fliSlwiW'j 3tl>ata. New gnrk ROLLIN ARTHUR HARRIS MATHEMATICAL LIBRARY THE GIFT OF EMFLY DOTY HARRIS 1919 arW3906 Of motion Cornell University Library ,. 3 1924 031 364 429 olin.anx Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031364429 OF MOTION, AN ELEMENTABY TREATISE BY JOHN ROBEET LUNIST, M.A. FELLOW AMD LADY SADLEIR'S LECTURER OF S. JOHN'S COT^LEG-B. CAMBEIDGE: DEIGHTGJSr, BELL, AND CO. LONDON: BELL AND DALDY. (iTantlirtlrge : PRIK.TED BT C J. CLAY, M.A. AT THE. UNIVEESITY PItESS, PREFACE. II TY object in the following pages has been to put forth the -^ -*- principles of the Science of Motion in their true geome- trical form, postponing the consideration of force (the pro- perties of which are presumed to have been fully investi- gated in Statics) until the reader may be able to separate in his mind the geometrical ideas from the mechanical. To the fact that these ideas are not kept separate at the outset I apprehend that the want of clearness in the stu- dent's mind about the real investigation that does take place in any case may be attributed. Until a comparatively recent period all works on this subject have been concerned with answering the question, Given the force acting on any body, how will it move ? But how a motion is to be estimated, which of course is a pre- liminary question that should be ftdly investigated, on this point very little has been said. The first book, I think, in which the geometry of Motion was formally treated of, separate from the cause, wa^ Griffin's Dynamics of a Rigid Body; this of course could not be referred to tiU the elements of the subject had been mastered!. The same method of treatment was adopted in Sandeman's excel- lent treatise Of the Motion of a Single Particle; a work to which I am greatly indebted, as the reader wiU easUy see, the 6th chapter being very little else than a transcript VJ PREFACE. An Appendix is added, containing certain geometrical properties of the cycloid, which it was necessary to as- sume ; and a number of problems, selected principally from recent Examination Papers in the Senate-House and S. John's College. I have to thank those friends who have in many ways as- sisted mfe in this work, especially Professor Sandeman, MA. of Queens' College, who permitted me to make unlimited use of his valuable treatise mentioned above, Mr W. H. Besant, M.A. of S. John's College, Rev. N. M. Ferrers, M.A. of Caius College, and Mr E. Wilson, M.A, of Trinity College. CONTENTS. CHAPTER I. General Principles. — Of Velocity and Acceleration. ART. JPA8B 1. Definition of motion r 2. Velocity ib. 4. Determination of the unit of Telocity, and of the measure of any velocity ............ 2 6, Transformation of the measure of a velocity to other units of space and time ............ ib. 1, The signs + and — indicating the direction of a velocity • • • 3 8. The parallelogram of velocities ....... ib. 9. The parallelepiped of velocities ........ 5 10. Resolution of a velocity ib. 11. Composition of velocities .... .... 6 12. Eeaolved part of a given velocity in any given direction ... 7 13. Measure of a variable velocity 8 14. Kelation between space, time, and variable velocity .... ib. 17. The parallelogram of velocities, &c. is true for variable velocities . . 9 18. Acceleration and retardation ........ ib. 20. Determination of the unit of acceleration and of the measure of an acceleration 10 21. The sign — to an acceleration indicates retardation . . . iJ. 22. Variable acceleration ib. 23. Eelation between' velocity, time, and variable acceleration . . 11 26. Parallelogram of accelerations ib. 27. Transformation of the measure of an acceleration to other units of space and time 12 VllJ CONTENTS. CHAPTER II. Of the Motion op a Point in general. Analytical Expres- sions FOR Velocities and Accelerations in certain direc- tions. AUT. PAGE 29. Method to be pursued in order to determine any motion . . .14. 30. Kelations between acceleration, space, time and velocity . . . ib. 32. Expressions for -velocities and accelerations in direction of the axes of X, y, and z 15 33. Radial and transversal velocities ....... ib. 34. Badial and transversal accelerations 16 35- Tangential and normal velocities and accelerations . . . . 17 37. Dimensions of the measure of a velocity or acceleration . . . .19 CHAPTEE III. Op the Motion of a Point affected by a constant Accelera- tion, THE direction OF WHICH IS ALWAYS THE SAME. 38, Rectilinear motion ...... ... 20 39- «=-2 »*• 40. 1^=^as . . . . . . , . . . . .21 41. i=v't+— ; i^—v'^=2as ih. it 42. Parabolic motion .......... ih. 43. Position of the focus . .... . . 23 44. Position of the vertex jj. 45. Magnitude of the latus rectum 44 47. Velocity at any point jj. 48. Investigation of the above by means of the differential calculus . . 25 CHAPTER IV. Of the Motion of a' Point affected by an Acceleration, the direction of which always passes through a fixed point. 50. The two kinds of this motion 27 51. (I.) Rectilinear motion, acceleration « distance . . . . i6. 54. Time of oscillation 20 57. Acceleration 00 (distance) "' ji CONTENTS. IX AKT. PAGE 58. Time of oscillation in this case ....... 32 60. General case of rectilinear motion ib. 61. (II.) Curvilinear motion. Three pairs of equations to determine it . ib. 62. Equable description of areas 33 63. General properties of the velocity ... . . 34 66. Space due to velocity =J-x chord of curvature . . . -35 67. Differential equation to the path jS+"=iF> . . . ih. 68. Acceleration 00 distance : determination of the path . . -36 6^. Determination of the constants from the initial circumstances . 37 7 1. Determination of the constants in terms of the axes of the path . 38 72. Velocity =nj^ (semi-conjugate diameter) .... 39 74. Determination of the axes from the initial circumstances . . 40 75. Period ,= -j= ib. 76. Acceleration 00 (distance) ~' : determination of the path . . ib. 79. Determination of the constants fi-om the initial circumstances . 41 80. Determination of the axes ........ 42 9' =y?^ ^ 81. Velocity = V **• r a 83. Determination of the eccentricity ..... 43 27r ft 84. Period = -= a 44 86. Law of acceleration for a given orbit ib. 88. Time of motion 47 89. General case of motion ........ ib. 90. Normal acceleration measuring the tendency to proceed in a straight line ib. CHAPTER V. Op Matter and Fokce. New class of ideas, formed upon experiment 49 92. Distinction between material and immaterial things, by the possession of impenetrability .......... 5° 93. Idea of force : necessarily connected with matter .... 46. 94. The idea of motion a geometrical one ib. 95. Generalization of the idea of force: method on which a force must be estimated " • • • S' X CONTENTS. ART. PAGE 96. The science of force ia statics. Fundamental law of force ; by means of which statics is reduced into geometry . . . . . . S^ 97. Expei-imental laws necessary to reduce dynamics into the geometrical science of motion ib. 98. Axiom to detennine the motion of a particle . • ■ ■ 53 99. Object of the experimental laws of force, and manner of arriving at them ib. CHAPTEE VI. Of the Dynamical Laws of Force, commonly called the Laws of Motion. 100. Necessity of three laws of motion . . .... 55 loi. First law of motion ......... ib. 102. Experiments to prove it 56 (i) If a particle be at rest it will remain at rest. (2) If it be in motion, its path is a straight line. (3) In this case, its velocity is constant. 103. Special dynamical properties of force 57 104. Second law of motion H. 105. Experiments to prove it 58 (i) A constant force produces a constant acceleration in its own direction. (2) This is not altered by any initial motion in that direction. (3) Nor by any initial motion in any other direction. (4) The acceleration produced varies as the intensity of the force. (5) It is independent of any other force acting in the same direction. (6) And of any force acting in any other direction. (7) AU this is true for variable forces. 106. Mass 60 107. F—Ma.: determination of the unit of mass 5i 108. Mass 00 weight 62 109. Pefinition of the motional effect of a force, momentum, and vis viva . ih. no. Third law of motion ib. III. Experimental proof 63 J 12. Atwood's machine 64 CONTENTS. XJ CHAPTEE YII. Op cjERTAEsr CASES OP Free Motion in Nature. ART. PAGE 113. Falling bodies 65 116. Projectiles 66 118. Rectilinear motion about a centre of force 00 distance, or as (dis- tance)"^ , 67 121. Curvilinear motion about a centre of force oc distance, or as (dis- tance)"" ........... 69 125. Kepler's laws .......... i6. is6. Conclusions from these, respecting the forces causing planetary motions 70 CHAPTEE VIII. Op Constrained Motion op Particles, 130. Artifice employed to determine the motion ...... 72 131. Motion down an inclined plane ....... ij. 132. Change of velocity due only to the vertical space . . . .73 133. This same property is true for a curve ....... 74 134. Times of descent down chords of a circle through the highest or lowest point 7£ 135. Two cases of motion on inclined planes of bodies connected with strings ih. 137. General case of motion on a smooth plane curve . . . . 77 140. Motion on a cycloid ........ -79 142. Oscillations in a cycloid isochronous ...... So 143. Time of oscillation of a simple pendulum. Determination of the value oi g ............ ib. 144. Motion in a circle . . ....... 81 149. Rectilinear motion of a particle hanging by an elastic string . . 84 150. Motion of a particle suspended to a point, and revolving about a, vertica,l line through that point ... . . 85 CHAPTEE IX. Op Impulses and Collision op Particles. 151. Measure of an impulse 87 154. Elasticity 88 155. Collision of two inelastic balls ib. XIJ CONTENTS. AET. PAGE 156. No momentum lost by the impact 89 157. Collision of two elastic balls ib. 158. i^—i'=e(F—F'): this establishes the truth of the hypothesis i?' = eJ? . 90 159. No momentum lost 91 160. Motion of the centre of gravity of the two unaffected by the impact . t6. 161. Method of treating oblique impact 92 162. No vis viva lost by the impact when the balls are perfectly elastic . ih. 163. When they are imperfectly elastic, via viva is lost .... iJ. 164. Oblique impact of an elastic ball against a plane 93 APPENDIX. Of the Cycloid. , I . Definition of a cycloid p j 1. Tangent to a cycloid at any point H' 3. Length of arc, measured from vertex 96 4. Mode of describing a cycloid by a string ib, 5. Equations to the cycloid py 6. Involute to a cycloid ...... v . ■ 99 Peoblems loi OF MOTION. CHAPTER I. GENEEAL PEINCIPLES. — OP VELOCITY AND ACCELERATION. 1. If a point change its position in space it is said to move. 2. All motion has reference to space and time, and since a point may, under different circumstances, pass over different spaces in equal intervals of time, or require different intervals of time to pass over equal spaces, the mind necessarily conceives the idea of quickness or slowness of motion. The degree of this quickness or slowness is called velocity. 3. If the moving point pass over equal spaces in equal suc- gessive intervals of time, its velocity, is said to be uniform. It IS evident that the velocity of a moving point will be greater or less in exact proportion as the space it passes over in any given time is greater or less, or as the time required for the point to pass over any given space is less or greater ; so that the mea- sure of the velocity varies as the space passed over when the time is constant, and inversely as the time when the space is constant, i.e. if v be the measure of the velocity with which a s moving point describes a space s in a time t, v x-. L. B MEASURE OF VELOCITY. 4, If, with the unit of velocity, a space cr be described in time T, we shall have 1 * <^ T S « = -.-. As v"St ; .•. gr < v , and > v , ACCELERATION AND RETARDATION. 9 and making Ss, St indefinitely small, as v, v" have both the same limit v, we get ds 16. This result includes the case of a constant velocity, iause in that case - h s and t indefinitely small, because in that case - being constant is not altered by taking or - = Jjimit - , or = -^ . t «' dt 17. The propositions called the parallelogram and paral- lelepiped of velocities are equally true for varying velocities, as may be clearly seen by considering an indefinitely small parallelogram or parallelepiped similar to those in the figures of Arts. 8, 9, or by comparing the motion of the proposed point with that of another moving with constant velocities equal to those in question. 18. If the velocity of a moving point be continually in- creased, its motion is said to be accelerated ; and if diminished, retarded. The terms acceleration and retardation* are used to express the degree of this change of velocity. The acceleration or retardation is uniform when the velocity is uniformly increased or diminished, i. e. when equal velocities are acquired in equal successive intervals of time, in or opposite to the direction in which the motion is estimated. 19. An uniform acceleration is clearly greater or less in exact proportion as the velocity acquired by the moving point in a given time is greater or less, or as the time requisite for the moving point to acquire a given velogity is less or greater.^ Therefore if a be the measure of an acceleration, owing to which the moving point acquires a velocity v in time i, a oc - , • N.B. We are here ooncerned with acceleration and retardation merely as matters of fact, and not considered as resulting from any particular cause whatever. L. 10 MEASUEE OP ACCELERATION. and if with the unit of acceleration a velocity v^ is acquired in time tx'Bt ; Bv , \ „ ■'■ Yf'^'"'' ^ ^ °^ ' and as a.', a" have the same limit a when Bt is indefinitely diminished, we get dv 25. This result includes the case of a constant acceleration, for then - being constant, will = Limit 7 or = -^ . 26. The Parallelogram of Accelerations. " If two adjacent sides of a parallelogram represent in magnitude and direction two accelerations by which the motion of a point is simultaneously affected, the resulting acceleration will be represented both in magnitude and direction by the 12 CHANGE OP UNITS IN THE diagonal of the parallelogram passing through the point of intersection of those sides." Let AB, AC represent the two accelerations. Then AB, AC measure the velocities which would be acquired in an unit of time. And if ^^, J F measure J" the velocities which actually are acquired in a very small time, AE, AF are proportional to AB, A C, the more nearly as the time in which they are acquired is indefinitely diminished. The resulting velocity will be measured by AG. The resulting acceleration then must be in the direction of AG and be to AB or J. O in the same ratio rs AG is to AE or AF respectively; i.e. it will be represented in direction and magnitude by the diagonal AB. This may be extended to space of three dimensions as was done for velocities in Art. 9, and all the analytical formulae and remarks of Arts. 10, 11, 12, will equally hold good for accelera- tions. 27. As we have shewn (Art. 6) how to transform the mea- sm'e of a velocity from one set of units of space and time to another, we must now shew how to transform the measure of an acceleration. Let the new units of space and time be a and h times re- spectively as great as the original units. Then a, being the measure of the acceleration, is equal to the measure of the velocity acquired in an unit of time; therefore the measure of this velocity, when referred to the new unit of time, will be ba. (See Art. 6.) But this velocity is not acquired in the new unit of time, but only in the 6"" part of it; therefore the velocity acquired in the new unit of time is b times as great as this, i. e. it = b'a. MEASURE OF ACCELEEATION. 