, and whose semi-angle is 7.
The angular velocity of the body about the instantaneous
axis varies as the radius vector of the spheroid, and is therefore
constant.
CO,
. = — jP sin
MOTION WHEN A = B. 431
542. The rate of motion of the invariable line and the
instantaneous axis in the body may be found most readily by
referring to the original equations of motion in Art. 508. We have
in this case
■A-r^ — {A — G) 03^(0 cos i=
A-^+ {A- C) (o^co cos 1=0
Solving these by differentiating the first and eliminating a>^,
we find
)^ = F cos f — -^ — cot cos i+fj,
— ^ — C()Ccos*4-/|,
where i^ and /are arbitrary constants. Let the projection of either
the instantaneous axis or the invariable line on the plane per-
pendicular to the axis of unequal moment make an angle ^ with
any fixed straight line which may be taken as axis OA. Then
tan Y = — ^. Hence we find
dv A-G
-^ = — - — -. — &) cos %.
dt - A
543. To find the common rate of motion in space of the
instantaneous axis and the axis of unequal moment.
Let G be the extremity of the axis of figure of the momental
ellipsoid, and let fl be the rate at which the plane LOG is turning
round OL. Let GM, GN be perpendiculars on GL and GI.
Then since the body is turning round GI, the velocity of G is
GN.co. But this is also GM M. Since GM=OGsm%
GN= 00 sin i, we have at once
O sin 7 = ft) sin i,
whence O can be found.
544. Ex. 1, If a riglit circular cone whose altitude a is double the radius of
its base turn about its centre of gravity as a fixed point, and be originally set in
motion about an axis inclined at an angle a to the axis of figure, the vertex of the
cone will describe a circle whose radius is j a sin a. [CoU. Exam. ]
Ex. 2. A circular plate revolves about its centre of gravity as a fixed point. If
an angular velocity w were originally impressed on it about an axis making an angle
a with its plane, a normal to the plane of the disc will make a revolution in space in
time — -. [Coll. Exam.]
w^l + ijf^in^a
482
MOTION UNDEE NO FORCES.
Ex. 3. A body which can turn freely about a fixed point at which two of the
principal moments axe equal and less than the third, is set in rotation about any
axis. Owing to the resistance of the air and other causes, it is continually acted
on by a retarding coviple whose axis is the instantaneous axis of rotation and whose
magnitude is proportional to the angular velocity. Show that the axis of rotation
will continually tend to become coincident with the axis of unequal moment. In
the case of the earth therefore, a near coincidence of the axis of rotation and axis
of figure is not a proof that such coincidence has always held. Astronomical
Notices, March 8, 1867.
Motion when G' = BT.
545. The peculiarities of this case have been akeady alluded
to in Art. 508. When the initial conditions are such that this
relation holds between the Vis Viva and the Momentum of the
body the whole discussion of the motion becomes more simple*.
The fundamental equations of motion are
Solving these, we have
^ B-C G'-BW
(1).
A-C AB
A-B G'- BW
But
dt
A-C'
C-A
BG
(2).
don,
dt
-v
{A-B){B-G) O^-B'
AG ' B'
When the initial values of cDj and co^ have like signs, (G — A) (o^a\
is negative and therefore —~ must be negative, hence in this
expression the upper or lower sign is to be used according as the
initial values of 03 ^, Wg have like or unlike signs.
B'
' ' G'- B'a>: dt
^/
{A-B){B~G)
AG
* This case appears to have been considered by nearly every writer on this
subject. As examples of different methods of treatment the reader may consult
Legendre, Traite dcs Fonctions Elliptiqucs, 1825, Vol. I. page 382, and Poinsot
Theorie Nouvclle cle la Rotation des corps, 1852, page 104.
MOTION WHEN G'-BT, 433
If we put + n for the right-hand side and integrate we have
G + Ba>^ ^ _^^^.,,t
G- Bco^
where E is some undetermined constant.
2G
As t increases indefinitely, co^ approaches +-5 as its limit and
therefore by (2) a^ and a>^ approach zero.
The conclusion is that the instantaneous axis ultimately ap-
proaches to coincidence with the mean axis of principal moment,
but never actually coincides with it. It approaches the positive
or negative end of the mean axis according as the initial value
of {p—A) ty^, 0)3 is positive or negative.
546. To find what the cones traced out in the body hy the
invariable line and instantaneous axis become when G^ = BT.
Eliminating co^ from the fundamental equations of the last
Article we have
A[A-B)a>^^ = G{B-C)
co„
Taking the principal axes at the fixed point as axes of refer-
ence, the equations of the invariable line are —r— = ^^ = -p^ — .
^ Aoy^ B(o^ Coy^
Eliminating w^ and co^ the locus of the invariable line is one of
the two planes
/A-B , /B-G
X If
The equations of the instantaneous axes are — = -^ =
z
Eliminating Wj and Wg the locus of the instantaneous axis is one
of the two planes
^A{A-B) x = ± \/G{B-G) z.
In these equations since — follows the sign of — ^ the upper
or lower sign is to be taken according as the initial values of
Wj, 0)3 have like or unlike signs. These planes pass through the
mean axis, and are independent of the initial conditions except
so far that G^ = BT.
R. D. 28
484
MOTION UNDER NO FORCES.
The rolling and sliding cone is the reciprocal of that described
by the invariable plane, and is therefore the straight line perpen-
dicular to that plane which is traced out by the invariable line.
Ex. 1. Sliow that the planes described by the invariable line coincide with the
central circular sections of the ellii^soid of gyration and are perpendicular to the
asymptotes of that focal conic of the momental elUpsoid which lies in the plane of
the greatest and least moments.
Ex. 2. The planes described by the instantaneous axis are perpendicular to the
umbilical diameters of the ellipsoid of gyration and are the diametral planes of
the asymptotes of the focal conic in the momental ellipsoid.
547. The relations to each other of the several planes fixed
in the body may be exhibited by the following figure. Let
A, B, C be the points in which the principal axes of the body
cut a sphere whose centre is 0, and radius unity. Let BLK' ,
BIJ' be the planes traced out by the invariable line and the
instantaneous axis respectively. Then by the last Article
tan GK' =
Hence we find
A B-
C A-B
, tan CJ' =
B-G
A'A-B'
tan K'J' = tan LBI =
{B-C){A-B)
AG
This is the quantity which has been called n in Art. 54.5
Exactly as in Art. 528 the direction of motion of L is perpen-
dicular to IL and hence the angle ILB is a right angle. Thus
the spherical triangle ILB has one angle right, and another
constant and independent of all initial conditions.
Exactly as in Art. 528, the velocity of L along LB is equal to
MOTION WHEN G^ = BT, 435
T
(o sin IL which, by Art, 514, is equal to -^tan/Z. But from the
spherical triangle I LB
n sin BL = tan IL.
If then we put as before ^ = BL, we have
-YT = + -;^nsin a,
at ~ (x
If the initial values of w^, co^ have the same sign, the body
is turning round / from K' to B. Hence, since L is fixed in
space, BL is increasing and therefore the upper sign must be
used in this figure. See also Art. 545,
We may also find an expression for /3 in terms of the time.
Since cos /3= -~ we have, by Art. 545,
1 + COS 13 _ T^+^-nt
1 — cos j$ '
Z
Ex, Show that the eccentric line describes a gi'eat circle passing through B and
cutting 4 C in some point D' where tan^ CD' = tan CJ' tan CK'. If E be the inter-
section of the eccentric line with the sphere, show that the arcs BE and BL are
always equal.
548. To find the motion of the hodij in space.
"We have already seen that the motion is such that a plane
fixed in the body, viz. the plane BK', contains a straight line
fixed in space, viz. the invariable line OLj. Since the body is
brought from any position into the next by an angular velocity
T
0) cos IOL=^ about OL, and an angular velocity cosinlOL
about a perpendicular to OL, viz. OH, it follows that the plane
fixed in the body turns round the line fixed in space with a
T C
uniform angular velocity -^ or -^. At the same time the plane
moves so that the line fixed in space appears to describe the
plane with a variable velocity co sin lOL. If /3 be the angle BL,
T
this has been proved in the last Article to be y^ n sin ^.
549. The cone described by OH in the body is the reciprocal
cone of that described by OL, and from it we may deduce re-
ciprocal theorems. The motion is therefore such that a straight
line fixed in the body, viz. OIL, describes a plane fixed in space,
viz. the plane perpendicular to OL. The straight line moves
28—2
436 MOTION UNDER NO FORCES.
T G
along this plane with a uniform angular velocity equal to ^ or -^ ,
"vvhile the angular velocity of the body about this straight line
is ±-Y,n sin ^.
550. The motion of the principal axes may be deduced from
the general results given in Art. 537. But we may also proceed
thus. Since the body is turning about 01, the point B on the
sphere is moving perpendicularly to the arc IB. Hence the
tangent to the path of B makes with LB an angle which is the
complement of the constant angle IBL. The path traced out
by the axis of mean moment on a sphere whose centre is at is
a rhumb line which cuts all the great circles through L at an
angle whose cotangent is + n.
551. To find the motion of the instantaneous axis in space.
This problem is the same as that considered in Art. 532. We
may however deduce the result at once from Art. 548. The angle
ILB is always a right angle, it therefore follows that the angular
velocity of / round L is the same as that of the arc BL round L.
But the angular velocity of the latter is constant and equal to -^.
If then ^ be the angle the plane LOI containing the instanta-
neous axis and the invariable line makes with some fixed plane
passing through the invariable line, we have -j7=p'
552. To find the equation of the cone described by the
instantaneous axis in space, we require a relation between f and (f>,
where ^ is the arc IL on the sphere. From the right-angled
triangle ILB we have ?i sin /3 = tan ^, and by Art. 547,
cot| = \/£'e ^ •
Eliminating /S, we shall have an expression for ^ in terms of t.
We find
-^ = cot^+tan^ = V£'6"^ +7t,^ ^ •
tan f 2 2 \/ E
T
By the last Article (}) = -7yt+F, where F is some constant.
Let us substitute for t in terms of (^^and let us choose the plane
from which cf) is measured so that 's/ Ee^""^ = 1.
The equation to the cone traced out in space by the instan-
taneous axis is
2wcot^=e"* + e"'**.
CORRELATED AND CONTRARELATED BODIES. 437
When (f> = 0, we have ian^=n. Therefore the plane fixed in
space from which (p is measured is the plane containing the axes
of greatest and least moment at the instant when that plane
contains the invariable line.
On tracing this cone, we see that it cuts a sphere whose centre
is at the fixed point in a spiral curve. The branches determined
by positive and negative values of <^ are perfectly equal. As ^
increases positively the radial arc ^ continually decreases, the
spiral therefore makes an infinite number of turns round the
point L, the last turn being infinitely small.
Ex. In the lierpolhode — = e™^ + e-'"^, if the locus of the extremity of the
polar subtangent of this curve be found and another curve be similarly generated
from this locus, the curve thus obtained will be similar to the herpolhode. [Math.
Tripos, 1863.]
On Correlated and Contrarelated Bodies.
553. To compare the motions of different bodies acted on hy
initial couples whose planes are parallel.
Let a, 13,
A" ' B" ' C"
But remembering the condition (6) these give
_ T r
cos
■■(»-i)(»-i)(«-^)^o-
9. If a body move in any manner, and all the forces pass through the centre of
gravity, prove that
-y + 2 J^ (log -x) ^-^(log <^.) j^ (log C.3) = 0,
where w^, Wg, W3 are the angular velocities about the principal axes at the centre of
gravity, and w is the resultant angular velocity.
CHAPTER X.
MOTION OF A BODY UNDER ANY FORCES.
557. In this Chapter it is proposed to discuss some cases
of the motion of a rigid body in three dimensions as examples
of the processes explained in Chapter V. The reader will find
it an instructive exercise to attempt their solution by other
methods ; for example, the equations of Lagrange might be
applied with advantage in some cases.
Motion of a Top.
558. A body two of whose principal moments at the centre
of gravity are equal moves about some fixed point in the axis
of unequal moment under the action of gravity. Determine the
motion. See Art. 874.
To give distinctness to our ideas we may consider the body
to be a top spinning on a perfectly rough horizontal plane.
Let the axis OZ be vertical. Let the axis of unequal moment
at the centre of gravity be the axis OG and let this be called
the axis of the body. Let h be the distance of the centre of
gravity of the body from the fixed point and let the mass
of the body be taken as unity. Let OA be that principal axis
at which lies in the plane ZOG, OB the principal axis perpen-
dicular to this plane.
If we take moments about the axis 00 we have by Euler's
equations (Art. 230),
C^-{A-B)co,co^^ = N.
But in our case A = B, and since the centre of gravity lies
in the axis OG, we have N= 0. Hence (o^ is constant and equal
to its initial value. Let this be called n.
Let us measure along the axis OG in the direction 00 a
MOTION OF A TOP.
445
lenoih OP = -t . Then, by Art, 92, P is tlie centre* of oscillation
° h -^
of the body. This length we shall call I. Let 6 be the inclina-
tion of the axis OC to the vertical, -v/r the angle the plane ZOG
makes with some plane fixed in space passing through OZ. Then
by the same reasoning as in Art. 235 we find that the velocities
of F resolved
perpendicular to plane ZOC= — lo3^ = l sm 6 —r-
parallel to plane ZOO = Zwj = Z -^7
.(1).
T
W
--,.
rr---'''
M
' /
^
T
\
It is clear that the moment of the momentum about OZ
will be constant throughout the motion. Since the direction-
cosines of OZ referred to OA, OB, OG are — sin^, and cos ^,
this principle gives
- ^&)j sin ^ + (7/1 cos ^ = ^ (2),
where E is some constant depending on the initial conditions,
and whose value may be found from this equation by substituting
the initial valae of w, and 6.
The equation of Vis Viva gives
A {o>^ + «/) + Cn^=^F- 2ghcos 9 (3),
where F is some constant, whose value may be found by substi-
tuting in this equation the initial values of co^, oi^, and ^ t.
* To avoid confusion in the figure, the body wlaicli is represented by a top
is drawn smaller than it should be.
t If we eliminate Wj, ui„ from equations (1), (2), (3) we have two equations from
which 9 and i// may be found by quadratures. These were first obtained by
Lagrange in his Mecanique Anabjtique, and wei'e afterwards given by Poiason in
his Traits de Mecanique. The former passes them over with but slight notice,
and proceeds to discuss the small oscillations of a body of any form suspended
under the action of gravity from a fixed point. The latter limits the equations to
446 MOTION UNDER ANY FORCES.
559. Let us measure along the vertical OZ, in the direction opposite to gravity
" -pi 1 (P f^71^\
as the positive direction, two lengths 0Z7 = -^, OV— ^ , — -. These lengths
Vn 2gh
we shall write briefly OU=a, and OV=b. Draw through U and V two horizontal
planes, and let the vertical through P intersect these planes in M and N. Then
the equations (2) and (3) give by (1),
horizontal velocity) Cn , „„,, ,,,
ofP '\=^t^nPUM (4).
(velocity of P)^=^2gPN (5).
Thus the resultant velocity of P is that due to the depth of P below the horizontal
plane through V, and the velocity of P resolved perpendicular to the plane ZOP
is proiDortional to the tangent of the angle PU makes with a horizontal plane.
It appears from this last result that when P is below the horizontal plane
through U, the plane POF turns round the vertical in the same direction as the
body turns round its axis, i.e. according to the rule in Art. 199, OF and OP are
the positive directions of the axes of rotation. "When P passes above the horizontal
plane through U, the plane POF turns round the vertical in the opposite direction.
If P be below both the horizontal planes through and U these results are still
true, but if a top is viewed from above, the axis will appear to turn round the
vertical in the du-ection opposite to the rotation of the top. In all the cases
in which P is below the plane UM the lowest point of the rim of the top moves
round the vertical ia the same direction as the axis of the top.
If we substitute for w^, Wj, E and F in (2) and (3) their values, we easily obtain
(6).
hi sin^ 6-^+ On cos 6=^0)1^
at i
{(fy+=^"'«(i)>^' <'-'-»'
These equations give in a convenient analytical form the whole motion. We
see from the last equation, what is indeed obvious otherwise, that h~l cos 6 is
always positive. The horizontal plane through F is therefore above the initial
position of P and remains above P throughout the whole motion.
Ex. 1. If w be the resultant angular velocity of the body and v the velocity of P
show that w2=n' + (yj.
Ex. 2. Show that the cosine of the inclination of the instantaneous axis to the
,. ,. E+ {A- C) Ilea's e
vertical is ^-^^ .
Ata
560. As the axis of the body goes round the vertical its
inclination to the vertical is continually changing. These changes
the case in which the body has an initial angular velocity only about its axis, and
applies them to determine directly the small oscillations of a top (1) when its axis
is nearly vertical, and (2) when its axis makes a nearly constant angle with the
vertical. His results are necessarily more limited than those given in this
treatise.
MOTION OF A TOP. 447
may be found by eliminating -~^ between the equation (6). "We
thus obtain
/7^^\' o /7 7 ay C'n' fa -I cos e \' ,„.
It appears from this equation that 6 can never vanish unless
a = l, for in any other case the right-hand side of this equation
Avould become infinite. This may be proved otherwise. Since
y is equal to the ratio of the angular momentum about the vertical
to that about the axis of the body, it is clear the axis could not
become vertical unless the ratio is unity.
Suppose the body to be set in motion in any way with its
axis at an inclination i to the vertical. The axis will begin to
approach or to fall away from the vertical according as the initial
value of -jr or co^ is negative or positive. The axis will then
oscillate between two limiting angles given by the equation
= 2ghr (b - 1 cos d) (1 - cos'^) - CV {a - I cos Of (8).
This is a cubic equation to determine cos 0. It wall be neces-
sary to examine its roots. When cos 6 = — 1 the right-hand side
is negative; when cos ^ = cos/, since the initial value of f-i-j is
essentially positive, the right-hand side is either zero or positive ;
hence the equation has one real root between cos = — 1 and
cos ^=cos i. Again, the right-hand side is negative when cos ^= -hi
and positive when cos = ao . Hence there is another real root
between cos = cos i, and cos 0=1, and a third root greater than
unity. This last root is inadmissible,
561. These limits may be conveniently expressed geometrically. The equation
(7) may evidently be written in the form
:'^r=^— ??-(^)' <»)•
Describe a parabola with its vertex at U, its axis vertically downwards and its
latus rectum equal to r-yr . Let the vertical PMN cut this parabola in Ji, we then
have
M -=.A. + .i (10).
2(jMN
The point P oscillates between the two positions in which the harmonic mean
of PM and PR is equal to - 2 . MN. In the figure V is drawn above U, and in
this case one of the limits of P is above UM, and the other below the parabola. If
we take U as origin and UO the axis of x, we have PM = x, UM=y. Let 2^5/ be the
448 MOTION UNDER ANY FORCES.
latns rectum of the parabola, and UV=c, then the axis of the body oscillates
between the two positions in which P lies on the cubic curve
y^{x + c)=2plx^ (11).
When c is positive, i. e. when V is above U, the form of the curve is indicated
in the figure by the dotted line. The tangents at U cut each other at a finite
angle and the tangent of the angle either makes with the vertical is ( — ) . When
c is negative the curve has two branches, one on each side of the vertical, with a
conjugate point at the origin. It is clear from what precedes that the upper
branch will lie above, and the lower branch below, the initial position of P,
and ihat P must always lie between the two branches.
562. In the case of a top, the initial motion is generally given
by a rotation n about the axis. We have initially co^ = 0, ty^ = 0,
and therefore by (2) and (3) E= Cn cos ^, and F— Cn^ = 2gh cos i,
CV
This gives a = J = Ico^i. Putting ^-^^ =2pZ, as before, the roots
of equation (8) are cos 9 = cos i, and cos ^ =p — Vl — 2p cos i-\-p^.
The value cos 6 =p + Vl — 2j9 cost +p^ is always greater than
unity, for it is clearly decreased by putting unity for cost, and
its value is then not less than unity. The axis of the body will
therefore oscillate between the values of 6 just found.
Since a = S, the horizontal planes through 77 and 7 coincide, and c=0. The
cubic curve which determines the Umits of oscillation, becomes the parabola TJR
and the straight line UM. The axis of the body will then oscillate between the two
positions in which P lies on the horizontal through U and on the parabola.
Generally the angular velocity n about the axis of figure is
very great. In this case p is very great, and if we reject the
1
squares of - we see that cos 9 will vary between the limits cos i
, .1.2.
and cos I — jr- sm i.
2p
If the initial value of i is zero, we see that the two limits of
cos i are the same. The axis of the body will therefore remain
vertical.
6G3. Ex. 1. When the limiting angles between which 6 varies are equal to
each other, so that 6 is constant throughout the motion and equal to a, show that
tan^ - tan tan a -\ — -r— tan* a = 0, , .
where where h is the distance of the centre of gi'avity of the top
from 0, and C is the moment of inertia about the axis of figure. Show also that if
the top be initially placed with 00 nearly horizontal and if a very great angular
velocity be communicated to it about 00 without any initial angular velocity about
OA or OB, then 00 wUl revolve round the vertical remaining very nearly in a hori-
zontal plane with an angular velocity fj. given by the same formula as before, and
.„ , 2-rrA
the time of the vertical oscillations of 00 about its mean position will be —7— .
568. A body whose principal moments of inertia are not neces-
sarily equal has a point O fixed in space and, moves about O under
the action of gravity. It is required to form the general equations
of motion.
Let OA, OB, OC he the principal axes at the fixed point 0,
and let these be taken as axes of reference. Let h, Ic, I he the
co-ordinates of the centre of gravity G, and let the mass of the
body be taken as unity. Let F be drawn vertically upivards
29—2
452
MOTION UNDER ANY FORCES.
and let p, q, r be the direction-cosines of OF referred to OA,
OB, OC. Then we have by Euler's equations
A
B
C
d(o^
'dt
d(o^
dt
d(o^
~dt
-{B-C) w,0)3 = -g(kr- Iq)
-{G -A) cOgWj = -g{lp- hr)
- {A - B) (o^oi^ = - ^ {hq - kp)
.(1).
Also p, q, r may be regarded as the co-ordinates of a point
in OV, distant unity from 0. This point is fixed in space, and
therefore its velocities as given by Art. 248 are zero. We have
dp
^d-
oi^r
dq
di~
co^r —
«3P
dr
di~
(o,^p-
w^q
(2).
It is obvious that two integrals of these equations are supplied
by the principles of Angular Momentum and Vis Viva. These
give
Aoi^p ->r Bw^q + Gay^r = E,
A(o^ + B(o^ + Ga^ = F- 2g [ph +qh-\- rl),
where E and F are two arbitrary constants. The first of these
might also have been obtained by multiplying the equations (1)
by p, q, r respectively, and (2) by Am^, Bco^, Gco^, and adding all six
results. The second might have been obtained by multiplying
the equations (1) by «j, co^, co^ respectively, adding and simpli-
fying the right-hand side by (2).
569. A body whose principal moments of inertia at the centre of gravity G are
not necessarily equal, has a point in one of the principal axes at G fixed in space
and moves about under the action of gravity. Supposing the body to be performing
small oscillations about the 2}osition in which OG is vertical, find the motion.
Referring to the general equations of Art. 568, we see that in this case h=0,
Jc=0. Since OC remains always nearly vertical, w^ and u^ are small quantities, we
may therefore reject the product w^Wg in the last of equations (1). This gives W3
constant. Let this constant value he called n. For the same reason r — 1 nearly
and p, q are both small quantities. Substituting we get the following linear
equations,
di>
£'^^-(C-A)n<^,= -Igp
.(3),
dp
di'
dq
di'
■ qn-ws
-pn + Wi
.(4).
MOTION OF A TOP, 453
To solve these, assume
wj = F sin {\t +/) ) 'p = P^m {\t +/) )
w2=Gcos(\«+/)i' 2=Qcos(\«+/)i*
Substituting, we get '
A\F-{B-C)nG=glQ) . \P = Q,n-G
B\G-(A-C)nF=glp\ ^ '' \Q^Pn~F
Eliminating the ratios F : G : P : Qvfe have
W {A +B- Gf= {9l + A\^+ (B - C) n^} {gl + B\^ + {A - C)n''}.
If the values of X thus found should be real, the body will make small oscillations
about the position in which OG is vei-tical. If C be the greatest moment, and n^
sufficiently great to make both gl- (C-A) n^ and gl- {C - B)n^ negative, then all
the values of X are real and the body will continue to spin with OG vertical. If G
be beneath 0, I is negative and it will be sufficient that OC should be the axis of
greatest moment.
In order that the values of X^ may be real, we must have
{gl[A + B)+n^{AG^BC-2AB~C'')\^>i{{B-C)n'+gl]{{A~C)n^+gl)AB,
and in order that the two values of X^ may have the same sign we must have the
last term of the quadratic positive ;
.'. {{B-C)in?-\-gl]{{A~C)n'^-irgl\= a positive quantity,
and in order that the values of X^ may be both positive, we must have the coefficient
of X^ in the quadratic negative ;
.:gl{A-\-B) Kn-^iB-OiA-C).
In the particular case in which A = B, each side of the quadratic becomes a
perfect square and we have
A\^Jz(2A-C)n\+ (A~C)7i^ + gl=0;
With the reservations mentioned in Art. 434, the necessary and sufficient
^\/ Ar/l
condition of stabihty is in this case n> ■ ■ . By referring to equations (5) and
(6) it will be seen that when J. = 5 we have F=G and P^Q. If X^, \ be the two
values of X found above, we have
i) = Pi sin {\t +/i) + Pa sin {X^t+f^)
q^P^ COS (\t +/i) + Pg cos (\t +/a)
Let 9 be the angle 00 makes with the vertical, then r'=cos*6' = l - 6>^ and
lience
d' = iP + ./ ^ p^i + p^, ^ 2P,P, cos ! (Xi - X„) t +/i -f,l.
