LIBRARY \ UNIVERSITY OP CALIFORNIA A SAN OIE&O 1 presented to the UNIVERSITY LIBRARY UNIVERSITY OF CALIFORNIA SAN DIEGO by KARL DYK V^ |s% A orc.u;oRNi..si^.a'^i?iiiii ,. iiiiiiiiiiiii|S 3 1822 01223 7244 ■^ ^/ >, r- / f:^ ^ S^ INTRODUCTION TO ANALYTICAL MECHANICS •The THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS SAN FRANCISCO MACMILLAN & CO.. Limited LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, Ltd. TORONTO INTRODUCTION TO ANALYTICAL MECHANICS BY ALEXANDER ZIWET If PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN AND PETER FIELD, Ph.D. ASSISTANT PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN Neb) ^orlt THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd. 1912 Copyright, 1912, By the MACMILLAN COMPANY. Set up and electrotyped. Published March, 1912. Nortoooti ISrrss : Berwick & Smith Co., Norwood Mass , U.S A. PREFACE. Ov)( ws OeXo/xev, dAA' a>s SvvdfiiOa. The present volume is intended as a brief introduction to mechanics for junior and senior students in colleges and universities. It is based to a large extent on Ziwet's Theo- retical Mechanics; but the applications to engineering are omitted, and the analytical treatment has been broadened. No knowledge of differential equations is presupposed, the treatment of the occurring equations being fully explained. It is believed that the book can readily be covered in a three- hour course extending throughout a year. For a shorter course, requiring half this time, the following selection may be made: Chapters 1, 2, 3 (omitting Arts. 81-95), 4 (omitting Arts. 114-150), 5 to 12 (omitting Arts. 244-268), 13 and 14 (omitting Arts. 340-355). While more prominence has been given to the analytical side of the subject, the more intuitive geometrical ideas are generally made to precede the analysis. In doing this the idea of the vector is freely used; but it has seemed best to avoid the special methods and notations of vector analysis. This has been done with re.uctance; the time has certainly come for introducing these methods in the very elements of mechanics. But this must be left to another opportunity. That many important subjects had to be omitted is another restriction arising from the nature and purpose of thi.s volume. While the selection of topics has been considered most care- fully it can hardly be expected to meet everybody's approval. The aim has been not only to select material useful to the beginning student of mathematics and physical science, but VI PREFACE at the same time to give the reader a general view of the science of mechanics as a whole, a broad enough foundation for further study. References to other works have been used sparingly. It seemed hardly necessary to refer to such standard works as those of Thomson and Tait, Routh, Schell, Appell, Kirch- hoff, etc., which are found in any good college library. But it did seem desirable to refer in a few cases to works where fuller information can be found on subjects somewhat out of the range of the ordinary text-book on mechanics. The fourth volume of the Encyklopddie der mathematischen Wissenschaften, especially the articles by P. Stackel, should be consulted by the more advanced student. Alexander Ziwet, Peter Field. University of Michigan, February, 1912. CONTENTS. INTRODUCTION. Page 1 PART I: KINEMATICS. CHAPTER I: Rectilinear motion of a point. 1. Velocity and acceleration in rectilinear mo- tion 3 2. Examples of rectilinear motion 10 CHAPTER II: Translation and rotation 21 CHAPTER III: Curvilinear motion of a point. 1. Relative velocity; composition and resolu- tion of velocities 26 2. Velocity in curvilinear motion 30 3. Acceleration in curvilinear motion 35 4. Examples of curvilinear motion 42 (a) Constant acceleration 42 (5) The pendulum . 47 (c) Simple harmonic motion 54 (d) Compound harmonic motion 59 (e) "Wave motion 63 (/) Curvilinear compound harmonic mo- tion 67 (g) Central motion 72 CHAPTER IV: Velocities in the rigid body. 1. Geometrical discussion 83 2. Analytical discussion 91 3. Plane motion 98 CHAPTER V: Accelerations in the rigid body 107 CHAPTER VI: Relative motion 117 vii Vlll CONTENTS PART II: STATICS. Page CHAPTER VII: Mass; density 120 CHAPTER VIII: Moments and centers of mass 124 CHAPTER IX: Momentum; force; energy 132 CHAPTER X: Statics of the particle 142 CHAPTER XI: Statics of the rigid body. 1. Concurrent forces 149 2. Parallel forces 151 3. Theory of couples 158 4. Complanar forces 165 5. The general system of forces 170 6. Constraints; friction 178 CHAPTER XII: Theory of attractive forces. 1. Attraction 187 2. The potential 196 3. Virtual work 201 PART III: KINETICS. CHAPTER XIII: Motion of a free particle. 1. The equations of motion 207 2. Examples of rectilinear motion 217 3. Examples of curvilinear motion 229 CHAPTER XIV: Constrained motion of a particle. 1. Introduction 248 2. Motion on a fixed curve 250 3. Motion on a fixed surface 258 4. The method of indeterminate multipUers 259 5. Lagrange's equations of motion 263 CHAPTER XV : The equations of motion of a free rigid body 268 CHAPTER XVI: Moments of inertia and principal axes. 1. Introduction 280 2. ElUpsoids of inertia 287 3. Distribution of principal axes in space . . 297 CONTENTS IX Page CHAPTER XVII: Rigid body with a fixed axis 304 CHAPTER XVIII: Rigid body with a fixed point. 1. The general equations of motion 313 2. Motion without forces 320 3. Heavy symmetric top 327 CHAPTER XI X: Relative motion 335 CHAPTER XX: Motion of a system of particles. 1. Free system 346 2. Constrained system 348 3. Generalized co-ordinates; Lagrange's equations of motion ; Hamilton's principle. 352 ANSWERS 361 INDEX 375 INTRODUCTION. 1. The science of mechanics can be regarded as an exten- sion of geometry obtained b}' adjoining tlie ideas of time and mass to tlie idea of space whicli is fundamental in geometry. We are thus led to the study of motion and of forces as the subject-matter of mechanics. 2. By adjoining the idea of time alone we obtain a pre- liminary branch of mechanics, known as kinematics. It develops the ideas of velocity and acceleration of geometrical configurations without using the notion of mass. 3. The introduction of mass leads to numerous new ideas such as momentum, force, energy. Owing to the importance of forces in physics the mechanics of bodies possessing mass is often called dynamics. It may be divided into statics and kinetics. Statics is the science of equilibrium; it considers the con- ditions under which the action of forces produces no change of motion. Thus, if force be regarded as a fundamental concept, statics is independent of the idea of time. Kinetics treats in the most general way the changes of motion produced by forces. PART I: KINEMATICS. CHAPTER I. RECTILINEAR MOTION OF A POINT. 1. Velocity and acceleration in rectilinear motion. 4. Consider the motion of a point P along a fixed straight line (Fig. 1). If we take on this line an origin and a definite positive sense, say toward the right from 0, the "position of the point P on the line at any' time t can be as- signed by its QO-ordinaie, or abscissa, OP = s, which may be P Fig. 1. any real number. As P moves along the line its abscissa ■s varies with the time; to every value of t (at least within a certain interval of time) corresponds a certain value of s; in other words, s is a function of /. We assume that s is a continuous function of t; this implies that while P may move arbitrarily, back and forth, along the line, it does not make any jumps, suddenl}^ disappearing at one point and reappearing at another; the path of P is connected. 5. The time-rate of change of the abscissa of P, i. e. the ^derivative of s, is called the velocity of the point P; it is usually denoted by the letter v: ds '^df 4 laNEMATICS [6. As the idea of velocity is fundamental in mechanics it may be well to explain somewhat more in detail the genesis of this idea, the more so as the process is typical and recurs frequently. Let the point P move along the line, or any segment of the line, always in the same sense and so that equal distances are always de- scribed in e'qual times. Such a motion is called uniform, and the quotient sjt of any distance OP = s described, divided by the corre- sponding time t, is called the velocity of the uniform motion: Suppose next that the point P does not move uniformly. The same quotient, v = sjt, of any distance described, divided by the time used in describing it, is now called the average, or mean, velocity for that distance or time. This mean velocitj^ varies in general according to the distance or time selected; it does not characterize the motion as a whole. We can, however, attach a definite meaning to the expression velocity at a given point or instant if we define it as follows. Hf. s ->jAg P' Fig. 2. Let s = OP (Fig. 2) be the abscissa of the moving point at the time t, s + As = OP' its abscissa at the time t + A/, so that the distance As is described by P in the time At; and let At be taken so small that P moves always in the same sense as it describes the distance As. Then As/At is the mean velocity for the distance As or time At. The Umit approached by this quotient as At approaches zero. ,. As ds V = lira- - = — A/=o At dt is called the velocity at the point P, or at the time i. It is assumed that such a limit exists, i. e., that s is a differentiable function of t. The definition of velocity as the time-rate of change of the co-ordinate s applies even in the case of uniform motion; for in this case we have as stated above s = vt, 5.] RECTILINEAR MOTION OF A POINT 5 where ii is a constant, i. e. independent of t; hence (Is dt = "■ In non-uniform, or variable, motion the velocity v varies from point to point and from time to time; it can be regarded as a function of the distance s or of the time t. It should be observed that in this whole discussion of velocity it is not essential that the path be rectilinear, this assumption being made only for the sake of simplicity. The discussion applies without change when the point P describes a curve; the co-ordinate s then means the arc of the cm've measured along the curve from some origin on the curve, a definite sense of progression along the curve being taken as positive. 6. Velocity l)eing defined as the quotient of distance by time in uniform motion, and as the Hmit of such a quotient in any motion, the unit of velocity is the unit of length divided by the unit of time. Thus we speak of a velocity of so many centimeters per second (cm./sec), or feet per minute (ft./min.), or miles per hour (M./h.), etc. Denoting the units of time, length, and velocity by T, L, V, this is expressed symbolically by writing V = ^ = LT-^ and saying that the dimensions of velocitj^ (V) are 1 in length (L) and — 1 in time (T). The reader is supposed to be familiar with the C.G.S. (centimeter-gram-second) and F.P.S. (foot-pound-second) systems of measurement. It will suffice to mention that the second is the -g-g-iFo P^^t of the mean solar day which is the average, for one year, of the time between two successive passages of the sun across the meridian; and that the foot is i of a yard, the American yard being defined (by act of 6 KINEMATICS [7. Congress, 1866) as |f |f of a meter. We have therefore the exact relations ft. 1200 1 cm. = 0.3937 m., = oTT^, cm. 39.37 which give approximately: Im. = 3.2808 ft., 1 ft. = 30.48 cm., 1 in. = 2.54 cm. 7. Exercises. (1) Compare the following velocities by reducing all to ft. /sec: (a) man walking 4 M./h.; (b) horse trotting a mile in 2 min. 10 sec; (c) train running 40 M./h.; (d) ship making 15 knots, a knot being a sea- mile (= 6080 ft.) per hour; (e) sound in dry air at 0° C. 331.3 m./sec; (/) sun moving in space 25 km. /sec; {g) light 3 X 10^" cm. /sec (2) Two men starting (in opposite sense) from the same point walk around a block forming a rectangle of sides a, b; H their constant velocities are Vi, V2, when and where will they meet? (3) The mean distance of the sun being 923^ million miles, find the velocity of light if it takes light 16 min. 42 sec. to cross the earth's orbit: (a) in miles per second, (fe) in kilometers per hour. (4) Two trains, one 250, the other 420 ft. ^ong, pass each other on parallel tracks in opposite sense, with equal velocities. A passenger in the shorter train observes that it takes the longer train just 6 sec. to pass him. What is the velocity? (5) What is the distance from J^ to 5 if a man walking 5 M./h. can cover it in 10 min. less than one walking 3 M./h.? (6) What is the answer to the preceding problem if both men start from A at the same time and, when the one has reached B, the other is 7K miles behind him? (7) Two ships start from the same port, the second an hour later than the first. The velocity of the first is 16 knots, that of the second 14 knots. How many miles are they apart 3 hours after the first ship started, the angle between their paths being 60°? 8. The definition of velocity ds enables lis to find the velocity when the co-ordinate s is 10.1 RECTILINEAR MOTION OF A POINT 7 given as a function of t. Conversely, when v is given as a function of Tor of s (or of both s and t), the integration of the same equation gives a relation between s and t which deter- mines the position of the moving point at any time. Thus, if V is given as a function of t, we find by integrating the relation ds = vdt: — So = I vdt, where So is the position of the moving point at the time to, the so-called initial position. If v is given as a function of s, we find by integrating the relation dt = ds/v: J.s'o V 9. Exercises. (1) Find the velocity when: (o) s = at + h, (h) s = af -\-ht -\- c, (c) s = aVT, {d)s = avoakt, (r) s = aerf, (/) s = laic* + e-'), ig) « = ia(2<3 + 3/2 + 1), (/i) s = a{C- - 1)=, {i) s = af-ii - 1), (j) s = a{l^ - 2W — 1), (A-) s = o//(l —'P). Taking a as a positive constant, discuss the motion by determining when s and v have maxima or minima. The nature of the motion will be best understood by skettihing in each case the curve that represents s as a function of i, and then imagining this curve projected on the axis of s. Analytically, the sign of the velocity determines the sense of the motion; i. e. when v is > 0, s increases; when V < 0, s decreases; when v — and dv/dt 4= 0, « has an extreme value and the sense of the motion changes. (2) Find the distance s in terms of I when: (a) v = vo + gl. (h) Sot V = a(t- - 4), (c) V = a scd^t, {d) ?; = - ^ _ , {c) v = ac«' ' /3 ; with s "= So for / = 0. (3) Find t as a function of s and s as a function of I when: (a) v = V2gs, with s = so for i = 0; {h) v = Va^ -^', with s = when / = 0;. (c) V = T/a2~+ .s2, with s = for / = 0. 10. In reciilinenr motion, the time-rate of change of the velocity is called the acceleration; dcMioting it by the l(>tter j, 8 KINEMATICS [10. we have ■ _ dv _ dH -^ ~ dt~ dt-' We are led to the idea of acceleration by a process of reasoning strictly analogous to that followed in defining velocity (Ai't. 5). Among non-uniform motions, the most simple kind is that in which the velocity always increases (or always decreases) by equal amounts in equal times; it is called uniformly accelerated motion. In this -kind of motion, the quotient obtained by dividing the increase (or decrease) of the velocity in any time by this time is called the acceleration of the uniformly ac- celerated motion. If the motion is not uniformly accelerated the same quotient is called the average, or mean, acceleration for that time. Thus, if the velocity is V at the time t when the moving point has the position P, and reaches the value v + Aji at the subsequent time < + A/, when the point is at P', the mean acceleration in the time A; (or distance PP' = As)#is AvjAt. The limit of this quotient, as At approaches zero, i. e. ,. Av dv •^ At=oAt dt ' is called the acceleration at the time t (or at the distance s). It follows that in uniformly accelerated motion the acceleration is constant; and conversely, when the acceleration is constant, the motion is uniformly accelerated. 11. A rectilinear motion is called accelerated whether the velocity be increasing or decreasing. But sometimes the term acceleration is used in a more restricted sense, as opposed to retardation. The motion is then called accelerated or retarded according as the absolute value of the velocity is increasing or diminishing. This gives the criterion d(v^) > ^ . dv ^SO, ^.<■..J,SO. Thus the motion is accelerated (in the narrower sense) or retarded according as vdv/dt is > or < 0; if dv'dt = while 13.] RECTILINEAR MOTION OF A POINT 9 dhjdP 4= 0, the motion changes from being accelerated to being retarded, or vice versa. Acceleration being defined, for rectilinear motion, as the quotient of velocity by time or as the limit of such a quotient, the unit of acceleration J is the unit of velocity divided by the unit of time. With the notation of Art. 6, this is ex- pressed symbolically by writing J = ^ = ^ ^ Lr- hence the dimensions of acceleration are 1 in length and — 2 in time. Thus we speak of an acceleration of so many centimeters per second per second (cm. /sec. ^). 12. Exercises. (1) A point moving with constant acceleration gains at the rate of 30 M./h. in every minute. Express its acceleration in ft./sec.^. (2) At a place where the acceleration of gravity is ^ = 9.810 m./sec.^, what is the value of g in ft. /sec.-? (3) A railroad train, 10 min. after starting, attains a velocity of 45 M./li.; what is its average acceleration during these 10 min.? (4) How does the acceleration of gravity which is about 32.2 ft./sec.^ compare with that of the train in Ex. (3)? (5) Find the acceleration for the motions in Art. 9, Ex. (1); apply the rule of Art. 11 to determine where each motion is accelerated or retarded. (6) Discuss in the same way the acceleration of the motions in Art. 9, Ex. (2) and (3). 13. A rectilinear motion is fully characterized if its acceleration is given as a function of t, s, v, provided that the initial conditions are also given, viz. the position and velocity of the moving point at any instant. For we then have d^s -^2= j(t,s,v), (1) 10 KINEMATICS Fl3. where v = ds/dt, while j(t, s, v) is a given function. The solution of this differential equation, which is called the equation of motion, with the initial conditions s = So, V = I'o for t = to, gives s as a function of t. The solution of such a differential equation may be diffi- cult; nor can any general rule of procedure be given. We here confine ourselves to very simple cases, especially those where the acceleration j is either a constant or a function of s alone, these cases being most important. 2. Examples of rectilinear motion. 14. Uniformly Accelerated Motion. As in this case the acceleration j is constant (see Art. 10), the equation of motion (1) d-s . dv can readily be integrated: V =jt-^ C. To determine the constant of integration C, we must know the value of the velocity at some particular instant. Thus, \i V = vq when t = 0, we find Vq = C; hence, substituting this value for C, V - Vo = jt. (2) This equation gives the velocity at any time t. Substi- tuting ds/dt for V and integrating, we find s = Vot-\- ^jt^ + C, where the constant of integration, C, must again be deter- mined from given " initial conditions." Thus, if we know that s = So when ^ — 0, we find So — C; hence s — So ^ Vol -\- \jt~. (3) This equation gives the space or distance passed over in terms of the time. 17]. RECTILINEAR MOTION OF A POINT 11 Eliminating j between (2) and (3), we obtain the relation s — So = Hvo + v)t, which shows that in uniformly accelerated motion the space can be found as if it were described uniformly with the mean velocity ^(vo + v). 15. To obtain the velocity in terms of the space, we have only to eliminate t between (2) and (3) ; we find Kv'-vo') =j{s-So). (4) This relation can also be derived by eliminating dt between the differential equations v = ds/dt, dvjdt = j, which gives vdv = jds, and integrating. The same equation (4) is also obtained directly from the fundamental equation of motion d"s/dt- = j by a process very frequently used in mechanics, viz. by multiplying both members of the equation by ds/dt. This makes the left-hand member the exact deriv- ative of ^{ds/dty or i^y-, and the integration can therefore be performed. 