13 This velocity has now to be referred to the new unit of space, and therefore (see Art. 6) its measure is — . ^ ' a This, then, is the measure of the acceleration in terms of the new units. E. g. If the acceleration on the motion of a point be mea- sured by 20 when a foot and a second are the units, what will be its measure when a yard and a minute are the units ? The velocity acquired in 1" is 20 feet per 1", i.e. 20 X 60 feet per 1'; therefore the velocity acquired in 1' is 60 x (20 x 60) feet per 1', 20.60.60 , = g yards per 1 ; therefore the measure of the acceleration is 24,000. We have purposely chosen the same numbers as those in the transformation of the velocity given in Art. 6, in order that the distinction may be clearly seen. No errour can well arise if it be borne in mind constantly, that the measure of an acceleration is the velocity acquired in an unit of time, estimated jgm that unit of time. 28. All that has been said respecting constant accelerations will equally apply to varying accelerations, by means of consi- derations analogous to those employed in Art. 17, and by making the necessary changes in phraseology. CHAPTEE II. OF THE MOTION OP A POINT IN GENERAL. ANALYTICAL EXPEESSIONS FOE VELOCITIES AND ACCELERATIONS IN CERTAIN DIRECTIONS. 29. The motion of a point in space will be completely determined, if we know the law to which its velocity is subject throughout the motion ; and this is usually discovered by know- ing the law of the acceleration by which it is affected. Since the propositions called the parallelogram and paralle- lepiped of velocities (which equally hold good for accelerations) shew that we may consider that part of the motion which results from any one irrespectively of the others, it will be convenient to resolve the velocity and acceleration into directions at right angles to each other, and to consider them separately. A. 30. We shall now find certain relations between space, time, velocity, and acceleration, here denoted" by the letters s, t, V, a. US We have «' = ^ (Art 15), « = |(Art.24); ■'- "^ ~ ;;p ' if i be considered independent variable; ds dv dv .p , . , T.I 1 , . ■, 1 or " = ^s "3" = ^ ;r J u s be considered independent variable. The motion has here been supposed rectilinear : if it be cur- vilinear we shall still have v = -5- , as may be seen by comparing the motion of a. point along a curve with that of one along the tangent, and bearing in mind that ,the corresponding elements of the arc and tangent are coincident. The other equations EADIAL AND TEANSVEESAL VELOCITIES. 15 cannot be assumed to be true, because the change of the velocity is not entirely along the tangent, A. 31. If, in the enunciation of Newton's 10th Lemma, we read "point" instead of "body," and "acceleration" instead of "force," the reasoning still holds good, and we have a = 2 limit -j . If t be considered independent variable, this vanishing fraction, being evaluated in the usual way, becomes, after two cPs differentiations, -jp, as before obtained. A. 32, If X, y, z be the co-ordinates of the moving point at time t, the cosines of the angles which the direction of the ds velocity -^ makes with the rectangular axes of x, y, z, are dx dy dz _ ds'' ds^ ds'' and therefore (Art. 10) the resolved parts of the velocity in the directions- of the axes will be dx dy ds _ It' ~di' dt' and therefore the accelerations in the same directions will be ^ ^ ^ (Art 30^ df df df (^^t-3«o A. 33. It is sometimes convenient to consider the position of the miOving point as determined by polar co-ordinates. OP=r, POx^e, whence a? = »• cos ^, y = r sin Q, dx „dr • ndd -5- = cos y -5 — r si.nd -^, dt dt dt dy •/!«?»•, ndO dt at dt and the velocity in direction of OP dx a , ^y • a ^''' = di'''^^dt''''^=dt' 16 RADIAL AND TEANSVEESAL ACCELEEATIONS, and the velocity in direction FT at right angles to OP, and in direction of the increase of 6, dx . n . dy n ^^ = — T- sm ^ + -^ coa6 = r -J- . at at at A. 34. The accelerations may also be obtained in the above directions: for d'x „d'r ^ . „drd6 . „d'e .fdd^' and therefore the acceleration along OP d^x „ d^y . a ~df *'\dt) ' and the acceleration along PT d^x . n , d^y a = __sm0+^cos5, _^±d6 ^ dt dt^"" df 1 d I ,d6\ r dt\ dt) ' * The difference in form between these expressions and those for the accelerations in SB and y is owing to the fact that the directions of OP, PT axe Tariable, whereas those of x and y are fixed. That -^ cannot represent the acceleration in the direction of OP may be con- cluded from the consideration of the simple case of motion where r is constant, and d'r .'. -r-^ = : bnt this motion is circular, and there must be an acceleration along the d'r radios, otherwise the point would move in a straight line ; therefore when -ry = air d'r there is yet an acceleration existmg towards the pole, whence •^-= does not express dt* d'x d'y the acceleration in OP as -^ and ^ do those in the directions of x and y. A direct demonstration of these formulie may be found in Sandcman's treatise Of the Motion of a Single Particle. TANGENTIAL AND NORMAL ACCELERATIONS. 17 It must be borne in mind that the positive direction along the radius vector is measured away from the pole, and the posi- tive direction at right angles to this is measured in the same direction as that in which d increases. A. 35. It may be also advantageous to consider the velocity or acceleration as resolved along the tangent and normal. ^ ^/- ^ ^ <^ /(b 7£^ Then we have eosP?I« = ^, . as sin PTx = ^ , as and the velocity in direction of the tangent is -r , and that in direction of the normal is 0. Also the acceleration along the tangent _ dx cPx dy d'y ~"d? 'df'^ds W ?x dy d^y\ ydt df'^dt'^j _ 1 /dx d'x dy d'y\ ~^' V^ df'^dt 'd^l dt „ds' dt\dt\'^ dt )' dt dh ' df' dx\ dt/ L. 18 TANGENTIAL AND NORMAL ACCELEEATIONS. and the acceleration along the normal _ S?x dy d'y dx ~dfds~de"ds' " Is \df dt df dt) " \^t) ' p" p' dt if p = the radius of curvature*. The positive directions in this case are measured in the di- rection of increase of s,' and in that which is considered the posi- tive direction of curvature. In the last two sets of resolutions, we have for simplicity considered the motion as taking place in one plane. A. 36. If we know the conditions to which the motion is subject, we can equate the expressions (whether for velocity or acceleration) obtained in the preceding Articles to certain given quantities, and thus obtain sets of differential equations, the solution of which will determine the motion. As the expressions of the preceding Articles have all been derived from' those in * These formulse may be obtained ver; elegantly in the following manner : let the angle Pr* = 0, and let » be the Telocity at P, l which = ;^ ) • Then the acceleration in a? = "X (" *°^ <^) = cos <^ -=- - » sin -^ , ut ot at and the acceleration in y —Zii^^ ''" <#") = sin ^ "3; + " ""^ ~r > dt ctt at .-. the acceleration in TP = -t-j cos + -^ sin iJ> = — = — -^ , cPx (Py and the acceleration along the normal = -^ sin - -~ cos 0, (tZ Ctt _ d d4> ds """ 'dt~~^ ds'lt' 2 . * ~p ' for as (/) does not appear except in a differential coefficient, we may increase it by some constant angle, so as to pass to the angle required in the intrinsic equation to the curve: the negative sign has been rejected by considering the direction of the curvature. DIMENSIONS OP THE MEASURE OP VELOCITY, ETC. 19 Axt. 32, these sets of equations are equivalent to one another, and the only advantage of one particular set over another is, that the solution is effected with greater ease. In a few cases, the circumstances of the motion can be determined without the aid of the differential calculus ; and we shall now proceed to deter- mine certain particular cases of motion, when possible, without such assistance. 37. Eeferring to Arts. 4, 14, in which we have u=- or t limit - , we conclude that the measure of a velocity is of 1 di- t mension in (linear) space, and of — 1 in time. Also by Arts. 19, 23, in which a = t or limit - , the measure of an acceleration is 6 t of 1 dimension in velocity, and of — 1 in time; and therefore is, on the whole, of 1 dimension in space, and of — 2 in time. The measures of space and time of course are of no dimensions in time and space respectively. This evidently accounts for the fact that S" appears in the transformation of Art. 27, while b only appears in that of Art. 6, and for a appearing equally in both. The reader would do well to apply these considerations to the ana- lytical expressions in the subsequent Articles, by which many results, that at first sight might appear surprising, will be found to be perfectly consistent: and if they are carefully borne in mind, they will be very useful in preventing errour "in any inves- tigation. CHAPTEE III. OF THE MOTION. OF A POINT AFFECTED BY A CONSTANT ACCELERATION, THE DIRECTION OF WHICH IS ALWAYS THE SAME. 38. This is evidently the simplest kind of motion affected by acceleration conceivable, and it will subdivide itself into two heads, according as the direction of the acceleration is or is not coincident with the initial direction of motion. I. First, then, let the direction of the acceleration be coin- cident with the direction of motion : here it is plain that the path will be a straight line. 39, If the point move from rest and pass over a space s in time t, the measure of the acceleration being a, then will For if we divide the time « into n equal intervals t, the velo- cities at the end of these intervals- will be a.T, a.2T, a.3T, a.WT., (Art. 21.) Suppose now the point to move for each interval t with the velocity it has at the end of that interval : then the whole space passed over would be aT. T + a. 2t.t + a. 3r .T+ + a.wT.T = aT* (1+2 + 3 + +n) 2 2 V «/ KECTILINEAE MOTION. 21 Again, suppose the point to move for each interval t with the velocity it has at the 'beginning of that interval ; then the whole space passed over would be O.T + aT.T + a.2T.T + +a. (m— 1)t.t = a7^ {0+1 +2 + 3 +... + (»»- 1)} -^•^^=^(-3- But it is manifest that the space reaZZy passed over must lie between these two magnitudes ; and since when n is indefinitely increased, they both become — , we must have s = — . 40. The velocity at the end of this time t will be ai (see Art. 21). Also we shall have by eliminating «, v^ = 2as. 41. Next, suppose the point t be initially moving with a velocity v'. Then as the mode of reasoning used in the " parallelogram of velocities" will apply here, we shall have final velocity = initial velocity + that due to the acceleration, i. e. « = u' + ai. Also the space passed over = space due to the initial velocity + that due to the acceleration, , ai I.e. s = v< + — . And we shall also have «'' = w''+2ato' + aY; If the motion be retarded, the sign of a is changed. By means of these equations all the circumstances of the motion are determined. 42. II. The second case will be when the direction of the constant acceleration is not coincident with the initial direction of motion. 22 PAEABOLIC MOTION. Here it is evident that the motion will take place altogether in one plane, viz. that which passes through the initial direction of motion, and that in which the acceleration takes place : let this he the plane of the paper, and let A he the. initial position of the moving point, -4r the direction of the motion at A, yA the direction of the constant acceleration a, v' the initial velocity which is in direction oi AT. Also let TA make with Ax at right angles to Ay the angle TAx=i. Let P be the position of the moving point at any time t reckoned from the beginning of the; motion. Draw TP parallel to yA, Then we shall have TA = v't. TP— space due to the acceleration DETERMINATION OF THE FOCUS AND VEETEX. 23 Draw PJf parallel to TA meeting yA produced in M\ where- fore TAMP is a parallelogram, or PM^=— .^Jf. a TLil is evidently a tangent to the path at A^ therefore the ahove equation shews that the path is a parabola whose axis is parallel to yA the direction of the acceleration, the concavity being turned in that direction, and the distance of A from the . 1 l^v'\ v"' rocus =1 C^ ■^'\ a. . 2a 43. The position of the focus 8 is determined as follows ; 8AB=^ 90" - 8Ay = 90° - 2. TAy = 90° -2 (90°-t)=2t-90°; therefore if G8B be the axis, AB = 8 A cos 8AB = '"— sin 2t 2a B8= 8A sin 8AB = - 1^ cos 2i 2a As we have drawn the figure cos 2i is negative, ■.• 2t is > 90°. 44. If V be the vertex, the direction of motion there is parallel to Ax, and therefore the velocity in direction of Ay = 0. Therefore if i be the time which has elapsed when the moving point reaches V. we must have = v sin 1 — a.t'; , _ v' sin I ■'• *-~^' and ^Fwill = BC- CV^AOmxi,- OV ,,, . at" = vt. sm f — — = 2 a2 L = -5t7= = I ;^ sm 2t ) -. = ^- cos I. 24 DETERMINATION OF THE LATTJS EECTUM. This gives the distance of the vertex V from the line Ax, i. e. shews the greatest positive distance of the moving point from 45. If L he the latus rectum of the parabola AR^L.BV. -— sm 2t -i = — ( V2a / 2a a This might have been obtained in the following manner : If iSF be perpendicular to AC, SV_8Y , _ /W SY- SA' ^"* ••-y 8A- SY SV But ■g^ = sin(90°-t)=cosi; .-. ^ = <^os\. .•. S'F= — cos^i, or L = cos"* (see Art. 42). 2a a 46. The distance from A of the other point where the para- bola cuts 4^ will = 2AB= — sin2t, and the time which elapses till the moving point takes that position will be «'^sin2t , 2v' . 1- V cos I = — sm t. a a This value might have been obtained by considering that v' sin I , , the time in question = 2 . time to vertex = 2 . , and the distance from A of the other point where the parabola cuts Ax ^•w'sint , ■«"'•„ = 2 X V cos t = — sin 2t. a a The directrix of the parabola will be par'allel to Ax, and will cut Ay at a point E whose distance from A — AS= — . 47. The velocity at F in direction Ax will be u'cos o, and that ill 'direction Ay will = tj'sin i — at; therefore if v be this velocity, DETERMINATION OF VELOCITY AT ANY POINT. 25 v' = (u'cos if 4-(«'sin t - aif = v'^ - aaij'sin i.t + a^f = 2oi.8A-2a. {TN- TP) (Art. 42) = 2a.{AE-PN) = 10.. FF = 2a . (distance of P from directrix). Thus the motion is completely determined. A. 48. Both of the foregoing cases of motion may be inves- tigated by the help of the differential calculus, as follows : — (I) If the acceleration be in the direction of motion, we have dv du de. du 1 /^ „ X . taking the negative sign for convenience ; udu •••^ = ^-' /^^^ d.U-j ^/(?-S-(«■-?)■ = /3 + -cos- 2 C u^— — 1 „„-. 2_ V 4 A^ ELLIPTICAL MOTION ABOUT THE CENTER. 37 If we had taken the positive sign above, the only difference would have been that /S would have been changed. :u^ = ^ + a/^^,cos2{6-^) (b). 69. We must now determine G, /3, arid A. If the point be moving initially with a velocity v' in a direction inclined at an angle t to the prime radius vector r', then v'r' sin t = h, which is therefore known. Also in (a) we have the left-hand side = p , (Art. 64) ; ■ C- ^ I '^ this determines C. As ^ = initially, we have 1 G^ /G' \ ,„ which gives /S. 70. The equation (b) shews that the orbit is an ellipse whose center is at the pole, and the angle vector of its apse* is /S. If the acceleration were in the opposite direction, the sign of X would be changed, and the equation (b) would represent an hyperbola whose center is at the polef. • By an "apse" is meant any point in the curve where the radius sector is perpen- dicular to the tangent. In the ellipse, for example, when the origin is the center, the extremities of the axes are apses ; if the focus were the origin, the extremities of the major axis only are apses. + This case of motion may be easily solved in the following manner. We have the acceleration in of = - Xr cos = - Xx, and that in y = - X.r sin 6 = - Xy, supposing its direction to be always towards the origin ; 38 DETEEMINATION OF THE AXES, ETC. 71. It may be convenient to obtain a yalue for C in terms of the axes of the ellipse. therefore --j-^+\x = 0, and ■^ + ^y=0- .