■\ \\\ \ \ * \ ^ ■" i^ ^\^ ^ *"^'% >.\^^ \^ §0htKi ^mv^ MMv^tm 1903 Cornell University Library arV17832 Elementary text-book on physics / 3 1924 031 224 649 olin,anx Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031224649 ELEMENTARY TEXT-BOOK ON PHYSICS BY PROFESSOR WILLIAM A. ANTHONY OF CORNELL UNIVERSITY AND PROFESSOR CYRUS F. BRACKETT OF THE COLLEGE OF NEW JERSEY NEW YORK JOHN WILEY AND SONS 1884 Copyright by JOHN WILEY & SONS, 1884. D^F ELECTKOTVPED AND PRINTED BY RAND, AVERY, AND COMPANY, BOSTON. PREFACE. In the preparation of this work, the authors have had in view the presentation, to our own classes, of the Science of Physics in its present condition. We have undertaken the task because we have been unable to find among existing text-books any one suited to our needs. The demonstrations given presuppose a knowledge of mathe- matics through plane trigonometry. In two or three cases, to make a subject complete, results have been derived requiring a knowledge of analytical geometry for their full comprehen- sion. The discussions leading to these results may, however, be omitted without any breach of continuity of the work. In the selection of the matter to be presented in this " First Part " of the work, we have had in view the subjects. Electricity and Magnetism, and Acoustics and Optics, which are to follow ; and we have endeavored to give such a treatment of the laws of motion, and of the action of force, as would enable the stu- dent to master easily the mechanical problems presented in the action of electrical forces, and in the study of the vibratory motions constituting sound and light. In the treatment of the subject, we have endeavored to be concise and exact, and we have tried to present the matter in the simplest form that a rigid treatment would admit of. IV PREFACE. We are well aware that to master some of the subjects pre- sented, will require hard work on the part of the student ; but we have not for that reason deemed it advisable to omit those subjects, or to confine ourselves to a mere statement of results reached. In the preparation of the work, we have made free use of all accessible sources of information. We wish especially to acknowledge our indebtedness to the works of Thomson, Tait, Maxwell, Jamin, VioUe, Miiller, and Stewart. Part II., which will complete the work, is in progress, and will be issued as soon as circumstances will admit. WILLIAM A. ANTHONY. CYRUS F. BRACKETT. TABLE OF CONTENTS. BOOK I. INTRODUCTION. Page. Divisions of Natural Science 3 Methods of Study 4 Measurements 6 Vernier 7 Cathetometer 9 Dividing Engine 10 SPHEROMETER II Units 11 Dimensions of Units 13 CHAPTER I. MECHANICS OF MASSES. Kinematics, Dynamics 15 Motion, Velocity i6 Momentum, Acceleration 17 Composition and Resolution of Motions, Velocities, and Accelerations 19 Simple Harmonic Motion 21 Force 28 Field of Force 29 Newton's Laws of Motion 30 Collision of Bodies 32 Inertia 34 V VI CONTENTS. Page. Work and Energy 34 Difference of Potential 36 Theorems relating to Difference of Potential ... 40 Moment of Force 44 Couple 45 Centre of Inertia 46 Mechanical Powers 47 Angular Velocity 50 Composition of Angular Velocities 51 Foucault's Pendulum 54 The Gyroscope 55 Moment of Inertia 58 Central Forces 63 CHAPTER II. UNIVERSAL ATTRACTION. Historical 69 Kepler's Laws 70 Measurement of the Force of Gravity 72 The Balance 78 Density of the Earth 82 Projectiles 84 CHAPTER III, MOLECULAR MECHANICS. General Properties of Matter 87 Molecules gg Composition of Bodies gg States of Aggregation go Structure of Solids gn Crystal Systems oo Capillary Phenomena oj Surface Tension n. Equation of Capillarity og Angles of Contact qg Plateau's Experiments jqq Jurin's Law 102 Movements of Solids •••..... 106 CONTENTS. vii Page. Diffusion of Liquids 107 Co-efficient of Diffusion 108 Dialysis jjq Diffusion of Gases m Elasticity 112 Modulus of Elasticity u, Boyle's Law n. Shears u- Elasticity of Tension 118 Elasticity of Torsion 120 Elasticity of Flexure 122 CHAPTER IV. MECHANICS OF FLUIDS. Pascal's Law laq The Barometer 128 Archimedes' Principle 131 Specific Gravity lyz Motions of Fluids 13c Velocity of Efflux 136 Vortices 142 AiR-PuMPS 144 Manometers 148 Limitations to the Accuracy of Boyle's Law . . . 149 BOOK II. PRELIMINARY. General Effects of Heat 155 Production of Heat, Nature of Heat .... 156 CHAPTER I. THERMOMETRY. Temperature, Construction of Thermometers . . . 159 Air Thermometer 162 Registering Thermometers 164 VIU CONTENTS. CHAPTER II. CALORIMETRY. Page. Unit op Heat i66 Specific Heat 167 Ice Calorimeter 168 Method of Mixtures 17° Method of Comparison 172 Method of Cooling 172 Determination of the Mechanical Equivalent of Heat, 173 CHAPTER III. EFFECTS OF HEAT. — SOLIDS. — LIQUIDS. Expansion of Solids 177 Measurement of Co-efficients 178 Absolute Expansion of Mercury 179 Apparent Expansion of Mercury 181 Expansion of Water 183 Determination of Specific Gravity 184 Conduction, Flow of Heat Across a Wall . . . .185 t' — t Flow Proportional to 187 Conductivity 188 Measurement of Conductivity 189 Conductivity diminishes with Rise of Temperature . . 190 Conductivity of Liquids 191 Specific Heat varies with Change of Temperature . . 191 Specific Heat changes with Change of Physical State, 192 Atomic Heat 192 Fusion and Solidification 193 Heat Equivalent of Fusion ig6 Determination of the Heat Equivalent of Fusion . . 197 CHAPTER IV. GASES AND VAPORS. Vaporization loq Pressure of Vapors 201 Ebullition 201 CONTENTS. IX Page. Production of Vapor in a Limited Space .... 202 Spheroidal State 203 Non-saturated Gases and Vapors 205 Pressure at Maximum Density 206 Kinetic Theory of Gases 208 Kinetic Theory accounts for the Laws of Boyle and Gay-Lussac 212 Mean Velocity of Molecules 213 Elasticity of Gases 213 Specific Heats of Gases 214 The Two Specific Heats and the Two Elasticities have THE Same ILatio 219 Heat Equivalent of Vaporization 221 Dissociation 222 CHAPTER V, HYGROMETRY, Object of Hygrometry 224 Dew Point 225 Relative Humidity 226 CHAPTER VI. THERMO-DYNAMICS. First and Second Laws of Thermo-Dynamics . . .227 Reversible Engine 228 Carnot's Cycle 230 Efficiency = — — 233 Absolute Scale of Temperatures 235 The Steam-Engine 237 Hot-air and Gas Engines 238 Water-Power and Wind-Power 241 Energy of Fuel 241 Animal Heat and Work 242 Tidal Energy 244 Solar Energy 245 Dissipation of Energy 245 Book I. MECHANICS. INTRODUCTION. I. Divisions of Natural Science. — Every thing which can affect our senses, we call matter. Any limited por- tion of matter, however great or small, is called a body. All bodies, together with their unceasing changes, con- stitute Nature. Natural Science makes us acquainted with the prop- erties of bodies, and with the changes, or phenomena, which result from their mutual actions. It is there- fore conveniently divided into two principal sections, — Natural History and Natural Philosophy. The former describes natural objects, classifies them according to their resemblances, and, by the aid of Natural Philosophy, points out the laws of their pro- duction and development. The latter is concerned with the laws which are exhibited in the mutual action of bodies on each other. These mutual actions are of two kinds : {a) those which leave the essential properties of bodies unaltered, and ip) those which effect a complete change of prop- erties, resulting in loss of identity. Changes of the first kind are called physical changes : those of the sec- ond kind are called chemical changes. Natural Philoso- phy has, therefore, two subdivisions, — Physics and Chemistry. 4 ELEMENTARY PHYSICS. Physics deals with all those phenomena of matter that are not directly related to chemical changes. As- tronomy is thus a branch of Physics, yet it is usually excluded from works like the present on account of its special character. It is not possible, however, to draw sharp lines of demarkation between the various departments of Natu- ral Science, for the successful pursuit of knowledge in any one of them requires some acquaintance with the others. 2. Methods. — The ultimate basis of all our knowl- edge of nature is experience, — experience resulting from the action of bodies on our senses, and the conse- quent affections of our minds. When a natural phenomenon arrests our attention, we call the result an observation. Simple observations of natural phenomena, only in rare instances can lead to such complete knowledge as will suffice for a full understanding of them. An observation is the more complete, the more fully we apprehend the attending circumstances. We are generally not certain that all the circumstances which we note are conditions on which the phenomenon, in a given case, depends. In such cases we modify or suppress one of the circum- stances, and observe the effect on the phenomenon. If we find a corresponding modification or failure with respect to the phenomenon, we conclude that the cir- cumstance, so modified, is a condition. We may pro- ceed in the same way with each of the remaining circumstances ; leaving all unchanged, except the single one purposely modified, at each trial, and always observ- ing the effect of the modification. We thus determine. introduction: 5 the conditions on which the phenomenon depends. In other words, we bring experiment to our aid in dis- tinguishing between the real conditions on which a phenomenon depends, and the merely accidental cir- cumstances which may attend it. But this is not the only use of experiment. By its aid we may frequently modify some of the conditions, known to be conditions, in such ways that the phe- nomenon is not arrested, but so altered in the rate with which its details pass before us that they may be easily observed. Again : experiment often leads to new phe- nomena, and to a knowledge of activities before unob- served. Indeed, by far the greater part of our knowledge of natural phenomena has been acquired by means of experiment. To be of value, experiments must be con- ducted with system, and so as to trace out the whole course of the phenomenon. Having acquired our facts by observation and experi- ment, we seek to find out how they are related ; that is, to discover the laws which connect them. The process of reasoning by which we discover such laws is called induction. As we can seldom be sure that we have apprehended all the related facts, it is clear that our inductions must generally be incomplete. Hence it fol- lows, that conclusions reached in this way are at best only probable ; yet their probability becomes very great when we can discover no outstanding fact, and especially so when, regarded provisionally as true, they enable us to foresee what will occur in cases before unknown. In conducting our experiments, and our reasonings likewise, we are often guided by suppositions, suggested -by previous experience. If the course of our experi- 6 ELEMENTARY PHYSICS. ment be in accordance with our supposition, there is, s.o far, a presumption in its favor. So, too, in reference to our reasonings : if all our facts are seen to be consistent with some supposition not unlikely in itself, we say it thereby becomes probable. The term hypothesis is usually employed instead of supposition. Concerning the ultimate modes of existence or action, we know nothing whatever ; hence, a law of nature can- not be demonstrated in the sense that a mathematical truth is demonstrated. Yet so great is the constancy of uniform sequence with which phenomena occur in accordance with the laws which we discover, that we have no doubt respecting their validity. When we would refer a series of ascertained laws to some common agency, we employ the term theory. Thus we find in the " wave theory " of light, based on the hypothesis of a universal ether of extreme elas- ticity, satisfactory explanations of the laws of reflection, refraction, diffraction, polarization, etc. 3. Measurements. — All the phenomena of nature occur in matter, and are presented to us in time and space. Time and space are fundamental conceptions : they do not admit of definition. Matter is equally indefina- ble : its distinctive characteristic is its persistence in whatever state of rest or motion it may happen to have, and the resistance which it offers to any attempt to change that state. This property is called inertia. It must be carefully distinguished from inactivity. Another essential property of matter is impenetra- bility, or the property of occupying space to the exclu- sion of other matter. INTRODUCTION. 7 We are almost constantly obliged, in physical science, to measure the quantities with which we deal. We measure a quantity when we compare it with some standard of the same kind. A simple number expresses the result of the comparison. If we adopt suitable units of length, time, and mass (or quantity of matter), we can express the measure of all other quantities in terms of these so-called funda- mental units. The actual operation of comparing a length with the standard is often difificult of direct accomplishment. This may arise from the minuteness of the object or distance to be measured, from the distant point (at which the measurement is to end) being inaccessible, or from the difficulty of accurately dividing our stand- ard into very small fractional parts. In all such cases we have recourse to indirect methods, by which the difficulties are more or less completely obviated. It will readily occur to the student that the appliances of trigonometry are available. It is important that we be able to estimate small fractions of our units. The vernier enables us to- do this in the case of the unit of length with great conven- ience and accuracy. It consists of an accessory piece, fitted to slide on the principal scale, or arc, of the instru- ment to which it is applied. A portion of the accessory piece, equal to n divisions of the principal scale, is divided into n plus one or into n minus one divisions. In the former case the divisions are numbered in the same sense with those of the principal scale ; in the latter they are numbered in the opposite sense. In either case we can measure a quantity accurately to the one nth part of 8 ELEMENTARY PHYSICS. 10 S (, 1 one of the primary divisions of the principal scale. The cut (Fig. i) will make the construction and use of the vernier plain. . In Fig. I, let o, I, 2, 3 ... 10 be the divisions on the vernier ; let o, i, 2, 3 . . . lo be any set of consecutive divisions on the 3J| 1 principal scale. If we suppose the o of the vernier to be in coincidence with the limiting point of the magnitude to be measured, it is clear, that, from the position shown in the figure, we have 29.7, expressing that magnitude to the nearest tenth ; So\ 1 — . and since the sixth division of the ver- nier coincides with a whole division of the principal scale, we have ^^ of ^, or Y^^ of a principal division to be added : hence the whole value is 29.76. The micrometer screw is also much employed. It consists of a carefully cut screw, accurately fitting in a nut. The head of the screw carries a graduated circle, which can turn past a fixed point designated by a fiducial mark. This is frequently on a scale whose divisions are equal in mag- nitude to the pitch of the screw. These divisions will then show through how many revolutions the screw is turned in any given trial ; while the divisions on the graduated circle will show the fractional part of a revo- lution, and consequently the fractional part of the pitch that must be added. If the screw be turned through n revolutions, as shown by the fiducial scale, and through an additional fraction, as shown by the divided Fig. 1. INTROD VCTION. «e P circle, it will pass through n times the pitch of the screw, and an additional fraction of the pitch deter- mined by the ratio of the number of divisions read from o on the divided circle to the whole number, into which it is divided. The cathetometer is used for measuring differences of level. A graduated scale is cut on an upright bar, which can turn about a vertical axis. Over this bar slide two accurately fitting pieces, one of which can be clamped to the bar at any point, and it can serve as the fixed bearing of a micrometer screw. The screw runs in a nut in the second piece, which*has a vernier attached, and carries a horizontal telescope furnished with cross- hairs. The telescope having been made accurately horizontal by means of a delicate level, cross-hairs are made to cover one of the points whose dif- erence of level is sought, and the reading upon the scale is taken ; the fixed piece is then undamped, and the telescope raised or lowered until the second point is covered by the cross- hairs, and the scale reading is again taken. The differ- ence of scale reading is the difference of level sought. The dividing engine may be used for dividing scales or for comparing lengths. In its usual form it consists lO ELEMENTARY PHYSICS. essentially of a long micrometer screw, carrying a table, which slides, with a motion accurately parallel with itself, along fixed guides, resting on a firm support. To this table is fixed an apparatus for making succes- sive cuts upon the object to be divided. Fig. 3. The object to be graduated is fastened to the fixed support. The table is carried along through any re- quired distance determined by the screw, and the cuts are thus made at the proper intervals. The same instrument, furnished with microscopes and accessories, may be employed for comparing lengths with a standard. It may then be called a comparator. The spherometer is a special form of the micrometer screw. As its name implies, it is primarily used for measuring the curvature of spherical surfaces. It consists of a screw with a large head, divided into a great number of parts, — say, five hundred, — turning INTRODUCTION. II in a nut supported on three legs, whose pointed ends form the vertices of an equilateral triangle. The axis of revolution of the screw is perpendicular to the plane of the triangle, and passes through its centre. The screw ends in a point which may be brought into the same plane with the points of the legs. This is done by placing the legs on a truly plane sur- face, and turning the screw till its point is just in con- tact with the surface. The sense of touch will enable one to decide with great nicety when the screw is turned far enough. If, now, we note the reading of the divided scale, and also that of the divided head, and then raise the screw, by turning it backward, so that the given curved surface may exactly coincide with the four points, we can compute the radius of curvature from the difference of the two readings and the known length of the side of the triangle formed by the points of the tripod. 4. Units. — The unit of length is the metre. The standard is a certain piece of platinum, declared by legislative act to be the standard. It is preserved in the archives of France. This standard was designed to be equal in length to one ten millionth of the earth's quadrant. Its mul- tiples and submultiples are in accord with the decimal notation. The n'ames of the former are derived from the Greek, the latter from the Latin. (The table will Fig. 4. 12 ELEMENTARY PHYSICS. exhibit these terms, and also the relation of the metric to other systems.) It is often necessary to measure angles. For this purpose we employ the divided circle, on whose divis- ions are based the ordinary trigonometrical tables. For great accuracy in reading minute subdivisions, the ver- nier is attached. The telescope with cross-hairs is used, mounted on the axis of the circle, and turning with it, to secure allignment with the objects or points whose angular distance is to be found. The unit of time is the mean time-second, which is the •j^lxriS' °f ^ mean solar day. We employ the clock, regulated by the pendulum or the chronometer balance, to indicate seconds. This, while sufficient for ordinary use, must for accurate investigations be frequently cor- rected by astronomical observations. Smaller intervals of time than the second are meas- ured by causing some vibrating body, as a tuning-fork, to trace its path along some suitable surface, on which also the beginning and end of the event whose duration we wish to know are recorded. The number of vibra- tions traced while the event is occurring determines its duration in known parts of a second. In estimating the duration of certain phenomena giving rise to. light, the revolving mirror may be employed. By its use, with proper accessories, inter- vals as small as the yoTo o^'o oTo o^ ^ second have been estimated. — Rood. The unit of mass may be any arbitrarily chosen piece of matter suitable for convenient reference. Those usually adopted are such as have been set apart by legislative enactment, and declared to be standards. INTROD UCTION. 1 3 That one which is usually adopted in scientific work is the gram, which is the thousandth part of a standard kilogram, preserved in the archives of France. It is intended to be equal in mass to one cubic centimetre of water at its greatest density. The names of the divisions and multiples are anala- gous to those of the metre. For the relation of the gram to the masses of other systems still used in commerce and in the arts, see table. Masses are compared by means of the balance, the principles of whose construction will be discussed here- after. 5. Dimensions of Units. — Any derived unit may be represented by the product of certain powers of the symbols representing the fundamental units of time, space, and mass. Thus, the unit of area varies as the square of the unit of length ; hence its dimensional equation is area = L^. In like manner, the dimen- sional equation for volume is vol. = L^. Any equation showing what powers of the funda- mental units enter into the expression for the derived unit is called its dimensional equation. CHAPTER I. MECHANICS OF MASSES. 6. The general subject of motion is usually divided, in extended treatises, into two topics, — Kinematics and Dynamics. In the first are developed, by purely mathematical methods, the laws of motion considered in the abstract, independent of any causes producing it and of any substance in which it inheres ; in the second these mathematical relations are extended and applied, by the aid of a few inductions drawn from universal experience, to the explanation of the motions of bodies, and the discussion of the interactions which are the occasion of those motions. For convenience merely, the subject of Dynamics is further divided into Statics, which treats of forces as maintaining bodies in equilibrium and at rest, and Kinetics, which treats of forces as setting bodies in motion. In this treatise it has been found more convenient to make no formal distinction between the mathemat- ical relations of motion and the application of those relations to the study of forces and the motions of bodies. The subject is so extensive that only those fundamental principles and results will be presented 15 1 6 ELEMENTARY PHYSICS. which have direct application in subsequent parts of the work. 7. motion is the change in position of a material particle. It is recognized by a change in the configura- tion of the system containing the displaced particle ; that is, by a change in the relative positions of the par- ticles making up the system. Any particle in the system may be taken as the fixed point of reference, and the motion of the others may be measured from it. Thus, for example, high-water mark on the shore may be taken as the fixed point in determining the rise and fall of the tides ; or, the sun may be assumed to be at rest in computing the orbital motions of the planets. We can have no assurance that the particle which we assume as fixed is not really in motion as a part of some larger system ; indeed, in almost every case we know that it is thus in motion. As it is impossible to con- ceive of a point in space recognizable as fixed and determined in position, our measurements of motion must always be relative. One important limitation of this statement must be made : by proper experiments it is possible to deter- mine the absolute angular motion of a body rotating about an axis. 8. Path. — The moving particle must always describe a continuous line or path. In all investigations the path may be represented by a diagram or model, or by reference to a set of assumed co-ordinates. 9. Velocity. — The velocity of a particle is its rate of motion. It may be either constant or variable. If constant, it is measured by the space traversed by the particle in a unit of time. If variable, it is measured MECHANICS OF MASSES. I7 by the space which would be traversed by the particle in a unit of time if the velocity at the instant were to remain constant. The practical unit of velocity is the velocity of a body moving uniformly through one centimetre in one second. The velocity of a particle is given, therefore, by the quotient of the number of centimetres traversed divided by the number of seconds in which it is traversed. The formula expressing uniform velocity is - = 7 (I) " in which s represents space and t time. 1 ( 5 - The dimensions of velocity are LT~^. ^^";''\'^-' 10. Uomenium. — The momentum of a body is its quantity of motion. This varies with its mass and its velocity jointly, and is measured by their product. Thus, for example, a body weighing ten grams, and having a velocity of ten centimetres, has the same momentum as a body weighing one gram, and having a velocity of one hundred centimetres. The practical unit of momentum is that of a gram of matter moving with the unit velocity. The formula is I «^v, (2) 1^^^^ where m represents mass. The dimensions of momentum are ML Z'"\ 11. Acceleration. — When the velocity of a particle varies, its rate of change is called the acceleration of the particle. Acceleration is either positive or nega- tive, according as the velocity increases or diminishes. 1 8 ELEMENTARY PHYSICS. Acceleration, if constant, is measured by the change in the velocity of the particle in a unit of time : if variable, it is measured by what would be the change in the velocity if from that instant the acceleration were to remain constant. The practical unit of acceleration is that of a particle whose velocity changes one unit of velocity in one second. Acceleration is therefore measured by the quotient of the number of units of change of velocity divided by the number of seconds during which the change has occurred. The formula expressing a constant acceleration is /= ^, (3) in which v„ and v represent the initial and final veloci- ties respectively. The dimensions of acceleration are LT''^. The space traversed by a point moving with a con- stant acceleration /, during a time /, is determined by considering, that, since the acceleration is constant, the average velocity for the time t, multiplied by t, will represent the space traversed : hence s = l±I^t, (4) 2 or since - = — — —, we have, in another form, 2 2 ' s = Vol^-\ff. (4) Multiplying equations (3) and (4), we obtain v^ == v^ -\- ifs. (5) MECHANICS OF MASSES. I9 When the point starts' from rest, v„=.o, and equations (3), (4), and (5) become v ^^ft, j = - ffi, and v^ = zfs. Formula (4) may also be obtained by a geometrical construction. At the extremities of a line AB (Fig. 5), equal in length to t, erect perpendiculars A C and BD, equal in length, proportional to the initial and final velocities of the mov- ing-point. For any interval of time Aa so short that the velocity during it may be considered constant, the space described is rep- resented by the rectangle Ca, and the space described in the whole time t, by a point moving with a velocity increasing by successive in- *"*"'' crements, is represented by a series of rectangles, eb, fc, gd, etc., described on ab, be, cd, etc. If ab, be . . . are diminished indefinitely, the sum of the areas of the rectangles can be made to approach as nearly as we please the area of the quadrilateral ABCD. This area, therefore, represents the space traversed by the point, having the initial velocity z/„ and moving with the acceleration /, through the time t. But ABCD is equal to AC • t-\- {BD — AC) i -^ 2 ; whence s = vj+l/i'. (4) 12. Composition and Resolution of Motions, Velocities, and Accelerations, — If a point «i move with a constant velocity relative to another point a^, and this point 122 move with a constant velocity relative to a third point ^8, then the motion, in any fixed time, of «i relative to «a may be readily found. 20 ELEMENTARY PHYSICS. Represent the motion, in a fixed time, of a-^ relative to «2 (Fig. 6) by the line z'ai and of a^, relative to a^ by the line v^. Now, it is plain that the motions v^ and ^8, whether acting successively or simultaneously, will bring the point a-y to B ; and also that if any portions of these motions Ab and be, occurring in any small portion of time, be taken, they will, because the veloci- ties of a-, and a« are constant or B proportional to v^ and z/„ bring the point aj to some point c lying on the line joining A and B. Therefore the diagonal AB of the parallelogram having the sides v^ and v^ fully represents the motion of a-^ relative to as- The line AB is called the resultant, of which the two lines Vi and v^ are the components. This proposition may now be stated generally. The resultant of any two simultaneous motions, represented by two lines drawn from the point of reference, is found by completing the parallelogram of which those lines are sides, when the diagonal drawn from the point of reference represents the resultant motion. The resultant of any number of motions may be found by obtaining the resultant of any two of the given com- ponents, by means of the parallelogram as before shown, using this resultant in combination with another component to obtain a new resultant, and proceeding in this way till all the components have been used. Or, represent the given motion by one side of a poly- gon. Its components may be represented by the remaining sides. And conversely, if it be required to MECHANICS OF MASSES. 21 find the resultant of a number of components, they may be laid off as the consecutive sides of a polygon when the line required to complete the polygon is the result- ant sought. Motions are usually resolved along three rectangular axes by means of the trigonometrical functions. Thus, if a be the line representing the motion, and Q, , and ^ the angles which it makes with the three axes, the com- ponents along those axes are a cos 6, a cos <^, and a cos = a cosf ^ \ = a cost -=^ ), the projection of B on the diameter OY also has a simple harmonic motion, differing in epoch from that in diameter OX by \T. It follows immediately, that the composition of two simple harmonic motions at right angles to one another, having the same amplitude and the same period, and differing in epoch by one quarter the period, will produce a motion in a circle of radius a with a constant velocity. More generally the co-ordinates of a point moving with two simple harmonic motions at right angles to one another are x = a cos( — e), and y = d cos ' are commensurable, that is, if ^' = ncfi, the curve is re-entrant. Making this supposition, X = a cos (j) cos £ -f- a sin ^ sin e, and y — b cos n^. 26 ELEMENTARY PHYSICS. Various values may be assigned to a and to b and to n. Let a and b be equal and « = i ; then a; = jv cos £ + {a^ — /)- sin e ; from which ^2 _ 2xy cos £ + ycos^f = a%in^£ — ysin^£, or, x^ — 2X)' cos £ + y = fl^sin^e. This becomes, when £ = 90°, x^ -\-y'^=- a^, the equation for a circle ; and if £ = 0°, the equation for a straight line through the origin, making an angle of 45° with the axis of X. With intermediate values of £, it is the equation for an ellipse. If we make ;? = 2, we obtain, as special cases of the curve, a parabola and a lemnis- cate, according as £ = 0° or 90°. If a and b are unequal, and ;? = i, we get, in general, an ellipse. If a line in which a point is describing a simple harmonic motion be made to move in a direction per- pendicular to itself, the moving-point will describe an harmonic curve, called also a curve of sines. It is in outline a simple wave. Examples of such curves are found in the waves of the ether which constitute light. If the ordinates of the curve be taken, not to represent displacement transversely from a fixed line, but dis- placement longitudinally from points of equilibrium along a fixed line, the curve may be employed to repre- sent the longitudinal vibrations of the air when trans- mitting sound. The length of the wave is the distance traversed by the moving line during the period of the simple harmonic motion, and is, therefore, the distance between any two identical conditions of points on the line of progress of the wave. The amplitude of the wave is the amplitude of the simple harmonic motion. MECHANICS OF MASSES. 27 ^ If we assume the origin of co-ordinates such that the epoch of the simple harmonic motion at the axis of ordinates is o, the displacement from the line of prog- ress of any point on the wave is represented by s ■= a cosi {-l^} The displacement due to any other wave differing from the first only in the epoch is represented by s, = a cosi (3.1.-.). We shall now show, in the simplest case, the result of compounding two wave motions. The displacement due to both waves is, of course, j-f jj = a!l c0S25r-^-f COsf 27r-^ — ej / i t . t . \ = al cos 2w-^-+- cos 27r:=,C0S e -f Sm 2ir ™Sin ej = a cos 27r^(i -t- cos e) -f- sin 2ir-=,sin e 1. If we assume a value A, such that A cos \ From the two equations containing A, we obtain, by adding the squares of the values of A sin <^ and A cos , A — {20^ -h 2 Aos e)* J 28 ELEMENTARY PHYSICS. and, by dividing the value of A sin ^ by that of A cos ^, we obtain 1 sin c i> = tan~^ — ; ^ I + cos £ The displacement thus becomes 6'= a(2 + 2 cose)'cos( 27r^ — tan"^ — ; )• (9) ^ ' \ T I + cos €/ This equation is of great value in the discussion of problems in optics. The principle suggested in the above discussion, that the resultant of the composition of two simple harmonic motions is an harmonic motion whose elements depend on those of the components, was proved by Fourier to hold generally. Fourier's theorem maybe stated as follows: "Any complex periodic function may be resolved into a num- ber of simple harmonic functions whose periods are commensurable with that of the original function." As an example, any wave not simple may be decom- posed into a number of simple waves whose lengths are to each other as \, \, \, etc. The number of these simple waves is, in general, infinite, but in special cases determinate both as to number and to period. 14. Force. — Whenever any change occurs, or tends to occur, in the momentum of a body, we ascribe it to a cause called a force. Whenever motions of matter are effected by our direct personal effort, we are conscious, through our muscular sense, of a resistance to our effort, involving an expendi- ture of muscular energy to overcome it. The concep- tion of force to which this consciousness gives rise, we MECHANICS OF MASSES. 29 transfer, by analogy, to the interaction of any bodies which is or may be accompanied by change of motion. The question whether this analogy is or is not valid, is not involved in a purely physical discussion of the sub- ject. A force, in the physical sense, is the assumed cause of an observed change of motion. It is known and measured solely by that change. It is, therefore, if constant, fully expressed by the acceleration which it will impart to unit mass. If momentary or variable, it is measured, at any instant, by the change of momen- tum which it would cause in unit time if during that time it were to remain constant. The formula for force is, therefore, m — - — = mf. (10) Its dimensions are ML T~^. As acceleration is always referred to some fixed direc- tion, it follows that force is a quantity having direction. The product of the time during which a force acts by its mean intensity is called the impulse of the force. The practical unit of the force is the dyne, which is the force that can impart to a gram of matter one unit of acceleration ; that is to say, one unit of velocity in one second. 