454 MOTION UNDER ANY FORCES.
Also if, as in Art. 235, we let
,
where a is some constant, depending on the position of the arbitrary plane from
which i/* is measured* .
* In order to understand the relation which exists between these results
and those of Art. 565, it will be necessary to determine the oscillations by some
process which holds both when a is large and very small. This may be done as
follows. We have by Vis Yiva the equation (see Art. 558)
(de\^ ( E - Cn cosO y _ F'- 2gh cos 9
[dtj'^y Asind )~ A ^^>'
where F' has been put for F- Cn"^. If we put 2 = cos Q, this takes the form
^'(iO +(^-<^«-)'=^(^'-v^^)(i-^') (2)-
Let us assume as the solution of this equation
2 = cos a + Pcos (Xi+/) (3),
where P is so small that on substituting in the above equation we may neglect P^.
Substituting and equating to zero the coefficients of the several powers of cos (\i+/)
we get
A'^P-\^ + (£' - Cn cos af =A(F" - 2(jJi cos a) (1 - cos^ a) ^
- (£■- Owcos a) C?i= -c/7i-^ -^P' cosa + 3(/A^ cos^a v. (4).
- ^2\2 4. (?2^2 =^ _ AF+ 6(/hA cos a )
Now let us change the constant E into another jj. by putting — - — r-^ =M-+yP^,
° A sm^ a
where 7 is to be so chosen as to remove the term A"P^X^ in our first equation.
Since
dx}/ _E- Cn cos 9
'di~ A sin2 e ^ ''
we see that, when 9 is not small, 11 differs from the constant part of -^ only by
quantities depending on the squares of the small oscillation, and which are
neglected in the text. Substituting for E and eliminating F' between the first
and second equations we get Cn/ji,=A cos a/x^ + gh.
Eliminating F' between the first and third of equations (4) and substituting for n
we get
^ , fi^A^- 2gliA cos a/x^ + g^h^
^^ IV *
This process gives the period of the small oscillation in cos 9. When 9 is finite
this is the same as the oscillation in 9, since cos ^ = cos a - sin ad'. When 9 is very
smaU, cos^=l — and the time of osciUatiou in cos ^ is the same as that in 9^.
With this understanding it will be seen that there is a perfect agreement between
the results of Ai'ts. 565 and 569, when a is put equal to zero.
MOTION OF A TOP. 455
570. A body lohose principal moments at the centre of gravity are not necessarily
equal is free to turn about a fixed point 0, and is in equilibrium under the action of
gravity. A small disturbance being given, find the oscillations,
Eeferring to the general equations in Art. 568 we see that in this case Wj, Wj, Wg,
are small, hence in equations (1) we may omit the terms containing the products
WiWjj WjWg, Wjw^. Also since in equilibrium OG is vertical, p, q, r are always
nearly in the ratio h:k:l; hence if OG = a, we may write-, -, -forjo, q, r on the
right-hand sides of equations (2). The sis equations are now all linear. To solve
these we put
ui=Hsin(\t + /j.) and p = - + P COS (\t + /J.) (3),
w^, W3, q and r being represented by similar expressions with K and L written for
H; Q, k and R, I written for P and h. Substituting these in the equations we get
six linear equations. Eliminating. P, Q, E we have
-- hkH+
- IhH- lkL+( - C\'^ + h^ + k^
f-AX^ + k'~ + l-\ H- hkK-lhL =
(-BX^ + P + hAK~lkL = ►
"' " f-C\^ + h'' + kAL = 0\
.(4).
Eliminating the ratios of H, K, L we have an equation to find X^. One root is
X^=0, the others are given by the quadratic
.4 , f^'^ + l'' ■ l' + 1^^ , ^^ + ^^\ 9.,. , „, Ah'^ + Bk^+GP
^ +(,-T- +-:B^ + -0-) a^ +^ ABG— =^ (^)-
To ascertain if the roots are real we must apply the usual criterion for a quad-
ratic. This requires that
{A{B-C)K'' + B{C-A)k^-C{A-B)m'' + iAB{B-C){A-C)h''k" (6)
should be positive. Since A, B, C can be chosen to be in descending order, we see
that the condition is satisfied. See also Art. 448.
If G is above 0, a is positive and the values of X^ are both negative. The equi-
librium is therefore unstable. If G is below 0, a is negative and the values of X^
are both positive. If the roots are equal, the two positive terms in (6) must be
separately zero, this gives h=(i and A [B - C)h" = C {A - B)l'^, i.e. the centre of
gravity hes in the asymptote to the focal hyperbola of the momental ellipsoid. In
^_ag
Bidered in Art. 564.
this case we find \^= -^, The case in which k=0, l-O, B=C has been con-
If the values of X'^ are written 0, X^', \^ we have
Wi = Ho + H^'t + H-^ sin {\t + fi-^ + H^ sin (X„< + ^Uj),
with similar expressions for w^, wg. Equations (2) then give p, q, r. But substitut-
ing in (1) we find that all the non-periodic terms which contain t are zero.
Remembering that x'^ + 3^ + r^ == 1 we have finally
Wj = n - + 7/j sin (Xjt -t- fx^) + IIj sin {\t + mj),
456 MOTION UNDER ANY FORCES.
Wg and W3 being represented by similar expressions with Tc, K and I, L written for
h, H. Tlie values of K^^ L^ and K^, L^ are determined by equations (4) in terms of
H^ and H2 respectively. We also liave
P = ~ + -^ <5os {\t + /^i) + -^^^ ? cos (Xgi + M2) .
with similar expressions for q and r. Tliere remain five constants viz. 0, H^, H^,
fJi-i, M2 to be determined by the initial values of w^, w^, w^, r and q.
When the roots are equal the equations depending on p, r, Wg separate from those
depending on q, w^, u^, forming two sets; we find
Wj = i2- + Zfsin (Xt + yUj) | 0.^= ^sin(\«H-^2)
'* \ h I
a gill
H —J cos (\t + n^)
r — --K--- cos [Xt + u„)
A solution of this problem conducted in a totally different manner has been
given by Lagrange in his ilecanique Analytique. His results do not altogether
agree with those given here.
If we substitute the values of w^, Wg, W3, p, q, r in the equation of angular
momentum of Ai't. 568 and neglect the squares of small quantities, we evidently
obtain
iAh'^ + £k^+ Cl^) Q = Ea^, A Hh + £Rk + CLI=0.
The first of these equations shows that fl vanishes when the initial conditions
are such that the angular momentum about the vertical is zero. In tliis case the
problem reduces to that considered in Art. 455.
571. A body whose principal moments of inertia are not necessarily equal has a
point fixed in space and moves about under the action of gravity. It is required
to find what cases of steady motion are possible in which one principal axis 00 at
describes a right cone round the vertical while the angular velocity of the body about
00 is constant; and to find the small oscillations.
Referring to the general equations of Art. 568, we see that r and wg are given to
be constants. In this case the first two equations of (1) and (2) form a set of linear
equations to find the four quantities p, q, w^, Wg. The solution of these equations
is therefore of the form
a}^=Fo + Fi sin {\t+f)) p=P(, + P^smiXt+f))
W2 = (?o + (?iCOs(X«+/)i ' 3 = Qo + QiCos(Xi+/)i*
But these must also satisfy the last of equations (1). Substituting we see that
there will be a term on the left side of the form
-l{A-J3)F^G^sin2{Xt+f).
But there wiU be no such term on the right side. Hence we must have either
A=B, Fj^ — or Cri = 0. The motion in the case in which A — B has already been
considered in Art. 564. Again, substituting in the last of equations (2) and equat-
ing to zero the coefficient of sin 2 [Xt +/) we find
P,G,-F,Q,^0.
MOTION OF A SPHERE. 457
Substituting m the first two of equations (1) and equating to zero the coefficients
of cos {\t+f) and sin (\t +/), we find
from these equations we have F-^, G-^, P^, Q^ all equal to zero and therefore Wj, Wg,
p, q are all constant as well as the given constants Wj and r.
In this case the equations (2) give
p 2 r '
so that the axis of revolution must be vertical. Let w be the angular velocity about
the vertical. Then w^=pw, w.^ — qu, u.^ — ro}. Substituting in equations (1) we get
P~ 9 ~<1 9 "r g ^ ^'
Unless, therefore, two of the principal moments are equal, it is necessary for
steady motion that the axis of rotation should be vertical and the centre of gravity
{hU) must lie in the vertical straight Hue whose equations are (3).
This straight line may be constructed geometrically in the following manner.
Measure along the vertical a length V- ~ and draw a plane through V perpeu-
dicular to F to touch an ellipsoid confocal with the ellipsoid of gyration. The
centre of gravity must lie on the normal at the point of contact.
To find the small oscillations about the steady motion, i.e. to determine whether
this motion be stable or not,- we must put
p — cos a + Pfl sin \t + P^ cos X*,
with similar expressions for q, r, w^, wg, ojy Substituting we shall get twelve linear
equations to determine eleven ratios. Eliminating these we have an equation to
find \. It is sufficient for stabihty that all the roots of this equation should be real.
Motion of a Sphere.
572. To determine the motion of a sphere on any perfectly rough surface under
the action of any forces ivhose resultant passes through the centre of the sphere.
Let G be the centre of gTavity of the body and let the moving axes GO,GA, GB
be respectively a normal to the surface and some two lines at right angles to be
afterwards chosen at our convenience. Let the motions of these axes be de-
termined by the angular velocities 6^, 6^, 6^ about their mstautaneous positions
in the manner explained in Art. 243. Let u, v, w he the velocities of G resolved
parallel to the axes so that w=0, and w^, w^, Wj the angular velocities of the body
about these axes. Let F, F' be the resolved parts of the friction of the perfectly
roiigh surface on the sphere parallel to the axes, GA, GB, and let B be the normal
reaction. Let X, Y, Z be the resolved parts of the impressed forces on the centre
of gravity. Let k be the radius of gyration of the sphere about a diameter, a its
radius, and let its mass be unity. The equations of motion of the sphere arc by
i^ts. 254 and 245,
458
MOTION UNDER ANY FORCES.
doj-i „ „ F'a 1
'^^'i o ^ Fa .
dt
du „
dv „
•(1).
and since tlie point of contact of the sphere and surface is at rest, we have
u - aco^ = )
V + aw^ = )
Eliminating F, F', co^, Wj from these equations, we get
.(2).
.(3).
dii
dv „
,X +
¥•
r+-.
p
i ^l«W3
^„aw.
.(4).
dt^^ a' + k' aHi'
573. The meaning of these eqtiations may be found as follows. They are the
two equations of motion of the centre of gravity of the sphere, which we should
have obtained if the given surface had been smooth and the centre of gravity had
been acted on by accelerating forces -5 — =-5 d^au^ and ^ ,3 ^j^'^a along the axes
a?
GA, GB, and by the same impressed forces as before reduced in the ratio ~ — — .
The motion therefore of the centre of gravity in these two cases with the same
initial conditions will be the same. More convenient expressions for these two
additional forces may be found thus. The centre of gravity moves along a surface
formed by producing all the normals to the given surface a constant length equal
to the radius of the sphere. Let us take the axes GA, GB to be tangents to
the lines of curvature of this surface and let pj, p^ be the radii of curvature of the
normal sections through these tangents respectively. Then
h
.(5).
If G be the position of the centre of gravity at the time t, the quantity 6.^dt is
the angle between the projections of two successive positions of GA on the tangent
plane at G. Let Xi> X2 ^^ *^^ angles the radii of the curvature of the lines of
curvature at G make with the normal. The centre of the sphere may be brought
from G to any neighbouring position G' by moving it first from G to H along one
hne of curvature and then from H to G' along the other. As the sphere moves
from G to II, the angle turned round by GA is the product of the arc GJI into
the resolved curvature of GH in the tangent plane. By Meunier's theorem, the
curvature is , multiplying this by sin xi to resolve it into the tangent plane
Pi cos Xi
we find that the part of 63 due to the motion along GH is - tan Xv Treating the
Pi
MOTION OF A SPHERE. 459
arc HG' in the same way, we have
6'3 = ^tanxi + ^tanx2 (6).
We have also an expression for Wj given by equations (1). Substituting for
w^, Wj from the geometrical equations (3) we get
a-y/ = z(v ) 7.
The solution of the equations may be conducted as follows. Let [x, y, z) be the
co-ordinates of the centre of the sphere. Then u, v may be found from the
equation to the surface in terms of — , -^ , T ^^ resolving parallel to the axes
ctz dv cit
of reference. If we eliminate u, v, d^, $2, 63 by means of (4), (5), and (6), we shall
get three equations containing x, y, z, cog, and their differential coefficients with
respect to t. These together with the equation to the sm-face will be sufficient to
determine the motion at any time. One integral can always be found by the
principle of Vis Viva. Since the sphere is tm-ning about the point of contact as an
instantaneously fixed point we have
where is the force function of the impressed forces. This is the same as
«^+^^+a^S-^'^3^ = \-^^ («)•
and the right-hand side of this equation is twice the force function of the altered
impressed forces.
574. It will sometimes be more convenient to take the axis GA to be a tangent
to the path. Then v = and therefore w.^ = 0. If U be the resultant velocity of
the centre of the sphere we have u = U. Also if JR be the radius of torsion of a
geodesic touching the path at G and p the radius of curvature of the normal
section at G through a tangent to the path, we have 6^— -^ and 6„= —. In these
K p
expressions, as elsewhere, R is estimated positive when the torsion round GA is
from the positive direction of GB to the positive direction of GC. If x he the
angle the radius of curvature of the path makes with the normal, we have as before
^3=— tan X- The equations (4) become
'di~a" + k^ ^ "^ ilFTWT'^^ I , .
U\ a^ ,^ Jc^ U \ ^^ ''
— tanx=- o , 7 Y+ ., , ,„ - au^j
The expression for W3 given by equations (1) now takes the form
dt^3 U^ , ,
""-di^-'M ("")•
It may be shown by geometrical considerations that this form is identical with
that given in (7).
575. To ^ud f^e 2"'^ssMre on t/ie sur/acc we use the last of equations (2). This
may be written in either of the forms
/■/2 ,,3 „2
^^'L+1 = -Z-Ji (9).
P Pi Pi
4G0 MOTION UNDER ANY FORCES.
The sphere will leave the sirrface when R changes sign. This will generally
occur when the velocity of the centre of the sphere is that due to one half of the
projection of the radius of curvature of the normal section on the direction of the
resultant force.
576. Ex. 1. Show that the angular velocity of the sphere about a normal to
the surface, viz. W3, is constant when the direction of motion of the centre of
gravity is a tangent to a line of curvatm-e, and only then.
Ex. 2. A sphere is projected without initial angular velocity about the radius
normal to the surface, so that its centre begins to move along a line of curvature.
Show that it will continue to describe that line of curvature if the force transverse
to the line of curvature and tangential to the surface is equal to seven-fifths of the
centrifugal force of the whole mass collected into the centre, resolved in the tangent
plane to the sui-face.
Ex. 3. If the sphere be homogeneous and be not acted on by any forces, show
that
Z7^( tan* X + 7 ) = constant, awg = „ ^ ^^^ X)
llog^tan^'x+D^-l-iJtnx.
Show also that the path will not be a geodesic unless the path is a plane curve.
577. If the given surface on which the sphere rolls be a plane, we have p-^ and p^
both infinite, hence ^^ 0^ are both zero. If therefore a homogeneous sphere roll
on a perfectly rough plane under the action of any forces whatever of which the
resultant passes through the centre of the sphere, the motion of the centre of
gravity is the same as if the plane were smooth, and all the forces were reduced in
ratio - . And it is also clear that the plane is the only surface which possesses this
property for all initial conditions.
Ex. A homogeneous sphere is placed upon an inclined plane sufficiently rough
to prevent shding and a velocity in any direction is communicated to it. Show
that the path of its centre will be a parabola, and if V be the initial horizontal
velocity of the centre of gravity, a the inclination of the plane to the horizon, the
14 F2
latus rectum will be -^ — : — .
5 g sm a
578. If the given surface on ichich the sphere rolls he another sphere of radius
t- cf, we have p^ = p.-^ = b. Hence W3 is constant ; let this constant value be called n,
and let U be the velocity of the centre of gravity. Since every normal section is
a principal section, let us take GA a tangent to the path. Hence the motion of
the centre of gravity is the same as if the whole mass collected at that point were
jf.2 anU
acted on by an accelerating force ^ —r- in a direction perpendicular to the
a*
path, and all the impressed forces were reduced in the ratio -^^^2 • According to
the usual convention as to the relative positions of the axes GA, GB, GO it is
clear that if the positive direction of GA be in the direction of motion, the angular
velocity n should be estimated positive when the part of the sphere in front is
moving to the right of GA and the additional force when positive will also act
MOTION OF A SPHERE. 461
toward tlie riglat-liand side of the tangent. Since this additional force acts per-
pendicular to the path, it will not appear in the equation of Vis Viva. Hence the
velocity of the centre of gravity in any position is the same as if it had arrived
there simply under the action of the reduced forces. Let be the centre of the
fixed sphere, 6 the angle OG makes with the vertical OZ, and \p the angle the plane
ZOG makes with any fixed plane passing through OZ. Then by Vis Viva we have
#Y^^_%
rnCOSf^,
dt) b a^ + k'^
where F is some constant to be determined from the initial conditions. This also
follows from equation (8).
Also taking moments about OZ, we have
h d f . ^ ^df\ P dd
— — -— (sin^^^)=-^ — —,an-r-,
emddtX dt I a^ + T,
we have in this case two linear equations to find u and awj. If X be zero, and
2
F = -, we find
aw3=^ sm(^^~(P + £j, u=Aa^ ^ cos f-s/ ^0 + -SJ,
where A and B are two arbitrary constants to be determined by the initial values of
u and W3.
If X be not the same for all positions of the sphere on the same generator, let ^
be the space traversed by the sphere measm-ed along a generator. Then
_dl__dlv_
dt dtp P2
Substituting this value of u, we have two equations to find ^ and awj in terms
of such that d is the
angle between two consecutive positions of the distance r, dtp being taken as positive
when the centre moves in the positive du-ection of GB. If the cone were developed
on a plane it is clear that r and ^ would be the ordinary polar co-ordinates of a
point G. We have
dd> dr dd)
0o = —-, u=-~-, v—r~^.
^ dt dt dt
The ec[uations (4) and (7) become therefore
dh- fd(t>y a^ „ k^ r
dt'- '''\dt) ~a^ + k^-^ a^ + k^p^^'^^ dt
Id. „_, „ ^
\ dt) a^
r dt\ dt) a^+W^
d (aw.j) _ r dcf> dr
dt p^ dt dt
If the impressed forces have no component perpendicular to the normal plane
through a generator, r=0, and we have r^ ~T- = h, where h is some constant dei^end-
dt
ing on the initial values of r and v.
If also the component X of the forces along a generator be a function of r
only, another integral can be found by the principle of Vis Viva, viz.
464 MOTION UNDER ANT FORCES.
m^'i
where h' is another constant depending on the initial values of it, v and r.
If, further, the cone be a right cone, p^ = r tan a where a is the semi-angle, and we
have
h cot a , ,,
awg = \-h' , ■
where h" is a third constant depending on the initial values of w, and r. The equa-
tions of the motion of the centre of the sphere resemble those of a particle in central
forces. Hence r and will be found as functions of the time if we regard them as
the co-ordinates of a free particle moving in a plane under the action of a central
force represented by
a^^!^-""3Si.
where w^ has the value just found.
Ex. A sphere roUs on a perfectly rough cone such that the equation to the cone
T
on which the centre G always lies is — = F (). If the centre is acted on by a force
P2
tending to the vertex, find the law of force that any given path may be described.
If the equation to the path be - =/ {(p), prove that the force X is
X=.fcV^^ + -'''^'
where Wg is given by
d(p a d, \p in the manner explained in
Art. 235. The Yis Yiva of the sphere may then be found as iu Art, 349, Ex. 1. If
MOTION OF A SPHERE. 467
■we put sin B cos !/■ = ^, sin ^ sin i/' = 17, + i/' = x, and reject all small quantities above
the second order, we find that the Lagrangian function is
i = I (^'^ + 2/'-) + 1 ^M x'^ - x' (%V - s^''?) + r + 'j'n + ^ J7 (^' + ^J .
It is easy to see by reference to the figure in Art. 235 that ^ and r} are the cosines
of the angles the diameter GC makes with the axes Ox, Oy.
If Wj., Uy, Wj are the angular velocities of the sphere about parallels to the axes
fixed in space, the geometrical equations are
\ ' P-2/ I,
2/' + a ^wa;-w^- j=0j
••\
These are found by making the resolved velocities of the point of contact in the
directions of the axes of x and y equal to zero ; see Art. 219. The angular velocities
«^, wy, Wg may be expressed in terms of 9, (p, \p by formulas analogous to those in
Art. 235. See also the note. Thus
0}^= - 0' sin ■^+(j>' sin 6 cos 1,^1
w^= 0' cos i/' + ^'sin ^ sini/'i
0)^= (p'cosd + ^j/'
Substituting and expressing the result in terms of the new co-ordinates ^ ij, Xi t^ie
geometrical eqiiations become
The equations of motion are given by
cl dL dL_^ dL^ dL.y
cTtd^~ di~ W^^ f¥' '
where q stands for any one of the five co-ordinates x, y, f, i}, x- The steady motion
is given by x, y, |, 1; all zero and ■jd^n. Taking q = x and q = y and giving the
several co-ordinates their values in the steady motion, we find that X and /n are both
zero in the steady motion.
To find the oscillations, we write for q in turn x, y, %, f and 77, and retain the
first powers of the small quantities. Remembering that X and jj. ai'e small quanti-
ties (Art. 461), we find
X -rj^ +
Pi
^==ol
„ y , A, ^^Mr+xV)-x=o. .
A:V'=OJ
These and the two geometrical equations L^ and Z/.^ are all linear, and may be
solved in the manner explained in Art. 432. If we put x' = w and eliminate first X
and ij, and then ^ and -q we get two equations to find x and y, which are the same as
those marked (iv) in the first solution.
80—2
468 MOTION UNDER ANY FORCES,
Ex. A perfectly rougli sphere is placed on a perfectly rotigli fixed sphere
near the highest point. The upper sphere has an angular Telocity n about the
diameter through the point of contact; prove that its equilibrium will be stable
iin^> „ , where h is the radius of the fixed sphere, and a the radius of the
moving sphere.
583. A perfectly rough surface of revolution is placed with its axis vertical.
Determine the circumstances of motion that a heavy sphere may roll on it so that its
centre describes a horizontal circle. And this state of steady motion being disturbed,
find the small oscillations.
In this ease we must recur to the equations of Art. 581, and let us adopt the
notation of that article, except that to shorten the expressions we shall put for Jifi
2
its value z. a^.
o
d^
dt
a, n and n be the constant values of 6, ~ and W3. Then we have m = 0, v — h/x,
T/here b is the constant value of r. The equation (i) becomes
— cos a/j.^ — - g sm a - ^ an sm a/j,.
The other dynamical equations are satisfied without giving any relation between
the constants. If the motion be steady, we have therefore
5 g 7b ^
n—- — + jr - /icota;
thus for the same value of n we have two values of /x, which correspond to different
initial values of v.
We have the geometrical relation au^ = — v, so that w^ and n have opposite
signs. Hence the axis of rotation which necessarily passes throiigh the point of
contact of the sphere and the rough surface makes an angle with the vertical less
than that made by the normal at the point of contact.
By inspecting the expression for n, it will be seen that it is a minimum when
5 g 7 bfi .
~ " =^ - ~ cot a,
2 a/* 2 a
and therefore n^ — 35 -v, cot a, u^ = =~tana.
a^ lb
To find the small oscillation.
Put 6 = a+6', -J— M+ 7 > where a and fjt, are supposed to contain all the con-
stant parts of 6 and -j- , so that 6' and -|- only contain trigonometrical terms. Let
d/t ctt
c - a be the radius of curvature of the surface of revolution at the point of contact
of the sphere in steady motion, so that p differs from c only by small qviantities,
and may be put equal to c in the small terms. Also we have r — i + c cos a . $'.
MOTION OF A SPHERE. 469
Now by equations (iv) and (v) of Art. 581 we have
dbi3
dt
dd d\f/ p sin 9 - r dO' c sin a -
dt dt a dt ^ a
c sin o - 6 „,
•*• W3=i" 6' + n,
-h
c d^d' b + c cos a0'
a d^
wliere n is the whole of the constant part of Wg.
Again, from equation (ii), we have
ld( df\ pd9 eft/' k^ dd_
"adty dtJ~adt'^°^^di'^^^TF^''^~dt~'^'
/x, dff hd^' ccosaudd' 2 dff
.: --ccosa-T- ~ ^3r + -« ^-=0:
a dt a dv a dt j dt
integrating we have
/2 2/j.c cos a\ h d\p'
\7^ a J ~alU'
the constant being put zero because 6' and ^' only contain trigonometrical terms.
Thirdly, from equation (i), we have
Id f d9\ r fdxl^y .2 . ^d^p 5g . „
adtydtj a\dtj 7 •* dt 7 a '
- (cos a - sin a9') f/j.^ + 2fj, . -^ J
+ -(sina + cosa^) (m+^J K + m ^^^^"" gM = ^-(sma + cos a(?').