•. as in Art. 61, j; = 4cos(;y/\. J-/3), and y = A'cos{iJ\,t—P'), where A, A', p, P', are certain constants. Between these 2 eqnations t must be eliminated. The result is clearly cos-'-j-oos-'-j;=/3'— ;8: take the cosine of each side and write Cfor cos (/3'— /3). Then ^4.yi":|Vl4^=C; ■ A^ A''"^ A^A" \ Aa) ' which is a conic section whose center is at the origin. To determine whether it is an ellipse or an hyperbola we have V AAV A^ A'^ A'A'''^ '' which is negative, •.■ C, being = cos (/3' -/3), is < 1; .-. the path is an ellipse. If the direction of the acceleration were away from the origin, then we should have — -X* = 0, and ^-Xy=0; .-. ^^Ae'^^-' + A-r'^'^-', and y=B,''^' + BT"^^'; . : B'x - A'y = (AB' - AB)e'^'^-\ and Bx-Ay = {A'B -lAB')e"^ ' ' ; .-. BB'a'' + AAY -iA'B + AB')xy = - {A'B - AB')'. And as {A'B + AB'f-iBB'.AA', which = (X'B -AB')', is positive, the path is an hyperbola. DETEEMINATION OP THE AXES, ETC. 39 If these be 2a and 2b, as «» = 4|i + i,+ (i,-l,)cos2(^-/3)}, we have from (b) 11 ^ J 1 1 fZ 4X -,+^=C, and ^-^_y G --^; \_ 1 72. Substituting this value of G in the equation (a), we obtain (velocity)'' =\ (a'' + &''-/•"), = \ (semi-conjugate diameter)^ 73. This might have been obtained without reference to (b); thus: If v^, «2) fee the velocities at the extremities of a and h, v^a = h = vj) (Art. 63); also from (a) v.^=h''C-Xa'] , ,. , v^^h'G- \&4 ' ■•■'■ = ''' ^' respectively. Then we have !!j! _<_< + %'_ 2^!^zM^+Z) . i. nen we nave -j«- — — y — 7,2 , 2 — 2 , ra > a +a a + ' also each =-\ — 12- = —^ — w'> a —b a — 40 DETERMINATION OP THE PERIOD. .•. by equating these we get -§ — p — \ = \, or h^C = 'K{a^ + y); .-. (velocity)^ = h^C—\r' = \ {a' + lf — r'), as before. 74. The axes of the ellipse are determined by the conditions X. \ \r' sin I ■ The first of these might have been got at once from the results of the last two Articles. 75. For the 'determination of the period, i.e. the time required to describe the whole ellipse, we haye - (period) = area of ellipse (Art. 62) ; . -, 2'7rab 27r .•. period =-=== = —= , ^ Jx^ ^/x which is independent of the form of the ellipse, and is therefore the same for all ellipses described under accelerations subject to the same law as above, and of the same magnitude at all equal distances. (Newton-, Prop. 10.) 76. (ij) Let the acceleration vary as the inverse square of the distance from the iixed point. Then a = Xm*; (Pu a X •■ dff'^'' h'u'~h:" w= 1 + del) '[hv'^[de + i)"'-(o), = ji + Acos{e-^), MOTION IN A CONIC SECTION ABOUT THE FOCUS. 41 or — M = 1 + — — cos {6 — ^). This is the equation to a conic section whose focus is at the pole, the angle vector of the apse being j8, the latus rectum ■—— , and A. the eccentricity — — ; and therefore it is an ellipse, parabola, or A¥ hyperbola, according as — — is< = or > 1 . 77. This might have been obtained, as in Art. 52, by putting M — p = w, whence -^ + w = ; .•. w = Acos{6 — ^). 78. We have also (S) + **' " ^' ''*'' (^ - /3) + {p + ^ cos {e - ;8)|' 2\ = j, + A'+ ■^^cos(0-/S); .-. (by Art. 64) (velocity)' = ^ + A^h^ + 2XA cos {6 - 0) 79. The constants A, /3, and h, are determined by the initial circumstances of the motion, viz. by the following equations: h' 1 .^Ah" - ^"=v-$-^''') v'r sini = h using the notation of Art. 69. Y, 42 DETERMINATION OF THE AXES. 80. If a be the semiaxis major of the conic section, and e the eccentricity, a (1 — e') = semi-latus rectum ^¥ \ AK' and e=-zr- , (Art. 76). A* V 1 1 .-. (velocity)', which = 2Xm — ^ ( p r- ) _2\_\ r a ' whence also - = -i — ■^- , a r A which is independent of the initial direction of motion ; and J= = a'(l-e')=a.^' A. = a. r 2\ \ &1, The expression «"= might have been obtained thus: multiplying the equation js + w = ^by2 ^ and integrating, we have or (velocity)" = 2X.m -K'O; DET3ERMINATI0N OP THE ECCENTRICITY. 43 .'. if i?j, «j, be the velocities at the extremities of the major axis, V, . (1 + e) .a = A = Va . (1 — e) . a, 2X a (1+ e) 2\ a (1 - e) -VG. .: ^(i+e)-A*C7(l + e)= = ^(l-e)-A'C?.(l-e)'; 2X 2e=h'C.ie, or h'G = -; \ (velocity)' = 2Xm = . ^ ■" a r a 82. In the two preceding Articles the conic section has been treated as if it were an ellipse : if e were > 1, i. e. if it were an hyperbola, we should have a (e' — 1) = semi-latus rectum ; the only efifect this has is to change the sign of a : then we shall have r a If e were = 1, i. e. if the curve were a parabola, we should have v' = - 2\ ^- --f^^K-'-^^) (^-"i- v'V 2^' ~ V x7"^ v'V sin't 2v"r' sin't + 1 = l+-'^^{v"r'-2X], 44 DETEEMINATIOK OP THE LAW OP which is = 1, according as w' =—?-; < < »• i.e. according as the space due to the initial velocity, which (see v'^ > ■ Art. 66) = — , is =/. 84. If the path be an ellipse, the period will he 2'jra5 _ 'j.irah a/V^' •.• — == A latus rectum = ^ ; .". the period = —= a*. 85. If in the preceding cases the direction of the accelera- tion had heen away from the fixed pdint, the sign of a would have been changed, and the motioQ would be determined in an exactly similar manner : if we change the sign of \ in the two preceding cases of motion, the equation (b) of Art. 68 would determine the path to be an hyperbola, in which 1 1 0,1 \ /'^~\ and the equation of Art. 76 would become •^ M = - 1 + -r— cos ip-^), which represents an hyperbola, because by Art. 83, e would then be always > 1. (Newton, Props. 11 — 15). 72 86. The equation -^ + m = , j-^ may be made use of to determine the law of variation of the acceleration towards a fixed point, in order that the moving point may trace out a given curve. ACCELERATION FOB A GIVEN OKBIT. 45 For a relation is given between u and 6, from which we obtain -^ , and then a. = AV [ -g^ + it\ gives a, which is usually required in terms of u. E.g. Let the given curve be an ellipse: the acceleration being towards the center. rrn 2 COS'^ sax' 6 Then u = — 5— + i .2 > = Ki4)4(74)««-''^ <" d-u fduY /I 1\ „/i /„\ =&+^)"^-"*-i^ (^) Also from (1) and (2), we get d'u^^,^(duV 11. = -jT3m"^j from (3); a - - - za = \.^ (distance), where \ = -^ . 46 DETERMINATION OF THE LAW, ETC. The velocity at any point will = h a/m' + (^) , (Art. 64.) = h^u^+{^,+^,-u^--l^,u'), from (3), -T, /l , l_Zl = VX (a" +&'-»•'■) = Vx . (semi-diameter conjugate to r), a3 before, (Art. 72). For an hyperbola we have bnly to write — V for V, and we get a similar result, the only difference being that the sign of \ is changed, i. e. that the direction of the acceleration is away from the fixed point. 87. Next, let the given curve be a conic section, the acce- leration being always towards the focus. Then - = 1 + e cos 5, r 1 + e cos 8 or w = , c where c = the semi-latus rectum, and e = the eccentricity; c^u _ e cos , , . fdPu . \ h , i» = X (distance) ■*, where \ = — . c And the velocity at any point = AA/w' + f^j -V(^^^7+(^ DETERMINATION OF THE TIME OP MOTION. 47 - 2e COS 6 ■ 2e cos d = a/ , as before, (Art. 80). 88. In these and other like cases of motion, the time of describing any part of the path will be found from the equation "" dt "' or Jit r«2 where ^, 6^ are the values of 6 at the beginning and end of the time of description that is under consideration, and r is deter- mined in terms of 6 from the equation d'u _ a 89. Any other cases of motion that may arise are investi- gated in a similar manner : the most general equations being d'x _ cPy _ d'z _ W^"' '^~"''' d?~'^''' where Oxaya, represent the resolved parts of the acceleration in the directions of the co-ordinate axes. The particular artifices to be used for the solution will of course depend on the forms in which flj; Oy a, appear. 90. In investigating any motion, if we use tangential and normal resolutions, so that our equations of motion are (see Art, 35) -jfi = Oi, and — = «', the former expression (which is equivalent to -t^ = «. oi' '^'T^^) ^^ ^°V concerned for the 48 NORMAL OR CENTRIFUGAL ACCELERATION. instant with changing the magnitude of the velocity without altering its direction; while the other, without causing any- instantaneous alteration in its magnitude, changes its direction. This normal acceleration may he taken to measure the tendency to proceed in a straight line. It is sometimes called centrifugal acceleration : the term is not a good one, because the direction of a' is estimated towards the center of curvature, not from it ; but whenever this, or any other equivalent term is used, let the v' reader know that — is all that is meant. CHAPTEE V. OF MATTER AND FORCE. 91. We have hitherto considered motion in itself, irrespec- tively of any cause: we shall now consider it with reference to the cases of nature, and apply the foregoing articles to the solution of such questions as may arise. It will be necessary to make some remarks on the diiferent class of ideas that will now be called up in the mind. All ideas with which we are here concerned are formed upon obser- vation or upon experiment: to the former class rigid mathe- matical reasoning can be applied, but not to the latter, at least to any extent; owing to the fact that in the former kind all adventitious circumstances attending the formation of the idea can at once be detected by the mind and separated fi.-om what is necessary to such formation, but in the latter case they cannot ; indeed, doubts may arise as to whether any such circumstance is really, adventitious or necessary. The idea of space or exten- sion, whether linear, superficial, or solid; the idea of time or succession of events, and those of number and magnitude, are obtained from observation. From experiments on thiags in the natural world we obtain the ideas of matter and force, of which we shall say more hereafter. The evidence of experiment is, so to speak, not so independent of ourselves as that of observation, and therefore the conclusions we draw from the ideas obtained from the former are not so abstractedly true as those drawn from the other kind of ideas. For example, the theorems of geometry, which merely depend on the idea of space, and the theorems relating to numbers, are so true that it is impossible to conceive any other circumstances existing under which they should cease to be true; but any conclusions obtained from experiments, which of course involve the nature of the things experimented upon, are not so : in fact, it is quite possible to h. H 50 IMPENETRABILITY OF MATTER. imagine that circumstances may exist under which they will be false. 92. We shall not attempt to define matter and force, be- cause no satisfactory definition can be given; we shall however make some remarks on the distinction between material and immaterial things, and on the connection of force with matter. If we observe objects around us, we immediately obtain the idea that they possess finite extension and a certain form: this is no more than the idea of a geometrical figure. But by eteperimentinff we find another property existing in them all; viz. that if we take two of them and endeavour to make one occupy any of the space taken up by the other, it is neces- sary first to displace this: in other words, that no two such things can occupy the same space at the same time. This pro- ■perty is called impenetrability. The possession of this property then, is that which fundamentally distinguishes a material thing (or a " body," as it is called) from a geometrical figure: for two geometrical figures can always be made to coincide, or occupy the same space at the same time, as, for instance, the two triangles in Euclid I. 4; also a geometrical figure can be made to coincide with a material body, as a shadow cast from an opaque body does with any object that may be situated in it, as far as it goes. 93. Another idea that is called up in the mind by the same experiment of displacing, in the case when the displacement is made in order that a man's hand may occupy the space taken up by any material body, is that of force or exertion required to perform this displacement. The ideas of force and matter are therefore necessarily connected, and cannot be considered in mathematical reasoning without reference to each other; in other words, the action of force follows fi:om the possession of impene- trability. These ideas cannot be obtained by any amount of observation whatever. 94. But motion was defined in Art. 1 to be change of position in space, and it was said that all motion had reference to space and time: it may further be said that the idea of FOECE. 51 motion involves no other ideas than those of space and time : it belongs therefore to the class of ideas ohtained from observation alone, which may be called geometrical ideas; and, consequently, the conclusions that have been arrived at in the preceding chapters (seeing that motion has been treated geometrically in them) are as true as any other geometrical propositions; and therefore will be more abstractedly true than the propositions in the subsequent articles, which will involve the ideas of matter and force. 95. It was stated above that the idea of force is obtained from the experiment of a body being displaced by ourselves. We can conclude, as before, that our own bodies consist of mat- ter, and as we also perceive that force is called into action merely by the bodies possessing impenetrability, we are able to generalize this notion, and infer that force is exerted when the displacement is made by any other material body than our own. However many experiments may be made on things in the world around us, they all call up the same ideas of matter and force as those above described, and no other. Force may clearly be looked upon in two ways : (1) that exertion we experience when we displace a material body; (2) that unknown something which goes on between us and that body, or between two bodies when one is displaced by the other (caused by the fact that both possess impenetrability), owing to which the displacement is effected. These of course are connected together, but the latter is the only way in which force will be regarded. The idea of the existence of force then is got from its dis- placing, and therefore causing motion in, a material body; another experiment enables us to conclude that force wiU change the motion of a body that already is in motion. Since in perfonning the experiments above mentioned we sometimes experience more exertion and sometimes less, it is clear that one force may be greater than another. These ideas however are but vague; and in order to get a clearer notion, a force must be estimated by what it wiU just do, or what it will just not do, which is much the same thiiig. We shall, then, by means of experiments relating to forces (for observations cannot be made here), and by making necessary 52 INERTIA OF MATTER. conventions regarding the measures thereof, be able to make our propositions depend on geometrical or algebraical truths. 96. All our experimraits on matter lead us to conclude that it is purely passive and does not of itself change its condition, but requires a force to act upon it to do so. Now it is observed that all material bodies will, if placed at rest and allowed to be free, fall towards the earth, but that this can be prevented by the action of a force : we infer then that it is a force which would cause all bodies to fall towards the earth. This force is called weight or gravity: a certain body is taken and the weight of it is called 1 lb.: the force which is to be exerted so as just to prevent it from falling is called a force of 1 lb. ; another such body would require another force of 1 lb. to support it, and the two together would require a force which would be called 2 lbs., and so on. This is the only method on which forces can be measured. And as this supposes a state of rest in the bodies on which they act, all investigations regarding forces as such must be pursued under the hypothesis of their producing rest; the science which treats of forces then is Statics. In works on this subject will be found a convention of representing forces by straight lines, and a law of force, tlie truth of which rests solely on experiment, viz. that the point of application of a force may be transferred to any point in its line of action without alter- ing the effect. From these is deduced the proposition called "the parallelogram of forces," and then the whole science is reduced into a geometrical science, to which of course the ordinary rea- sonings will apply. Thus after the above law and convention are taken for granted, we find that the resolved part of a force F in a direction inclined at an angle 6 to its direction is i''cos 6, simply because the length of the side of a rectangular parallel- ogram inclined at an angle d to its diagonal = length of diagonal X cos 6 ; and for no other reason. The experimental laws then are for the purpose of getting rid of forces; and in all state- ments respecting forces that occur in Statics the word "force" may be replaced by the word "line." ,97. We shall now treat of the effect of forces when causing liiotion in material bodies. The science which refers to this is AXIOM POK PARTICLE MOTION. 