15. Field of Force. — A field of force is a region such that a particle constituting a part of a mutually inter- acting system, placed at any point in the region, will be acted on by a force, and will move, if free to do so, in the direction of the force. The particle so moving would, if it had no inertia, describe what is called a line of force, the tangent to which, at any point, is the direction of the force at that point. The strength of 30 ELEMENTARY PHYSICS. the field at a point is measured by the force developed by unit quantity at that point, and is expressible, in terms of lines of force, by the convention that each line represents a unit of force, and that the force acting on unit quantity at any point varies as the number of lines of force which pass perpendicularly through unit area at that point. Each line, therefore, represents the direction of the force, and the number of lines in unit area the strength of field. An assemblage of such lines of force considered with reference to their bounding-sur- face is called a tube of force. i6. Newton's Laws of Motion. — We are now ready for the consideration of the laws of motion, first formally enunciated and successfully applied by Newton, and hence known by his name : — Lex I. — Corpus omne perseverare in statu suo qui- escendi vel movendi uniformiter in directum, nisi qua- tenus illud a viribus impressis cogitur statum suum mutare. Lex II. — Mutationem motus proportionalem esse vi motrici impressae & fieri secundum lineam rectam qua vis ilia imprimitur. Lex. III. — Actioni contrariam semper & aequalem esse reactionem ; sive corporum duorum actiones in se mutuo semper esse aequales & in partes contrarias dirigi. The subjoined translations are given by Thomson and Tait : — Law I. — Every body continues in its state of rest or of motion in a straight line, except in so far as it may be compelled by force to change that state. Law II. — Change of motion is proportional to force MECHAmCS OF MASSES. 3 1 applied, and takes place in the direction of the straight line in which the force acts. Law III. — To every action there is always an equal and contrary reaction : or, the mutual actions of any two bodies are always equal and oppositely directed. 17. Discussion of the Laws of Motion. — (i) The first law is a statement of the important truths implied in our definition of force, — that motion, as well as rest, is a natural state of matter ; that moving bodies, when entirely free to move, proceed in straight lines, and describe equal spaces in equal times ; and that force is the cause of any deviation from this uniform rectilinear motion. That a body at rest should continue indefinitely in that state seems perfectly obvious as soon as the propo- sition is entertained ; but that a body in motion should continue to move in a straight line is not so ob- vious, since motions with which we are familiar are frequently arrested or altered by causes not at once apparent. This important truth, which is forced upon us by observation and experience, may, however, be presented so as to appear almost self-evident ; for if we conceive of a body moving in empty space, we can think of no reason why it should alter its path or its rate of motion in any way whatever. (2) The second law presents, first, the proposition on which the measurement of force depends ; and, sec- ondly, states the identity of the direction of the change of motion with the direction of the force. Motion is here synonymous with momentum as before defined. The first proposition we have already employed in de- riving the formula representing force. The second. 32 ELEMENTARY PHYSICS. with the further statement that more than one force can act on a body at the same time, leads directly to a most important deduction respecting the combination of forces ; for the parallelogram law for the resolution and composition of motions being proved, and forces being proportional to and in the same direction as the motions which they cause, it follows, if any number of forces acting simultaneously on a body be represented in direction and amount by lines, that their resultant can be found by the same parallelogram construction as that which serves to find the resultant motion ; whence we derive the parallelogram and the polygon of forces. In case the resultant of the forces acting on a body be zero, the body is said to be in equilibrium. (3) When two bodies interact so as to produce, or tend to produce, motion, their mutual action is called a stress. If one body be conceived as acting, and the other as being acted on, the stress, regarded as tending to produce motion in the body acted on, is a force. The third law states that all interaction of bodies is of the nature of stress, and that the two forces into which the stress can be resolved are equal and oppositely directed. From this follows directly the deduction, that the total momentum of a system is unchanged by the inter- action of its parts ; that is, the momentum gained by one part is counterbalanced by the momentum lost by the others. This principle is known as the conserva- tion of momentum. 18. Collision of Bodies. — If two bodies, m-^^ and m^, with velocities v^ and v^ in the same line, impinge, their velocities after contact are found, in two extreme cases, as follows : — MECHANICS OF MASSES. 33 (i) If the bodies are perfectly inelastic, there is no tendency for them to separate, their final velocities will be equal, and their momentum will be equal to the sum of their separate momenta; hence m^^ + m^^ = (wj + m^x, (11) where x is the velocity after impact. (2) If the bodies are perfectly elastic, they tend to separate with a force equal to that by which they are compressed. Let V represent their common velocity just at the instant when the resistance to compression balances the impulsive force. Then the change in momen- tum in each body up to this instant is m-^iy-^ — v), or m^i^v — ■z/g) ; and the further change of momentum, by reason of the elasticity of the bodies, is the same, whence the whole momentum lost by the one is 2m-^{v-^ — v) and that gained by the other zm^i^v —v^ ; that is, /Wj^i — m^x = 2m^^ — 2m-^, if X represent the final velocity of m^. From this X = 2V — v^. In like manner, y representing the final velocity of m^, we find y = 2V ~ v^. From the formula for inelastic bodies, which is applica- ble at the moment when both bodies are moving with the same velocity. V ■■ m^ + m^ 34 ELEMENTARY PHYSICS. whence, finally, X = ; > (12) 19. Ineriia. — The principle of equality of action and re-action holds equally well when we consider a single body as acted on by a force. The resistance to change of motion offered by the inertia of the body is equal in amount and opposite in direction to the acting force. Inertia is not of itself a force, but the property of a body, enabling it to offer a resistance to a change of motion. 20. Work and Energy. — When a force causes motion through a space, it is said to do work. The measure of work is the product of the force and the space traversed by the body on which the force acts. The formula expressing work is evidently w = Fs = mfs. (13) Its dimensions are MLP'T~^. It appears, then, that; in the defined sense of the term, no work is done upon a body by a force unless it is accompanied by a change of position ; and, further, that the amount of work is independent of the time taken to perform it. Both of these statements need to be made, because of a natural tendency to confound work with conscious effort, and to estimate it by the effect on our system. A body may, in consequence of its motion or position with respect to other bodies, have a certain capacity MECHANICS OF MASSES. 35 for doing work. This capacity for doing work is its energy. Energy is of two kinds, usually distinguished as potential and kinetic. The former is due to the position of the body, the latter to its motion. If a mass m be moving with a velocity v, its capacity for doing work may be determined from the considera- tion, that, if the motion be opposed by a force F equal to mf, the mass suffers a negative acceleration, and is finally brought to rest after traversing a space s equal to — , m opposition to the force. (See Eq. 5, p. 19). Multiplying both sides of this equation \>y F=. mf, we have Fs = But Fs is the work done by the body against the force F, and is, therefore, the capacity which the body originally had for doing work. This capacity — that is, the kinetic energy of the body — is then represented by the expression It can also be shown that the difference between the kinetic energy of a body at the beginning and at the end of a given space is equal to the work done in traversing that space. For, if we consider the mass m having an acceleration /, and moving through a space s, so small that the acceleration may be assumed constant, we have, from Eq. 5, p. 18, v^ = v^ ± 2fs, whence ^mv^ = ^mv^ ± mfs, the plus or minus sign being used according as the work is done by or against the force. Since any motion whatever may be divided into portions in which the 36 ELEMENTARY PHYSICS. above conditions hold true, it follows that we have finally, for any motion, \mv^ = \mv^ ± mf^s^ ± mfs" ± . . . = \inv^ ± %mfs. Since \mv^, or the initial kinetic energy, is a con- stant quantity, it follows that \mv^ =F Sm/j, or the sum of the kinetic and potential energies, is a constant quantity in any system where the total energy exists in these two forms. In other words, a body, by losing potential energy, gains an equal amount of kinetic energy ; and the kinetic energy, being used to do work against acceleration, places the body in a position where it again possesses its original amount of poten- tial energy. So, evidently, if a body start from rest, the kinetic energy acquired by motion through a certain distance is equal to the potential energy of the body at rest due to the possibility of its traversing that distance. Since the dimensions of energy are expressed in every case by ML^T-^, and since the square of a length can- not involve direction, it follows that energy is a quantity independent of direction. The practical unit of energy is the erg, and is the energy of a body so conditioned that it can exert a force of one dyne through a space of one centimetre. 21. Difference of Potential. — In the discussion which follows, we deal with forces which vary directly as the product of the quantities acting, and inversely as the squares of the distances which separate them. The difference of potential between two points in a field of force is measured by the work done in moving a test unit of the quantity to whose presence the force is due, from one point to the other. MECHANICS OF MASSES. 37 If the field is due to the presence of an acting quan- tity m, which repels the test unit, at a distance s, with a force expressed by Km the difference of potential between two points distant, r and R, from the acting quantity m is expressed by Vr-Vn = Kmil-'^. The symbol K represents the force with which two unit masses at unit distance repel one another. To obtain this formula in the simplest case, let us sup- pose a body at the point (Fig. 9) acting upon a unit at P TS RQP I U l_U Fig. 9. TCwi with a force equal to -7y^ If the body move to Q, the force at Q is pr-^ ; and the average force acting while the body is moving in the path PQ, provided this path be taken small enough, is Km OP- C(2" The work done through PQ is The work done in moving through any other small space QR towards S is, similarly, 38 ELEMENTARY PHYSICS. The last value obtained by moving from 5 to T is The sum of these values, gives the work done in passing over the space PT. Designating OT 2ls r and OP as R, and performing the operation indicated by the formula for every other acting quantity at O, we obtain the expression, %Km (-:-i) for the difference of potential between P and T, where S is a summation sign. Since the point T can be considered as on the sur- face of a sphere of which O is the centre, and since the force Ym ^^^^ along TO perpendicular to that surface, and cannot, therefore, have a component tending to produce motion on that surface, there is no work done in moving the unit at T over the surface "of the sphere to any other point ^ on it : it follows that the difference of potential between P and any other point at distance r from is the same as that between P and T. V.-V^ = Xm(l - ^. (14) Such a surface as the one described, to which the lines of force are perpendicular, is called an equipotential surface. MECHANICS OF MASSES. 39 It is further evident, that the amount of work done upon the unit to move it from P to 2" is independent of the path. For, if this were not so, it would be possible, by moving from P to T on. one path,. and returning on another, to accumulate an indefinite amount of energy ; which the doctrine of the conservation of energy shows to be impossible. If the point P be supposed to be at a distance from O so great that the force at that distance vanishes, it is then at zero potential. R becomes indefinitely large, and the absolute potential at T becomes Vr = S (15) r This formula expresses the work necessary to move the unit against the repulsion of the system at O up to the point T from an infinite distance. If the acting quantity attract the unit, the work is done by the attraction upon the body in so moving up to 7", and the potential is negative. From the definitions, it is plain that the difference of potential between P and T equals the difference of the potential energies of a unit at those points. The amount of work done in moving a unit from P to Q, and hence the difference of potential between P and Q, is represented by where ^^ represents the average force. From this relation we have Km _ Vp- Vq OP- 0Q~ PQ ' 40 ELEMENTARY PHYSICS. or, if PQ be taken small enough, Km _Vp- Vq OP'' PQ whence the force in any one direction at a point in the field of force is equal to the rate of change of potential in that direction, at that point, with respect to space. 22. Theorems relating to Difference of Potential. — (i) The force at any point within a spherical shell of uni- form thickness and density of acting material is zero. For, at the point O (Fig. lo) within the sphere, the force, if an attraction, acting towards ra2.sscdi ais mz.ss ab-. ,^ , i — ^ „ \ and towards ci&- Ofi Fig. 10. expressed by so that the efficient force tending to produce motion, say, towards a, is mass a(5j mass cd^ Oa^ Oc'' Now, if ab-^, cd-^, be taken small enough, they will be frusta of similar cones, and, as a consequence, from which, since the density of the shell is uniform, mass iZi5j mass cd.^ Oc' = o. Since the whole surface of the shell may be cut by similar cones, for which similar equations will hold, the total force exerted by the shell on a particle within it becomes zero. This being so, it follows that the poten- tial throughout the sphere is constant; for no work is MECHANICS OF MASSES. 41 required to move the unit from one point to another in the interior. (2) The potential, and therefore the force at a point due to the presence of a uniform spherical shell of act- ing material, depends only on the quantity of material and on the distance of the point considered, from the centre of the sphere. In the figure (Fig. 1 1), let CKL Ffg. 11. represent a central section of the shell, of which is the centre. Let d represent the mass of unit of area. The potential at B, due to the element s of the sphere ds at ^, is V,^= ^-^ ; and the potential due to the whole UK. sphere is the summation of that due to all the similar elements making up the sphere. Take a point A on the line OB, such that OA- OB = R^, where R is the radius of the sphere, and draw AK, produce it to L, and draw OK and OL. Now, if we represent the angle OKA by a, and the solid angle subtended by the element s, as seen from A, by w, we may express s AK^ in other terms as cos a V,= hence dAK^m BKco^a 42 ELEMENTARY PHYSICS. Now, since, by construction, OB : R = R : OA, and the angle KOB is common to the two triangles KOA and BOK, these triangles are similar ; hence BK:AK= OB: OK, whence AK _ R BK~ OB The value of the potential due to K may be written ^. ,AK R cos a OB The value for the potential of the element at L is, similarly, COS a OB Adding these values, we obtain r, rr .R fAK+AL\ V, + V„ = d-^ ■ J ). OB \ cos a / But AR+AZ KL cos a cos a hence we obtain, finally, = 20K= 2R; R V, + V„=2d OB Now the sphere can be divided into two portions, made up of elements similar to K and L, by a plane passing through A normal to OB. We obtain the whole poten- tial, therefore, by summing all the potential values due to these pairs of elements, whence R^ V=2d-^%.. MECHANICS OF MASSES. 43 The sum of all the elementary solid angles on one side of the plane from the point A in it is, of course, 27r; hence, finally, ^^ R^d M where M is the mass of the spherical shell. Since the force at the point B depends on the rate of change of potential at that point, it varies inversely as the square of the distance OB. Represent OB by /. In the expression V=^^, let / change by a small incre- ment A/, and denote the corresponding change in the potential by A V; then F/-f- FA/+ AF/+AF-A/=»«. As A F arid A/ are both very small quantities, their product may be neglected. We then have r-A/-|-AF-/=o, or AF__ F_ _m The force, therefore, at a point outside a sphere of uniform density varies inversely as the square of its distance from the centre of the sphere. This result enables us to deal with spheres of gravitating matter, or spherical shells, upon which is a uniform distribution of electricity, as if they were gravitating or electrified points. (3) If, in the last proposition, we let / = 7?, we obtain for the value of the force just outside the shell 44 ELEMENTARY PHYSICS. Since the force just inside the shell vanishes, in con- sequence, as we have seen, of the equal and opposite actions of the portions of the sphere ab and acb (Fig. 12), and since the total force at the point P outside the sphere is if-^d, it follows that the force at P, due to ab, is 2-ird. If the radius be taken large enough, ab may be considered as fiat, and constituting a disk : hence the force at a point near a flat disk of density d is 2Trd. Since the force at a point near one Fig. 12. surface of the disk is ZTrdm one direc- tion, and near the other surface zt^d in the other direc- tion, it is clear, that, in passing through the disk, the force changes by oprd. 23. Moment of Force. — The moment of force about a point is defined as the product of the force and the perpendicular drawn from the point upon the line of direction of the force. The moment of a force, with respect to a point, measures the value of the force in producing rotation about that point. If momentum be substituted for force in the fore- going definition, we obtain the definition of moment of momentum. In order to show that the moment of a force meas- ures the value of that force in producing rotation, we will find the direction and amount of the resultant of two forces acting on a rigid bar, but not applied at the same point. Let BD (Fig. 13) be the bar, ZJi^and BG the forces. Their lines of direction will, in general, meet at some MECHANICS OF MASSES. 45 point, as O. Moving the forces up to O, and applying the parallelogram law, we obtain the resultant OJ, which cuts the bar at A. If we re- solve both forces separately, parallel to OJ and BD, this resultant equals in amount the sum of those compo- nents taken parallel to OJ. Hence the components EF and CG, taken parallel to ^^ ,^ DB, annul one another's ac- tion, and, being in opposite directions, are equal. Now, by similarity of triangles, OA:AB=BC: CG, and OA:AD=DE: EF; whence AB-BC=^AD-DE; and resolving both DE and BC perpendicular to DB, we see that the moments of force about A are equal. Now, if the resultant OJ be antagonized by an equal and opposite force applied at A, there will be no motion. Hence the tendencies to rotation due to the forces are equal, — a result which is in accord with our statement that the moment of force is a measure of the value of the force in producing rotation. 24. Couple. — The combination of two forces, equal and oppositely directed, acting on the ends of a rigid bar, is called a couple. By the preceding proposition, the resultant of these forces vanishes, and the action of a couple does not give rise to any motion of transla- tion. The forces, however, conspire to produce rota- 46 ELEMENTARY PHYSICS. tion about the mid-point of the bar. It follows from the fact that a couple has no resultant, that it cannot be balanced by any single force. 25. Moment of Couple. — The moment of a couple is the product of either of the two forces into the perpen- dicular distance between them. It follows from what has been already proved, that this measures the value of the couple as respects rotation. 26. Centre of Inertia. — If we consider any system of equal material particles, the point whose distance from any plane is equal to the average distance of the several particles from that plane is called the centre of inertia. This point can be shown to be perfectly definite for any system of particles. It follows from the definition, that, if the plane pass through the centre of inertia, the sum of the distances of the particles on one side of the plane, from the plane, will be equal to the sum of the distances of the particles on the other side : hence, if the particles are all moving with a common velocity parallel to the plane, the sum of the moments of momentum on the one side is equal to the sum of the moments of momentum on the other side. And, further, if the particles all have a common acceleration, or are each acted on by equal and similarly directed forces, the sum of the moments of fOrce on one side is equal to the sum of the moments of force on the other side. If we combine the forces acting on two of the parti- cles, one on each side of the plane, we obtain a result- ant equal to their sum, whose distance from the plane is determined by the distances of the two particles from the plane. Combining this resultant with the force on MECHANICS OF MASSES. 47 another particle, we obtain a second resultant ; and, by- continuing this process until all the forces have been combined, we obtain a final resultant, equal to the sum of all the forces lying in the plane, and passing through the centre of inertia. This resultant expresses, in amount, direction, and point of application, the force which, acting on a mass equal to the sum of all the particles situated at the centre of inertia, would impart the same acceleration to it as the conjoined action of all the separate forces on the separate particles imparts to the system. When the forces do not act in parallel lines, the proposition just stated does not hold true, except in special cases. Bodies in which it still holds are for that reason called centrobaric bodies. The centre of inertia can be readily found in most of the simple geometrical figures. For the sphere, ellipsoid of revolution, or parallelopiped, it evidently coincides with the centre of figure ; for a plane pass- ing through that point in each case cuts the solid symmetrically. 27. Mechanical Powers. — The preceding definitions and propositions find their most elementary application in the so-called mechanical powers. These are all designed to enable us, by the applica- tion of a certain force at one point, to obtain at another point a force, in general not equal to the one applied. Six mechanical powers are usually enumerated, — the lever, pulley, wheel and axle, inclined plane, wedge, and screw. (i) The Lever is any rigid bar, whose weight may be neglected, resting on a fixed point called a fulcrum. 48 ELEMENTARY PHYSICS. From the proposition under "Moments of Force," it may be seen, that, if forces be applied to the ends of the lever, there will be equilibrium when the resultant passes through the fulcrum. In that case the moments of force about the fulcrum are equal ; whence, if the forces act in parallel lines, it follows that the force at one end is to the force at the other end in the inverse ratio of the lengths of their respective lever-arms. If / and l-^ represent the lengths of the arms of the lever, and P and P-^ the forces applied to their respective extremities, then PI = P^l-^. The principle of the equality of action and re-action enables us to substitute for the fulcrum a force equal to the resultant of the two forces. We have then a combination of forces as represented in the diagram (Fig- 14)- Plainly any one of ^ these forces may be considered '^'^' '*■ as taking the place of the ful- crum, and either of the others the power or the weight. The lever is said to be of the first kind if R is fulcrum and P power, of the second kind if P^ is ful- crum and P power, of the third kind if P is fulcrum and R power. (2) The Pulley is a frictionless wheel, in the groove of which runs a perfectly flexible, inextensible cord. If the wheel is on a fixed axis, the pulley merely changes the direction of the force applied at one end of the cord. If the wheel is movable and one end of the cord fixed, and a force be applied to the other end parallel to the direction of the first part of the cord. MECHANICS OF MASSES. 49 the force acting on the pulley is double the force applied : for the stress on the cord gives rise to a force in each branch of it equal to the applied force ; each of these acts on the wheel, and, since the radii of the wheel are equal, the resultant of these two forces is a force equal to their sum applied at the centre of the wheel. From these facts the relation of the applied force to the force obtained in any combination of pulleys is evident. (3) The Inclined Plane is any frictionless surface, making an angle with the line of direction of the force applied at a point upon it. Resolving the force P (Fig. 15), making an angle ^, with the normal to the plane, into its components P cos ^ and P sin ^ perpendicular to and parallel with the plane, P sin ^ is alone effective to produce motion. Consequently, ^^^' '^" a force P sin acting parallel to the surface will balance a force P, making an angle ^, with the normal to the surface. If the plane be taken as the hypothenuse of a right-angled triangle ABC, of which the base AB is perpendicular to the line of direction of the force, then, by similarity of triangles, angle BA C = : whence the force obtained parallel to AC is equal to the force ap- plied multiplied by the sine of the angle of inclination of the plane. If the components of the force applied be taken, — the one, as before, perpendicular to the plane A C, and the other parallel to the base AB, — the force obtained parallel to AB is equal to the force applied multiplied by the tangent of the angle of inclination of the plane. 50 ELEMENTARY PHYSICS. (4) The Wheel and Axle is essentially a combination of levers. (5) The Wedge is made up of two similar inclined planes set together, base to base. (6) The Screw is a combination of the lever and the inclined plane. The special formulas expressing the relations of the force applied to the force obtained by the use of these combinations, are deduced from those for the more elementary mechanical powers. It may be seen, in general, in the use of the mechani- cal powers, that the force applied is not equal to the force obtained. A little consideration will show, how- ever, that the energy expended is always equal to the work done. Any arrangement of the mechanical powers designed to transmit work, undiminished, is called a machine. The more nearly this design is realized in actual com- binations of materials, the more closely the machine approaches perfection. The elasticity of the materials we are compelled to employ, friction, and other causes which modify the conditions required by theory, make the attainment of such perfection impossible. The ratio of the useful work done to the energy ex- pended is called the efficiency of the machine. Since in every actual machine there is a loss of energy in the transmission, the efficiency is always a proper fraction. 28. Angular Velocity. — The angle contained by the line passing through two points, one of which is in motion, and any assumed line passing through the fixed point, will, in general, vary. The rate of its change is called the angular velocity of the moving-point. If ^ MECHANICS OF MASSES. 51 denote the angle swept out in the time t, then the angular velocity o. = f 06) The angular acceleration is the rate of change of angular velocity, and, if constant, is measured by ^ = fi:p. (x7) If the radian, or that arc of a circle which is equal to the length of its radius, be taken as the unit of angle, the dimensions of angle become arc L radius ~ L~ Hence the dimensions of angular velocity are ^, and of angular acceleration, -=^. If any point be revolving about a fixed point -as a centre, its velocity in the circle varies as its angular velocity and the length of the radius jointly. Angular velocities may be compounded by a process similar to that employed for the compositions of motions. Let OA and OB (Fig. 16) rep- resent two axes of rotation, about which points are revolving with angular velocities, aij and og re- spectively ; both rotations being clockwise when seen from the point O. The velocity at a point L, at unit distance from O, due to the motion about OA, is Mjsin a, and that DB 52 ELEMENTARY PHYSICS. due to motion about OB is wjSin ^ in the opposite direc- tion. The whole velocity of L is, therefore, (Oisin a — lOjSin /8. There must be some position of L for which this velocity becomes zero. Then w^sin a.=: ug • sin ^. It follows at once that every point on the line OL is at rest. If we consider OL as the axis of rotation, and suppose the angular velocity of every point of the system about this axis to be o), such that m sin a := o)2sin(a + /3), this angu- lar velocity will give the actual velocity of any point. For example, . - sin(a -f- /8) . 0) sin fi = o)» - — ^H sm a '^ •* smtt ^ is the velocity at B ; for the velocity at B is only due to rotation about OA, and is therefore given by o)iSin(a -{- ^). From our previous equation, (iJiSin a =■ cojsin ^ ; hence • / , o\ sin(a -I- j8) . _ Now . „ sin(a -t- j8) . ft) sm a = m„ — ^ -^ sm a, and the equality of the expressions is shown. In a similar manner it may be shown that the value of the angular velocity at any point N as given by the expres- sion repre- g sent the angular velocity of the earth, t the duration of the experi- ment, and ^ the latitude, the angle a between the directions of the first and last swing, as marked on the earth's surface, is expressed by a = (o^sin (^. The truth of this formula may be seen from the following demonstration : In the figure (Fig. 17), let the pendulum be supposed to be at A. Let NS be the earth's axis. MECHANICS OF MASSES. 55 Now, the angular velocity w, represented by OC, may be resolved into two components, OD in the direction of the force acting on the pendulum, and DC at right angles to it. The angular velocity DC has no influ- ence in changing the relations of the pendulum and the earth ; but the angular velocity OD = OC sin <^ = w sin <^ is made evident by the rotation of a fixed line on the earth's surface, cutting the invariable plane of oscillation at the point of equilibrium of the pendulum. The plane of oscillation of the pendulum consequently appears to rotate in the opposite direction with an angular velocity 0) sin <^, and the angle swept out in any time t is wt sin <^. By such an apparatus has been determined, not only the fact of the earth's rotation, but even an approximate value of the length of the day. (2) The phenomena presented by the gyroscope also offer an example of the application of the foregoing principles. The construction of the apparatus can best be understood by the help of the diagram (Fig. 18). The outer- most ring rests in a frame, and turns on the points a, a^. The inner rests in the outer one, and turns on the pivots f d, 6p at right angles to^ the line of aa^. Within this ring is mounted the wheel G, the axle of which is at right angles to the line i>d-^, and therefore in the plane Fig. 18. 5 6 ELEMENTARY PHYSICS. of aa-^. At the point e is fixed a hook, from which weights may be hung. It is evident at once, that, if the wheel is mounted on the middle of the axle, the equilibrium of the apparatus is neutral in any position, and that a weight hung on the hook e would bring the axle of the wheel vertical, without moving the outer ring. If, however, the wheel be set in rapid rotation, with its axle horizontal, and a weight be hung on the hook, the whole system will revolve with a constant angular velocity about the points a, a^, and the axle of the wheel will remain horizontal. The explanation of this phenomenon follows from the principles which we have already discussed. The con- ditions given are, that a body rotating with an angular velocity in one plane is acted on by a force tending to produce rotation in a perpendicular plane. Let the plane of the paper represent the horizontal f^ plane, and the line AB (Fig. 19) represent the direction of the axle at any moment. Lay off on OA a length OP proportional to the D angular velocity of the wheel. If B be the point of application of the weight, gravity tends to turn the system about an axis CD at right angles to AB. Let us sup- pose, first, that the system has an angular velocity about CD proportional to OQ. Com- pounding the two angular velocities OP and OQ, we obtain the resultant OR. Now, resolving OQ parallel and at right angles to OR, we see that the parallel com- ponent has been efficient in determining the length of MECHANICS OF MASSES. 57 OR, the component at right angles the direction of OR. In the limit, if OQ be made indefinitely small, the resolved component Ox parallel to OR vanishes in com- parison with OQ; because from the triangles we have 7^-H = 7v?^- The effect will be a change of direction of UK U(J, the axle AB in the horizontal plane, without a change in the angular velocity of the wheel. This change is the equivalent of the introduction of a new angular velocity about an axis perpendicular to the plane of the paper. This new angular velocity, compounded with the angular velocity about OA, gives rise, as before, to a change in the direction of the axis without a change in the angular velocity of the wheel ; and this change in direction is such as to oppose the angular velocity about CD, introduced by the weight at B. The system will revolve in a horizontal plane about O as a. centre. Another explanation, leading to the same results, has been given by Poggendorff. As we have already shown, it requires the application of a force to change the direction of the axis of a rotating body. This force is expended in changing the direction of motion of the component parts of the body. Poggendorff's explanation of the move- ments of the gyroscope is based on the action of couples formed by these separate p'\ forces. Let Fig. 20 represent the rotat- ing wheel of the former diagram, the axle being supposed to be nearly horizontal. If the weight be hung at the point ^, gravity tends to turn the wheel about a horizontal axis CD. The parti- 58 ELEMENTARY PHYSICS. cles moving at A and at B in the plane CD offer no resistance to this change. Those at C moving down- wards, and those at D moving upwards, act otherwise. The forces expressed by their momentum in the direc- tions Cp and Dq are resolved into two each, — one of them in the new plane assumed by the wheel, and the other at right angles to it. It will be seen that the latter com- ponent acts at C towards the right, and at D towards the left. There is thus set up a couple acting to turn the system about the axis AB counter-clockwise, as seen from A. As soon as this rotation begins, the particles moving at A out of the paper, and at B through the paper, are turned out of their original directions, and there arises another couple, of which the component at A is directed towards the left, and at B towards the right. This couple tends to cause the system to rotate about the axis CD counter-clockwise, as seen from C, and thus to oppose the tendency to rotation due to the weight at e. All other points on the wheel, except those in the lines AB and CD, are turned out of their paths by both rotations ; and therefore components of the forces due to their motions appear in both couples in the final summation of effects. The result of the existence of these couples is a movement such as has already been described. 