16. The three equations (2), (3), (4) contain the complete solution of the problem of uniformly accelerated motion. For uniformly retarded motion, j is a negative number. If the spaces be counted from the position of the moving point at the time f = 0, we have Sq = 0, and the equations become V = Vo-\- jt, s = vd + ijt^, i{v^ - vo") =js. If in addition the initial velocity Vo be zero, the point starting from rest at the time t — 0, the equations reduce to the following: V = jt, s = ijt-, iv"^ = js. 17. The most important example of uniformly acceler- ated motion is furnished by a body falling in vacuo near 12 KINEMATICS [18. the earth's surface. Assuming that the body does not rotate during its fall, its motion relative to the earth is a mere translation, i. e. the velocities of all its points are equal and parallel; and it is sufficient to consider the motion of any one point of the body. It is known from observation and experiment that under these circumstances the acceleration of a falling body is constant at any given place and equal to about 980 cm., or 32.2 ft., per second per second; the value of this so-called acceleration of gravity is usually denoted by g. In the exercises on falling bodies (Art. 19) we make through- out the following simplifying assumptions: the falling body does not rotate; the resistance of the air is neglected, or the body falls in vacuo; the space fallen through is so small that g may be regarded as constant; the earth is regarded as fixed. 18. The velocity v acquired by a falling body after falling from rest through a height h is found from the last equation of Art. 16 as V = V2gh. This is usually called the velocity due to the height (or head) h, while h = v'^/2g is called the height (or head) due to the velocity v. 19. Exercises. (1) A body falls from rest at a place where g = 32.2. Find (a) the velocity at the end of the fourth second; (6) the space fallen through in 4 seconds; (c) the space fallen through in the fifth second. (2) A train, starting from the station, acquires a velocity of 30 M./h.: (a) in 8 min.; (b) in 2 miles; what was its acceleration (regarded as constant)? (3) Galileo, who first discovered the laws of falling bodies, ex- pressed them in the following form: (a) The velocities acquired at the end of the successive seconds increase as the natural numbers; (6) the spaces described during the successive seconds increase as the 19.J RECTILINEAR MOTION OF A POINT 13 odd numbers; (c) the spaces described from the beginning of the motion to the end of the successive seconds increase as the squares of the natural numbers. Prove these statements. (4) A stone dropped into the vertical shaft of a mine is heard to strike the bottom after I seconds; find the depth of the shaft, if the velocity of sound be given = c. Assume < = 4 s., c = 332 meters,V g = 980. ^ (5) A railroad train in approaching a station makes half a mile in the first; 2,000 ft. in the second, minute of its retarded motion. If the motion is uniformly retarded: (a) When will it stop? (6) What is the retardation? (c) What was the initial velocity? (d) When will the velocity be 4 miles an hour? (6) A body being projected vertically upwards with an initial velocity vo, (a) how long and (6) to what height will it rise? (c) When and (d) with what velocity does it reach the starting-point? (7) A bullet is shot vertically upwards with an initial velocity of 1200 ft. per second, (a) How high will it ascend? (b) What is its velocity at the height of 16,000 ft.? (c) When will it reach the ground again? ((/) With what velocity? (e) At what time is it 16,000 ft. above the ground? Explain the meaning of the double sign in (e). Use g = 32. (8) With what velocitj^ must a ball be tlirown vertically upwards to reach a height of 100 ft.? (9) A body is dropped from a point J5 at a height AB = h above the ground; at the same time another body is thrown vertically up- ward from the point A, with an initial velocity ;'o. (a) When and (b) where will they collide? (c) If they are to meet at the height ^h, what must be the initial velocity? (10) The barrel of a rifle is 30 in. long; the muzzle velocity is 1300 ft./ sec; if the motion in the baTcl bo uniformly acc(>lerated, what is the acceleration and what tlie lime? (11) If a stone dropped from a balloon while ascending at the rate of 25 ft. /sec. reaches the ground in 6 seconds, what was the height of the balloon when the stone was dropped? (12) If the speed of a train increases uniformly after starting for 8 minutes while the train travels 2 miles, what is the velocity acquired? (13) Two j)articles fall from rest from the same point, at a .short interval of time r; find how far they will be apart when the first par- 14 KINEMATICS [20. tide has fallen through a height h. Take e. g., h = 900 ft., t = ^^ second. 20. Acceleration inversely proportional to the square of the distance, i. e. j = /x/s^ where ju is a constant (viz. the acceleration at the distance s = 1) and s is the distance of the moving point from a fixed point in the line of motion. The differential equation (1) becomes in this case the first integration is readily performed by multiplying both members by ds/dt so as to make the left-hand member the exact derivative of ^{dsjdtY or ^v"^. Thus we find "r/.s ^v- /^-:+<^' («) where the constant of integration, C, must be determined from the so-called initial conditions of the problem. For instance, if v = v^ when s = So, we have ^V(f = — fx/so + C; hence, eliminating C between this relation and (6), \S So ) K„=-.V) = -m(^^-^J. (7) To perform the second integration solve this equation for V and substitute dsldi for v. di \j '' + so ~ s ' or putting Vo^ + 2/i/so = 2)u//i', '^^^. (8) -^4'j' ds dt "Mm' N/ Here the variables s and t can be separated, and we find if 5 = So for ^ = 22.] RECTILINEAR MOTION OF A POINT 15 ^==^j2'i.X J,-:^'^^- (9) To integrate put s — /x' = x"^. The result will be different according to the signs of /jl, ix\ and v, which must be deter- mined from the nature of the particular problem. It is easily seen that the methods of integration used in this problem apply whenever j is given as a function of s alone. 21. Whenever in nature we observe a motion not to remain uni- form, we try to account for the change in the character of tlie motion by imagining a special cause for such change. In rectiUnear motion, the only change that can occur in the motion is a change in the velocity, i. e., an acceleration (or retardation). It ia often convenient to have a special name for this supposed cause producing acceleration or retarda- tion; we call it force (attraction, repulsion, pressure, tension, friction, resistance of a medium, elasticity, cohesion, etc.), and assume it to be proportional to the acceleration A fuller discussion of the nature of force and its relation to mass will be found in Arts. 171-188. The present remark is only intended to make more intelligible the physical meaning and application of the problems to be discussed in the follow- ing articles. 22. It is an empirical fact that the acceleration of bodies falling in vacuo on the earth's surface is constant only for distances from the surface that are very small in comparison with the radius of the earth. For larger distances the acceleration is found inversely proportional to the square of the distance from the earth's center. By a bold generalization Newton assumed this law to hold generally between any two particles of matter, and this assumption has been verified by subsequent observations. It can therefore be regarded as a general law of nature that any particle of matter produces in every other such particle, each particle being regarded as concentrated at a point, an acceleration inversely proportional to the square of the distance between these points. This is known as Neivton's law nf universal gravi- tation, the acceleration being regarded as caused by a force of attraction inherent in each particle of matter. It is shown in the theory of attraction (Art. 253) that the attraction of a spherical mass, such as the earth, on any particle outside the sphere 16 KINEMATICS [23. is the same as if the mass of the sphere were concentrated at its center. The acceleration produced by the earth on any particle outside it is therefore inversely proportional to the square of the distance of the particle from the center of the earth. 23. Let us now apply the general equations of Art. 20 to the particular case of a body falling from a great height towards the center of the earth, the resistance of the air being neglected. Let be the center of the earth (Fig. 3), Pi a point on its surface, Po the initial position of the moving point at the time t = 0, P its posi- tion at the time t; let OPi = R, OPo = So, OP — s; and let g be the accel- eration at P\,j the acceleration at P, both in absolute value. Then, ac- cording to Newton's law, j -.g = R- :s^. This relation serves to determine the value of the constant fx in (5) ; for since the acceleration is to have the value g when s = R we have JL R^ 9> the minus sign being taken because the acceleration is directed toward the origin 0. We have therefore i" = - gR\ so that (5) becomes in our case dt^ gR' (10) 24.] RECTILINEAR MOTION OF A POINT 17 the minus sign indicating that the acceleration tends to diminish the distances counted from as origin. The integration can now be performed as in Art. 20. Multiplying by dsjdt and integrating, we find ^v- = gR^js + C. If the initial velocity be zero, we have y = Ofors = s^; hence C = — gR^/so, and „=_ffv^ JI3=_K J§ N^Z^. (11) \ S .So \ So \ s Here again the minus sign before the radical is selected since the velocity v is directed in the sense opposite to that of the distance s. Substituting ds/dt for v and separating the variables t and s we have dt = -i^^[o"a.| ds; R\2g\so — s hence, integrating as indicated at the end of Art. 20: the constant of integration being zero since s = So for f = 0. The last term can be slightly simplified by observing that sin~i Vl — u~ = cos~Ht, whence finally: 24. Exercises. (1) Find the velocity with which the body arrives at the surface of the earth if it be dropped from a height equal to the earth's radius, and determine the time of falling through this heisrht. Take R = 4000 miles, g = 32. 3 18 KINEMATICS [25. (2) Show that formula (11) reduces to v = V2gh (Art. 18) with s = R'\i So — s = his small in comparison with R. (3) Show that when so is large in comparison with R while s differs but sUghtly from R, the formula; (11) and (12) reduce approximately to , — R , irsoi ^^ = - V 2o _, i = ----— . 1 s 2V2g R Hence find the final velocity and time of fall of a body falling to the earth's surface (a) from an infinite distance; (6) from the moon (so = mR). (4) Derive the expressions for v and I corresponding to (11) and (12) when the initial velocity is ro (toward the center). (5) A particle is projected vertically upwards from the earth's sur- face with an initial velocity v^. How far and how long will it rise? (6) If, in (5), the initial velocity be ?'o = VgR^ how high and how long will the particle rise? How long will it take the particle to rise and fall back to the earth's surface? 25. Acceleration directly proportional to the distance, i. e j = KS, where k is a constant. The equation of motion can be integrated by the method used in Art. 20. The result of the second integration will again be different according to the sign of k. We shall study here only a special case, re- serving the general discussion of this law of acceleration until later. 26. It is shown in the theory of attraction (Art. 251) that the attraction of a spherical mass such as the earth on any point within the mass produces an acceleration directed to the center of the sphere and proportional to the distance from this center. Thus, if we imagine a particle moving along a diameter of the earth, say in a straight narrow tube 26.] RECTILINEAR MOTION OF A POINT 19 passing through the center, we should have a ease of the motion represented by equation (13). To determine the value of k for our problem we notice that at the earth's surface, that is, at the distance OPi = R from the center (Fig. 4), the acceleration must be g. If, there- fore, j denote the numerical value of the acceleration at any distance OP = s(< R), we have j : g = s : R, or j ^ gs/R. But the acceleration tends to dimin- ish the distance s, hence d^s/dt^ = — {glR)s. Denoting the posi- tive constant g/R by /x^, the equation of motion is d s I n = — /x-s, where /* = ^f dt R (14) Integrating as in Arts. 20 and 23, we find If the particle starts from rest at the surface, we have v = when s = R; hence = — ifi^R^ + C; and subtracting this from the preceding equation, we find V = - IX V/e^ - s2, (15) where the minus sign of the square root is selected because s and V have opposite sense. Writing dsjdt for v and separating the variables, we have dt = -- ds M ^jR^ - s2' whence 20 KINEMATICS [27 jx K As, s = R when ^ = 0, we have = ^ cos~i 1 + C , or C = 0. Solving for s, we find s = R cosfxt. (16) Differentiating, we obtain v in terms of t: V = — fxR smut. (17) 27. The motion represented by equations (16) and (17) belongs to the important class of simple harmonic motions (see Arts. 71 sq.). The particle reaches the center when s = 0, 1. e. when nt = -kI'I, or at the time t = 7r/2^t. At this time the velocity has its maximum value. After passing through the center the point moves on to the other end, P2, of the diameter, reaches this point when s = — R, i. e. when p.t = TV, or at the time t = tt/ijl. As the velocity then vanishes, the moving point begins the same motion in the opposite sense. The time of performing one complete oscillation (back and forth) is called the period of the simple harmonic motion; it is evidently T = 4-— = ~. 2/x /x ' 28. Exercises. (1) Equation (14) is a differential equation whose general integral is known to be of the form s = Ci sirifit + C2 cosfit; determine the constants Ci, C2 and deduce equations (16) and (17). (2) Find the velocity at the center and the period, taking (7 = 32 and.i2 = 4000 miles. (3) A point whose acceleration is proportional to its distance from a fixed point O starts at the distance so from O with a velocity Va directed away from O; how far will it go before returning? CHAPTER II. TRANSLATION AND ROTATION. 29. In kinematics, the term rigid body is used to denote a figure of invariable size and shape, or an aggregate of points whose distances from each other remain unchanged. The position of a rigid body is given if the positions of any three of its points, not in a straight Hne, are given; when three such points are fixed the body is fixed. The kinematics of rigid bodies will be discussed more fully later on (Arts. 114-150); it will here suffice to mention two particular types of motion of a rigid body: translation, and rotation about a fixed axis. 30. The motion of a rigid body is called a translation when all points of the body describe equal and parallel curves. This will be the case if any three points of the body, not in a straight line, describe equal and parallel curves. Owing to the rigidity of the body, i. e. the invaria])ility of the mutual distances of its points, the velocities and accelerations of all points at any given instant must then be equal; thus, in translation, the motion of the whole body is given by the motion of any one of its points. 31. When a rigid body has two of its points fixed, the only motion it can have is a rotation about tlu^ line joining the fixed points as axis. Thus, in rotation about a fixed axis, all points of the body excepting those on the axis describe arcs of circles whose centers lie on the axis and whose planes are perpendicular to the axis; all points on the axis are at rest. The position of a rigid ])()dy wilh a fix(^d axis I is given by 21 22 KINEMATICS (32. the position of any one of its points P, not on the axis. This position is most conveniently assigned by tlie dihedral angle 6, made by the plane (/, P) of the body with a fixed plane through I. If a definite sense of rotation about the axis is assumed as positive, say the counter-clockwise sense as seen from a marked end of the axis (Fig. 5), the angle 6, expressed in radians, is a real number and serves as co-ordinate to de- termine the position of the body. 32. As the body turns about the axis I in any way, the angle 6 varies with the time; the co-ordinate d can be regarded as a function of the time, just as in the case of the rectihnear motion of a point (and hence (Art. 30) also in the rectili- near translation of a rigid body) the co- ordinate s is a function of the time. The rotation is called uniform if equal angles are always described in equal times. In this case the quotient Ojt of the angle 6 described in any time t, divided by this time, is called the angular velocity, co, of the uniform rotation: Fis. 5. t ' If, in particular, the time of a complete revolution be denoted by T, we have for uniform rotation: 27r In the applications, angular velocity is often measured by the number of complete revolutions per unit of time. Thus, if n be the number of revolutions per second, A^ that per minute, we have* 34.1 TRANSLATION AND ROTATION 23 CO = ZTrn = — ^ . oU 33. When the rotation is not uniform, the quotient ob- tained by dividing the angle of rotation by the time in which it is described, gives the viean, or average, angular velocity for that time. The rate of change of the angle of rotation with the time at any particular moment is called the angular velocity at that moment: (Id The rate at which the angular velocity changes with the time is called the angular acceleration; denoting it by a, we have "^ ~ dt ~ dt" 34. The most important special case of variable angular velocity is that of uniformly accelerated (or retarded) rota- tion when the angular acceleration is constant. The formulae for this case have precisely the same form as those given in Arts. 14-lG for uniformly accelerated rectilinear motion. Denoting the constant linear acceleration by j, we have, when the initial velocity is 0, FOR translation: FOR rotation: ^ = j, a constant; d-'d ^ = a, a. const; V = jt, o) = at. s = ^3t\ 6 = haf". W = js; W = ad', 24- KINEMATICS l35. and when the initial velocities are Vo and coo, respectively: FOR translation: for rotation: V = Vo -}- jt, CO = too + Oct, S = Vot-i- ijt\ = coo« + iar~, iy- — ho^ = js; ico^ — iwo^ = ad. 35. Let a point P, whose perpendicular distance fronn the axis of rotation is OP = r, rotate about the axis with the angular velocity co = dd/dt. In the element of time, dt, it will describe an element of arc ds = rdd = roodt. Its velocity V = ds/dt (frequently called its linear velocity to distinguish it from the angular velocity) is therefore related to the angular velocity of rotation by the equation V = cor. The close analogy between rectilinear translation and rotation about a fixed axis, which is not confined to uniform or uniformly accelerated motion and arises from the fact that in each of the two cases the position of the body is determined by a single co-ordinate, can be illustrated by laying off on the axis of rotation a length measuring the angle of rotation. The rectilinear motion of the extremity of this vector along the axis gives an exact representation of the rotation. 36. Exercises. (1) If a fly-wheel of 10 ft. diameter makes 30 revolutions per minute, what is its angular velocity, and what is the linear velocity of a point on its rim? (2) Find the constant acceleration (such as the retardation caused by a Prony brake) that would bring the fly-wheel in Ex. (1) to rest in Js minute. How many revolutions does the fly-wheel make during its retarded motion before it comes to rest? (3) A wheel is running at a uniform speed of 32 turns a second when a resistance begins to retard its motion uniformly at a rate of 8 radians 36.] TRANSLATION AND ROTATION 25 per second, (a) How many turns will it make before stopping? (6) In what time is it brought to rest? (4) A wheel of 6 ft. diameter is making 50 rev./min. when thrown out of gear. If it comes to rest in 4 minutes, find (a) the angular retardation; (o) the linear velocity of a point on the rim at the be- ginning of the retarded motion; (c) the same after two minutes. CHAPTER III. CURVILINEAR MOTION OF A POINT. 1. Relative velocity; composition and resolution of velocities. 37. It is often convenient to think of the velocity of a point not as a mere number, but as a vector, i. e. a segment PQ of a straight line (Fig. 6), drawn from the point P in the direction of motion and repre- ' p ~^^ senting by its length the mag- jp- Q nitude of the velocity, by its direction the direction of mo- tion of P, and by an arrowhead the sense of the motion. 