53 called Dynamics, It must be remembered that it is only in material bodies, this takes place, for in the motion of a shadow, or in that of a ghost (which in popular belief possesses finite extension and capability of motion, but is without impenetra- bility), no force can act at all. The science of motion being a purely geometrical one, though force be the cause of motion in material bodies, it wiU be necessary for us in this case also to ascertain certain experimental laws by which we may get rid of force, and reduce Dynamics into the geometrical science of motion, in, a similar manner to that in which Statics is reduced into Geometry. 98. A quantity of matter of the smallest possible dimen- sions is called a particle, and may be regarded approximately as like a geometrical point. The fundamental proposition by which the motion of a particle is determined is the following: — If a material particle and a geometrical point be initially coincident in space, and be moving with the same velocity in the same direction, and if forces act on the particle so as to change its velocity in the same way as that of the point is changed, then the particle and the point will always be coinci- dent in space. " - This is obviously axiomatic ; and hence the motion of the particle is deduced from that of the point, which can be deter- mined by the methods of the foregoing chapters. From this we shall see that the proposition called " the parallelogram of velocities" will be true for a particle, though the reasoning used in Art. 8 to prove it will not hold good, owing to the fact that the particle cannot be compelled to move along the line AB (see the figure of Art. 8), except under the application of a force to compel it: we cannot then conclude that that hypothesis will properly represent the motion of the particle. 99. Since the eficct of a force is to produce or change motion, i. e. to generate velocity, and therefore to produce an ac- celeration, the laws we must seek for must be to connect the measures of the forces in any case with those of the accelerations they produce: these can only be determined by experiment. They ought to be obtained from the most simple cases, and 54 NATUEE OF THE LAWS OF FOECE. therefore the forces ought to be supposed to act on material particles : we can consequently make approximate experiments only, from a number of which we draw our conclusions regard- ing the laws. Their truth is established in the following manner : (1) an hypothesis is taken which is a priori probable, as far as we can judge; (2) experiments are made in which the conditions of the hypothesis are approximately satisfied, and observations are taken by which we see that the more nearly the above conditions are satisfied, the more nearly does the result agree with what we should expect it to be; (3) calculations are made of complicated cases of motion, under the assumption of the truth of the hypothesis, and the result of calculation is found to agree with the case of nature. The next chapter will be devoted to the discussion of these laws. CHAPTEE VI. OP THE DYNAMICAL LAWS OF FORCE, COMMONLY CALLED, THE LAWS OF MOTION. 100. In investigating the connection between forces and the motions, produced, it will be clearly necessary and suflScient to obtain definite statements on the three following points: (1) the general effect of a force on a particle: (2) what that effect is: (3) when there are several particles influencing each other's motion, what connection there is between the forces called into action. We shall therefore have three laws of mo- tion. It will be necessary to have a law to satisfy us on the first point, because force does not admit of a satisfactory defi- nition, as the idea of it is got from experiment. The formal statement on this point must then be taken together with our notions of force, thus supplying the place of a definition. The idea of force was that it caused or changed motion. The first law of motion then will be for the purpose of setting forth this idea definitely, i.e. stating a necessary connection (under the present state of nature) between force and motion in material bodies: this is best done negatively, by considering the state of the case where no force acts. If there is such a necessary connection, no motion will then be caused, or if the particle be in motion, its motion will not be changed ; therefore it will either be at rest, or move in a straight line with uniform velocity. Since forces that statically balance each other have no resultant, the same ought to be true when the particle is supposed to be acted upon by forces in statical equilibrium. 101. The First Law of Motion then will be: — "If a material particle be acted upon by no external force, or by 56 FIRST LAW OF MOTION. forces which statically balance each other, it will either be at rest, or be moving uniformly in a straight line." 102. We shall now have to test the truth of this by experiments. (1) If the particle be at rest it will remain at rest. This is a reasonable hypothesis from the ideas we have of the passive nature or inertia of matter ; for tliere seems to be no reason why a body at rest should begin to move in any one direction rather than any other : moreover, in all cases when a body at rest begins to move, we find from experience that some force has always acted to cause this motion. (2) If the particle be in motion it will proceed in a straight line. There is a priori no reason why it should deviate on one side rather than another. And in all cases of curvilinear motion we find that external forces act. If a stone be thrown in a direction inclined to the vertical, its path is curved; but the force of weight has been continually acting which would make the position of the stone at any instant lower than what it would otherwise be. When a stone is thrown along the ground, sup- posed horizontal, its path is nearly straight; and considering the asperities and unevennesses of the surface, there will be a number of forces called into action at the stone's contact with them, that may account for its deviation from a rectilinear path : moreover, the more we do away with these forces, which is done by making the surface smoother, the more nearly do we find the path become a straight line, thus leading us to suppose that the above forces do account for the deviation. When a carriage in motion is suddenly turned to one side, a person in it feels a tendency towards the other side of the carriage, i. e. to proceed in space in the same direction as before. (3) If the particle be in motion its velocity is constant. There is a priori no reason why of itself the particle should increase its velocity rather than diminish it, or vice versd. And we find from experience that when the velocity is changed, forces have acted : for example, in the case of the stone thrown SECOND LAW OF MOTION. 57 along the ground, the velocity certainly is diminished, but this may be due to the friction and the resistance of the air ; and it is also found that the more these are diminished, as in the case of a smooth level sheet of ice, the less is the diminution of the velocity. If a carriage in motion be suddenly stopped, a person in it is thrown forward, i. e. his body has a tendency to proceed with its previous motion. From such experiments as the above we conclude that the first Law of Motion is true. 103. Having settled that the invariable effect of a force is motion, we proceed to inquire what motion does a force cause? To obtain an answer that we may expect to be true, we must consider that the idea of force is one in itself, whether force be considered statically or dynamically: therefore the special pro- perties of force must be intrinsically the same in both subjects. These have been investigated^ in Statics : therefore we must state them now in a dynamical form. The special properties are the following: that a force is independent of any particular point in its line of action ; that it is independent, as far as regards its line of action, of any other force (this latter is got feom the parallelogram of forces): and as time is not involved in Statics, because we have there not instantaneous but permanent rest, tiiat force is si\.m independent of time. From this we should infer that the special dynamifial pro- perties of a force would be that its effect is independent of any velocity already existing (for that only depends on space and time); and independent, as far as regards its own direction, of any other force. If with these we combine the consideration of the measure of a force arising from the addition of units, we should expect the effect of a force, i. e. the acceleration it pro- duces, to be proportional to its magnitude. 104. We state all this in the following manner as The Second Law of Motion. *' When any number of forces act aa a material particle,, the acceleration which any one of tiiem produces on. the motion is the same, both in direction and magnitude, as if it had acted on. 58 SECOND LAW OF MOTION. the particle ai rest, and the other forces had not acted at all, being proportional to the intensity of the force." 105. In establishing the truth of this, we find (1) that a force acting constantly in the same direction and with the same intensity on a particle at rest, will produce an uniformly accele- rated motion in its own direction. As an example of such a force we may take the weight of the body, which acts always vertically downwards, and is the same for all moderate heights above the earth's surface ; therefore if our statement is true, the motion of a body dropped from rest ought to be uniformly acce- lerated. If observations be made on such motions, it is found that they very nearly agree with what they ought to be, and the resistance of the air seems quite sufficient to account for the discrepancy. If a body were let fall down an inclined plane, the force on it would be its weight x sine of the inclination of the plane to the horizon, and therefore is constant, and the motion ought to be of the same kind as before. This motion is easier to observe than the preceding, because by diminishing the inclination of the plane the acceleration may be made small, but there is friction as well as the resistance of the air to cause a dis- crepancy. (2) The same is true if the particle have an initial motion in the direction in which the force acts. This may be tested by observing the motion of a body thrown vertically upwards, or that of a body projected directly up or down an inclined plane; and we can also investigate whether the acceleration in this case is the same as in the pre- ceding : we find that it is so, and hence conclude that the effect of a constant force is independent of any velocity in its own direction. (3) This is also tnie if the body have a velocity in any other direction. As an example of such motion we may take the case of a body projected^ in a direction not vertical : then we ought to have the case of a point whose motion is affected by a constant SECOND LAW OF MOTION. 59 acceleration vertically downwards, and equal to the acceleration due to its weight (Art, 42), and therefore the path of such a body ought to be a parabola whose axis is vertical, and con- cavity downwards. The resistance of the air will of course cause a deviation from this motion, but the deviation is so small as to lead us to conclude that this statement is true. This might also be tested on an inclined plane, as in the two former cases. From all this it appears that the effect of a force is independent of any velocity the body may have. (4) The accelerations produced by constant forces are pro- portional to their intensities. If a body of weight Whe placed successively on two planes inclined to the horizon at angles i, il, then the forces which cause its motion are in the respective cases TFsin «, PFsin i; therefore if this statement be true, the accelerations of the mo- tion in these cases are in the ratio of TFsin t, : Wsva. i, i. e. sin t : sin C, or if the planes be of the same length, in the ratio of their heights. This can easily be tested from observations of the motions; and it may be observed that the friction may be made very small, and if the inclinations of the planes be small, the velocity of the body will never be veiy great, so that the resistance of the air will have much less effect than if the body were moving with a great velocity, and the observations on the motion can be more conveniently taken. (5) If any number of constant forces act in the same di- rection, the acceleration produced will be equal to the sum of the accelerations which they would separately produce. Let the forces be 2^, F^, &c. and the accelerations they would separately produce be «!, «,, &c. ; also let J' be the result- ing force, and a the acceleration produced by it. Then jPis in the same direction as F^, F^, &c., and therefore a is in the same direction as a,, o^, &c. Also, by the preceding case, ^=^ = ^-&c., a a, Kj F -vF + ... and therefore each = — *— — —, . a, + «,+ ... 60 SECOND LAW OF MOTION. But i^=*i?; + i^j+...; .-. a = a, + aj+... !From this it appears that we can correctly obtain the motion by considering the component forces, taking their effects sepa- rately, and then combining them. (6) The effect of any constant force acting on a particle is independent of any other con- as b stant force that may be acting. /N^ / V Let AB, ^C represent two con- 'A -Xf / stant forces in direction and / n. / magnitude, then completing the / ^\/ parallelogram ^0, their result- c c ant is represented by AD. The acceleration produced will consequently bfe in the direo- tion of AD^ and let it be represented by Ad. Draw dh, dc parallel to DB, DO. Then Ah : Ag\ Ad :'. AB: AG \ AD; and therefote Al, Ac represent the accelerations due to AB, A C. Whence it appears that we may take the separate effects of the component forces and combine them by the parallelogram of accelerations, and thus correctly investigate the motion. This can be extended to more than two forces, after the manner of Art. 9. (7) The foregoing statements will also be true for any forces. I'or the effect of a variable force at the instant in question will be measured by the effect of an equal force continued con- stant for a certain time, and therefore the measures of the accele- rations will be the same (for the instant) as if the forces were continued constant, and consequently will be subject to the above laws. On this plan the truth of the Second Law of Motion is established. 