29. Moment of Inertia. — The moment of inertia of any body about an axis is defined as the summation of the products of the masses of the particles making up the body and the squares of their respective distances from the axis. This product is the measure of the importance of the body's inertia with respect to rotation, and is propor- MECHANICS OF MASSES. 59 tional to the kinetic energy of the body having a given angular velocity about the axis ; for if any particle m, at a distance r from the axis, rotate with an angular velocity w, its velocity is rw and its kinetic energy is J»2toV^. The whole kinetic energy of the body is, therefore, ^u?%mr^ ; and since we have assumed ^0? to be constant, ^Sitnr^ is proportional to the kinetic energy of the rotating body. If we can find a distance k such that ^k^tji^^m = ^dfi'Smr^, k is called the radius of gyra- tion, and is the distance at which a mass equal to that of the whole body must be concentrated to possess the same moment of inertia as the body possesses. The formula for moment of inertia is, of course, %mr\ (18) and its dimensions are ML?. The moment of inertia of a body with reference to an axis passing through its centre of inertia being known, its moment of inertia with reference to any other axis parallel to this is found by adding to the mo- ment of inertia already known the product of the mass of the body and the square of the distance of its centre of inertia from the new axis of rotation. For if the centre of inertia of the body whose moment of inertia we know be C, and if m be any particle of that body, and if be the new axis to which the moment of inertia is to be referred, making the construction as in the figure (Fig. 21), we have r" = r^ -F 2r^b + P + c\ a^ = b^ + c^; whence 6o ELEMENTARY PHYSICS. Multiplying all through by the mass m, and performing a similar operation for every particle of the body, and summing the results, we have Fig. 21. The term 2%mr-J? on the right vanishes, for we may write it 2,r{%mb ; and, since C is the centre of inertia, 5(»2<^) is zero. Therefore /j = / + Mr^. (19) Thus the proposition is proved. The moment of inertia of the simple geometrical solids may be found by reckoning the moments of inertia for the separate particles of the body, and sum- ming the results. We will show how this may be done in a few simple cases. (i) To find the moment of inertia of a very thin rod AB, length 2/' and mass 2m', about an axis xx!, passing through the middle point : — Suppose the half-length to be divided into a very large number n of equal parts. The mass of each will m' . I' be — . The distance of the first from the axis is — , of n n 2/' the second — , etc. Their moments of inertia are n m' l"" m' l'^ m' J"^ — X -5, —X 4-5 . . . —X n^—„, n rr n rr n rr MECHANICS OF MASSES. 6l and the moment of inertia of the half-rod is But (1+4 + 9 //'2 + «^), where n is very large, -is — ; hence /' : 3 3 If / equals the whole length of the rod, m the whole mass, and / the entire moment of inertia. /= 12 (20) Fig. 22. Fig. 23. (2) To find the moment of inertia of a thin plate AB (Fig. 23), length / and breadth 2b', about an axis per- pendicular to it and passing through its centre : — Suppose the half-plate to be divided into n rods, par- allel to its length : each rod will have a length / and a breadth — . Their distances from the axis are — , — , n 11 n etc. Let m be the mass of the plate. The moment of inertia of each rod, with respect to an axis passing through its centre of inertia and perpendicular to its length, is — X — • The moments of inertia of the sev- 272 1 2 62 ELEMENTARY PHYSICS. eral rods about the parallel axis xx' are mil'' b"\ mil-" , b'-\ ^ —\ h -2 . — ( H 2 — ), etc., 2ra\I2 «7 2n\\2 «7 and the moment of inertia of the half-plate, m l^ , mb'\ ^ ^ ^ ,- m(l^ ^b'^\ — X «— H -Ji +4 + 9 • • • +«) =- h-— ' 2« 12 2« M^^ T 7 / 2\l2 3/ or, for the whole plate, equals w— — ^- (21) 12 A parallelepiped whose axis is xxf may be supposed to be made up of an infinite number of plates, such as AB. Its moment of inertia will be the moment of inertia of one plate multiplied by the number of plates ; or, if M is the mass of the parallelepiped, its moment of inertia is — (/' + '5'). (22) The moment of inertia of any body, however irregu- lar in form or density, may be found experimentally by the aid of another body whose moment of inertia can be computed from its dimensions. We will anticipate the law of the pendulum, which has not been proved, for the sake of clearness. The body whose moment of inertia is desired is set oscil- lating about an axis under the action of a constant force/! Its time of oscillation is, then. 4, where /" is the moment of inertia. MECHANICS OF MASSES. 63 If, now, another body, whose moment of inertia can be calculated, is joined with the first, the time of oscil- lation alters to A=V^' / where /j is the moment of inertia of the body added. Combining the two equations, we obtain, as the value of the moment of inertia desired, 30. Ceniral Forces. — If the velocity or direction of motion of a moving body in any way alter, we conceive it to be acted on by some force. In certain cases the direction of this force, and the law of its variation with the position of the body, may be determined by consid- ering the path or orbit traversed by the body and the circumstances of its motion. We shall illustrate this by a few propositions, selected on account of their applicability in the establishment of the "Theory of Universal Gravitation." The proofs are substantially those given by Newton in the "Principia." Proposition I. — If the radius vector drawn from a fixed point to a body moving in a curve describe equal areas in equal times, the force which causes the body to move in the curve is directed towards the fixed point. Let us suppose the whole time divided into equal periods, during any one of which the body is not acted on by the force. It will, in the first period, move over a space represented by a straight line, as AB (Fig. 24). In the second period, it would, if unhindered, move 64 ELEMENTARY PHYSICS. Fig. 24. over an equal space BD and in the same line. It, how- ever, is deflected by a force acting instantaneously at the point B, and moves in a .line BC such that, by hypothesis, triangle OB A — triangle OBC. Now, tri- angle ODB also = triangle OAB, therefore triangle OCB = triangle ODB, and CD is parallel to OB. Com- plete the parallelogram CDBE; then it is evident that the motion BC is compounded of the motions BD and BE; and since forces are proportional to the motions they occasion, the force act- ing at B is proportional to BE, and is directed along the line BO. If, now, the period of time becomes indefi- nitely small, the broken line ABC approaches indefinitely near to a curve, the force which causes the motion in the curve being always directed to the centre 0. Proposition II. — If a body move uniformly in a circle, the force acting upon it varies as its mass and the square of its velocity directly, and as the radius of the circle inversely. If the body m move in a circle (Fig. 25) with a constant angular velocity, and pass over, in any very, small time t, the arc ad, which is so small that it may be taken equal to its chord, the motion may be resolved into two components ab ''■ * and ac, one tangent and the other normal to the arc. Now /, the acceleration towards O, being constant for that small time, we have s = ac = \ft'^. MECHANICS OF MASSES. 65 By Euclid, iii. 31, ade is a right angle, and therefore, by similar triangles, , ac But ae = 2r; ad may be taken equal to ab, which repre- sents the space traversed in the time t ; and from the previous reasoning ac represents \ffi : whence and J. mzF mf = . r Corollary I. — If two bodies are revolving about the same centre, and the squares of their periodic times are in the same ratio as the cubes of the radii of their respective orbits, the forces acting on them will be inversely as the squares of their radii, and conversely. For, if T and 7\ represent the periodic times of the two bodies moving in circles of radii r and r-^, with velocities v and v-^, then, by hypothesis, „ ™ 27rr 2-nr^ % a whence \ \ v:v-^=r.{ -.r . Now whence f:A=r^:r\ Corollary II. — The relation of Corollary I. holds with reference to bodies describing similar parts of any simi- 66 ELEMENTARY PHYSICS. lar figures having the same centre. In the application of th» proof, however, we must substitute for uniform velocity the uniform description of areas ; and instead of radii we must use the distances of the bodies from the centre. The proof is as follows : — If D and D^ represent the radii of curvature of the paths of the two bodies, R and i?i the distances of the bodies from the centre of force, then, by hypothesis (letting A represent the area described in one period of time), T: from the similarity of figures. Now hence and v:Vi = R}:R^=:n^:D^, ir V- ^■■f^=i>--%r^^^--^- Proposition III. — If a body move in an ellipse, the force acting upon it, directed to the focus of the ellipse, varies inversely as the square of the radius vector. Suppose the body moving ' in the ellipse to be at the point P (Fig. 26), and the force to act upon it along the radius vector SP. At the point P draw the tan- gent PR, and from a point Q on the ellipse draw the Fig. 26. MECHANICS OF MASSES. 6j chord Qv, cutting SP in x, and complete the parallelo- gram PRQx. From Q draw QT perpendicular to SP. Also draw the diameter GP and its conjugate DK. The force which acts on the body, causing it to leave the tangent PR and move in the line PQ, acts along SP, and in a time t (supposed very small) causes the body to move in the direction SP over the space Px; and since, in the small time considered, it may be assumed constant, whence /• Again : the area described by the radius vector in the SP- me t is equal to described in unit time. SP- QT time t is equal to — — ; and if A represent the area 2 Equating these values of /, we obtain 4A^ - / ' whence 8A'-Px I From Proposition I., the value of A is constant for Px any part of the ellipse. We shall now show that jyv^ is also constant. From similar triangles, Px:Pv = PE: PC; or, since by a property of the ellipse PE = AC, Px:Pv = AC:PC. 68 ELEMENTARY PHYSICS. Again, by another property of the ellipse, Gv-Pv: Qf = PC: CD\ If, now, we consider the time t to become indefinitely small in the limit, P and Q approach indefinitely near ; whence Qv =■ Qx and Gv = 2PC. The last proportion then becomes PC-Pv:Qx^ = PC^:2CD\ Again, from similar triangles, Qx : QT= PE : PF = AC:PF; and from another property of the ellipse, AC:PF= CD-.CB; whence Qx:QT= CD:CB. Combining these proportions, Px :Pv =AC -.PC, Pv : Qx^ = PC : 2 C£>% Qx^:Qr^= cr^-.CB", we obtain, finally, Px:QT''=AC:2CB^; that is, smce-^Cand CB are constant, ^^^ is constant. Px . We have now shown, that, in the expression for the value of the force on the body at any point in the ellipse, all the factors are constant except -^p^- The force, therefore, varies inversely as the square of the radius vector. UNIVERSAL ATTRACTION. 6g CHAPTER II. UNIVERSAL ATTRACTION. 31. Universal Attraction. — The law of universal attrac- tion was the first generalization of modern science. In its most complete form it may be stated as follows : — Between every two material particles in the universe there is a stress, of the nature of an attraction, which varies directly as the product of the masses of the particles, and inversely as the square of the distance between them. Some of the ancient philosophers had a vague belief in the existence of an attraction between the particles of matter. This hypothesis, however, with the knowl- edge which they possessed, was not susceptible of proof. The geocentric theory of the planetary system, which obtained almost universal acceptance, offered none of those beautiful and simple relations of the planetary motions upon which the law was finally estab- lished. It was not until the heliocentric theory, advo- cated by Copernicus, strengthened by the discoveries of Galileo, and systematized by the labors of Kepler, had been fully accepted, that the discovery of the law became possible. In particular, the three laws of planetary motion pub- lished by Kepler in 1609 and 1619 laid the foundations for Newton's demonstrations. The laws are as follows : — 70 ELEMENTARY PHYSICS. I. The planets move in ellipses of which one focus is situated at the sun. II. The radius vector drawn from the sun to the planet sweeps out equal areas in equal times. III. The squares of the periodic times of the planets are proportional to the cubes of their distances from the sun. Kepler could give no physical reason for the existence of such laws. Later in the century, after Huyghens had discovered the theorems relating to centrifugal force, it was seen that the third law would hold true for bodies moving in concentric circles, and attracted to the common centre by forces varying inversely as the squares of the radii of the circles. Several English philosophers, among them Hooke, Wren, and Halley, based a belief in the existence of an attraction between the sun and the planets upon this theorem. The demonstration was by no means a rigorous one, and was not generally accepted. It was left for New- ton to show that not only the third, but all, of Kepler's laws were completely satisfied by the assumption of the existence of an attraction acting between the sun and the planets, and varying inversely as the square of the distance. His propositions are substantially given in the section on " Central Forces." Newton next showed that the attraction holding the moon in its orbit, which is presumably of the same nature as that existing between the sun and the planets, is the same as that which causes heavy bodies to fall to the earth. This he accomplished by showing that the deviation of the moon from a rectilineal path is such as should occur if the force of gravity at the UNIVERSAL ATTRACTION. 7 1 earth's surface were to extend outwards to the moon, and vary in intensity inversely as the square of the distance. Two further steps were necessary before the final generalization could be reached. One was, to show the relation of the attraction to the masses of the attract- ing bodies ; the other, to show that this attraction exists between all particles of matter, and not merely, as Huyghens believed, between those particles and the centres of the sun and planets. The first step was also taken by Newton. By means of pendulums having the same length, but with bobs of different materials, he showed that the force acting on a body at the earth's surface is proportional to the mass of the body, since all bodies have the same accel- eration. He further brought forward, as the most satisfactory theory which he could form, the general statement that " every particle of matter attracts and is attracted by every other particle." The experiments necessary for a complete verifica- tion of this last statement were not carried out by Newton. They were performed in 1798 by Cavendish. His apparatus consisted essentially of a bar furnished at both ends with small leaden balls, suspended hori- zontally by a long fine wire, so that it turned freely in the horizontal plane. Two large leaden balls were mounted on a bar of the same length, which rotated about a vertical axis coincident with the axis of rota- tion of the suspended bar. The large balls, therefore, could be set and clamped at any angular distance de- sired from the small balls. The whole arrangement was enclosed in a room, to prevent all disturbance. The 72 ELEMENTARY PHYSICS. motion of the suspended system was observed from without by means of a telescope. Neglecting as unes- sential the special methods of observation employed, it is sufficient to state that an attraction was observed between the large and small balls, and was found to be in accordance with the law as above stated. 32. Measurement of the Force of Gravity. — When two bodies attract one another, their relative motions are determined by Newton's third law. In the case of the attraction between the earth and a body near its sur- face, the velocity acquired by the earth in a unit of time is so small that it may be neglected, and the acceleration of the body alone need be considered. Since the force acting upon it varies with its mass, and since its gain in momentum also varies with its mass, it follows that its acceleration will be constant, however its mass may vary. We may, therefore, obtain a direct measure of the earth's attraction, or of the "force of gravity," by allowing a body to fall freely in vacuo, and determining its rate of acceleration. It is found that a body so falling at latitude 40° will describe in one second about 16.08 feet, or 490 centimetres. Its accel- eration is therefore 32.16 in feet and seconds, or 980 in centimetres and seconds. We denote this acceleration by the symbol g. The force acting on the body, or the weight of the body, is seen at once to be ing, where m is the mass of the body. On account of almost insuperable difficulties in the employment of this method, various others are used to obtain the value of g indirectly. For example, we may allow bodies to roll down a smooth inclined plane, and UNIVERSAL ATTRACTION. 73 observe their motion. The force effective in producing motion on the plane is g sin <^, where <^ is the angle of the plane with the horizon ; the space traversed in the time t, s ^ ^^^^sin <^. By observing s and /, the value of g may be obtained. The motion is so much less rapid than that of a freely falling body that tolera- bly accurate observations can be made. Irregularities, due to friction upon the plane and the resistance of the air, however, greatly vitiate any calculations based upon these observations. This method was used by Galileo, who was the first to obtain a measure of the acceleration due to the earth's attraction. The most exact method for determining the value of g is based upon observations of the oscillation of a pendulum. A pendulum may be defined as a heavy mass, or bob, suspended from a rigid support, so that it can oscillate about its position of equilibrium. In the simple, or mathematical, pendulum, the bob is assumed to be a material particle, and to be sus- pended by a thread without weight. If the bob be stationary and acted on by gravity alone, the line of the thread will be the direction of the force. If the bob be withdrawn from the posi- tion of equilibrium (Fig. 27), it will be acted on by a force at right angles to the thread, ex- pressed by ^ sin ^, where ^ is the angle between the perpendicular and the new position of the pig. 27. thread. The force acting upon the bob at any point in the circle of which the thread is radius, if it be released, and allowed to swing in that circle, varies as the sine 74 ELEMENTARY PHYSICS. of the arc cut off by the perpendicular and the radius drawn to that point. If we make the oscillation so small that the arc may be substituted for its sine with- out sensible error, the force acting on the bob varies as the displacement of the bob from the point of equi- librium. Now a body acted on by a force varying as the displacement of the body from a fixed point will have a simple harmonic motion about its position of equilibrium. As a consequence, it may be noted that the oscilla- tions of the pendulum are symmetrical about the position of equilibrium ; and hence the bob will have an ampli- tude on the one side of the vertical equal to that which it has on the other, and the oscillations, once set up, will continue forever unless modified by outside forces. The importance of the pendulum as a means of de- termining the value of g consists in this : that, instead of observing the space traversed by the bob in one second, we can observe the number of oscillations made in any period of time, and determine the time of one oscillation ; from this, and the length of the pendulum, we can calculate the value of g. The errors in the necessary observations and measurements are very slight in comparison with those of any other method. 33. Formula for Simple Pendulum. — The formula con- necting the time of oscillation with the value of g is obtained as follows : The acceleration of the bob at any point in the arc is, as we have seen, g sin <^, or g^ if the arc be very small. The acceleration in a simple harmonic motion is , Air^ 2Trt ± ^flCOS-^. UNIVERSAL ATTRACTION, 75 Since the bob has a simple harmonic motion, we may- equate these expressions : hence 47r^ ■Ztrt si- = 7^« cos But a cos -=- is the displacement of the point having the simple harmonic motion, and is therefore equal to /. The kinetic energy of the pen- dulum at the lowest point of its arc is, therefore, This must be equal to the potential energy of the pendulum at the highest point of its arc; that is, to UNIVERSAL attraction: yy ^MRg^^, where ^Mgtf) represents the average force acting on the centre of gravity, from the highest to the lowest point of the arc, to produce rotation ; and, R being the distance of the centre of gravity from the point of suspension, ^<^ is the distance over which the average force acts. We therefore have whence ■=2^\/ MR-g (25) This is the time of oscillation of a simple pendulum whose length is • Therefore the moment of in- ertia of any physical pendulum divided by its static moment gives the length of the equivalent simple pen- dulum. The point on the line joining the point of suspension with the centre of gravity of the pendulum, and distant from the point of suspension, is called mR the centre of oscillation. A pendulum consisting of a heavy spherical bob sus- pended by a cylindrical wire was used by Borda in his determinations of the value of g. The moment of inertia and the centre of gravity of the system were easily calculated, and the length of the simple pendulum to which the system was equivalent was thus obtained. (2) We may determine the length of the equivalent simple pendulum directly by observation. The method depends upon the principle, that, }i the centre of oscil- lation be taken as the point of suspension, the time of 78 ELEMENTARY PHYSICS. oscillation will not vary. The proof of this principle is as follows : — Let r and / — r represent the distances from the centre of gravity to the point of suspension and centre of oscillation respectively, m represent the mass of the pendulum, and T its moment of inertia about its centre of gravity. Then, since the moment of inertia about the point of suspension is /" + mr'', we have When the pendulum is reversed, we have I+m{l-ry h = m (J — r) From the first equation we have / = mr{l — r), which value substituted in the second gives, after reduc- tion, /^ = /; that is, the length of the equivalent simple pendulum, and consequently the time of oscillation when the pendulum swings about its point of suspension, is the same as that when it j - is reversed, and swings about its former centre of oscillation. A pendulum (Fig. 28) so constructed as to take advantage of this principle was used by Capt. Kater in his determination of the value of g; and this form is known, in consequence, as Kater's O pendulum. 35. The Balance. — The comparison of masses Fig. 28. is of such frequent occurrence in physical investi- gations that it is important to consider the theory of the balance and the methods of using it. UNIVERSAL ATTRACTION. 79 To be of value, the balance must be accurate and sensitive ; that is, it must be in the position of equi- librium when the scale-pans contain equal masses, and it must move out of that position on the addition to the mass in one pan of a very small fraction of the original load. These conditions are attained by the application of principles which have alrea:dy been developed. The balance consists essentially of a regularly formed beam, poised at the middle point of its length upon knife edges which rest on agate planes. From each end of the beam is hung a scale-pan, in which the masses Fig. 29. to be compared are placed. Let O (Fig. 29) be the point of suspension of the beam ; A, B, the points of suspen- sion of the scale-pans ; C the centre of gravity of the beam, whose weight is W. Represent OA = OB by /, OC by d, and the angle OAB by a. If the weight in the scale^pan at A be P, and that in the one zX. B \>q P -\- p, where / is a small additional weight, the beam will turn out of its original horizontal position, and assume a new one. Let the angle COC^, through which it turns, be designated by ^. Then the moments of force about O are equal ; that is, (/'-F/)/-cos(a 4- j8) = Pl-co%[a. - fi) + Wd-sin/S; from which we obtain, by expanding and transposing, //cos a . - ^^ ^ = i2P + J>ysma+Wa- ('^) 8o ELEMENTARY PHYSICS. The conditions of greatest sensitiveness are readily deducible from this equation. As long as cos « is less than unity, it is evident that tanyS, and therefore /3, increases as the weight 2P of the load diminishes. As the angle a becomes less, the value of ^ also increases, until, when A, O, and £ are in the same straight line, it depends only on -^73, and is independent of the load. In this case tan ^ increases as d, the distance from the point of suspension to the centre of gravity of the beam diminishes, and as the weight of the beam W dimin- ishes. To secure sensitiveness, therefore, the beam must be as long and as light as is consistent with stiff- ness, the points of suspension of the beam and of the scale-pans must be very nearly in the same line, and the distance of the centre of gravity from the point of suspension of the beam must be as small as possible. Great length of beam, and near coincidence of the cen- tre of gravity with the axis, are, however, inconsistent with rapidity of action. The purpose for which the balance is to be used must determine, therefore, the extent to which these conditions of sensitiveness shall be carried. Accuracy is secured by making the arms of the beam of equal length, and so that they will perfectly balance, and by attaching scale-pans of equal weight at equal distances from the centre of the beam. In the balances usually employed in physical and chemical investigations, various adjustments are pro- vided, by means of which all the required conditions may be secured. The beam is poised on knife edges ; and the adjustment of its centre of gravity is made by UNIVERSAL ATTRACTION. 8l changing the position of a nut which moves on a screw, placed vertically, directly above the point of suspen- sion. Perfect equality in the moments of force due to the two arms of the beam is secured by a similar hori- zontal screw and nut placed at one end of the beam. The beam is a flat rhombus of brass, large portions of which are cut out so as to make it as light as possible. The knife edge on which the beam rests, and those upon which the scale-pans hang, are arranged so that, with a medium load, they are all in the same line. A long pointer attached to the beam moves before a scale, and serves to indicate the deviation of the beam from the position of equilibrium. If the balance be accu- rately made and perfectly adjusted, and equal weights placed in the scale-pans, the pointer will remain at rest, or will oscillate through distances regularly diminishing on each side of the zero of the scale. If the weight of a body is to be determined, it is placed in one scale-pan, and known weights are placed in the other until the balance is in equilibrium or nearly so. The final determination of the exact weight of the body is then made by one of three methods : we may continue to add very small weights until equilibrium is established ; or we may observe the deviation of the pointer from the zero of the scale, and, by a table pre- pared empirically, determine the excess of one weight over the other ; or we may place a known weight at such a point on a graduated bar attached to the beam that equilibrium is established, and find what its value is,, in terms of weight placed in the scale-pan, by the relation between the length of the arm of the beam and the dis- tance of the weight from the middle point of the beam. 82 ELEMENTARY PHYSICS. If the balance be not accurately constructed, we can, nevertheless, obtain an accurate value of the weight desired. The method employed is known as Borda's method of double weighing. The body to be weighed is placed in one scale-pan, and balanced with fine shot or sand placed in the other. It is then replaced by known weights till equilibrium is again established. It is manifest that the replacing weights represent the weight of the body. If the error of the balance consist in the unequal length of the arms of the beam, the true weight of a body may be obtained by weighing it first in one scale- pan and then in the other. The geometrical mean of the two values is the true weight ; for let /j and l^ represent the lengths of the two arms of the balance, P the true weight, and P-^ and P^ the values of the weights placed in the pans at the extremities of the arms of lengths /j and 4, which balance it. Then Pl^ = P-J.-^, and Pl-^ = PJ-^ ; from which P=-^ is a very small angle, Ts 2l or Ts —^. The 2 2j?i opposite side gives a similar compo- nent ; the side s^ and the opposite s give each a component Ts^ —^. The total force along the normal at P is then Fig. 33. ^ From a theorem of Euler's we know that the sum -^-\--^is constant at any point for any position of the rectangular normal plane sections : hence the expression we have obtained fully represents the pressure at P. If the surface is convex, the radii of curvature are positive, and the pressure is directed towards the liquid ; if concave, they are negative, and the pressure is di- rected outwards. This pressure is to be added to the 98 ELEMENTARY PHYSICS. constant molecular pressure which we have already- seen exists everywhere in the mass. If we denote this constant molecular pressure by K, the expression for the total pressure within the mass is (^H> K^T\ where the convention with regard to the signs of R-^ and R must be understood. For a plane surface, the radii of curvature are infinite, and the pressure under such a surface reduces to K. 52. Angles of Contact. — Many of the capillary phe- nomena appear when different fluids, or fluids and solids, are brought in contact with one another. It becomes, therefore, necessary to know the relations of the surface tensions and the angles of contact. They are deter- mined by the following considerations : — Consider first the case when three fluids meet along a line. Let represent the point where this line cuts a plane drawn at right angles to it. Then the tension T^i of the surface of separation of the fluid a from the fluid b, acting normal to this line, is counterbalanced by the ten- and Tic of the surfaces of separation of a and These tensions are always the same for the Fig. 34. sions Tat c, b and c. three liquids under similar conditions of temperature and purity. Knowing the value of the tensions, the angles which they make with one another are deter- mined at once by the parallelogram law; and these angles are always constant. MOLECULAR MECHANICS. 99 If Tab is greater than the sum of Tac and Ti^, the angle between Tac and Ti^ becomes zero, and the fluid C spreads itself out in a thin sheet between a and b. Thus, if a drop of oil be placed on water, the tension of the surface of separation between the air and water is greater than the sum of the tensions of the surfaces between the air and oil, and between the oil and water : hence the drop of oil spreads out over the water until it becomes almost indefinitely thin. In the case of two fluids in contact with a plane solid (Fig. 35), it is evident that when the system is in equi- librium, the surface of separation between the fluids a and b, making the angle 6* with the solid c, Tac= Tic+ Tai cose. The angle of contact is then deter- mined by the equation Tac— Tic cos 6 = Tab If Tac be greater than Tat + Tic, the equation gives an impossible value for cos 0. In this case the angle becomes evanescent, the fluid b spreads itself out, and wets the whole surface of the solid. In other cases the value of is finite and constant for the same sub- stances. Thus, a drop of water placed on a horizontal glass plate will spread itself over the whole plate ; while a small quantity of mercury placed on the same plate will gather together into a drop, whose edges make a constant angle with the surface. lOO ELEMENTARY PHYSICS. S3. Plateau's Experiments. — The preceding principles will enable us to deduce a few of the most important experimental facts of capillarity. A very interesting series of results was obtained by Plateau from the examination of the behavior of a mass of liquid removed from the action of gravity. His method of procedure was to place a mass of oil in a mixture of alcohol and water, carefully adjusted, to have the same specific gravity as the oil. The oil then had no tendency to move as a mass, and was free to arrange itself entirely under the action of the molecular forces. Referring to the equation of Laplace, already obtained, it is evident that equilibrium can exist only when the f — + — j is constant for every point on the sur- face. This is manifestly a property of the sphere, and is true of no other finite surface. Plateau found, accord- ingly, that the freely floating mass at once assumed a spherical form. This result we had previously reached by another method. If a solid body — for instance, a wire frame — be introduced into the mass of oil of such a size as to reach the surface, the oil clings to it, and there is a break in the continuity of the surface at the points of contact. Each of the portions of the surface divided from the others by the solid then takes a form which fulfils the condition already laid down, that equals a constant. Plateau immersed a wire sum ik^i) ring in the mass of oil. So long as the ring nowhere reached the surface, the mass remained spherical. On withdrawing a portion of the oil with a syringe, that which was left took the form of two equal calottes, or MOLECULAR MECHANICS. lOI sections of spheres, forming a double convex lens. A mass of oil, filling a short, wide tube, projected from it at either end in a similar section of a sphere. As the oil was removed, the two end surfaces became less curved, then plane, and finally concave. Plateau also obtained portions of other figures which fulfil the requisite condition. For example, a mass of oil was made to surround two rings placed at a short distance from one another. Portions of the oil were then gradually withdrawn, when two spherical calottes formed, one at each ring, and the mass between the rings became a right cylinder. It is evident that the cylinder fulfils the required condition for every point on its surface. Plateau also studied the behavior of films. He de- vised a mixture of soap and glycerine, which formed very tough and durable films ; and he experimented with them in air. Such films are so light that the action of gravity on them may be neglected in comparison with that of the surface tension. If the parts of the frame upon which the film is stretched be all in one plane, the film will manifestly lie in that plane. When, how- ever, the frame is constructed so that its parts mark the edges of any geometrical solid, the films which are taken up by it often meet. Any three films thus meet- ing so arrange themselves as to make angles of 120"* with one another. This follows as a consequence of the proposition which has already been given to deter- mine the equilibrium of surfaces of separation meeting along a line. If four or more meet, they always meet at a point. Plateau also measured the pressure of air in a soap- I02 ELEMENTARY PHYSICS. bubble, and found that it differed from the external press- ure by an amount which varied inversely as the radius of the bubble. This follows at once from Laplace's equation. This measurement also gives us a means of determining the surface tension ; for, from Laplace's equation, the pressure inwards, due to the outer surface, 2 is T. -^, and the pressure in the same direction due to .A. 2 the inner surface is also T-=, for the film is so thin that we may neglect the difference in the radii of cur- vature of the two surfaces : hence the total pressure inwards is -=- ; and if this be measured by a manometer, we can obtain the value of T. 54. Liquids influenced by Gravity. — Passing now to con- sider liquid masses acted on by gravity, we shall treat only a few of the most important cases. Jurin's Law. — We notice first, that, if a glass tube having a narrow bore be immersed perpendicularly in water, the water rises in the tube to a height inversely proportional to the diameter of the tube. This law is known as Jurin's law. Let Fig. 36 represent the section of a tube of radius r immersed in a liquid, the surface of which makes an angle 6 with the wall. Then if T^is the surface tension of the liquid, the tension acting upward is the component of this surface tension parallel to the wall, exerted all around the circumference of the tube. This is expressed by 2irrTcos 6. This force, for each unit area of the tube, is ZTrrTcos 6 MOLECULAR MECHANICS. 