This expression must be expanded and expressed in the form
In this case, since 6' contains only trigonometrical expressions, we must have B=0.
Putting ^'=0 in the above expression, we find the same value for n as in steady
motion. After expanding the preceding equation we find
^=/A^ ( -cos^a + ^sin^a ) + ^2 __ — ( 2cos2a + ssin*a )
• V 7 J CBma\ 7 /
25 o2 sin a 10 cf . 10 o
+ -ui—Tj — '- V rsmacoga + -— ^cosa.
49 iJ?bc 7 6 7 c
In order that the motion may be steady, it is sufficient and necessary that this
value of A should be positive. And the time of oscillation is then -^ .
It is to be observed that this investigation does not apply if a and therefore b be
small, for some terms which have been rejected have 6 in their denominators, and
may become important.
584. The general equations of the motion of a sphere on an imperfectly rough
surface may be obtained on principles similar to those adopted in Art. 306. The
difference in the theory will be made clear by the following example, in which a
method of proceeding is explained which is generally applicable, whenever the
integrations can be effected.
470 MOTION UNDER ANY FORCES.
585. A homogeneous sphere moves on an imperfectly rough inclined plane with
any initial conditions, find the direction of the motion and the velocity of its centre
at any time.
Let G be the centre of gravity of the sphere. Let the axes of reference GA, GB,
GC have their directions fixed in space, the first being directed down the inclined
plane and the last normal to the plane. Let u, v, w be the velocities of G resolved
parallel to these axes, and w^, W2) '"'3 the angular velocities of the body about these
axes. Let F, F' be the resolved parts of the frictions of the plane on the sphere
parallel to the axes GA, GB, but taken negatively in those directions. Let k be the
radius of gyration of the sphere about a diameter, a its radius, and let the mass bo
unity.
Let a be the inclination of the plane to the horizon. The equations of motion
will then be
-'^-« J " %-— \ "■
EUminating F and F' from these equations and integrating we have
ft + -2 aoij = t^o + 6^ sin af j
:- , \ - (')■
where Z/q and Vq are two constants determined by the initial values of u, v, w^, Wj.
The meaning of these equations may be found as follows. Let P be the point
of contact of the sphere and plane, let Q be a point within the sphere on the normal
at P so that PQ—- , so that Q is the centre of oscillation of the sphere when
suspended from P. It is clear that the left-hand sides of the equations (3) express
the components of the velocity of Q parallel to the axes. The equations assert that
the frictional impulses at P cannot affect the motion of Q, and this readily foUows
from Art. 119, because Q is in the axis of spontaneous rotation for a blow at P.
586. The friction at the point of contact P always acts oppositp to the direction
of sliding and tends to reduce this point to rest. When sliding ceases the friction
(see Art. 148) also ceases to be hmiting friction and becomes only of sufficient mag-
nitude to keep the point of contact at rest. If sliding ever does cease, we then have
u-aw^ = 0, v + ao)j^ = (4).
The equations (3) and (4) suffice to determine these final values of 11, v, w^ and
Wjj. Thus the du-ection of the motion and the velocity of the centre of gravity after
sliding has ceased have been found in terms of the time. It appears that both these
elements are independent of the friction.
If the equations (4) hold initially the sphere will begin to move without sliding
if the friction foitnd from the equations (1), (2) and (4) is less than the limiting
friction. As in Art. 147, this requires that the coefficient of friction fi> -^ — ^„ tan a.
Supposing tliis inequality to hold, the friction called into play will be always less
than the limiting friction and therefore equations (3) and (4) give the whole motion.
MOTION OF A SPHERE. 471
587. if tlie equations (4) do not hold initially or if the inequality just men-
tioned is not satisfied, let S be the velocity of sliding and let be the angle the
direction of sliding makes with GA. To fix the signs we shall take S to be positive
while may have any value from - ■jt to t. Then
Scos 9 = u-aufi, S sin d = v + aw^^ (5).
The friction is equal to ng cos a and acts in the direction opposite to sliding,
hence
F = /t(jf cos a cos ^, F'=/Mgcosa sin ^.
The equations (1), (2) and (5) therefore give
d (S cos 6) f, a^\ a , • 1
-^-^- i-=-( l + -^j Mucosa cos 5+5? sm a
Expanding we find
— = - I 1 + T-^ 1 ytt^ cos a + (7 sm a cos 5
,.. ^ . ^ ("•
. iS — = - ^ sm a sm ^
If Q be not constant, we may eliminate t and integi-ate with regard to Q, this
gives
5rsin^ = 2^ f*™!)" (®)'
where n = ( 1 + rs ] ^ cot a, and A is the constant of integration. If S^ and ^o ^'^ t^iQ
initial values of S and determined by equations (5), we have
24=SoSm6lo(cot^)" (9).
Substituting the value of /S given by (8) in the second of equations (7) and inte-
grating we find
ftan^y ' ft
L^-HL-ii^^V ^/ +V_^ 9_^, (10),
n-\ ?i + l n-1 n + l A ^ "
;t»|)"« (.an|)-' (.an|)-«
the constant of integration being determined from the condition that 9 = 0o when
t = 0. The equations (8), (9) and (10) give S and ^ in terms of t. The equations
(3) and (5) then give u, v, w^ and w^ in terms of t.
df)
The second of equations (7) shows that — has an opposite sign to 9, hence 9 be-
ginning at any initial value except ±7r continually approaches zero. It follows
that, unless a is zero, 9 will be constant only when 9^ = ox iTr.
If n > 1, i.e. fi > — — J- tan a, we see from (8) that sliding will cease when 9
vanishes. This, by (10) will occur when
flsin a ^w -1 71 + 1 ' '
t
9
The subsequent motion has already been found.
472 MOTION UNDER ANY FORCES.
If 71 <1 we see by (8) that S iucreases as 6 decreases, so tliat sliding ■will never
cease. It also follows from (10) tliat 6 vanislies only at the end of an infinite time.
If Sf^—0, sliding will never begin if ?i > 1, but will immediately begin afld never
cease if »i< 1.
588. The theory of the motion of a sphere on an imperfectly rough Jwrizontal
plane is so much simpler than when the plane is inclined or when the sphere rolls
on any other surface, that it seems unnecessary to consider this case in detail. At
the same time the game of billiards supplies many problems which it would be
unsatisfactory to pass over in silence. The following ex-amples have been arrasged
so as both to indicate the mode of proof to- be adopted and to supply some results
which may be submitted to experiment.
The result given in Ex. 1, was first obtained by J. A. Eulor the son of the cele-
brated Euler, and published in the Mem. de. VAcad. de Berlin, 1758. Most, possibly
aU, of the other results may be found in the Jeu de Billard par G. Coriolis, pub-
lished at Paris in 1835.
Ex. 1. A bilUard-ball is set in motion on an imperfectly rough horizontal
plane, show that the direction and magnitude of the friction are constant through-
out the motion. The path of the centre of gravity is therefore an arc of a parabola
while sliding continues, and finally a straight line. The parabola is described with
the given initial motion of the centre of gravity under an acceleration equal to /xg
tending in a direction opposite to the initial direction of sliding.
Ex 2. If So be the initial velocity of sliding prove that the parabolic path lasts
2 S 1
for a time - — . From some experiments of Coriolis it appears that /u=^ nearly.
7 ixg 5
If the initial velocity of sliding be one foot per second, the parabolic path lasts
therefore less than a twentieth part of a second.
Ex. 3. If P be the point of contact in any position and Q the centre of oscilla-
tion with regard to P, prove that the velocity of Q is always the same in direction
and magnitude. Thence show that the final rectilinear path of the centre of gravity
is parallel to the initial direction of the motion of Q and the final velocity of the
5
centre of gravity is ^ of the initial velocity of Q. If PP' be the initial direction of
motion and V the initial velocity of the centre of gravity and t the time given by
Ex. 2, prove that the final rectilinear path of the centre of gravity intersects PP' in
a point P' so that PP'=~ F^
Ex. 4. A billiard-ball, at rest on ah imperfectly rough horizontal table, is struck
by a cue in a horizontal dii'ection at any point whose altitude above the table is h,
and the cue is withdrawn as soon as it has delivered its blow. Supposing the cue
to be sufficiently rough to prevent sliding, show that the centre of the ball will
move in the direction of the blow and that its velocity will become uniform and
eaual to — £ after a time — ^ • where B is the ratio of the blow to the mass
^ 7 a la ixg
of the sphere and a is the radius.
In order that there should be no sliding the distance of the cue from the centre
of the ball must be less than a sin e where tan e is the coefficient of friction between
the cue and ball.
MOTION OF A SOLID BODY ON A PLANE.
473
Ex. 5. A billiard-ball, initially at rest and touching the table at a point P, is
struck by a cue making an angle j3 with the horizon. Show that the final recti-
linear motion of the centre of gravity is parallel to the straight hne PS joining P
to the point S where the direction of the blow meets the table, and the final velocity
5 PS
of the centre of gravity is - — ;;- .E sin jS in the direction of the projection of the blow
on the horizon.
7 a
It will be noticed that these results are independent of the friction.
Ex. 6.
Measure ST—-=acoip along the projection of the blow oh the horizon-
tal table, then TS measures the horizontal component of the blow referred to a
unit of mass, on the same scale that PS measui'es the final velocity of the centre of
gravity. Prove that during the impact and the whole of the subsequent motion the
friction acts along PT and that the whole friction called into play will be measured
5 PT
by PT on the scale just mentioned. Thence show that unless A* < ^ ^^^ parabolic
arc of the path wiU be suppressed. Show also that PT is the direction in which
the lowest point of the ball would begin to move if the horizontal plane were smooth
and the ball were acted on by the same blow as before.
Motion of a Solid Body on a jplane,
589, A solid of revolution rolls on a perfectly rough horizontal plane under the
action of gravity. To find the steady motion and the small oscillations.
Let be the centre of gravity of the body, GO the axis of figure, P the point of
contact. Let GA be that principal axis which Ues in the plane PGO and GB the
axis at right angles to GA, GC. Let GM be a perpendicular from G on the hori-
474 MOTION UNDER ANY FORCES.
zontal plane, and PN a perpendicular from P on GO. Let 6 be the angle GCmsikes
with the vertical, and i/* the angle MP makes with any fixed line in the horizontal
plane. Let R be the normal reaction at P; F, F' the resolved parts of the frictions
respectively in and perpendicular to the plane PGO. Let the mass of the body be
unity.
Let us take moments about the moving axes GA, GB, GO according to Art. 253.
As in the second case of Art. 254, we put d^^w^, dj-u^ and 6*3 = ^008 d. Remem-
bering that A/=^Wi, h^'=A(>}^, h^ = Cu}^ we have
A^-Aw^^^co^e + Co,^o3,^-F'.GN (1).
A ^2 _(7w3«, + lwi^cose= -F. GM-R.MP (2).
C^^F'.PN (3).
at
The geometrical equations are
§=". w- -»^=-''. (=)•
Let u and v be the velocities of the centre of gravity respectively along and per-
pendicular to MP, both being paraUel to the horizontal plane. The accelerations
of the centre of gravity along these moving axes will be
-r- -v-r- —F .■•(6),
dt dt ^ '
dv dip -r-,, ,rn
—-+u^^F' ....(7).
dt dt ^ -
And if z be the altitude of G above the horizontal plane, we have
£=-^+^ '«)■
Also since the point P is at rest, we have
u-GMo}2 = (9),
v + PNb,s-GNo}-, = (10),
z^-G'iV^cos^ + PiVsinfi (11).
These are the general equations of motion of a solid of revolution moving on a
perfectly rough horizontal plane. If the plane is not perfectly rough the first eight
equations wiU still hold, but the remaining three must be modified in the manner
explained in the next proposition.
When the motion is steady, wo have the surface of revolution rolling on the
plane so that its axis makes a constant angle with the vertical. In this state of
motion, let e = a, ^=/*, <^3-=n, GM^p, MP^q, GN^^, NP=v, and let /) be the
radius of curvature of the rolling body at P. Then the relations between these
quantities may be foimd by substitution in the above equations.
Suppose it were required to find the conditions that the surface may roll with a
given angular velocity n with its axis of figure making a given angle with the verti-
cal. Here n and a are given, and p, q, ^, V, P may be found from the equations to
MOTION OF A SOLID BODY ON A PLANE. 475
the surface. We have to find p., Wj, Wg, m, v and the radius of the circle described
by 6" in space. Then eUminating F, F', E, we get
Ij.^ sin a (A cos a -p^)-nfJL{C sin a + pri)-gq = 0,
u-^^= — /x. sin a, Wj = 0,
•u = 0, v= -717]- ^/JL sin a.
Let r be the radius of the cii-cle described by G as the surface rolls on the plane.
Since G describes its circle with angular velocity fi, we have r/x=u, and hence
r= 1 sm a.
Eliminating n we have
fi" {At) sin a cos a+C^ sin^ a + r (C sin a +2^v)} = ffiV-
For every value of n and a there are two values of y«, which however correspond
to different initial conditions. In order that a steady motion may be possible, it
is necessary that the roots of this quadratic should be real. This gives
(Csin a + prj)^ n^ + igq sin a {A cos a-p^) = a positive quantity.
If the angular velocity n be very great, one of these values of fi is very great
and the other small. If the angular velocity be communicated to the body by
unwinding a string, as io a top, the initial value of w^ will be small. In this case
the body will assume the smaller value of p^, and we have approximately
91
ix=
n[Csuia-^py})'
To find the small oscillation, vte put 6 = a^-d , ^-1^^ -^^ (a^-n + u^.
we have by geometry,
t^GM=p + -^Jag. Let the hoop be a disc, then
/eg
V ¥■
C= -^ , and we have F>
Ex. 2. A cii-cular disc is placed with its rim resting on a perfectly rough
horizontal table and is spun with an angular velocity fl about the diameter through
the point of contact. Prove that in steady moti£»n the centre is at rest at an
altitude above the horizontal plane, where k is the radius of gyration about a
9
diameter; and, if a be the inchnation of the plane to the horizon, the point of
27r
contact has made a complete circuit in the time -- sin a. If the disc be slightly
disturbed from this state of steady motion, show that the time of a small oscillation
ljfc2 (jL--^ + gg) sin g )i
iga 3k^ cos^ a + a^ sm^ a )
Ej:. 3. An infinitely thjn circular disc moves on a perfectly rough horizontal
plane in such a manner as to preserve a constant inclination a to the horizon.
Find the condition that the motion may be steady and the time of a small oscillation.
Let the radius of the disc be a, and the radius of gyration about a diameter k.
Let W3 be the angular velocity about the axis, fi the angular velocity of the centre
of gravity about the centre of the circle described by it, r the radius of this circle,
then in steady motion
{2F- + Or) W3 = Tc^iJ. CQSOL-— cot o, {21:- + aP-) r = - k"a cos a + ^ cot a.
If The the time of a small oscillation
/'^y(;t2 + a') = M-{/^-(l + 2cos2a)+a2sin2a}-«,acosa(6P + a2) + 2ft2{2Z;2 + a^)-i7asina.
Ex. 4. A heavy body is attached to the plane face of a hemisphere so as to form
a solid of revolution, the radius of the hemisphere being a and the distance of the
centre of gravity of the whole body from the centre of the hemisphere being h. The
body is placed with its spherical siirface resting on a horizontal plane, and is set
in motion in naij manner. Show that one integral of the equations of motion is
AsvD?9~ + Co}^ I cos^ + - )=constant whether the pla:iie bo si?io.oth, imperfectly
dt ^ \ aj
rough, or perfectly rough.
It is clear that the first two terms on the left-hand side of this equation is the
angular momentum about the vertical through G. Let this be called /. Since we
may take moments about any axis through G as if G' were fixed in space, we have
*'—=¥'. PM. But PM= -PN .-, hence eliminating F' by equation (3) and in-
tegrating, we get the required result.
Ex. 5. A surface of revolution roUs on another porfectly rough surface of
revolution with its axis vertical. The centre of gravity of the rolling surface lies
in its axis. Find the cases of steady motion in which it is possible for the axes of
both the surfaces to he in a vertical plane throughout the motion.
Let 9 be the inclination of the axes of the two surfaces, P the point of
contact, GM a perpendicular on the tangent plane at P, PN a perpen-
dicular on the axis GCoI the rolling body; F the friction, E the reaction at P;
478 MOTION UNDER ANY FORCES.
n the angular velocity of the rolling body about its axis GC, fi the angular rate at
which G describes its circular path in space, r the radius of this circle. Then in
steady motion
Mil sin e{Cn-AixcoBe) = -F .GM-E. MP,
Iv = - Mr/j? sin a + Mg cos a,
F= - Mr/jL^ cos a - Mg sin a,
n. PN+/xsm0 .GN— -r/x,
where M is the mass of the body,
591. A surface of any form rolls on a fixed horizontal plane under the action of
gravity. To form the equations of motion*.
* The motion of a heavy body of any form on a horizontal plane seems to have
been studied first by Poisson. The body is supposed to be either bounded by a
continuous surface which touches the plane in a single point or to be terminated
by an apex as in a top, while the plane is regarded as perfectly smooth. Poisson
uses Euler's equations to find the rotations about the principal axes, and refers
these axes to others fixed in space by means of the formulae of Art. 235. He finds
one integral by the principal of vis viva and another by that of angular momentum
about the vertical straight line tkrough the centre of gravity. These equations are
then applied to find how the motion of a vertical top is distm-bed by a slow move-
ment of the smooth plane on which it rests. 'See the Traite de Meeanique.
In three papers in the fifth and eighth volumes of Crelle's Journal (1830 and
1832) M. Cournot repeated Poisson's equations, and expressed the corresponding
geometrical conditions when the body rests on more than one point or rolls on an
edge such as the base of a cylinder. He also considers the two cases in which the
plane is (1) perfectly rough, and (2) imperfectly rough. He proceeds on the same
general plan as Poisson, having two sets of rectangular axes, one fixed in the body
and the other in space connected together by the formulse usually given for
transformation of co-ordinates. As may be supposed, the equations obtained are
extremely complicated. M. Cournot also forms the corresponding equations for
impulsive forces. Those however which include the effects of friction do not
agree with the equations given in this treatise.
In the thirteenth and .seventeenth volumes of Liouville's Journal (1848 and
1852) there will be found two papers by M. Puiseus. In the first he repeats
Poisson's equations and applies them to the case of a solid of revolution on a
smooth plane. He shows that whatever angle the axis initially makes with the
vertical, this angle will remain very nearly constant if a sufficiently great angular
velocity be communicated to the body about the axis. An inferior limit to this
angular velocity is found only in the case in which the axis is vertical. In the
second memoir he applies Poisson's equations to determine the conditions of
stability of a solid of any form placed on a smooth plane with a principal axis at
its centre of gravity vertical and rotating about that axis. He also determines
the small oscillations of a body resting on a smooth plane about a position of
equilibrium.
In the fourth volume of the Quarterly Journal of Mathematics, 1861, Mr G. M.
Slesser forms the equations of motion of a body on a perfectly rough horizontal
plane and applies them to the problem considered at the end of Art. 597. He uses
moving axes, and his analysis is almost exactly the same as that which the author
had adopted.
MOTION OF A SOLID BODY ON A PLANE.
479
Let GA, GB,GC, the principal axes at the centre of gravity, be the axes of
reference and let the mass be unity. Let ^ (f, 77, f) = be the equation to the
bounding surface, (f, rj, f) the co-ordinates of the point P of contact. Let {p, q, r)
be the direction-cosines of the outward direction of the normal to the surface at
the point ^, rj, f, then
d(j> d(p d(j> '
d^ d-r} d^
Firstly, let the plane be perfectly rough. Let X, Y, Z be the resolved parts
along the axes of the normal reaction and the two frictions at the point $, 77, f, and
let the mass of the body be unity. By Evder's equations we have
^^1 -(^-C) 0^20,3= TjZ-i-r"
Also the equations of motion of the centre of gravity are by Art. 245,
du „
-J- - t'Wj + ww.2^=gp + X
dv
— -ww-^ + uu}.^=gil-\-Y >
dxo „ I
— - uu2 + vw^=gr + Z J
Also since the line {p, q, r) remains always vertical,
.(1).
.(2).
dp
dt
— 5W3 -rw„
dq I
^ = m, -2,0,3 y
■{3),
dr
ji=p^t-q^i J
and since the point (f, rj, f) is at rest we have
TF= w - ^^2 -I- i?Wi =
where U, V, ir are the resolved parts of the velocity of the point of contact P
the positive directions of the axes.
(4),
m
480 MOTION UNDER ANY FORCES.
592. Secondly, let the plane he perfectly smooth. The equations (1), (2), (3),
apply equally to this case, but equations (4) are not true. Since the resultant of
X, Y, Z is a reaction R normal to the fixed plane, we have
X^-pE, Y=-qR, Z=-rM (5),
The negative sign is prefixed to R because (p, q, r) are the direction-cosines of
the outward direction of the normal, and it is Qlear that when these are taken posi-
tively, the components of E are all negative. If at any moment R vanishes and
changes sign the body will leave the plane.
Since the velocity of 6 parallel to the fixed plane is constant in direction and
magnitude, it will ustially be more convenient to replace the equatiops (2) by tlje
following single equation. I^et GM be the perpendicular on the fi?,ed plane and let
MG-z, th.en
S-"'^ • <^''
It is necessary that the velocity of the point of contact resolved uormal to the
plane should be zero, this condition may be written in either of the equivalent
forms
Up + rq+Wr = >
593. Thirdly, let the body slide on an imperfectly rough plane. The equa-
tions (1), (2), (3) and (7) hold as before. If fx be the coefficient of friction the
resultant of the forces X, Y, Z naust make an angle tan-^ fi with the normal at the
point of contact, hence
(Xp+Yq + Zry^^ 1 ,g>
X^+Y-^+Z^ 1 + m'
Also since the resultant of {X, Y, Z), the normal at P and the direction of slid-
ing must lie in one plane, we have the determinantal equation
X(qW~rV}+Y{rU-pW) + Z{2)V-qU)=0 (9).
gjnce the friction must act opposite to the direction of sliding, we must have
XU-i- YV+ZW negative. When this vanishes and changes sign, the point of con-
tact ceases to slide.
If the body start from rest we must use the method explained in Art. 14G to
determine whether the point of contact will begin to slide or not. Assume X, Y, Z
to be the forces necessary to prevent sliding. Then since u, v, iv, w^, u^, w? ^^^ ^^^
initially zero, we have by differentiating {4) and eliminating the differential coeffi-
cients of M, V, IV, Wj, w.,, Wg three linear equations to find X, Y, Z, in terms of the
known initial values of {p, q, r) and (|, 77, f ). The point of contact will slide or not
according as these values make the left-hand side of equation (8) less or greater
than the right-hand side.
The equations to find X, Y, Z may be obtained by treating the forces as if they
were indefinitely small impulses. In the time dt, we may regard the body as acted
on by an impulse gdt at G and a blow whose components are Xdt, Ydt, Zdt at P.
By Art. 296 we may consider these in succession. The effect of the first is to com-
municate to P a velocity gdt in a direction normal to the fixed plane and outwards.
If P does not sli(Je, the effect of the blow at P must be to destroy this velocity.
Hence X, Y, Z may be found from the equations of Art. 304 if we write %ii-=pg,
MOTION OF A SOLID BODY OX A PLANE. 4S1
Vi — qff, w-^^ — rg and 11.2, v^, w.^ all equal to zero ou tlie left-hand sides and (to suit the
notation of this article) change p, q, r on the right-hand sides into fi '"1^ f-
Geometrically the point of contact will not slide if the diametral line of the fixed
plane with regard to the ellipsoid called E in Art. 304 makes a less angle with the
normal than tan~i fx.
In any of these cases when p, q, r have been found, the inclinations of the prin-
cipal axes to the vertical are known. Their motion round the vertical may then be
deduced by the rule given in Art. 249. When u, v, w and the motions of the axes
have been found, the velocity of the centre of gravity resolved along any straight
line fixed in space may be found by resolution,
594. Some integrals of these equations are supplied by the principles of angular
momentum and Vis Viva. If the plane is perfectly smooth we have
A oi}^p + Bojoq + CcoaV — a,
where a and /3 are two constants.
If the plane is perfectly rough we have
A WjS 4. Bu)„" + Cwg" + u^ -tv^ + 10^ = /3 - 2gz.
595. Ex. 1. A body rests with a plane face on an imperfectly rough horizon-
tal plane whose coefficient of friction is fj,. The centre of gravity of the body is
vertically over the centre of gravity of the face, and the form of the face is such
that the radius of gyration of the face about any straight line in its plane through
its centre of gravity is 7. The body is now projected along the plane so that the
initial velocity of its centre of gravity is Vq and the initial rotation about a vertical
axis through its centre of gravity is Wq. If wq be very small, prove that the centre
of gravity moves in a straight line and its velocity at the end of any time t ia Vq- /j.gt,
/" lJ.qt \ ^^"^
Show also that the angular velocity at the same time is w^ ( 1 1 , where Tc is
the radius of gyration of the body about a vertical through the centre of gravity.
\Poisson, Traite de Mecanique.]
Ex. 2. A body of any form rests with a plane face in contact with a smooth
fixed plane so that the perpendicular from the centre of gravity G on the plane falls
within the face. If the body is then struck by a blow which passes through or
begins to move from rest under the action of any finite forces whose resultant
passes through G, prove that it will not turn over, but will begin to shde along the
plane, even if the line of action of the force cuts the plane outside the base.