38. Consider a point P (Fig. 7) moving along a straight line I with constant velocity Vr, while the line I moves in a fixed plane with a con- stant velocity Vb in a di- rection making an angle a with the line I. Then the vector PQ = Vr is called the relative veloc- ity of P with respect to I; the vector PS = Vb may be called the body velocitij, or the velocity of the body of reference (here the hno I). With respect to the fixed plane, the point P has not only the velocity Vr, but it participates in the motion of I. Its absolute velocihj v, i. e. its velocity with respect to the fixed plane, is therefore represented in magnitude, direction, and 26 39.1 CURVILINEAR MOTION OF A POINT 27 sense by the vector PR, i. e. by the diagonal of the parallelo- gram constructed on the vectors Vr and I'b. This vector PR = y is called the resultant, or geometric sum, of the vectors PQ = Vr and PS = v^. It is easy to see that this result will hold even when the motions are not uniform, provided we mean by Vr the instan- taneous relative velocity of P and by Vb the simultaneous velocity of that point of the body of reference with which P happens to coincide at the instant. We have thus the general proposition that the absolute velocity v of a point P is the resultant, or geometric sum, of its relative velocity Vr and the body velocity Vb- 39. The term " geometric sum," of the vectors Vr = PQ and Vb = PS may be justified by observing that (Fig. 7) QR = PS; hence the resultant PR = i; is obtained simply by adding the vectors Vr and Vb geometrically, i. e. l^y drawing first the vector PQ = Vr and then from its extremity Q the vector QR = Vb. Conversely, the relative velocity PQ = Vr is found by geo- metrically subtracting the body velocity Vb from the absolute velocity v; i. e. by drawing the vector PR = ?;,. and from its extremity R the vector RQ equal and opposite to the vector PS = Vh- This result can be interpreted as follows: In the example of Art. 38 of a point moving with velocity Vr along the line I while I moves with velocity Vb in a fixed plane, let us superimpose the velocity — Vb, i. e. a velocity equal and opposite to the body velocity, on the whole system, formed by the line and the point; the line is thereby brought to rest while the point will have the velocities v and — Vb whose resultant is the relative velocity Vr. Hence the relative velocity is found as the resultant of the absolute velocity and the body velocity reversed. 28 KINEMATICS [40. 40. It is this idea of relative motion that leads to the so- called parallelogram of velocities, i. e. to the proposition that a point whose velocity is v = PR (Fig. 8) can be regarded as pos- sessing simultaneously any two velocities, such as vt = PQ, v^. = PS = QR, whose geometric sum is i; = PR. For we can always regard V\ as the relative velocity of the point along the line PQ and V2 as the body velocity, i. e. as the velocity of the line PQ. 41. Finally, if in the example of Art. 38 we suppose the plane tt in which the line I moves to have itself a velocity v„, Fig. 8. Fig. 9. it is clear that the absolute velocity v of the point will be the resultant, or geometric sum, of the three velocities v,-; Vb, v„; i. e. it will be represented by ihe diagonal of the paral- lelepiped that has the vectors Vr, Vh, v„ as adjacent edges. It then follows that the velocity v = PR of a point (Fig. 9) can be regarded as equivalent to any three simultaneous velocities Vi = PQi, Vo = PQ2, Vs = PQ3, whose geometric 42.] CURVILINEAR MOTION OF A POINT 29 sum is V = PR. This proposition is known as the paral- lelepiped of velocities; Vi, Vo, vz are called the components of v. The corresponding propositions for forces in statics will be familiar to the student from elementary physics. But it will be seen later that these propositions in statics are really based on the more elementary propositions for velocities. 42. Exercises. (1) The components of the velocity of a point are 5 and 3 ft. /sec. and enclose an angle of 135°; find the resultant in magnitude and direction. Check the result by graphical construction. (2) Find the components of a velocity of 10 ft. /sec, along two hnes inclined to it at 30° and 90°. (3) A man jumps from a car at an angle of 60°, with a velocity of 9 ft. /sec. (relatively to the car). If the car is running 10 M./h., with what velocity and in what direction does the man strike the ground? (4) Two men, A and B, walking at the rate of 3 and 4 M./li., respec- tively, cross each other at a rectangular street corner. Find the relative velocity of A with respect to B in magnitude and direction. (5) How must a man throw a stone from a train running 15 M./h, to make it move 10 ft. /sec. at right angles to the track? (6) The velocity of light being 300,000 km. /sec, the velocity of the earth in its orbit 30 km. /sec, determine approxunately the con- stant of the aberration of the fixed stars. (7) A man on a wheel, riding along the railroad track at the rate of 9 M./h., observes that a train meeting him takes 3 sec to pass him, while a train of equal length takes 5 sec. to overtake him. If the trains have the same speed, what is it? What is the length of the train? (8) A swimmer starting from a point A on one bank of a river wishes to reach a certain point B on the opposite bank. The velocity Vb of the current and the angle 6( < I^tt) made by AB with the current being given, determine the least relative velocity v,- of the swimmer in magnitude and direction. (9) A straight line in a x'lane turns with constant angular velocity w about one of its points O, while a point P, starting from 0, moves along the line with constant velocity Vo. Determine the absolute path of P and its absolute velocity v. 30 KINEMATICS [43. Fig. 10. (10) Show how to construct the tangent and normal to the spiral of Archimedes, r = aB, where d = wt. 2. Velocity in curvilinear motion. 43. If on the curve described by the moving point we select an origin Po, and take a definite sense of progression T along the curve as positive, the position P of the point at any time t is given by the arc PqP = s, which might be regarded as the co-ordi- nate of P (Fig. 10). As s is a function of the time t, its time-derivative ds '=dt gives the magnitude of the velocity of the point at P, or at the instant t, in its curvilinear motion (comp. Art. 5). To incorporate in the definition of velocity the idea of the varying direction of the motion, which at any instant t is that of the tangent to the path, we lay off from P, on this tangent, a segment PT oi length v = dsjdt, in the sense of the motion, and define the vector PT as the velocity of the point in its curvilinear motion (comp. Art. 37). 44. When the motion of the point P is referred to fixed rectangular axes Ox, Oij, Oz, the co-ordinates x, y, z oi P (Fig. 11) are functions of the time: X = x{t), y = y(t), z = z{t). Now the a:-co-ordinate of P is at the same time the co- ordinate of the projection Px of P on the axis Ox on this axis. As the point P moves in space, its projection Px moves 45. CURVILINEAR MOTION OF A POINT 31 along the axis Ox, and the velocity of Px in its rectilinear motion is dx Vx = dt' Similarly the velocities of the projections Py, Pz of P on Oy, Oz are _ dy _dz The rectilinear motions of P^, Py, Pz along the axes Ox, Oy, Oz, respectively, fully determine the curvilinear motion of P{x, y, z) in space. 45. On the other hand, the velocity-vector PT = v can, by Art. 41, be resolved into its three components along the Fig. 11. axes; if the tangent to the path at P makes the angles a, iS, 7 with Ox, Oy, Oz, respectively, these components are V COSa, V COSjS, V COS7. It is easy to show that these components of v are equal, respectively, to the velocities dx/dt, dy/dt, dz/dt of the projections 32 KINEMATICS [46. Px, Py, Pz of P on the axes. For we have, if As is the arc described by P in the time A^ : dx ,. Ax ,. Ax As ds — = lim -— = hm -— -.^ = cosa-r; = v cosa, dt A<=o At At=o As At at since at any ordinary point of the curve (i. e. at any point at which the curve possesses a definite tangent) we have A^^ Um -7— = coso:. As Similarly for dy/dt, dzldt. We shall therefore henceforth denote by v-c, Vy, v^ not only (as in Art. 44) the velocities of Px, Py> Pz, but also the components of the velocity v along the axes Ox, Oy, Oz. Thus we have: dx ^ dy dz Vx = V cosa = -rr , Vy = V cos/3 = 37 > ^^ = i' COS7 = ^t", In the language of infinitesimals we may say that the velocity is found by dividing the element of arc ds = Vdx''- -\- dy- + dz~ by dt. 46. In polar co-ordinates OP = r, xOP = 9, yOQ = (Fig. 12), the rectangular components Vr, vg, v^ of the velocity v along OP, at right angles to' OP in the plane xOP, and at right angles to this plane are readily found from the last remark in Art. 45, by observing that ds'' = dr^ + {rdey + (r sin0 ^0)^, whence dr de ■ n(^ '^ = Jt''' = 'dt'''='''''^dt' 47 CURVILINEAR MOTION OF A POINT 33 Fig. 12. 47. If the path of P is a plane curve we have in rectangular cartesian co-ordinates dx ^'~ dt' ^" ~ dt' dy ds V(l) 2 /d^J\^ ■^ ^ dt and in polar co- ordinates dr do ds "' = If "' "-'df " = di dry idey \dt As the point P moves in the plane curve its radius vec- tor OP sweeps out the polar area S of the curve, i. e. the area bounded by any two radii vectores and the arc of the curve between their ends. If AS be the increment of this area in the time At, the limit of the ratio AS/At, as At ap- proaches zero, is called the sectorial velocity dS/dt of the point P (about the origin 0) : dS ,. AS -7- = lim ^^ . dt yt=o At It follows from the well-known expression for the element of polar area that in polar co-ordinates 34 KINEMATICS i48. dS dt = i7- dB dt' and in rectangular cartesian co-ordinates dy dx\ rf8 ^ ^ _^ _ dt ~ ' \^dt '^dt /' 48. Exercises. (1) If the point P describos a circle of radius a about the origin 0, with angular velocity CO, the linear velocity of P is /,' = oco (Art. 35); its components along rectangular axes through the Drigin are ^Fig. 13): Fig. 13. Vx = aw cosCiTT -\- 6) = — aoi sin0 = — uy, Vy = aoi sin(2 7r -\- 0) = ow cosO = wx. Obtain these results by differentiating the equations of the circle X = a cos^, y — a sinO with respect to the time. (2) Show that the velocity of a point describing a cycloid passes through the highest point of the generating circle. (3) The ellipse being defined as the locus of a point such that the sum of its distances from two fixed points is constant; show that the normal bisects the angle between the focal radii n, r-i. In bilinear co-ordinates the equation of the ellipse is simply n + r-i = 2a. 49.] CURVILINEAR MOTION OF A POINT 35 Differentiating with respect to t and denoting time-derivatives by dots, we find h + h = 0; i. e. the rate of increase of one focal radius is equal to the rate of de- crease of the other. Notice, however, that 7\ and h are not the com- ponents of the velocity of the describing point P along the focal radii, but the projections of this velocity on these radii. For, the velocity voi P can be resolved : (a) into fi along ri and a component perpendicular to n; (b) into r2 along r2 and a component perpendicular to rz. Both resolutions arise from the sam'e vector v; hence perpendiculars erected at the extremities of ri and h (laid off from- P along ri, n in the proper sense) must meet at the extremity of v. As rz = — ri, v bisects the angle between r\ (produced) and r2. (4) Find a construction for the tangent to any conic given by directrix, focus, and eccentricity. (5) Derive the expressions for Vr and vq in Art. 46 by the method of limits. 3. Acceleration in curvilinear motion. 49. As the moving point describes its path the velocity vector V = PT (Art. 43) will in general vary both in mag- nitude and in direction. To compare the velocities v = PT at the time t and v' = P'T' at the time t -{- At (Fig. 14) Fig. 14. we must draw these vectors from the same origin, say from the point P. Making PT" = P'T' = v', it appears that 36 KINEMATICS [50. the vector v' can be obtained from the vector v by adding to it geometrically the vector TT" which represents the geometrical increment of the velocity in the time interval This vector TT", divided by M, is the average accelera- tion in the time M. As M approaches zero, the vector TT" approaches zero; but its direction will in general ap- proach a definite direction as a limit, and the ratio of its length to M will approach a definite number as limit. A vector (generally drawn from the point P) having this limiting direction as its direction and a length rp rplf j = lim -—-— '' st=o At is defined as the acceleration of the moving point at P, or at the time t. It follows from this definition that the acceleration vector lies in the osculating plane of the path at P, this plane being the limiting position of the plane determined by the tangent at P and any near point P' of the curve as P' approaches P along the curve. 50. Acceleration being defined as a vector can be resolved into components by the parallelogram or parallelepiped rules (Arts. 40, 41). Thus, in particular, the acceleration j, since it lies in the osculating plane, can be resolved into a tangential component jt along the tangent, and a normal component jn along the principal normal at P, the principal normal being the inter- section of the normal plane with the osculating plane. If \p (Fig. 14) is the angle between the velocity and the acceleration these components are jt = j cos;/', jn = j sinr/'. 51.] CURVILINEAR MOTION OF A POINT 37 51. If from any fixed point we draw vectors OQ equal and parallel to the velocity vectors PT oi the moving jioint P, the extremities Q lie on a curve called the hodograph of the path of P] and it follows from Art. 49 that the accel- eration vector of P is equal and parallel to the velocity vector in the motion of Q along the hodograph. Hence the tangential and normal components of the acceleration of P are equal, respectively, to the components of the velocity' of Q along the radius vector OQ and at right angles to it. Observing that the acceleration lies in the osculating plane we have therefore by Art. 47 . _ dv . _ dd ^' ~ dV ^''~''dt' where d is the angle made by OQ, i. e. by the velocity vec- tor at P, with any fixed direction in the osculating plane. Now if ds be the element of arc of the path of P we have (cbmp.* below. Art. 54) ^ - 1 ^ ^ ds~ p' ' • vr^^**V^ where p is.the radius of (first) curvature of the path at P; hence . _ ddds _ v^ •^" ~ ^ dsdt ~ p' Thus we have for the tangential acceleration ji and the normal acceleration j„ of a moving point . _ dv . _ v"^ ^' ~ dt' •^" " p- 52. When the rectangular cartesian co-ordinates of the ^ moving point are given as functions of the time, X = x(t), y = y(t), z = z{t), their first derivatives with respect to the time are on the 38 KINEMATICS [52 one hand the velocities of the projections Px, Py, Pz of P on the axes in their rectihnear motions, on the other the components Vx, Vy, Vz of the velocity v = dsldt of P in its ciirvilineap motion (Art. 44). Thus, using dots to denote time-derivatives, we have Vx = X, Vy = y V2 = z. It will now be shown that the second lime-derivatives x, y, z of a;, y, z, which are the accelerations of Px, Py, Pz in their rectilinear motions, are at the same time the components jx, jy, jz of the acceleration vector along the axes of co-ordinates. 53. We have . _ dx _ dxds _ dx dt ds dt ds ' whence, differentiating with respect to /, ^ ~ dt^ " dt\7h) "Itds^^ds'^dt'^ds "^*^ ds2' Writing down the corresponding expressions for y, z by cyclic permutation of a;, ?/, z we find : . dx , „ d?-x . dij , „ dhi y-'i + '-'d^ ■ . dz , . dh ds as^ Now if a, /3, 7 are the direction cosines of the velocity vector we have _ dx o _ dy _ dz " ~ ds' '^ ~ ts' ^ ~ ds' hence the first terms in the expressions found for x, ij, z are 54.] CURVILINEAR MOTION OF A POINT 39 the components along the axes of a vector, parallel to the velocity and of length v, i. e. of the tangential acceleration i^Art. 51). To see that the second terms are the components of the normal acceleration j'„ = v'^jp (Art. 51) we have only to remember that the direction cosines X, ix, v of the principal normal of any curve are ^ (Px dry dh ^^'ds^'^^'ds^' ' = 'ds^' a proof of this fact is supplied in Art. 54 Thus it appears that x, y, z are the components along the axes of the total acceleration j of the moving point. ~r 54. To determine the (first) curvature 1/p and the direction cosines X, fjL, V of the principal normal of any curve imagine the curve described by a moving point P with constant velocity 1. The hodograph con- structed at the origin of co-ordinates, is then a spherical curve, called the spherical indicatrix, and the co-ordinates of the point Q of this indicatrix, corresponding to the point P of the given curve are a, /3, y. Hence, if ds' is the element of arc QQ' of the indicatrix corresponding to the arc PP' — ds of the given curve, we have ^ _ (/« d dx_ _ d-x ds ds' ds' ds ds- ds' ' But as the radii vectores of the indicatrix are parallel to the tangents of the given curve we have (Art. 51) ds' ^ 1. ds p' hence . d^x 'ds^' and similar expressions for p, p. 55. When the path of P is a pla7ie curve we have as com- ponents of the acceleration j along rectangular cartesian axes in the plane of motion: 40 KINEMATICS [55. df" When polar co-ordinates r, 6 are used we may resolve the acceleration j into a component jV along the radius vector OP = r and a component je at right angles to r (Fig. 15). Fig. 15. They are found by projecting jx = x and jv = ^ on these directions. Differentiating the relations x = r cos6, y = r siiid twice with respect to t we find X = r cos^ — rd sin0, y = r s\n9 + rd cos0, X = (r - re^~) COS0 - (2fd + rd) sm9, y = 0' - rd") sin0 + {2fd + rd) cosd. These expressions show directly that jr = r - rd^, je = 2rd + rd = rdt 56. Exercises. (1) Show that the velocity of a moving point is increasing, con- stant, or diminishing according to the value of the angle f between v and J (Fig. 14). (2) Show that in plane motion the sectorial velocity (Art. 47) is constant if je — 0, and vice versa. 56.1 CURVILINEAR MOTION OF A POINT 41 (3) Show that the normal component of the acceleration is the product of the radius of curvature into the square of the angular velocity about the center of curvature. (4) If the acceleration of a point P be always directed to a fixed point 0, show that the radius vector OP describes equal areas in equal times. (5) Show that in uniform circular motion the acceleration is directed to the center and proportional to the radius. (6) For motion in the circle x = a cos9, y = a sin0 find jx and jy, jr and jg, jt and _/». (7) A wheel rolls on a straight track; find the acceleration of any point on its rim, and in particular that of its lowest and highest points. (8) What is the hodograph (a) for any rectilinear motion? (6) for any uniform motion? (c) for uniform circular motion? (d) What can be said about the acceleration of any uniform motion? (9) The spherical, or polar, co-ordinates of a point are the radius vector r = OP (Fig. 16), the polar distance or colatitude d = xOP, and Fig. 16. the longitude <}> = yOQ. The cylindrical co-ordinates of the same point are r' = RP = r sin0, <^ = jjOQ, x = QP = r cos0. Find the cylindrical components of the acceleration (along PP, normal to xOP, and along QP), and hence show that the spherical componenti^^long OP, per- pendicular to OP in the plane xOP , and normal to xOr) arc jr = f — rd^ — r(^2 gin20, jg = rd -\- 2rd + r^- sin» cos^, ./ 4,= r4> sine + 2f<^ sinfl + 2r(?<^ COS0. 42 IvINEMATICS [57. 57. The fundamental problem of the kinematics of the point consists in determining the motion of the point when the acceleration is given. In cartesian co-ordinates this requires the solution of the simultaneous differential equa- tions d~x _ . d-y _ . d-z _ . dt^ ~ ^" dt^ ~ ^"^ dt^ ~ ^" jx, jy, jz being given functions of t, x, y, z, dx/dt, d.y/dt, dz/dt. A first integration would give the components of the velocity ; a second integration should give the co-ordinates x, y, z as functions of the time, and hence also the path of the moving point. It may often be more convenient to use polar co-ordinates; in the case of plane motion, we have then the equations at the end of Art. 55, with jr and je as given functions of t, r, 6 and their first time-derivatives. If the tangential and normal components of the accelera- tion are given we can use the equations (Art. 51) : dv _ . v^ _ • A number of simple illustrations will be found in the following articles. 4. Examples of curvilinear motion. (a) Constant acceleration. 58. Motion on a straight line under gravity. Let a point P move along a line inclined at the angle 6 to the horizon, under the acceleration g of gravity. The motion is rectilinear; the component of the acceleration along the line is g sin9; hence the motion is uniformly accelerated. The equations 60.1 CURVILINEAR MOTION OF A POINT 43 are the same as those for falhng bodies (Arts. 14, 15) except that g is replaced by g sin0. A particle placed on a smooth inchned plane will have this motion if its initial velocity is zero or directed along the greatest slope of the plane. 59. Exercises. (1) Show that the final velocity is independent of the inclination; in other words, in sliding down a smooth inclined plane a body acquires the same velocity as in falling vertically througli the "lieight" of the plane. (2) Show that it takes a body twice as long to slide down a plane of 30° inclination as it would take it to fall through the height of the plane. (3) At what angle 6 should the rafters of a roof of given span 2b be inclined to make the water run off in the shortest time? (4) Prove that the times of sliding from rest down the chords issuing from the highest (or lowest) point of a vertical circle are equal. (5) Show how to construct geometrically the line of quickest (or slowest) descent from a given point: (a) to a given straight line, (b) to a given circle, situated in the same vertical plane. (6) Analytically, the line of quickest or slowest descent from a given point to a curve in the same vertical plane is found by taking the equation of the curve in polar co-ordinates, r = f{d), with the given point as origin and the axis horizontal. The time of sliding down the radius vector r is ; = i/2r/{(j s'm9). Show that this becomes a maximum or minimum when tanO = f{d)/f'{d), according as/(0) + f"{e) is negative or positive. (7) Show that the line of quickest descent to a parabola from its focus, the axis of the parabola being horizontal and its plane vertical, is inclined at 60° to the horizon. 