103 The downward force, at the level of the free surface, making equilibrium with this, is due to the weight of the liquid column. If we neglect the weight of the meniscus, this force per unit area is expressed by hdg. y Fig. 36. ^ where h is the height of the column and d the density of the liquid. We have, accordingly, .Tco's.Q^hdg; whence h = zTcosO rdg and the height is inversely as the radius of the tube. If the liquid rises between two parallel plates of length / at distance r, the upward tension is determined by the 2/ expression j-T'cos^, and the downward pressure by hdg; whence 2 r cose rdg arid the height to which the liquid rises between two I04 ELEMENTARY PHYSICS. such plates is equal to that to which it rises in a tube whose radius is equal to the distance between the plates. If the two plates are inclined to one another so as to touch along one vertical edge, the elevated sur- face takes the form of a rectangular hyperbola; for let the line of contact of the plates be taken as the axis of ordinates, and a line drawn in the plane of the free surface of the liquid as the axis of abscissae, the elevations corresponding to each abscissa are in- versely as the distance between the plates at that point, and are therefore inversely as the abscissae : hence the product of any abscissa by its corresponding ordi- nate is a constant. The extremities of the ordinates then mark out a rectangular hyperbola referred to its asymptotes. 55. Liquid Drops in Capillary Tubes. — When a drop of liquid is placed in a conical tube, it moves, if the Fig. 37. surfaces are concave, towards the smaller end ; if con- vex, towards the larger end. The explanation of these movements follows readily from the foregoing results. In case the surfaces are concave, letting 6 (Fig. 37) be the angle of contact and a the angle of inclination of the wall of the tube to the axis, r and r^ the radii of the tube at the extremities of the drop, r being the smaller of the MOLECULAR MECHANICS. lOS two, then the expressions for the tensions acting in both cases outwards are respectively f- C0S(6' — a) and -—V COS ((9 + a). Of these two expressions the former is manifestly- greater than the latter : hence the tendency of the drop is to move towards the smaller end of the tube. If we assume that the concave surfaces are portions of spheres, of which R and R^ are the respective radii of curvature, it follows very simply that r-=-R cos(^ — a), and r-^ = R-^o%{B + a) : hence the expressions for the 2T 2T tensions become -^ and -^-. These are the values of R Rj^ the tensions as determined by Laplace's equation, and the movements of the drop might have been inferred di- rectly from this equation by making the same assumption. If a drop of water is introduced into a cylindrical capillary tube of glass, and if the air on the two ends of the drop has unequal pressures, the concavities thereby become unequal, the one on the side of the greater pressure presenting the greater concavity. The bead so circumstanced offers a resistance to this press- ure ; and it may, if the pressure be not too great, entirely counterbalance it. It is also evident, that, if several such beads are introduced successively, with intervening air-spaces, the pressure which they can unitedly sustain is equal to that which one can sustain multiplied by their number. Jamin found, that, with a tube containing a large number of beads, a pressure I06 ELEMENTARY PHYSICS. of three atmospheres was maintained without diminu- tion for fifteen days. 56. Movements of Solids. — In certain cases the action of the capillary forces produces movements in solid bodies partially immersed in a liquid. For example, if two plates, which are both either wetted or not wetted by the liquid, are partially immersed vertically, and brought so near together that the rise or depression of the liquid due to the capillary action begins, then the plates will move towards one another. In either case this movement is explained by the inequality of pressure on the two sides of each plate. When the liquid rises between the plates, the pressure is zero at that point in the column which lies in the same plane as the free external surface. At every internal point above this the molecules of the liquid are in a state of negative pressure or tension, and the plates are consequently drawn together. When the liquid is depressed between the plates, they are pressed together by the external liquid above the plane in which the top of the column between the plates lies. When one of the plates is wetted by the liquid and the other not, the plates move apart. This is explained by noting, that, if the plates are brought near together, the convex surface at the one will meet the concave surface at the other, and there will be a consequent diminution in both the eleva- tion and the depression at the inner surfaces of the plates. The elevation and depression at the outer sur- faces remaining unchanged, there will result a pull outwards on the wetted plate and a pressure outwards on the plate which is not wetted ; and they will conse- quently move apart. Laplace showed, however, as the MOLECULAR MECHANICS. 1 07 result of an extended discussion, that, though seeming repulsion exists between two plates such as we have just considered, yet, if the distance between the plates is diminished, this repulsion changes to an attraction. This prediction has been completely verified by the most careful experiments. 57. Porous Bodies. — Porous bodies may be considered as assemblages of more or less irregular capillary tubes. Thus the explanation of many natural phenomena — as the wetting of a sponge, the rise of the oil in the wick of a lamp — follows directly from the preceding discussion. DIFFUSION OF LIQUIDS. 58. Free Diffusion. — When two liquids which are miscible are so brought together in a common vessel that the heavier is at the bottom and the lighter rests upon it in a well-defined layer, it is found that after a time, even though no agitation occur, they become uni- formly mixed. Molecules of the heavier liquid make their way upwards through the lighter ; while those of the lighter make their way downwards through the heavier, in apparent opposition to gravitation. Diffu- sion is the name which is employed to designate this phenomenon and others of a similar nature. If one of the liquids is colored, — as, for example, solution of cupric sulphate, — while the other is color- less, the progress of the experiment may easily be watched and noted. If both liquids are colorless, small glass spheres, adjusted and sealed so as to have different but determined specific gravities between those of the liquids employed, may be placed in the vessel used in I08 ELEMENTARY PHYSICS. the experiment, when they will show by their positions the degree of diffusion which has occurred at any given time. 59- Co-efficient of Diffusion. — Experiment shows that the amount of a salt which at a given temperature passes, in unit time, through unit area of a horizontal surface, depends upon the nature of the salt and rate of change of concentration at that surface, — that is, the quantity of a salt that passes a given horizontal plane in unit time is kCA, where A is the area, C the rate of change of concentration, and k a co-efHcient that depends upon the nature of the substance. By rate of change of concentration is meant the difference in the quantities of salt in solution measured in grams per centimetre cube, at two horizontal planes one centi- metre apart, supposing the concentration to diminish uniformly from one to the other. It is plain, that, if C and A in the above expression are unity, the quan- tity of salt passing in unit time is k. k, or the co- efficient of diffusion, is, therefore, the quantity of salt that passes in unit time through unit area of a horizon- tal plane when the difference of concentration is unity. Co-efficients of diffusion increase with the temperature, and are found not to be entirely independent of the degree of concentration. As implied above, the units of mass and length employed in these measurements are respectively the gram and the centimetre ; but, since in most cases the quantity of salt that diffuses in one second is extremely small, it is usual to employ the day as the unit time. The following examples will give an idea of the values of these co-efficients upon the basis of these units : — MOLECULAR MECHANICS. 1 09 Cane sugar at 15° temperature 0-3144 Common salt at 5° temperature 0.7650 Common salt at 15° temperature 0.9370 Common salt at 20° temperature 1.1330 Zinc sulphate at 24° temperature 0.2401 Albumen at 13° temperature 0.0630 60. Diffusion t/irough Porous Bodies. — It was found by Graham that diffusion takes place through porous solids, such as unglazed earthenware, plaster, etc., almost as though the liquids were in direct contact, and that a very considerable difference of pressure could be estab- lished between the two faces of the porous body while the rate of diffusion remains nearly constant. 61. Diffusion througfi Membranes. — If the membrane through which diffusion occurs is of a type represented by animal or vegetable tissue, the resulting phenomena, though in some respects similar, are subject to quite different laws : thus, a certain class of non-crystallizable substances, represented by albumen, gelatine, etc., hence called colloids, pass through the membrane only to a very small extent ; while those bodies capable of crystallization, called crystalloids, pass more freely. It is to be noted that the membrane is not a mere pas- sive medium, as is the case with the porous substances already considered, but takes an active part in the process ; and consequently one at least of the liquids frequently passes into the other more rapidly than would be the case if the surfaces of the liquids were directly in contact. The explanation of these facts follows if we suppose that diffusion of a liquid through a continuous mem- brane can occur only when the liquid is capable of tem- no ELEMENTARY PHYSICS. porarily uniting with the membrane, and forming a part of it. Diffusion occurs, therefore, by the union of the liquid with the membrane on one face, and the setting free of an equal portion on the other. If the membrane separate two crystallizable sub- stances, it often happens that both substances pass through, but at different rates. In accordance with the usage of Dutrochet, who carefully studied the phe- nomena, we may say there is endosmose of the liquid, which passes more rapidly to the other liquid, and ex- osniose of the latter to the former. The whole process is frequently called osmosis. If the membrane be stretched over the end of a tube, into which the • more rapid current sets, and the tube be placed in a vertical position, the liquid will rise in the tube until a very considerable pressure is attained. Dutrochet called such an instrument an endosmometer. Graham made use of a similar instrument, which he ■ called an osmometer, by means of which he studied, not only the action of porous substances, such as are men- tioned above, but' also that of various organic tissues ; and he was able to reach quantitative results of great value with respect to the substances employed. Pfeffer has more recently made an extended study of the phe- nomena of osmosis, especially in those aspects relating to physiological phenomena, in which he has shown that colloid membranes produced by purely chemical means — for example, ferrocyanide of copper — are even more efficient than the organic membranes employed by Graham. 62. Dialysis. — Upon the principles just set forth Gra- ham has founded a method of separating crystalloids MOLECULAR MECHANICS. Ill from any colloid matters in which they may be con- tained, which is often of great importance in chemical processes. The apparatus employed by Graham con- sists of a hoop, over one side of which parchment paper is stretched so as to constitute a shallow basin. In this basin is placed the mixture under investigation, and the basin is then floated upon pure water contained in an outer vessel. If crystalloids are present, they will in due time pass through the membrane into the water, leaving the colloids behind. The process is often employed in toxicology for separating poisons from ingesta or other matters suspected of containing them. It is called dialysis, and the substances that pass through are said to dialyse. DIFFUSION OF GASES. 62,. Laws of Diffusion of Gases. — Diffusion of gases obeys the same elementary laws as liquids. The rate of diffusion varies inversely as the pressure, directly as the square of the absolute temperature (see Book II.), and inversely as the square root of the density of the gas referred to its normal standard. A gas diffuses through porous solids according to the same laws. An apparatus by which this may be conveniently illustrated consists of a porous cell, whose open end is closed by a stopper, through which passes a long tube. This is placed in a vertical position, with the open end of the tube in a vessel of water. If, now, a bell-jar containing hydrogen be placed over the porous cell, hydrogen passes into the cell more rapidly than the air escapes from it : the pressure inside is increased, which is shown by the escape of bubbles from the end of the tube. If, now. 1 1 2 ELEMENTAR Y PHYSICS. the jar be removed, diffusion outward occurs more rap- idly than diffusion inward: the pressure within soon becomes less than the atmospheric pressure, as is shown by the rise of the water in the tube. ELASTICITY. 64. stress and Strain. — When a body is made the medium for the transmission of force, the application of Newton's third law shows that there is a stress in the medium. This stress is always accompanied in natural bodies by a corresponding change of form of the body, called a strain. In some bodies equal stresses applied in any direc- tion produce equal and similar strains. Such bodies are isotropic. In others the strain alters with the direction of the stress. These bodies are eolotropic. 65. Molecular Theory. — According to the molecular theory of matter, a body's form is permanent so long as the resultant of the stresses acting on it from with- out, with the attractions and repulsions existing between the individual molecules, reduces to zero. The repul- sions and attractions vary by different laws as respects distance : consequently there is a certain form of the body for every external stress in which its molecules are in equilibrium. Any change of the stress in the body is accompanied by a re-adjustment of the mole- cules, which is continued until equilibrium is again established. If the stress tends only to increase or diminish the distance between the molecules, it is called a tension or a pressure respectively ; if it tends to slide one line or sheet of molecules past another tangentially, it is MOLECULAR MECHANICS. II3 called a shear or a shearing-stress. All stresses can be resolved into these two forms. The corresponding changes of shape are called dilatations, compressions, and shearing-strains. 66. Modulus of Elasticity. — If for a given amount of stress between certain limits a body is deformed by a definite amount, which is constant so long as the stress remains constant, and if, when the stress is removed, the body regains its original condition, it is said to be perfectly elastic. Any body only partially fulfilling these conditions is said to be imperfectly elastic. The definition of elasticity in its physical sense, as a property of bodies, has been already given. As a measurable quantity, it may be defined as the rate of change, in a unit of the body, of the stress with respect to the strain. Thus, for example, the voluminal or bulk elasticity of a fluid is the ratio of any small change of pressure to the corresponding change of unit volume. The tractional elasticity of a wire under tension is the ratio of any small change in the stretchingrweight to the corresponding change in unit length. This ratio expressed numerically is called the modulus, and its reciprocal the co-efficient, of elasticity. 6^. Modulus of Voluminal Elasticity of Gases. — All gases within certain limits of temperature and pressure obey very closely the following law, discovered by Boyle in 1662, and afterwards fully proved and discussed by Mariotte : — Boyle's Law. — The temperature remaining the same, the volume of any gas is inversely as the press- ure upon it. 114 ELEMENTARY PHYSICS. Thus, if P and P-^ represent different pressures, V and Fj the corresponding volumes of any gas, then whence p.p --L.jL- PV = P^V^. (n) Now, Pj f^ is a constant which may be determined by choosing any pressure P^ and the corresponding volume V^ as standards : hence we may say, that, at any given temperature, the product PV \s, z. constant. The limi- tations of this law will be noticed later. If we draw the curve marked out by a point whose ordinate and abscissa are so related that xy equals a constant, we obtain a rectangular hyperbola referred to its asymptotes. > Let x represent the volume and y the pressure of a quantity of gas. Then this curve shows the relation of pressure and volume in all their combi- nations. Draw the lines as in Fig. 38, letting AC,JD, repre- sent volumes differing only by a small amount. It can be shown that AE is numeri- cally equal to the modulus of elasticity, CG for — =, is the voluminal compression per unit volume for the increment of pressure GD: hence, by definition, GD ■jT^ is the modulus of elasticity. 'AC AE:GD = AC:CG, But MOLECULAR MECHANICS. IIS from similarity of triangles : hence GD AE = 777T = the modulus of elasticity. CO- 'AC Further, since the rectangles JG and CK are equal, DG-GA = CG-GK: whence CG:DG = GA: GK. By similar triangles, CG:DG = CA:AE; whence GA:GK = CA: AE. Now, if the increment of pressure be made very small, so that D and C ultimately coincide, the line CE becomes a tangent, and GA, GK, are respectively equal to CA, CB. CB therefore equals AE from the last proportion : hence, in the case of a gas obeying Boyle's law, the modulus of elasticity is numerically equal to the pressure. The discussion of the experimental facts in connec- tion with the elasticity of gases, and the explanation of the apparatus founded upon it will be resumed in a future chapter. G&. Modulus of Voluminal Elasticity of Liquids. — When liquids are subjected to voluminal compression, it is found that their modulus of elasticity is much greater than that of gases. For at least a limited range of pressures the modulus of elasticity of any one liquid is constant, the change in volume being proportional to the change in the pressure. The modulus differs for different liquids. Il6 ELEMENTARY PHYSICS. The instrument used to determine the modulus of elasticity of liquids is called a piezometer. The first form in which the instrument was devised by Oersted, while not the best for accurate determinations, may yet serve as a type. The liquid to be compressed is contained in a thin glass flask, the neck of which is a tube with a capillary bore. The flask is immersed in water contained in a strong glass vessel fitted with a water-tight metal cap, through which moves a piston. By the piston, pressure may be applied to the water, and through it to the flask and to the liquid contained in it. The end of the neck of the small flask is inserted downwards under the surface of a quantity of mercury which lies at the bottom of the stout vessel. The pressure is registered by means of a compressed-air manometer (see § 86) also inserted in the vessel. When the apparatus is arranged, and the piston depressed, a rise of the mercury in the neck of the flask occurs, which indicates that the water has been compressed. An error may arise in the use of this form of appa- ratus from the change in the capacity of the flask, due to the pressure. Oersted assumed, since the pressure on the interior and exterior walls was the same, that no change would occur. Lam6, however, showed that such a change would occur, and gave the formulas by which it might be calculated. By thus introducing the proper corrections. Oersted's piezometer may be used with success. A different form of the instrument, employed by Regnault, is, however, to be preferred. In it, by an arrangement of stop-cocks, it is possible to apply the MOLECULAR MECHANICS. WJ pressure upon either the interior or exterior wall of the flask separately, or upon both together, and in this way to experimentally determine the correction to be applied for the change in the capacity of the flask. It is to be noted that the modulus of elasticity for liquids is so large, that, within the ordinary range of pressures, they may be regarded as incompressible. Thus, for example, the alteration of volume for sea- water by the addition of one atmosphere's pressure is 0.000044. The change in volume, then, at a depth in the ocean of one kilometre, where the pressure is 93 atmospheres nearly, is 0.0041, or about 2^-^ of the whole volume. 6g. Modulus of Voluminal Elasticity of Solids. — The modulus of voluminal elasticity of solids is believed to be generally greater than that of liquids, though no reliable experimental results have yet been obtained. The modulus, as is the case with liquids, differs for different bodies. 70. Shears. — A strain in which parallel planes or sheets of molecules are moved . „ , r Ci D E D , tangentially over one another, ^'^ each plane being displaced by an amount proportional to its distance from one of the planes assumed as fixed, is called a shear. ''"'■ ^^■ To illustrate this definition, let us consider a paral- lelopiped, whose cross-section made at right angles to its sides is a rhombus, and let AB DC in the diagram (Fig. 39) represent that cross-section. If the rhombus ABJDC is deformed so as to become Il8 ELEMENTARY PHYSICS. ABD^C], that deformation is a simple shear. It is plain that a simple shear is equivalent to an extension in lines parallel to AD, and a contraction in those at right angles to AD. The directions AD and CB are called the principal axes of the shear. The amount of the shear is the displacement of the planes per unit of dis- tance from the fixed plane ; that is, -=^ is the amount of the shear. The stresses that give rise to a simple shear can plainly be conceived of as consisting of two equal couples, the forces comprising which act tangentially upon parallel planes which are moved over oije another, and make equal angles with the axes of the shear. The forces making up these couples may be compounded two and two, a and b, a-^ and b^ (Fig. 40), making up a tension normal to the diminished axis ; a-^ and b, a and b-^, making up a pressure normal to the increased axis. These stresses are measured per unit of area of the unde- formed sides or sections of the solid. The resistance offered by a body to a shearing-stress is called its rigidity, and the ratio of a very small change in the stress to the corresponding increment in the amount of the shear is called the modulus of rigidity. 71. Elasticiiy of Tension. — The first experimental determinations of the relations between the elonga- tion of a solid and the tension acting on it were made by Hooke in 1678. Experimenting with wires of dif- erent materials, he found that for small tensions the MOLECULAR MECHANICS. II9 elongation is proportional to the stress. It was after- wards found that this law is true for small compres- sions. The ratio of the stress to the elongation of unit length of a wire of unit section is the modulus of trac- tional elasticity. For different wires it is found that the elongation is proportional to the length of the wires, and inversely to their section. The formula embodying these facts is SI e = —, (28) where e is the elongation, / the length, s the section of the wire, S the stress, and /t the modulus of tractional elasticity. A method of expressing the modulus of elasticity, due to Thomas Young, is sometimes valuable. "We may express the elasticity of any substance by the weight of a certain column of the same substance, which may be denominated the modulus of its elas- ticity, and of which the weight is such that any addition to it would increase it in the same proportion as the weight added would shorten, by its pressure, a portion of the substance of equal diameter." For example, considering a cubic litre of air at 0° C. and 760 milli- metres of mercury pressure, and calling its weight unity, we find, from the fact that the weight of one litre of mercury is 105 17 times that of a litre of air, that the pressure of the column of air upon a square decimetre is 79929 units. If we conceive the air as of equal density throughout, this pressure is equivalent to the weight of a column of air one square decimetre in section and 7992.9 metres high. The weight of this I20 ELEMENTARY PHYSICS. column is the modulus of elasticity for air ; for we know, by Boyle's law, that if the column be altered in length, and its weight therefore correspondingly altered, the volume of the cubic litre of air under consideration will also alter inversely. The height of such a column of air as we have assumed is called the height of the homogeneous atmosphere. 72. Elasticity of Torsion. — When a cylindrical wire clamped at one end is subjected at the other to the action of a couple whose axis is the axis of the cylinder, it is found that the amount of torsion (measured by the angle of displacement of the arm of the couple) is pro- portional to the moment of the couple, to the length of the wire, and inversely to the fourth power of the radius. It also depends on the modulus of rigidity. The formulated statement of these facts is ^ = — i> (29) where ^ is the angle of torsion, / the length, r the radius of the wire, C the moment of couple, and n the modu- lus of rigidity. No general formula can be found for wires whose sections are of variable form. The laws of torsion in wires were first investigated by Coulomb, who applied them in the construction of an apparatus of great value for the measurement of small forces. The apparatus consists essentially of a small cylindri- cal wire, suspended firmly from the centre of a disk, upon which is cut a graduated circle. By the rotation of this disk any required amount of torsion may be given to the wire. On the other extremity of the wire MOLECULAR MECHANICS. 121 is fixed, horizontally, a bar, to the ends of which the forces constituting the couple are applied. Arrange- ments are also made by which the angular deviation of this bar from the point of equilibrium may be deter- mined. When forces are applied to the bar, it is brought back to its former point of equilibrium by rota- tion of the upper disk. Let ® represent the moment of torsion (that is, the couple which, acting on an arm of unit length, will give the wire an amount of torsion equal to a radian), C the moment of couple acting on the bar, t the amount of torsion measured in radians ; then C = ®T. We may find the value of ® in absolute measure by a method of oscillations analogous to that used to deter- mine g with the pendulum. A body whose moment of inertia can be determined by calculation is substituted for the bar, and the time t of one of its oscillations about the position of equilibrium observed. We then obtain © from the formula -Vi' for the amount of torsion is proportional to the mo- ment of couple, — a condition giving a simple harmonic motion to the oscillating body. The velocity of any point of distance r from the axis of rotation, therefore, is -^ at the point of equilibrium, and its angular velocity is -^, for a = rr. The kinetic energy of a body rotating about a centre is i/w^; and the kinetic 122 ELEMENTARY PHYSICS. energy of the body considered at the point of equilib- rium is, therefore, ^ 2 2 The potential energy due to the torsion of the wire is \@T^, where 4®t is the average moment of couple, and T the distance through which this couple acts. These expressions are necessarily equal : hence ^^ = ie.^ 2/^ or ® = -^. We may use a single instead of a double oscillation, when we may write the formula ® = ^- (3°) This apparatus was used by Coulomb in his investi- gations of the law of electrical and magnetic actions. It was afterwards employed by Cavendish, as has been already noticed, to determine the constant of gravita- tion. 73. Elasticity of Flexure. — If a rectangular bar be clamped by one end, and acted on at the other by a force normal to one of its sides, it will be bent or flexed. The amount of flexure — that is, the amount of displacement of the extremity of the bar from its original position — is found to be proportional to the force, to the cube of the length of the bar, and in- versely to its breadth, to the cube of its thickness, and MOLECULAR MECHANICS. 1 23 to the modulus of tractional elasticity. The formula therefore becomes 74. Limits of Elasticity. — The theoretical deductions and empirical formulas which we have hitherto been considering are strictly applicable only to perfectly elastic bodies. It is found that the voluminal elasticity of fluids is perfect, and that within certain limits of deformation, varying for different bodies, we may con- sider the elasticity of solids to be practically perfect for every kind of strain. If the strain be carried be- yond the limit of perfect elasticity, the body is perma- nently deformed. This permanent deformation is called set. Upon these facts we may base a distinction between solids and fluids : a solid always requires the stress act- ing on it to exceed a certain limit before any permanent set occurs, and it makes no difference how long the stress acts provided it lies within the limits. A fluid, on the contrary, may be deformed by the slightest shearing-stress, provided time enough be allowed for the movement to take place. The fundamental differ- ence lies in the fact that fluids offer no resistance to shearing-stress other than that due to internal friction or viscosity. A solid, if it be deformed by a slight stress, is soft ; if only by a great stress, is hard or rigid. A fluid, if deformed quickly by any stress, is mobile ; if slowly, is viscous. It must not be understood, however, that the behavior of elastic solids under stress is entirely independent of 124 ELEMENTARY PHYSICS. time. If, for example, a steel wire be stretched by a weight which is nearly but not quite sufficient to pro- duce an immediate set, it be found, that, after some time has elapsed, the wire acquires a permanent set. If, on the other hand, a weight is put upon the wire some- what less than is required to break it, by allowing inter- vals of time to elapse between the successive additions of small weights, the total weight supported by the wire may be raised considerably above the breaking-weight. Again : if the weight stretching the wire be removed, the return to its original form is not immediate, but gradual. If a wire carrying a weight is twisted, and the weight set oscillating by the torsion of the wire, it is found that the oscillations die away faster than can be explained by any imperfections in the elasticity of the wire. These and similar phenomena are manifestly depend- ent upon peculiarities of molecular arrangement and motion. The last two are exhibitions of the "viscosity of solids." The molecules of solids, just as those of liquids, move among themselves, but with a certain amount of frictional resistance. This resistance causes the external work done by the body to be diminished, and the internal work done among the molecules be- comes transformed into heat. MECHANICS OF FLUIDS. 1 25 CHAPTER IV. MECHANICS OF FLUIDS. 