106. We have yet to determine the coefficient of the pro- portionality between the measures of a force and the acceleration MASS. 61 produced by it: this evidently depfehds oh the nature of the particle aeted upon, for the same force acting upon differeM particles is foimd to produce diflfereut accelerations^ It is evident that if any numher of particles be taken with equal (q[uantities of matter in thenis the same force must be e&erted on each to make them move in the same way: and if they be connected together, the effect of the above system of forces 'Will not be altered : i. e. if F b& the fOrCe acting on each particle to produce an acceleration a in its "motion, there being n particles; when they are connected together, or formed into one particle, all the forces, i.e. wi^, must act in order to produce the same acceleration a as before. Now the quantity of matter in this last particle iS n times as great as the quan- tity of matter in one of the original particles: wherefore, in order that the accelerations on different particles may be the Same, the intensity of the farces acting 6n them must be prb- portional to the quantity of matter in the particles. This is the only plan on which we can prCceed to estimate the quan- tity of matter in bodies. The measure of the quantity of matter in a body is called its mass. 107. The force then acting on a particle varies as the ac- celeration produced as long as the mass of the particle is the same, and as the mass when the acceleration is the same ; there- fore generally the force varies as the mass of the particle and the acceleration produced jointly; i. e. if forces F, F' acting on particles whose masses are M, M', produce accelerations a, a, then rps^fc-T^.— . We aiu8t assume the unit of mass: let the mass of the second particle M be the unit of mass, then F_M a a' F F'~ 1 •«" """^-a'F'- F', a' are at present undetermined : let them both = 1, rp thea M^-^ or F=Ma. a This assumption fixes the unit of mass to be that quantity of matter in which the unit of force produces the unit of acCelc- 62 THIRD LAW OP MOTION. ration, and then we have the numerical measure of the intensity of a force equal to the product of the numerical measures of the mass moved and the acceleration produced. 108. The mass of a hody is proportional to its weight, for the acceleration of gravity is the same on all bodies, as is established by the experiment of letting fall at the same instant two bodies of very different weights, such as a sovereign and a feather from the same height within the exhausted receiver of an air-pump, when they are found to reach the bottom at the same instant. Therefore, as the accelerations on these are equal, the force causing motion is proportional to the mass moved (Art. 106), i. e. the mass of a body is proportional to its weight. 109. The expression Ma. is called the moving or motional effect of a force, and My. (the measure of the velocity) is called the momentum or quantity of motion of a body. Also My (velo- city)' is called the vis viva of the body. 110. It now remains to determine what effects particles pro- duce an the motion of each other, and we must be guided by the analogous case in Statics, which is that of bodies pressing against each other, or exerting forces by means of strings or rods : in this case the forces exerted by any two particles are equal in magnitude and opposite in direction, and we should therefore expect the same to hold good in Dynamics. Since under the conventions we have adopted the intensity of the force is measured by the product of the measures of the mass and the acceleration, i. e. by the motional effect, we state in the following manner The Third Law op Motion. " If one particle act on another particle, the motional effect produced by the first on the second is equal in magnitude and opposite in direction to that produced by the second on the first." Gr concisely thus: "Action and reaction are equal and oppo- site." THIRD LAW OF MOTION. 63 111. We may test this Iby observing the motion of two bodies of different weights hanging by a fine inextensible string oyer a pulley. Let W, W be the weights of the bodies, {W> W), and g the acceleration due to gravity, i. e. the acceleration of motion in a body falling freely under its own weight. Also let T be the tension of the string ; then the force downwards on the heavier particle is W— T, and that upwards on the other is T— W, if the law be true. Now on the motion of the heavier particle the force W, and on the other the force W, produces an accelera- tion g, therefore (by Art. 105, (4)) the accelerations on these are respectively W- T T-W ^ W •^' W But the string being always stretched, the downward motion of the heavier particle is identical with the upward motion of the other, and consequently the above accelerations are equal, . W-T T-W W—W and therefore each of these = ^^^^ — ij« a. W+W " This is the acceleration on the motion of each particle ; and as it is a constant acceleration, the motion possesses the proper- ties investigated in Arts. 39, 40, 41. In making experiments we can see whether these properties are possessed in any case, and as it is found that they are, we conclude that the law is true. If M, M" be the masses of the particles, these accelerations can be represented J, ^ itf' AT' ^y9-L%,-g- (Art. 107.) 64 THIRD LAW OF MOTION. k" 112. This mation is shewn in a maehine ealled from its in- Yentor Atwood'a Machine, whieh C(eaisis.ta of a graduated pillar, on the top of which is a pulley whose a^is regts on. wheels in order to diminish friction. 4 is a swaU atage wh,ich can be, fixed at any point alqng the graduated scale. The, maejiine is ftirnished with. clockw being expressed in circular measure); .*. {v^ + v^ + ...+ v„^) ain" 6 is a positive quantity less than one that becomes indefinitely small, and therefore it vanishes. Whence v" - v"' = 2g . AL, and AL is the vertical space descended. 134. The times of descent of a heavy body initially at rest down all chords of a vertical circle through the highest or lowest points are the same. Let A be the lowest point of a circle, and let t be the time of descent down any chord A 0. Also let AB be the diameter, and the aag\QBAO=d. Then the force down AO is Wcoad, and therefore the acceleration is g cos 6; .-. AC=\gcQ&e.f (Art, 39); 2AB [Tag' _ /2I. ■' y ^w^~V g which is independent of 0. The same reasoning will apply, if the highest point of the circle were taken instead of the lowest point. 135. A case of constrained motion was noticed in the de- scription of Atwood's machine (Arts. Ill, 112): the following are of a somewhat similar nature. Let the body A draw the bodies 5 and up the planes whose inclinations are t, i, by means of two inextensible strings. 76 MOTION OP BODIES CONNECTED WITH STEINGS. Then whatever space A descends through, B and C pass each of them through an equal space in the same time, therefore the motion of each of the bodies is the same. Let The the tension of the string connecting A and B, and T that of the string connecting A and C: also let W, w, w', be the weights oi A, B, d then the force causing A to move downwards is W— T— T'; and the forces causing B and to move up the inclined planes are respectively T—wsini, T-w sin i. But on the body A the force W, on B the force w, and on the force w', produces an acceleration g; therefore the accelera- tions on A, B, are respectively W- T- T T- w sin l T-w' sin i W w -sr> -9- all these three are equal, therefore each of them W— w sin i — v) sin i ~ W+w + w' ^' which is constant. The motion of each body then is that in Arts. 38—41. T and T are easily obtained by equating the accelerations on B and C to the expression last found. 136. Suppose there to be only one string, and A to be suspended by a pulley: then the tension throughout is the same, T; and the accelerations on ^, j5, are respectively W-2T T-w am I T-w' sin i W -9, w MOTION ON A PLANE CURVE. 77 and we have to consider the eflfect of the particular kind of con- straint in this case. If B were to move over any space, and G were immoveable, A would move over half that space, by the property of the sim- ple moveable pulley ; and so if G were to move, and B were fixed. Therefore A moves over half the sum of the spaces pstssed over by B and G, whatever these may be ; therefore the acceleration on ^ = half the sum of the accelerations on B and G; 1 fT—wsini T—w'smi' \ ' = *( w ^+ w' V- W-2T , /■T-wsini T-'. This equation will give T, and then the accelerations on A, B, G will be determined. In both these cases the pressures on the planes can be found, as in Art. 131, to be w cos t, w' cos i. A. 137. The general case of motion on a smooth plane curve is determined in the following manner : — Take a pair of rectangular axes in the plane of the curve, and let a^, Oy, be the accelerations on the motion in directions of X and y, arising from the external forces, and f the acceleration due to the force of constraint, which is in the direction of the normal at any point : then if m be the mass of the particle, ma^, moy, are the forces acting on it, and m^ is the force of constraint (Art. 107) ; and we have d?ii fc dx these, together with the equation to the curve, determine the motion. We obtain immediately dxd^x dyS^y _ dx dy ^ di'Se ■*■ dt'dF''^^'' dt + ''"'' dt ' 78 MOTION ON A PLANE CURVE. dt) this gives the velocity at any point. then We might have used tangential and normal resolutions, and d^s _ dx dy •k? ^ dy dx which would give the same result more easily. From the second of these, ^, and therefore the force of constraint m%, is determined. A. 138. If the only force acting be the weight of the particle, then taking the axis of x vertically downwards, the above equations become d'^y _ f.dx ds dy^ J' cPs _ dx v^ J. dy\' whence f;^) = O-^r^gx. If the initial value of x be a, and the initial velocity be v', this becomes v'-v"' = 2g{x-a), or the change of velocity is only due to the vertical space de- scended, as before (Art. 133). MOTION ON A CYCLOID. 79 A. 139. The pressure of the particle on the curve is -r m^, the positive direction of ^ being towards the center of cnrvaturei When the particle is merely on the curve, since the reaction can never be in the direction from the particle to the curve, whenever ^ takes the value zero, the particle will leave the curve. If, however, the reaction be capable of being exerted in any direction, as in the case of a particle in a tube, there will be no such limitation. 140. To determine the motion of a heavy particle on a cycloid whose axis is vertical and vertex downwards. Let P be the position of the particle at any time, A the vertex of the cycloid, BPT the generating circle, whose di- ameter MT is vertical and = 2a. Then PT is a tangent to the cycloid (Appendix, Art. 2). The acceleration on the particle in the direction of its motion PT n AP = g cos PTR = g J± = 1 1^ (App. Art. 3). = 3-AP: 4a this varies as the distance from A. The other part of the force and the constraint are only con- cerned with changing the direction of the motion, i. e. keeping the particle on the curve. This kind of motion has been investigated in Arts. 51 — 54, and in Newton, § ij. The particle will oscillate between two points equally dis- tant from A, the time of a complete oscillation being — ti= . The velocity at any point is determined from the considera- tions in Art. 133. 80 PENDULUM. A. 141. This may be easily got from the equations to the cycloid a; = a(l -cos^), 1/ =a {0 + sind); (App. Art. 5), where the axis of x is measured vertically upwards. Then -jp = — g-r-ia the equation of motion. . ■, dx . „ ds ^ And ^ = aBm0; ^=2acos-; doi . e ^=-^sm- = -^ 4a- Therefore from Arts. 51 — 54, the motion is oscillatory and the time of a complete oscillation is , 142. It must be observed that in this case of motion the time of one oscillation is independent of the initial distance, or the oscillations in a cycloid are isochronous. 143. By means of two vertical semi-cycloidal cheeks con- nected so as to form a cusp, a particle suspended by a string can be made to oscillate in a cycloid, as indicated in App. Art. 4 : the string will manifestly be always stretched. This forms a " simple pendulum." The length of this string is 4a = Z say, and therefore the time of a complete oscillation = 27r a/-. In pendulums the time of an oscillation is generally taken to be the time from rest to rest, and therefore it = tt a/ - . This formula is made great use of for finding the value of g : for the length Z of a seconds' pendulum can be determined with great accuracy*, and then we have l = ir a/ - , whence g — irH : * The manner in which this is ascertained will be found in Griffin's Dynamics of a Rigid Body. MOTION IN A CIRCLE. 81 the units in which g is expressed are seconds, and whatever unit of length is assumed in I. The length of a seconds pendulum = 39"1392 inches, very nearly, i.e. 3-2616 feet; and putting 7r = 3-1416, we have g = 32-19 feet per second, nearly. 144. To determine the motion of a heavy particle in a vertical circle. If the motion is very small, it will consist of small oscil- lations about the lowest point, which can be deduced from the case of the cycloid, the time of one of these complete oscillations being i-jr/u - , where I is the radius of the circle. In other cases we must proceed as follows : Let C be the initial position of the particle, P its position at any time, A the lowest point of the circle, its center, a = its radius, v, v' = the veloci- ties at P, G respectively, angle A 0P= 9, AOC=a.; and CM, PN horizontal lines. Then the change of velocity from C to P is due to the ver- tical space descended (Art. 133) ; .'. v" - w" = 2g . MN= 2ga (cos - cos a). Let ^ be the acceleration due to the force of constraint in the normal, considered positive if towards 0. Then — = acceleration in PO = f —g cos (Newton, § ij) ; .-. P=qcos0 + — v"' =gcos0 + 2g (cos ^ — cos a) H =g {Scos0—2cosa.) ■] ; M 82 MOTION IN A CIRCLE. therefore if It be the force of constraint, and W the weight of the particle, -I' ^=^ (Art. 105(4)); .-. R = wO. V 3 cos — 2 cos a + - ga. If the particle be on the outside of the circle, B must never be positive ; if on the inside, or if suspended from by a string, E must never be negative ; but if the particle be moving in a circular tube or groove, or be connected with by a rigid rod, B may be either positive or negative. 145. If the particle be initially at rest, v =0; and then v' = 2ga (cos 6 — cos a) 1 _ B = W{3cos9-2 cosa)j ' whence cos > cos a, or 6 lies between + a and — a ; therefore the motion is oscillatory between the points for which 6 = + a and — a. The greatest value of B in any case is when cos is greatest, i.e. when ^ = 0, or at the lowest point. If the particle start from rest at the highest point, a = tt, and the greatest value of -B is 5 W. 146. If the particle be suspended by a string, making complete revolutions, and if the velocity at the highest point be just sufficient to keep the string stretched, 2 then f , which = gco30 + — , must = 0, when = Tr, v = v' ; .-. v'^—ga. And we have -B= 'Pr(3cos^— 2cosa+ 1) = 3Tr(l+cos0), for a=7r. The greatest value of this is 6 W\ i. e. the string must be able to bear six times the weight of the particle without breaking, in order that this motion may continue. MOTION IN A CIRCLE. 83 147. The motion of Art. 145 will not apply to a particle suspended hj a string unless B is always positive: the least value of It corresponds to the least value of cos d, i. e. when d = a, and i? then becomes Wcos a. If this is positive, a must not exceed — : therefore oscillatory motion of a particle sus- pended by a string is not possible unless the extent of oscillation be not greater than a semicircle. A. 148. All this might have been obtained by means of the equations in Arts. 34 or 35 ; which, remembering that in a circle r is constant and = a, and s = a6, become If be so small that its cube and higher powers may be neglected, the second equation becomes df~ a ' whence, by Arts. 51 — 54, the motion will be oscillatory, and the time of a complete oscillation will be 27r a/ - ; therefore a pen- dulum oscillating in a circle may be considered as one oscillating in a cycloid, when the arc of vibration is so small that its cube and higher powers may be neglected. T .1 ^'^ 9 ■ a In other cases -^rs = — - sm t/ ; at a dQ e?e ^g . adO .-. 