[Cournot,]
596. Whatever the shape of a body may be we may suppose it to be set in
rotation about the normal at the point of contact with an angular velocity n.
If this angular velocity be not zero, the normal must bo a principal axis at the
point of contact, and yet it must pass through the centre of gravity. This cannot
be unless the normal be a principal axis at the centre of gi-avity. If however n = 0,
this condition is not necessary. There are therefore two cases to be considered.
Case 1. A body of any form is placed in equilihrium resting ivith the point C ou
a rough horizontal plane, ivith n principal axis at the centre of gravity vertical, and
R. D. 31
482 MOTION UNDER ANY FORCES.
is then set in rotation with an angular velocity n about GO. A small disturbance
being given to the body, it is required to find the motion.
Case 2. A body of any form is placed in equilibrium on a rough horizontal plane
with the centre of gravity over the point of contact. A small disturbance being given
to the body, to find the motion.
597. Case 1. Supposing the body not to depart far from its initial position,
we have p, q, u, v, vj, Wj^, W2 all small quantities and r=l nearly. Hence by (2),
■when we neglect the squares of small quantities, we see that X, Y are also small,
and Z = -g nearly. It follows by(l) that Wg is constant and .•. =n. Also f and rj
are small and f = h nearly, where h is the altitude of the centre of gravity above the
horizontal plane before the motion was disturbed. The equation to the surface
may, by Taylor's theorem, be written in the form
f-''-2U ^^J'
where {a, h, c) are some constants depending on the curvatures of the principal
sections of the body at the point 0.
The squares of all small quantities being neglected, the preceding equations
become
^t-fi'-c)
71^2= -gv- ^^^
,
B'-g-iO-A)
nw^ = hX + g^
du , ^
--nv=gp + X,
dv ,,
- + nu=gq + Y,
dp
dq
'ei-nr] + hu^ = 0,
V - hu^ + n^ = 0,
P
a b
b c
Eliminating X, Y, u, v, w^, Wg ^om these equations, we get
(A + h^)^+{A+B + 2h-'-C)n^^-{{B-C)n^ + hg + hhi^\q==-{g + hn-')v + hn^
~{B + h^)^ + {A + B + 2/t2 - C) n |f + {(^ - C)n^ + hg + hhi^p = [g + hn^) ^ + hn-^.
It will be found convenient to express |, tj in terms of p, q. The right-hand
sides of each of these equations will then take the form
To solve these equations, we must then assume p, q to be of the form
p = Po cos Xi + P^ sin X< )
q = Qo cos \t + Qi sin Xi ' '
MOTION OF A SOLID BODY ON A PLANE. 483
If the tangents to the lines of curvature of the movhig body at C be parallel to
the principal axes at the centre of gravity, these equations admit of considerable
simplification. In that case the equation to the surface may be written in the form
2 \ a c /
where a and c are the radii of curvature of the lines of curvature. The right-hand
sides of the equations then become respectively
- (^f + h-n?) cq + hna-j- and {g + hi?) ap + hnc j- .
ctt vie
To satisfy the equations, it will be sufficient to put
i? = Fcos(Xf+/), 3 = (? sin (\f +/).
This simplification is possible, because we can see beforehand that -^ = — ^ .
rt
Substituting and eliminating the ratio — , we get the following quadratic to de-
G
termine X^.
[{A + h^) \^ + {B -C+h {h - c)} n^+g{h - c)][{B + h^)r- + {A - C+ h{h-a)]nP+g{h-a)]
= Xhi^ {A+ S + 2/j2 - c- ha} {A +B + 2h"- - C- he).
If \, Ag be the roots of this equation, the motion is represented by the
equations
p = F^ cos {\t +/i) + f 2 cos (Xgi 4-/2)
g = (?i sin [\t +/i) + G.^ sin (Xg* +f^)
C Q
where ~ , -^ are known functions of X^, \ respectively, and F-^, -fj' /i' A ^^'^
constants to be determined by the initial values oi p, q, -f- ^ -^ .
In order that the motion may be stable, it is necessary that the roots of this
quadratic should be real and positive. These conditions may be easily expressed.
Ex. 1. A solid of revolution is placed with its axis vertical on a perfectly rough
horizontal plane and is set in rotation about its axis with an angular velocity n.
If c be the radius of curvature at the vertex, h the altitude of the centre of
gravity, i: the radius of gyration about the axis, k' that about an axis through the
vertex perpendicular to the axis of figure, show that the position of the body will be
I' s/g{h-c)
stable if 71 > 2 ■
k'' + he
Ex. 2. An ellipsoid is placed with one of its vertices in contact with a smooth
horizontal plane. What angular velocity of rotation must it have about the vertical
axis in order that the equihbrium may be stable ?
Eesult. Let a, h, c be the semi-axes, c the vertical axis, then the angular
/5o sJ C'^ — a'^ + tJ c'^ — 6*
velocity must be greater than . / — . — v—^t. . \Puiseux. ]
\/ c d^ + b^ ■■ -^
Ex. 3. A solid of any form is placed in equilibrium with the point C on a
smooth horizontal plane, a principal axis GO at the centre of gravity being vertical,
and an angular velocity n is then communicated to it about GO. A small disturb-
31—2
484 MOTION UNDEK ANY FORCES,
anee being given, show that the harmonic periods may be deduced from the quad-
ratic
{A\^ + E) [B\^ + F) = {A +B- 0) n^\" + g^ (/)' - p)^ sin^ 5 cos^ 5,
where
E = {B-C)n^+g{{h-p) sin^ d + {h - p') cos^ S\,
F={A - C) n^+g {{h~p) cosn + Qi- p') shx^ 8}.
Also h is the altitude of the centre of gravity, p, p' are the principal radii of
curvature at the vertex, and 5 is the angle the principal axis GA makes with the
plane of the section whose radius of curvature is p. [Pitiseua;.]
598. Case 2. Supposing the disturbance to be small, we have w^, w^, Wj,
u, V, w all small quantities. Hence when we neglect the squares of small quantities
the equations (1) and (2) become respectively,
■ ^'^'='^-f^. ^t=f-^-«^' «t=^^-'^ «•
du „ dv ^^ dw ^ ....
_=^p+x, ^=^2+r, ^=6"-+z (u).
Let ^0, ■)7o, fo ^® *^^^ co-ordinates of the point of contact in the position of equili-
brium, and let ^=?o + ?'> ''? = % + '?'> f=fo + f'' Then in the small terms of
equation (4) we may write |o, i?,,,^ fo ^^^ ?> '7) ?• Hence differentiating these and
eliminating X, Y, Z, u, v, lo by help of equations (i) and (ii), we get
ON dio^ . dwi, . . do}., , . . ,....
{^ + Vo' + ^o')j~-^oVoj^'-^oh^f--9{vr-k) (m),
and two siuiilar equations.
Let Po> Q'o) ''o ^e the values of p, q, r in the position of equilibrium. Then
-^ = ^ = ^ = p, where p is the radius vector from G to the point of contact. Now
in the small terms of equations (3) we may write ^oi Qo' ''o ^or p, q, r. Hence equa-
tions (iii) become by substitution
. du-. d^r . d\ , . . ...
^irh''^f'df-^"'di^-^^'"-^^^- (^^^'
and two similar equations. At the time t \e%p=P(j+p', q — q^i + q', and ?-=7"o + r'.
Then since (PQ-\-p'f + {qo + q')'^ + [i'o + i'T = '^y ^^ ^^"^^ i'oP' + g'o2' + V'=0. The
form of the surface being known we can find p', q', r' in terms of |', t)', f , and thus
express Tjr - fg, ^p - ^r, ^q - rip in the form -g {vr-^q) — Lp' + Mq'.
The equations (iv) now become
and two similar equations.
Differentiating equations (3), and substituting for — -^, — ^ ^ Iff ' '"'' ^^^ wTa '
we get equations of the form
-0 (vi),
MOTION OF A ROD. 485
To solve these we put p' = P cos {\t+f), ^' = Qcos (\t+f), substituting and
p
eliminating the ratios -^ , we have the quadratic
i FX^ + R, G\^ + K
\f'X^ + H', 0'\^ + K'
to determine X^.
Thus by virtue of the relation existing between p', q', r', each of these may be
represented by an expression of the form
Pi cos {\t +/,) + P., cos (KJ +f.,).
Substituting these values in equations (v) we see that w^, w^, W3 can each be
represented by an expression
fi^ + E^ cos {\t +/i) + E.2 cos {\t H-/^),
•where E-^, E^ are known functions of P^, Pj ... and X^, \, but 0^, S2.,, Q.^ are small
arbitrary quantities. By substituting in equations (3) and equating the coefficients
of cos (X-it+f-i) and cos (X^i+Zg), we may find the values of E-^^ and E.^ without diffi-
culty. And we also see that we must have
u^l a&o ^lo
so that, of the three fi-,, fig. ^3' only one is really arbitrary. -^We have therefore
but five arbitrary constants, viz. P^ P.^, f^, f.^, and fi^. These are determined by
the initial values of w^, w.^, Wj, p' and q'.
To find the motion of the principal axes round the vertical, let be the angle
the plane containing GC and the vertical makes with the plane of A G. Then by
drawing a figure for the standard case in which p, q, r are all positive, it will be
seen that if /t be the rate at which OG goes round the vertical,
, . Po^h±M?
/x^l-r-^^WjCOS
,f^^ap^=.-Tm y ^'^-
df dt^
0=9- T J
By D'Alembert^s Principle the equation of moments round x will be
^du {y~^-z^y^=^du{yZ-zY)=^du{:yg).
By equations (1) this reduces to
£du j - {h + u) (i^V t*g)| -2acj[bm^aq).
Integrating, we get
„ ^ [ , dhn d^q\ _, „dhn Qa? d^q „ ,, .
which by equations (2) reduces to
, d^m 4 d'^q
Therefore the four equations of motion are
, dH d^p , ,dH 4 dhj
^di^'-^'d^--^^' ^d^ + l^dt^-^P^ (^)'
and two similar equations for m, q. These equations do not contain m or q, and
on the other hand the equations to find m and q do not contain I or p. This shov/s
that the oscillations in the plane xz are not affected by those in the perpendicular
plane yz. See Art. 450.
To solve these equations, jrat l = F&hi (\t + a), p — G sin (Xt + a),
we get bX^F + a\^G = (jF, lX^F-\-~ay-G=gG;
...X^-'-^gX^+'-^^O,
ah ab
and the values of X may be found from this equation.
EXAMPLES. 487
In order to make a comparison of different methods, let us deduce the motion
from Lagrange's equations. In this case we must determine the semi vis viva T
true to the squares- of the small quantities p, q, I, m, we cannot therefore put r=l,
n = l. Since p^ + q^ + r^^l, l^i-m^ + 71^=1, we have
.._i -c t^ n = l-
2 ' 2 •
we must therefore replace the third of equations (1) by
- , J^ + m^ p2 + gs
z = bn + ur=o + u-o —^ w — - — .
If accents denote diSerential coefficients with regard to t, as in Lagrange's
equations we have
Smx'2 = 2m {hH'^ + 2U'p'u+^"^vP) = M{ bH'^ + 2biya + -^P'-)-
The value of 'Zmy'^ may he found in a similar manner. The value of Sms'^ is of
the fourth order and may he neglected. Hence we have
2T=h"- il"^ + m"') + 2ab [Vp' + vi'q') + -g- (p'^ + 2'^).
f P + nv^ p^ + q\
Also we have U= - g ( 6 — ^ + a^- „ ) + constant.
d dT dT dU , „„ , „ ,
The equation -r -rr, - -3-1 = in- becomes bi'+ap'= -gl;
dt dl dl dl
ia
similarly we get ^^""*" T ^"~ ~ ^^'
These are the same equations which we deduced from D'Alembert's Principle,
and the solution may be continued as before.
EXAMPLES*.
1. A uniform rod, moveable about one extremity, moves iu such a manner as
to make always nearly the same angle a with the vertical ; show that the time of a
cos a
3cf ' 1 4-3cos2 a '
2. If a rough plane inclined at an angle a to the horizon be made to revolve
with uniform angular velocity n about a normal Oz and a sphere be placed at rest
upon it, show that the path in space of the centre will be a prolate, a common, or a
curtate cycloid, according as the point at which the sphere is initially placed is with-
out, upon, or within the circle whose equation is x'^ + y^— — '^-^ — x, the axis Oy
being horizontal.
When the sphere is placed at rest on the moving plane, it should be noticed
that a velocity is suddenly given to it by the impulsive frictions.
V2(x ^ ^
a~ • 1 I ^u"Sa.2 _ > " being the length of the rod.
* These Examples are taken from the Examination Papers which have been
set in the University and in the Colleges.
488 MOTION UNDER ANY FORCES.
3. A circular disc capable of motion about a vertical axis through its centre
perpendicular to its plane is set in motion with angular velocity ft. A rough
uniform sphere is gently placed on any point of the disc, not the centre, prove that
the sphere will describe a circle on the disc, and that the disc will revolve with
angular velocity ^^^rr^ — s — -„ ft, where Mk" is the moment of inertia of the disc
about its centre, 7n is the mass of the sphere and r the radius of the circle traced
out.
4. A sphere is pressed between two perfectly rough parallel boards which are
made to revolve with the uniform angular velocities ft and ft' about fixed axes per-
pendicular to their planes. Prove that the centre of the sphere describes a circle
about an axis which is in the same plane as the axes of revolution of the boards and
whose distances from these axes are inversely proportional to the angular velocities
about them.
Show that when the boards revolve about the same axi?, their points of contact
will trace on the sphere small circles, the tangents of whose angular radii will be
- . —^ — -, , a being the radius of the sphere and c that of the circle described by its
tt ft + ft
centre.
5. A perfectly rough circular cylinder is fixed with its axis horizontal. A
sphere being placed on it in a position of unstable equilibrium is so projected
that the centre begins to move with a velocity V parallel to the axis of the cyhnder.
It is then slightly disturbed in a direction perpendicular to the axis. If be
the angle the radius through the point of contact makes with the vertical, prove
that the velocity of the centre parallel to the axis at any time t is Fcos*/ ~d
and that the sphere will leave the cylinder when cos ^=— .
6. A uniform sphere is placed in contact with the exterior surface of a perfectly
rough cone. Its centre is acted on by a force the direction of which always meets
the axis of the cone at right angles and the intensity of which varies inversely as
the cube of the distance from that axis. Prove that if the sphere be properly
started the path described by its centre will meet every generating line of the cone
on which it lies in the same angle.
See the Solutions of Camhridge Prohlems for 1860, page 92.
7. Every particle of a sphere of radius a, which is placed on a perfectly rough
sphere of radius c, is attracted to a centre of force on the surface of the fixed siAere
with a force varying inversely as the square of the distance ; if it be placed at the
extremity of the diameter through the centre of force and be set rotating about that
diameter and then slightly displaced, determine its motion; and show that when it
leaves the fixed sphere the distance of its centre from the centre of force is a root of
the equation 20ic-* - Id (2c + a) x^ + 7a {2c + af ^0.
8. A perfectly rough plane revolves uniformly about a vertical axis in its own
plane with an angular velocity n, a sphere being placed in contact with the plane
rolls on it under the action of gravity, find the motion.
Take the axis of revolution as axis of z, and let the axis of x be fixed in the
plane. Let a be the radius, vi the mass of the sphere ; F, F' the frictions resolved
EXAMPLES. 489
parallel to the axes of x and z and R the normal reaction. The eqnations of
dP'x F dx R d-z F'
motion are therefore by Art. 11% -r- - n^x^ - , -an^ + 2n — = - and — == - gf+ — .
(XZ 7Tt CI Till CI C" ^Th
The equations of rotation by Art. 255 are /^ -no}y= - — , — " + nu^ = 0,
J^ = — , Since the point of contact has the same motion as the plane the
dt h?
geometrical equations by Art. 244 are -- - an + aw^^O, — -aw^=0. Solving these
equations we find that the sphere will not fall down. If the sphere start from
relative rest at a point in the axis of x, we have z— ^ tan^ i {1-cos (nf cosi)}
where sin i = \j -. The sphere will therefore never descend more than — ^ below
its original position.
9. A perfectly rough vertical plane revolves with a uniform angular velocity n
about an axis perpendicular to itself, and also vnth a uniform angular velocity fi
about a vertical axis in its own plane which meets the former axis. A heavy uni-
form sphere of radius c is placed in contact with the plane ; prove that the position
of its centre at any time t, will be determined by the equations
7g-5fff-2,|=0,
'S+^^'^^-GI-^)-"-
z denoting the distance of the centre from the horizontal plane through the hori-
zontal axis of revolution, and f that from the plane through the two axes.
Prove also that 7u = 7c^ + 2fj.h, 7v + 2/xa = 0, if a and b be the initial values of ^
and 2, u and v those of — and — .
at dt
10. A hoop ^Gi?F revolves about 45 its diameter as a fixed vertical axis. GF
is a horizontal diameter of the same circle which is without mass and which is
rigidly connected to the circle ; BO is a smaller concentric hoop wliich can turn
freely about GF as diameter. If 12, 0', w, w', be the greatest and least angular
velocities about A£, GF respectively, prove that i2 . ^' — la" - co''^.
11. OA, OB, 00 are the principal axes of a rigid body which is in motion
about a fixed point 0. The axis 00 has a constant inclination a to a line OZ
fixed in space, and revolves with uniform angular velocity fi round it, and the
axis OA always Lies in the plane ZOO. Prove that the constraining couple has its
axis coincident with OB, and that its moment is - {A- C) fl^ sin a, cos a.
CHAPTER XI.
PEECESSION AND NUTATION,
&c, &c.
On the Potential.
GOO. To find the potential of a body of any form at any
extei^nal distant point.
Let the centre of gravity G of the body be taken as the origin
of co-ordinates and let the axis of x pass through S the external
point. Let the distance GS= p. Let {x, y, z) be the co-ordinates
of any element dm of the body situated at any point P and let
GP = r, then P/S' = p' -h r' - 2px. The potential of the body is
V— V ^^'^ ir— V ^^* f 1 ^P^ ~ ''^1 "^
arranging these terms in descending powers of p, we get
^^ ^ dm r, X ^x' - r' 5x' - dxr' , Sox' - dOxV + 3r* ,
p [ p 2p 2p 8p
Let M be the mass of the body, then "^dm = M. Also since the
origin is at the centre of gravity, we have Xxdm = 0.
Let A, B, G be the principal moments of inertia at the centre
of gravity, / the moment of inertia about the axis of x, which in
our case is the line joining the centre of gravit}?- of the body to
the attracted point. Then
tdmr' = \{A-\-B+G),
^dm x' = ^dm {r' -f-z')=\[A + B+C)-I.
ON THE POTENTIAL. 491
Let I be any linear dimension of the body, tlien if p be so
great compared with Z that we may neglect the fraction f-j of
the potential, we have
^^ M A + B + C-SI
p ^P
If we wish to make a nearer approximation to the value of V,
we must take account of the next terms, viz.
V •
Let {^, 7), ^) be the co-ordinates of m referred to any fixed
rectangular axes having the origin at G, and let (a, /3, 7) be the
angles GS makes with these axes. Then
07 = 1^ cos a + ?; cos /3 + ^cos 7 ;
. • . "Zmaf = cos^ a Sm^^ + 3 cos^a cos /S ^m^i] +
If the body be symmetrical about any set of rectangular axes
meeting at G, we have Xm^^ = 0, Xm^^ij = 0, &c. = 0, so that this
next term in the expression for the potential vanishes altogether.
Thus the error of the preceding expression for V is comparable
to only the fraction (- j of the potential. This is the case with
the earth, the form and structure of which are very nearly sym-
metrical about the principal axes at its centre of gravity.
This theorem is due to Poisson, but it was put into the con-
venient form just given by Prof. MacCullagh. See Royal Irish
Transactions for 1855, page 387.
GOl. In the investigation of this value for the potential, S
has been supposed to be at a very great distance. But the ex-
pression is also very nearly correct wherever the point 8 be
situated, provided the body be an ellipsoid whose strata of equal
density are concentric ellipsoids of small ellipticity.
To prove this, we may use a theorem in attractions due to
Maclaurin, viz. The potentials of confocal ellipsoids at any ex-
ternal point are proportional to their masses. Let us first con-
sider the case of a solid homogeneous ellipsoid. Describe an
internal confocal ellipsoid of very small dimensions and let a', b', c
be its semi-axes. Then because the ellipticity is very small, we
can take a, b', c so small that S may be regarded as a distant
point with regard to the internal ellipsoid. Hence the potential
due to the internal ellipsoid is
iir A'+J^'j-C'-dF
P -P
492 PRECESSION AND NUTATION.
where accented letters have the same meaning relatively to the
internal ellipsoid that unaccented letters have with regard to the
given ellipsoid. The error made in this expression is of the
order f — j V. Hence, by Maclaurin's theorem, the potential V
of the given ellipsoid is
M M A' + B'+C'-Sr
p '^ M' 2p^
and the error is of the order ( — IF.
If a, b, c be the semi-axes of the given ellipsoid, we have
Similarly, B=^,B' + ^ MX\ G=~G' + ~ M\\
''Mo Mb
Also if {% /9, 7) be the direction-angles of the line G8 with
reference to the principal axes at O, we have
/=^cos^a + 5cos'/3+acos^7 = -|^7' + |ifA,^
Hence, substituting, we have
^^ M A+B+C-SI
P ^P
If a, h, c be arranged in descending order of magnitude, we
can by diminishing the size of the internal ellipsoid make c as
small as we please. In this case we have ultimately a = Va^ — c\
Let e be the ellipticity of the section containing a and c the
greatest and least semi-axis. Then a = a V2e, and the error of
the above expression for V is of the order 4 f - J e^V.
The theorem being true for any solid homogeneous ellipsoid
is also true for any homogeneous shell bounded by concentric
ellipsoids of small ellipticity. For the potential of such a shell
may be found by subtracting the potentials of the bounding
ellipsoids, A +B+ C by Art. 5 being independent of the direc-
tions of the axes.
Lastly, suppose the body to be an ellipsoid whose strata of
equal density are concentric ellipsoids of small ellipticity, the
external boundary being homogeneous. Then the proposition
being true for each stratum, is also true for the whole body.
ON THE POTENTIAL. 493
This theorem was first given by Prof. MacCullagh as a
problem, and was published in the Dublin University Calendar for
1834, page 268. Some years after, about 1846, he gave his proof
of the theorem in his lectures, which is substantially the same
as that given in this Article. See the Transactions of the Royal
Irish Academy, Vol. xxii., Parts I. and II., Science.
602. The following geometrical interpretation of the formula of Art. 600 is
also due to Prof. MacCullagh. His demonstration and another by the Eev. E.
Townsend may be found in the Irish Transactions for 1855.
A system of material points attracts a point S whose distance from the centre
of gravity G- of the attracting mass is very great compared ivith the mutual
distances of the particles. If a tangent plane he drawn to the ellipsoid of gyration
perpendicular to GS, touching the ellipsoid in T and cutting GS in U, then the
resultant attraction on S lies in the plane SGT. The component of the attraction
3M
on S in the direction TU= 2" GrU . UT. The component of the attraction on
c ■ r ^- • Tin ^^ 3A + B + C-3I
S m the direction UG= -s + jr — t .
P^ 2 p*
These theorems are also true if we replace the ellipsoid of gyi-ation by any
confocal ellipsoid. Let a, b, c be the semi-axes of this confocal, and let^ be the
perpendicular GU on the tangent plane. Since by Art. 26, A = Ma^ + \, B—Mh'^ + \,
^ M M{a^ + h^ + c^-Sp^)
&c. where X is some constant, we have V = — 1 pr-s — •
P 2/)3
To prove that the resultant force on S lies in the plane SGT, let us displace
S to S' where SS' is perpendicular to this plane and is equal to pd\l/. By Art. 326
1 dV
the force on S in the direction SS' is - -— . But after this displacement the tan-
p axj/
gent plane perpendicular to GS intersects along TU the former tangent plane, hence
|f = 0,and...§r=o.
dxp d\p
To find the force P acting at S in the direction TU, let us displace S to S" where
*S't B -\- G—^I (. /terms depending on theN)
2R^ I V squares of x, y , z J) '
Hence the required force-function is
,, MM' ,^A' + B'+G'-SI',.,A + B+C-SI
^=="ir"^^^ 2m + ^^ 2W~ •
The error of this expression is of the order ( d^ ) ^; where I, V
are any linear dimensions of the two bodies respectively.
608. To find the moment of the attraction of the sun and
moon about one of the 'princiixd axes of the earth at its centre of
gravity.
Let the principal axes of the earth at its centre of gravity be
taken as the axes of reference, and let a, j3, 7 be the direction-
angles of the centre of gravity 0' of the sun. Then if Fbe the
potential of the sun or moon on the earth, we have
,^ MM' , ^.A' + B'-^ 0'-3J' u'^^rB±G-M
^=-R-^^ — -^w ^^ — m — '
where unaccented letters refer to the earth, and accented letters
to the sun or moon. Let 6 be the angle the plane through the
dV .
sun and the axis of y makes with the plane of xy, then -^ is the
required moment in the direction in which we must turn the body
to increase 6. From the above expression, since 9 enters only
through /, we have
dV_ _SMdI
W^ 2R' d0'
. ON THE POTENTIAL. 497^
Now I = Acos^a + Bco&^^+Ccos^j, and by Spherical Trigo-
uometry, we have
cos 7 = sin /3 sin 6)
cos a = sin /3 cos 6} '
.■.^=-2(A-G)sin'^smdcose;
da
• .•. the moment required] _ ii' ^ ..