75. Pascal's Law. — A perfect fluid may be defined as a body which offers no resistance to shearing-stress. No actual fluids are perfect. Even those which approxi- mate that condition most nearly, offer resistance to shearing-stress, due to their viscosity. With most, how- ever, a very short time only is needed for this resist- ance to vanish; and all mobile fluids at rest can be dealt with as if they were perfect in determining the conditions of equilibrium. If they are in motion, their viscosity becomes a more important factor. As a consequence of this definition of a perfect fluid, follows a most important deduction. It was flrst made by Pascal, and is known as Pascal's law. "In a fluid in equilibrium, not acted on by any outside forces except the pressure of the containing vessel, the pressure at every point and in every direction is the same." By pressure at a point is meant the ratio of the pressure on an area to that area taken with its centre of inertia at the point, and normal to the direction of the press- ure, when the area is diminished indefinitely. The truth of Pascal's law appears, if, in a fluid fulfill- ing the conditions indicated, we consider a cube of the fluid to become solidified. "Then if the law as just 126 ELEMENTARY PHYSICS. stated were not true, there would be an unbalanced force in some direction, and the cube would move, which is contrary to the statement that the fluid is in equilibrium. If a vessel filled with a fluid be fitted with a number of pistons of equal area A, and a force Ap be applied to one of them, acting inwards, a press- ure Ap will act outwards upon the face of each of the pistons. These pressures may be balanced by equal forces applied to each piston. If n-\-\ be the number of the pistons, the outward pressure on n of them, caused by the force applied to one, is npA. The fluid will be in equilibrium when a force p is acting on unit area of each piston. It is plain that the same reasoning will hold if the area of one of the pistons is A and of another is nA. A pressure Ap on the one will balance a pressure of nAp on the other. This prin- ciple governs the action of the hydrostatic press. "jS. Relaiions of Fluid Pressures due to Outside Forces. — If forces, such as gravitation, act on the mass of a fluid from without, Pascal's law no longer holds true. For consider the unit cube of solidified fluid to be acted on by gravity ; then the pressure on the upper face must be less than that on the lower face by the weight of the cube, in order that the fluid may still be in equilibrium. As the cube may be made as small as we please, it appears that in the limit the pressure on the two faces only differs by an infinitesimal ; that is, the pressure in a fluid acted on by outside forces is the same at one point for all directions, but varies continu- ously for different points. The surface of a fluid of uniform density acted on by gravity, if at rest, is everywhere perpendicular to the MECHANICS OF FLUIDS. 1 2/ lines of force ; for, if this were not so, the force at a point on the surface could be resolved into two compo- nents, one perpendicular and the other tangent to the surface. But from the nature of a fluid the tangential force would set up a motion of the fluid, which is contrary to the statement that the fluid is at rest. If a surface be drawn through the points in the field at which the pressure is the same, that surface will be perpendicular to the lines of force. For consider a filament of solidified fluid lying in the surface : its two ends suffer equal and opposite pressures ; hence, since by hypothesis the fluid is in equilibrium, the force act- ing upon it, due to gravity, can have no component in the direction of i,') = {V,-V)+ ^-^, (33) or, again, since Av = A-^Vy Applying equation (34) to the case of a liquid flowing freely into air from an orifice at C, we have 138 ELEMENTARY PHYSICS. where h is the height of the surface above C, and h-^ is that of the surface above B ; Px=Pa->rdgh.^, where pa is the atmospheric pressure. At the orifice p =pa- We have then ¥'{^ - ^) = g{^ - K) + gh, = gh, whence 2 ^g^ If, now, A is very small as compared with A], the velo- city at C becomes V = '{Tgh; (3s) that is, the velocity of efflux of a small stream issuing from an orifice in the wall of a vessel is independent of the density of the liquid, and is equal to the velocity which a body would acquire in falling freely through a distance equal to that between the surface of the liquid and the orifice. This theorem was first given by Torricelli on experi- mental considerations, and is known by his name. We may apply the general equation to the case of the eflSux of a liquid through a siphon. A siphon is a bent tube, one limb of which is longer than the other. It is filled with liquid, stopped with the finger, inverted, and the end of the shorter limb immersed in some of the liquid contained in a vessel. On removing the finger the liquid runs from the longer limb. A^ In this case, as before, -^-^ = 0, 2/^ = 0, / and /^ MECHANICS OF FLUIDS. 139 both ■=-pa, and (Fj — V) ^gl, where / is the distance between the surface level and the orifice of the longer limb. The velocity becomes v = S2gl. The siphon, therefore, discharges more rapidly the greater the dis- tance between the surface level and the orifice. It is manifest that the height of the bend in the tube cannot be greater than that at which atmospheric pressure would support the liquid. The ordinary lifting-pump may also be discussed by the same equation. This pump consists essentially of a tube, fitted near the bottom with a partition, in which is a valve opening upwards. In the tube slides a tightly fitting piston, in which is a valve, also opening upwards. The piston is first driven down to the par- tition in the tube, the enclosed air escaping through the valve in the piston. When the piston is raised, the liquid in which the lower end of the tube is immersed passes through the valve in the partition, and fills the tube. When the piston is again lowered, the space above it is filled with the liquid, which is lifted out of the tube at the next up-stroke. To determine the velocity of the liquid following the piston, we notice that in this case p^ =/« and / = o if the piston moves upward very rapidly, ( F^ — F) ^ — gh, where h is the height of the top of the liquid column above the free surface in the reservoir, and ^-^ again = o. We then have ,-^'- The velocity when ^ := o is I40 ELEMENTARY PHYSICS. and at other points it can be found by suitable substitu- tions. When h is such that dgh^^pa, v^^o, which expresses the condition of equilibrium. Torricelli's theorem is shown to be approximately- true by allowing liquids to run from an orifice in the side of a vessel, and measuring the path of the stream. If the theorem be true, this ought to be a parabola, of which the intersection of the plane of the stream and of the surface of the liquid is the directrix ; for each portion of the liquid, after it has passed the orifice, will behave as a solid body, and move in a parabolic path. *The equation of this path is found, as in § 37, to be — X =y . g Now, by Torricelli's theorem, we may substitute for v^ its value 2gk, . •. j/^ = /^x ; whence, since the initial movement of the stream is supposed to be horizontal, the perpendicular line through the orifice being the axis of the parabola, and the orifice being the origin, h is the distance from the orifice to the directrix. Experiments of this kind have been frequently tried, and the results found to approximate more nearly to the theoretical as various causes of error were removed. When, however, we attempt to calculate the amount of liquid discharged in a given time, there is found to be a wider discrepancy between the results of calcula- tion and the observed facts. Newton first noticed that the diameter of the jet at a short distance from the orifice is less than that of the orifice. He showed this to be a consequence of the freedom of motion among the particles in the vessel. The particles flow from all MECHANICS OF FLUIDS. 141 directions toward the orifice, those moving from the sides necessarily issuing in streams inclined towards the axis of the jet. Newton showed, that by taking the diameter of the narrow part of the jet, which is called the vena contractu, as the diameter of the orifice, the calculated amount of liquid escaping agreed far more closely with theory. If the orifice is fitted with a short cylindrical tube, the interference of the different particles of the liquid is in some degree lessened, and the quantity discharged increases nearly to that required by theory. 83. Diminution of Pressure. — The Sprengel air-pump, an important piece of apparatus, to be described here- after, depends for its operation on \ ^ the diminution of pressure at points along the line of a flowing column of liquid. Let us consider a large reservoir filled with liquid, which runs from it by a vertical tube en- tering the bottom of the reservoir. Transposing equation (34) to find the value of /, we have The ratio -j-^ is again o. If A (Fig. 42) represent the height of the upper surface above the point in the tube at which we desire to find the pressure, then (F^ — V) ^=gh. We then have/ =/j -|- dgh — \dv^. If the tube is always filled with the liquid, Av = AqVq, where A and Aq are the areas of the cross-sections of the tube at the point h \ K Fig. 42. 142 ELEMENTARY PHYSICS. we are considering and at the bottom of the tube, and V and Vf^ are the corresponding velocities. Further, v^ = 2gh^ if h^ is the distance from the upper surface to the bottom of the tube. We obtain, by subtitution, p=p^ + dg{h-'^,h^. (36) A ^ If k^^-J^h^, p=ip^z=pa; and if an opening were made in the wall of the tube, the moving liquid and A 2 the air would be in equilibrium. If h<. —^ h^, the press- ure p would be less than pa, and air would flow into the tube. Since this inequality exists when A^ = A, it follows, that, if a liquid flow from a reservoir down a cylindrical tube, the pressure at any point in the wall of the tube is less than the atmospheric pressure by an amount equal to the pressure of a column of the liquid the height of which is equal to the distance between the point considered and the bottom of the tube. 84. Vortices. — A most interesting series of results have been obtained, by Helmholtz, Thomson, and others, from the discussion of the rotational motions of fluids. Though the proofs are of such a nature that they can- not be presented here, the results are so important that they will be briefly stated. A vortex line is defined as the line which coincides at every point with the instantaneous axis of rotation of the fluid element at that point. A vortex filament is any portion of the fluid bounded by vortex lines. A vortex is a vortex filament which has " contiguous to it over its whole boundary irrotationally moving fluid." MECHANICS OF FLUIDS. 143 The theorems relating to this form of motion as first proved by Helmholtz, in 1 868, show that, — (i) A vortex in a perfect fluid always contains the same fluid elements, no matter what its motion through the surrounding fluid may be. (2) The "strength" of a vortex, which is the product of its angular velocity by its cross-section, is constant : therefore the vortex in an infinite fluid must always be a closed curve, which, however, may be knotted and twisted in any way whatever. (3) In a finite fluid the vortex may be open, its two ends terminating in the surface of the fluid. (4) The irrotationally moving fluid around a vortex has a motion due to its presence, and transmits the influence of the motion of one vortex to another. (5) If the vortices considered are infinitely long and rectilinear, any one of them, if alone in the fluid, will remain fixed in position. (6) If two such vortices be present parallel to one another, they revolve about their common centre of gravity. (7) If the vortices be circular, any one of them, if alone, moves with a constant velocity along its axis, at right angles to the plane of the circle, in the direction of the motion of the fluid rotating on the inner surface of the ring. (8) The fluid encircled by the ring moves along its axis in the direction of the motion of the ring, and with a greater velocity. (9) If two circular vortices move along the same axis, one following the other, the one in the rear moves faster, and diminishes in diameter ; the one in advance 144 ELEMENTARY PHYSICS. moves slower, and increases in diameter. If the strength and size of the two are nearly equal, the one in the rear overtakes the other, and passes through it. The two now having changed placets, the action is repeated in- definitely. (lo) If two circular vortices of equal strength move along the same axis toward one another, the velocities of both gradually decrease and their diameters increase. The same result follows if one such vortex moves toward a solid barrier. The preceding statements apply only to vortices set up in a perfect fluid. They may, however, be illustrated by experiment. To produce circular vortices in the air, we use a box which has one of its ends flexible. A cir- cular opening is cut in the opposite end. The box is filled with smoke or with finely divided sal-ammoniac, resulting from the combination of the vapors of am- monia and hydrochloric acid. On striking the flexible end of the box, the rings are at once sent out. The ring is easily seen to be made up of particles revolving about a central core in the form of a ring. With such rings many of the preceding statements may be verified. An illustration of the open vortex is seen when an oar-blade is drawn through the water. By making such open vortices, using a circular dish, many of the obser- vations with the smoke-rings may be repeated in another form. 85. Air-Pumps. — The fact that gases, unlike liquids, are easily compressed, and obey Boyle's law under ordi- nary conditions of temperature and pressure, underlies the construction and operation of several pieces of ap- MECHANICS OF FLUIDS. 1 45 paratus of great value in physical investigations. The most important of these is the air-pump. The working portion of the air-pump is constructed essentially like the common lifting-pump already de- scribed. The valves must be light and accurately fitted. The vessel from which the air is to be exhausted is joined to the pump by a tube, the orifice of which is closed by the valve in the bottom of the cylinder. A special form of vessel much used in pneumatic experiments is called the "receiver." It is usually a glass cylinder, open at one end, and closed by a hemi- spherical portion at the other. The edge of the cylinder at the open end is ground perfectly true, ^so that all points in it are in the same plane. This ground edge fits upon a plane surface of roughened brass, or ground glass, called the "plate," through which enters the tube which joins the receiver to the cylinder of the pump. The joint between the receiver and the plate is made tight by a little oil or vaseline. The action of the purhp is as follows : as the piston is raised, the pressure on the upper surface of the valve in the cylinder is diminished, and the air in the vessel expands in accordance with Boyle's law, lifts the valve, and distributes itself in the cylinder, so that the press- ure at all points in the vessel and the cylinder is the same. The piston is now forced down, the lower valve is closed by the increased pressure on its upper sur- face, the valve in the piston is opened, and the air in the cylinder escapes. At each successive stroke of the pump this process is repeated, until the pressure of the remnant of air left in the vessel is no longer suffi- cient to lift the valves. 146 ELEMENTARY PHYSICS. The density of the air left in the vessel after a given number of strokes, is determined, provided there be no leakage, by the relations of the volumes of the vessel and the cylinder. Let F represent the volume of the vessel, and Cthat of the cylinder when the piston is raised to the full extent of the stroke. Let d and d^^ respectively repre- sent the density of the air in the vessel before and after one stroke has been made. After one down and one up stroke has been made, the air which filled the volume Fnow fills V-\-C. It follows that d~ V+C' As this ratio is constant no matter what density may be considered, it follows, that, if d„ represent the density after n strokes, ^=f_^Nr (37) d \V+C) ^^" As this fraction cannot vanish until n becomes infinite, . it is plain that a perfect vacuum can never, even theo- retically, be obtained by means of the air-pump. If, however, the cylinder be large, the fraction decreases rapidly, and a few strokes are sufficient to bring the density to such a point that either the pressure is in- sufficient to lift the valves, or the leakage through the various joints of the pump counterbalances the effect of longer pumping. In the best air-pumps the valves are made to open automatically. In Fig. 43 is represented one of the methods .by which this is accomplished. They can then be made heavier and with a larger surface of con- MECHANICS OF FLUIDS. 147 tact, so that the leakage is decreased, and the limit of the useful action of the pump is much extended. With the best pumps of this sort a pressure of one-half a millimetre of mercury is reached. ^fe S Fig. 43. The Sprengel air-pump depends for its action upon the principle discussed in § 83, — that a stream of liquid running down a cylinder diminishes the pressure upon its walls. In the Sprengel pump the liquid used is mercury. It runs from a large vessel down a glass tube, into the wall of which, at a distance from the bottom of the tube of more than 760 millimetres, enters the tube which connects with the receiver. The lower end of the vertical tube dips into mercury, which pre- vents air from passing up along the walls of the tube. When the stream of mercui'y first begins to flow, the air enters the column from the receiver, in consequence 148 ELEMENTARY PHYSICS. of the diminished pressure, passes down with the mer- cury in large bubbles, and emerges at the bottom of the tube. As the exhaustion proceeds, the bubbles become smaller and less frequent, and the mercury falls in the tube with a sharp, metallic sound. It is evident, that, as in the case of the ordinary air-pump, there cannot be a perfect vacuum secured. There being no leakage, however, in this form of the air-pump, a very high degree of exhaustion can be reached. The Morren or Alvergniat mercury-pump is in princi- ple merely a common air-pump, in which combinations of stop-cocks are used instead of valves, and a column of mercury in place of the piston. Though slow in its operation, it is sometimes used, its particular excellence being that there need be scarcely any leakage. The compressing pump is used, as its > name implies, to increase the density of air or any other gas within a receiver. The receiver in this case is generally a strong metallic vessel. The working parts of the pump are precisely those of the air-pump, with the exception that the valves open downwards. As the piston is raised, air enters the cylinder, and is forced into the receiver at the down-stroke. 86. Manometers. — The manometer is an instrument used for measuring pressures. One variety depends for its performance upon the regularity of change of volume of a gas with change of pressure. This in its typical form consists of a heavy glass tube of uniform bore, sealed at one end, with the open end immersed in a basin of mercury. The pressure to be measured is applied to the surface of the mercury in the basin. As this pressure increases, the air contained in the tube MECHANICS OF FLUIDS. I49 is compressed, and a column of mercury is forced up the tube. The top of this column serves as an index. We know, from Boyle's law, that, when the volume of air has diminished one-half, the pressure is doubled. The downward pressure of the mercury column makes up a part of this pressure ; and the pressure acting on the surface of the mercury in the basin is greater than that indicated by the compression of the air in the tube, by the pressure due to the mercury column. For many purposes the manometer tube may be made very short, and the pressure of the mercury column that rises in it may be neglected. 87. Aneroids. — The aneroid is an instrument used to determine ordinary atmospheric pressures. On account both of its delicacy and its easy transportability, it is often used instead of the barometer. It consists of a metallic box, the cover of which is made of thin sheet- metal corrugated in circular grooves. The air is par- tially exhausted from the box, and it is then sealed. Any change in the pressure of the atmosphere causes the corrugated top to move. This motion is very slight, but is made perceptible, either by a combination of levers, which amplifies it, or by an arm rigidly fixed on the top, the motion of which is observed by a micro- scope. The indications of the aneroid are compared with a standard mercurial barometer, and an empirical scale is thus made, by means of which the aneroid may be used to determine pressures directly. 88. Limitations to the Accuracy of Boyle's Law. — In all the previous discussions, we have dealt with gases as if they obeyed Boyle's law with absolute exactness. This, however, is not the case. In the first place, some gases ISO ELEMENTARY PHYSICS. at ordinary temperatures can be liquefied by pressure. As these gases approach more nearly the point of lique- faction, the product PV oi the volume and pressure becomes less than it ought to be in accordance with Boyle's law. Secondly, those gases which cannot be liquefied at ordinary temperatures by any pressure, however great, show a different departure from the law. For every gas, except hydrogen, there is a minimum value of the product PV. At ordinary temperatures and small pressures the gas follows Boyle's law quite closely, becoming, however, more compressible as the pressure increases, until the minimum value of PV i^ reached. It then becomes gradually less compressible, and at high pressures its volume is much greater than that de- termined by Boyle's law. If the temperature be raised, the agreement with the law is closer, and the pressure at which the minimum value oi PV occurs is greater. Hydrogen seems to differ from the other gases, only in that the pressures at which the observations upon it were made were probably greater than the one at which its minimum value of PVozcwxs. The volume of the compressed hydrogen is uniformly greater than that required by Boyle's law. Important modifications are introduced into the be- havior of gases under pressure by subjecting them to intense cold. It is then found that all gases, without exception, can be liquefied, and even solidified. Though no observations have been made upon their adherence to Boyle's law under the conditions of the experiments, it is highly probable, that, as they approach the point oi liquefaction, they behave as those gases which can be MECHANICS OF FLUIDS. 151 liquefied at ordinary temperatures, and become more compressible than they should be according to the law. The subject is intimately connected with the subject of "critical temperature," and will be again discussed in the chapters on " Heat." Book II. HEAT. PRELIMINARY. 89. General Effects. — First, Bodies are warmed, or their temperature is raised, by heat. The sense of touch is often sufficient to show difference in temperature; but the true criterion is the transfer of heat from the warmer to the colder body when the two bodies are brought in contact, and no work is done by one upon the other. This transfer is known by some of the effects described below. Second, Bodies, in general, expand when heated. Experiment shows that different substances expand dif- ferently for the same rise of temperature. Gases, in general, expand more than liquids, and liquids more than solids. Expansion, however, does not universally accompany rise of temperature. A few substances contract when heated. Third, Heat changes the state of bodies, as solids into liquids, and liquids into gases. The melting of ice and the conversion of water into steam are familiar examples. Fourth, Heat breaks up chemical compounds. The compounds of sodium, potassium, lithium, etc., give to the flame of a Bunsen lamp the characteristic color of '55 156 ELEMENTARY PHYSICS. the metallic vapor, — evidence that the heat has sepa- rated the metal from the element with which it was combined. Fifth, Heat produces electric currents. Sixth, Heat performs mechanical work. The heat produced in the furnace of the steam-boiler drives the engine. 90. Production of Heat — Heat is produced by the re- verse of the above processes. Bodies that expand by being heated are warmed by compression. The fire- syringe illustrates this. Heat is generated when a liquid becomes solid or a gas becomes liquid. Experi- ments in proof of this will be adduced later. Heat is generated when substances combine chemi- cally. The heat of combustion is one example. Heat is generated by electric currents. The flow of electricity through a conductor always generates heat, greater in quantity as the resistance of the conductor is greater. The electric light is the result of heat so produced. Heat is produced by mechanical energy. Buildings are sometimes warmed by the heat produced by the friction of iron plates made to revolve under pressure by means of water-power. Mills have been set on fire by heat generated by the friction of the machinery not properly lubricated. 91. Nature of Heat. — Heat was for a long time con- sidered to be a substance which passed from one body to another, lowering the temperature of the one and raising that of the other, which combined with solids to form liquids, and with liquids to form gases or vapors. But the most delicate balances fail to show any change PRELIMINAR Y. 1 5 7 of weight when heat passes from one body to another. Count Rumford was able to raise a considerable quantity of water to the boiling-point by the friction of a blunt boring-tool within the bore of a cannon. It was evident that the heat manifested in this experiment could not have come from any of the bodies present. It was evident, also, that heat would continue to be developed as long as the borer continued to revolve ; that is, the supply of heat was practically inexhaustible. The heat must have been generated by the friction. That ice is not melted by the combination with it of a heat substance was shown early in the present cen- tury by an experiment by Sir Humphry Davy. He caused ice to melt by frjction of one piece upon another in a vacuum, the experiment being performed in a room where the temperature was below the melting-point of ice. There was no source from which heat could be drawn. The ice must, therefore, have been melted by the friction. The explanation of such experiments is very simple upon the hypothesis that heat is a form of energy. Count Rumford was convinced that the heat in his experiment was only transformed mechanical energy ; but to demonstrate this it was necessary to prove that the quantity of heat produced was always proportional to the quantity of mechanical energy employed. This was done in the most complete manner by Joule in a series of experiments extending from 1842 to 1849. He showed, that, however the heat was produced by me- chanical means, whether by the agitation of water by a paddle-wheel, the agitation of mercury, or the friction of iron plates upon each other, the same expenditure of IS8 ELEMENTARY PHYSICS. mechanical energy always developed the same quantity of heat. Joule also proved the perfect equivalence of heat and electrical energy. There are many reasons for believing that "heat is a mode of motion." Upon this hypothesis the molecules of all bodies are in constant motion. Upon the energy of this motion depends the temperature of the body. When a body is warmed, the motion of its particles becomes more energetic ; when it is cooled, the energy of the motion diminishes. CHAPTER I. THERMOMETRY. 92. Temperature. — Two bodies are said to be at the same temperature, when, if they be brought into inti- mate contact, no heat is transferred from one to the other. A body is at a high temperature relatively to other bodies when it gives up heat to them. The fact that it gives up heat may be shown by its own contrac- tion. A body is at a low temperature when it receives heat from surrounding bodies. It is understood, of course, in what is said above, that one body has no action upon the other ; in other words, no work is done by one body upon the other when they are brought in contact. 93. Thermometers. — Experiments show, that, in gen- eral, bodies expand, and their temperature rises pro- gressively, with the application of heat. An instrument may be constructed which will show at any instant the volume of a body selected for the purpose. If the volume increases, we know that the temperature rises ; if the volume remains constant or diminishes, we know that the temperature remains stationary or falls. Such an instrument is a thermometer. The thermometer most in use consists of a glass bulb with a fine tube attached, the bulb and part of the tube 159 l60 ELEMENTARY PHYSICS. containing mercury. In order that the thermometers of different makers may give similar readings, it is necessary to adopt two standard temperatures which can be easily and certainly reproduced. The tempera- tures adopted are the melting-point of ice, and the tem- perature of steam from boiling water under a pressure equal to that of a column of mercury 760 millimetres high at Paris. The instrument having been filled with mercury, it is plunged in melting ice, from which the water is allowed to drain away, and the place occupied by the top of the column is marked upon the stem. It is then placed in an apparatus consisting of a vessel in which water is boiled, the steam rising through a tube, surrounding the thermometer, and then descending by an annular space between the inner and outer tubes, and escaping at the bottom. The thermometer does not touch the water, but is entirely surrounded by steam. The point reached by the column of mercury is marked on the stem, as before. The space between these two points is now divided into a number of equal parts. But, while all makers of thermometers have adopted the same standard temperatures for the fixed points of the scale, they differ as to the number of divisions between these points. The thermometers used for scientific purposes, and in general use in France, have the space between the fixed points divided into a hundred equal parts or degrees, the melting-point of ice being marked 0°, and the boiling-point 100°. This is called the Centigrade or Celsius Scale. The Reaumur Scale, in use in Germany, has eighty degrees between the freezing and boiling points, and the boiling-point is marked 80°. THERMOMETRY. l6l The Fahrenheit Scale, in general use in England and America, has a hundred and eighty degrees between the freezing and boiling points ; the former being marked 32°, and the latter 212°. The following formulas represent the relations be- tween these scales : — tc = temperature Centigrade. tf =■ temperature Fahrenheit. tr = temperature Reaumur. tf=%tc + Zi=^lU + Z2. (38) 4 = K = f(-f/-32)- The divisions in all these cases are extended below the zero point, and are numbered from zero downward. Temperatures below zero must, therefore, be read and treated as negative quantities. A few points in the process of construction of a ther- mometer deserve notice. It is found that glass, after being heated to a high temperature, does not return to its original volume for some time after being cooled. The bulb of a thermometer must be heated in the pro- cess of filling with mercury, and it will not return to its normal volume for some months. The construction of the scale should not be proceeded with till the reser- voir has ceased to contract. For the same reason, if the thermometer is used for high temperatures, even the temperature of boiling water, time must be given for the reservoir to return to its original volume before using it for the measurement of low temperatures. It is essential that the diameter of the tube should be nearly uniform throughout, and that the divisions of 1 62 ELEMENTARY PHYSICS. the scale should represent equal capacities in the tube. To test the tube, a thread of mercury of say 50 milli- taietres length is introduced, and its length measured in different parts of the tube. If the length varies by more than a millimetre, the tube should be rejected. If the tube be found to be suitable, a reservoir is at- tached, mercury is introduced, and the tube sealed after the mercury has been heated to expel the air. When it is ready for graduation, the fixed points are deter- mined ; then a thread of mercury of about ten degrees in length is detached from the column, and its length measured in all parts of the tube. By reference to these measurements, the tube is so graduated that the divisions represent parts of equal capacity, and are not necessarily of equal length. If such a thermometer indicate a temperature of 10° (Centigrade), this means that the thermometer is in such a thermal condition that the volume of the mercury has increased from zero one-tenth of its total expansion from zero to 100". There is no reason for supposing that this represents the same proportional rise of tem- perature. If a thermometer be' constructed in the manner described, using some liquid beside mercury, it would not in general indicate the same temperature as the mercurial thermometer, except at the two standard points. It is plain, therefore, that a given fraction of the expansion of a liquid from zero to 100° cannot be •taken as representing the same fraction of the rise of temperature. 94. Air Thermometer. — If a gas is heated, and its volume kept constant, its pressure increases. The so- called permanent gases — those that are liquefied only THERMO ME TRY. 1 63 With great difficulty — are each found to increase in pressure by almost exactly the same amount for the same increase of temperature. This is itself a reason for supposing that the increase of pressure is pro- portional to the increase of temperature. There are theoretical reasons, as will be seen later, for the same supposition. If air be substituted for mercury in the thermome- ter, and means are provided for maintaining its volume constant, and measuring its pressure, the instrument becomes an air thermometer. For the reasons given above, the air thermometer is taken as the standard instrument for scientific purposes. Its use, however, involves several careful observations and tedious com- putations. It is, therefore, mainly employed as an instrument with which to compare other instruments. By making such a comparison, and constructing a table of corrections, the readings of any thermometer may be reduced to the corresponding readings of the air thermometer. 95. L'imiis in the Range of the Mercurial Thermometer. — The range of temperature for which the mercurial ther- mometer may be employed is limited by the freezing of the mercury on the one hand, and its boiling on the other. For temperatures below the freezing-point of mercury, alcohol thermometers may be employed. For the measurement of high temperatures, several different instruments have been made use of. One depends upon the expansion of a bar of platinum, another upon the variation in the electrical resistance of platinum wire, another upon the strength of the electric current gen- erated by a thermo-electric pair ; another consists of 1 64 ELEMENTARY PHYSICS. a globe of glass or porcelain, having a short tube with a small orifice containing a small quantity of mercury. The whole is placed in the furnace or other place whose temperature is to be measured. The mercury boils, the air and the excess of mercury are expelled, and we have finally the globe full of mercury vapor at the tempera- ture of the furnace. The globe is cooled, and the weight of the mercury left in it determined. Knowing this and the volume of the globe, the temperature can be computed. 