60. Free motion under gravity. The motion of a point, when subject only to the constant acceleration of gravity is necessarily in the vertical plane determined by the initial velocity and the direction of gravity. Taking the hori- zontal line in this plane through the initial position of the 44 KINEMATICS [60. point as axis of x, and the vertical upwards as positive axis of y (Fig. 17), the components of acceleration along Fig. 17. these axes are evidently and — g, so that the equations of motion (Arts. 55, 57) are d; = 0, ij = - g. The first integration gives X = ci, y = - gt ■{- Ci. To determine the constants Ci, Ci we must know the initial velocity in magnitude and direction. If the point starts at the time from with a velocity ^o, inclined to the horizon at an angle e, the angle of elevation, we have for t = 0: X = Vo cose, y = Vo sine. Substituting these values we find Ci = Vo cose, C2 = Vo sine, so that the velocity components at any time t are : X = Vo cose, y = Vo sine — gt. Integrating again we find X = Vo cose-t, y = sine-^ — igf^, the constants of integration being since x = and y = for t = 0. 61]. CURVILINEAR MOTION OF A POINT 45 These equations show that the horizontal projection of the motion is uniform, while the vertical projection is uni- formly accelerated, as is otherwise apparent from the nature of the problem. Eliminating t between the last two equations we find the equation of the path y = tane-o; - ^r— ^ -x"^, 2vo cos^e which represents a parabola passing through the origin. To find its vertex and latus rectum, divide by the coefficient of x^ and rearrange: X" sme cose -a; = cos^e-w; g y completing the square in x, the equation can be written in the form [ X — ^^ sm2e = — cos-e [ y — -^ sm-^e I . \ 2g / g V 2g J The co-ordinates of the vertex are therefore a = (yoV26f)sin2e, /3 = (yoV2{/)sin2e; the latus rectum 4a = {2vo'^lg)coQ^t; the axis is vertical, and the directrix is a horizontal fine at the distance a = (?'o^/2g) cos^e above the vertex. 61. Exercises. (1) Show that the velocity at any time is w = Vv^'^ — 2gy. (2) Prove that the velocity of the projectile is equal in magnitude to the velocity that it would acquire by falling from the directrix: (a) at the starting point, {h) at any point of the path (see Art. 18). (3) Show that a body projected vertically upwards with the initial velocity vo would just reach the common directrix of all the parabolas described by bodies projected at different elevations e with the same initial velocity !'o. (4) The range of a projectile is the distance from the starting point to the point where it strikes the grovmd. Show that on a horizontal plane the range is i? = 2a = {vi?l(j) sin2€, 46 KINEMATICS [61. (5) The lime of flight is the whole time from the beginning of the motion to the instant when the projectile strikes the ground. It is best found by considering the horizontal motion of the projectile alone, which is uniform. Show that on a horizontal plane the time of flight is T = i2vo/g) sine. (6) Show that the time of flight and the range, on a plane through the starting point incUned at an angle d to the horizon, are ™ 2!iosin(e — e) IT, 2ro^6in(e — 0)cose Tq = — ■ — - , and Re = — • ;:. • (7) What elevation gives the greatest range on a horizontal plane? (8) Show that on a plane rising at an angle d to the horizon, to obtain the greatest range, the direction of the initial velocity should bisect the angle between the plane and the vertical. (9) A stone is dropped from a balloon which, at a height of 625 ft., is carried along by a horizontal air-current at the rate of 15 miles an hour, (o) Where, (6) when, and (c) with what velocity will it reach the ground? (10) What must be the initial velocity ro of a projectile if, with an elevation of 30°, it is to strike an object 100 ft. above the horizontal plane of the starting point at a horizontal distance from the latter of 1200 ft? (11) Whsit must be the elevation e to strike an object 100 ft. above the horizontal plane of the starting point and 5000 ft. distant, if the initial velocity be 1200 ft. per second? (12) Show that to strike an object situated in the horizontal plane of the starting point at a distance x from the latter, the elevation must be € or 90° — e, where t = J sin"' (gx/vo^). (13) The initial velocity Vo being given in magnitude and direction, show how to construct the path graphically. (14) The solution of Ex. (11) shows that a point that can be reached with a given initial velocity can in general be reached by two different elevations. Find the locus of the points that can be reached by only one elevation, and show that it is the envelope of all the parabolas that can be described with the same initial velocity (in one vertical plane). (15) If it bo known that the path of a point is a parabola and that 63.J CURVILINEAR MOTION OF A POINT 47 the acceleration is parallel to its axis, show that the acceleration is constant. (16) Prove that a projectile whose elevation is 60° rises three times as high as when its elevation is 30°, the magnitude of the initial velocity being the same in each case. (17) Construct the hodograph for the motion of Art. 60, taking the focus as pole and drawing the radii vectores at right angles to the velocities. (IS) A stone slides down a roof sloping 30° to the horizon, through a distance of 12 ft. If the lower edge of the roof be 50 ft. above the ground, (o) when, (6) where, (c) with what velocity does the stone strike the ground? (19) If a golf ball be driven from the tee horizontally with initial speed = 300 ft. /sec, where and when would it land on ground 16 ft. below the tee if resistance of air and rotation of ball could be neglected? (20) A man standing 15 ft. from a pole 150 ft. high aims at the top of the pole. If the bullet just misses the top where will it strike the ground if vo= 1000 ft. /sec? 62. While the type of motion discussed in Art. 60 is commonly spoken of as projectile moliori, it should be kept in mind that it takes no account of the resistance of the air; it gives the motion of a projectile in vacuo. Owing to the very high initial velocities of modern rifle bullets, the range may be only about one tenth of what it would be according to the formula? given above. The study of the actual motion of a projectile in a resisting medium, such as air, forms the subject of the science of ballistics. See for instance C. Cranz, Lehrbuch der BalUstik, Vol. I, 2te Auflage, Leipzig, Teubner, 1910. (b) The pendulum. 63. The mathematical pendulum is a point constrained to move in a vertical circle under the acceleration of gravity. Let be the center (Fig. 18), A the lowest, and B the highest point of the circle. The radius OA = I oi the circle is called the length of the pendulum. Any position P of the moving point is determined by the angle AOP = 6 48 KINEMATICS [64. counted from the vertical radius OA in the positive (counter- clockwise) sense of rotation. If Pq be the initial position of the moving point at the time t = 0, and 2^ AOPo = do, then the arc PoP = s de- scribed in the time ^ is s = ^(^0 — d); hence v = ds/dt — - Idd/dt, and dv/dt = - Id^d/dt^, the negative sign indicating that 6 diminishes as s and t increase. Resolving the acceleration of gravity, g, into its normal and tangential components g COS0, g sin^, and considering that the former is without effect owing to the condition that the point is constrained to move in a circle, we obtain the equation of motion in the form dv/dt = g sin0, or .d'9 Fig. 18. mg J^ + <, sine = 0. (1) 64. The first integration is readily performed by multiply- ing the equation by dd/dt which makes the left-hand member an exact derivative, dt [l(^) "''"'^1' hence integrating, we obtain dd dt U — g COS0 = C, or considering that v = — Idd/dt, 65.] CURVILINEAR MOTION OF A POINT 49 ^v^ — gl COS0 = CI. To determine the constant C, the initial velocity Vq at the time t = must be given. We then have i^o" — gl cos^o = CI; hence |y2 = ^^^2 _ gl COS0O + gl COS0 'vq"^ \ (2) — I COS^o + I COS0 .2^ The right-hand member can readily be interpreted geo- metrically; ^'o^/2gr is the height by falling through which the point would acquire the initial velocity Va (see Art. 18); I COS0 — I cos^o = OQ — OQq = QoQ, if Q, Qo are the pro- jections of P, Po on the vertical AB. If we draw a hori- zontal line MN at the height vo'^/2g above Po and if this line intersect the vertical AB at R, we have for the velocity V the expression: h' = g-RQ. If the initial velocity be zero, the equation would be h' = g-QoQ. At the points M, N where the horizontal line MN inter- sects the circle the velocity becomes zero. The point can therefore never rise above these points. Now, according to the value of the initial velocity Vo, the line AIN may intersect the circle in two real points M, N, or touch it at B, or not meet it at all. In the first case the point P performs oscillations, passing from its initial position Po through A up to M, then falling back to A and rising to A'', etc. In the third case P makes complete revolutions. 65. The second integration of the equation of motion cannot be effected in finite terms, without introducing elliptic functions. But for the case of most practical importance, 5 50 KINEMATICS 1 66. viz. for very small values of 6, it is easy to obtain an ap- proximate solution. In this case 6 can be substituted for sin0, and the equation becomes: or, putting g/l = m" f = - "'»■ ^ (3) This is a well known differential equation (compare Art. 26, eq. (14), and Art. 28, Ex. 1), whose general integral is 6 = Ci cos/if + C2 sin/jLt. The constants Ci, Co can be determined from the initial conditions for which we shall now take 6 = 60 and v = when ^ = 0; this gives Ci = 60, Co = 0; hence 1 6 = 60 cosfj.t, t = - cos"^ — . The last equation gives with 6 = — 60 the time ti of one swing or beat, that is, half the period: ^.-" = .rJl ■ (4) M \g The time of a small oscillation or swing is thus seen to be independent of the arc through which the pendulum swings; in other words, for all small arcs the times of swing of the same pendulum are very nearly the same; such oscillations are therefore called isochronous. 66. The formula (4) shows that for a pendulum of given length h the time of one swing /i varies for different places owing to the variation of g. As h and 'i can be measured very accurately, the pendulum can be used to determine g, the acceleration of gravity at any place; (4) gives : 67.] CURVILINEAR MOTION OF A POINT 51 Now let lo be the length of a pendulum which beats seconds, i. e., makes just one swing per second; by (4) and (5) we find for the length lo of such a seconds pendulum: 'o = ^2 = ry • (6) The length k of the seconds pendulum is therefore found by measuring the length h and the time of swing ti of any pendulum. This length k is very nearly a meter; it varies sUghtly with g; thus, for points at the sea level it varies from ^o = 99.103 cm. at the equator to lo — 99.610 at the poles. If go be the value of g at sea level, i. e., at the distance R from the center of the earth, gi the value of g at an elevation h above sea level in the same latitude, it is known that fir„ ^ {R + hY gi R' ' Hence, if go be known, pendulum experiments might serve to find the altitude of a place above sea level; but the observations would have to be of very great accuracy. 67. Let n be the number of swings made by a pendulum of length I in any time T so that h = T/n. Then, by (4), T 7 If T and one of the three quantities n, I, g in this equation be re- garded as constant, the small variations of the two others can be found approximately by differentiation. For instance, if the daily number of oscillations of a pendulum of constant length be observed at two different places, T and I keep the same values while n and g vary by small amounts, say An and Ag. Now the differentiation of (7) gives or, dividing by (7) : T, TvVldg -n^^''=--~2 gl dn _ J dg n ~ g ' We have therefore approximately, for small variations An, Ag: ^^i,^^. (8) n ' g 52 KINEMATICS 168. 68. Exercises. (1) Find the number of swings made in a second and in a day by a pendulum 1 meter long, at a place where g = 980.5. (2) Find the length of the seconds pendulimi at a place where g = 32.17. (3) Find the value of gr at a place where a pendulum of length 3.249 ft. is found to make 86522 swings in 24 hours. (4) A chandelier suspended from the ceiling is seen to make 20 swings a minute; find its distance from the ceiling. (5) A pendulum of length 1 meter is carried from the equator where g = 978.1 to another latitude; if it gains 100 swings a day find the value of g there. (6) Investigate whether the approximate formula (8) is sufficiently accurate for Ex. (5). (7) If the length of a pendulum be increased by a small amount Al, show that the daily number of swings, n, will be diminished by An so that approximately An _ J AZ (8) A clock beating seconds is gaining 5 minutes a day; how much should the pendulum bob be screwed up or down? (9) A clock beating seconds at a place where g = 32.20 is carried to a place where g = 32.15; how much will it gain or lose per day if the length of the pendulum be not changed? (10) A pendulum of length 100.18 cm. is foimd to beat 3585 times per hour; find the elevation of the place if in the same latitude g = 981.02 at sea level. 69. When the oscillations of a pendulum are not so small that the angle can be substituted for its sine, as was done in Art. 65, an expression for the time h of one swing can be obtained as follows. We have by (2), Art. 64. iw^ — ivo^ = gl{cosd — cos^o). Let the time be counted from the instant when the moving point has its highest position {N in Fig. 18), so that vo = 0. 69. CURVILINEAR MOTION OF A POINT 53 Substituting v = — Idd/dt and applying the formula cos6 = 1 — 2 sin^i^ we find : il(§Y = 2?(sin2i5o - sin^ie), whence . Pf dd dt ■*i^ g /sin^l-^o - sin^^^ ' Integrating from 9 = to 6 ^ do and multiplying by 2 we find for the time h of one swing: I r^« dd h = Vli g Jo Vsin^i^o — sin^^-^ * As 6 cannot become greater than Oq we may put sini^ = sini^o sin0, thus introducing a new variable <^ for which the limits are and 7r/2. Differentiating the equation of substi- tution, we have i cos^O dd = sini^o cos0 d(f), or, as cosi^ = VI — sin^i^o sm^^, 2 sini^o coS(/> dcj) dd = V 1 — sin'-i^o sinV Substituting these values and putting for the sake of brevity sini^o = K, we find for the time ti of one swing: d(j) ti ■'M g Jo Vl — K^ sin^^ The integral in this expression is called the complete elliptic integral of the first species and is usually denoted by K. Its value can be found from tables of elliptic integrals or by expanding the argument into an infinite series by the binomial theorem (since k sin0 is less than 1 ) , and then performing the 54 KINEMATICS [71. integration. We have 1-3 (1 - K^ sin2)-J = 1 + i/c^sinV + — ^ k' sinV + 2-4 hence ^'-4b^(Xf--{m^--} If H be the height of the initial point N{d = 6o) above the lowest point A of the circle, we have 2 . ^1 . 1 — cos^o H ,^ = ,n,-^0o= = -^, so that the expression for ti can be written in the form 70. Exercises. (1) Show that /. = TT ;/r/^(l + Jj + To':ri + tgVjt + • ' ) if the angle 20o of the swing is 120°. (2) Show that as second approximation to the time of a small swmg we have h = irVllffil + tV^o")- (3) Find the time of oscillation of a pendulum whose length is 1 meter at a place where (j = 980.8, to four decimal places, the amplitude Oa of the swing being 6°. (4) Denoting bj^ /o the first approximation, irVl/g, to the time h of one swing, the quotient (U — to)/io is called the correction for mnplilude. Show that its value is 0.0005 for do = 5°. (5) A pendulum hanging at rest is given an initial velocity vu Find to what height hi it will rise. (6) Discuss the pendulum problem in the particular case when MA'' (Fig. 18) touches the circle at B, that is when the initial velocity is due to falling from the highest point of the circfle. (c) Simple harmonic motion. 71. Simple harmonic motion is that kind of rectilinear motion in which the acceleration is proportional to the dis- tance of the moving point P from a fixed point in the 72.] CURVILINEAR MOTION OF A POINT 55 line of motion and is always directed toward this fixed point (Fig. 19). An example of simple harmonic motion was discussed in Arts. 26, 27. We now resume its study from a more general point of view, owing to its great importance. It naturally P 1 1 ^_ — ^_^ Pg .? Pi *■ Fig. 19. leads to the study of certain important motions known as coinpound harmonic, which may be curvilinear. By definition, the differential equation of simple harmonic motion is X = — fi-X, where ju is a constant, /x^ being evidently the absolute value of the acceleration at the distance x = 1 from the origin 0. The equation has the form of the pendulum equation (3), Art. 65, except that 6 is replaced by x. Its general integral is therefore X = Ci coS)u/ + C2 smut. Differentiating, we find the velocity V = — CiM ^in/jLt + C2M cosnt. If a: = a;o and i^ = i^o for i = we find Ci = Xo, C2 = Vq/ij.', hence X = Xo cosfit -\ — sinut, v ^ — x^p. sin/xi + vo cosut. 72. The expression found for x can be given a more con- venient form by observing that if we construct a right-angled triangle (Fig. 20) with xo and Voffx as sides and call a its hypotenuse, e its angle adjacent to Xo, we have 56 KINEMATICS 1 72. Xo = a cose, — = a sine: substituting these values we find X = a cose cos/if + sine sinjui = a cos(fjLt — e). Hence, in simple harmonic motion we have X = a cos(ixt — e), V = — ajx sin()uf — e), where ■=J xo- +-T> ^ "= tan 1 -. The motion is clearly periodic since both position and velocity regain the same values when the angle jui — e is increased by any integral multiple n of 2t, i. e. if the time t is increased by n times 27r/ju. The time J" between any two successive equal stages of the motion is called the period; the length a, which is evidently the greatest distance on either side of the origin reached by the point, is called the amplitude of the simple harmonic motion. The angle fxt — e \s called the phase-angle, e the epoch- angle of the motion. The point oscillates between the positions Pi and P2 (Fig. 19) whose abscissas are =ta. It is at Pi (at elongation) at the time ^0 = e//x (and also at the times to + n- 27r//x = (e + 2mr)j p) ; it reaches the position at the time fi = (e +-2-7r)/ju, so that the time of passing from Pi to is The time of passing from to the other elongation P2 is 73.] CURVILINEAR MOTION OF A POINT 57 easily shown to be equal to this; so that the time of one swing (from Pi to P2) is The backward motion from P2 to Pi takes place in the same time so that the period, that is the time of a double (forward and backward) swing, is, as shown above, T = ^ 73., An instructive illustration is obtained by observing that any simple harmonic motion can he regarded as the 'projection of a uniform circular motion on a diameter of the circle. In other words, it is the apparent motion of a point describing a circle uniformly, as seen from a point in the plane of the circle (at an infinite distance). For, let a point Q (Fig. 21) describe a circle of radius a with constant angular Fig. 21. velocity w, say in the counterclockwise sense. If Qo is the position of the point at the time i = 0, we have QoOQ = oil, so that the projection of Q on the diameter OQa has, for the center O as origin, the abscissa 58 KINEMATICS [74. a coswt. And if P be the projection of Q on a diameter OA making with OQo the angle e, the abscissa of P will be X = a cos{u}i — e). Hence the motion of P is a simple harmonic motion for which the acceleration at unit distance from is m^ = t^^- 74. Notice that the linear velocity v = ace oi Q has along OA the component Vz= X = —aw sin(co/ — e), which is the velocity of P; and the acceleration of Q, j = ow^ along QO, has along OA the component jx = x = — aw^ cos(w/ — e) = — w^x, which is the acceleration of P. The projection of the uniform circular motion of Q on the diameter OB, perpendicular to OA, gives also a simple harmonic motion, viz. y = a sin(aj< — t) = a cos[co/ — (e + lir)\, which merely differs by Itt in phase from the motion along OA. The period of the simple harmonic motion of P along OA is (Art. 72) : T = litjix), i. e., it is equal to the time in which Q makes one revolution on the circle. The fact that this period depends only on the angular velocity and not on the radius a, i. e. on the amplitude, is expressed by saying that simple harmonic motions of the same ^ or w are isochronous. If Q describes the circle p times per second so that P makes p com- plete (forth and back) oscillations per second, we have w = 27rp, so that T = 1/p; i. e. the number of oscillations per second, the so-called frequency, is the reciprocal of the period. 