, , ,, . \ ^ = -3 -^((7- J.) cos a cos 7.
about the axis 01 y ) M
In this expression the mass of the attracting body is measured
in astronomical units. We may eliminate this unit in the fol-
lowing manner. Let n be the mean angular velocity of the sun
about the earth, R^ its mean distance, so that if M be the mass
M' + M
of the earth, we have — p-3— = 71'^. Now 31 is very small com-
-"-0
If
pared with M', so small that -jr, is of the order of terms already
M'
neglected. Hence we may in the same terms put -^ = n'^, and
therefore
the moment of the sun's at-) ^ _ 3^^, (^_ ^) ^os a cos 7 f^Y-
traction about the axis of 3/ J \Jti /
Let n" be the mean angular velocity of the moon about the
earth, so that, if M" be the mass of the moon, R'^ the mean dis-
tance, we have — „, 3 = n"^ Let v be the ratio of the mass of
M"{1 + v) ,„ ,
the earth to that of the moon, then we have -^ — -=n , and
therefore if R' be the distance of the moon
the moment of the moon's \ ^_ ^ (^c - A) cosacosy f f?' "
attraction about the axis of 3/ J 1 + i^ \-ti
In the same way the moments about the other axes may be
found. Putting k for the coefficient, we have
moment about axis of ic = - 8/c (5 - C) cos ^ cos 7,
moment about axis 0? z = -Zk{A- B) cos a cos ^.
609. Ex. 1. A body free to move about its centre of gravity is acted on by any
number of attracting particles arranged in any way at a constant distance p from
the centre of gravity. If J.,, B^, C^, D^, E^, F^ be the moments and products of
inertia of the body referred to any rectangular axes meeting in the centre of gravity,
498 PRECESSION AND NUTATION.
and if accented letters represent corresponding quantities for the particles referred
to tlie same axes, prove that the mutual potential of the body and the particles is
-—+ - V
where M' is the mass of all the particles. If the axes of reference be principal
axes for either body, this result admits of considerable simplification.
Show that the numerator of the second term may be expressed in terms of the
invariants of the momental ellipsoids of the body and of the system of particles.
Ex. 2. The force function between a body of any form and a uniform circular
ring whose centre is at the centre of gravity of the body and whose mass is M' ia
„ MM' ^^,A+B+C-SJ
V= - M' 3-3 »
where / is the moment of inertia of the body about an axis through its centre of
gravity perpendicular to the plane of the ring, and A, B, G are the principal
moments of inertia at the centre of gravity.
This follows from Ex. 1.
Ex. 3. Thence show that Saturn's ring supposed uniform will have the same
moments to turn Saturn about its centre of gravity as if half the whole mass were
collected into a particle and placed in the' axis of the ring at the same distance
from Saturn, provided the particle repelled instead of attracted Saturn.
Ex. 4, If the earth be formed of concentric spheroidal strata of small but
different ellipticities and of different densities, show that
C-A fi
da
da
where e is the ellipticity and p the density of a stratum, the major-axis of
which is a ; the square of e being neglected. It follows that if e be constant,
C — A
is independent of the law of density.
C
If we assume the law of density and the law of ellipticity given in the Figure of
C-A
the Earth, this formula gives ■ :=-00313593. See Pratt's Figure of the Earth.
Ex. 5. A body free to turn about a fixed straight line passing through the
centre of gravity is in equihbrium under the attraction of a distant fixed particle.
( Bp^ \ i
Show that the time of a small oscillation is 2 tt L --.,.,, .^ ^..^ — =- J , where the
{3M'^{{C - A)£, + FfiW
fixed straight hne is the axis of y, the plane of xy in equilibrium passes through the
attracting particle, and §, i? are the co-ordinates of the particle. Also A, B, C, D, E,F
are the moments and products of inertia of the body about the axes.
If the straight line did not pass through the centre of gravity show that the
time would be proportional to p.
MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 499
Motion of the Earth about its Centre of Gravity.
610. To find the motion of the pole of the earth about its
centre of gravity when disturbed by the attraction of the sun and
moon, the figure of the earth being taken to be one of revolution.
Let us consider the effect of these two bodies separately.
Then, provided we neglect terms depending on the square of
the disturbing force, we can by addition determine their joint
effect.
The sun attracts the parts of the earth nearer to it with a
force slightly greater than that with which it attracts the parts
more remote, and thus produces a small couple which tends
to turn the earth about an axis lying in the plane of the equator
and perpendicular to the line joining the centre of the earth
to the centre of the sun. It is the effect of this couple which
we have now to determine. It clearly produces small angular
velocities about axes perpendicular to the axis of figure. We
shall also suppose that the initial axis of rotation so nearly coin-
cides with the axis of figure, that we may regard the angular
velocities about axes lying in the plane of the equator to be small
compared with the angular velocity about the axis of figure.
Let us take as axes of reference in the earth, OG the axis
of figure, GA and GB moving in the earth with an angular
velocity 6^ round GG. Then following the notation of Art. 252,
we have
A/ = ^0),, h^ = A(o^ , A/ = Cw^ ,
The equations of motion are therefore
a^-^-a^A+Gco,o^.=l'
A^^-Go>,co,^A -0
dt
(1).
The last of these equations shows that Wg is constant. Let
this constant be denoted by n.
The other two angular velocities are to be found by solving
the other two equations. This solution must be conducted by
the method of continued approximation, — y^ — sm cos 21
, , Sk G-A zi /, 1 . „,x
vr = const. — ^ — -, — 77— cos u U — z; sm zl)
Znn U ^ 2 V
613. We may also solve equations (2) in tlie foUowing manner. Since we
reject the squares of the small quantities to be found, we may in calculating the
values of L and M to a, first approximation suppose 6 to be constant and I to be
measured from a fixed point in space. We then have by the theory of elliptic
motion
l=n't + e' + P^ sin {p-^t + q^j) + P^ sin {p^t + §2) + &e.,
where the coefficients of the trigonometrical terms are all known small quantities,
and aU the coefficients of t are very small compared with n. In the case of the
sun the coefficient of t in the greatest of the trigonometrical terms is ^^ n and in
1
the case of the moon — n.
We may also include in this formula the secular ineqtialities in the value of I.
For, we shall presently find that 6 has no secular inequalities, and that the first
point of Aries from which I is measured has a very slow motion which is very
nearly uniform on the plane of the orbit of the disturbing body. This slow motion
may obviously be included in the n'.
If we eliminate Wg between equations (2) we have
d'^u}, CV 1 dL Cn^^
-W^-^^'^^A-dt-T^^'
The first term on the right-hand side we have already agreed to neglect. Sub-
stituting in the expression for M given in (5) the value of I, suppose we have
lf=:SFcos(\t+/),
where the constant part of M is given by \ = and all the other values of \ are
very small. Then solving, we find
FCn ,,, , .V
'^^--^CS^^^ri2^.cos(Xi+/).
Since P and X^ are both very small we may reject the small term X" in the
denominator, we then have
1 M
This result is strictly true for the constant term and very nearly true for the
periodical terms. In the same way we may prove that u).j= -7-.
504! ' PRECESSION AND NUTATION!
When we proceed to find d and ^ from the values of w^ and w^ by the help of
equations (3), it wiU be seen that no term will rise on integration in which \ is not
ismall. These rejected terms will not therefore afterwards become important.
614. The integration of equation (7) may be effected without neglecting the
terms containing the powers of e' in the expression for I. By the theory of
elliptic motion we have
R^ |^= constant = Bq^u'Ji - e'^,
where a very small term has been rejected on the left-hand side depending on the
motion of Aries. Substituting for k its value given in Art. 608 we find
— =-„— Tj — ■ — ^7 ■ " sing Bin 2Z
dl 2n 1 + v C Rjl-e'^ \
d^p 3n' 1 C-A Rq -,, -,,
-^= -■— -_ ; " ■ ■ cos ^ (1 - COB 20
dl 2nl + p R.JT^'2 ^
where v is to be put equal to zero when the disturbing body is the sun. From
the equation to the ellipse, we have
^^^^ = 1 + e' COS [l-L).
If this value of R be substituted in the equations, the integrations can be effected
without difficulty. But it is clear that all the terms which contain e' are periodic
and do not rise on integration so as to become equally important with the others.
Since then e' is small, being equal in the case of the sun to about 7r„, it will be
needless to calculate these terms.
615. Let us now examine the geometrical meaning of the
equations (8). For the sake of brevity, let us put S= ^ — > — p^— ,
A . ^Ao a SO -An' ^ 2C-An" 1
so that by Art. 60SJS=-^ — -p^ or = 77 — j^ -^— — ac-
*' 2 G n 2 V n 1 + v
cording as the sun or moon is the disturbing body, the orbit of the
disturbing body being in both cases regarded as circular.
Let us consider first the term —S cos 6 1 in the value
of y}r. Let a point G^ describe a small circle round Z the pole
of the orbit of the disturbing planet, the distance G^ being
constant and equal to the mean value of 6. Let the velocity
be uniform and equal to Sn cos 6 sin &, and let the direction of
motion be opposite to that of the disturbing body. Then G^
represents the motion of the pole of the earth so far as this
term is concerned. This uniform motion is called Precession.
Next let us consider the two terms
hO=^ SsmO cos 21, S-\lr=^ r, Scos 6 sin 2i.
2 ' ^ 2
MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 505
If we put a; = sin ^ S-v/^, y = W, we have
JT" T^ + /i V ^
which is the equation to an ellipse.
Let us then describe round C^ as centre an ellipse whose
semi-axes are ^ S cos 6 sin d and ~ S sin 6 respectively perpen-
dicular to and along ZC; and let a point (7^ describe this
ellipse in a period equal to half the periodic time of the dis-
turbing body. Also let the velocity of C^ be the same as if
it were a material point attracted by a centre of force in the
centre varying as the distance. Then G^ represents the motion
of the pole of the earth as affected both by Precession and the
principal parts of Nutation,
If we had chosen to include in our approximate values of
6 and i^ any small term of higher order, we might have re-
presented its effect by the motion of a point C^ describing an-
other small ellipse having (7^ for centre. And in a similar manner
by drawing successive ellipses we could represent geometrically all
the terms of 6 and y^r.
616. In this solution we have not yet considered the Com-
plementary Functions. To find these we must solve
A^+Cnco^^O, A'^-Cnco, = 0.
at at ^
We easily find (o^ = H sin ( -j- t + KY w^ = — ^cos {-~t i + ^j -
The quantities H and K depend on the initial values of &)j &),.
As these initial values are unknown H and K must be de-
termined by observation. If H had any sensible value it would
be discovered by the variations produced by it in the position
in space of the pole of the earth. The period of these would
• 27r A
be — y^ , as J. and C are nearly equal in the case of the earth,
this period is nearly equal to a day. No such inequalities have
been found. If however any such inequality existed we might
consider these two terms together as a separate inequality to
be afterwards added to that produced by the other terms of co^ co.^
whose period is half a year.
The effect of the complementary function on the motion
of the pole of the earth has been already considered. The
motion is the same as if the earth were at any instant set iu
506 PRECESSION AND NUTATION.
rotation about an axis whose direction-cosines are proportional
to Hsml-^t + Kj, -Ecosf-^t + Kj and n and then left
to itself. The instantaneous axis will describe a right cone of
small angle round the axis of figure and also a right cone of
small angle in space. Hence from this cause there can be no
permanent change in the position in space of the axis of the
earth. See Art. 522.
617. The preceding investigations are of course approxima-
tions. In the first instance we neglected in the differential equa-
tions the squares of the ratios of co^ and co^ to n, and afterwards
Bome periodical terms which are an — th of those retained. We
see by equations (3) and (8) that the second set of terms rejected
is much greater than the first, and yet when the sun is the dis-
turbing body these terms are only about -^-^1^ th part of those
retained, and when the moon is the disturbing body these
are only ^ th part of terms which themselves are imperceptible.
We have also regarded the earth as a solid of revolution so
that A — B may be taken zero, a supposition which cannot be
strictly correct.
S G — A n
618. In the case of the sun we have ^=s — tt , so that
2 6 n
the precession in one year is ^ — 7T~~ cos O^ir. It is shown in
treatises on the Figure of the Earth that there is reason to put
(1 — A n' ^
^ =-0031. Also we have -=~^, and 6' =23°. 8'. This
gives a precession of about 15" '42 per annum. Similarly the
coefficients of Solar Nutation in a/t and 6 are respectively found
to be 1"'23 and 0""53. If we supposed the moon's orbit to be
fixed, we could find in a similar manner the motion of the pole
produced by the moon referred to the pole of the moon's orbit.
In this case /S' = ^ — t^ ^;i • The value of 6 varies be-
2 G n 1+v
Th 1
tween the limits 23" ± 5". Putting -=-^,v = 80, 6 = 23°, we
find a precession in one year a little more than double that pro-
duced by the sun. But the coefficients of what would be the
nutations are about one-sixth of those produced by the sun.
619. We have hitherto considered the orbit of the disturbing
body to be fixed in space. If it be not fixed, we must take the
MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 507
plane GA perpendicular to its instantaneous position at the
moment under consideration. The quantity 6^ will not be the
same as before*, but if the motion of the orbit in space be very-
slow, ^3 will still be very small. We may therefore neglect the
small terms 6^(Oj^ and d^co^ as before. The dynamical equations
will not therefore be materially altered. With regard to the
geometrical equations (3) it is clear that co^, co^ will continue to
express the resolved parts of the velocity of G in space along and
perpendicular to the instantaneous position of ZG. To this degree
of approximation therefore, all the change that will be necessary is
to refer the velocities as given by equations (7) to axes fixed in
space and then by integration we shall find the motion of G. This
is the course we shall pursue in the case of the moon.
The attractions of the planets on the earth and sun slightly
alter the plane of the earth's motion round the sun, so that the
position of the ecliptic in space varies slowly. It can oscillate
nearly five degrees on each side of its mean position. If the earth
were spherical there would be no precession caused by the at-
tractions of the sun and moon. The direction of the plane of the
equator would then be fixed in space, and the changes of its
obliquity to the ecliptic would be wholly caused by the motion of
the latter, and would be very considerable. But, as Laplace re-
marks, the attractions of the sun and moon on the terrestrial
spheroid cause the plane of the equator to vary along with the
ecliptic so that the possible change of the obliquity is reduced
to about one and a third degrees which is about one-quarter of
what it would have been without those actions.
At present the obliquity is decreasing at the rate of about
48" per century. After an immense number of years, it will begin
to increase and will oscillate about its mean value. These in-
equalities we do not propose to discuss in this treatise. We must
refer the reader to the second volume of the Mecanique Geleste,
livre cinqui^me. He may also consult the Gonnaisscmce des Temps
for 1827, page 234.
620. Ex. 1. If the earth were a homogeneous shell bounded by similar elhpsoids,
the interior being empty, the precession would be the same a3 if the earth were
solid throughout.
* The value of 6^ may be found in the following manner. The orbit at any
instant is turning about the radius vector of the planet as an instantaneous axis.
Let u be this angular velocity which we shall suppose known. Let Z, Z'; B, B' be
two successive positions of the pole of the orbit and the extremity of the axis of B
respectively. Then ZB = a. right angle = Z' 5'. Hence the projections of ZZ', BB',
on ZB are equal. This gives, since ZB is at right angles to both CZ and SB,
BSB' Bin BS=ZC'Z' Bin ZG. Now the angle ZCZ'-= - S0^ and the angle BSB'=u,
hence o9n . sin ^= - n sin I. The value of dO:, must bo added to the former value of On.
508 ^ PRECESSION AND NUTATION. '
Ex. 2. If the earth were a homogeneous shell bounded externally by a spheroid
and internally by a concentric sphere, the interior being filled with a perfect fluid
of the same density as the earth, show that the precession would be greater than if
the earth were solid throughout.
Let (a, a, c) be the semi-axes of the spheroid, r the radius of the sphere. Then
C — A
since the precession varies as — -^^ by Art. 615, the precession is increased in the
ratio a^c : a'^c — r^.
Ex. 3. If the sun were removed to twice its present distance show that the
solar precession per unit of time would be reduced to one-eighth of its present
value; and the precession per year to about one-third of its present value.
Ex. 4. A body turning about a fixed point is acted on by forces which tend to
produce rotation about an axis at right angles to the instantaneous axis, show that
the angular velocity cannot be uniform unless the momental ellipsoid at the fixed
point is a spheroid.
The axis about which the forces tend to produce rotation is that axis about
which it would begin to turn if the body were placed at rest.
Ex. 5. A body free to turn about its centre of gravity is in stable equilibrium
under the attraction of a distant fixed particle. Show that the axis of least
moment is turned toward the particle. Show also that the times of the
principal oscillations are respectively 27r L„,vi,_ > and 27r L„,/l_ .A .
If the body be the earth and M' be the sun, show that the smaller of these two
periods is about ten years.
621. To giv6 a general explanation of the manner in which
the attraction of the Sun causes Precession and Nutation.
If a body be set in rotation about a fixed point under the
action of no forces, we know that the momenta of all the particles
are together equivalent to a couple which we shall represent by G
about an axis called the invariable line. Let T be the Vis Yiva
of the body. If a plane be drawn perpendicular to the axis of G-
sj MT
at a distance — y^ ^ from the fixed point, then the whole motion
is represented by making the momental ellipsoid whose parameter
is e roll on this plane. In the case of the earth, the axis 01 of
instantaneous rotation so nearly coincides with OG the axis of
figure that the fixed plane on which the ellipsoid rolls is very
nearly a tangent plane at the extremity of the axis of figure.
This is so very nearly the case that we shall neglect the squares
of all small terms depending on the resolved part of the angular
velocity about any axis of the earth perpendicular to the axis of
figure.
Let us now consider how this motion is disturbed by the action
of the sun. The sun attracts the parts of the earth nearer to it
with a slightly greater force than it attracts those more remote.
MOTION OF THE- EARTH ABOUT ITS CENTRE OF GRAVITY. 509
Hence wlien the sun is either north or south of the equator its
attraction will produce a couple tending to turn the earth about
that axis in the plane of the equator which is perpendicular to
the line joining the centre of the earth to the centre of the sun.
Let the magnitude of this couple be represented by a, and let us
suppose that it acts impulsively at intervals of time dt.
At any one instant this couple will generate a new momentum
adt about the axis of the couple a. This has to be compounded
with the existing momentum G, to form a resultant couple G'.
If the axis of a were exactly perpendicular to. that of G we should
have G' = V 6^' + {adtf = G ultimately.
Let 6 be the angle that the axis of G makes with OC, then
6 is a quantity of that order of small quantities whose square is
to be neglected. Taking the case when OG, OC and the axis of a
are in one plane, for this is the case in which G' will most differ
from G, we have
G" = (G cos ey+{G sin 9 -l- adtf
= G'+2Goism6dt (1).
Then a and 6 being of the same order of small quantities, the
term a sin 6 is to be neglected. Hence we have G' = G. But the
axis of G is altered in space by an angle — ?^ in a plane passing
through G and the axis of a.
Next let us consider how the Vis Viva T is altered. If T' be
the new Vis Viva we have
T' — T = twice the work done by the couple a
= 2a ((o cos /3)df (2),
where co cos /3 is the resolved part of the angular velocity about
the axis of a. For the same reason as before the product of this
angular velocity and a is to be neglected. Hence we have T' = T.
ij MT
It follows from these results that the distance ^ of the fixed
G
plane from the fixed point is unaltered by the action of a.
Thus the fixed plane on which the ellipsoid rolls keeps at the
same distance from the fixed point, so that the three lines OG,
01, OG being initially very near each other will always remain
very close to each other. But the normal OG to this plane has
a motion in space, hence the others must accompany it. This
motion is what we call Precession and Nutation.
Lastly these small terms which have been neglected will not
continually accumulate so as to produce any sensible effect. As
the earth turns round in one day, the axis OG will describe
510
PRECESSION AND NUTATION.
a cone of small angle 6 round G. The axis about which the sun
generates the angular velocity a is always at right angles to the
plane containing the sun and OG. Hence, regarding the sun as
fixed for a day, the angle 6 in equation (1) changes its sign every
half day. Thus 0' is alternately greater and less than 0. Simi-
larly since the instantaneous axis describes a cone about OG it
may be shown that T' is alternately greater and less than T.
622. Let us trace the motion of the axis 00 through a whole
year. Describe a sphere whose centre is at and let us refer the
motion to the surface of this sphere. Let K be the pole of the
ecliptic and let the sun 8 describe the circle DEFH of which K
is the pole. Let DF be a great circle perpendicular to KG, then
since OG and the axis of figure of the earth are so close that we
may treat them as coincident, D and ^will be the intersections of
the equator and ecliptic. When the sun is north or south of the
equator, its attraction generates the couple a, which will be
positive or negative according as the sun is on one side or the
other. This couple vanishes when the sun is passing through the
equator at D or F. If the sun be anywhere in DEF, i.e. north
of the equator, G is moved in a direction perpendicular to the
arc 08 towards D. If the sun be anywhere in FED, a has the
opposite sign and hence G is again moved perpendicular to the
instantaneous position oi G8 but still towards D. Considering
the whole effect produced in one year while the sun describes the
circle DEFH, we see that G will be moved a very small space
towards D, i.e. in the direction opposite to the sun's motion.
Resolving this along the tangent to the circle centre K and radius
KG, we see that the motion of G is made up of (1) a uniform
MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 511
motion of G along this circle backwards, which is called Preces-
sion and (2) an inequality in this uniform motion which is one
part of Solar Nutation. Again as the sun moves from D to E, G
is moved inwards so that the distance KQ is diminished, but as
the sun moves from E to F, KG is as much increased. So that
on the whole the distance KG is unaltered, but it has an in-
equality which is the other part of Solar Nutation.
It is evident that each of these inequalities goes through its
period in half a year.
623. To explain the cause of Lunar. Nutation.
The attraction of the sun on the protuberant parts at the
earth's equator causes the pole C of the earth to describe a small
circle with uniform velocity round K the pole of the ecliptic with
two inequalities, one in latitude and one in longitude, whose period
is half a year. These two inequalities are called Solar Nutations.
In the same way the attraction of the moon causes the pole of the
earth to describe a small circle round M the pole of the lunar
orbit with two inequalities. These inequalities are very small
and of short period, viz. a fortnight, and are therefore generally
neglected. All that is taken account of is the uniform motion
of C round M. Now K is the origin of reference, hence if M
were fixed the motion of C round M would be represented by a
slow uniform motion of G round K together with two inequalities
whose magnitude would be equal to the arc MK, or 5 degrees, and
whose period would be very long, viz. equal to that of G round K
produced by the uniform motion. But we know by Lunar Theory
that M describes a circle round K as centre with a velocity much
more rapid than that of G. Hence the motion of G will be repre-
sented by a slow uniform motion round K, together with two
inequalities which will be the smaller the greater the velocity
of M round K, and whose period will be nearly equal to that
of M round K. This period we know to be about 19 years.
These two inequalities are called the Lunar Nutations. It will
be perceived that their origin is different from that of Solar
Nutation.
624. To calculate the Lunar Precession and Nutation.
Let K be the pole of the ecliptic, M that of the lunar orbit,
C the pole of the earth. Let KX be any fixed arc, KG = 6,
XKG = '^, then we have to find 6 and i/r in terms of t. By
Art. 615 the velocity of G in space is at any instant in a direction
perpendicular to MG, and equal to
8w"^ G-A 1 ,,„ . ,^„
- -TT ?=Y— ^; cos MG sm MC.
2n G l + v
512 PRECESSION AND NUTATION.
For the sake of brevity let the coefficient of cos MC sin MC
be represented by P. Then resolving this velocity along and
perpendicular to KG, we have
ja
^ = -P&mMC cos MC sin KCM
at
sine^ = -P sin MC cos MC cos KCM
at J
By Lunar theory we know that 31 regredes round K uniformly,
the distance K3I remaining unaltered. Let then KM=t, and
the angle XKM- -mt + a.
Now by spherical trigonometry,
cos MC = cos I cos + sin t sin $ cos MKC,
. Trrmr COSZ — COS MC COS
sua MC COS KCM = -. — 7,
sm &
= cos i sin — sin i cos cos MKC,
sin MG . sin KCM- sin i sin MKC.
Substituting these we have
^ = - P jsin ^• cos i cos 6 sin i/ZC + ~ sin%' sin ^ sin 2MKc\ ,
sin ^ -^ = — P -{sin ^ cos ^ [cos^ t — ^ sin^f j
— sin 1 cos « cos 20 cos MKC— ^ sin'^i sin ^ cos ^ cos 2MKGy .
For a first approximation we may neglect the variations of
and i/r when multiplied by the small quantity P. Hence -j-
contains only periodic terms, and the inclination has no per-
manent alteration. But -~ contains a term independent of
MKC ; considering only this term, we have
i/r = constant — Pcos [ cos^« — -= sin^ i\t.
This equation expresses the precessional motion of the pole
due to the attraction of the moon. We may write this equation
in the form ^1^ = "v/^o —pt.
To find the nutations, we must substitute for MKG its approxi-
mate value
MKG={-m+p)t+a-yfr^.
MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 513
We then have after integration
^ , Psini cos i'cos^ nrr^^ Psin'^^'sin^ ^T,,T^r-i
6 = const. cos MKG , r cos 23/A C.
m — p 4 (in — j))
The second of these two periodic terms being about one-
fiftieth part of the first, which is itself very small, is usually
neglected. Also p is very small compared with m, hence we have
- ^ Psin t cost cos ^ ,tt^^
6 = 6^ cos MKG.