96. Registering Tfiermometers. — Maximum and mini- mum thermometers are employed to register the highest and lowest temperatures reached during a given period. By a change in construction, the ordinary mercury be- comes a self-registering maximum thermometer. This change consists in contracting the tube just above the reservoir to such an extent, that, though the mercury is pushed through it as the temperature rises, it does not return as the temperature falls. It thus serves as an index to show the highest temperature reached during the period of its exposure. After an observation has been made, the thermometer is re-adjusted for a new observation by allowing the instrument to swing out of the horizontal position in which it usually rests, about a point near the upper extremity of the tube. In the construction of the minimum thermometer, alcohol is the liquid employed. Before sealing, an index of glass, smaller in diameter than the bore of the tube, is inserted. When the instrument is adjusted for use, this is brought in contact with the extremity of the column, and the tube is placed in a horizontal position. If, now, the alcohol expand, it will flow past the index THERMOMETRY. l6S without moving it ; but, if it contract, it will, by adhe- sion, draw the index after it. Thus the minimum tem- perature is registered. Registering thermometers have been made to give a continuous record of changes of temperature. One method of effecting this is to produce an image of the thermometer tube, which is strongly illuminated by a light placed behind it, upon a screen of sensitized paper which moves continuously by means of clockwork. Light is excluded from the whole of the paper, except the* part that corresponds to the image of the tube above the mercury. This part of the paper is blackened by the light ; and, as the paper moves, the edge of the black- ened portion will present a sinuous line corresponding to the movements of the mercury of the thermometer. l66 ELEMENTARY PHYSICS. CHAPTER II. CALORIMETRY. 97. Unit of Heat. — It is evident that more heat is required to raise the temperature of a large quantity of a substance t degrees than to raise the temperature of a small quantity of the same substance by the same amount. More heat is generated by a gas-flame in one hour than in half an hour. More heat is generated by burning a ton of coal than by burning a less amount. Heat is therefore a quantity the magnitude of which may be expressed in terms of some convenient unit. The unit of heat generally adopted is the heat required to raise the temperature of one kilogram of water from zero to one degree. It is called a calorie. It is sometimes convenient to employ a smaller unit, — the quantity of heat necessary to raise one gram of water from zero to one degree. This unit is desig- nated the lesser calorie. It is one one-thousandth of the larger unit. It may, therefore, be called a mille- calorie. 98. Heat required to raise tiie Temperature of a Ufass of Water. — It is evident that to raise the temperature of m kilograms of water from zero to one degree will require m calories. If the temperature of the same quantity of water fall from one degree to zero, the same quantity of heat is given to surrounding bodies. CALORIMETRY. 1 67 Experiment shows, that, if the same quantity of water be raised to different temperatures, quantities of heat nearly proportional to the rise in temperature will be required : hence, to raise the temperature of vi kilo- grams of water from zero^to t degrees requires mt calories very nearly. This is shown by mixing water at a lower with water at a higher temperature. The temperature of the mixture will be almost exactly the mean. Regnault, trying this experiment with the great- est care, found the temperature of the mixture a lit- tle higher than the mean, and concluded that the quantity of heat required to raise the temperature of a kilogram of water one degree increases slightly with the temperature ; that is, to raise the temperature of a kilogram of water from twenty to twenty-one degrees requires a little more heat than to raise the temper- ature of the same quantity of water from zero to one degree. Rowland found, by mixing water at various tempera- tures, and also by measuring the energy required to raise the temperature of water by agitation by a paddle- wheel, that, when the air thermometer is taken as a standard, the quantity of heat necessary to raise the temperature of a given quantity of water one degree diminishes slightly from zero to thirty degrees, and then increases to the boiling-point. 99. Specific Heat — Only one-thirtieth as much heat is required to raise the temperature of a kilogram of mercury from zero to one degree as to raise the tem- perature of a kilogram of water through the same range. Other substances require, to raise their temperature, quantities of heat peculiar to each substance. l68 ELEMENTARY PHYSICS. The quantity of heat required to raise the tempera- ture of one kilogram of a substance from zero to one degree is called its specific heat. The quantity of heat required to raise one kilogram of a substance from tX.Q t-\-\ degrees is called its spe- cific heat at temperature t.* The specific heats of substances are generally nearly constant between zero and one hundred degrees. The mean specific heat of a substance between zero and one hundred degrees is the one usually given in the tables. The measurement of specific heat is one of the im- portant objects of calorimetry. lOO. Ice Calorimeter. — The ice calorimeter of Black consisted of a block of pure ice having a cavity in its interior covered by a thick slab of ice. The body whose specific heat was to be determined was heated to T degrees, then dropped into the cavity, and immediately covered by the slab. After a short time the tempera- ture of the body falls to zero, and in doing so converts a certain quantity of ice into water. This water is re- moved by a sponge of known weight, and its weight is determined. It will be shown, that to melt a kilogram of ice requires 80 calories ; if, then, the weight of the body be P, and its specific heat x, it gives up, in falling from T degrees to zero, PxT calories. On the other hand, if / kilograms of ice be melted, the heat required is- Bop. Therefore PxT=z Sop. ^ = ^- (39) * Or, more exactly, the ratio of the small quantity of heat dQ to the corre- sponding rise of temperature di is the specific heat at i degrees. CALORIMETRY. 169 Bunsen's ice calorimeter (Fig. 44) is used for deter- mining the specific heats of substances of which only a small quantity is at hand. The apparatus is entirely of glass. The tube B is filled with water and mercury, the latter extending into the graduated capillary tube C. To use the apparatus, alcohol which has been artificially cooled to below zero is passed through the tube A, caus- ing a layer of ice to form around the outside of this tube. As water freezes, it expands. This causes the mercury to advance in the capillary tube C. When a sufficient quan- tity of ice has been formed, the alcohol is removed from A, the apparatus is surround- ed by melting snow or ice, and a small quantity of water is introduced, which soon falls in temperature to zero. The position of the mercury in C is now noted ; and the substance, at the temperature of the surrounding air — say, fifteen or twenty degrees — is dropped into the water in A. Its temperature quickly falls to zero, its excess of heat being entirely employed in melting the ice which surrounds the tube A. As the ice melts, the mercury in the tube C retreats. The change of position is an indication of the quantity of ice melted, and the quan- tity of ice melted measures the heat given up by the substance. The number of divisions of the tube C cor- responding to one calorie can be determined by direct experiment. Perform the experiment as described above. 170 ELEMENTARY PHYSICS. using a substance whose specific heat c is known. If its mass be p and its temperature t, it gives up, in cool- ing to zero, cpt calories. If the mercury retreat at the same time n divisions, to one division corresponds — n calories. If, now, a mass /' of a substance at a tempera- ture f be introduced, and the mercury fall ;/ divisions, the number of calories which must have been given to 11'cpt the ice is — ^—, and the specific heat of the substance n n'cpt ' / >. " = ^- (40) lOi. Method of Mixtures. — This consists in bringing together at different temperatures the substance whose specific heat is desired and another whose specific heat is known, and noting the change of temperature which each undergoes. The water calorimeter consists of a vessel of very thin copper or brass, highly polished, and placed within another vessel upon non-conducting supports. A mass P of the substance whose specific heat is to be deter- mined is brought to a temperature 7" in a suitable bath, then plunged in water whose temperature is t, contained in the calorimeter. The whole will soon come to a common temperature 6. The substance loses Px(T — 0) calories. The heat gained by the calorimeter consists of the heat gained by the water, p{Q — t) ; heat gained by the vessel, p'c'{9 — t) ; heat gained by the glass-stirrer and the glass of the thermometer, p"c"{d — t) ; heat gained by the mercury of the thermometer p"'c"\6 — t) ; where CALORIMETRY. 171 f represents the mass of the water, p' the mass of the vessel, /" the mass of the glass of the stirrer and of the thermometer, /'" the mass of the mercury of the thermometer, d the specific heat of the material of the vessel, d' the specific heat of glass, and d" the spe- cific heat of mercury. If no heat is lost or gained by radiation, the heat lost by the substance is equal to that gained by the calorimeter: whence Px(T — 0) ■= {p +/V +/V' +/V")(^ - t). To determine x from this formula, d, d', and d", must be known. Approximate values of these may be obtained and used in the formula, but it is better to determine the value of /5/); (44) and, since density is inversely as the volume, For a homogeneous solid, the co-efficient of voluminal expansion is three times that of linear expansion ; for if a cube, the length of whose edge is _unity, is raised one degree in temperature, its edge becomes \ -\-a and its volume i + 3« -|- 3^^ + <^- Since a is very small, its square and cube may be neglected ; and its volume at one degree is, therefore, i + 3«. 31a: is, therefore, the co-efficient of voluminal expansion. 106. Measurement of Co-efficienis of Linear Expansion. — Co-efficients of linear expansion are measured by com- paring the lengths, at different temperatures, of a bar of the substance whose co-efficient is required, with the length, at constant temperature, of another bar. The constant temperature of the latter bar is secured by EFFECTS OF HEA T. — SOLIDS. — LIQUIDS. 1 79 immersing it in melting ice. The bar whose co-efficient is sought may be carried to different temperatures by immersing it in a liquid bath ; but it is found better to place the bar upon the instrument by means of which the comparisons are to be made, and leave it for several hours exposed to the air of the room, which is kept at a constant temperature by artificial means. Of course several hours must elapse between any two comparisons by this method, and its application is restricted to such ranges of temperature as may be obtained in occupied rooms ; but within this range the observations can be made much more accurately than where the bar is im- mersed in a bath, and it is within this range that an accurate knowledge of co-efficients of expansion is of most importance. 107. Expansion of Liquids. — In studying the expan- sion of a liquid, it is important to distinguish its abso- lute expansion, or the real increase in volume, and its apparent expansion, or its increase in volume in com- parison with that of the containing vessel. 108. Absolute Expansion of Mercury. — A knowledge of the co-efficients of expansion of mercury is of the greatest importance in physical investigations, since mercury is made use of for so many purposes in physical research. Various experimenters have occupied them- selves with this problem, but Regnault has made the most elaborate and accurate experiments. To determine the absolute co-efficient, the experi- ment must be so made that the expansion of the vessel shall not influence the result. Regnault's experiment consisted in comparing the lengths of two columns of mercury, which produced i8o ELEMENTARY PHYSICS. the same pressure, though of different temperatures. Two vertical tubes, ab, a'b' (Fig. 47), were connected at the top by a horizontal tube aa', and at the bottom by a tube bcdd'db', a part of which was of glass, and shaped like an inverted U. The top of the inverted U was con- nected by a tube e with a vessel /, in which air could be maintained at any desired pressure. Filling these tubes with mercury, it flows freely from one to the other at the top ; but the flow of mercury between them at the bottom is pre- vented by air imprisoned in the inverted U, while the pressure is transmitted undiminished. The pressure at each end of the imprisoned air must, therefore, be the same ; and, since a and ffl' are connected by a horizontal tube, the pressures at those points are the same also : hence the difference of pressure be- tween a and d must be equal to the difference between a' and d' ; and, from "Mechanics," § ^6, it follows that the lengths of the columns (without regard to the diameters of the tubes) pro- ducing this difference, are inversely proportional to their densities. If, now, one branch be raised to tempera- ture t, while the other remains at o, the mercury in the inverted U will assume different levels. Measuring the height of each column from the surface of the mer- cury in the U to the horizontal tube at top, we have, Fig. 47. EFFECTS OF HEAT. — SOLIDS. — LIQUIDS. l8l formula (45), if h and k' equal the height of the cold and warm columns respectively, h _d' _ I h'~ d~ 1+ kt' hkt=h' — h, h'-h k = ht ' (46) 109. Apparent Expansion of Mercury. — If a glass bulb (Fig. 48), furnished with a capillary tube, be filled with mercury at 0°, and heated to t°, some of the mercury runs out. The amount which overflows, evidently depends upon the difEerence of JT u expansion between the mercury and the f^ glass. Let P equal mass of mercury that fills the bulb and tube at 0°. After heating, there remains in the bulb a mass P — p oi mercury, which at 0° occupies the volume t^y-^ ab. The mercury which runs over, p, would at the same temperature fill the remainder of the bulb p J) and tube. Volume ab = — -—-, where d equals the den- a i) sity of mercury. Volume above ^ = ^- The mercury m ab, when heated to t°, just fills the tube ; and its apparent p volume is -z- If »• equal co-efficient of apparent expan- d sion. or f = ^(.+..), (47) 1 82 ELEMENTARY PHYSICS. Having found «, the instrument may be used as a ther- mometer; for, suppose it filled at o°, and subjected to an unknown temperature t-^, we shall have, if /j equal the mercury that then runs over, ^=(^-A)(i + «0. / A . (48) The instrument is therefore called a weight thermome- ter. The difference between the value k, found above, and the absolute co-efficient k, is due to the expansion of the glass. And if M be the co-efficient for glass, for, referring again to Fig. 48, the volume of the glass at P-P, P P 0° is -J, and at i° is -7(1 -j- i'i), which, from (47), equals a -(l-f K/)(l-f->^0- The real volume at i° of the mercury remaining in the tube is ^(i-f^.). Hence Since «■ and 1^ are small quantities, their product may be neglected : hence EFFECTS OF HEAT. — SOLIDS. — LIQUIDS. 1 83 1 10. Determination of Voluminal Expansion of Solids. — The weight thermometer may be used to determine the co-efficient of expansion of solids. For this purpose, the solid, whose volume at zero is known, must be introduced into the bulb by the glass-blower. If the bulb containing the solid be filled with mercury at 0°, and afterward heated to f, it is evident that the amount of mercury that will overflow will depend upon the co-efficient of expansion of the solid, and upon the ap- parent co-efficient of mercury. The latter having been determined for the kind of glass used, the former can be deduced. By this means the voluminal co-efficients of some solids have been determined ; and the results are found to verify the fact, deduced from theory (§ 105), that the voluminal co-efficient is three times the linear. 111. Absolute Expansion of Liquids other than Mercury. — The weight thermometer may also serve to determine the co-efficients of expansion of liquids other than mer- cury; for, kf having been found as described above, the instrument may be filled with the liquid whose co- efficient is desired, and the apparent expansion of this liquid found exactly as was that of mercury. The abso- lute co-efficient for the liquid is then the sum of the apparent co-efficient and the co-efficient for the glass. 112. Expansion of Water. — The use of water as a standard with which to compare the densities of other substances makes it necessary that we know, not merely its mean co-efficient of expansion, but its actual expan- sion, degree by degree. This is the more important since water expands very irregularly. The best series of determinations of the volumes of water at different temperatures are those of Matthiessen, who weighed 1 84 ELEMENTARY PHYSICS. in water a mass of glass whose co-efficient of expansion had been previously determined. Water contracts, instead of expanding, from o° to 4° : from that temperature to its boiling-point it expands. 113. Determination of Specific Gravity. — Water at its maximum density at 4° is the standard to which is re- ferred the specific gravities of solids and liquids. Since it is seldom practicable to make the determinations at that temperature, corrections must be made as fol- lows : — Let /^(i be the density of a substance at t° compared to water at the same temperature. Let A^ be the density of the substance at t° compared to water at 4°. Then A^ = A^^ X df, where dt is the density of water at f ; for, if W = mass of substance, Wi = mass of an equal volume of water at i°, W^ = mass of same volume of water at 4°, If Aq equals the density of a substance at 0°, A^ = A^(i -\- at), where a equals the co-efficient of voluminal expansion of the substance. 1 14. Convection. — If a vessel of water be heated at the bottom, the bottom layers become less dense than those above, producing a condition of instability. The lighter liquid is forced to rise, and the heavier layers from EFFECTS OF HEAT. — SOLIDS. — LIQUIDS. 1 85 above, coming to the bottom, are in their turn heated ; causing continuous currents. This process is called convection. By this process, masses of liquids, although poor conductors, may be rapidly heated. Water is often heated in a reservoir at a distance from the source of heat by the circulation produced in pipes leading to the source of heat and back. The winds and the great currents of the ocean are convection currents. An interesting result follows from the fact that water has a maximum density. When the water of lakes cools in winter, currents are set up, and maintained as long as the surface water becomes more dense by cooling, or until the whole mass reaches 4°. Any further cooling makes the surface water lighter, — it remains at the surface, and its temperature rapidly falls to the freezing- point, — while the great mass of the water remains at the temperature of its maximum density. 115. Conduction. — If one end of a metal rod be heated, it is found that the heat travels along the rod ; since those portions at a distance from the source of heat finally become warm. This process of transmission of heat from molecule to molecule of a body, while the molecules themselves retain their relative places, is called conduction. 1 16. Flo\N of Heat across a Wall. — To study the trans- fer of heat by this process, we will consider what takes place in a wall of homogeneous material, whose exposed surfaces, assumed to be of indefinite extent, are main- tained at a constant difference of temperature. Suppose the wall to be cut by a series of planes parallel to the exposed surfaces : there must in the fixed state of the body "be the same flow of heat across all sections ; also 1 86 ELEMENTARY PHYSICS. there must be a uniform fall of temperature from one side of the wall to the other, — that is, if t' — t is the difference of temperature between the two exposed /— t surfaces, and d the thickness of the wall, — ;— is the d fall of temperature per unit thickness, and / - i:-t d' d is the temperature at a distance d' from the warmer surface. To demonstrate this, suppose A, Fig. 49, be one ex- ^ P posed surface at temperature /, and B the other surface at temperature t: let a, a, a", be three surfaces parallel to the faces of the wall, and at very small equal distances from each other. Suppose the tempera- tures to exist according to the law stated in the proposition : then the difference of tem- perature between a and a will be the same as between a' and a". Experiment shows that the flow of heat between two points in a body depends chiefly upon the difference of temperature, the distance between them, and the nature of the material. It is also found to depend, to a very limited extent, upon the temperature itself. The effect of this factor for the pairs of surfaces considered may, however, be neglected, since they are nearly at the same temperature. The other factors being the same for both pairs, it follows that there will be the same flow of heat from a' to a" as from a to a'. The same will hold true for any other set of surfaces parallel to the faces of the wall : hence the molecules in any surface such as a' receive and part with equal ig. 49. EFFECTS OF HEAT. — SOLIDS. — LIQUIDS. 1 87 amounts of heat, and can neither rise nor fall in tem- perature. Hence the temperatures, once established in accordance with the law enunciated, could never change. On the other hand, if the difference of temperature be- tween a and a' were greater than between a' and a", the molecules in a' would receive more heat than they part with : their temperature would rise, and tend to equal- ize these differences. The proposition is therefore demonstrated. i! — t 117. f/oiy Proportional to — -j-. — It can be further shown that the flow of heat across walls of the same material is directly proportional to the differences of temperature, and inversely proportional to the thick- nesses of the walls. Let there be two walls, A and B, of thickness d and S, and let the temperatures of the exposed surfaces of .^4 be / and t, and of B, 0' and 0. Assume two planes in each wall parallel to the exposed surfaces, at very small equal distances apart, and simi- larly situated. The flow of heat in each wall, from the molecules in one plane to those in the other, will be proportional only to the differences of temperature, since all other things are equal. If e be the common distance between the planes, -^ (/ — /) will be the differ- ence of temperature between the planes in A, and ^(^ — 6) will be the same in B : hence flow of heat between the planes tn A _ d^ _ d flow of heat between the planes in B e ^ . ff —6 which proves the proposition, since the heat which flows 1 88 ELEMENTARY PHYSICS. across the wall is the same as flows between any two planes. if — t It will be seen that — =— is the rate of fall of tem- a perature at the section considered ; and it follows, finally, that the flow of heat across any section parallel to the exposed surfaces of a wall is proportional to the rate of fall of temperature at that section. 1 1 8. Conductivity. — If, now, we consider a prism ex- tending across the wall, bounded by planes perpendicular to the exposed surfaces, the area of its exposed bases being A, the quantity of heat which flows in a time T through this prism may be represented by Q = K^-^AT, (so) where ^ is a constant depending upon the material of which the wall is composed. K is the condiictivity of the substance, and may be defined as the quantity of heat which in unit time flows through a section of unit area in a wall of the substance whose thickness is unity, when its exposed surfaces are maintained at a difference of temperature of one degree ; or, in other words, it is the quantity of heat which in unit time flows through a section of unit area in a substance, where the rate of fall of temperature at that section is unity. In the above discussions the temperatures / and t are taken as the actual temperatures of the surfaces of the wall. If the colder surface of the wall be exposed to air of tempera- ture T, to which the heat which traverses it is given up, t will be greater than T. The difference will depend upon the quantity of heat which flows, and upon the facility with which the surface parts with heat. EFFECTS OF HEAT. — SOLIDS. — LIQUIDS. 189 119. Flow of Heat along a Bar. — If a prism of a sub- stance have one of its bases maintained at a tempera- ture t, while the other base and the sides are exposed to air at a lower temperature, the conditions of uniform fall of temperature no longer exist, and the amount of heat which flows through the different sections is no longer the same ; but the amount of heat that flows through any section is still proportional to the rate of fall of temperature at that section, and is equal to the heat which escapes from the portion of the bar beyond the section. 120. Measurement of Conductivity. — The bar heated at one end furnishes the best means of measuring con- ductivity. In Fig. 50 let AB represent a bar heated at A. Let the ordinates aa', bl/, cd, represent the excess of tem- peratures above the tempera- ture of the air at the points from which they are drawn. These temperatures may be determined by means of thermometers inserted in cavities in the bar, or by means of a thermopile. Draw the curve dUdd' . . . through the summits of the ordinates. The inclination of this curve at any point represents the rate of fall of tempera- ture at that point. The tangent Um, drawn to the point V, shows what would be the temperatures at various points of the bar if the fall were uniform and at the same rate as at U. It shows, that, at the rate of fall at U, the bar would be at the temperature of the air at m ; or, in the length bm, the fall of temperature would equal the amount represented by bU. The rate of fall is, there- 19° ELEMENTARY PHYSICS. fore, Y— . If Q represent the quantity of heat passing the section at b in the unit time, we have, from (ii8), Q = JS y. rate of fall of temperature X area of section. Q is the quantity of heat that escapes from all that portion of the bar beyond b. It may be found by heat- ing a short piece of the same bar to a high temperature, allowing it to cool in the same room under the same conditions that surround the bar AB, and observing its temperature from minute to minute as it falls. These observations furnish the data for computing the quantity of heat which escapes per minute from unit length of the bar at different temperatures. It is then easy to compute the amount of heat that escapes per minute from each portion, be, cd, etc., of the bar beyond b; each portion being taken so short that its temperature throughout may, without sensible error, be considered uniform and the same as that at its middle point. Sum- ming up all these quantities, we obtain the quantity Q which passes the section b in the unit time. Then ^^ Q rate of fall of temperature at b X area of section 121. Conductivity diminishes as Temperature rises. — By the method described above. Principal Forbes deter- mined the conductivity of a bar of iron at points b, c, d, etc., at different distances from the heated end, and found that the conductivity is not the same at all tem- peratui'es, but is greater as the temperature is lower. 122. Conductivity of Crystals. — The conductivity of crystals of the regular system is the same in all direc- EFFECTS OF HE A T. — SOLIDS. — LIQ UIDS. 1 9 1 tions, but in crystals of the other system it is not so. In a crystal of Iceland spar the conductivity is greatest in the direction of the axis of symmetry, and equal in all directions in a plane at right angles to that axis. 123. Conductivity of Non-liomogeneous Solids. — De la Rive and De Candolle were the first to show that wood conducts heat better in the direction of, than at right angles to, the fibres. Tyndall, by experimenting upon cubes cut from wood, has shown that the conductivity has a maximum value parallel to the fibres, a minimum value at right angles to the fibres and parallel to the annual layers, and a medium value at right angles to both fibres and annual layers. Feathers, fur, and the materials of clothing, are poor conductors because of their want of continuity. 124. Conductivity of Liquids. — The conductivity of liquids can be measured in the same way as that of solids, — by noting the fall of temperature at various distances from the source of heat in a column of liquid heated at top. Great care must be taken in these ex- periments to avoid the existence of convection currents. Liquids are generally poor conductors. 125. Radiation of Heat. — We have now considered those cases in which there is a tendency to equalization of temperature between bodies in contact. The same tendency exists between bodies not in contact. This is effected by a process called radiation, which will be considered in another part of this work. 126. Effect of Variation of Temperature upon Specific Heat. — It has already been stated (§ 98) that the spe- cific heat of bodies changes with temperature. With 192 ELEMENTARY PHYSICS. most substances specific heat increases as temperature rises. The mean specific heat Of iron Between 0° and 100° is 0.1088 Between 100° and 200° is . . . . . 0.1228 Of glass Between 0° and 100° is 0.1770 Between 100° and 200° is 0.1900 Of the diamond the true specific heat At 0° is 0.0947 At 50° is 0.1435 At 100° is 0.1905 At 200° is 0.2719 These examples are sufficient to show the fact stated. 127. Effect of Change of Pliysical Siate upon Specific Heat. — The specific heat of a substance is not the same in the solid and liquid states. In the solid state it is generally less than in the liquid. For example : — Mean Specific heat in the Sohd. Liquid. Water 0.504 i.ooo Mercury 0.0314 0.333 Tin 0.056 0.0637 Lead 0.0314 0.0402 128. Atomic Heat. — Dulong and Petit have discovered the following law : Tke product of the specific heat by the atomic weight of any simple body is a constant quan- tity. EFFECTS OF HEAT. — SOLIDS. — LIQUIDS. I93 This is equivalent to saying, that to raise the tempera- ture of an atom of any simple substance one degree requires an amount of heat which is the same for all substances. The experiments of Regnault show that this law may be extended to compound bodies ; that is, for all com- pounds of similar chemical composition, the product of the total chemical equivalent by the specific heat is the same. The following table will illustrate the law of Dulong and Petit. The atomic weights are those given by Clarke. ELEMENTS. Specific heat of equal weights. Atomic weight. Product of specific heat into atomic weight. Iron . . Copper . Mercury- Silver . Gold. . Tin . . Lead. . Zinc . . O.II4 0.095 0.0314 (solid) 0.057 0.0329 0.056 0.0314 0.0955 S5-9 63.17 199.71 107.67 196.15 1 17.7 206.47 64.9 6.372 6.001 6.128 6-137 6.4 S3 6.591 6.483 6.198 129. Fusion and Solidification. — When ice at a tem- perature below zero is heated, its temperature rises to zero, then it begins to melt ; and, however high the temperature of the medium that surrounds it, its tem- perature remains constant at zero as long as it remains in the solid state. This temperature is the melting- point of ice, and because of its fixity it is used as one of the standard temperatures in graduating thermometric scales. Other bodies melt at very different but at fixed 194 ELEMENTARY PHYSICS. and definite temperatures, as shown in the following table : — Substance. Melting-point. Carbonic acid — 78 Mercury — 39 Water o Phosphorus 44 Beeswax (white) 69 Sodium go Sulphur Ill Tin 228 Antimony 432 Silver 1000 Gold 1250 Platinum (about) 2000 Many substances cannot be melted, as they decom- pose by heat. Alloys often melt at a lower temperature than either of their constituents. An alloy of lead i, tin i, bismuth 4, melts at 94°; while the lowest melting-point of its constituents is that of tin, 228°. An alloy of lead, tin, bismuth, and cadmium, melts at 62°. If a liquid be placed in a medium whose temperature is below its melting-point, it will, in general, begin to solidify when its temperature reaches its melting-point, and it will remain at that temperature till it is all solidi- fied. Under certain conditions, however, the tempera- ture of a liquid may be lowered several degrees below its melting-point without solidification, as will be seen below. 1 30. Change of Volume with Change of State. — Sub- stances are generally more dense in the solid than in EFFECTS OF HEAT. — SOLIDS. — LIQUIDS. 195 • the liquid state, but there are some notable excep- tions. Water, on solidifying, expands : so that the density of ice at zero is only 0.9167, that of water at 4° being i. This expansion exerts considerable force, as evidenced by the bursting of vessels and pipes contain- ing water. 131. Change of Melting and Freezing Points. — If water be enclosed in a vessel sufficiently strong to prevent its expansion, it cannot freeze except at a lower tempera- ture. The freezing-point of water is, therefore, lowered by pressure. On the other hand, substances which con- tract on solidifying, have their solidification hastened by pressure. The lowering of the melting-point of ice by pressure explains some remarkable phenomena. If pieces of ice be pressed together, even in warm water, they will be firmly united. Fragments of ice may be moulded, under heavy pressure, into a solid, transparent mass. This soldering together of masses of ice is called regelation. If a loop jof wire be placed over a block of ice and weighted, it will cut its way slowly through the ice: but regelation will occur behind it ; and, after passing through, the block will be found one solid mass, as- before. The explanation of these phenomena is, that the ice is partially melted by the pressure : the hquid, as will be seen below, is colder than the ice ; it finds its way to points of less pressure (for it is not to be sup- posed that over the entire surface between two masses of ice the pressure is the same) ; and there, because of its low temperature, it congeals, firmly uniting the two masses. Water free from air, and kept perfectly quiet, will not 19^ ELEMENTARY PHYSICS. • form ice at the ordinary freezing-point. Its tempera- ture may be lowered to — io° or —12° before solidifi- cation commences. In this condition a slight jar, or the introduction of a small fragment of ice, will cause a sudden congelation of part of the liquid, accompanied by a rise in temperature in the whole mass to zero. A similar phenomenon is observed in the case of several solutions, notably sodic sulphate and sodic acetate. If a saturated hot solution of one of these salts be made, and allowed to cool in a closed bottle in perfect quiet, it will not crystallize. Upon opening the bottle and admitting air, crystallization tommences, and spreads rapidly through the mass, accompanied by a considerable rise of temperature. If the amount of salt dissolved in the water be not too great, the solution will remain liquid when cooled in the open air, and it may even suffer considerable disturbance by foreign bodies without crystallization ; but crystallization begins immediately upon contact with the smallest crystal of the same salt. 132. Heat Equivalent of Fusion. — Some facts that have appeared in the above account of the phenomena of fusion and solidification require further Study. It has been seen, that, however rapidly the temperature of a solid may be rising, the moment fusion begins, the rise of temperature ceases. Whatever the heat to which a solid may be exposed, it cannot be made hotter than its melting-point.