75. Exercises. (1) Integrate the equation x = — \iH, by multiplying it by x, and determine the constants of integration if x = xo, w = Vq for t = 0. (2) Show that the period T can be expressed in the form li^V — x/x; also find the velocity in terms of x. 77.] CURVILINEAR MOTION OF A POINT 59 (d) Compound harmonic motion. 76. Apart from the initial conditions, a simple harmonic motion is fully determined by its line I, its center 0, and its period (or frequency) , which determines the constant fx. The amplitude a and the phase e depend on the initial conditions (see Art. 72). Let a point P have a simple harmonic motion of period T = 2t/ijl along a line I, about the center 0; and let the line I have a motion of rectilinear translation in a fixed plane TT (comp. Art. 38). If the motion of I is likewise a simple harmonic motion, al^out as center, in a direction U, the absolute motion of P in the plane tt is called a compound harmonic motion. This is in general a curvilinear motion; but it becomes rectilinear when the direction V is parallel to I. We proceed to examine in some detail the most important cases of this composition of two or more simple harmonic motions, beginning with those cases in which the resultant motion is rectilinear. As, according to Hooke's law, the particles of elastic bodies, after release from strain within the elastic limits, perform small oscillations for which the acceleration is pro- portional to the displacement from a middle position, the motions under discussion find a wide application in the theories of elasticity, sound, light, and electricity, and form the basis of the general theory of wave motion in an elastic medium. 77. Two simple harmonic motions in the same line, of equal period T, hut differing in amplitude and phase, compound into a single simple harmonic motion in the same line and of the same period. For, by Art. 72, the component displacements can be written 60 KINEMATICS [78. Xi = ai COs(co/ + ei), Xo = 02 COs(cof + €2), and being in the same line they can be added algebraically, giving the resultant displacement X = Xi -\- X2 = Qi cos(coi + ei) + a2 COs(w^ + eo) = (tti cosei + 02 COS62) coscof — (ai sinei + 02 sin€2) sinwi. Putting (comp. Art. 72) ai costi + «2 cose2 = a cose, ai sinei + 02 sineo = a sine, we have r-- X = a cose coscof — sine smut = a cos(coi + e), where a^ = (ai cosei + a2 cose2)^ + (oi sinei + 02 sine2)^ = ai^ + fl2" + 2aia2 cos(e£ — ei) and tti sine] + a2 sine2 tane = ai cosei + Go cose2 78. A geometrical illustration of the preceding proposition is ob- tained by considering the uniform circular motions corresponding to the two simple harmonic motions (Fig. 22). Fig. 22. Drawing the radii OPi = oi, OP2 = (h so as to include an angle equal to the difference of phase £2 — ei and completing the parallelo- gram OP1PP2, it appears from the figure that the diagonal OP of this parallelogram represents the resulting amplitude a. 80. CURVILINEAR MOTION OF A POINT 61 As PiP is equal and parallel to OP 2, we have for the projections on any axis Ox the relation OPx^, + OPx^ = OP , or Xi + 0:2 = x. If now the axis Ox be drawn so as to make the angle xOPi equal to the epoch- angle €1, and hence xOPi = €2, the angle xOP represents the epoch e of the resulting motion. We thus have a simple geometrical construction for the elements a, e of the resulting motion from the elements ai, ei and a^, ez of the com- ponent motions. As the period is the same for the two component motions, the points Pi and P2 describe their respective circles with equal angular velocity so that the parallelogram OP1PP2 does not change its form in the course of the motion. 79. The construction given in the preceding article can be de- scribed briefly by saying that two simple harmonic motions of equal period in the same line are compounded by geometrically adding their amplitudes, it being understood that the phase-angles determine the directions in which the amplitudes are to be drawn. Analytically, this appears of course directly from the formulse of Art. 77. It follows at once that not only two, but any number of simple har- monic motions, of equal period in the same line, can be compounded by geometric addition of their amplitudes into a single simple harmonic mo- tion in the same line and of the same period. Conversely, any given simple harmonic motion can be resolved into two or more components in the same line and of the same period. 80. Exercises. (1) Find the resultant of three simple harmonic motions in the same line, and all of period T = 12 seconds, the amplitudes being 5, 3, and 4 cm., and the phase dififerences 30° and 60°, respectively, between the first and second, and the first and third motions. (2) If in the proposition of Art. 77 the amplitudes are equal, ai = 02 = a, while the phase-angles differ by e2 — ei = 5, show that the re- sulting motion has the amplitude 2a cos|5 and the phase-angle \5: (a) directly, (6) from the formula; of Art. 77, (c) by the geometric method of Art. 78. (3) Find the resultant of two simple harmonic; motions in the same line and of equal period when the amplitudes are equal and the phases differ: (a) by an even multiple of w, (b) by an odd multiple of tt. (4) Resolve a; = 10 cos{o)t + 45°) info two components in the same 62 KINEMATICS [81. line with a phase difference of 30°, one of the components having the epoch 0. (5) Trace the curves representing the component motions as well as the resultant motion in Ex. (1), taking the time as abscissa and the displacement as ordinate. (6) Show that the resultant of 7i simple harmonic motions of ec[ual period T in the same line, viz. Xi = Oi cos f *^i + «' ) , is the isochronous simple harmonic motion X = o cos f "^ < 4- e j, where " 2^ai smei o^ = f ^a; COSei ) + ( YL^^ ^i^^^' ) ' ^ane ^aiCOSe; 81. The composition of two or more simple harmonic mo- tions in the same line can readily be effected, even when the components differ in period. But the resultant motion is in general not simple har7nonic. Thus, with two components Xi = tti cos(coi^ + ei), X2 = a2 cos(co2^ + €2), putting oo2t -{- eo — coit -{- (u2 — o:i)t -{- €2 = wit + ei -j- 8, say, where 5 = (co2 — u)i)t -\- eo — ei is the difference of phase at the time t, we have for the resulting motion X = Xi -{- X2 = ai cos(a)ii + ei) + a2 COs(a;i^ + ei + 5) ; = (oi + 02 cos8) cos(aji^ + ei) — 02 sinS sin(a)ii + ei), or putting ai + 02 cos5 = a cose, 02 sin5 = a sine: X = a COs(coi^ + ei + e), where 02 sinS Oi^ + a2- + 2aia2 cos5, tane «! + 02 cos5' 5 = (coo — coi)^ -|- eo — ei. 82.] CURVILINEAR MOTION OF A POINT 63 It can be shown that this represents a simple harmonic motion only when C02 = ^ o}\. The formulae can be interpreted geometrically by Fig. 22 as in Art. 78. But as in the present case the angle 5, and consequently the quantities a and e in the expression for x, vary with the time, the parallelogram OP1PP2 while having constant sides has variable angles and changes its form in the course of the motion. (e) Wave motion. 82. To show the connection of the present subject with the theory of wave motion, imagine a flexible cord AB oi which one end B is fixed, while the other A is given a sudden Fig. 23. jerk or transverse motion from A to C and back through A to D, etc. (Fig. 23). The displacement given to A will, so to speak, run along the cord, travelling from A to B and producing a wave, while any particular point of the cord 64 KINEMATICS [83. has approximately a rectilinear motion at right angles to AB. The figure exhibits the successive stages of the motion up to the time when a complete wave A'K has been produced. The distance A'K = \ is called the length of the wave. Let T be the time in which the motion spreads from A' to K, that is, the time of a complete vibration of the point A, from A to C, back to D, and back again to A ; then T is called the velocity of propagation of the wave. 83. Suppose now that the vibration of ^ is a simple har- monic motion, say y = a smcot. As the time of vibration of A is T we must have w = 27r/T', and hence 27r CO = — V . X If we assume that the vibrations of the successive points of the cord differ from the motion of A only in phase, the dis- placements of all points of the cord at any time t can be represented by y = a sm{wt — e), where e varies from to 27r as we pass from A' to K. If we further assume that the phase-angle e of any point of the cord is proportional to the distance x of the point from A' we have e = kx, or since e = 27r for x = X: 2t e = -x. Substituting the values of co and e we find 2x y = a sm ^ (Vt - x) (9) The assumptions here made can be regarded as roughly 85.] CURVILINEAR MOTION OF A POINT 65 suggested by the experiment of Art. 82 or similar observa- tions. The motion represented by the final equation (9) may be called simple harmonic wave motion. 84. To understand the full meaning of the equation (9) it should be observed that, as (in accordance with the assump- tions of Art. 83) the quantities a, X, V are regarded as constant, the displacement ?/ is a function of the two variables t and X. If t be given a particular value h, equation (9) represents the displacements of all points of the cord at the time h. The substitution for x oi x -\- n\, where n is any positive or negative integer, changes the angle (2x/X) {Vt — x) by 2x71 and hence leaves y unchanged. This means that the dis- placements of all points whose distances from A differ by whole wave-lengths are the same; in other words, the state of motion at any instant is given by a series of equal waves. If, on the other hand, we assign a particular value Xi to x and let t vary, the equation represents the rectilinear vibra- tion of the point whose abscissa is Xi. By substituting for t the value t + nT = t -^ nK/V, the angle (27r/X)(F^ - x) is again changed by 2Tn, so that y remains unchanged. This shows the periodicity of the motion of any point. 85. It may be well to state once more, and as briefly as possible, the fundamental assumptions that underlie the important formula (9); The idea of simple harmonic wave motion implies that the dis- placement y should be a periodic function of x and t such as to fulfil the following conditions: y must assume the same value (a) when x is changed to x + nX, (h) when t is changed to t + nT, (c) when both changes are made simultaneously; the constants X and T being con- nected by the relation X = VT. The condition {c) rociuires y to be of the form y = f{Vt — x); for Vt — X remains unchanged when x is replaced by x + nX and at the same time ^ by ^ + nT. 6 66 KINEMATICS [86. A particular case of such a function is y = a sinc(T'/ — x). As y should remain unchanged when t is replaced by I + T, we must have c = 2ir/FT = 27r/X. Thus the function y = a sin "";^ {Vt — x) fulfils the three conditions (a), (b), (c). Putting 2irxJ\ = — 6 we have y = a sinl'^^ t +eY The importance of this particular solution of our problem lies in the fact that, according to Fourier's theorem, any single-valued periodic function of period T can be expanded, between definite limits of the variable, in a series of the form: fit) = ao + ai sin (-^-t + eij + 02 sin f ^ ■ 2/ -|- €2 j + Ui sin (-1, ■ Zt + i3 ) + ■■ ■ . As applied to the theory of wave motion this means that any wave motion, however complex, can be regarded as made up of a series of superposed simple harmonic wave motions of periods T, \T, \T, . . ., or since T = X/V, of wave-lengths X, IX, |X, .... For, if the point A (Fig. 23) be subjected simultaneously to more than one simple harmonic motion, the displacements resulting from each can be added algebraically, thus forming a compound wave which can readily be traced by first tracing the component waves and then adding their ordinates. The motion due to the superposition of two or more simple harmonic waves may be called compound harmonic irave motion. 86. Exercises. (1) Trace the wave produced by the superposition of two simple harmonic wave motions in the same line of equal amplitudes, the periods being as 2 : 1, (a) when they do not differ in phase, (6) when their epochs differ bj^ 7/16 of the period. (2) In the problem of Art. 81, determine the maximum and mini- mum of the resulting amplitude a and show that the number of maxima 87.] CURVILINEAR MOTION OF A POINT 67 per second is equal to the difference of the number of vibrations per second. (f) Curvilinear compound harmonic motion. 87. An important and typical case is the motion of a point P whose acceleration j is directed toward a fixed center y !\ ^ Py ^^ VP ^ Tj^ J i> X Px Fig. 24. and proportional to the distance OF = r from this center (Fig. 24). If the initial velocity is =1= and does not happen to pass through the center 0, the motion is curvilinear. But it is confined to the plane determined by the center and the initial velocity since the acceleration j = yih hcs in this plane. Taking the center as origin and any rectangular axes Ox, Oy in this plane, we have for the direction cosines of OP: x/r, y/r, for those of the acceleration : — x/r, — y/r, so that the equations of motion are x = n'x, y fj^^y- These equations show that the projections Px, Py of P on the axes have each a simple harmonic motion, of the same center and period. The motion of P is the absolute motion of a point having a simple harmonic motion of period 27r/jLt 68 KINEMATICS [88. along the axis Ox, about 0, while this axis itself has a simple harmonic motion of the same period about along the axis Oy. Each of the two equations is readily integrated, and by eliminating t it is found that the path is an ellipse, with as center. See Arts. 298-302. 88. To corn-pound any number of simple harmonic motions not in the same line observe that the projection of a simple harmonic motion on any hne is again a simple harmonic motion of the same period and phase and with an amplitude equal to the projection of the original amplitude. For the sake of simplicity we confine ourselves to the case of motions in the same plane and with the same center 0. Projecting all the simple harmonic motions on two rectangular axes Ox, Oy, we can, by Arts. 77, 79, compound the components in each axis; it then only re- mains to find the resultant of the two motions along Ox and Oy. 89. Just as in Arts. 77, 81, we must distinguish two cases: (o) When the given motions have all the same period, and (6) when they have not. In the former case, by Art. 77, the two components along Ox and Oy will have equal periods, i. e. they will be of the form X = a coscot, y = b cos(wt + 5). The path of the resulting motion is obtained by eliminating t be- tween these equations. We have cosut cos5 — sinut sin6 ^-cos5- Jl-^^ a \ a^ Writing this equation in the form -?- C085 + f, = sin^S, (10) ab b^ 11 cos^S / sinS \2 see that it represents an ellipse (since ^ ^ ' 1,2 ~ 2i,f ~ \ fT ) 91.] CURVILINEAR MOTION OF A POINT 69 is positive) whose center is at the origin. The resultant motion is therefore called elliptic harmonic motion. We have thus the general result that any number of simple harmonic motions of the same -period and in the same plane, whatever may be their directions, amplitudes, and phases, cornpoiutd into a single elliptic har- monic motion. 90. A few particular cases may be noticed. The equation (10) will represent a (double) straight line, and hence the elliptic vibration will degenerate into a simple harmonic vibration, whenever sin^S = 0, i. e. when 8 = nir, where n is a positive or negative integer. In this case cos5 is + 1 or —1, and (10) reduces to and to '^ = 0, if 5 = 27mr, - + ■' = 0, if 5 = (2/« + 1)^. a b Thus two rectangular vibrations of the same period compound into a simple harmonic vibration when they differ in phase by an integral multiple of it, that is when one lags behind the other by half a wave- length. Again, the ellipse (10) reduces to a circle only when cosS = 0, i. e. 6 = {2m. + l)7r/2, and in addition a = b, the co-ordinates being as- sumed orthogonal. Thus two rectangular vibrations of equal period and amplitude com- pound into a circular vibration if they differ in phase by 7r/2, i. e. if one lags behind the other by a quarter of a wave-length. This circular harmonic motion is evidently nothing but uniform motion in a circle; and we have seen in Art. 73 that, conversely, uniform circular motion can be resolved into two rectangular simple harmonic vibrations of equal period and amplitude, but differing in phase by ir/2. 91. It remains to consider the case when the given simple harmonic motions do not all have the same period. It follows from Art. 81 that in this case, if we again project the given motions on two rec- tangular axes Ox, Oy, the resulting motions along Ox, Oy are in general not simple harmonic. The elimination of t between the expressions for x and y may present difficulties. But, of course, the curve can always be traced by points, graphically. 70 KINEMATICS [92. We shall here consider only the case when the motions along Ox and Oy are simple harmonic. 92. If two simple harmonic motions along the rectangular directions Ox, Oy, viz.: X = ai cos(wi< + ei), y = 02 cos(co2< + €2), of different amplitudes, phases, and periods are to be compounded, the resulting motion will be confined within a rectangle whose sides are 2ai, 2a;, since these are the maximum values of 2x and 2y. The path of the moving point will be a closed curve only when the quotient C02/C01 = Ti/T-, is a rational number, say = 7n/?i, where m is prime to n. The x co-ordinate of the curve will have m maxima, the y co-ordinate n, and the whole curve will be traversed after m vibrations along Ox and n along Oy. The formation of the resulting curve will best be understood from the following example. 93. Let fli = a2 = a, «i = 0, €2 = 5, and let the ratio of the periods be T1IT2 = 2/1. The equations of the component simple harmonic vibrations are X = a coswt, y = a cos(2a)i -\- 8). Here it is easy to eliminate t. We have y = a cos2co< cos5 — a sm2wt sinS 2 ' , — 1 ) cos5 — 2a - \1 — - , sinS. a^ J a \ a' Hence the equation of the path is: ay = (2z' — a^) cos5 — 2x Va- — x^ sinS. If there be no difference of phase between the components, i. e. if 5=0, this reduces to the equation of a parabola: x^ = ia(y -f- a). For 5 = ir/2, the equation also assumes a simple form: (j2y2 — 4^2 (a2 _ 2;2)_ 94. It is instructive to trace the resulting curves for a given ratio of periods and for a series of successive differences of phase {Lissajous's Curves) . Thus in Fig. 25, the curve for TJTo = H, and for a phase differ- ence 5 = is the heavily drawn curve, while the dotted curve repre- 93. CURVILINEAR MOTION OF A POINT 71 sents the path for the same ratio of the periods when the phase difference is one-twelfth of the smaller period. The equations of the components are for the heavy cm-ve X = D cos /, y = o cos ~~.~ t, and for the dotted curve „ / 2x 27r V 2x X = 6 cos ( ^ / + ^.^ 1 , 2/ = 5 cos — /. In tracing these curves, imagine the simple harmonic motions re- placed by the corresponding uniform circular motions (Fig. 25). E^ / y ^ ^ .^--' ---,,^ ^::>^;:^ >< / \ '"^•v,^ ^^' ^>< X i \ JSvI ^xT^ 0.-^" \ X X I / ><^ "\. y\ i \ / ,.'•'' '""*^,^ ^x^ X / X, ~^---, ^>-' to a 0-derivative since r, and hence u, is now a given function of 6. As du/dr = .. . _ c^ J. 2 _ ^^^" ^^^ — 4j1^ ^^ ^^ •'^^^ ~ ~ dr^^' ~ ~ ~dd dr~ ~ dd dudr _ ,^dd d ^ ^ _ ^ ^dd / du d'hi du ^ "" dadd'^'^ ~ ^'''* dM\dddd'^ ""dd- hence Kr) = c^w^(^^ + t.). (16) This important relation can also be obtained directly from the equations of motion in polar co-ordinates which are (see Art. 55) f - rd^ = - f{r), ^ 4r(r4) = 0. For, with r — 1/w we have since the second equation gives (11): dr ■ c dr du .. dhi : . Su ' = de^^r^de^-'W '=-'de'^^ = -'''de-^' substituting these values in the first equation we find (16). 