2 -7- -TT = - 2 - sin -jr ; dt dt a dt dt •j =2^ (cos ^- cos a) ; de\ whence f = 5'cos0+ o , ^^ = 5- (3 cos ^ — 2 cos a) : 84 CASES OF MOTIOK OP A PAETICLE the particle having been supposed to start from rest at a point for which 6= a. A. 149. To determine the motion of a heavy particle hang- ing by an elastic string, it having been vertically displaced from its position of rest. Since the only forces acting are the weight of the particle and the tension of the string, the directions of which are both vertical, the motion is rectilinear. Let W be the weight of the particle, a the natural length of the string : then in the position of equilibrium the length of the string = a f 1 +^ ) , E being the weight which will stretch the string to twice its natural length. Let all + -=• j + a; be the length of the string at any time t, and T its tension. Then a(l +^^+a:=-a(l +^j. whence T=W+E.-. a Off The downward force on the particle is W— T,ox —E.- ; E X therefore the acceleration downwards is — ^. - .g; d dt ■f (i+f)+4' cZ'aj E X ''^-de-=-w-a3' which varies as x. The motion then is that investigated in Art. 51, and there- fore the particle makes oscillations about its position of statical equilibrium, the time of each being ■ V W'a E. g. If the natural length of the string be 4 yards, and if it be stretched J inch by a weight of 1 lb. ; the weight being slightly depressed, to find the time of an oscillation. SUSPENDED BY A STRING. 85 Take 1 foot, 1 second, and 1 lb. for the units of length, time, and force. Then a= 12, W= 1, g = 32-2 ; and to find E we have 1 : E :: finch : 4 yards; .-.^=576. And the time of a complete oscillation = 217- V 576 X -— seconds la i T Y g^gg.g = '16 seconds, nearly. A. 150. To determine the motion of a particle suspended by an inelastic string to a point, and revolving in space with an uniform angular velocity about a vertical line through that point. Let be the fixed point, P the position of the particle at time t, OA the vertical line, PN perpendicular to OA, NM parallel to the initial position of PN, PON=^, OP=a, ON==z; PN=r, MNP =6, the polar co-ordinates of P in the plane perpendicular to OA. Then r = a sin <^, z=a cos fj), 6 = at (a being the angular velocity about OA). The forces acting on P are its weight downwards, i. e. in the direction of z, and the tension of the string in the direction PO; which produce accelerations g and f say: therefore the acceleration in » is ,9- f cos ^, that in r is - ^ sin ^, and that at right angles to r and » is ; 1 d r dt ^=^-^osf (Arts. 32, 33). 86 MOTION OP A PARTICLE SUSPENDED, ETC dt dO Since -5- = o) a constant, the second equation shews that r 7* is constant ; and therefore sin ^, which = — , is constant, and Qi consequently a, which = a cos ^, is constant. Therefore from the first equation, taV = ^ sin ^, or ^ = m^a; and from the third, .9' — |cos ^ = 0, or cosrf> = -^. From these it appears that the particle describes a circle - whose center is N, a fixed point ; and that the string is inclined to the vertical line OA at the same angle fcos"^ -^j throughout the motion. This motion is similar to that of the ' governor' of a steam- engine. CHAPTER IX. OP IMPULSES AND COLLISION OP PARTICLES. 151. The forces we have hitherto considered have required a finite time in which to act in order that they might generate an appreciable velocity in a particle, but cases occur in which forces are brought into action for an inappreciably small time and yet a finite velocity is generated: such forces are called impulses. Of course they will be measured in an analogous way to that in which all other forces are measured, and therefore their measures will be proportional to the accelerations they would produce, i. e. to the velocities they would generate in an unit of time, supposing them to be constantly acting for that unit. But these would be infinitely great, therefore we must seek for some other measure of an impulse. Now with an uniform acceleration, we have v = at (Art. 21) ; therefore if t be the same for a set of accelerations, v will vary as a, and there- fore the measure of an impulse will vary as the velocity it generates in any assigned time. We shall assume that the in- appreciably small times for which impulses act are appreciably equal ; then the measure of an impulse will vary as the velocity it actually generates. This is as long as the same quantity of matter is acted on: and when different particles are acted on and the same velocity is generated, the measm-e of an impulse will of course vary as the measure of the quantity of matter, as in Art. 106. Then if with the unit of impulse acting on a particle of mass M' a velocity v is generated, and with the impulse B, acting on the mass M a velocity v is generated, we have E : 1 :: Mv : M'v. As 1^' and v are arbitrary, let them both = 1 ; this fixes the unit of impulse to be that which generates the unit of velocity 88 IMPULSES. when acting on the unit of mass, and then we have B = Mv: therefore the measure of an impulse is the momentum produced by it (Art. 109). 152. No confiision can be introduced by this modification, for the effects of finite forces can never appear as long as the effects of impulses are being considered; and after that the impulses have ceased to act, and we have only certain velocities affected by certain finite forces. 153. The mutual action at the surfaces of two smooth bodies in contact is normal to both surfaces, for otherwise it would have a tangential component which would make one body slip along the other. So also the mutual action between a par- ticle and a smooth surface with which it is in contact, is normal to the surface. In the collision of smooth balls (treated as particles) we shall ■have to consider them as elastic or inelastic ; moreover the di- rections of their motions may be coincident with the line in which the impulse takes place or not. 154. If a ball of ivory be thrown against a plane surface smeared slightly with some discolouring matter, it is found that the ball has, not a point, but a finite portion of its surface dis- coloured; its spherical form however remains as before: this shews that it has been compressed and that its original form has been regained. This projperty is called elasticity, and it exists more or less in all bodies with which we are acquainted. The force called into action to restore the form is evidently that which causes the rebound in such an experiment as the above, so that a perfectly inelastic body (if we could take such an one) would not separate from anything after an impact with it. 155. To determine the effect of the collision of two inelastic balls moving with given velocities in the line of impact. This is a case of direct impact. Let M, M' be the masses of the balls A and B respectively ; V, V their velocities, taking account of algebraic signs for direction ; IMPACT OP INELASTIC BALLS. 89 B the impulse between them, measured byl;he momentum generated (Art. 151), which must he the same for each, and in opposite directions, by the Third Law of Motion {Art. 110) ; so that if A impinges on B, the im- pulse B increases B's motion and diminishes -4's. As the balls are inelastic, they do not separate after the impact, therefore they move with a common velocity; let this be V. «» >- Before the impact the momenta of A and B are respectively MV, M'V; therefore after the impact, the momentum of -4 = MV— B, and ths.toiB = M'V' + B; .". the velocity of -4 = ^ — ii> 7? and the velocity of ^ = F' 4- -tt, (Art. 109) ; whence and MM' MV+M'V ~ M+M' ■ This gives the common velocity after impact. 156. From the equation Mv + M'v = MV+ M'V we see that the momentum of the system before impact is equal to the momentum afterwards. 157. If the balls be elastic, we have to consider (1) the circumstances of their mutual compression, (2) the circumstances of their restitution of figure. We have then two impulses, L. N 90 IMPACT OP ELASTIC BALLS. It for the compression, and B' for the restitution of figure, the whole impulse being B + R', which increases jB's motion and diminishes A'a. During the compression the impulse B' has not been acting, and therefore the circumstances are the same as if the balls were inelastic ; therefore B is the same in this case as MM' in the former, and consequently = -rj^ — =^(F— F'). Let V, v' be the velocities of A and B after impact ; therefore their momenta after impact are Mv, M'v' ; .-. Mv = MV-{B + B'),) M'v' = M'V'+{B+B').] It is also supposed that B' bears to i? a ratio depending only upon the nature of the substances that impinge, so that B' = eB, where e depends only upon the nature of the materials of which A and B are composed, e is called the modulus of elasticity, and lies between and 1 : if e = 1, the elasticity is said to be perfect, but we know of no such case in nature. Then we have v = V — {l + e)-jr^ v'=V' + {l + e) M MM' v=V -{1 + e) M' M+M' M Av-V) which equations determine the subsequent motion. 158. Also we have v'-v^-{V-V') + {l + e){V-V') = eiV-r), or the velocity of separation of the balls after impact : the velocity of approach before impact :: e : 1. IMPACT OP ELASTIC BALLS. 91 This is a property that can be tested by observation, and establishes the correctness of our hypothesis that R' — eB. For if we had started with the equations Mv = MV-{R + R') \ M'v' = M'V' + {B + It') MM' and assumed that v' — v = e{V— V), we should directly have obtained B' = eB. 159. We have also Mv + M'v = MV+ M V in this case as well as in the preceding, so that no momentum is lost by the impact. 160. If X, x' be the distances of the centers of the balls from any fixed point in the line of impact at time t after the collision, we have X =a +vt\ x'^a' + v't]' a, a being the initial values of x, x'. And if X be the distance of the center of gravity of the two from the same point, - _ Mx + Mx' *~ M+M Ma + M'a + {Mv + M'v) t M+M' ■ Ma + Ma' MV+M'V ~ M+M' ^ M+M' ' therefore the velocity of the center of gravity of the two balls MV+ M' V after impact is „ „, , the same as its velocity before impact. 92 OBLIQUE IMPACT. 161. If the impact be not direct, i. e. if the balls be initially- moving in directions not coincident with the line of impulse, we must resolve their velocities into the directions of the line of impulse and perpendicular to it. The resolved parts in the direction of impulse will be affected by the impact after the manner of the preceding investigations : those perpendicular to that direction will not be affected at all, because the balls are . smooth. Then the velocities of either ball in the direction per- pendicular to the line of impulse before and after the impact are equal ; and the equations of the preceding Articles will be true in this case, wherein V, V, v, v', represent the resolved parts of the velocities in the direction of impact. 162. In the collision of two perfectly elastic balls, the vis viva of the system after impact is the same as that before impact. By the vis viva of the system is meant the sum of the vires vivce of the balls. (See Art. 109.) We have, using the notation of Art. 157, v-v' = -e{V-V) =-{v-r), because the elasticity is perfect; and Mv + M'v' = MV+M'V'. Whence M{v-V)=M'{V'-v'), and w+ F= V' + v'; .-. M{v^-r)=M'iV"-v"'), or Mv' + M'v" = ]\fr+M'r''. 163. If the elasticity be imperfect, vis viva is lost by the collision. For v-v' = -e{V-V')' Mv + M'v'^MV+M'V, .: M{v-V)=M{r-v'), and v+V=v'+V' + {l-e){V-V); IMPACT AGAINST A PLANE, 93 M{v^- r) = M'{V"-v") + {l-e)M'{V-v'){V- V). or V {l+e)M M+M' .-. M{v'-V')=M'{r'-v'')-{l-e)M'.^^^^{V-ry, MM' or Mv'' + M'v'^= MT + M' V" - (1 - e') -^^ {V-V) whicli is less than MV' + M'V" e<]. 164. An elastic ball strikes a smooth plane ohliquely: to determine the motion. If a plane be drawn through the initial direction of motion and the normal to the plane against which the ball impinges, there is no velocity and no impulse perpendicular to this plane, and therefore after impact the ball will still move in it. Let this plane be that in which the annexed figure is drawn, so that the plane against which the ball impinges is perpendicular to that of the paper. Let V, v be the velocities of the ball before and after impact, M its mass, {l+e)E the impulse, 6, ^, the angles which the directions of motion before and after impact make with the normal to the plane. Then as the impulse does not aifect the velocity resolved in the direction of the plane, we have « sin ^ = «;' sin . For the motion perpendicular to the plane, we have v COS (ji = v cos 0— (l + e) -Tj.. 94 IMPACT AGAINST A PLANE, B is determined by the snppoaition of inelasticity, i. e. that an inelastic body would after the impact have no velocity per- pendicular to the plane; .*. = V cos 6 — Tfj.. M And then we immediately get v' cos = — ev cos 6. Combining these results we get cot ^ = — e cot ]■ These determine the direction of motion and the velocity after impact. It appears from them that the ball rebounds on the opposite side of the normal to that on which it impinges, the direction of its motion making a greater acute angle with it after the impact than before; its velocity is diminished by the impact. In the case of perfect elasticity, the velocities before and after impact are equal, and in directions equally inclined to the normal on opposite sides of it. APPENDIX. OP THE CYCLOID. 1. If a circle roll on a straight line, any point in its cir- cumference will trace out a cycloid. It is evident that the form of the curve will be such as that in the figure, P being the point in the generating circle which traces it, and GC the line on which the circle rolls. The curve may be continued to an unlimited extent, as indicated in the figure, where the beginning of the portion beyond C" is shewn; but we shall only consider the one portion CA C, with which all the others will be identical. If AB bisect CC at right angles, the curve will be sym- metrical with respect to AB, which is called the axis of the cycloid. The point A is called the vertex and 0, C are cusps. 2. In the generating circle, if P be joined with the ex- tremities of the diameter BOH, which passes through the point of contact B of the circle with the straiglit line OC, the line PS will be perpendicular to PB. And at the instant of time when the circle is in the position HPB, since it rolls along GBC the point B is at rest, and therefore P is moving at that 96 LENGTH OF ARC. instant as if describing a circle about R ; therefore the motion of P is for the instant perpendicular to PB, and therefore is in the direction PH. Therefore P^is a tangent to the cycloid. 3. The arc of a cycloid measured from the vertex to any point is equal to_ twice the length of the portion of the tangent at that.point intercepted by the generating circle passing through that point. {AP='2.PE). Take a point P very near to P. Draw Fp parallel to GB and join Hp. Then ultimately Up is parallel to the tangent at P', and HP is the tangent at P; therefore if Pifbe drawn parallel to Pp, the figure Pi^P' is ultimately a parallelogram*, and pM= PP'. Also ultimately pH is perpendicular to PR, i. e. the angles PNp, PNM are ultimately right angles. And the chord PR of the circle makes equal angles with the tangents at its extremities, therefore the angles pPR, CRP are ultimately equal; i. e. pPN, NPM axe ultimately equal. Therefore ultimately the triangles pPN, MPN have two angles equal each to each, and the side PN common ; therefore ultimately pN= NM, or pM= 2 .pN. And ultimately HN= HP; therefore ultimately PF = 2.pN=2{Hp-HP), i. e. the increment of the arc AP is twice the corresponding in- crement of the chord HP. This arc and chord begin simultaneously from zero ; therefore the arc AP = twice the chord HP. 4. If two equal cycloids be placed with their axes parallel and a cusp of the one coincident with the vertex of the other, their concavities being turned towards the same parts ; and if a • The chords PP", Pp (which are ultimately coincident with the arcs PP, Pp) have not been drawn, to aroid complicating the figure. MODE OP DESCRIBING A CYCLOID BY A STRING. 