* When ice is melted by pressure, its temperature is lowered. When a liquid, is cooled, fall * Some experiments of Carnelley, however, seem to indicate, that, under some circumstances, this is not true. EFFECTS OF HEAT. — SOLIDS. — LIQUIDS. I97 of temperature ceases when solidification begins; and if, as may occur under favorable conditions, a liquid is cooled below its melting-point, its temperature rises at once to the melting-point, when solidification begins. Heat, therefore, disappears when a body melts, and is generated when a liquid becomes solid. It was seen (§ 90) that ice can be melted by fric- tion ; that is, by the expenditure of mechanical energy. Fusion is, therefore, work which requires the expendi- ture of some form of energy to accomplish it. Experi- ment shows that to melt a given mass of a substance always requires the same amount of energy. The heat required to melt unit mass of a substance is the heat equivalent of fusion of that substance. When a sub- stance solidifies, it develops the same amount of heat as was required to melt it. 133. Nature of the Energy stored up in the Liquid. — From the facts given above, as well as from the prin- ciples of the conservation of energy, it appears that the energy expended in melting a body is stored in the liquid. It is easy to see what must be the nature of this energy. When a body solidifies, its molecules assume certain positions in obedience to their mutual attractions. When it is melted, the molecules are forced into new positions in opposition to the attractive forces. They are, therefore, in positions of advantage with respect to these forces, and possess potential energy. 1 34. Determination of the Heat Equivalent of Fusion. — The heat equivalent of fusion may be determined by the method of mixtures (§ 100), as follows : a mass P of the substance, ice, for example, at temperature t be- 1 98 ELEMENTARY PHYSICS. low its melting-point, to insure dryness, is plunged into a mass P' of warm water at temperature T. When the ice is all melted, the resulting temperature is B. lip be the water equivalent of the calorimeter, {P -\-p) {T—&) is the heat given up by the calorimeter and its contents. Let c be the specific heat of ice, and x the heat equiva- lent of fusion. The ice absorbs, to raise its temperature to zero, Ptc calories ; to melt it, Px calories ; to warm the water after melting, Pd calories. We then have this equation, Fi+Pd + Px= (P+/) {T- 6), from which x may be found. Other calorimetric methods may be employed. The best experiments give, for the heat equivalent of fusion of ice, very nearly eighty calories. GASES AND VAPORS. 199 CHAPTER IV. GASES AND VAPORS. 135. The Gaseous State. — A gas may be defined as a highly compressible fluid. A given mass of gas has no definite volume. Its volume varies with every change in the external pressure to which it is exposed. A vapor is the gaseous state of a substance which at ordi- nary temperatures exists as a solid or a liquid. 1 36. Vaporization is the process of formation of vapor. There are two phases of the process, — evaporation, in which vapor is formed at the free surface of the liquid, and ebullition, in which the vapor is formed in bubbles in the mass of the liquid, or at the heated sur- face with which it is in contact. 137. Nature of the Process of Evaporation. — It has been seen (§91) that there are many reasons for believ- ing* that the molecules of solids and liquids are in a state of continual motion. It is not supposed that any one molecule maintains continuously the same condition of motion ; but in the interaction of the molecules the motion of any one may be more or less violent, as it receives motion from, or gives up motion to, its neigh- bors. It can easily be supposed, that, at the exposed surface of the substance, the motion of a molecule may at times be so violent as to project it beyond the reach 200 ELEMENTARY PHYSICS. of the mutual attractions. If this occur in the air or in a space filled with any gas, the molecule may be turned back, and made to rejoin the molecules in the liquid mass ; but many will find their way to such a distance that they will not return. They then constitute a vapor of the substance. As the number of free molecules in the space above the liquid increases, it is plain that there may come a time when as many will rejoin the liquid as escape from it. The space is then saturated with the vapor. The more violent the motion in the liquid, — that is, the higher its temperature, — the more rapidly will the molecules escape, and the greater must be the number in the space above the liquid before the return- ing will equal the outgoing molecules. In other words, the higher the temperature, the more dense the vapor that saturates a given space. If the space above a liquid be a vacuum, the escaping molecules will at first meet with no obstruction, and, as a consequence, the space will be very quickly saturated with the vapor. Experiment verifies all these deductions. Evapora- tion goes on continually from the free surfaces of many liquids, and even solids. It increases in rapidity as the temperature increases, and ceases when the vapor has reached a certain density, always the same for the s9me temperature, but greater for a higher temperature. It goes on very rapidly in a vacuum ; but it is found that the final density of the vapor is no greater, or but little greater, than when some other gas is present. In other words, while a foreign gas impedes the motion of the outgoing molecules, and causes evaporation to go on slowly, it has very little influence upon the number of molecules that must be present in order that those that return may equal those that escape. GASES AND VAPORS. 20I 138. Pressure of Vapors. — As a liquid evaporates in a closed space, the vapor formed exerts a pressure upon the enclosure and upon the surface of the liquid, which increases as long as the quantity of vapor increases, and reaches a maximum when the space is saturated. This maximum pressure of a vapor increases with the tem- perature. When evaporation takes place in a space filled by another gas which has no action upon the vapor, the pressure of the vapor is added to that of the gas, and the pressure of the mixture is therefore the sum of the pressures of its constituents. The law was an- nounced by Dalton, that the quantity of vapor which saturates a given' space, and consequently the maximum pressure of that vapor, is the same whether the space be empty or contain a gas. Regnault has shown, that, for water, ether, and some other substances, the maxi- mum pressure is slightly less when air is present. 139. Ebullition. — As the temperature of a liquid rises, the pressure which its vapor may exert increases, until a point is reached where the vapor is capable of form- ing, in the mass of the liquid, bubbles which can with- stand the superincumbent pressure of the liquid and the atmosphere above it. These bubbles of vapor, escaping from the liquid, give rise to the phenomenon called ebullition, or boiling. Boiling may, therefore, be defined as the agitation of a liquid by its own vapor. Generally speaking, for a given liquid ebullition always occurs at the same temperature for the same pressure ; and, when once commenced, the tempera- ture of the liquid no longer rises, no matter how in- tense the source of heat. This fixed temperature is called the boiling-point of the liquid. It differs for dif- 202 ELEMENTARY PHYSICS. ferent liquids, and for the same liquid under different pressures. That the boiling-point must depend upon the pressure is evident from the explanation of the phenomenon of ebullition above given. Substances in solution, if less volatile than the liquid, retard ebullition. While pure water boils at loo", water saturated with common salt boils at 109°. The material of the containing vessel also influences the boiling-point. In a glass vessel the temperature of boiling water is higher than in one of metal. If water be deprived of air by long boiling, its temperature may be raised con- siderably above the boiling-point before ebullition com- mences. Under these conditions, the first bubbles of vapor will form with explosive violence. The air dis- solved in water separates at a high temperature in minute bubbles. Into these the water evaporates, and, whenever the elastic force of the vapor is sufficient to overcome the superincumbent pressure, enlarges them, and causes the commotion that marks the phenomenon of ebullition. If no such openings in the mass of the fluid exist, the cohesion of the fluid, or its adhesion to the vessel, as well as the pressure, must be overcome by the vapor : hence the rise in temperature when air is absent. 140. Produciion of Vapor in a Limited Space. — When a liquid is heated in a limited space the vapor generated accumulates, increasing the pressure, and causing the temperature to rise above the ordinary boiling-point. Cagniard-Latour experimented upon liquids in spaces but little larger than their own volumes. He found, that, at a certain temperature, the liquid suddenly disap- peared ; that is, it was converted into vapor in a space GASES AND VAPORS. 2O3 but little larger than its own volume. It is supposed, that above the temperature at which this occurs, which is called the critical temperature, the substance cannot exist in the liquid ^state. 141. Spheroidal State. — If a liquid be introduced into a highly heated capsule, or poured upon a very hot plate, it does not wet the heated surface, but forms a flattened spheroid, which presents no appearance of boiling, but evaporates only very slowly. Boutigny has carefully studied these phenomena, and made known the following facts. The temperature of the sphei'oid is below the boiling-point of the liquid. The spheroid does not touch the heated plate, but is separated from it by a non-conducting layer of vapor. This accounts for the slowness of the evaporation. To maintain the spheroid, the temperature of the capsule must be much above the boiling-point of the liquid ; for water it must be at least 200° C. If the capsule be allowed to cool, the temperature will soon fall below the limit necessary to maintain the spheroid, the liquid will moisten it, and there will be a rapid ebullition, with disengagement of a large amount of vapor. If a liquid of very low boiling- point, as liquid nitrous oxide, which boils at — 88°, be poured into a red-hot capsule, it will assume the sphe- roidal state ; and, since its temperature cannot rise above its boiling-point, water, or even mercury, plunged into it will be frozen. 142. Liquefaction. — Only a certain amount of vapor can exist in a given space at a given temperature. If a space saturated with vapor be cooled, some of the vapor must condense into the liquid state. It is not necessary that the whole space be cooled ; for, the vapor in the 204 ELEMENTARY PHYSICS. cooled portion being condensed, its pressure is dimin- ished, the vapor from the warmer portion flows in, to be in its turn condensed, and this continues till the whole is brought to the density and pressure due to the cooled portion. Any diminution of the space occupied by a saturated vapor will cause some of the vapor to become liquid, for, if it do not condense, its density and pressure must increase; but a saturated vapor is already at its maximum density and pressure. If the vapor in a given space be not at its maximum density, its pressure will increase when its volume is diminished, until the maximum pressure is reached ; when, if the temperature remain constant, further re- duction of volume causes condensation into the liquid state, without further increase of density or pressure. This statement is true of several of the gases at ordi- nary temperatures. Chlorine, sulphurous acid, ammonia, nitrous oxide, carbonic acid, and several other gases, become liquid under sufficient pressure. Andrews has found, that, at a temperature of 30.92", pressure ceases to liquefy carbonic acid. This is the critical tempera- ture for that substance. The critical temperatures of oxygen, hydrogen, and the other so-called permanent gases, are so low that it is only by methods capable of yielding an extremely low temperature, combined with great pressure, that they can be liquefied. By the use of such methods any of the gases may be made to assume the liquid state. In the case of hydrogen, how- ever, the low temperature necessary for its liquefaction has only been reached by allowing the gas to expand suddenly from a condition of great condensation, in which it has already been cooled to a very low point. CASES AND VAPORS. 205 143. Pressure and Density of IVon-saturated Gases and Sapors. — If a gas or vapor in the non-saturated condi- tion be maintained at constant temperature, it follows very nearly Boyle's law (Mechanics, ^^67 and 88). If its temperature be below its critical temperature, the product of volume by pressure diminishes, and near the point of maximum density the departure from the law may be considerable. At this point there is a sud- den diminution of volume, the vapor assuming the liquid state. This liquid is at first much more compressible than ordinary liquids, but its compressibility diminishes with the pressure. The less the pressure and density in the gaseous state, the more nearly is Boyle's law fulfilled. It has been seen already (§ 89) that gases expand as the temperature rises. Gay-Lussac was one of the first to measure accurately the expansion. He announced the following law, which bears his name : For each increment of temperature of i" a gas expands by a con- stant fraction of its volume at zero. This is equivalent to saying that a gas has a constant co-efficient of expan- sion. Hence the following formulas. Let Vq, Vi, represent the volumes at zero and f re- spectively, and u the co-efficient of expansion. Then, the pressure remaining constant, V, = V,{i+a(). (51) If (/g, di, represent the densities at the two temperatures, v/e have, since densities are inversely as volumes, 1 + at Charles discovered another very important law, — 206 ELEMENTARY PHYSICS. that all gases expand equally for equal increments of temperature ; in other words, the co-efficients of expan- sion of all gases are the same. Later investigations, especially those of Regnault, show that these simple laws, like the law of Boyle, are not rigorously true, though' they are very nearly so for all gases and vapors when not too near their points of maximum density. The common co-efficient of expan- sion is a = 0.00367 = ^^3 very nearly. From the law of Boyle we have, for a given mass of gas, temperature remaining constant, Vj,p = Vp,J>' = volume at pressure unity, where Vp, V^, represent the volumes at pressure / and /' respectively. From the law of Gay-Lussac we have, pressure re- maining constant, \-\-at I + af V^ = ^J^ = ^J^,. (53) If the temperature and pressure both vary, we have Vfip___Vjwf_. u.\ \ -{■ o.t I + a/ that is, if the volume of a given mass of gas be mul- tiplied by the corresponding pressure and divided by the factor of expansion, the quotient is constant. The symbol Vpt may be read, volume at pressure p and tem- perature t. T/)/[ Pressure of a Gas or Vapor at its Maximum Density. — It has been seen, that, for each gas or vapor at a temperature below the critical temperature, there is a maximum density, and consequently a maximum press- GASES AND VAPORS. 20J ure which it can exert at that temperature. To each temperature, there corresponds a maximum pressure, which is higher as the temperature is higher. A gas in contact with its liquid in a closed space will exert its maximum pressure. The relation between the temperature and the cor- responding maximum pressure of a vapor is a very important one, and has been the subject of many in- vestigations. The vapor of water has been especially studied, the most extensive and accurate experiments being those of Regnault. Two distinct methods were employed, — one for tem- peratures below 50°, and the other for higher tempera- tures. The first consisted in observing the difference in height of two barometers placed side by side, the vacuum chamber of one containing a little water. The temperature was carried from zero to about 50° ; both barometers being surrounded by the same medium, and in every way under the same conditions, except the presence of water and its vapor in one, and their absence from the other. The difference between the heights of the two, therefore, gave the pressure of the vapor at the temperature of the experiment. The second method was founded on the principle that the vapor of a boiling liquid exerts a pressure equal to that of the atmosphere above it. The experiment con- sisted in boiling water in a closed space in which the air could be rarefied or condensed to a known pressure, and noting the temperature of the boiling liquid and the vapor above it. To prevent the accumulation of the vapor and the consequent change of pressure, a part of the space communicating with the boiler, was a tube 2o8 ELEMENTARY PHYSICS. surrounded by a larger tube, forming an annular space, through which a stream of cold water was kept flowing. By this means the vapor was condensed as fast as formed, the water from its condensation flowing back into the boiler. By rarefying or compressing the air in the closed space, an artificial atmosphere of any desired pressure could be obtained, and maintained constant as long as was necessary for making the observations. The temperature was determined by means of four thermometers placed in the boiler, two of them in the liquid and two in the vapor. The bulbs of the thermome- ters were in metal tubes, which protected them from the pressure, which, by compressing the bulb, would cause the thermometer to register too high a tempera- ture. The results of Regnault's observations may be repre- sented graphically, as in Fig. 5 1, where pressures are measured in the vertical, and temperatures in the horizontal, direction. It is seen that the pressure varies very rapidly with the tem- perature. 145. Kineiio Theory of Gases. — According to the ki- netic theory, a perfect gas consists of an assemblage of free, perfectly elastic molecules in constant motion. Each molecule moves in a straight line with a constant velocity, untd it encounters soiTie other molecule, or the Temperahires. Fig. 51. GASES AND VAPORS. 209 sides of the vessel. The impact of the molecules upon the sides of the vessel are so numerous that their effect is a continuous constant force or: pressure. The entire independence of the molecules follows from the fact, that, when gases or vapors are mixed, the pressure of one is added to that of the other ; that is, the pressure of the mixture is the sum of the pressures of the separate gases. It follows from this independ- ence that no energy is required to separate the mole- cules ; in other words, no internal work needs be done to expand a gas. This was demonstrated experimentally by Joule, who showed, that, when a gas expands without performing external work, it is not cooled. Let us consider the laws and phenomena of gases from the standpoint of the kinetic theory. The action between two molecules, or between a molecule and solid wall, must be of such a nature that no energy is lost ; that is, the sum of the kinetic energies of all the mole- cules must remain constant. Whatever be the nature of this action, it is evident, that, when a molecule strikes a solid stationary wall, it must be reflected back with a velocity equal to that before impact. If the velocity be resolved into two components, one parallel and the other normal to the wall, the parallel component remains unchanged, while the normal component is changed from -f- V before impact, to — v after impact. The change of velocity is, therefore, 2v ; and if 6 be the duration of impact, the mean acceleration is — -, and the mean force V of impact / = m—, where m is the mass of the mole- cule : hence pB=2mv; (ss) 2IO ELEMENTARY PHYSICS. that is, the product of the mean force by the duration of impact equals the change of momentum of the par- ticle, and the total force or pressure exerted upon a given area, multiplied by a time, will equal the sum of the changes of momentum of the molecules impinging upon that area during that time : for the number of molecules, even in the smallest mass of gas upon which we can experiment, is so great that the impacts will be continuous, and the same at any one instant as at any other. The number of impacts occurring in unit time may be determined from the following considerations : Suppose the molecules between two parallel walls at distance s from each other : any molecule may be sup- posed to suffer reflection from one wall, pass across to the other, be reflected back to the first, and so on. Whatever may be the effect of the mutual collisions of the molecules, the number of impacts upon the surface considered will be the same as though each one pre- served its rectilinear motion unchanged, except when reflected from the solid walls. The time required for a molecule to pass across the space between the two walls and back is, if v be the component of the velocity normal to the surface, — ; and the number of impacts v upon the first surface in unit time is — • The sum of the changes of the momentum of this molecule is 2mv X — = 2S s The total pressure P upon unit area, multiplied by the time, one second, equals the sum of the changes GASES AND VAPORS. .211 of momentum of all the molecules in the space ; that is, /' X I = 5 s The sum of is proportional to the number, n', of molecules in the space. These molecules must be con- sidered as moving in all directions and with various velocities. But the velocity of any molecule may be resolved in the direction of three rectangular axes, one normal to the surface and the other two parallel to it ; and, since the number of molecules in any finite volume of gas is practically infinite, the effect upon the wall due to their real motions would be the same as would result from a motion of one-third the molecules in each of the three directions with the mean velocity. Now, the molecules moving in the two directions parallel to the wall have no influence upon the pressure. We have, n' then, the pressure upon unit surface, P = i—mv\ n' But — = the number of molecules in unit volume s = n: hence P= \nmv^ (s6) and the pressure upon unit area is equal to one-third the number of molecules in unit volume at that pressure multiplied by twice the kinetic energy of each molecule. Suppose, now, the volume of the gas be changed to V, without change of temperature. The number of ft molecules in unit volume is now -=^, and the pressure 212 ELEMENTARY PHYSICS. Pi = ij^mv\ whence /'jF= Inmv^, equals a constant, since n and m are constant for the same mass of gas, and V is constant if there is no change of temperature. But PF" equal to a constant is Boyle's law. From the law of Gay-Lussac we have, if /* =: pressure at t°, and P^ = pressure at zero, P=P,{l+C.t). a = 0.00397 = g^3 very nearly : hence If ( = - 273°, that is, at 273° below zero Centigrade the pressure is null ; and since P = inmv^, it follows, that, at this tem- perature, V ^o, or the molecules are at rest. This is therefore called the absolute zero. In studying the expansion of gases, it is very conven- ient to use a scale of temperatures whose zero-point is at the absolute zero. Temperatures reckoned upon this scale are called absolute temperatures. Let T represent a temperature upon the absolute scale : then 7'=jf-l-273, and formula (57) becomes T °273 Substituting the value of P from (56), \nmv'' = /*(, T 273' ■^ GASES AND VAPORS. 213 That is, the absolute temperature of a gas is propor- tional to the kinetic energy of the molecules. It has been seen already (§ go), that, when- a gas is compressed, a certain amount of heat is generated. Suppose a cylinder with a tight-fitting piston. As long as the piston is at rest, each molecule that strikes it is reflected with a velocity equal to that before im- pact : but if the piston be forced into the cylinder, each molecule, as it is reflected, has its velocity increased ; and, as was shown above, this is equivalent to a rise in temperature. It can be shown that the increase of kinetic eflergy in this case is precisely equal to the work done in forcing the piston into the cylinder against the pressure of the gas. On the other hand, if the piston be pushed backward by the force of the impact of the molecules, there will be a loss of velocity by reflection from the moving-surface, kinetic energy equal in amount to the work done upon the piston disappears, and the temperature falls. 146. Mean Velocity of Molecules. — Equation (56) enables us to determine the mean velocity of the molecules of a gas whose density and pressure are known, since nm is the mass of the gas in unit volume. Solving the equation with reference to v, and substitut- ing the known values for hydrogen, namely, P^ioiss/s dynes per square centimetre, and nm, or density, r= 0.00008954 grams per cubic centimetre, we have 184260 centimetres per second, or a little more than one mile per second. 147. Elasticity of Gases. — It has been shown (Mechan- ics, § 6y) that the modulus of elasticity of a gas, obeying Boyle's law, is numerically equal to the pressure. This 214 ELEMENTARY PHYSICS. is the elasticity for constant temperature. But, as was seen (§ 145), when a gas is compressed, it is heated ; and heating a gas increases its pressure. Under ordinary conditions, therefore, the ratio of a small increase of pressure to the corresponding decrease of the unit vol- ume (= elasticity) is greater than when temperature is constant. It is important to consider the case when all the heat generated by the compression is retained by the gas. The elasticity is then a maximum, and is called the elasticity .when no heat is allowed to enter or escape. Let mn (Fig. 52) be a curve representing the relation between volume and pres- sure for constant tempera- ture, X representing volume and y pressure. Such a curve is called an isothermal line. It is plain that to each temperature must cor- respond its own isothermal line. If, now, we suppose the gas to be compressed, and no heat to escape, it is plain, that, if the volume di- minish from OC to OG, the pressure will become greater than GD ; say, GM. If a number of such points as M be found, and a line be drawn through them, it repre- sents the relation between volume and pressure when no heat enters or escapes. It is called an adiabatic line. It evidently makes a greater angle with the hori- zontal than the isothermal. 148. Specific Heats of Gases. — In § 145 it is seen that Fig. 52. GASES AND VAPORS. 21$ the temperature of a gas is proportional to the kinetic energy of the molecules. To warm a gas without change of volume is, therefore, only to add to this kinetic en- ergy. If, however, the gas be allowed to expand when heated, the molecules lose energy by impact upon the receding surface ; and this, together with the kinetic energy due to the rise in temperature, must be supplied from the source of heat. It has been seen that the loss of energy resulting from impact upon a receding surface is equal to the work done by the gas in expanding. The amount of heat necessary to raise the tempera- ture of unit mass of a gas one degree, while the volume remains unchanged, is called the specific heat of the gas at constant volume. The amount of heat necessary to raise the temperature of unit mass of a gas one degree when expansion takes place without change of pressure, is called the specific heat of the gas at constant pressure. From what has been said above, it is evident that the difference between these two quantities of heat is the equivalent of the work done by the expanding gas. The determination of the relation of these two quan- tities is a very important problem. The specific heat of a gas at constant pressure may be found by passing a current of warmed gas through a calorimeter, — method of mixtures. There are great difficulties in the way of an accurate determination, because of the small density of the gas, and the time required to pass enough of it through the calorimeter to obtain a reasonable rise of temperature. The various sources of error produce effects which are sometimes as great as, or -even greater than, the quantity to be measured. It is beyond the scope of this work to 2l6 ELEMENTARY PHYSICS. describe in detail the means by which the effects of the disturbing causes have been determined or elimi- nated. The specific heats of a few gases are given below. Air 0.23741 Oxygen 0.21851 Hydrogen 34090 Carbonic acid 0.2169 The specific heat of a gas at constant volume is generally determined from the ratio between it and the specific heat at constant pressure. The first determi- nation of this ratio was accomplished by Clement and Desormes. The theory of the experiment may be understood from the following considerations : Let a unit mass of gas at any temperature t and volume V be raised in temperature 1° at constant volume. It absorbs heat, which we will represent by c, the specific heat at constant volume. Now let the gas expand against a mean press- ure H, maintaining its temperature at 1° until it returns to its original pressure. It absorbs an additional quan- tity of heat equivalent to the work done in expanding, r=Zf X the increase in volume. The increase in volume is the same as would have occurred had the gas been raised in temperature 1°, and allowed to expand from the beginning ; that is, — - — = « f^, where V^ = vol- ume at zero. The work done is Ha-V^, and the heat equivalent is — y^, where J has its usual signification. The heat absorbed is then C=c + ^, (58) GASES AND VAPORS. 217 which is the specific heat at constant pressure. Hence 7=x+--^. (59) If, in the case considered above, the gas had expanded without receiving any heat, the work Ha. Vq would have been done at the expense of its own internal energy, and the temperature would have fallen. The perform- ance of this work is equivalent to abstracting the quan- tity of heat, — jr-^, which would lower the temperature 1 HaVr, _, j-^ degrees, smce a quantity c of heat raises the temperature one degree. Represent this change of temperature by 0. It corresponds to a change in the unit volume of — ^ — . Substituting 6 for - tl^ [^ C equation (59), we have - = 1 ■\- 6. It is the object of the experiment to find B. The method of Clement and Desormes is as follows : — A large flask is furnished with a stop-cock having a large opening, and a very sensitive manometer which shows the difference between the pressure in the flask and the pressure of the air. The air in the flask is first rarefied, and left to assume the temperature of the surrounding atmosphere. Suppose the pressure now to be .^ — h, H being the height of the barometer, and h the difference between the pressure in the flask and the pressure of the atmosphere, as shown by the manometer. The large stop-cock is then suddenly opened for a very short time only ; the air rushes in. 2l8 ELEMENTARY PHYSICS. re-establishing the atmospheric pressure, compressing the gas, and raising its temperature. The volume of the air becomes i — , where the original volume is taken as unity, and ^ represents its reduction ; and, if there were no change of temperature, the pressure would be . If the temperature increase 6'°, and become t -{■ 6', the pressure will be H-k .^..it^p ^ (60) I— ^ 1+"^ the atmospheric pressure. The flask is now left till it returns to the temperature of the atmosphere, t, when the manometer shows a fall of pressure h', and we have ^^^=^B-h'. (6i) I -^ From these two equations we have h-h! {i-\-af)h' Suppose, now, the change of volume had been I 4-ajf' then the change of temperature would have been 6; and, since change of volume is proportional to change of temperature, hence '^ = 7+T.= ^' = ^' ff- " e- ' + "'■ or, substituting the values of <^ and &, we have h' H-h' h' ^ = ^7 — ^x H-K k-h' h-h' GASES AMD VAPORS. 219 Now, C , h h rr \ It is evident that the specific heat may be computed from formula (58). 149. The Two Specific Heats of a Gas have the Same Ratio as the Two Elasticities. — Suppose a gas whose mass is unity and volume Fto rise in temperature at constant pressure from t° to (t-\-At)°, Ai representing a very small increment of temperature. The heat con- sumed will be CA.i, and the increase of volume ij/At, where \j/ represents Va. Now, suppose the volume to remain constant, the amount of heat required to cause the rise of temperature Ai will be cAt: hence the amount of C heat, CAt, will cause a rise of temperature —At; and the same rise of temperature will occur if the gas, after iirst being allowed to expand, be compressed to its initial volume. Such a compression would be attended by an increase of pressure, which we will call A/>. The ratio between this and the corresponding change of volume is where E^ is the elasticity under the condition that no heat enters or escapes. If, now, the heat produced by compression be allowed to escape, there will remain the quantity cAi, and the increment of pressure will be reduced to 8^ = Aj>-^. 220 ELEMENTARY PHYSICS. This is the increase of pressure that will occur if the gas be compressed by the amount ^bJ. without change of temperature : hence where Et is the elasticity for constant temperature. Dividing (63) by (64), we have A/ ^ _ ^A£ _ A^ _ _A£_ _ C. that is, the two elasticities have the same ratio as the two specific heats of a gas. In acoustics it will be seen that the velocity of sound in any medium is equal to the square root of the quotient of the elasticity divided by the density of the medium ; that is, velocity = \^ ^- , (65) It will be seen also, that, in the progress of a sound- wave, the air is alternately compressed and rarefied, the compressions and rarefactions occurring in such rapid succession that there is no time for any transfer of heat. If the above equation, (65), be applied to air, the E becomes E^, or the elasticity under the condition that no heat enters or escapes. Since we know the density of the air and the velocity of sound, E^ can be com- puted. In " Mechanics," § 6"], it is shown that Et is numerically equal to the pressure : hence we have the values of the two elasticities of air, and, as seen above, their ratio is the ratio of the two specific heats of air. GASES AND VAPORS. 221 150. Examples of Energy absorbed by Vaporization. — When a liquid boils, its temperature remains constant, however intense the source of heat. This shows that the heat applied to it is consumed in producing the change of state. Evaporation absorbs heat. By pro- moting evaporation, intense cold may be produced. In a vacuum, water may be frozen by its own evaporation. If a liquid is heated to a temperature above its ordinary boiling-point under pressure, relief of the pressure is followed by a very rapid evolution of vapor and a rapid cooling of the liquid. Liquid nitrous oxide at a tem- perature of zero is still far above its boiling-point, and its vapor exerts a pressure of about thirty atmospheres. If the liquid be drawn off into an open vessel, it at first boils with extreme violence, but is soon cooled to its boiling-point for the atmospheric pressure, about — 88°, and then boils away slowly, while its tempera- ture remains at that low point. 