107. Kepler in his second law had cstal)lished the empirical fact that the orbits of the planets are ellipses, with the sun at one of the foci. 108.] CURVILINEAR MOTION OF A POINT 79 From this Newton concluded that the law of acceleration must be that of the inverse square of the distance from the sun. Our equation (16) enables us to draw this conclusion. The polar equation of an ellipse referred to focus and major axis is r = -— -, I.e. u=Y + rCosd, 1 + e cos^ I I where I = h^/a = a(l — e^); a, b being the semi-axes, I the semi-latus rectum, and e the eccentricity. Hence d~u e „ dhi , 1 . so that we find p2, pi 1 108. The third law of Kepler, found by him likewise as an empirical fact, asserts that the squares of the periodic times of different -planets are as the cubes of the major axes of Iheir orbits. From this fact Newton drew the conclusion that in the law of acceleration, the constant n has the same value for all the planets. Our formulse show this as follows. Let T be the periodic time of any planet, i. e. the time of describing an ellipse whose semi-axes are a, b. Then, since the sector described in the time T is the area irab of the whole ellipse, we have by Art. 99 -wab = icT. Substituting in (17) the value of c found from this equation we have 80 KINEMATICS [109. Hence is constant by Kepler's third law, 109. Planetary motion in its simplest form is that particular case of aentral motion in which the acceleration is inversely proportional to the square of the distance from the center so that where m is a constant, viz., the acceleration at the distance r = 1 from 0. The equations of motion (12) are in this case, with O as origin, df= - ^r^' d^^ ~''r'y (^^) Combining these by the principle of energy (Arts. 103, 104), we find Integrating the differential equation d^v- = — (jj.lr'^)dr we find lj,2_^j,.2=if _if . (19) r ro 110. To find the equation of the path, or orbit, write the equations (IS) in the form X = — -„ cos9, ij = — „ sin9 r- r- and eliminate r^ by means of (11): X = - - cose -e, y = - - sine • d. c c Each of these equations can be integrated by itself: X -V, = - :^ sine, ij - V, =^ (cose - 1), (20) where vi, V2 are the components of the velocity when e = 0. 112.] CURVILINEAR MOTION OF A POINT 81 Multiplying by y, x, subtracting, and integrating, we find by Art. 102: i~-Vi\x + Viy-\-c=-~{x cose + y amd) = ~ Vx^ + y"". (21) 111. The geometrical meaning of this equation is that the radius vector r = Vx"^ + y~ drawn from the fixed point to the moving point P is proportional to the distance of P from the fixed straight line {j-^ - v^Yx + viy + c = Q. (22) It represents, therefore, a conic section having O for a focus and the line (22) for the corresponding directrix. The character of the conic depends on the absolute value of the ratio of the radius vector to the distance from the directrix; according as this ratio .S'(^ --)'+».'. is < 1. = 1, or > 1, the conic will be an ellpise, a parabola, or a hyper- bola. This criterion can be simplified. Multiplying by njc and squaring, we have or since v^ + vn} — vi and c = nvo sin>/'o = rav^: t;o2^-''.1 (23) 112^ If polar co-ordinates be introduced in (21), the equation of the orbit assumes the form 1 =M + fL^_ii')cos0--'6in5, T & \ C C-' ) C or putting {cv^ — ii)lc^ = C cosa, th/c = C sina, 1 = -^ + C cos(9 + a). (24) r c^ This equation might have been obtained dhectly by integrating (16), which in our case, with/(r-) = iJi/r-, reduces to ^^ 1 , 1 ^ M . dm r^ r c^ ' 82 KINEMATICS (113. the general integral of this differential equation is of the focm (24), C and a being the constants of integration. Equation (24) represents a conic section referred to the focus as origin and a line making an angle a. with the focal axis as polar axis. 113. Exercises. (1) A point moves in a circle; if the acceleration be constant in direc- tion, what is its magnitude? (2) A point describes a circle; if the acceleration be constantly directed towards the center, what is its magnitude? (3) A point has a central acceleration proportional to the distance from the center and directed away from the center; find the equation of the path. (4) A point P is subject to two accelerations, ^^ - 0\P directed toward the fixed point 0\, and ^t^ o^P directed awaj^ from the fixed point O2. Show that its path is a parabola. (5) A point P describes an ellipse owing to a central acceleration ^('■) = i"/^^ directed toward the focus S. Its initial velocity i^o makes an angle yp^ with the initial radius vector Td. Determine the semi-axes a, h of the ellipse in magnitude and position. (6) Find the law of acceleration when the equation of the orbit is j.n = 5''/(l 4- e cosnO), e being positive, and investigate the particular cases n = I, n = 2, n = — 1, 71 = — 2. (7) Find the law of the central acceleration directed to the origin under whose action a point will describe the following curves: (a) the spiral of Archimedes r = ad; (6) the hyperbolic spiral Or = a; (c) the logarithmic or equiangular spiral r = ae"^; (d) the curve r = a cosnd. (8) A point moves in a circle and has its acceleration directed towards a point on the circumference. Find the law of acceleration. (9) The acceleration of a point is perpendicular to a given plane and inversely proportional to the cube of the distance from the plane. Determine its motion. (10) A point moves in a semi-ellipse with an acceleration perpen- dicular to the axis joining the ends of the semi-ellipse. Determine the law of acceleration and the velocity. CHAPTER IV. VELOCITIES IN THE RIGID BODY. 1. Geometrical discussion. 114. The velocities of the various points of a rigid body, at any instant, are in general different, both in magnitude and direction; i. e. they are different vectors; but they are not independent of each other In particular, it is clearly possible (i. e. compatible with the rigidity of the body) that the velocities of all points, at the given instant, are equal vectors. The instantaneous state of motion of the body is then called a translation; it is fully determined by the velocity vector of an}^ one point of the body, and this is called the velocity of translatio7i, or linear velocity, of the body. Comp. Art. 30. The ideas of absolute and relative velocity and of composi- tion and resolution of velocities apply to the velocity of trans- lation of a rigid l^ody just as they apply to the linear velocity of a point (comp. Arts. 38, 40, 41). 115. As the position of a rigid body is fully determined by the positions of any three of its points, Oi, O2, O3, not in a straight line, it is clear that if any three such points have zero velocity at any instant, all points of the body must have zero velocity at that instant. The body is then said to be instantaneously at rest. It may also be regarded as geometrically obvious that if any two points 0^, O2 of a rigid body have zero velocity, all points of the line I joining Oi and Oj must have zero velocity and hence (unless all points of tlie ])ody have zero velocity) 83 84 KINEMATICS [il6. the velocity of every point P of the body is normal to the plane {I, P) and proportional to the distance of P from I (comp. Art. 31). The instantaneous state of motion of the body is then called a rotation; the line I is called the instan- taneous axis of rotation; and the common factor of propor- tionality CO of the velocities is called the angular velocity. It is convenient to think of the rotation as represented geometrically by a vector of length w, laid off on the axis of rotation /, in a sense such that the rotation appears counter- clockwise as seen from the arrowhead of the vector (Fig. 5, Art. 31). Such a vector confined to a definite straight line is called a localized vector, or rotor. The rotor co fully char- acterizes the instantaneous state of motion of the body since the velocity of every point of the body can be found from it as we shall see in Art. 118. 116. The instantaneous state of motion of a rigid body one of whose points is fixed, if not a state of rest, is a rotation. For, it can be shown that if one point of the body has zero velocity there exists a line I through all of whose points have zero velocity. An analytical proof is given in Art. 128. Geometrically the proposition can be proved as follows. Observe first that in any motion of a rigid line the velocities of all points of the line must have equal projections on the line; this follows directly from the rigidity of the line. Hence if the velocity of any point of the line is normal to the line or zero the velocities of all points of the line must be either normal to the line or zero. Now consider a rigid body of which one point has zero velocity, and let Pi, P^ be any tw^o points of the body, not in line with 0. The velocities of Pi, P2 must be normal to OPi, OP2, respectively. If the velocity of either of these points were zero, the line joining this point to would be 117.] VELOCITIES IN THE RIGID BODY 85 the required axis of rotation. We assume therefore that these velocities are both different from zero. We can also assume that these velocities are not parallel; for if they happened to be so we could replace one of the two points by a point whose velocity is not parallel to those of Pj and P^] otherwise the motion would be a translation which is im- possible for a body with a fixed point. It follows that the planes through Pi, Po, normal respec- tively to the velocities of Pi, P2, must intersect in a line I which of course passes through 0; this line I is the axis of rotation. For, any point P of I must have a velocity normal to PO, and at the same time normal to both PPi and PP2; this means that the velocity of P is zero. 117. Composition of intersecting rotors. A rigid body C may have, at a given instant, an angular velocity w, about an axis li, while the body of reference B to which li belongs rotates at the same instant with angular velocity C02 about an axis h belonging to a fixed body A. We then say that, with respect to A, the body C has the simultaneous angular velocities wi about U and coo about h. If the axes U, k intersect, say at 0, the instantaneous motion of C with respect to A is a rotation about an axis I passing through such that sinlj. _ smlh _ sinZiZ2 002 COi CO with an angular velocity CO = VcOi^ + C02^ -|- 2c0lC02 COSZ1Z2. This proposition, known as the parallelogram of angular velocities, means simply that two simultaneous angular velocities coi, coo, al)out intersecting axes h, k, are together y r- / / .0^ / _ / 86 KINEMATICS [118. equivalent to a single angular velocity co about I, whose rotor CO is the geometric sum of the rotors coi, a;2 (Fig. 27). The proof is as follows. The linear velocity of any point P of the body has two components, coiri and co2r2, where ri, r2 are the perpendiculars let fall from P on the axes h, lo. These components lie in the same line only for the points of the plane (^i, k) ; and they are equal and opposite only for the points on the di- agonal of the parallelogram constructed on co], co2 as sides. All the points of this diagonal "■ ■ having the velocity zero, this line is the axis of rotation. The above equations follow at once from the parallelogram construction. The proposition is readily extended to the composition of three or more angular velocities about axes passing through the same point. Conversely, the proposition is used to resolve a rotor oj along lines through any point of its line I. The resolution of co along three rectangular lines through such a point into the components co^, coy, Wz, so that co"^ = cox^ + <^y'^ + ^z^, is used very often. 118. In the case of rotation, of angular velocity co, about an axis I, if the motion be referred to rectangular axes, with the origin on I, we can find the components Vx, Vy, Vz of the velocity V of any point P{x, y, z) of the body, by replacing oo by its com- ponents co^, coy, Wz (Art. 117). It then follows from Art. 48, Ex. 1, that Ux produces at P a velocity whose components are 0, — WxZ, coxy, similarly, coy gives the components UyZ, 119. VELOCITIES IN THE RIGID BODY 87 0, — coyo;; and co, gives — ui,y,w,x, 0. Adding the components having the same direction, we find WijZ — co^y, Vy = oiz^ — <^xZ, v. <^xy (jOyX. 119. By Art. 115, the velocity y of P (Fig. 28) is normal to the plane (I, P) and equal to wCP, where C is the foot of the perpendicular let fall from P on I. Putting OP - r, 4 COP = (/>, so that CP = r sinyi -f- 6-32:1. Notice m particular that if the motion of the body is a translation, the direction cosines of the moving axes arc con- stant so that di, • • • C3 arc zero; all points of the body have 94 KINEMATICS [127. then the same velocity (i;o, yo, zq). Again if the point Oi of the body is fixed, Xq, yo, Zq are zero, and the velocity com- ponents are linear liomogeneous functions of Xi, yi, Zi. 127. The velocity of any point P of the bod}- relative to the ijoint Oi has along the fixed axes the components: X — ±0 = diXi + d2?/i + ('s^i, y - yo = fei.Ti + JMji + 6321, i — io = ciXi + c-iiji + i'zZi. To find the components along the moving axes of this same relative velocity of P we have only to project the components along the fixed axes on the moving axes, which is readily done by means of the scheme of direction cosines in Art. 124. The resulting expressions (aidi + 6161 + CiCi)xi + {aid2 + biL + CiCo)?/! + (aids + 61&3 + CiCsjZi, (Oidi + 62^1 + C2Ci)xi + (0002 + 62^2 + c-2C2)yi + (fl2d3 + &2^3 + CoCs)^!, (Osdl + fesi*! + C3Cl)Xl + (0302 + hh + CsCo)//! + («3d3 + bshs + CsCs)^! can be simplified very much l)y means of the identities (2) which give upon differentiation with respect to t: difli + bihi + CiCi = 0, 0202 + ^2^2 + <^2C2 = 0, d3a3 + hsbs + C3C3 = 0, .^^ d2a3 + 62^3 + C2C3 = — ,(o2d3 + hobs + C2C3), dsOi + ^3^1 + (Vi = — («3di + bzbi + C3C1), dia2 + ^1^2 + C1C2 = — (aid2 + 6162 + C1C2). Denoting, for the sake of brevity, the left-hand members of the last three equations by coi, un, 0^3 (we shall find very soon that these are precisely the components along the moving 129.] VELOCITIES IN THE RIGID BODY 95 axes of the rotor w) we find for the cotnjjonents along the moving axes of the velocity of P relative to Oi, the simple ex- pressions iC'zZi — wziji, cosXi — wiZi, coiiji — 0022:1, which agree (considering our present notation) with the values found in Art. 118. 128. The locus of those points of the body whose velocity relative to Oi is zero is given by cooSi — CO32/1 = 0, cosXi — oji^i = 0, a)i2/i — co2a;i = 0, i. e. by Xi ^ yi ^ Zi COi CO2 0)3 This is a straight line I through Oi whose direction cosines are proportional to coi, W2, 003. Hence the motion of the body relative to Oi is a rotation about the line I. To see that ui, 0)2, C03 are the angular velocities about the axes OiXi, Oiyi, OiZ\, respectively, take Oi as origin and the line I as axis OiZi] then the velocity of any point in the Xiyi- plane has the components — (^^yi, oo^Xi, 0; i. e. (Art. 48, Ex. 1) W3 is the angular velocity about OiZi; similarly for coi, Wo. Cornp. Art. 118. By Art. 117, the three angular velocities coi, aj2, W3 about OiXi, Oiyi, OiZi are together equivalent to the single angular velocity a? = Vcoi^ + C02- + cos^ about the line through Oi whose direction cosines are proportional to coi, C02, C03. 129. If, as in Art. 120, we denote by u the velocity of the point Oi and l)y Wi, Vo, V3 its components along the moving axes, we have for the components Vi, V2, Vz of the absolute velocity of any point P (xi, 7/1, Z}) of the body along the moving axes: 96 KINEMATICS 1 129. Vi = Ui + U2Z1 — a)3?/i, Vo = U2 -jr 0)3^:1 — wi2i^ . (5) vz = lis + carji — 0:2X1; or in vector notation V = u -{- CO X r. On the other hand, referring the motion to the fixed axes Ox, Oy, Oz, let Xo, jjo, Zo be (as in Arts. 125, 126) the co- ordinates with respect to these axes of any point 0] of the body, and let coi, o)y, Uz be the components of oj along the fixed axes; then the components of the absolute velocity of any point P{x, y, z) of the body along the fixed axes are X = Xo + coy{z - Zo) - o},{y - yo), y = yo -\- o)z{x — Xo) — o)x{z — Zo), i = io + ojx(.y — t/o) — o:y{x — Xo). If in these formuljfi we put a; = 0, ?/ = 0, 2; = we obtain the components Ux, Uy, Uz (along the fixed axes) of the velocity of the origin 0, regarded as a point of the moving body, viz. Ux = Xo — C0y2o + Oizyo, Uy = yo — 0)zXo + WxZo, Uz ^ Zo — coxyo + Wy.ro. B}^ introducing these components in the preceding formula? we obtain for the components of the velocity v of any point P {x, y, z) of the body along the fixed axes the simple expressions Vx = X = Ux -\- ooyZ — cjzy, Vy — y = Uy + OJzX — COxZ, (6) Vz. = Z = Uz + 0>xy — OJyX. Thus the velocity v is the resultant of the velocity u of {i. e. of that point of the rigid body which at the instant considered happens to coincide wnth the fixed origin 0) and of the linear velocity arising from the rotation of angular 131.] VELOCITIES IN THE RIGID BODY 97 velocity co about the line through parallel to the instan- taneous axis. The equations (5) and (6) are of exactly the same form; each of these sets of equations is equivalent to the vector relation V = u -]r uXr] (5) arises by projecting on the moving axes, (6) by projecting on the fixed axes. 130. We have seen (Art. 123) that the instantaneous state of motion of a rigid bod}' is in general a twist about the central axis (in the exceptional case of translation this line lies at infinity). In the course of the motion the central axis changes its position both in space {i. e. relatively to the fixed trihedral Oxyz) and in the body (relatively to the moving trihedral 0\Xiy\Z-^. If the motion is continuous, the succes- sive positions of the central axis in space will be the generators of a ruled surface S fixed in space; and the successive posi- tions of the central axis in the body, i. e. the various lines of the body which in the course of time become central axes, will be the generators of a ruled surface Si, fixed in the body and moving with it. At any given instant these surfaces S and Si have the central axis corresponding to this instant in common; it can be shown that they are in contact along this common gen- erator, so that the motion consists in a rotation about, and a sliding along, this generator. Two particular cases, that of the body with a fixed point and that of plane motion, deserve special mention. 131. Body with a fixed point. As the fixed point has zero velocity the central axis is the instantaneous axis at 0, and the velocity of translation is zero. The surfaces S, Si 8 98 KINEMATICS [132. are cones with as common vertex; and the motion can be shown to consist in the roUing of »Si over *S., This motion will be studied more fully in Chapter XVIII. 3. Plane motion. 132. If the velocities of all points of a rigid Vjody remain parallel to a fixed plane, the motion of the body is fully determined by the motion of the cross-section made by the bodj^ in this plane. This case might he regarded as the limiting case of the motion of a body with a fixed point as this point is removed to infinity. But it is more in- structive to study it directly. Taking in the plane of the motion a set of fixed axes Ox, Oy (Fig. 32) and a set of moving axes OiXi, Oiiji we have Fig. 32. if .To, i/o are the co-ordinates of Oi and d is the angle between Ox and Oi^r. X = Xq -\- Xi cos9 — iji sin0, y = ijo + xi sin0 + 2/i cos^. 133. Differentiating with respect to t we find for the components along the fixed axes of the velocity v of P{xi, yi) (xi smd + yi cos^)^ = Xo — w{y — yo), (7) Xq y = i/o -\- (xi COS0 — yi sin0)^ = 2/o + w(x — Xo), (8) 134.] VELOCITIES IN THE RIGID BODY 99 where w = d. The velocity of Oi has the components Xq, i/o; the velocity of P relative to Oi has the components — oo(y — yo), ui{x — Xo), i. e. it can be regarded as due to a rotation of angular velocity co about Oi (Art. 48, Ex. 1). The instantaneous motion of the plane section of the body consists therefore of a translation of velocity u{xq, yo), equal to the velocity of Oi, and a rotation about Oi of angular velocity co = 6. Now, excluding the case of pure translation when CO = 0, we can find in the plane, at any instant, a point C of zero velocity, i. e., such that X = Xo — u){y — ?/o) = 0, 7/ = 2/0 + o){x — Xo) = 0. This point C, the intersection of the central axis with the plane, is called the instantaneous center; its co-ordinates X, y are evidently X = Xo , y = yo-{ . (9) CO CO Hence, the instantaneous state of motion, in the case of 'plane motion, is either a -pure translation or a pure rotation about the instantaneous center. It follows that (excepting the case of translation), at any instant, the velocity of every point P is normal to the radius vector CP and equal to co times CP. Conversely, if the directions of motion of any two points Pi, P^ are known, the instantaneous center C can in general be found as the intersection of the perpendiculars through Pi, P^ to these directions. 134. In ]-)lane motion the ruled surfaces aS, >Si (Art. 130) are cylind(Ts. Instead of tlicse cylinders it suffices to consider their curves of intersection, s, Si with the plane. The curve s is called the j&xed, or space, centrode (path of 100 KINEMATICS 1135. the center); the curve Si which, as will be proved in Art. 135, rolls over s is called the moving, or body, centrode. Thus any plane motion consists in the rolling of the body centrode Sx over the space centrode s (except in the case of translation). It is fully determined if, in addition to any particular position of these centrodes, the angular velocity 0} is given as a function of the time. The equation of the space centrode, referred to the fixed axes, is found by eliminating t between the equations (9). That of the body centrode, i. e. of the locus of those points of the moving figure which in the course of the motion become instantaneous centers, must be referred to the moving axes OiXi, Oiyi. Substituting in (7) for x, y the values (9) and solving for Xi, yi we find the co-ordinates Xi, y\ of the instantaneous center with respect to the moving axes: Xi = —(xo sin0 — vo cos9), 1 (10) yi = (.to COS0 + yo sin5) ; CO the elimination of t gives the body centrode. 