97 tangent drawn at any point of the one within the concavity of the other be produced to meet it, it will be a normal to it, and will be equal in length to the arc of the first cycloid fi:om the vertex to the point of contact. Take the generating circle in the position HPR: then PH is a tangent to the cycloid A G. Produce BS to K, make KH=BH, and describe the semicircle KQH, which •m\\. = BPH. Produce PH to meet the semicircle jK'^^in Q, and join QK. Then the angles PHB, KHQ are equal: therefore, since the circles are equal, arc HQ = arc PH = axe HPB- axe PB = BG — BG, by the mode of generation of the cycloid, = AH. Therefore Q is in the cycloid A' A : and^ff^jSTis its generat- ing circle ; therefore KQ is the tangent at Q, and consequently HQ is tlie normal. Also as PH=^HQ, PQ=:2PH=aTcAP. From this it is evident that if a string fastened at G equal in length to GA be unwrapped from the cycloid GA^ its extreptity will trace out an equal cycloid in the position AA\ A. 5. This may be investigated by means of the differen- tial calculus. Take the vertex of the cycloid for origin, and its axis for the axis of x ; and for any point P(_x, y) let the angle POH = 0, HPB being the generating circle. Let the radius^ of this circle be a, 98 EQUATION TO THE CYCLOID. y H A ■h B \ c B X then a! = S"0 + POcosPO^ = a-acos^, y = BR->rPOwa.POR = arc J5P4- PO sin POE = a5 + a sin ^ ; .'. a;=a(l — cos^)) a;=a(l — cos^)) , . 3^ = a(^ + sine)j ^"^' X • a ■" versin v = - , and *^ = ^ + sin a = ver8m - + ^l-(^-— j or y = a versin"* - + \/2, which is the intrinsic equation to the cycloid. A. 6. To find the involute of a cycloid produced by unwrapping a string from it whose length = the semi-cycloid, the extremity of the string being initially at the vertex. If A' be perpendicular to OB we a' shall have A'0 = axcAG=2x chord of gene- rating circle touching the cycloid at C = 4a; and P^ = arc.^P = ia sm - . 2 And referring to the figure of Art. 1, angle LPQ = PUB = i (ir - 0) ; .-. QL = PQ am LPQ = PQcos'^ = 2a sin 0. Also 0M= OB- BM= to - a (^ + sin 6) by the second of equations (a) ; .-. QN=QL+OM=a{7r-0 + sin 0). Also ON=PM+PQcoaLPQ = AB-x + PQ sin 2 = 2a — a{l- cos 0) + 4a sin" - = a (3 — cos 0) ; The positive sign is taken, because t increases with 6. 100 INVOLUTE OF A CYCLOID. /. A'N=A'0- ON These values of AN, QN can be got from those of x and y in equations (a) by writing ir — d instead of 6, and therefore we see that the involute is an equal cycloid whose vertex is A' and axis A'G. By means of the intrinsic equation this may very easily be obtained : for let A' be the origin of measurement for the curve A'Q, then ^TTj = radius of curvature at Q = PQ evidently, = e, and <2PL = ^' = |-0; .'. -3-3-, = 4a cos 9 ; whence s' = 4a sin ^', which represents an equal cycloid. PKOBLEMS. The usual notation has been adopted in these Problems, excepting where it dififera from that in the preceding chapters ; it ia therefore thought unne- cessary to explain the meaning of the symbols in every case. CHAPTERS I, II. 1. The measures of an acceleration and a velocity when referred respectively to (a + b) ft., (m + «)", and (a — h) ft., (m — n)" as units, are in the inverse ratio of their measures when referred to (a — h) ft., {rn — n)", and (a + 6) ft., (jn + n)"\ their measures when referred to a ft., m", and h ft., n", are as ma : nb. Shew that — = a. / 1 — i • Let a, V, be their measures when referred to 1 ft., 1" Then, by Arts. 6, 27, (m + 7if m — n m + n (m — rif i j^a ; rV :i j-w : ^ r" o a+h a — o a + o a — o m n , — a : fV : : ma : no . a From (1), {m'-n')a' = v', . a mb from (2), j« = — °, ■(1), .(2), ] whence [m - «■') tj = ^-p- ; a 102 PROBLEMS. 2. Two candles, which will bum for 4 and 6 hours respectively, are placed in candlesticks 1 foot high and 1 foot apart, and are lighted simultaneously. The shadow of the shorter is received on the table on which they stand: that of the longer on a wall 10 feet dis- tant from it, and perpendicular to the plane of the candles. Each candle is originally 1 foot long : find the velocity of the extremity of the shadow of the longer.. Find also the mean velocity of the ex- tremity of the shadow of the shorter during the last hour in which it is burning. Let ABC, DBF represent the candles at any time, and GH the wall. Then the motion of the extremity of the shadow on the wall consists of two parts^ (1) that due to the burning <£. A,.D being stationary; (2) that due to the burning of D, A being stationary. F a (1) If A passes over a space a in any time, the extremity of the GF shadow moves downward over jj= a in the same time. (2) If B passes over a' in any time, the shadow moves up- , GO , ward over Trp'^- If then the time assumed in the above be the unit of time, the velocity of the extremity of the shadow downwards is GF GG M i- / j-^n / ^rt\ CF"'~GF"'' ^^ Q^{«':<^^- /^Y d^ ds_ dy d's\dt) ' °'' di'di'^'dt' , iton-li I C dg n a +c dt dip c ds s' + e dt ' or 6 + iff=tan~' -. ^ ^ c Suppose that the initial values of <^ and a are both 0, then 8 = c tan i therefore the path is a catenary. 4. A person travelling eastward at the rate of 4 miles an hour, observes that the wind seems to blow from the north ; on doubling his speed, the wind seems to blow from the north-east. Determine the direction and the velocity of the wind. [4 1J2 miles per hour, from N.W.] 5. If U.J, Oj, be the measures of an acceleration referred to (m + n)", aft., and (m-n)", 5ft., as units; then its measure when referred to (2m)", eft., is c 6. If the acceleration of gravity (32 feet per sefcond) be the unit of acceleration, and the rate of 10 miles per hour be the unit of velo- city, what must be the units of space and time? [6ffc. 8|in.; |lsec.] A. 7. If the axes of x and y turn about the origin with the same uniform angular velocity to, find expressions for the accelera- tions parallel to them. rd'x . „ ay dy J „ dx -\ PKOBLEMS. 105 OHAPTEE III. 8. If a straight line be "drawn in a given direction from the initial position of a point whose motion is affected by a constant acceleration in another given direction : find the condition that the point may cross the line at right angles, and the distance from its initial position at which it will do so. Let the straight line be inclined at an angle -^ + ^S to the direc- tion of the constant acceleration a, and let the initial direction of motion be inclined at the angle 6 to this line, which take for the axis of X, Then the acceleration in a; is — a sinyS, and that in y is — o cos )8. And we have x = vcos6.t „ — .t" ... acos^ . y = v sm9.t ^r-!—. t' velocity in a; = i? cos 6 — a sin ^ . i 1 2/ = «sin6 — acos)8. t j Then by qiiestion, when y = 0, the velocity ia a; = ; .-. ■wcos^— asin/8.< = j . . acos/8 [j 1) sin H-^ . < = I I cos 2v sin 6 a sin /8 a cos j3 Whence cot 8 = 2 tan j8, the condition required. And for this value of f, I ■ o ^\^ a sin/? Z 2«'sin;8 1 a ■ 3 sin"^ + 1 ■ 9. A number of points, moving from the same initial position in the same plane, describe equal parabolas, their motion being L, P 106 PEOELEMS. aflfected by the same constant acceleration in the same direction: prove that the locus of the vertices of these parabolas mil be an equal parabola. 10. Shew that the space passed over in any time by a point, there being a constant acceleration in the direction of motion, is equal to the space which it woiild describe uniformly in the same time with the mean between its greatest and least velocities during that time. 11. A number of points start from the same position with different velocities, the directions of which are in one plane: there is a constant acceleration in the direction perpendicular to this plane. Shew that the extremities of the latera recta of the parabolas described lie on a cone whose vertical angle = 2 tan"' 2. 12. Two points begin to move with equal velocities in parallel directions : the line joining, their initial positions is the direction of a constant acceleration affecting the motion of both. Shew that tan- gents drawn to the path of the one will cut off from the path of the other arcs described in equal times. CHAPTEE IV. 13. A point is moving in an ellipse, the acceleration being con- tinually towards the focus S. When it arrives at £ (the extremity of the minor axis) , the acceleration becomes constantly directed to a point S' in SB, the law of its variation being the same, and its mag- nitude at all equal distances being one-eighth of what it was before; having given S'B = J SB, shew that the periodic time is unchanged, and find the minor axis of the new orbit. ■ This is a case of the motion in Art. 76. In the original orbit, at B, ,_2\_\ " "SB a' In the new one, v' = -^-g -, . o Ji a And we have v' = v, S'B = ^.SB, X' = |x, SB = a. PROBLEMS. 107 Whence -we obtain a' = = . Z A J j.1. • • , 2ira'f 2irai , And tne penod = ' /-, ~ ~~r~' ^'^^ ^ "^ orbit. The other focus of the new orbit evidently lies in BE; let it be 5"'. Then S'B^BH'=%a: = a; S'B=^%; coslsBH=-; 5' 2 a' S'H'", or 4 (a" - 6"') = S'B' + BE" - 2S'B . BE' cos SBE, ■whence b' = -=b. 14. If a point describe an ellipse, the acceleration being towards the center, the locus of the middle points of the chords of all arcs described in equal times, is a similar ellipse. Let PQ be one of the arcs, and let the ordinate of F, Q meet the auxiliary circle in P', Q'. Let + ^ j cos ^ y = 2 (si^ <^ + sin ^) = 6 sin (if> + ^ j cos | therefore the locus of x, y is an ellipse whose axes are 2a cos 5 , 26 cos 5 . 108 PKOBLEMS, 15. If a point be describing an ellipse, tlie acceleration being towards the focvisj its velocity at any point may be resolved into two constant velocities in directions perpendicular to tte radius vector and the major axis, and respectively proportional to the major axis and the distance between the focL Let SP, HZ meet in W when produced. The velocity at P = -_=. , and.-. t wiU lie as the dotted curve. Using the notation in Arts. 68 and following, and employing suffixed letters for the new orbit, we have PROBLEMS. 109 ^\{a'+b'-r') = v, >JX{a^'+W-7^)=(l + ^v (1); vvsmi = h, (l+-jv.rsini = \ •••«A = (l + ^)»6 (2). 008"^ sm'(9_ 1 _cos^ Bin' g, ,. a r - ttj o' these equations determine a^, b^ and ^,, And we must bear in mind that - is small, and . •. 0,~ 0, a, — a, 6,-6 are small. Second part. Here we have tan 6 = -, whence r = ^ — s — ; and let a, - a, b^ — b, 6^ — = a, p, tji. Then from (1) we have From (2) we have Neglecting squares and products of the small quantities a, p, - , these become aa + bfi = ^{a'+b^ aP + ba = — ab IV a'+b" whence a = £, p=^. 110 PEOBLEMS. 17 /Qs , 2 cos''(^ + . cos 6 sin 6 { 75 5 ) = -^ cos*5 + & sin' 6 ; ■■■ *.i. ab 17. If a point describe a parabola, the acceleration being towards the focus ; shew that the time of describing any arc bounded by a focal chord « (length of chord)*. Here §F' = 4^i'.PF, and SP = P7; .:SP or Fr=^. And area QPQ = ^Q(/:pr.^nprQ -3^"- 2 -SP therefore time of description 2area©Pe' 8 QV.ST 8 QV h 3" 3 ' velocity at P = |.-2^, which cc«2F*- PKOBLEMS. Ill A. This might be solved as ia Art. 88. c h' For -we have — = 1 + cos ^, X = — . r c 1 [^ And the time reqiiired = r I r'dO, where ASQ = P, 4 a ■which = ^= c* (sin j8)"'. And the chord QQ' = + 1^ , = Jf- • ^^ 1 + cos^ l+C08(/3-ir) sin»/3' therefore the time o= (chord)*- A. 18. A number of points move in hyperbolas, starting from their vertices at the same instant j the directions of the accelera- tions pass through the common center, and their magnitudes are equal at aU equal distances. Shew that if the major axes coincide, the points wUl always lie in a common ordinate : and if the asymp- totes coincide, they will always lie in a straight line through the center. From Art. 70, note, we have, for any one of the moving points, -.£e^>^ ■t + A'e-'^-t) Initially x = a, y = 0, -t- =0, -^= sfx. . t ; x = :(£Vr.« + £-VX.r;^ --^(e''>^-(-e-''>^-t) At any assigned instant of time, let x^t/^, x^y^, &c. be the co- ordinates of the points j afi^, a^b^, &c. being the elements of their Then as \ is the same for all. If «, = «, = &c., we shall have x,—x^- &c. Aoidif ^=^ = &c., ^=-» = &c. 112 PEOBLEMS. A. 19. To find the law of acceleration towards the node of a lemniscate, in order that a point may move in that curve. Here r' = a" cos 26, or m = a"' >Jaeo2d; .-. ^ = a- x/iec2^(3 sec' 20 - 1) = i4(3aV-l); .: a = h'u' (** + jZ5 ) = 3a*AV, which ec (distance)"^. A. 20. The acceleration towards a fixed point is X (r' — a*r) 3,t distance r : a point is initially moving with velocity \/| A«° in a direction at right angles to its initial distance (a) from the fixed point. Find the orbit described. Here h = a>J^\a", Initially -^2=0) w = -; .•. (7=0; du _ 3 /a* 1 \ , ■'■ de~2^''\u'~3^y~'^ ' j-ySaV- l-2aV ^-3 = sin4(e-/3). PEOBLEMS. 113 Initially »• = a, 6 = 0; .■. l = -sin4^j whence — f- =3 + sin (4& + -^j, a* or -j = cos* 6 + am*0 : r .•. x* + y* = a'^; the equation to the path. A. 21. Acceleration = Xr H 5—.* A point is initially moving ■with a velocity 2a VSA. at right angles to its initial distance a : shew that it will come to a second apse at distance 3a. Here h = (2a \/3X) a, 1 A ^''^AVU f- + 2Xa'u'\ InitiaUy^. = i, ^ = 0; .-.0=^,. For any other apse ts = ; therefore the apsidal distances are determined by the equation ■e^(-?*H _3_ '^ 4ta' ™4/..4 1 a a i* s 2 . '■ or a'M' - TT »"■!*■' - T aV + th = Oj 4""" '12 whence om = 1, or — j o the first of which gives the original apsidal distance, the other gives an apsidal distance 3a. * In such cases as this, the acceleration is supposed to be towards the origin unless the contrary be distinctly expressed. L. Q 114 PROBLEMS. A. 22. The motion, of a point is affected by an acceleration in an Tinvarying direction, whose magnitude varies as the chord of cur- vature of the path drawn in that direction; shew that the normal acceleration is constant, anid determine the path. Take the given direction to be parallel to the axis of x, ., da's /, dy\ dx' '^'^de = {^pi)Ts\ p \ as J ds ...v=x(,f),- ■, dv , , „ and -J- = sm <^, ds and if we choose the axes properly we may have ^ initially = -= . It is evident moreover that <^ diminishes with an increase of s; . .'. 1) ^ - tJ'K. p sin = - \/a -J- sin (^ : .-. ^=^->/A:.sin^ ds Then ^j- = - n/\ . sin d 7- : '" ^ d^ dt .-. ^ = Xsm.A(sm^^+cos.^^j, and the right-hand member of the first equation is It- dx ■ ds = "Jk .V -r-= six ^ cos ^ = — X sin <^ cos tjt whence we have -=--5-= - 2 cotd-^- ; d^' ^d ds d^' .: J- = — c coseo'A; a(p .: s^ — c cot ^; PROBLEMS. 115 the constant being (Jetennined by the condition that 8 = 0, \rhen <^ = „ . This shews that the path is a catenary. The normal acceleration =^(f) %Y_ , ds which is constant. 23. If a point be moving in an ellipse, the acceleration being towards the focus, shew that the mean proportional between its velo- cities at the extremities of any diameter is constant. 24. If the velocities at any number of points of an ellipse de- scribed about the focus be in Arithmetical progression, the velocities at the opposite extremities of the diameters passing through those points will be in Harmonic progression. 25. Two parabolas have the same axis, and the vertex of one of them lies half way between the focus and vertex of the other, which is intersected by the first at the extremities LL of its latus rectum. If the accelerations affecting the motions of points that descxibe these parabolas be towards the respective foci, compare the times of moving from L to L. ^ i 26. Acceleration cc (distance)~^ A point is describing a circle with velocity Fin a periodic time P: if a velocity nVhe communi- cated to it in the direction towards the center, shew that a diameter of the circle is the latus rectum of the new orbit, and that the period in the new orbit is P(l — n^)~i. 27. A point is describing an ellipse, the acceleration being towards the center; when it is at its greatest distance, its velocity is suddenly increased in the ratio of m : 1 ; find the eccentricity of the new orbit, and explain the result when the eccentricity of the original orbit is < — . 