151. Heat Equivalent of Vaporization. — It is plain, from what has preceded (§ 137), that the formation of vapor is work requiring the expenditure of energy for its accom- plishment. Each molecule that is shot off into space obtains the motion which projected it beyond the reach of the molecular attraction, at the expense of the en- ergy of the molecules that remain behind. A quantity of heat disappears when a liquid evaporates ; and experi- ment demonstrates, that to evaporate a kilogram of a liquid at a given temperature always requires the same amount of heat. This is the heat equivalent of vaporiza~ tion. When a vapor condenses into the liquid state, heat is generated the same in amount as disappears when the liquid assumes the state of vapor. The heat 222 ELEMENTARY PHYSICS. equivalent of vaporization is determined by passing the vapor at a known temperature into a calorimeter, there condensing it into the liquid state, and noting the rise of temperature in the calorimeter. This, it will be seen, is essentially the method of mixtures. Many experimenters have given attention to this determina- tion ; but here, again, the best experiments are those of Regnault. He determined what he called the total heat of steam at various pressures. By this was meant the heat required to raise the temperature of a kilogram of water from zero to the temperature of saturated vapor at the pressure chosen, and then convert it wholly into steam. The results of his experiments give, for the heat equivalent of vaporization of water at ioo°, 537 calories. That is, he found, that by condensing a kilo- gram of steam at 100° into water, and then cooling the water to zero, 637 calories were obtained. But almost exactly 100 calories are derived from the water cooling from 100° to zero : hence 637— 100 = 537 calories is the heat equivalent of vaporization at 100°. 152. Dissociation. — It has already been noted (§ 89), that, at high temperatures, compounds are separated into their elements. To effect this separation, the powerful forces of chemical affinity must be overcome, and a considerable amount of energy must be consumed. 153. Heat Equivalent of Dissociation and C/iemical Union. — From the principle of the conservation of energy, it would be supposed that the energy required for dissociation is the same as that developed by the re-union of the elements. The heat equivalent of chemi- cal union is not easy to determine, because the process is usually complicated by changes of physical state. GASES AND VAPORS. 223 We may cause the union of carbon and oxygen in a calorimeter, and, bringing the products of combustion to the temperature of the elements before the union, measure the heat given to the instrument ; but the carbon has changed its state from .a solid to a gas, and some of the chemical energy must have been consumed in that process. The heat measured is the available heat. Several experimenters have made determinations of the available heat of chemical union. Among these, Andrews, and Favre and Silberman have performed experiments with great care, and obtained results, a few of which are given in the following table, which gives the number of calories produced by the combus- tion of one gram of the substance in oxygen : — Hydrogen 344^2 Carbon (wood charcoal) 8080 Native sulphur 2261 224 ELEMENTARY PHYSICS. CHAPTER V. HYGROMETRY. 154. Objeci of Hygrometry. — Hygrometry has for its object the determination of the state of the air with regard to moisture. 155. Amount of Vapor in a Given Volume of Air. — This may be determined directly by passing a known volume of air through tubes containing some substance which will absorb the moisture, and determining the increased weight of the tubes and their contents. The quantity of vapor in grams contained in a cubic metre of air is called its absolute humidity. Methods of deter- mining this quantity indirectly will be given below. 156. Pressure of tlie Vapor. — It has been seen (§ 138), that, when two or more gases occupy the same space, each exerts its own pressure independently of the others. If we can determine this pressure, it is easy to compute the quantity of moisture in the air. It appears, also, that the pressure exerted by the vapor in the air would at a certain temperature be its maximum pressure. Now, if any small portion of the space be cooled till its temperature is below that at which the pressure exerted is the maximum pressure, a portion of the vapor will condense into the liquid form. If, then, we determine the temperature at which HYGROMETRY. 225 condensation begins, the maximum pressure of the vapor for this temperature, which may be found from tables, is the real pressure of the vapor in the air. The mass of vapor in a metre cube of air may then be computed as follows : A metre cube contains 1293.2 grams of air at zero and pressure 760 millimetres. At the pressure / of the vapor, and temperature t of the air at the time of the experiment, the same space would contain / I 1293-2 X -^ X 760 I + a^ grams of air ; and, since the specific gravity of vapor of water in relation to air is 0.623, a metre cube would contain 1293.2 X -4- X — ; — ^ X 0.623 (66) '•^ 760 X ■\-at "^ ^ ' grams of vapor. 157. Devi Point. — The temperature at which the vapor of the air begins to condense is called the dew point. It is determined by means of instruments called dew- point hygrometers, which are instruments so constructed that a small surface exposed to the air may be cooled till moisture deposits upon it, when its temperature is accurately determined. The Alluard hygrometer con- sists of a metal box about one and a half centimetres square and four centimetres deep. Two tubes pass through the top of the box, — one terminating just inside, and the other extending to the bottom. One side of the box is gilded and polished, and is so placed that the gilded surface lies on the same plane with, and in close proximity to, a gilded metal plate. The box is 226 ELEMENTARY PHYSICS. partly filled with ether, and the short tube connected with an inspirator. Air is thus drawn through the longer tube, and, bubbling up through the ether, causes rapid evaporation, which soon cools the box, and causes a deposit of dew upon the gilded surface. The presence of the gilded plate forming an extension of the surface of the box helps very much in recognizing the beginning of the deposit of dew, by the contrast between it and the dew-covered surface of the box. A thermometer plunged in the ether gives its temperature, and another outside gives the temperature of the air. 158. Relative Humidity. — The amount of moisture that the air may contain depends upon its temperature. The dampness or dryness of the air does not depend upon the absolute amount of moisture it contains, but upon the ratio of this to the amount it might contain if saturated. The relative humidity is the ratio of the amount of moisture in the air to that which would be required to saturate it at the existing temperature. Since non-saturated vapors follow nearly Boyle's law, this ratio will be very nearly the ratio of the actual pressure to the possible pressure for the temperature. Both these pressures may be taken from the tables. One corresponds to the dew point, and the other to the temperature of the air. ia£.tiMU-U YJMAMICS. 227 CHAPTER VI. THERMO-DYNAMICS. 159. First Law of Thermo-dynamics. — This law may be thus stated : Where heat is transformed into work, or work into heat, the quantity of work is mechanically equivalent to the quantity of heat. The experiments of Joule and Rowland establishing this law, and deter- mining the mechanical equivalent, have already been described (§ 104). 160. Second Lam of Thermo-dynamics. — When heat is converted into work under the conditions that exist on the earth's surface, only a comparatively small pro- portion of the heat drawn from the source can be transformed : the remainder is given up to a refrigerator, which in some form must be an adjunct of every heat- engine, and still exists as heat, which, so far as any useful effect is concerned, is wasted. It can be shown that the heat which can be converted into work, bears to that which must be drawn from the source of heat a certain simple ratio depending upon the temperatures of the source and refrigerator. The second law of thermo- dynamics asserts this relation. The ratio between the heat converted into work and that drawn from the source is called the efficiency of the engine. To convert heat into mechanical work, it is neces- 228 ELEMENTARY PHYSICS. sary that the heat should act through some substance called the working substance ; for instance, steam in the steam-engine, air in the hot-air engine. In studying the transformation of heat into work, it is an essential condition, which must not be lost sight of, that the work- ing substance must, after passing through a cycle of operations, return to the same condition as at the begin- ning; for, if the substance is not in the same condi- tion at the end as at the beginning, internal work may have been done, or internal energy expended, which would increase or diminish the work apparently devel- oped from the heat. To determine the laws governing the conversion of heat into work, recourse is had to a conception of Car- not, — that of an engine completely reversible in all its mechanical and physical operations. Such an engine must, when running forward, receive from a source a cer- tain quantity of heat H, transfer to a refrigerator a certain quantity of heat h, and perform a certain amount W of mechanical work. If it be perfectly reversible, it will, by the performance upon it of the amount of work W, take from the refrigerator the quantity of heat h, and restore to the source the amount H. Such an engine will convert into work, under given conditions, as large as possible a proportion of the heat taken from the source. For, let there be two engines, A and B, of which ^,is reversible, working between the same source and refrigerator. If possible let A perform more work than B, while taking from the source the same amount of heat. If W be the work it performs, and w the work B performs, B will, from its reversibility, by the performance upon it of the work w (less than W), THERMO-DYNAMICS. 229 restore to the source the amount of heat, H, which it takes away when running forward. Let A be employed to run B backward : A will take from the source a quan- tity of heat, H, and perform work, W. B will restore the heat H to the source by the performance upon it of work, w. The system will then continue running, developing the work W — w; while the source loses no heat. It must be, then, that A gives up to the refriger- ator less heat than B takes away ; and the refrigerator must be growing colder. For the purposes of this dis- cussion, we may assume that all surrounding bodies, except the refrigerator, are at the same temperature as the source: hence the work W — w, performed by the system of two engines, must be performed by means of heat taken from a body colder than all surrounding bodies. Now, this is contrary to all experience. Heat caniiot be drawn from a cold body, and made to do work. The hypothesis with which we started must, therefore, be false ; and we must admit that no engine, no con- trivance for converting heat into work, can under simi- lar conditions, and while taking the same heat from the source, perform more work than a reversible engine. It follows that all reversible engines, whatever the work- ing substance, have the same efficiency. This is a most important conclusion. In view of it, we may, in study- ing the conversion of heat into work, choose for the working substance the one which presents the greatest advantage for the study. Of all substances, the prop- erties of gases are best known. These, then, are best adapted to this purpose. Assuming a perfect gas as the working substance, let us see how the results sought may be reached. Carnot employed a cycle of four 230 ELEMENTARY PHYSICS. operations, perfectly reversible, returning the substance at the end, to the same condition as at the beginning. The effect of every operation of this cycle, when applied to a gas, may be completely predicted. Let the gas be enclosed in a cylinder having a tightly fitting piston. Suppose the cycle to begin by a depres- sion of the piston, compressing the gas, without loss or gain of heat, till the temperature rises from ^ to t ; t being the temperature of the source, and Q that of the refrigerator. In Fig. 53, let Oa represent the volume, and Aa the pressure at the beginning. If the gas be compressed till its vol- ume becomes Ob, its pressure will be bB, where AB is an adia- batic line. This is iho. first oper- ation. For the second operation, let the piston rise, increasing the volume from 3 to c at the constant temperature of the source. The pressure will fall from bB to cC, BC being the isotherm for temperature /. During this operation, heat H must be taken from the source, to maintain the constant temperature t. For the third operation, let the piston still ascend, increasing the volume from c to d without loss or gain of heat, till the temperature falls from t to 6, the temperature at which the cycle began. CD, representing the fall of pressure, is an adiabatic line. For the fourth operation, let the piston be depressed to the starting-point, and the gas main- tained at the constant temperature of the refriger- ator. The volume becomes Oa, and the pressure a A, as at the beginning. Now let us consider the work THERMO-DYNAMICS. 23 1 done in each operation. While the piston is being depressed through the volume represented by ab, work must be performed upon it equal to ab X the mean pressure exerted upon the piston. This mean pressure lies between Aa and Bb, and the product of this by ab is evidently the area ABba. In the same way it is shown, that, when the gas expands from b to c, it per- forms work represented by the area BCcb ; and again, in the third operation, it performs work represented by CDdc. In the fourth operation, when the gas is com- pressed, work must be done upon it represented by the area ADda. During the cycle, therefore, work is done by the gas represented by BCDdb, and work is done upon the gas represented by BADdb. The difference repre- sented by the area A BCD is the work done by the engine in one revolution. Since the gas is in all respects in the same condition at the end as at the beginning of the cycle, no work could have been developed from it ; and the work which the engine does must have been derived from the heat communicated to the gas during the second operation. Now, it has been shown, that, when a gas expands, no internal work is done in separating the molecules, and when it expands at constant temperature, no change occurs in the internal kinetic energy : the heat which is imparted to the gas during the second operation is, therefore, the equivalent to the work done by the gas upon the piston, and may be represented by the area BCcb. It will be seen, also, that the heat given up to the refrigerator during the fourth operation is repre- sented by the area ADda, and that heat, the equivalent of the work performed by the engine, represented by the 232 ELEMENTARY PHYSICS. area ABCD, has disappeared. Of the heat withdrawn from the source, then, only the fraction ^^^^ ^£5^ is area BCcb converted into work. This fraction is the efficiency of the engine. Now let the operations of the cycle be reversed. Starting from a, the gas expands at temperature 6, ab- sorbs heat h the same as it gave up when compressed, and performs work represented by ADda; next, it is compressed, without loss of heat, till its temperature rises to t, and work represented by DCcd is done upon it ; next, it is still further compressed at temperature t, till its volume becomes Ob, and its pressure, Bb. During this operation, it gives up the heat H which it absorbed during the direct action, and work represented by CBbc is done upon it. Lastly, it expands to the starting- point, and falls to its initial temperature. It will be seen, that each operation is the reverse in all respects of the corresponding operation of the direct action, and that, during the cycle, work represented by the area ABCD must be performed upon the engine ; while heat h is taken from the refrigerator, and heat H is transferred to the source. Such an engine is, therefore, a reversible engine ; and it converts into work as large a proportion of the heat derived from the source as is possible under the circumstances. An inspection of the figure shows, that, since the Ime BC remains constant as long as the amount of heat. H and the temperature t of the source remain constant, the only way to increase the propor- tion of work derived from a given amount of heat H is to increase the difference of temperature between the source and the refrigerator; that is, to increase the THERMO-DYNAMICS. 233 area ABCD, the line AD must be taken lower down. The proportion of heat which can be converted into work depends, therefore, upon the difference of tempera- ture between source and refrigerator. To determine the nature of this dependence, suppose the range of temperature so small that the sides of the figure ABCD may be considered straight and parallel. Produce AD to e, and draw gh representing the mean pressure for the second operation. Now ABCD = eBCf ^ Be X be =: gi X be. Also BCcb = gh X be. Then area ABCD _ giy,bc_gi_H— h area BCcb gh y. be gh H But gh is the pressure corresponding to volume Oh and temperature t, hi is the pressure corresponding to the same volume and temperature B. These pressures are proportional to the absolute temperatures (§ 145) ; that is, if t and B are temperatures on the absolute scale. (67) (68) ih 6 and gh~~t gi H-h t-e. hence gh- H - t effici area ABCD t—6 ■^'^"^ area ABcb t This proportion has been derived upon the supposition that the range of temperature was very small : but it is equally true for any range ; for, let there be a series of engines of small range, of which the second has for a source the refrigerator of the first, the third has for 234 ELEMENTARY PHYSICS. a source the refrigerator of the second, and so on. The first takes from the source the heat H, and gives to the refrigerator the heat h, working between the tempera- tures t and B. The second takes the heat h from the refrigerator of the first, and gives to its own refrigerator the heat h], working between the temperatures B and ^p and similarly for the others ; then, from (67), H-h _t-9 h ~ 6 ' h-K ._ ^1 - h h„_i — hn 6n-\ — 6„ or h 6 H -1' h„ On h„_i 6„.i multiplying. jxM- or hn 6„ H t ^1 vy ^2 N/ _^ and H-h„ ^ t-en H t THERMO-DYNAMICS. 235 Hence it appears, that, in a perfect heat-engine, the heat converted into work is to the heat received, as the difference of temperature between the source and the refrigerator is to the absolute temperature of the source. This ratio can become unity only when = 0°, of when the refrigerator is at the absolute zero of temperature. Since the difference of temperatures between which it is practicable to work is always small compared to the absolute temperature of the source, a perfect heat-engine can convert into work only a small fraction of the heat it receives. 161. Absolute Scale of Temperatures. — An absolute scale of temperatures, formed upon the assumed prop- erties of a perfect gas, has already been described (§ 1.45) No such substance as a perfect gas exists ; but, since (§ 160) any two temperatures on the absolute scale are to each other as the heat taken from the source at the higher temperature is to the heat transferred to the re- frigerator at the lower, by a perfect engine, any substance whose properties we know with sufficient exactness to draw its isothermal and adiabatic lines, may be used as a thermometric substance, and, by means of it, an abso- lute scale of temperatures may be constructed. For, in Fig. 54 let BB' be an isothermal line, corresponding to temperature /, say, of boiling water at a standard press- ure. Let ^fi' be the isotherm for the temperature t^ of melting ice, and let bb' be an isotherm for an inter- mediate temperature. Let BP, B'fi', be adiabatic lines, such that, if the substance expand at constant tempera- ture t from the condition B to the condition B', the equivalent in heat of one mechanical unit of energy will be absorbed. If, now, the figure BB'^'P represent Car- 236 ELEMENTARY PHYSICS. not's cycle, the heat given to the refrigerator at tem- perature t^, measured in mechanical units, is less than the heat taken from the source at temperature t, by the Fig. 54. energy represented by the area BB'p'P, or, the heat given to the refrigerator = i — area B^' : hence and /„ I — area Bji'' t— t^ _ area B^ Now, if t — tfj= 1 00°, as in the Centigrade scale, we have 100 areaBji' __ _ , / = 100 area B^' (69) If B be the temperature corresponding to the isotherm bU, we have, as above, THERMO-DYNAMICS. 237 / Q I — area Bb' 6 = i(i— area Bb') = — ^^^,(1 — cirea Bb') . (70) ^ ' areaBp^ ' ^' ' If, now, it be proposed to use the substance as a ther- mometric substance by noting its expansion at constant pressure, take Om to represent that pressure, and draw the horizontal line mnop ; mn is the volume of the sub- stance at temperature t^, mo the volume at temperature B, and mp the volume at temperature t. This method of constructing an absolute scale of temperature was proposed by Sir William Thomson. 162. The Steam-Engine. — The steam-engine in its usual form consists essentially of a piston, moving in a closed cylinder, which is provided with passages and valves by which steam can be admitted and allowed to escape. A boiler heated by a suitable furnace supplies the steam. The valves of the cylinder are opened and closed automatically, admitting and discharging the steam at the proper times to impart to the piston a re- ciprocating motion, which may be converted into a circular motion by means of suitable mechanism. There are two classes of steam-engines, — condensing and non-condensing. In condensing engines the steam, after doing its work in the cylinder, escapes into a con- denser, kept cold by a circulation of cold water. Here the steam is condensed into water ; and this water, with air or other contents of the condenser, is removed by an "air-pump." In non-condensing engines the steam escapes into the open air. In the latter case the tem- perature of the "refrigerator" must be considered at 238 ELEMENTARY PHYSICS- least that of saturated steam at the atmospheric press- ure, or about 190°, and the temperature of the source must be taken as that of saturated steam at the boiler pressure. Applying the expression for the efficiency (§ 160), t-6 it will be seen, that, for any boiler pressure that it is safe to employ in practice, it is not possible, even with a perfect engine, to convert into work more than about fifteen per cent of the heat used. In the condensing engine the temperature of the refrigerator may be taken as that of saturated steam at the pressure which exists in the condenser, say, 30'' to 40° : hence t — 6 is a much larger quantity for con- densing than for non-condensing engines. The gain of efficiency is not, however, so great as would appear from the formula, because of the energy that must be expended to maintain the vacuum in the condenser. 163. Hot-air and Gas Engines. — Hot-air engines con- sist essentially of two cylinders of different capacities, with some arrangement for heating air in, or on its way to, the larger cylinder. In one form of the engine, an air-tight furnace forms the passage between the two cylinders, of which the smaller may be considered as a supply-pump for taking air from outside, and forcing it through the furnace into the larger cylinder, where, in consequence of its expansion by the heat, it is enabled to perform work. On the return stroke, this air is ex- pelled into the external air, still hot, but at a lower temperature than it would have been had it not ex- panded and performed work. This case is exactly THERMO-DYNAMICS. 239 analogous to that of the steam-engine, in which water is forced by a piston working in a small cylinder, into a boiler, there converted into steam, and then, acting upon a much larger piston, performs its work, and is rejected. In another form of the engine, known as the "ready motor," the air is forced into the large cylinder through a passage kept supplied with crude petroleum. The air becomes saturated with the vapor, forming a combustible mixture, which is burned in the cylinder itself. The Stirling hot-air engine and the Rider " compres- sion engine " are interesting as realizing an approach to Carnot's cycle. These engines, like those described above, consist of two cylinders of different capacities, in which work air- tight pistons ; but, unlike those, there are no valves communicating with the external atmosphere. Air is not taken in and rejected ; but the same mass of air is alternately heated and cooled, alternately expands and contracts, actuating the piston, and performing work at the expense of a portion of the heat imparted to it. It is of interest to study a little more in detail the cycle of operations in these two forms of engines. The larger of the two cylinders is kept constantly at a high temperature by means of a furnace, while the smaller is kept cold by the circulation of water. The cylinders communicate freely with each pther. The pistons are actuated by cranks set at an angle of about 90°, so that twice during the revolution both pistons are moving for a short time in the same direction. Let us begin the study of the cycle at the instant that the small 240 ELEMENTARY PHYSICS. piston reaches the top of its stroke. The large piston will then be toward the bottom of the cylinder, and descending. The small piston now descends, as well as the larger, the air in both cylinders is compressed, and there is but little transfer from one to the other. There is, therefore, comparatively little heat given up. The large piston, reaching its lowest point, begins to ascend, while the descent of the smaller continues. The air is rapidly transferred to the larger heated cyhnder, and expands while taking heat from the highly heated surface. After the small piston has reached its lowest point, there is a short time during which both the pistons are rising and the air expanding, with but little transfer from one cylinder to the other, and with a relatively small absorption of heat. When the descent of the large piston begins, the small one still rising, the air is rapidly transferred to the smaller cylinder : its volume is dimin- ished, and its heat is given up to the cold surface with which it is brought in contact. The completion of this operation brings the air back to the condition from which it started. It will be seen, that there are here four opera- tions, which, while not presenting the simplicity of the four operations of Carnot, — since the first and third are not performed without^ransfer of heat, and the second and fourth not without change of temperature, — still fur- nish an example of work done by heat through a series of changes in the working substance, which brings it back, at the end of each revolution, to the same condition as at the beginning. Gas-engines derive their power from the force devel- oped by the combustion, within the cylinder, of a mixture of illuminating gas and air. THERMO-D YNAMICS. 24 1 As compared with steam-engines, hot-air and gas engines use the working substance at a much higher temperature, t — 6 is, therefore, greater, and the theo- retical efficiency higher. There are, however, practical difficulties connected with the lubrication of the sliding surfaces under such high temperatures that have so far prevented the use of large engines of this class. 164. Water-Power. — Water flowing from a higher to a lower level furnishes energy for driving machinery. Many waterfalls are thus utilized. The energy theo- retically available in a given time is -the weight of the water that flows during that time multiplied by the height of the fall. If this energy is not utilized, it develops heat by friction of the water or of the mate- rial that may be transported by it. But water-power is only possible so long as the supply of water continues. For the supply oy^gmker we are dependent upon the rains ; for the rains, we depend upon evaporation ; and evaporation is maintained by solar heat. The energy of water-power is, therefore, transformed solar radiation. -, 165. Wind-Power. — A moving mass of air possesses energy equal to the mass multiplied by half the square of the velocity. This energy is available for propelling ships, for turning windmills, and for other work. Winds are due to a disturbance of atmospheric equilibrium by solar heat ; and the energy of wind-power, like that of water-power, is, therefore, derived from solar energy. 166. Fuel. — By far the larger part of the energy employed by man for his purposes is derived from the combustion of wood and coal. This energy exists as the potential energy of chemical separation of oxygen from carbon and hydrogen. Now, we know that vege- 242 ELEMENTARY PHYSICS. table matter is formed by the action of the solar rays through the mechanism of the leaf, and that coal is the carbon of plants that grew and decayed in a past geo- logical age. The energy of wood and coal is, therefore, the transformed energy of solar radiations. 167. Animal Heat and Work. — It is well known, that, in the animal tissues, the same chemical action takes place as in the burning of a candle. The oxygen taken into the lungs and absorbed by the blood combines with the fatty matters and starch, producing carbonic acid and water, as in combustion of the same substances elsewhere. Lavoisier was the first to attempt a meas- urement of the heat developed by the animal. He compared the heat developed with that due to the formation of the carbonic acid exhaled in a given time. Despretz and Dulong made similar experiments with more perfect apparatus, and found that the heat pro- duced by the animal was about one-tenth greater than would have been produced by the formation by combus- tion of the carbonic acid and water exhaled. These and similar experiments, although not taking into account all the chemical actions taking place in the body, leave no doubt that animal heat is due to atomic and molecular changes within the body. The work performed by men and animals is also the transformed energy of food. Rumford, in 1798, saw this clearly ; and he showed, in a paper of that date, that the amount of work done by a horse is much greater than would be obtained by using its food as fuel for a steam-engine. Mayer, in 1845, held that an animal is a heat-engine, that every motion of the animal is a transformation into work of the heat developed in the tissues. THERMO-DYNAMICS. 243 Him, in 1858, tried a very interesting series of ex- periments bearing upon this subject. In a closed box was placed a sort of treadmill, which a man could cause to revolve by stepping from step to step, as though to ' go up stairs. He thus performed work which could be measured by suitable apparatus outside the box. The treadwheel could also be made to revolve backward by a motor placed outside, when the man descended from step to step, and work was performed upon him. Three distinct experiments were performed ; and the amount of oxygen consumed by respiration, as well as the heat developed, was determined. In the first experiment the man remained in repose ; in the second he performed work by causing the wheel to revolve ; in the third the wheel was made to revolve backward, and work performed upon him. In the sec- ond experiment, the amount of heat developed for a gram of oxygen consumed was much less, and in the third case much greater, than in the first ; that is, in the second case, a portion of the chemical action, indi- cated by the absorption of oxygen, went to perform the work, and hence the failure to develop the due amount of heat, while in the third case the motor causing the treadwheel to revolve, performed work, which produced heat in addition to that due to the chemical action. It has been thought that muscular energy is due to the waste of the muscles themselves : but experiments show that the waste of nitrogenized material is far too small in amount to account for the energy developed by the animal ; and we must, therefore, conclude that the principal source of muscular energy is the oxidation of the non-nitrogenized material of the blood by the oxygen absorbed in respiration. 244 ELEMENTARY PHYSICS. An animal is, then, a machine for converting the po- tential energy of food into mechanical work : but he is not, as Mayer supposed, a heat-engine ; for he performs far more work than could be performed by a perfect heat-engine working between the same limits of tem- perature, and using the food as fuel. The food of animals is of vegetable origin, and owes its energy to the solar rays. The energy of men and animals is, therefore, the transformed energy of the sun. 1 68. Ocean Currents. — These are maintained by solar heat. 169. Tidal Energy. — The tides are mainly caused by the attraction of the moon upon the waters of the earth. If the earth did not revolve upon its axis, or, rather, if it always presented one face to the moon, the elevated waters would remainstationary upon its surface, and fur- nish no source of energy. But as the earth revolves, the crest of the tidal wave moves apparently in the oppo- site direction, meets the shores of the continents, and forces the water up the bays and rivers, where energy is wasted in friction upon the shores, or may be made use of for turning mill-wheels. It is evident that all the energy derived from the tides comes from the rotation of the earth upon its axis ; and a part of the energy of the earth's rotation is, therefore, being dissipated in the heat of friction they cause. 170. Internal Heat of the Earth. — This and a few other forms of energy, such as that of native sulphur, iron, etc., are of little consequence as sources of useful energy. They may be considered as the remnants of the original energy of the earth. 171. The Sun. — It has been seen, that, except the THERMO-D YNAMICS. 245 energy of the tides, the sun's rays are the source of all the forms of energy practically available. It has been estimated that the heat received by the earth from the sun each year would melt a layer of ice over the entire globe a hundred feet in thickness. This represents energy equal to one-horse power for each fifty square feet of surface. The heat which reaches the earth is 0"^^ ^f U'So'o ooT °^ the heat that leaves the sun ; and the question naturally arises, What is the origin of this enormous store of energy .' Helmholtz and Sir William Thomson have shown that the nebular hypothesis, which supposes the solar system to have originally existed as a chaotic mass of widely separated gravitating particles, presents to us an adequate source for all the energy of the system. As these particles rush together by their mutual attractions, heat would be generated ; and, as the con- densation continues, the result would be highly heated suns and worlds. 172. Dissipation of Energy. — It has been seen that only a fraction of the energy of heat is available for transformation into other forms of energy, and that such transformation is possible only when a difference of temperature exists. Every conversion of other forms of energy into heat puts it in a form from which it can be only partially recovered. Every transfer of heat from one body to another, or from one part to another of the same body, tends to equalize temperatures, and diminish the proportion of energy available for transformation. Such transfers of heat are continually taking place ; and, so far as our present knowledge goes, there is a tendency toward an equality of temperature, or, in other words, 246 ELEMENTARY PHYSICS. a uniform molecular motion, throughout the universe. If this condition of things were reached, although the total amount of energy existing in the universe would remain unchanged, the possibility of transformation would be at an end, ana all activity and change would cease. This is the doctrine of the dissipation of energy to which our limited knowledge of the operations of nature leads us ; but it must be remembered that our knowledge is very limited, and that there may be in nature the means of restoring the differences upon which all activity depends. l0b«j;t genrji ®fettt;;pit0tt ^ mi to 19D3 r^4>Mll.i m