° in
This term expresses the Lunar Nutation in the obliquity.
In the same way by integrating the expression for >/r, and
neglecting the very small terms, we have
f I D af 2- 1 • 2^^ „sin2i cos2^ . ,,i^^
Y = '\lr^- P cos 6 { cos^ i—- sm^4 ]t — P -rq — . -: — tt Sin MKG.
\ z J zm sin U
The angle MKG is the longitude of the moon's descending
node, and the line of nodes is known to complete a revolution
in about 18 years and 7 months. If we represent this period by
27r
T we have MKC= — ij, t-{- constant.
The pole M of the lunar orbit moves round the point of re-
ference K with an angular velocity which is rapid compared with p,
but yet is sufficiently small to make the Lunar Nutations greater
than the Solar. We may also notice that if M had moved round
K with an angular velocity more nearly equal to p the Nutations
would have been still larger. This may explain why a slow motion
of the ecliptic in space may produce some corresponding nutations
of very long period and of considerable magnitude.
II. D. 33
514 PRECESSION AND NUTATION.
Motion of the Moon about its centre of gravity.
625. In discussing tlie precession and nutation of the equinoxes, the earth has
been regarded as a rigid body two of whose principal moments at the centre of
gravity are equal to each other. One consequence of this supposition was that the
rotation about the axis of unequal moment is not directly altered by the attraction
of the disturbing bodies. As an example of the elfect of these forces on the
rotation when all the three principal moments are unequal, we shall now consider
the case of the moon as disturbed by the attraction of the earth. As our object is
to examine the mode in which the forces alter the several motions of the moon
about its centre of gravity rather than to obtain arithmetical results of the greatest
possible accuracy, we shall separate the problem into two. In the first place we
shall suppose the moon to describe an orbit which is very nearly circular in a plane
which is one of the principal planes at its centre of gravity. In the second case we
shall remove the latter restriction and examine the effects of the obliquity of the
moon's orbit to the moon's equator.
628. The moon describes an orhit about the centre of the earth which is very
nearly circular. Supposing the plane of the orbit to be one of the principal planes
of the moon at its centime of gravity, find thetnotion of the moon about its centre of
gravity.
Let GA, GB, GC be the principal axes at G the centre of gravity of the moon,
and let GC be the axis perpendicular to the plane in which G moves. Let A, B, C
be the moments of inertia about GA, GB, GC respectively, and let M he the mass
of the moon, and let accented letters denote corresponding quantities for the
earth.
Let be the centre of the earth, and let Ox be the initial line. Let OG = r,
GOx = d. Let us suppose the moon turns round its axis GC vn the same direction
that the centre of gravity describes its orbit about 0, and let the angle OGA = (j>.
The mutual potential of the earth and moon is by Art. 607
^^ ^^^j^A'^B'^C^-,1' A+B^^O-SI^
r 2r^ Sr-*
Here I=A cos'^ + ^sin^^ and therefore the moment of the forces tending to
turn the moon round GC is
dV 3 M' ,„ ,, . „ ,i\
MOTION OF THE MOON ABOUT ITS CENTRE OF GRAVITY. 515
Since ^4- ^ is the augle which GA, a line fixed iu the body, makes with Ox, a
line fixed in space, the equation of the motion of the moon round GC is
The motion of the centre of gravity of the moon referred to the centre of the
earth as a fixed point is found in the Liinar Theory. It is there shown that r and
6 may be expressed in the form
r=c{l + icos {pt+ a) +&c. [,
dd
= M + /3M- M« cos (_p< + a) + &c. ,
where /3f is a very small term which represents a secular change in the moon's
angular velocity about the earth, and is really the first term of the expansion of a
trigonometrical expression.
If we substitute the value of -r- in equation (2) we have the following equation
to determine cp,
j~- -^q^sin2(p-fi + npMBin {pt + a}-\-&c (3)^-
S B — A q^
where for the sake of brevity we have put n^ ^ — ~^— = -^r- .
Now we know by observation that the moon always turns the same face towards
the earth, so that amongst the various motions which may result from different
initial conditions, the one which we wish to examine is characterized by (p being
nearly constant. Let us then introduce into this equation the assumption that
is nearly constant; we may then deduce from the integral how far this assumption
is compatible with any given initial conditions which we may suppose to have been
imposed on the moon. Putting (p = cp^ + cp', where cp^, is sujiposed to contain all the
constant part of (j>, we easily find
L/ sin 2^0= -/3 1
^.^ , (4).
-j^- + q"^ cos 2(p^(p' = np3{ sin {pt + a) +&C.I
The numerical value of q depends on the structure of the moon and can there-
fore only be found by comparing the results of this investigation or some other
results -with observation. The first of equations ^4) shows that 2^3 must be less
than (/-. But for various reasons, though q is very small, we must yet suppose that
— is also extremely small. Assuming this, we see that i^^ must also be very small.
q'
It follows also that we may write 2(/)g for sin 2^^ and unity for cos 2 may represent the actual motion it is
necessary and sufficient that H when found from the initial conditions should
be small. We see, by differentiation, that i/g is of the same order of small
quantities as ~. Hence ^ will be small if at any instant the angular velocity,
viz. — + -- , of the moon about GO were so nearly equal to the angular velocity ,
at at
do
viz. — , of its centre of gravity round the earth, that the ratio of the difference to
g is very small.
If therefore we suppose the moon at any instant to be moving with its axis of
least moment pointed towards the earth and its angular velocity about its axis of
rotation to be nearly equal to that of the moon round the earth, then the axis of
least moment will continue always to point very nearly to the earth. The mean
angular velocity of the moon about its axis will immediately become equal to that
of the moon about the earth and will partake of all its secular changes. This is
Laplace's theorem. It shows that the present state of motion of the moon is
stable, rather than explains how the angular velocity about the axis came to be so
nearly equal to the angular velocity about the earth.
627. By comparing the value of the angular velocity of the moon about its
axis obtained by theory with the restdts of observation, we may hope to obtain
B — A
some indications of the value of q^ and thence of — j^ . If the term HqsxQ.{qt + K)
B - A
could be detected by observation, we should deduce the value of — ^r ^^*^^ ^*^
period.
Among the other terms of the expression for the angular velocity of the moon
about its axis, those will be best suited to discover the value of q which have the
largest coefficients, that is those in which either the numerator M is the greatest
or the denominator q^ -^'^ the least possible. By examining the numerical value of
B- A
their coefficients Laplace has shown that if — ^— were as great as -03 the elliptic
inequality could be recognized by observation, and if it were between -OOld and '003
the annual equation could be observed.
628. We may also calculate by the help of Art. 326 the radial and transverse
forces which act on the centre of gravity of the moon due to the mutual
attractions of the earth and moon. Since the principal nioments of the moon
are nearly equal and its linear size small compared with its distance from
the earth, these forces are very nearly the same as if the moon were collected
MOTION OF THE MOON ABOUT ITS CENTRE OF GRAVITY. 517
into its centre of gravity. The effect of the small forces neglected by
this assiimption mil be insignificant compared with the other forces which act on
the centre of gi-avity of the moon. The motion of the centre of gravity of the
moon is therefore very nearly the same as if the whole mass were collected into its
centre of gravity.
Since however there are no other forces which have a moment round GC besides
those found above, the effect of these may be perceptible. The effects of tidal
friction on the rotation of the moon may be omitted, at least at the present time.
Ex. The centre of gravity (? of a rigid body describes an orbit which is
nearly circular about a very distant fixed centre of force attracting according
to the Newtonian law and situated in one of the principal planes through G. If
r — c{\ + p), 6 = nt + n\l/ be the polar co-ordinates of G referred to 0, show that the
equations of motion are
S-^">-2""-'^ = -!"V-|n^7Cos2^^
TJ-A , 20- A- B
where 7 = ,, ., , 7 = — „ , , -^ — •
We may notice that the values of 7 and 7' are much smaller than c^^ and might
therefore be rejected in a first approximation.
If the bddy always turns the same face to the centre of force so that cj> is
nearly constant and is small, show that there will be two small iuecjualities in the
value of (p of the form L sin (pi + a), where p is given by
one of these periods being nearly the same as that of the body roimd the centre
of force and the other bein'g very long.
If the body turns very nearly uniformly roiind its axis GC, so that (p = 7i't + e'
nearly, show that there will be two small inequahties in the value of ^, one iu
which 2} = 71 and another iu which p — 2n'.
629. Ex. 1. Show that the moon always turns the same face very nearly to that
focus of her orbit in which the earth is not situated. [Smith's Prize.]
Ex. 2. If the centre of gravity G of the moon were constrained to describe a
circle with a uniform angular velocity n about a fixed centre of force attracting
according to the Newtonian law ; show that the axis GA of the moon will oscillate
on each side of GO ov will make complete revolutions relatively to G'O according
as the angular velocity of the moon about its axis at the moment when GA and GO
coincide in direction is less or greater than n + q. Find also the extent of the
oscillations.
Ex. 3. A particle m moves without pressure along a smooth circular wire of
mass M with uniform velocity under the -action of a central force situated in the
centre of the wire attracting according to the law of nativre. Show that this system
of motion is stable if ^. > ..C^- • The disturbance is supposed to be given
to the particle or t!ie wiro, the centre of force remaining fixed in space.
518 PRECESSION AND NUTATION.
Ex. 4. A uniform riug of mass M and of very small section is loaded with a
heavy particle of mass m at a point on its circumference, and the whole is in
uniform motion about a centre of force attracting according to the law of nature.
Show that the motion cannot be stable unless ^, lies between -815865 and
M + m
•8279.
This example shows (1) that if a ring, such as Satiirn's ring, be in motion
about a centre of force, its position cannot be stable, if the ring be uniform ; and
(2) that if, to render the motion stable, the ring be weighted, a most delicate
adjustment of weights is necessary. A very small change in the distribution of
the weights would change a stable combination to one that is unstable. This
example is taken from Prof. MaxiveWs Essay on Saturii's Rings.
Ex. 5. The centre of gravity of a body of mass M, symmetrical about the plane of
xy, is G; and is a point such that the resultant attraction of the body on is
along the line GO. Then if the body be placed with coinciding with a fixed
centre of force S, and be set in rotation about an axis through perpendicular to
the plane of xy with an angular velocity w, G will, if undisturbed, revolve uniformly
in a circle, always turniug the same face towards 0, provided MauP' is equal to the
resultant attraction along GO, where a is the distance GO. It is required to
determine the conditions that this motion should be stable.
The motion being disturbed, will no longer coincide with the centre of force
S. Let two straight lines at right angles revolving uniformly round S as origin
with an angular velocity w be chosen as co-ordinate axes, and let x be initially
parallel to OG. Let [x, y) be the co-ordinates of 0, 4> the angle OG makes with
the axis of x, then x, ?/, 4lAC. This pro-
position is due to Sir W. Thomson and is given in Prof. MaxioeWs Essay on Saturn's
Eings.
630. The motion of a rigid body about a distant centre of force has been
investigated on the supposition that the motion takes place entnely in one
plane. We see by equation (2) of Art. G26 that the case in ^\hich the centre
MOTION OF THE MOON ABOUT ITS CENTEE OF GRAVITY. 519
of gravity describes a circular orbit, aud the rigid body always tiirns the axis
of least moment towards the centre of force, is one of steady motion. The
preceding investigation also shows that this motion is stable for all distm-bances
which do not alter the plane of motion. It remains now to determine the effect of
these disturbances in the more general case when the motion takes place in three
dimensions.
The whole attraction of the centre of force on the body is equivalent to a single
force acting at the centre of gravity, and a couple. If the size of the body be small
compared with its distance from the centre of force, we may neglect the effect of the
motion of the body about its centre of gi'avity in modifying the resultant force.
The motion of the centre of gravity will then be the same as if the whole were
collected^ into a single particle. The problem is therefore reduced to the following.
A rigid body turns about its centre of gravity G, and is acted on by a centre of
force E which moves in a given manner. In the case in which the rigid body is
the moon, this centre of force, i.e. the earth, moves in a nearly circular orbit in a
plane which itself also has a slow motion in space. This motion is such that a
normal GM to the instantaneous orbit describes a cone of small angle about a
normal GX to the ecliptic. The two normals maintain a nearly constant in-
clination of about 5". 8'; and the motion of the normal to the instantaneous orbit is
nearly uniform.
631. It will clearly be convenient to refer the motion to axes GX, GT, GZ
fixed in space such that GZ is normal to the ecliptic. Let GA, GB, GO he the
principal axes of the moon at the centre of gravity G. Let {p, q, r) be the direction-
cosines of GZ referred to the co-ordinate axes GA, GB, GO. Then we have, since
GZ is fixed in space,
dt
dq
dt
dr
Now our object is to find the small oscillations about the state of steady motion
in which GZ, GO, GM all coincide. We shall therefore havep, q, w^, Wg all small,
and r very nearly equal to unity. The equations (I) will therefore become
dp
where n is the mean value of w^.
Let X, fi, V be the direction-cosines of the centre of force E as seen from G.
Then we have by Euler's equations and Ai't. G08,
A '^2 - ( 5 - C) 0^2 wg = - 37i'= iB-C)ixv
B-''^-{C-A)oi.,w^=-^n'"-{0-A)v\ i- (II).
W32 + o}^r=0 j
-Wir+W3p = y (I).
dt
dui^
dt
C ^'^f -{A-B) w^w^ = - 3h'^ {A - B)\/M
520
PRECESSION AND NUTATION.
In the case of steady motion, the rigid body always turns the axis (GA) of least
moment towards the centre of force, and u^=n'. We have then both fi and v small
quantities, so that in the first equation we may neglect their product /xv, and in
the second equation we may put i'X = ;'. Also, we may put o}^=n=n' in the small
terms.
If I be the latitude of the earth as seen from the moon, we have
sin I — cos ZE=p\+ q/x + rf^p + v nearly.
Hence the two first of Euler's equations take the form
A^-(B-C) 710,^ =
If the earth, as seen from the moon, be supposed to move in a circular orbit in
a plane making a constant inclination tan"^ h with the ecliptic, and the longitude
of whose node is - gt + /3, we shall have
sin Z = ^ sin {n't + gt- jS).
In this expression g measures the rate at which the node regredes, and is about
the two hundred and fiftieth part of n. We shall therefore regard - as a small
n
quantity.
.(III).
To solve these equations, it will be foimd convenient to substitute for w^, w^
their values in terms of p, q. We then have
A^[l+{A + B-C)nf^-nHB-Oq^0 1
dV''
dq
dt
MOTION OF THE MOON ABOUT ITS CENTRE OF GRAVITY, 521
To find p, q, let us put
p = P sin {{n'+g)t-p}, q==Qeos {(n' +g)t- p],
where P, Q are some constants to be determined by substitution in the equation.
We have
Q{A{n+g)^+ (B - C)nn=.P(A + B-C)n{n+g) )
P{B{n + gy--i{C-A)n"}-Q{A+B-C)n{n + g)^-dn'-]c{C-A))' '
We may solve these equations to find P and Q accurately. In the case of the
moon the ratios — -; — , , ■ — - — and - are all small. If then we neglect the
C A B n
products of these small quantities, the first equation gives us ^=:1- -. The
second equation will then give
5nh{C-A )
dnlC-A)-2Bg'
As g is very small compared with n, we may regard P and Q as equal.
632. The complementary functions may be found in the usual manner by
assuming
p = F sin {st + 11), 2 = G cos (si + if),
on substituting we have the quadratic
ABs^ - {{A +B - C)^- BiB - C) -4:A [A - C)]n"~s" + 4:{A - C) {B-C)n*=0,
G (A+B-C)ns
to find s^, and
F As^-i-{B-C)ii^'
to find the ratio of the coefficients of corresponding terms in p and q. If the roots
of this equation were negative p and q would be represented by exponential values
of t, and thus they would in time cease to be small. It is therefore necessary for
stability that the coefficient of s^ should be negative and the product (A - C)(B- C)
positive. Both these conditions are probably satisfied in the case of the moon.
For since B- C and A - C are both small, the term {A + B- C')^ is much greater
than the two other terms in the coefficient of s^. Also, since the moon is flattened
at its poles, we shall probably have A and B both less than C.
633. Let M be the pole of the moon's orbit, which is the same as that of the
earth's orbit as seen from the centre of the moon. Then M is the pole of the
dotted line in the figure of Art. 631. Therefore the angle EZM measured by
turning ZE in the positive direction round Z until it comes into coincidence with
ZM,is= —--{{n-\-g)t-^\. Again, if the angle £"^(7 be measured in the same
direction, we have
* ^ COS EC- cos CZ COS ZE v-r(p\ + qn+rv) ~p ,
cos EZC= . „„ . „„ = -y — ^^ ^ ' = ■ , nearly.
sm CZ sm ZE Jps + ^a gjn ZE Jp' + q^
-1
Hence we easily find &\vlEZC= i r~~:.
But sin (JZM = sin EZM cos EZU- cos EZM sin EZG
_cos{(;j + r/)t •-/3)jp-sin{(« + r/)<-/3}(2'
522 PRECESSION AND NUTATION.
If now we substitute for p and g their values, it is clear that the terms in p and
q, whose argument is n + g, disappear. So that if F and G were zero, the sine of
the angle CZM would be absolutely zero. In this case the three poles C, Z, M
must lie in an arc of a great circle, or, which is the same thing, the moon^s equator,
the moon^s orbit, and the ecliptic must cut each other in the same line of nodes.
If however F and G be not zero, but only very small, we have
7:F'sm{s't+H')
sin CZM--
Vi" + 2(?'='sin(s'«+i^')'
where F', G' contain either jF or G' as a factor, and are therefore small. If then F
and G be both small compared with P, the angle CZM wUl remain either always
small or always nearly equal to tt.
The intersection of the moon's equator with the ecliptic will then oscillate about
the intersection of the moon's orbit with the ecliptic as its mean position. Since
these oscillations are insensible, it follows that in the case of nature, the com-
plementary functions must be extremely small compared with the terms depending
directly on the disturbing force.
634. If we disregard the complementary functions we have p = P sin 4>,
q = P cos = -r- with
respect to t;
, d(b du ...
■■■-'''^'^dt'ds (*)•
Similarly, by differentiating cos "^ — rf- ^^^ cos % = ^7- , we
get two other similar equations for i/r and ;)^. Taking these six
equations in conjunction with the following
COS'^ (ji + cos^ yjr + cos^ % = 1 (5),
we have seven equations to determine u, v, w, (f), yp-, % and T.
THE EQUATIONS OF MOTION.
525
If the motion takes place in one plane, these reduce to the
four followiDg equations :
''^:^* = ;7::(^^cos<^) + 7?iX
dt ds
dv _ d
dt ds
dv d ,rr, ■ ,s T-
m^n =-1- [J- sm 1 d ,^ .
-f- = - sm V^ + - — f T cos (p),
da ^ dt Kdt ■ ^ '
■with similar expressions for v and ic.
638. When the motion of the string takes place in one plane,
it is often convenient to resolve the velocities along the tangent
and normal to the curve.
Let u', v be the resolved parts of the velocity of the element
ds along the tangent and normal to the curve at that element.
Let <^ be the angle the tangent to the element ds makes with
the axis of x. Then by Art. 179 or 252, the equations of motion
are
du , d(b ^^, dT ^
—, V ^} =A -\ ,- '
di dt mas
dv ,d(^ ^^, T
at dt mp
^ (1).
The geometrical equations may be obtained as follows. We
have
u = u' cos (p — v' sin (p.
Differentiating with respect to s, we have by Art. 636,
dd> . , /du v'\ , (dv , u'\ . ,
ds
where p is the radius of curvattire, and is equal to -y- . Since
the axis of x is arbitrary in position, take it so that the tangent
during its motion is parallel to it at the instant under considera-
tion ; then ^ = and we have
= ^-^ (2).
ds p
Similarly, by taking the axis of x parallel to the normal,
dcf) _ dv' u' ,Q^
dt ds p
These four equations are sufficient to determine u', v , _ dT 1
dt dt mdcr i
dv' , d4> ,„ T ds\'
dt dt mp d(T J
To find the geometrical equations, vre may differentiate
u — u' cos, rp-v' B\n [^ T\ du' u' /, T
-£v+x)=d.-'-p['^x
639. The equations (2) and (3) may aJso be obtained in the
following manner. The motion of the point P of the string being
represented by velocities u and v' along the tangent FA and
normal FO at F, the motion of a consecutive point Q will be
represented by velocities u + du and v' + dv' along the tangent
QB, and normal QO at Q. Let the arc FQ = ds, and let QNhe
a perpendicular on FA. Since the string is inextensible, the
resultant velocity of Q resolved along the tangent at F must be
ultimately the same as the resolved part of the velocity of F in
the same direction. Hence
(u + du) cos dcj) — (v + dv) sin d<^ = it,
or, proceeding to the limit,
du — V dcp = ; . . — , = 0.
dio V
P
Again, -^ is the angular velocity of FQ round F. Hence
the difference of the velocities of F and Q resolved in any direc-
tion which is ultimately perpendicular to FQ must be equal to
.'. (u + du) sin d(p + {v + dv) cos d(^ — v'= ds --p ,
or in the limit
d(^ _ dv u'
dt ds p
640. Ex. 1. An clastic ring without weight, whose length when unstretched is
given, is stretched round a circular cylinder. Tho cylinder is suddenly annihilated.
528 MOTION OF A STRING. .
show that the time which the ring will take to collapse to its natural length is
iMair . , , .
\/ —IT—, where ill is the mass of the string, X its modulus of elasticity, and a is
V oA
the natural radius.
Ex. 2. A homogeneous light inextensible string is attached at its extremities
to two fixed points, and turns about the straight line joining those points with uni-
form angular velocity. Find the form of the string, supposing its figure per-
manent.
Result. Let the straight line joining the fixed points be the axis of x, then the
form of the string is a plane curve whose equation is 1 + ( -~ | —I ' | , where a
and b are two constants.
On Steady Motion.
641. Def. When the motion of a string is such that the
curve which it forms in space is always equal, similar, and simi-
larly situated to that which it formed in its initial position, that
motion may be called steady.
642. Prop. To investigate the steady motion of an inexten-
sible string.
It is obvious that every element of the string is animated with
two velocities, one due to the motion of the curve in space, and
the other to the motion of the string along the curve which it
forms in space. Let a and h be the resolved parts along the axes
of the velocity of the curve at the time t, and let c be the velocity
of the string along its curve.
Then, following the usual notation, we have
w = a + ccos^] , .
■y = 6 + csin(jf)i
Now a, h, c are functions of t only, hence -^ = — cs'ind) -r .
•^ ds ^ as
Therefore by equation (7) of Art. G36 we have
-csmc/,-J = A + ^(-cosc^J
dh dc . , , d(b „ d fT . ,\
> .
ON STEADY MOTION. 529
Substituting for -— , these equations reduce to
da ( ^^ do ,\ d {(T \ ,1
- i:r\^-'dt'''V'^ds\^i-'r''^\
{--%^^-IMr^^^
(3).
The form of the curve is to be independent of t\ hence, on
eliminating T, the resulting equation must not contain t. This
will not generally be the case unless -^ , ~j i ~t1 ^^^ ^ ^*^^"
stants. In any case their values will be determined by the known
circumstances of the Problem. The above equations must then
be solved, s being supposed to be the only independent variable,
and t being constant.
643. If the string move uniformly in space, and the elements
of the string glide uniformly along the string, -r.—^i 'df~^'
dc
-j7 = 0. It will then follow from the above equations, that the
Civ
form of the string will be the same as if it was at rest, but the
tension will exceed the stationary tension by mc".
644. Ex. 1. Let an electric cable be deposited at the bottom of a sea of uniform
depth from a ship moving with uniform velocity in a straight line, and let the cable
be delivered with a velocity c equal to that of the ship. Find the equation to the
curve in which the string hangs.
The motion may be considered steady, and tlie form of the curve of the string
will be always the same.
If the friction of the water on the string be neglected, gravity diminished by the
buoyancy of the water will be the only force acting on the string, let this be repre-
sented by g'. Hence the form of the travelling curve will be the common catenary,
and the tension at any point will exceed the tension in the catenary by the weight
c2
of a length of string equal to -. .
g
Next let the friction of the water on any element of the cable be supposed to
vary as the velocity of the element, and to act in a direction opposite to the direc-
tion of motion of the element*. Let /i be the coefficient of friction.
Let the axis of x be horizontal, and let x' be the abscissa of any point of the
cable measured from the place where the cable touches the ground, in the direction
* Each element of the string has a motion both along the cable and trans-
versely to it. The coefficients of these frictions are probably not the same, but
they have been taken equal in the above investigation.
R. D. 34
530 MOTION OF A STRING.
of the ship's motion. Also let s' be the length of the curve measured from the same
point. Then x=x' + ct, and s=s' + ct.
Following the same notation as before, we have
A' = - fill, Y= - g' - fiv.
But M = c-ccos0, v—-csm(p.
Hence the equations (3) become
0= -/UC + /XC cos + -T- I ( — c^Jcos^l 1
O=-f/' + .^esiJi,6+|jg-c^)sin0J 1
To integrate these put sin ^ = -^ , cos = -r- . Hence,
CIS G/S
ff'A — ~ /xcs + ficx +{ — c^ I cos (p 1
^"' ^ I (1),
g'B= - g's + ixcy + ( - - cM sm ^ I
where A and B are two arbitrary constants.