135. To prove that, as stated in Art. 134, the body cen- trode Si rolls over the space centrode s it suffices to show that these curves have at the instantaneous center C not only a common point but a common tangent ; in other words, that the slopes m, m\ of s, si at C are equal. These slopes can be found from the equations (9) and (10). From (9) we find by differentiating with respect to / : y co.ro + i^'yo — <^Xq m = ~ = ' — X — oiijo -j- co^^o + coyo Without loss of generahty we may, at the instant considered, 136.) VELOCITIES IN THE RIGID BODY 101 let the moving axes coincide with the fixed axes and take the origin at the instantaneous center so that xo, yo, Xo, yo are zero; we then find : •To m = . yo From (10) we find similarly co(xo COS0 + ijo sin0) + co-(— Xo miO + ijo cos6) ^ ^Ul = -co(.tocosg + yosme) _ Xi co(xo sine — ijo COS0) + co'^{xo cos0 + tjo sin0) ' — 6}(xq sine — 2/0 cos^) and, taking the axes as above, since io, ijo, Q are zero: OCo nil = . 2/0 Hence w = Wi, i. e. the curves s, Si have a common tangent at the instantaneous center. It appears, moreover, that this tangent is norrnal to the acceleration of the instantaneous center. Thus, in the case of a circle rolling over a straight line, where s is the line, Si the circle, the acceleration of the point of contact is normal to the lino. It should be observed that the equations (9) are the parameter equations of the fixed centrodc, the parameter lacing t; hence the ^-derivatives .f, y, used above in forming m, are not the components of the velocity of the instantaneous center as a point of the moving figure (these velocities are zero), but those of the velocity, say w, with ivhich the instantaneous center proceeds along the curve s. Similarly the quantities Xi, yi, used in forming nii, are the components, along the moving axes, of the same velocity w. 136. This velocity w with which the instantaneous center C changes its position along the centrodes s, Si is connected 102 KINEMATICS [136. by a simple relation with the angular velocity co and the radii of curvature p, pi of s, Si at C, viz. w p pi' To prove this let C (Fig. 33) be the position of the instan- taneous center at the time /, C its position in the fixed plane Fig. 33. and Ci its position in the moving figure at the time t -\- At. Then, denoting by As, Asi the equal arcs CC\ CCi, we have as definition of w : ,. As ,. Asi ic = hm— = lim -— . A<=o At A/=o At On the other hand, if Ad is the angle through which any line of the figure turns in the time A^ we have ,. A9 dd CO = inn rr = 37 • st=oAt at The motion that takes place in the interval At carries the 137.] VELOCITIES IN THE RIGID BODY 103 point Ci to the position C and brings the normal to Si at Ci' to coincidence with the normal to s at C; these normals include therefore the angle Ad. Hence if A(y?, Atpi are the angles that these normals make with the common normal at C we have Ad = A(p — Acpi; dividing by As = Asi and passing to the limit we find for the left-hand member ,. Ad ,. AdAt 0} lim — = hm — — = - , At=o As st=o At As w provided lim As/At = w is 4= 0. In the right-hand member, the limits of AipjAs and A^fijAsi are clearly the curvatures of s and Si at C; hence CO ^ 1 _ 1^ w p pi * It is easily seen that this formula holds even when the centers of curvature lie on opposite sides of the tangent, provided we take pi then negative. The counterclockwise sense of co is taken as positive, and id is taken positive if the normal at C turns counterclockwise in passing to its new position through C 137. Exercises. (1) A plane figure moves in Us plane so that two of its points A, B (Fig. 34) move along two perpendicular straight lines Ox, Oy. By Art. 133, the instantaneous center C is found as the intersection of the perpendiculars at A to Ox and at B to Oy. As yli? is of constant length it follows readily that the space centrode is a circle of radius AB = 2a about 0. As OC = AB it follows that the body centrode is a circle of diameter OC = 2a. Hence the motion can also be brought about by the rolling of a circle of radius a within a circle of twice this radius. Taking the midpomt Oi oi AB as origin and OiA as axis OiXi of the set of moving axes, and denoting by the angle BAO, we have for the co-ordinates of any point P{x-i, y\) of the moving figure: X = {a -\- Xi) cos we find as equation of the path of P, referred to the fixed axes: f yix - (g + Xi)y Y , V ViV - (a - X i)x V ^ [{a - XiY + yi2]x2 - 402/1X2/ + [(a + x^r- + 2/i=]2/- = (xi= + 2/i' - a')'- This is an ellipse referred to its center. Show that Oi describes a circle, and that every point on the circle about AB as diameter describes a Fig. 34. straight line through 0. Show that the velocity of P is w = [o- + xx"^ + 2/1^ — 2a(x: cos2<^ + 2/1 An24i)]h4>; hence find the velocities of B and Oi when A moves uniformly. (2) A 'point A of the figure moves along a fixed straight line I ivhile a line of the figure, U, containing the point A, alioays passes through a fixed point B (Fig. 35). The fixed point B may be regarded as the limit of a circle which the line li is to touch. The instantaneous center is therefore the inter- section C of the perpendiculars erected at .A to Z and at B ioli. The fixed centrode is a parabola whose vertex is B. To prove this take the fixed line 7 as axis Oy, the perpendicular OB to it drawn through 137.] VELOCITIES IN THE RIGID BODY 105 the fixed point B as axis Ox. Then, putting OBA = 4> and OB = a, we have for C: X = a -\- y tanyZ — oizy, y = Uy + O^zUx — COxUz + 0}y{oOxX + 0}yy + OizZ) — whj -f dzX — ojxZ, Z = Uz -\- UxUy — (Jiylix + Oiz{r cos^. no KINEMATICS [141. In vector analysis, the product ah cos^ of any two vectors a, b into the cosine of the angle between them is called the dot-product (scalar or internal product) of the vectors a and h and is written briefly a ■ b (read : a dot b) . If the rectangular components of a vector are denoted by subscripts (as in Art. 119) we have a-6 = Uxbx + Oyby + 0^62. Hence in our case ajj-X + onylj + (jOzZ = oj-r. Thus the first of the three partial accelerations, ja, has along the fixed axes the components oj^wr costp, ccycor cos may be called the angular accelera- tion of the body; its components along the fixed axes are coj, (Jiy, w,. The body has therefore the infinitesimal angular Fig. 37. 112 KINEMATICS [144. velocities 6)xdt, dydt, 6i,dt about the axes Ox, Oy, Oz, respec- tively. These produce at P(x, y, z) the infinitesimal linear velocities 0, — CjjZcU, o^xydt; Uyzdt, 0, — oiyxdt; — Uzydt, 03zxdt, ; dividing by dt and collecting the terms we find the accelerations (hyZ — Ci^y, Oi^X — Wx2, 6ixy — (JiyX. which are the components of jc- 144. Plane motion. Taking the plane of the motion as rcy-plane we have to put co^ = 0, Wy = 0, co^ = co. d^ = 0, Uy = 0, ojj = w, Uz = 0; Jig = in the equations (2) of Art. 138 so that we find X = Ux — wUy — orX — CO?/, i) = Uy + coUx — CO-?/ — cox, while 2 = 0. As Ux = Xo + ojyo, Uy = yo — ojXq and hence Ux = Xq -\- on/o + co?/o, iiy = yo — wxo — wxo, we find as com- ponents of the acceleration of P(x, y) along the fixed axes: X = xo - co^ix - xo) - 6i{y - yo), (5) 7j = i/o - co2(?/ - ?/o) + (^{x - Xo). These equations are also obtained directly by differentiating the components of the velocity in plane motion, (8), Art. 133, which express that the instantaneous state of motion (unless a translation, co = 0) can be regarded as a rotation of angular velocity co about the instantaneous center (xo — 2/o/co, yo + io/co). The equations (5) show that (excepting the case of transla- tion when CO = 0, CO = 0) there exists at every instant a point /, the center of acceleration, whose acceleration is zero; its co-ordinates are co^.fo — coj/o , o}-yo + cbi'o .^s 146. ACCELERATIONS IN THE RIGID BODY 113 145. If this point I of zero acceleration be taken as origin of the moving axes 0\Xi, Oiiji (Fig. 38), the components along y > ^r ^ \ 1 1 «i^ _o ^ .^ 1 X ^ Fig. 38. the fixed axes of the acceleration of any point P(a:, y) are by (5) - by{x - x^ - cJ(2/ - ?/o), - co2(?/ - ?/o) + ^{x - a;o). The form of these expressions shows that if we put IP = r, the acceleration of P can be resolved into a component coV along PI and a component ur at right angles to IP; and the total acceleration of P is j = r Vo)^ + oj2. Hence, at any instant, all points on a circle about I as center have accelerations of equal magnitude and are equally inclined to their radii vectores r = IP; all points on a straight line through I have parallel accelerations, propor- tional to their radii vectores r = IP. 146. If any point Oi different from I be taken as origin of the moving axes (Fig. 39) we have simply to superimpose its acceleration jo(xo, yo); and it appears from (5) that the acceleration of every point P can be regarded as having the three components: 9 114 KINEMATICS ll47. jo = the acceleration of Oi, ji = oj-r along POi, jo = cjr at right angles to OiP, where r = OiP. Fig. 39. 147. If, in particular, we take as origin of the moving axes the instantaneous center C and as axis OiXi the common tangent of the centrodes (Fig. 40), the acceleration j of C Fig. 40. is normal to this tangent (Art. 135), and as CP is the normal to the path of P (Art. 133), the normal and tangential com- 148.] ACCELERATIONS IN THE RIGID BODY 115 ponents of the acceleration of P are: Jn^ oi^r - 3-, Jt = <^r+j—, (7) where r — CP. Hence the loci of the points having only tangential and only normal acceleration are the circles: o^Kxi' + yr) - hi = 0, w{xi^ + ^1^) + jxi = 0. (8) Finally, it can be shown that the acceleration of the instan- taneous center C is j = uw, where w is the velocity with which the instantaneous center travels along the centrodes (Art. 135). For, just as in Art. 135, we find by differentiating the equation (9) of Art. 133 and putting io = 0, ?/o = that the components of w along the fixed axes are X ^ , 1/ =- - . CO CO whence ^0 = — co.f, .t'o = ooy. The acceleration of C is therefore j = V.'Co^ + 2/0^ = <^ Vi" + y^ = om. (9) 148. Exercises. (1) A wheel of radius a rolls on a straight track; find the center of acceleration : (a) when the velocity v of the axis of the wheel is constant ; (b) when the axis is uniformly accelerated, as when the wheel rolls down an inclined plane. (2) Find the locus of the points of equal tangential acceleration. (3) Show that the components, along the axes Cxi, Cyi of Fig. 40, of the acceleration of any point arej'i = — co^Xi — o:y\,ji = — co^^/i + wxi + ;■; and hence the co-ordinates of I are — wj/(w< + 6r), co^j/Cw + w^). Verify that these co-ordinates satisfy the equations (8) ; this shows that the center of acceleration is the intersection (different from C) of the circles (8). 116 KINEMATICS [148. (4) Show that the resultant of j and cir in Fig. 40 is an acceleration wr', perpendicular to r' = HP, where H, the center of angular acceleration, is the intersection of the circle of no tangential acceleration (second of the equations (8)) with the common tangent of the centrodes at C; it lies at the distance CH = jjw from C. It follows that the acceleration of any point P can be resolved into two components, uh along PC and cor' normal to HP = r'. (5) The first of the circles (8) is called the circle of inflections; why? (6) Show that the diameter of the circle of inflections is the recip- rocal of the difference of the curvatures of the centrodes at their point of contact. (7) Determine the locus of the points whose acceleration at any instant is parallel: (a) to the common normal, (6) to the common tan- gent, of the centrodes. CHAPTER VI. RELATIVE MOTION. 149. In studying the motion of a point P relatively to a rigid body of reference B which is itself in motion we use, just as in Art. 124, two rectangular trihedrals, one Oxyz fixed in space, the other OiXiyiZi fixed in the body B and moving with it. The absolute co-ordinates x, y, z oi P are connected with its relative co-ordinates Xi, yi, Zi by the relations (1), Art. 125; but now not only the absolute co- ordinates X, y, z but also the relative co-ordinates Xi, yi, Zi of P are functions of the time. Hence, differentiating the equations (1), Art. 125, we find for the components, along the fixed axes, of the absolute velocity V of P: X = xo -\- diXi + (ky^ + d^Zi + aiXi + aniji + as^i, y = yo-\- hiXi + hojji + 632:1 + biXi + bojji + 63^1, (1) i = io + ciXi + 62/1 + C3Z1 + CiXi + C2^i + csii. If the point P were rigidly connected with the body B the last three terms would be zero ; hence the first four terms represent the components along the fixed axes of the so-called body-velocity Vb, i. e. the velocity of that point of the rigid body with which the point P happens to coincide at the instant considered. This also follows from the equations (3) of Art. 126. As ±1, yi, Zi are the components along the moving axes of the relative velocity Vr of P with respect to B, the last three terms of (1) are the components along the fixed axes of this 117 118 KINEMATICS 1150. same velocity Vr (comp. the scheme of direction cosines in Art. 124). Thus the equations (1) are merely the analytical expression of the vector equation V = Vb -{- Vr', i. e. the absolute velocity v of a point P is the geometric sum, or resultant, of the body-velocity Vb and the relative velocity Vr', comp. Art. 38. 150. Differentiating the equations (1) again with respect to t we find the comjionents, along the fixed axes, of the absolute acceleration j of P. X = Xo-\- ciiXi + doyi + d^Zi + 2(oiii + (yji + Osii) + aj-i + a.2yi + a^z,, y = yo-\- bxXi + 622/1 + 63^1 + 2(6i.ri + hill + 6321) ,^. + biXx + b^iji + 63^1, z = Zo + CiXi + Mji + C3Z1 + 2(^i.ri + c.iji + f'3ii) + Cii-i + cMji + CiZi. The first four terms on the right represent, by (1), Art. 138, what we may call for the sake of brevity the body-acceleration jb, i. e. the acceleration of that point of the body of reference B with which the point P happens to coincide at the instant considered. The last three terms are the components along the fixed axes of the relative acceleration jr of P whose com- ponents along the moving axes are Xi, iji, Zi, i. e. of the acceleration of P relatively to the moving body B. To interpret the middle terms, those with the factor 2, observe that by comparing Arts. 119 and 127 it appears that the velocity v of any point P of a rigid body wath a fixed point 0, which is a vector of length v = cor sin^?, perpendicular to the rotor w and the radius vector r = OP, has along the fixed axes the components 150.] RELATIVE MOTION 119 diXi + diiji + daZi, biXi + biUi + 63^1, CiXi + Ciiji + CiZi. The vector that we wish to interpret has along the fixed axes the components di-2ii + d2-2?/i + d3-2ii, 6i-2ii + h2-2yi + 63'2ii, ci-2xi + C2'2yi + C3-2ii; it differs from the preceding vector merely in having Xi, iji, Zi replaced by 2ii, 2?/i, 2ii. It represents therefore a vector of length oi-2vr sin(aj, Vr), at right angles to the rotor co and the relative velocity Vt{xi, yi, ii), drawn in a sense such that CO, Vr, and this vector form a right-handed set. More briefly we may say (Art. 119) that this acceleration jc, which is called variously compound centripetal, complementary, or acceleration of Coriolis, is twice the cross-product of the angular velocity co of the body B and the relative velocity Vr oi P: jc = 2coX Vr. Thus, the absolnte acceleration j of a point P whose motion is referred to a moving body of reference B, is the geometric sum of three accelerations, the body-acceleration jh, the com- plementary acceleration jc, and the relative acceleration jr.' (3) j = jh + jc + jr. This proposition is known as the theorem of Coriolis. Appli- cations will be given in Chap. XIX. PART II: STATICS. CHAPTER VII. MASS; DENSITY. 151. Physical bodies are distinguished from geometrical configurations by the property of possessing mass; and the way in which this property affects their motions is studied in that part of mechanics which is called dynamics. We may think of the mass, or quantity of matter, in a physical body as a certain indestructible content in the portion of space occupied by the body. By the methods of weighing explained in physics we can compare these con- tents of different ])odies; and, taking the mass content of some particular body as the standard unit we can express the mass of every body by a single real number. We here confine ourselves to so-called gravitational masses ; the num- ber that expresses such a mass is always positive, and it remains constant in whatever way the body may move. The student must be warned not to confound mass with weight. The weight of a body, as we shall see later, is the force with which the body is attracted by the earth; it varies, therefore, with the distance of the body from the earth's center, and would vanish completely if the earth were sud- denly annihilated; while the indestructibility of mass is the first fundamental principle of chemistry and physics. The modern developments in the theory of electricity may, and probably will, lead to a better understanding of 120 153.] MASS; DENSITY 121 the intimate nature of mass or matter. But this would hardly affect ordinary mechanics which will always retain a wide range of applicability. 152. The unit of mass in the C.G.S. system (Art. 6) is the gram, in the F.P.S. system the 'pound. The American pound is defined (by act of Congress, 1866) as 2^.2^^t6 of ^ kilogram : 1 lb. - 453.597 gm., 1 gm. = 0.002 204 6 lb. The three units of s-pace, time, and mass are called the fundamental units of mechanics, because with the aid of these three, the units of all other quantities occurring in mechanics can be expressed. Thus we have seen how the units of velocity and acceleration are based on those of space and time, and we shall have many more illustrations in what follows. Any unit that can be expressed mathematically by means of one or more of the fundamental units is called a derived unit. 153. A continuous mass of one, two, or three dimensions is said to be homogeneous if the masses contained in any two equal lengths, areas, or volumes (as the case may be) are equal. The mass is then said to be distributed uniformly. In all other cases the mass is said to be heterogeneous. The whole mass ilf of a homogeneous body divided by the space V it fills is called the density of the mass or body; denoting density by p we have therefore M P = y , for homogeneous bodies. It follows from the definition of homogeneity that the density of a homogeneous mass can 122 STATICS [154. also be found by dividing any portion Ailf of the whole mass M by the space AV occupied by AM. In a heterogeneous body, the quotient AM/AV is called the average, or 'mean, density of the portion AM. The limit of this average density as the space AV approaches zero while always containing a certain point P is called the density of the mass M at the point P: ,. AM dM Ar=oAK dV 154. The unit of density is the density of a substance such that the unit of volume contains tlie unit of mass. If the units of volume and mass are selected arbitrarily, there need not of course necessarily exist any physical substance having unit density exactly. Thus in the F.P.S. system, unit density is the density of an ideal substance one pound of which would just fill a cubic foot. As a cubic foot of water has a mass of about 62 H pounds, or 1000 ounces, the density of water is about 623^ times the unit density. The specific density, or specific gravity, of a substance, is the ratio of its density to that qf water at 4° C. Let p be the density, pi the specific density, M the mass, V the volume of a homogeneous mass, then in British units M = pV = 62.5pi7. In the C.G.S. system, the unit of mass has been so selected as to make the density of water equal to 1 very nearly; in other words, the unit mass (1 gram) of water, at the temperature of 4° C, fills one cubic centimeter. In the metric system, then, there is no difference between density and specific density or specific gravity. 