116 PROBLEMS. [Eccentricity = s/m'e' - {m' - 1). When e < , the maior-axis of the new orbit is in m •' the direction of the minor-axis of the original one, and ., J. . ■, /m'' — 1 — m") its eccentricity = . / - rn' (1 - e") _ ■ 28. Acceleration oc (distance)"'. A point is initially at a dis- tance of 32 feet, and is moving -with a velocity of 100 feet per minute : the velocity necessary for it to describe a circular orbit is 80 feet per minute. Find the periodic time. [8-68488. ..minutes.] 29. Acceleration <» (distance)~^ A point describing an ellipse is at one extremity of the minor-axis ; shew how the velocity must be changed that the point may proceed to describe a parabola; and shew that the axis of the parabola will pass through the other ex- tremity of the minor-axis. [The velocity must be increased in the ratio of v2 : l.J 30. A point is describing an ellipse, the acceleration being towards the focus : if the magnitude of the acceleration and of the square of the velocity be doubled, what is the effect on the periodic time ? [It is doubled.] 31. A point is describing a parabola about the focus, and when it arrives at L, the extremity of the latus rectum, its velocity is diminished in the ratio of \' 2 : 1 ; find the position and axes of the - new orbit. \L is the extremity of the minor-axis, which is normal to the parabola.] 32. A point is describing a parabola about the focus S, and when it arrives at a given point Q in its path, the direction of the acceleration is suddenly reversed so as to be always away from ^S* : determine the nature, position, and dimensions of the new orbit. [It is an hyperbola, S being the exterior focus : the other focus lies in the diameter of the parabola at Q, and its dis- tance from Q = \. SQ.^ PROBLEMS, 11 f A. 33. A point moving from the origin in the direction of the axis of y with a velocity c describes a parabola y^ — iax, its motion being effected by an acceleration — Xy in the direction of y, and by another in the direction of x ; shew that its velocity at any point = yx(l + ^,)(c='-2/^). A. 34. A curve y =/{x), (which touches the axis of y at the origin) is described by a point, there being a constant acceleration a in the direction of x, and another in the direction of y : shew that this acceleration : 2a Jx . -r- I Jx -T- ) • ax \ ax J A. 35. If the motion of a point be affected by two equal accelerations \y in directions of x and y, find the path. [2/ = ^6~ '"""""' +^^"'^''"°"'".] A. 36. The acceleration affecting the motion of a point P varies as the distance from a point Q, which moves uniformly in the cir- cumference of a circle, being constantly directed from Q : shew that when the point Q arrives successively at any given position on the circle, the point P will be always situated on a certain hyperbola. A. 37. The acceleration towards a fixed point, at distance r = — J- + — — . A point is initially moving in a direction making an angle tan~'| with the initial distance a, and with a velocity equal to that with which, affected by the above acceleration, it would describe a circle of radius a : determine its path. r = a cot ( 6 + '■ ^8a' ■'Hy A. 38. Acceleration =X(— j jj. A point is initially at an apse at distance a, the space due to its initial velocity being ^f '■ determine the path. [r = a cos 2$.] A. 39. Acceleration cc (distance)""^ A point is initially mov- ing in a direction inclined at an. angle of 45° to its distance (a) with 118 PROBLEMS. 2 a velocity = —=. x that whicli it would acquire in moving from an infinite distance to its initial distance, starting from rest, if affected by tlie above acceleration. Find the orbit described. I _ 1 ■^=(V2+l)cVi''-(V2-l)/v.''.J CHAPTERS VI. VII. 40. If the unit of force be that which will just support a weight of 5 lbs., and the unit of acceleration be that which, if re- ferred to 1 foot and 1 second as units, is 3 : find the unit of mass. The unit of acceleration is that of 1 yard per second. The unit of mass is that on which the unit of force (5 lbs.) produces the unit of acceleration (1 yard per second) ; see Art. 107; therefore the unit of mass is that on which the force of 1 lb. would produce the acceleration of -= yard per second. But on the mass weighing 1 lb., the force of 1 lb. produces an acceleration gf, or 3 2 '2 feet per second; therefore on the mass weighing 32-2 lbs., the force of lib. pro- duces an acceleration of 1 foot per second ; g therefore on the mass weighing -^ of 32'21bs., the force of lib. 3 1 produces an acceleration of -pft., or ^yd. per second; therefore the unit of mass is that quantity of matter, the weight of wHch is I (32-2) lbs., or 53|lbs. 41. The period of the moon round the earth is 27| days, nearly : her mean distance from the earth's center is nearly = 60 x earth's radius. Calculate approximately the value of the accelera- tion of gravity at the earth's surface. PROBLEMS. 119 Tke earth's attraction oc (distance)"' ; therefore th.e motion is that in Art. 76, and the period = -j=a'. (Art. -84, or Newton, § iij.. Prop. 15). Let the earth's radius be the nnit of space, and 1 day the unit of time : then a = 60 ; X = acceleration produced by the earth's attrac- tion on a body placed at the unit of distance from its center, i. e. on a body at the earth's surface, and therefore \ measures the accelera- tion of gravity. And we have 27^ = -^ 60*- , 10 . 9 . 60' ,,. ,_ Whence A = j=i > putting ■n- = ^10. If we wish this to be expressed in terms of feet and seconds, we must transform it after the manner of Art. 27, and putting the earth's radius = 4000 miles nearly, we have ^10.9.60»\ 4000.1760.3 „„ „ _ /lO . 9 . ^Q '\ (24 . 60 . 60)' 42. If a body be shot up vertically at a place in latitude I, it will describe a curve of which the equation is 2u? ooa'l.^ — 2Fo) cos l,x^ + go? = 0, where x, y are the horizontal and vertical co-ordinates from the point of projection, co the earth's angular velocity, V the velocity of pro- jection j and all variation of gravity in magnitude and direction is neglected. The linear velocity of the point A on the earth's surface is tor cos I, r being the earth's radius ; therefore the body has initially a horizontal velocity wrcoal, and a vertical velocity V; therefore if P be its position in space at time t, A its initial posi- tion, AM vertical, PM horizontal, AM=Vt-^4-'> PM=mrcoal.t. 120 PROBLEMS. Draw FN towards the earth's center; then if the point of pro- jection A be transferred to A' at the instant under consideration, AA' =tjirco%l.t, A'N=x, NP = y; .: x = AA' — . PM = — ^ (or cos I . t, r + y r + y and y=Vt-^^=Vx — ^. j-'Vl — -) 2 y larco&l 2 \ y J and putting ~ = 1, this becomes 2u?oos'l.'i/'- 2Vtoooal.xy + gx' = (i. 43. In what time will a force which will just support 5 lbs. weight, move a mass of 10 lbs. weight through 50 feet on a hori- zontal plane, and what wUl be the velocity at the end of that time? [2^ sec. ; 40 feet per sec. ; if we consider g = B2 feet per sec] 44. Of all comets moviag in the ecliptic in parabolic orbits, that which has the latus rectum of its orbit equal to the diameter of the earth's orbit, considered circular, will remain within the latter for the longest period. 45. From a given point particles are projected in all directions so as to describe parabolas about a center of force, the attraction to which 05 (distance)"'. Find the Jocus of the vertices. r ^ r = c cos-^. CHAPTER VIIT. 46; A pendtilum is found to make 640 vibrations at the equator in the same time as that ra which it makes 641 at Greenwich: if a string will just sustain 80 lbs. at Greenwich, how many such lbs. will it sustain at the equator ? Let g be the acceleration of gravity at Greenwich, and g' that at the equator; 3f, M' the quantities of matter which the string will sustain at Greenwich and the equator respectively. PROBLEMS. 121 Then Mg, M'g are the forces exerted in these cases by the string (Art. 107) ; .-. ifjr = J/y = 80 lbs. Also 640 . IT J- = 641 . IT J- (Art. 143); M' _g__ /641Y. whence M" g'~ \UQJ ' \U0J M'g = TE^V 80 lbs. = 80 y^ lbs. 1281 *5120 A. 47. A particle starts from rest at the top of a sphere of radius a, acted on by its weight and an attractive force varying as the distance towards the lowest point of the sphere : find the pressure on the sphere; and if -at the distance a the force produces an accele- ration equal to g, when will the particle leave the sphere ? The motion is evidently in a vertical plane : let this be the plane of the paper, and let P be the position of the particle at time t, AB a vertical diameter, AOP = 6, \ the acceleration at distance unity to- wards B. Then a-y^ =g sinO + \. PB ainOPB = {g + Xtt) sin 0, a (|^'= g COB 6 + \.PB cos OPB ~ i = {g + A.fls) cos + \a — i. From the first, (|)"= 2 ^-^ (1 - cos 6); .: i = \a+(g + Xa)(3cose-2). This multiplied by the measure of the mass of the particle gives the pressure on the sphere. L. R 122 PROBLEMS. The particle leaves the sphere when ^ = 0, i. e. -when 3 cos = 2- g + \a Also we have \a = g; .: cos = ^. A. 48. Two equal weights (W) are connected by an extensible string whose natural length is a, and are placed, the one on an in- clined plane of angle i, and the other hanging vertically over the top. If the system be left to itself, shew that at the time t the ten- sion of the string is 2W cos^ fj - ^ sia^ f ^ ^ . t\ ; the string being such that either of the -weights suspended at one end would stretch it to a length 2a, it being initially stretched without tension in a vertical plane. Sere W=S; and if f be the acceleratioii due to the tension at time t, the length of the string = a (1-1--=) = a (l + -j . Let c be the length initially on the plane, s the length on the plane at time t ; then the distance of the hanging particle from the top of the plane is ll + -j -s, =x say. Wherefore -^=g-^; and for the motion of the other -5-j = gr sin t — f . cPi 2g ^ g' ,^ . , ... 1 = ^-^(1. .in.)4^..^cos(yf.-^), PROBLEMS. 123 T ... „ dx . ds . initially x = a-c, s = e, -^ = 0, 3-= ; .-. «=o, 1 = 0, ■•• l = |(l+sizi6)-|(l+sini)cos(^|^.«); A. 49., A straight tube of length I revolves with an angular velocity ^ j about a vertical axis through its lowest point, being inclined to it at the angle , . Shew that if a particle be placed in o the tube just above its position of relative equilibrium, the latus 13 rectum of the parabola described after leaving the tube = -^ I. o Let OF be the tube, OS the vertical axis, ■ ON=z, NP=r, which together with fl (the angle through which NP has revolved in a horizontal plane) detemiine the position of P j f = the acceleration in PR, t, = that per- pendicular to PR and PO, due to the con- straint. r dt\ dt) ^ d'z -^ . TT Also i£ OF = s, r = « sin ^ , » = s cos ^ , ^ = /v/ T • *> d's % g whence ja-T7* = -o- dt^ M 2 124 PROBLEMS. For relative equilibrium s is constant, and .-. = -^ . For the motion {j\- ^s'=-gs + G. Initially ^^ = 0, s = -; and at the end of the constrained motion s=l; therefore at this instant ^ = kI tt^qI; this is in the direction OP. The horizontal and vertical components of this velocity will he There is also a velocity perpendicular to PR and PO, which M ~'^ dt ' /3 and therefore = ^ / -r ql &t the end of the constrained motion : V 4" this is horizontal; therefore the whole horizontal velocity ^Vl^^^ + i^^^Vll^^' 13 therefore the latus rectum of the parabola described = -g- ?. (Ajt. 45.) 50. How must the earth's angular velocity be changed that a body may just lose its weight at the equator ? To ^ a/ -qq- (circular measure) per hour. | 51. A right-angled triangle has its hypotenuse vertical : if three particles slide from rest down the three sides, the velocities acquired will be proportional to the sides. 52. Two bodies are dropped from points P, ^ on to a smooth incUned plane, and reach the bottom of it with the same velocity : prove that the line PQ is perpendicular to the plane. PEOBLEMS. 125 53. A double hollow cone is formed by the revolution of a right- angled triangle about its hypotenuse which is vertical. If any num- ber of particles be let fall at the same instant from different points in the interior surface of the upper cone, and run down the surface of the lower, they will all arrive simultaneously at its vertex. 54. A. heavy body is projected up a rough plane of inclination 60°, with the velocity due to falling freely through 12 feet, and just reaches the top of the plane : given that the body will rest on the plane if it be inclined at any angle not exceeding 30°, find its altitude. [9 feet.] 65. In Atwood's machine, if at the end of each second from the beginning of the motion W be increased, and W diminished by — th of their original difference, then at the end of (w+ 1) seconds, the system will be at rest. 56. A string charged with n + m + l equal weights at equa.1 intervals a, which would rest on an inclined plane with m weights hanging over the top, is placed on the plane, passing over a smooth pulley at the top, with the {m + l)**" weight just over : shew that the velocity when the last weight leaves the plane = si nag. 57. A string loaded with n equal particles at equal distances a, is placed at rest on a smooth horizontal table with one particle just over the edge, where there is a pulley over which the string passes : find the velocity when the last particle leaves the table. \jj{n - 1) ga.] Hence shew that if a heavy string of length c be so placed, its velocity on falling off = \/gc. 58. A seconds pendulum was too long on a certain day by a quantity a, it was then over-corrected so as to be too short by a during the next day. Shew that the nimiber of minutes gained in the two days = 1080 ^ nearly, when I = the proper length of the pendulum. 126 PROBLEMS. 59, A seoaa4a pendulum tanging against the smooth face of a slightly inclined wall and swinging in its plane, ia observed to lose s seconds in t, hours : find the inplination of the w*ll to the horizon. t'^^"0-i8w«)^^'^^y-] 60. A smooth tube is bent into the form of a circular arc greater than a semicircle, and placed in a vertical plane with the open ends upwards and in the same horizontal line. Find the velo- city -with which a ball that exactly fits the tube must be projected from the low«st point so as to pass out at one end and re-enter at the other. [J gd {sec a + 2 -I- 2 cos a), the length of the tube being 2a (ir — a).] 61. A rigid framework consists of a square base ABC J), two uprights £F, QH, rising from E, &, the middle points of AB, CD, and a cross beam FS, the uprights being of such height that AFB, GH.B are equilateral triangles : the framework rests on a rough hori- zontal table, and from FH a heavy body is suspended by a fine string, and oscillates through a semicircle in a vertical plane perpendictdar to WS. Prove that the framework will be tilted if its weight do not 3 exceed n x weight of the suspended body, A. 62. An elastic string has its length increased by h, when a weight W is suspended from it at rest, and the greatest weight which it can bear is (w -i- 1) IT: if TF be let fall from any height not exceeding ^ (^ + 1)4 above it^ pogition of equUibrinm, tie string will not be broken. A- 63. A center of attractive force revolves in a horizontal circle of radius a, with the velocity due to falling from the center to the circumference under the action of gravity ; its intensity varies as the distance and at a distance a producea an acceleration g,. A smooth ring slides on a concentric horizontal circle of radius 2a, being initially at rest and at its, least distance from the center of force j if at time i the distance of the ring from the center of forqe gubtends 2,-0, angle ^ g.t the center of the circles, shew^ that tan-7- = 4 s/I., €" +1 PROBLEMS. 127 A. 64. A smooth, circular wire is made to revolve with an uniform angular velocity about ia vertical idiameter: detwmine the motion of a small heavy ring placed on it at a given point and initi- ally at rest. Find also the strain on the wire, and the condition that the ring may oscillate. [For oscillation