At the point where the cable meets the ground, we must have either T-Q or
0=0, For if (j) be not zero, the tangents at the extremities of an infinitely small
portion of the string make a finite angle with each other. Then, if T be not zero,
resolving the tensions at the two ends in any dii-ection, we have an infinitely small
mass acted on by a finite force. Hence the element will in that case alter its posi-
tion with an infinite velocity. First, let us suppose that ^ = 0. Also at the same
point, 2/=0 and s' = 0. Hence 5= -cf.
Putting — = e, we get by division
dy _ s' - cy
dx' A-ex' + es'
This is the differential equation to the curve in which the cable hangs.
To solve this equation*, let us find s' in terms of the other qiiantities,
, dy ,dy
(2).
Differentiating, we have
dx
\ dx' J
* The problem of the mechanical conditions of the deposit of a submarine cable
has been considered by the Astronomer Eoyal in the Fhil. Mag. July 1858. His
solution is different from that given above, but his method of integrating the differ-
ential equation (2) has been followed.
ON STEADY MOTION.
KOI
Put n for ^ where convenient, and put v for A -ex'+e^-i/; the equation then
dx
becomes
1 du _ (?a;'
V dx'~ 0_-ep)Jl+p^
in which the variables are separated, and the integrations can be effected. The
equation can be integi'ated a second time, but the result is very long. The arbitrary
constant A may have any value, depending on the length of the cable hanging from
the ship at the time i=0.
The cm-ve in its lower part resembles a circular arc or the lower part of a com-
mon catenary. But in its upper part the curve does not tend to become vertical,
but tends to approach an asymptote making an angle cot^^e with the horizon. The
asymptote does not pass through the point where the cable touches the ground but
below it, its smallest distance being — v_^=^ ; the asymptote also passes below the
^A^e^ + l
ship.
If the conditions of the question be such that the tension at the lowest point of
the cable is equal to nothing, the tangent to the cm-ve at that point will not neces-
sarily be horizontal. Let X be the angle this tangent makes with the horizon,
Referring to equations (1) we have simultaneously
x'=0, 7j = 0, s' = 0, T=0, and ^=X.
Hence ^ := - -. cos X, B= --. sin X - ct.
9 9
The differential equation to the curve will now become
, -. sin X + s' - ew
^-y g (3),
— -,cosX + (?s -ex'
9
which can be integrated in the same manner as before. One case deserves notice ;
viz. when e = cot X. The equation is then evidently satisfied by y = - x'. The two
constants in the integral of (3) are to be determined by the condition that when
dx'
yz=-x'. Hence this is the required integral. The form of the cable is therefore a
straight line, inclined to the horizon at an angle X = cot~^e; and the tension may be
found from the formula y^ ingy
1+cosX
Ex. 2. Let a cable be delivered with velocity c' from a ship moving with uni-
form velocity c in a straight line on the surface of a sea of uniform depth. If the
resistance of the water to the cable be proportional to the square of the velocity,
the coefficient B, of resistance for longitudinal motion being different from the
coefficient A , for lateral motion, prove that the cable may take the form of a
straight line making an angle X with the horizon, such that cot^X^ ^e^ + i - 1,
where e is the ratio of the speed of the ship to the terminal velocity of a length of
34—2
532 MOTION OF A STRING.
cable falling laterally in water. Prove also that the tension will be found from the
equation
I ^ ~ f '' ('c ~ '"''' ^)'siii ! "'^'' ^^^'^- ^^^^-^
0)1 Initial Motions.
6-l'5. A string, under the action of any forces in one plane,
begins to move from a state of rest in the form of any given curve.
To find the initial tension at any given point.
Let mPds, m Qds be the resolved parts of the forces respectively
along the tangent and normal to any element ds. The force P is
taken positively when it acts in the direction in which s is mea-
sured, and Q is positive when it acts in the direction in which
p is measured along the normal, viz. inwards. Let m be the mass
of a unit of length.
Let u, V be the velocities of the element along the tangent
and normal. Then the equations of motion are by Art. 638
du d(f) _ jy 1 dT .
~dt~'"dt~^m1E ' • ^ ^'
dv d(b ^ 1 T ,^.
S + "3* = « + ™^ (2)'
where T is the tension, p the radius of curvature, and (f) the angle
the tangent makes with any fixed straight line. The geometrical
equations are
du V ^ ._, dv u d(b ...
&-r° ^'^' ds+r^ -*'•
Differentiating (1) and multiplying (2) by - , we get
r
.(5).
d'u d^_dvd_dP 1 d'T
ds dt ds dt ds dt ds m ds^
Idv ud^_Q IT
p dt p dt p m p' J
But by differentiating (3) we have, since - = -^ ,
d^u d^d> 1 dv ^
ds dt dsdt p dt
Hence, subtracting the second of equations (5) from the first,
we have by (4) and (6)
l/d^_T\ dP_Q__(d4^
m \ ds p^J ds p \dt
ON INITIAL MOTIONS. 533
In tlie beginning of the motion just after the string has been
cut we may reject the squares of small quantities, hence (-^
maybe rejected. Hence we have
d'T T dP q'
^-p^^=-^^+^V ^'^-
This is the general equation to determine the tension of a
string just after it has been cut.
The two arbitrary constants introduced in the solution of this
equation are to be determined by the circumstances of the case.
If both ends of the string are free, we must have 2^= at both
ends.
Since the string begins to move from a state of rest we have
(J? f clij
initiallv ^i = 0, v = 0. At the end of a time dt, ~r. dt and t- dt
at dv
will be the velocities of any element of the string. Hence if -x/r
be the angle the initial direction of motion of any element of the
string makes with the tangent to the element, we have by equa-
tions (1) and (2)
Q + ~ -
tanf =— -—_ (8;.
m ds
It must be remembered that the constants of integration are
necessarily constant only throughout the length of the string at
the time ^ = 0. They may be functions of t and may be either
continuous or discontinuous. For example, if a point of the string
be absolutely fixed in space, the transverse action of the fixed
point on the string may cause the constants to become discon-
tinuous at that point. In this case equation (8) is not necessarily
true in the immediate neighbourhood of the fixed point.
646. If the string be heterogeneous we may easily show in
the same way, that the initial tension is given by
A (1 ^\ _ 1 ^ _ _ ^-P ^
ds \m ds J m p^ ds p '
647. A string is in eqidlihrium tinder the action of forces
in one ^iilane. Supposing the string to he cut at any given point,
find the instantaneous change of tension.
Let T^ be the tension at any point {x, y) just before the
string was cut. Then the forces P, Q satisfy the equations of
equilibrium
o = p+l^\ o=(2 + --".
m ds m p
534 MOTION OF A STRING.
Hence — — + -* = —5 « .
as p 711 as m p
If T'be the instantaneous change of tension, we have T'=T—T^.
The equation of the last article therefore becomes
d^r_r_
ds' p' ~
648. Ex. 1. A string is in equilibrium in the form of a circle about a centre of
repulsive force in the centre. If the string be noto cut at any point A, prove that the
tension at any point P is instantaneously changed in the ratio of
1-? — : 1,
e'^ + e"'^
where 9 is the angle subtended at the centre by the arc AP.
Let Fhe the central force, then P=0, and mQ= ~F. Let a be the radius of the
cu'cle. Then the equation of Art. 645 to determine T becomes
(FT _ r _ _ F
ds'^ a^ a
Let s be measured from the point A towards P, then s — ad; also F is independ-
ent of s. Hence we have
T=Fa + A€^ + Be-^.
To determine the arbitrary constants A and B we have the condition y^O when
^ = 0and^ = 27r;
T=FaAl-- i^
e" + e "
But just before the string was cut T—Fa. Hence the result given in the enuncia-
tion follows.
Ex. 2. A string is wound round the under part of a vertical circle and is just
supported in equilibrium at the ends of a horizontal diameter by two forces. The
circle being suddenly removed, prove that the tension at the lowest point is
instantly decreased in the ratio 4 : e^+ e~^ .
Ex. 3. The extreme links of a uniform chain can slide freely on two given
curves in a vertical plane, and the whole is in equilibrium under the action of
gravity. Supposing the chain to break at any point, prove that the initial tension
at any point is T — y (A(p i-B), where y is the altitude of the point above the direc-
trix of the catenary, (p the angle the tangent makes with the horizon, and A, B two
arbitrary constants. Explain how the constants are to be determined.
Ex. 4. A string rests on a smooth table in the form of an arc of an equiangular
spiral and begins to move from rest under the action of a central force F which
tends from the pole and varies as the n*'' power of the distance, show that the initial
tension is given by
™ „ n cos^ a H- sin- a ,
T= ~ rF -7--rr-— 2 ^»- + ^'^ + ^""''
n (n + 1) COS'' a - sin- a
ON INITIAL MOTIONS. 535
where a is tlie angle of the spiral, p and q are the roots of the quadratic
X (x-l)=tan*a.
Show that the solution changes its form when a is such that the first term is
infinite, and find the new form.
649. A string rests on a smooth horizontal table and is acted
on at one extremity by an impulsive tension, find the impulsive
tension at any point and the initial motion.
Let T be the impulsive tension at any point P, T + dT the
tension at a consecutive point Q, then the element PQ is acted on
by the tensions T and T+ dT at the extremities. Let <^ be the
angle the tangent at P to the string makes with any fixed line ;
u, V the initial velocities of the element resolved respectively
along the tangent and normal at P to the string. Then, resolving
along the tangent and normal, we have
7nuds = (T+dT) cos d(j>-T
mvds = {T+dT) sin d(f>
therefore proceeding to tlie limit
1 dT IT
m as m p
du V
But by Art. 63^9, we have ~r = - • Hence the equation to find
T becomes
d'T T __
ds' p'
This, as might have been expected from mechanical consi-
derations, is the same as the equation in Art. 647.
If the chain be heterogeneous we easily find in the same way
d_ (\^dT\^l_T
ds \m ds I m p^ "
The two results in this article appear to have been first given
in College Examination Papers.
650. Ex. If 7*1, Tg ^6 t^6 impulsive tensions at the extremities of any arc of
the chain, u\, u^ the initial velocities at the extremities resolved along the tan-
gents at the extremities, prove that the initial kinetic energy of the whole arc is
This readUy follows by integrating m (v? + v"-) ds along the whole length of the
arc. But it also follows at once from Art. 331, for the work done at either extre-
mity is the product of the tension into half the initial tangential velocity.
.(1).
:536 MOTION OF A STRING.
Small Oscillations of a loose chain.
651. A heavy heterogeneous chain is suspended hy one ex-
tremity and hangs in a straight line under the action of gravity.
A small disturbance being given to the chain in a vertical plane,
it is required to find the equations of motion*.
Let be the point of support, let tbe axis Ox be measured
vertically downwards and Oy horizontally in the plane of disturb-
ance. Let 7nds be the mass of any elementary arc whose length
PQ is c?s, and let T be the tension at P. Let I be the length of
the string, and let us suppose that a weight Mg is attached to the
lower extremity.
The equations of motion as in Art. 635 will be
d'^x \ d f rp dx\
df m ds \ dsj ^
df m ds \ ds
Since the motion is very small, the point P will oscillate in a
very small arc, the tangent at the middle point being horizontal.
Hence we may put -tt = 0. For a similar reason we may put
dx = ds. We therefore have by integrating the first of equa-
tions (1)
T= constant — g jmdx.
But T= Mg when x = l, hence we find
T=Mg+g\ mdx (2).
* In the Seventh Volume of the Journal Polytechiique, Poisson discusses the
oscillations of a heavy homogeneous chain suspended by one extremity. Putting
(l-x)^=iz^g^t equal to s or s' according as the upper or lower sign is taken, and
y'—y[l-x)i, he reduces the equation to the form ' , = ~l / 1 '\2 ' -^^ obtains
the integral by means of two definite integrals and two infinite series. After a
rather long discussion of the forms of the arbitrary functions which occur in the
integral, he finds that a solitary wave will travel up the chain with a uniform
acceleration and down with a uniform retardation each equal to half that of
gravity.
SMALL OSCILLATIONS OF A LOOSE CHAIN. 537
When the chain is homogeneous, this equation takes the simple
form
T=Mg + mg{l-x) (3).
It may be noticed that this expression is independent of the
time ; the tension at any point of the chain is equal to the total
weight of matter below that point.
The second equation may be written in either of the forms
df m dx\ dxj
(4),
ni dx^ m dx dx ^
where 2^ is a function of x given by the equations (2) or (3).
652. Let us suppose that the displacements of the particles
forming any finite portion of the chain during a finite time, are
represented by y = ^ {x, t), where (^ is a continuous function of x
and t. Let P be a geometrical point within this portion of the
dy
chain which moves so that the particle-velocity at P, i. e. -j- is
always equal to some constant quantity A. Let v be the velocity
with which P moves, then following in our mind the motion of P,
we have
Let Q be a point also within the portion, such that the tangent
to the chain at Q makes with the vertical an angle Avhose tangent,
i. e. -^ , is yp, where B is some constant quantity.
Let v' be the velocity with which Q moves, then
dxdt
Eliminating the second differential coefficients of y from equa-
tions (4), (5) and (6), we easily deduce that if P and Q coincide
at any instant,
T
vv =— {l).
m
This reasoning requires that all the second differential coeffi-
cients should be finite, and that y should be a continuous function
of X and t. It would not apply to any point P, if the discontinuous
extremities of two waves were passing over P in opposite direc-
tions. But the consideration of these exceptions is unnecessary
for our present purpose.
538 MOTION OF A STRING.
Let AB be a disturbed portion of the chain travelling in the
direction AB on a chain otherwise in equilibrium. At the con-
fines of the disturbance the two portions of the string must not
make a finite angle with each other. If they did, an element of
the string would be acted on by a finite moving force, which is the
resultant of the two finite tensions at its extremities. In such
a case the disturbance would instantly extend itself further along
the chain and take up some new form. Supposing we exclude
any such case as this, we must have, as long as the motion is
finite, both -jr — ^, and -j=0, at both the upper and lower ex-
tremity of the disturbance. If then P be a point at which —^ = 0,
and Q a point at which -— = 0, P and Q may be considered as
taken just within the boundary of the wave ; P and Q will there-
fore each travel with the velocity of that boundary. Hence
putting V = v', we find for the velocity of either point
T
v' = ^ (8).
It appears therefore that if a solitary wave travel up the chain,
the velocity increases as the wave approaches the upper extremity.
The upper end of the wave will travel a little quicker than the
lower end, because the tension at the upper end exceeds that at
the lower; thus the length of the wave will gradually increase.
When the wave travels down the chain, the velocity for the same
reason decreases.
653. Ex. 1. If the chain be homogeueons, show that the boundaries of a
solitary wave will travel up the chain with an acceleration equal to half that of
gravity, and down the chain with a retardation of the same numerical amount.
Ez. 2. Let the law of density he m = A{l\l' -'£)~'^ where I is the length of
the chain and A, V two constants. Also let a weight equal to '^AqsJV be fastened
to the lower extremity, prove that
This integration may be effected by writmg ^ = (Z + Tj^ - (? + T - x)^. The equation
d 11 cj (ill/
of motion then takes the form -r£ = •^- -r^, , which can be solved in the usual manner.
Ex. 3. The chain is said to sound an harmonic note when its motion can bo
represented by an expression of the form ?/ — ^ (x) sin («;« + o) ; so that the motion of
every element repeats itself at the same constant interval. Show that the harmonic
periods of the chain and weight are given by
Kl'''{&nK{{l + l'f-l'^\^l (1).
SMALL OSCILLATIONS OF A LOOSE CHAIN. 539
To prove this, we substitute y=f {0) sin {Kt + a) in the differential equation
obtained in the last Example; we thus find/(^) to be trigonometrical. Since y=0
when x=0 for all values of t, the expression for y reduces to
y--
■ siuKd Ll^sin/ct [f\ +Bi,cosKt (^Y i (2),
where Ak and Bk are two arbitrary constants. But when x=l, y must satisfy the
equation of motion of the weight, viz. —^ — -g -^ . Whence the result follows by
substitution.
Ex. 4. If the initial motion of the chain and weight be given by the equations
y=f{x), —=F{x) when t = 0, then y can be expanded in a series, the general term
of which is expressed by equation (2) of the last example. Find the values of
Ak and 3^.
"We notice that equation (1) of the last example may be written in the form
cos kO^ = kJv sin kO^,
where ^^ is the value of 9 when x = Z. We then easily find that
/ sin Kd sin K'9d6 = - ^V sin k6j^ sin /c'^^,
rOi 11
J^ I 2,
These results may be obtained by integrating the left-hand sides and substi-
tuting for cos Kd^ and cos /c'6'^ their values in terms of sin k9-^ and sin k'6^.
If we now multiply both sides of equation (2) by sin k9 and integrate from
^=0 to 6 — 9-^, we find by the use of these two results
s^K (6'i + sJfBin^ k9^ = f 'y sin k9cW +/(Z) Jv sin k9-^.
2i •'0
Differentiating (2) and performing the same process, we have
I J^ /|(6>i + v'Z'sin2/c6'i)=.J^'^sin/c^fZ(? + i^(0 ^VsinK9^.
654. An inelastic heterogeneous chain is suspended from two
fixed points under the action of gravity. Any small disturbance
being given in its oiun plane, it is required to find the small oscil-
lations.
Let the axis of x be horizontal and that of y vertical Let G
be any point on the chain when hanging in equihbrium, and let
the arc s be measured from G. Let [x, y) be the co-ordinates of
any point P determined by GP = s. Let T be the tension at P,
mgds the weight of an element ds situated at P. The equations
of equilibrium are
K^s)-'' K^'l)-""^-'-
540 MOTION OF A STRING.
Let a be the angle the tangent at P makes with the axis of x,
then we easily find
m v)q d!tana ,,\
T= — — , m = w—, — (1),
cos a as
where w is an undetermined constant.
When the chain is in motion, let (a; + 1^, y +'n) be the co-
ordinates of the position of the particle P at the time t, and let
the tension at that point be T' = T+ U. The equations of motion
will be
111 as [ \ds as/)
df mds\ \ds'^ ds!^ ^'
which, by subtracting the equations of equilibrium, reduce to
i (2),
df m ds \ ds dsj \
dSi^l d_f^
with a similar equation for 77. Thus the dynamical equations be-
come at the boundary
m) ds^ m ds ds
y,
f ^_T\d^i_]^dUdi
V w/ ds^ m ds ds ^
and the geometrical equation becomes
d~^ dx _ d\ dy
ds^ ds ds^ ds '
T . .
From these we easily get ■y^ = — . Substituting for T and m their
values, we have if p be the radius of curvature at P,
v = \/{gp cos a) (4) ,
so that the velocity of either boundary of the wave is that due to one
quarter of the vei'tical chord of curvature at that point.
Ex. 1. A chain is in equilibrium under the action of any forces which are
functions only of the position in space of the element acted on. Show that the
velocity of either boundary of a solitaiy wave is that due to one quarter of the chord
of curvature in the direction of the resultant force at that boundary.
656. To solve as far as lyossihle the equations of motion of a heavy slack
heterogeneous chain.
It wiU he convenient to express the unknown quantities ^, rj, Uin terms of
some one function (p.
Let a + cphe the angle the tangent at P makes with the horizon at the time t.
Then
, ^ dx + d^ . , ,^ dy + dr] ■
cos(a + 0)=-^, sin(a + .^)--^^^';
, • '^l , dri
.•.-/'(P) andx(Q) where i^-and x are arbitrary functions of two determinate
combinations P and Q of the variables. The arbitrary functions A and i? are not
independent of C and B, and the relations between them may be found by substi-
tuting in equations (8).
We have thus four arbitrary functions whose values have to be determined from
the conditions of the question. Let Oq, a^, be the values of a which correspond to
the two extremities of the string. Then the values of and -~ are given by the
question when t = for all values of a. from a — a^) to a = aj; also the initial values
of yi and i? are given. Thus the values of ^piP) andx(Q) are determined for all
values of P and Q between the two limits which correspond to a = a,„ i = and a = aj,
t = 0. The forms of 1/' and % for values of P and Q exterior to these limits, and the
values of A and B when t is not zero, are to be found from the conditions at the
extremities of the chain. If the extremities be fixed, we have both ^ and tj equal to
zero for all values of t when a = aQ and a = a^. It may thus happen that the
arbitrary functions A, B,\}/ and % f-i'e discontinuous.
In many cases the circumstances of the problem will enable us to determine at
once the form of C. Thus, suppose the string when in equilibrium to be
symmetrical about a vertical line, say the axis of y, and let the points of support be
SMALL OSCILLATIONS OF A LOOSE CHAIN. 543
fixed iu the same horizontal liue. Then if tlie initial motion be also symmetrical
about the axis of y, the whole subsequent motion will be symmetrical. Thus
must be a function of a, containing when expanded only odd powers of o. Sub-
stituting such a series in equation (10) we see that C must be zero.
658. There are several eases in which the equation to find the small motions
of a chain may be more or less completely integrated. One of the most interesting
of these is that in which the chain hangs in equilibrium in the form of a cycloid.
In this case we have, if h be the radius of the generating circle, p—M) cos a. The
w
density of the chain at any point is given by ni— tt —, so that all the lower
part of the chain is of nearly uniform density, but the density increases rapidly
higher up the chain and is infinite at the cusp.
The equation to find the oscillations now takes the simple form
d^cj) (J ( d'^4>
d^d> n (d-d) . , „^) ,, ,.
dhu\J^''^'-^^\ (^'^'
in which all the coefficients are constants.
There are two cases of motion to be discussed, (1) when the chain swings up
and down, and (2) when it swings from side to side. The results are indicated in
the two following examples.
Ex. 1. A heavy chain suspended from tivo points in the same horizontal line
hangs under gravity in the form of a cycloid. Find the symmetrical oscillations
of the chain, when the loiuest point moves only up and down.
In this case we have (7=0. To find the nature and time of a small oscillation,
we put
= SiZ sin Ki + Si2' cos Kt,
where S implies summation for all values of k, and JR, R' are functions of a only.
Substituting, we have
with a similar equation to find R'. Therefore
i?=Lsin2
y(-T>'
where L is an arbitrary constant, the other constant being determined by the
consideration that the motion is symmetrical about the axis of y. For the sake of
brevity, put X=:2. /[1 + — j. Substituting in (7), we find that the terms derived
from jR become
2&
^='ZL—-, — . {XcosXa sin 2a -2 sin Xa cos 2a} sinxi,
X^-4
r ■ 26 2h ~\
•)7 = 2 - I, -g'^i; (^cosXacos2a + 2suiXasin2a} -L Y^os Xa + 7/ sin /cf,
where II is a constant depending on the position of the points of supijort. The
terms derived from R' must be added to these, but have been omitted for the sake
of brevity. They may be derived from those just written down by writing cos Kt
for sin Kt and changing the constants Z, II into two other constants L', IF.
544 MOTION OF A STRING.
Let the length of the chain be 2 Z, then at either end sinao= jr. At both
extremities we must have ^ = 0, ij = 0. All these four conditions can he satisfied if
tan Xoq _ tan 2ao
~X 2 •
This equation therefore determines the possible times of symmetrical vibration
of a heterogeneous chain hanging in the form of a cycloid.
659. If a he not very large, the oscillations are nearly the same as those of a
uniform chain*. In this case since a,, is small but XaQ is not necessarily small,
the equation to determine X is approximately
tan \ag=\ao.
Stt
The least value of Xa which can be taken is a little less than — . Hence X
is great, and therefore '^ = a / ( JI ) ^ nearly. The expressions for ^ and i] now
take the simple forms
^ = SZ —2 {XacosXa-sinXa} sin ( x/ if. ^* + ^ ) I
r]=liL — {cosXtto - cosXa} sin ( x/jf X< + e ) J
The terms depending on cos Kt have been included in these expressions for ^ and
7] by introducing e into the trigonometrical factor.
The roots of the equation tan Xag^Xag ^ay be found by continued approxi-
mation. The first is zero, but since X occm's in the denominator of some of the
small terms, this value is inadmissible. The others may be expressed by the
formula Xoq ={2i + l)-^-6, where is not very large. This makes the time of
4 I
vibration nearly equal to ^. — ^ . — ,^= . Thus the times of vibration of the chain
are all short.
This result will explain why the marching of troops in time along a suspension
bridge may cause oscillations which are so great as to be dangerous to the bridge.
It is clearly possible that the "marching time" may be equal to, or very nearly
equal to some one of the times of vibrations of the bridge. If this should occur
it follows from Arts. 498 and 503 that the stability of the bridge may be severely
strained.
* The reader who may wish to see another method of discussing the small
oscillations of a suspension chain may consult a memoir by Mr Eohrs in the ninth
volume of the Cambridge Transactions. Mr Eolu'S considers the chain to be homo-
geneous, symmetrical about the vertical, and nearly horizontal from the beginning
of the process. In the second edition of this treatise the small oscillations were
also treated on the same hypothesis, but in a different manner. That method,
however, is not nearly so simple as the one here given in which the approximate
oscillations for a catenary are deduced from the accurate ones for a cycloid.
SMALL OSCILLATIONS OF A LOOSE CHAIN. 545
It should be noticed that the terms in the expression for f have the square of \
in the denominator, while those in the expression for ?; have the first power of X.
Since X is great we might as a first approximation reject the values of ^ altogether,
and regard each element of the chain as simply moving up and down.
660. Ex. 2. A heavy chain suspended from two points hangs under gravity in
the form of a cycloid. If it swings from side to side in its own plane so that the
middle point has only a lateral motion without any perceptible vertical motion,
find the times of oscillation.
As in the last example, we put
= 2E sin Kt + 'ZR' cos kI,
where R and R' are functions of a only. Substituting in equation (11) we see that
2C— "Zh ^hx Kt+Tik sin Kt where h and k are arbitrary constants. The equation to
find R becomes
d-^R . A h^-X „
1 + — J as before, we find i2= - — + Z sin (Xa+ M).
Thence taking the term of (f> which contains sin Kt,
P h'-hbcoB2a , 25 , ,,, .
-. 7= To hx/-^ — -. {Xcos,(Xa + iJ/^)siu2a-2 sm (Xa + M) cos 2a!,
sm Kt X' . X^-4 ^ ' "
where h' is an arbitrary constant introduced on integration. Substituting in
equation (8), we find h'= -h(b + ^\. Also, we have in the same wa-y
V Jib ,„ . „
= -r^(2a + sin2a)
sm Kt
26 '^b
-L- — J {X cos (Xa + if) cos 2a + 2 sin (Xa + iJ/) sin 2a} - L'^- cos{\a + M)+H.
X'' — 4 X
If we suppose the two supports to be on the same horizontal line, we must have
^=0 and 7j=0, when a=±ao. These conditions may be satisfied if we take
M=-, H—0, for then ^ becomes an even and rj an odd function of a. In this case
?7 = at the lowest point of the chain. We have then two equations to find — ,
h
equating these values, we have
tan Xao X^ - 4
2 tan 2an - X tan Xon -
" cos 2a X X tan Xa^ tan 2o(, + 2
^^+'^^'^-0 2cos«ao+^,^_^
661. If ag be small, this equation is very nearly satisfied by \aQ = iir where
i is any integer. In this case the complete expressions for t, and rj take the simple
forms
46,
^— SZ r7(cos XaQ - cos Xa - Xa sin Xci) sinf a/tj "Xt + '
»; = 2Z,^8inXasin(^|-X« + e) )
K. I). ;i5
546 MOTION OF A STRING.
6G2. Ex. 1. If we change the variables from a, t to p, q where
show that the general equation (10) of small oscillations takes the form
y^-ic
where /x* =' ^ and (j)-ix4>'.
P
Show also that the coeffieient of <(>' is a function of p + q, the form of the
function depending on the law of density of the chain.
This transformation may be useful, because it follows from Art. 655 that p is
constant for the boundaries of a solitary wave travelling in one direction, and q for
a wave travelling in the other direction.
Ex. 2. A heavy string hangs in equilibrium under gravity in such a form that
COS a J)'^
its intrinsic equation is = - sin^ (2a + c) where b and c are any constants.
P 9
. n . . . , &'* sin^ (2a 4- c) ^ , , ^ ■ i
Show that its law of density is given by m=iv —. . If such a chain be
•'*='' g CDS'* a
set in motion in any symmetrical manner, prove that its motion is given by
^ 7, • /o , \\nf^. cot(2a + c)\ ./ C0t(2a + c)\)
Ex. 3. If in addition to gravity, each element of the chain be acted on by a
small normal force whose magnitude is Fg, prove that the equation of motion
of the chain is
g cos a dt^ da?' cos a aa J cos a
If the chain is nearly horizontal, so that a is very small, and if i^=/sin {at - ca),
prove that the denominator of the corresponding term in the expression for
f/=o,
li' = f/.
?2' = ^3'.
^3=0,
^ dx ^ dx'
These give
ig sin M^ — L^ sin (n^Zj + M-^) )
E^n^L^coaM^ — E^Ti^L^cos (n^li + M-^)\
L3 sin M^ = Z2 sin {n^l^ + M^)
E.^n.^L^ cos M.^ = E.^n^L^ cos [njL^ + Mg) 1
= ig sin (WjZj + M3).
These give the following equations to find the M's ;
tan M^ _ tan (n^Zj + M-^ tan M^ _ tan [nJL^ + M^ _ tan [nj,^ + 71/ 3)
~ ^' E^n^ ~" J'jMj ' jBgWj ~ JBjWj ' ~ jE;3n3
Solving these we find
tan n^ri tan n^Zj tan WjZg _ tan»?jZj tan WgZg i&nn^l^
E-^n-y E^n^ E^n^ " '^ ^' EjU-y ' E^n^ ' E^n^
SMALL OSCILLATIONS OF A TIGHT STRING. 553
Substituting for nj, n„, n^ in terms of p we have an equation to find the har-
monics.
The values of p being known, it is clear that the preceding equations determine
all the constants except L^. We have therefore one constant undetermined for
each harmonic. To find these we must have recourse to the initial conditions.
The equations may be written in the forms
f/ — SP„cosnai, ^^'='ZQn cos nat, ^^' =^ 2 B^ cos nat,
where P,„ Q^ and E„ satisfy the equation -r-^ — "" ^^^-P* ^^ have therefore, after
integration by parts,
m-^fp^PJx^ -fpj^dx= -P^^ + ^ P^ + n^fp^P^d^.
Similar theorems apply to Q„ and i2„.
We also have the conditions
when x=0, x — l-^, x — l^^ + l^, x = l^ + l^ + l^,
P=0, P=Q, Q=R, R^O,
^ das ' dx' ^ dx ^dx'
whatever the suffixes may be, provided they are the same in each equation. If
then we put
4>{m,n)=^ E^PJPJx+ E^Q^QJx+ E^R^R^dx,
•/ ''h ''h + h
we have m?(j> {m, n) = ji^0 {m, n), and therefore each is necessarily zero when m and n
are different. A precisely similar theorem would apply if one or both ends of the
string were loose, or if the string were vibrating transversely instead of longitudinally.
Suppose now that we have initially ^i'=/i (x), ?/=/2(a;), i3=fi{^)- We easily
find
pli r^i+^2
/ jBi/i (x) sm {n-^x + M^ dx+ E^f^ {x) sin {% {x - l-^) + M^) dx
*'0 •'^i
"h + h + l;
-E3/3 (^) sin {«3 i^-h- h) + ^Q dx
+I0
= E^L-^ / sin^ (n^x + M^j dx + EJ^^ / sm^ {n^ [x - l^ + M^ dx
•'0 ''h
+ -^3^3 ^1/ ' sin2 [n^ {x-\- l^) + M^] dx,
these integrations may be easily effected and give an additional equation to find the
L, which corresponds to any value otp.
If the strings did not start from rest, we should merely have to add to the
expressions for ^/, Ig', I3' similar functions of x but with sin nat written for cos nat.
670. Ex. 1. If the three strings vibrate transversely, and a-^, a^, a^ be the
velocities of a wave along them measured in units of length of unstretched string,
prove that the periods of the notes are given by the equation
tan ri^l-^ tan n^l^ tan 713^3 _ ^iaxniyl^ tann^^j ian njt^
554 MOTION OF A STRING.
2'jr
where n-^aj^=n^a.2 = n^a.^ = — . If the initial disturtance is given show how to find
the subsequent motion.
Ex. 2. Two heavy strings AB, BC of different materials are attached together
at B and suspended under gravity from a fixed point A. Prove that the periods of
the vertical oscillations are given by the equation
tan^-^Man^-:l-|l^%
the notation being the same as before. If the two strings be initially unstretched,
find their lengths at any time.
671. An elastic string is stretched between two fixed points A and B' and is set
in vibration, it is required to find the energy.
Let the notation be the same as that used in Arts. 663 and 664.
First let the vibrations be longitudinal. The equation of motion is
dt^ dx^
Hence we have
^= -j—x+'2[A sin {n {at - x) + a) + B sin {n [at + x)+^}].
Since | must vanish when a;=0 and be equal to I'-l when x = l we find, as
in Art. 666,
I' -I
^— —J- x+'^Csinnx sin (nat + y),
where nl=i'rr and S imphes summation for all positive integer values of i. The
letters O and y are constants which may be different in every term and which de-
pend on the initial disturbance.
The kinetic energy of the whole string is
rii fdtY r^i
= / ^vidx[j\ —\ - mdx{2(7«asinna;cos(»ia< + 7)}'.
rl
Now / sin nx sin n'xdx = when n and n' are numerically unequal since nl and
n'l are both integer multiples of tt. Hence, when the square of the series is ex-
panded, the integi-al of the product of any two terms is zero.
Also r
Jo
si-Qp nxdx=^ I, hence the kinetic energy becomes
= 2 mla"^ 'LC^n^ cos'' [nat + 7).
To find the potential energy; we notice that the work done in stretching an
element from its unstretched length dx to its length dx + d^ is, by Art. 327, equal
1 fd^\^
io ~E\-f-\ dx. Hence the whole work done in stretching the string is
2 \dxj
7 1
Now f cos nx cos n'xdx -Q or - I according as n and n' are numerically unequal
Jo ^
SMALL OSCILLATIONS OF A TIGHT STRING. 555
or equal to each other ; also / cos nxdx = 0. Hence as before, the integral becomes
^\e ^^LJi31 + \ ElZChi^ sin2 hiat + y).
The first term is the work done in stretching the string from the unstretched
length I to the stretched length V. If we refer the potential energy to the position
of the string when stretched in equihbrium between the extreme points A and £'
as the standard position, we retain the latter tenn only.
The energy is the sum of the kinetic and potential energies. Since jB = ma',
this becomes
energy = j mla'^1,C'ti^,
This result might have been deduced more simply from Art. 458, where it
is shown that the energy of a compound vibration is the sum of the energies of the
simple vibrations into which it may be resolved. See also Art. 451. The kinetic
energy of any single harmonic is easily seen by integration to be
2 mla^Chi^ cos^ [nat + y).
Hence the whole energy is t mla^ZC^n^.
We may also notice that, as in Art. 457, the mean kinetic energy is equal to the
mean potential energy, the means being taken for any very long period.
672. Next, let the vibrations he transversal.
Following the notation of Art. 664, the motion is given, as before, by
y' — 1,0 sin nx sin [nat + 7),
where nl = iir and S impKes summation for all positive integer values of t.
The kinetic energy by the same reasoning as in Art. 671 is equal to
J mla'^'Z CPn" cos^ [nat + 7) .
To find the potential energy, we notice that the work done in stretching an
element from its unstretched length dx to its stretched length ds' is by Art. 327
1 fds' \^
equal io -E{- — 1 J dx. Now
{ds'Y = [dx'Y- + [dy'Y^ (j dxy + dy\
• '■ -T- = 7 J 1 +^ T7o ^ I t nearly.
dx I 2 1'^ \dxj "^
Eemcmbering that, by Art. 664, ma^ — E—jj- ; we find that the whole work done
in stretching the string is
Substituting for y' and integrating we find that the work is equal to
1 U'-])^ 1
- E ^ — j-^ + -mlu'^C-n^Bin- (mU + y).
556 MOTION OF A STRING.-
If we take the position of equilibrium of the string when stretched between the
extreme points A and £' as the position of reference, we find that the
energy = - mZa^S C V.
This we may call the energy of the disturbance.
Prof. Donkin in his treatise on Acoustics, page 128, has found the energy of a
string vibrating transversely, by an ingenious appUcation of the method of sub-
tractions.
Ex. 1. An elastic rod AB has the end A fixed and B free. Being placed on a
perfectly smooth table, it vibrates longitudinally. Show that the energy of a disturb- ^
IT 1
ance represented by f = SC sin nx sin {nat + 7) where nl:={2i + 1) ^ is ^ mla^20^n*.
NOTES.
On D'Alemherfs Principle, by Sir G. B. Airy.
I HAVE seen some statements of or remarks on this principle which
appear to me to be erroneous. The principle itself is not a new physical
principle, nor any addition to existing physical principles ; but is a con-
venient principle of combination of mechanical considerations, which
results in a comprehensive process of great elegance.
The tacit idea, which dominates through the investigation, is this : —
That every mass of matter in any complex mechanical combination may
be conceived as containing in itself two distinct properties : — one that of
connexion in itself, of svisceptibility to pressure-force, and of connexion
with other such masses, but not of inertia nor of impressions of momen-
tum : — the other that of discrete molecules of matter, held in their places
by the connexion-frame, susceptible to externally impressed momentum,
and possessing inertia. The union produces an imponderable skeleton,
carrying ponderable particles of matter.
Now the action of external momentum-forces on any one particle
tends to produce a certain momentum-acceleration in that particle,
which (generally) is not allowed to produce its full effect. And what
prevents it from producing its full effect ] It is the pressure of the
skeleton-frame, which pressure will be measured by the difference be-
tween the impressed momentum-acceleration and the actual momentum-
acceleration for the same. Thus every part of the skeleton sustains a
pressure-force depending on that difference of momenta. And the whole
mechanical system, however complicated, may now be conceived as a
system of skeletons, each sustaining pressure-forces, and (by virtue of
their combination) each impressing forces on the others.
And what will be the laws of movement resulting from this connexion ?
The forces are pressure-forces, acting on imponderable skeletons, and
they must balance according to the laws of statical equilibiinm. For if
they did not, there would be instantaneous change from the understood
motion, which change would be accompanied with instantaneous change
of momentum-acceleration of the molecules, that would produce different
pressures corresponding to equilibrium. (It is to be I'emarked that
momentum cannot be changed instantaneously, but momentum-accelera-
tion can be changed instantaneously.)
558 NOTES.
"We come thus to the conclvision, that, taking for every molecule the
difference between the impressed momentum-acceleration and the actual
momentum-acceleration, those differences through the entire machine
will statically balance. And — combining in one group all the impressed
momentum-accelerations, and in another group all the actual momentum-
accelerations — it is the same thing as saying that the impressed momen-
tum-accelerations through the entire machine will balance the actual
momentum-accelerations through the entire machine. This is the usual
expression of D'Alembert's principle.
Elder s Geometrical JEquations.
Art. 235. It is sometimes necessary to express the angular veloci-
ties of the body about the Jixed axes OX, OY, OZ in terms of 6, (f>, if/.
This may be effected in the following manner. Let w^:, «,y, «>. be the
angular velocities about the fixed axes, O the resultant any velocity. If
we impress on space and also on the body in addition to its existing
motion, an angular velocity equal to — O about the resultant axis of
rotation, the axes OA, OB, OC will become fixed, and the axes OX, OY,
OZ will move with angular velocities —m^, — o>y, — w^- Hence, in the
formulae of the text, if we change
, i{/ being measured as indicated
in the figure after this change, the relations connecting them with the
angular velocities about the axes fixed in space, are obtained from those
in the text by simply changing Wj, w^, Wg into — w^, —Wy, —to,- If we
choose to measure 6 in the ojiposite direction to that indicated in the
figure, the expressions for coj, w^, become identical with those for to^, o)„,
in the text.
NOTES. 559
On the Impact of Bodies.
Arts. 156 and 305. The problem of the impact of two smooth
inelastic bodies is considered by Poisson in his Traite de Mecanique,
Seconde Edition, 1833, The motion of each body just before impact
being supposed given, he forms six equations of motion for each body to
determine the motion just after impact. These contain thirteen un-
known quantities, viz. the resolved velocities of the centre of gravity of
each body along three rectangular axes, the three resolved angular
velocities of each body about the same axes, and lastly the mutual
reaction of the two bodies. Thus the equations are insufficient to
determine the motion. A thirteenth equation is then obtained from the
principle that the impact terminates at the moment of greatest compres-
sion, i. e. at the moment when the normal velocities of the points of con-
tact of the two bodies which impinge, are equal.
When the bodies are elastic, Poisson divides the impact into two
periods. The first begins at the first contact of the bodies and termi-
nates at the moment of gTeatest compression. The second begins at the
moment of greatest compression and tei-minates when the bodies sepai-ate.
The motion at the end of the fii'st period is found exactly as if the bodies
were inelastic. The motion at the end of the second period is found
from the principle that the whole momentiim communica,ted by one body
to the other during the second period, bears a constant ratio to that com-
municated during the first period of the impact. This ratio depends on
the elasticity of the two bodies and can be found only by experiments
made on some bodies of the same material in some simple cases of
impact.
"When the bodies are rough and slide on each other during the impact,
Poisson remarks that there will also be a frictional impulse. This is to
be found from the principle that the magnitude of the friction at each
instant must bear a constant ratio to the normal pressui^e and the direc-
tion must be opposite to that of the relative motion of the points in
contact. He applies this to the case of a sphere, either inelastic or
perfectly elastic, impinging on a rough plane, the sphere turning before
the impact about a horizontal axis perpendicular to the direction of
motion of the centre of gravity. He points out that there are several
cases to be considered; (1) when the sliding is the same in direction
during the whole of the impact and does not vanish, (2) when the sliding
vanishes during the impact and remains zero, (3) when the sliding
vanishes and changes sign. This third case, however, contains an un-
known quantity and his formulae therefore fail to determine the motion.
Poisson points out that the problem would be veiy complicated if the
sphere had an initial rotation about an axis not perpendicular to the
vertical plane in which the centre of gravity moves. This case he does
not attempt to solve, but passes on to discuss at greater length the im-
pact of smooth bodies.
M. Coriolis in his Jeu de Billard (1835) considers the imjiact of two
roiir/li, spheres sliding on each other during the whoh* of the impact. Ho.
obtains the i-e^ult given in Art. 312, Ex. 3.
560 NOTES.
M. Ed, Phillips in tlie fourteenth volume of Liouville^s Journal, 1849,
considers the problem of the impact of two rough inelastic bodies of any
form when the direction of the friction is not necessarily the same
throvighout the impact, provided the sliding does not vanish during the
impact. He divides the period of impact into elementary portions and
applies Poisson's rule for the magnitude and directien of the friction to
each elementary period. He points out how the solution of the equa-
tions may be effected, and in particular he discusses the case in which the
two bodies have their principal axes at the point of contact parallel each
to each and also each body has its centre of gravity on the common
normal at the point of contact. He deduces from this the two results,
given in Ai-t. 312, Ex. 4 and 5.
M. Phillips does not examine in detail the impact of elastic bodies,
though he remarks that the period of impact must be divided into two
portions which must be considered separately. These however, he con-
siders, do not present any further peculiarities.
The case in which the sliding vanishes and the friction becomes
discontinuous, does not appear to have been examined by him.
Sir W. R. Hamilton s Equations.
Art. 378. The demonstration as given by Sir W. R. Hamilton
requires that I' should be a homogeneous quadratic function of the
accented letters and this is generally the case in dynamics. The exten-
sion to the case in which the geometrical equations do not contain the
time explicitly is due to Prof Donkin. Prof. Donkin has made a
further extension of this theorem which is sometimes useful. If T^ be
a function of any other letter, say $, as well as 0, ^, &c., then we shall
dT dT
have —rj'^ j^ , the differentiation with respect to ^ being in each
db. dt,
case performed only so far as ^ appears explicitly. The theorem may be
demonstrated as in the note to page 374.
On the Principle of Least Action.
The argument in Art. 394 shows that 8 j Tdt = under certain
conditions. According to the usual phraseology it is asserted that
/ Tdt is either a maximum or a minimum. But this is not strictly
correct. It seems clear that since the Vis F^va cannot be negative, there
must be some mode of motion from one given position to another, for
which the action is the least possible. When, therefore, the equations
supplied by the Calculus of Variations lead to but one possible motion,
that motion must make / Tdt a minimum. But when there are several
NOTES. 561
possible modes of motion, tlioiigli none can Le a maximum for the
reason given in the text, some of these may be neither maxima nor
minima.
To determine whether the integral is a maximum or a minimum or
neither, we must examine the terms of the second order in the variation
of the integral to ascertain if their sum keeps one sign or not for all
variations of the independent variables. This is a very troublesome
process, and we do not propose to discuss it. It will be sufficient to call
the reader's attention to some remarks of Jacobi, given in the seven-
teenth volume of Grelle's Journal, 1837, and translated in Mr Tod-
hunter's History of the Calculus of Variations, page 250.
Suppose a dynamical system to start from any given position which
we shall call A, and to arrive at some position B. If the time be
given, the motion is found by making 8 j Lclf — ; if the energy be given,
by making S / Tdt = 0. The constants which occur in integrating the
differential equations supplied by the Calculus of Variations are to be
determined by means of the given limiting values ; but as this involves
the solution of equations there will in general be several systems of
values for the arbitrary constants, so that several possible modes of
motion from A to B may be found which satisfy the same differential
equation and the same limiting conditions. Now let one of these modes
of motion be chosen, and let the position B approach A, so as to be
always on this chosen mode of motion. Suppose that when B reaches
the position C another possible mode of motion from A to B is indefi-
nitely near to the chosen motion. Then C determines the boundary up
to which or beyond which the integration must not extend if the inte-
gral is to be a minimum.
The reason seems to be as follows. If U be equal to the integral
under consideration, we have along each of the motions from A to B
811=0. Hence when B coincides with C, we have both 8U=0 and
8 (U+SU) = 0. It easily follows that the terms of the second order can
be made to vanish by a proper variation. When the limits of integra-
tion are more extended than AC, it is not difficult to show that the
terms of the second order can be made not merely to vanish, but to
change sign.
Jacobi illustrates his rule by considering the principle of least action •
in the elliptic motion of a planet. Let /S be the sun, and let the particle
start from A towards aphelion to arrive at a point B. The path is
known to be an ellipse with S for focus. Since we use the principle of
least action, the energy of the motion is given : hence the major axis of
the ellij)se is known, let this be 2a. The other focus H of the ellipse is
the intersection of two circles described with centres A and B and radii
2a -SA, 2a — SB respectively. The two intersections giA^e two solutions
which only coincide when the circles touch, that is when the line AB
passes through the focus //. Thus if we draw a chord AC through II
to cut the ellipse described by the particle in C, then the terminal posi-
tion B must fall between A and C if the integral which occurs in the
principle of least action is really to be a minimum for this ellipse. If ^
R. D. 3G
562
NOTES.
coincide with C, then the second variation cannot become negative, but
it can become zero, so that the variation of the integral is then of the
third order, and may thei-efore be either positive or negative. If B be
beyond G the second vaiiation itself can become negative.
If the particle start from A towards perihelion, then the extreme
point G is determined by drawing a chord AC through the focus S to
cut the ellipse in G. For if A and G are the limits we can obtain an
infinite number of solutions by the revolution of the ellipse round AG.
If then in the last case the second limit B fall beyond G there wUl be a
curve of double curvature between the two given points for which
/
Tdt is less than it is for the ellipse.
On Spheiv- Conies.
The following properties of a sphero-conic will be found useful in
connexion with the theorems of Art. 527. They appear to be new. The
curve is represented by the line DED'E'. As in the text, the eye is
supposed to be situated in the radius throiigh A, viewing the sphere
from a considerable distance. The three principal planes of the cone
intersect the sphere in the three quadrants AB, BG, GA, and any one of
the three points A, B, G might be called the centre. The arcs AD and
AU are represented by a and b.
I. Equation to the conic. Draw the arc PjN' perpendicular to AD
and let BN-y, AN-x. Let BP produced cut the small circle de-
scribed on DD' as diameter in P', let NP' be called the eccentric
ordinate and be represented by y' . We then have
tan y . , tan 5
—. = constant =
tan y tan a
cos a = cos y' cos x
NOTES. 563
2. The projection of the normal PG on the focal radius vector SP,
i. e. PL, is constant and equal to half the latus rectum.
tan^ b
If 11 be the latus rectum, then tan I = .
tana
. , tan GL
Also -: — Kirp = constant.
sm PN
3. If QAF be an arc cutting PG at right angles, QA may be called
the semi-conjugate of A P. Then
tanPG^.tanP^ = tan'6.
4. The length PK cut off the focal radius vector by the conjugate
diameter is constant and equal to a. This follows from (2) and (3).
5. If 1 — e"= -r-^; — , e may be called the eccentricity of the sphe ro-
sin^ «' ^ J r
conic. Then
t3ia AG =e'i?in AN.
6 . Also S being a focus
tan {SP -a) = e tan AN.
7. Polar equation to the conic
tanZ ^ e %.
;= 1 ^ cos PSA.
tan SP cos'
8 . If p be the radius of curvature at P, then
tan^ n
tan p = - — 5-T •
tan" t
9. Regarding AP, AQ as conjugate semi-diameters,
sin^ AP + sin" ^^ - sin^ a + sin^S )
sin AQ .sin PF- sin a . sin 6 j '
10. If ^; be the perpendicular from the centr-e A on the tangent
atP,
tan^ a tan^ h
tsin^p
= tan^ a + tan' b - tan^ AP.
1 1 . Also tan'' PG - tan" I = -^ sin' PiV^.
cos 6
12. sin" a -sin- -4 P) e" . 2 n^ir
' :sm PiV.
= sin'-4^-sin"&j l-e
Cor. tan' PG^ — ff - 2 (cos' AP- cos," a cos' b).
cos sin a ^
564 NOTES.
If sin AM = sin Alf = - — , tlie planes of the arcs BM and Bid' are
sma
parallel to the circtilar sections of the cone. Some of the properties of these
arcs resemble those of asymptotes when B is regarded as the centre of
the conic. The properties which connect the sphero-conic with the arcs
BM and BM' will be found in Dr Salmon's Solid Geometry.
Many other properties of sphero-conics will also be found in Mr Frost's
Solid Geometry.
Miscellaneous Notes.
Art. 3. The term moment of inertia with regard to a plane seems to
have been first used by M. Binet in the Journal Poly technique, 1813.
Arts. 19 and 182. So much has been written on the ellipsoids of
inertia and on the kinematics of a solid body that it is difficult to
determine what is due to each of the various authors. The reader will
find much information on this point in Prof Cayley's report to the
British Association on the Special Problems of Dynamics, 1862.
CAMBEIDGB : PRINTED BY C. J. CLAY, M.A. AT THE UNIVEESITY PRESS.
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