155. We speak in mechanics not only of three-dimensional material bodies, or volume masses, but also of material surfaces, or surface masses, and of material lines, or linear masses, one or two of the spatial dimensions being made to approach zero while the mass content remains finite. Thus, in a surface mass, sometimes called a shell, lamina, or mem- brane, a finite mass content is assigned to every finite portion 156.] MASS; DENSITY 123 of a surface; in a linear mass, often designated as a rod, wire, or chain, a finite mass content is assigned to every finite arc of a curve. If d(x is the area element of the surface a, ds the length element of the curve s, the surface density p and the linear density p" are defined (comp. Art. 153) by P = dM ,, _ dM da ' ^ ~ ~di 156. Finally, letting all three dimensions of a physical body approach zero, while the mass content may remain finite, we arrive at the idea of the mass-point, or particle, viz. a geometrical point to which a definite mass is assigned. As a finite physical mass is always thought of as occupying a finite space, the particle, or geometrical point endowed with a finite mass, is a pure abstraction. The importance of this conception lies not so much in its relation to the idea that physical matter is ultimately an aggregation of such points or centers possessing mass (molecules, atoms), but in the fact that for certain purposes (viz. as far as translation only is concerned) the motion of a physical solid is fully determined by the motion of a certain point in it, called the center of mass or centroid, the whole mass of the body being regarded as concentrated at this point. CHAPTER VIII. MOMENTS AND CENTERS OF MASS. 157. The product of a mass m, concentrated at a point P, into the distance of the point P from any given point, Hne, or plane is called the moment of this mass with respect to the point, line, or plane. Thus, denoting by r, q, p the distance of the point P from the point 0, the line I, and the plane tt, respectively, we have for the moments of m with respect to 0, I, ir the expressions mr, mq, m/p. 158. Let a system of n points, or particles, Pi, P2, ■ ■ • Pn be given; let mi, mo, • • • w„ be their masses, and pi, p^, • ■ -pn their distances from a given plane tt. Then we call moment of the system with respect to the plane tt the algebraic sum Wipi + mnp2 + • • • + mnpn = 2?np, the distances pi, Pi, • • ■ Pn being taken with the same sign or opposite signs according as they lie on the same side or on opposite sides of the plane tt. It is always possible to determine one and only one distance p such that Xmp = Mp, where M = Sm = nii + ?W2 + • • • + Wn is the total mass of the system. If this distance p should happen to be equal to zero, the moment of the system would evidently vanish with respect to the plane tt. 159. Let us now refer the points P to a rectangular set of axes and let x, y, z be their co-ordinates. Then we have for the moments of the system with respect to the co-ordinate planes yz, zx, xy, respectively: 124 161.] MOMENTS AND CENTERS OF MASS 125 miXi + ^12X2 + • • • + ninXn = Smx = Mx, mii/i, + moijo + • • • + mnyn = ^my = My, niiZi + 'MnZo + • • • + ninZn = '^mz = AIz. The point G whose co-ordinates are _ _ I,mx _ _ I,my _ _ 'Emz is called the center of mass, or the centroid, of the system. The centroid is, therefore, defined as a point such that if the whole mass M of the system he concentrated at this jjoint, its moment with respect to any one of the co-ordinate planes is equal to the moment of the system. 160. It is easy to see that this holds not only for the co- ordinate planes but for any plane whatever. Jjct ax -\- ^y -h yz - po = be the equation of any plane in the normal form; p, pi, P2, ■ • • Pn, the distances of the points G, Pi, Po, ■ ■ ■ Pn from this plane. Then we wish to prove that ^mp = Mp. Now p = ax -\- ^ij -\- yz - Po, pi = axi + /3?/i + yZi - po, - - ■ ; hence Iimj) = aXmx + jSSm?/ -|- yZmz — poEm = M{ax + j8?7 + 72 - Po) - Mp. The centroid can therefore ]>e defined as a point such that its moment with respect to any plane is equal to that of the whole system, with respect to the same plane. It follows that the moment of the system vanishes for any plane passing through the centroid. 161. In the case of a continuous mass, whether it be of one, two, or three dimensions, the same reasoning will apply 126 STATICS [161. if we imagine the mass divided up into elements dM of one, two, or three infinitesimal dimensions, respectively. The summations indicated above by S will then become integra- tions, so that the centroid of a continuous mass has the co-ordinates fxdM CydM fzdM '''' JdM' y~ JdM' JdM' ^^^ According as the mass is of one, two, or three dimensions, a single, double, or triple integration over the whole mass will in general be required for the determination of the moments fxdM, CydM, fzdM of the mass with respect to the co- ordinate planes, as well as of the total mass JdM = M. Thus, for a mass distributed along a line or a curve we have, if ds be the line-element, dM = p"ds, where p" is the linear density (Art. 155); for a mass dis- tributed over a surface-area we have, with da as a surface- element, dM = pda, where p' is the surface (or areal) density; finally, for a mass distributed throughout a volume whose element is rfr, dM = pdT, where p is the volume density. If the mass be distributed along a straight line, the centroid lies of course on this line, and one co-ordinate is sufficient to determine the position of the centroid. In the case of a plane area, the centroid lies in the plane and two co-ordinates determine its position; we then speak of moments with re- spect to lines, instead of planes. 164.] MOMENTS AND CENTERS OF MASS 127 162. If the mass be homogeneous (Art. 153), i. e. if the density p be constant, it will be noticed that p cancels from the numerator and denominator in the equations (2), and does not enter into the problem. Instead of speaking of a center of mass, we may then speak of a center of arc, of area, of volume. The term ceniroid is, however, to be preferred to center, the latter term having a recognized geometrical meaning different from that of the former. The geometrical center of a curve or surface is a point such that any chord through it is bisected by the point; there are but few curves and surfaces possessing a center. The centroid (Art. 160) is a point such that, for any plane passing through it, the moment of the system is equal to zero. Such a point exists for every mass, volume, area, or arc. The centroid coincides, of course, with the center, when such a center exists and the distri- bution of mass is uniform. 163. As soon as p is given either as a constant or as a function of the co-ordinates, the problem of determining the centroid of a con- tinuous mass is merely a problem in integration. To simplify the integrations, it is of importance to select the element in a convenient way conformably to the nature of the particular problem. Considerations of symmetry and other geometrical properties will frequently make it possible to determine the centroid without rcsorling to integration. Thus, in a homogeneous mass, any plane of symmetry, or any axis of symmetry, must contain the centroid, since for such a plane or line the sum of the moments is evidently zero. It is to be observed that the whole discussion is entirely inde- pendent of the physical nature of the masses rn which appear here simply as numerical coefficients, or "weights," attached to the points. Some of the masses might even be negative provided the total mass is not zero. It will be shown later that the center of gravity, as well as the center of inertia, of a body coincides with its centroid. 164. In determining the centroid of a given system it will often be found convenient to break the system up into a num- ber of partial systems whose centroids are either known or can be found more readily. The ■moment of the whole system is obviously equal to the sum of the moments of the partial systems. 128 STATICS 1165. Thus let the given mass M be divided into k partial masses Ml, • • • Mk so that M = Ml + • • ■ -\- Mk] let G, Gi, - ■ ■ Gk be the centroids of M, Mi, • • • Mk and p, pi, • • • pk their distances from some fixed plane. Then we have Mp = Mipi + • • • + Mkpk- 165. The particular case of two partial systems occurs most frequently. We then have with reference to any plane Mp = Mipi + M2P2, M ^ Mi-\~ M2. Letting the plane coincide successively with each of the three co-ordinate planes it will be seen that G must lie on the line joining Gi, G2. Now taking the plane at right angles to G1G2 through Gi we have M-GiG = il/o-GA; similarly for a plane through G2 M-GG2 = Mi-GiG2U whence ljri(j \J\J2 yjc^i 1^2 ^ Wi ^ ~W '' i. e. the centroid of the whole system divides the distance of the centroids of the two partial systems in the inverse ratio of their masses. 166. Exercises. (1) Show that the centroid of two particles ?«i, '"2 divides their distance in the inverse ratio of the masses by taking moments about the centroid. Find the centroid: (2) Of three masses 5, 7, 23 on a line, the mass 7 lying midway between 5 and 23. (3) Of earth and moon, the moon's mass being 1/SO of that of the earth and the distance of their centers 240,000 miles, (4) Of three equal particles. 166.] MOMENTS AND CENTERS OF MASS 129 (5) Of a circular arc, radius r, angle at center 2a; in particular, of a semicircle. (6) Of the arc of a parabola, if = 4aa;, from vertex to end of latus rectum. (7) Of one arch of the cycloid x = a(d — sinO), y = ail — cos0). (8) Of half the cardioid r = a(l + cose). (9) Of an arc of the common helix x = r cos0, y = r sin5, z = krd, from 6 = to e = d. (10) Of a circular arc AB, of angle AOB = a, whose density varies as the length of the arc measured from A (11) Show that the centroid of a triangular area is the intersection of the medians. (12) From a square ABCD of side a one corner EAF is cut off so that AE = fa, AF = la; find the centroid of the remaining area. (13) An isosceles right-angled triangle of sides a being cut out of the area of its circumscribed circle, find the centroid of the remaining area. (14) Find the centroid of the surface area of a sphere between two parallel planes, by observing that this area is equal to the surface area of the circumscribed cylinder perpendicular to these planes. (15) Show that for an area a, bounded by a curve y — fix), the axis Ox and two ordinates, we have X = I xydx, r(! is closed l)y a circular lid of a material whose density is three tinu>s that of the bowl. Find the centroid. 10 130 STATICS [166. (19) The cissoid (2a — x)y^ = x^ can be represented by the equa- tions X = 2a sin^, y = 2a sin^^/cos^, where d is the polar angle, 2a the distance from cusp to asymptote. Show that the centroid of the area between the curve and its asymptote divides the distance between cusp and asymptote in the ratio 5:1. (20) The centroid of a rectilinear segment of length I whose linear density is proportional to the «th power of the distance from one end is at the distance {n + 1)1/ (n + 2) from that end. Hence show that (o) the centroid of a triangular area lies on the median at % the distance from the vertex to the base; (b) the centroid of the surface area of a cone or pyramid lies on the line joining the vertex to the centroid of the base, at % the distance from the vertex to the base; (c) the centroid of the volume of a cone or pyramid lies on the same line, at % the distance from the vertex to the base. (21) For a solid of revolution, generated by the revolution of the curve y = f(x) about the axis of x and bounded by planes perpendicular to the axis Ox, show that the centroids of the curved surface area a and of the volume t are given by: a-xa = 2w \ ' xy 1^1 + w'^ dx, T ■ Xt = IT \ xyHx. (22) Find the centroid of the segment of a sphere between two parallel planes; and hence (a) that of a segment of height A, cut off by a plane; (b) that of a hemisphere; (c) that of a spherical sector of ver- tical angle 2a. (23) Find the centroid of the paraboloid of revolution of height h, generated by the revolution of the parabola y~ = 4ax about its axis. (24) The area bounded by the parabola y- = 4ax, the axis of x, and the ordinate y = yi revolves about the tangent at the vertex. Find the centroid of the solid of revolution so generated. (25) The same area as in Ex. (6) revolves about the ordinate yi. Find the centroid. (26) Find the centroid of an octant of an ellipsoid xVa^ + yy¥ + zVc^ = 1. (27) The equations of the common cycloid referred to a cusp as origin and the base as axis of x are x = a(d — sinO), y = a{l — cos^). Find the centroid: (a) of the arc of the semi-cycloid {i. e. from cusp 166.1 MOMENTS AND CENTERS OF MASS 131 to vertex) ; (b) of the plane area included between the semi-cycloid and the base; (c) of the surface generated by the revolution of the semi- cycloid about the base; (d) of the volume generated in the same case. (28) Find the centroid of a solid hemisphere whose density varies as the nth power of the distance from the center. CHAPTER IX. MOMENTUM ; FORCE ; ENERGY. 167. Let us consider a point moving with constant accelera- tion from rest m a straiglit line. We know from Kinematics (Art. 16) that its motion is determined by the eauations V = jt, s = ^jr~, ^v'' - js, (1) where s is the distance passed over in the time t, v the velocity, and j the acceleration, at the time t. If, now, for the single point we sul^stitute an m-tuple point, i. e. if we endow our point with the mass ni, and thus make it a, particle (see Art. 156), the equations (1) must be multiplied by m, and we obtain mv = mjt, ms = h^njt}, \mv'^ = mjs. (2) The quantities mv, 7nj, iww" occurring in these equations have received special names because they correspond to certain physical conceptions of great importance. 168. The product mv of the mass m of a particle into its veloc- ity V is called the momentum, or the quantity of motion, of the particle. In observing the behavior of a physical body in motion, we notice that the effect it produces — for instance, when impinging' on another body, or more generally, whenever its velocity is changed — depends not only on its velocity, but also on its mass. FamiUar examples are the following : a loaded railroad car is not so easily stopped as an empty one; the destructive effect of a cannon-ball depends both on its velocity and on its mass; the larger a fly-wheel, the more difficult is it to give it a certain velocity; etc. It is from experiences of this kind that the physical idea of mass is derived. 132 I7i.[ MOMENTUM; FORCE; ENERGY 133 The fact that any change of motion in a physical body is affected by its mass is sometimes ascribed to the so-called "inertia," or "force of inertia," of matter, which means, however, nothing else but the property of possessing mass. 169. Momentum, being by definition (Art. 168) the product of mass and velocity, has for its dimensions (Art. 6), MV = MLT-\ The unit of momentum is the momentum of the unit of mass having the unit of velocity. Thus in the C.G.S. system the unit of momentum is the momentum of a particle of 1 gram moving with a velocity of 1 cm. per second. In the F.P.S. system, the unit is the momentum of a particle of 1 pound mass moving with a velocity of 1 ft. per second. To find the relations between these two units, let there be x C.G.S. units in the F.P.S. unit; then \ gm. cm. ., lb. ft. X • = 1 ; sec. sec. hence ^ Ib^ ft. gm. ' cm. ' or, by Art. 152 and Art. 6, X = 13,825.7; i. e. 1 F.P.S. unit of momentum = 13,825.7 C.G.S. units, and 1 C.G.S. unit = 0.000072 33 F.P.S. units. 170. Exercises. (1) What is the momentum of a cannon-ball weighing 200 lbs. when moving with a velocity of 1500 ft. per second? (2) With what velocity must a railroad-truck weighing 3 tons move to have the same momentum as the cannon-ball in Ex. (1)? (3) Determine the momentum of a one-ton ram after falling through 4 feet. 171. The 'product mj of the mass m of a particle into its acceleration j is called force. Denoting it by F, we may write our equations (2) in the form mv = Ft, s = * t^, hnv'^ = Fs. (3) " m 134 STATICS .173] As long as the velocity of a particle of constant mass remains constant, its momentum remains unchanged. If the velocity changes uniformly from the value v at the time t to v' at the time t', the corresponding change of momentum is my' — mv = mji' — mjt = F{t' — t); (4) hence „ mv' — mv ,^. Here the acceleration, and hence the force, was assumed constant. If F be variable, we have in the limit as t' — i approaches zero: F=^ = m^. (6) at at Instead of defining force as the product of mass and acceleration, we may therefore define it as the rate of change of momenturyi with the time. 172. Integrating equation (6), we find J^ Felt = mv' — mv. (7) The 'product F(t' — t) of a constant force into the time t' — t during which it acts, and in the case of a variable force, the time-integral J Fdt, is called the impulse of the force during this time. It appears from the equations (4) and (7) that the impulse of a force during a given time is equal to the change of momentum during that time. 173. The idea of force is no doubt primarily derived from the sensa- tion produced in a person by the exertion of his "muscular force." Like the sensations of Hght, sound, heat, etc., the sensation of exerting force is capable, in a rough way, of measurement. But the physiological and psychological phenomena attending the exertion of muscular force when analyzed more carefully are very complicated. 174-1 MOMENTUM; FORCE; ENERGY 135 In popular language the term "force" is applied in a great variety of meanings. For scientific purposes it is of course necessary to attach a single definite meaning to it. It is customary in physics to speak of force as ^producing or generating velocity, and to define force as the cause of acceleration. Thus obser- vation shows that the velocity of a falling body increases during the fall; the cause of the observed change in the velocity, i. e. of the ac- celeration, is called the force of attraction, and is supposed to be exerted by the earth. Again, a body falling in the air, or in some other medium, is observed to increase its velocity less rapidly than a body falling in vacuo; a force of resistance is therefore ascribed to the medium as the cause of this change. In a similar way we speak of the expansive force of steam, of electric and magnetic forces, etc., because it is convenient to think of such agencies as producing changes of velocity. Now, any change in the velocity v of a body of given mass m implies a change in its momentum mv; and it is this change of momentum, or rather the rate at which the momentum changes with the time, which is of prime importance in all the applications of mechanics. It is there- fore convenient to have a special name for this rate of change of mo- mentum, and that is what is called force in mechanics. Thus, in using this term "force," it is not intended to assert any- thing as to the objective reality or actual nature of force and matter in the popular acceptation of these terms. With the ultimate causes science has nothing to do; it can observe only the phenomena them- selves. 174. The definition of force (Art. 171) as the product of mass and acceleration gives the dimensions of force as F = MJ = MLT-^. The tmit of force is therefore the force of a particle of unit mass moving with unit acceleration. Hence, in the C.G.S. system, it is the force of a particle of 1 gram moving with an acceleration of 1 cm./sec.^. This unit force is called a dyne. The definition is sometimes expressed in a slightly different form. We may say the dyne is the force which, acting on a gram uniformly for one second, would generate in it a velocity of 1 cm. /sec; or would give it the C.G.S. unit of acceleration; or it is the force which, acting 136 STATICS [175. on any mass uniformly for one second, would produce in it the C.G.S. unit of momentum. That these various statements mean the same thing follows from the fundamental formulae F = mj, v = jl, if F, m, t, v, j be expressed in C.G.S. units. In the F.P.S. system, the unit of force is the force of a mass of 1 lb. moving with an acceleration of 1 ft./sec.^. It is called the poundal. 175. To find the relation between these two units, let x be the number of dynes in the poundal; then we have gm. cm. ^ lb. ft. X ■ ;: — = 1 860.-= sec.'' hence, just as in Art. 169, x = 13,825.7; i. e. 1 poundal = 13,825.7 dynes, and 1 dyne = 0.000 072 33 i)oundals. 176. Another system of measuring force, the so-called gravitation (or engineering) system, is in very common use, and must be explained here. Among the forces of nature the most common is the force of gravity, or the weight, i. e. the force with which any physical body is attracted by the earth. As we have convenient and accurate appliances for comparing the weights of different bodies at the same place, the idea suggests itself of selecting as unit force the weight of a certain standard mass. In the metric gravitation system the weight of a kilogram has been selected as unit force; in the British gravitation system the weight of a pound is the unit force. 177. The system in which the units of time, length, and mass are taken as fundamental, while the unit of force ( = mass times accelera- tion) is regarded as a derived unit (Art. 175), is called the absolute or scientific system, to distinguish it from the gravitation system (Art. 176) in which the units of time, length, and force are taken as funda- mental, while the unit of mass (= force divided by acceleration) is a derived unit. As the weight of a body varies from place to place with the variation of the acceleration of gravity g, the unit of force as defined in Art. 176 would not be constant. This difficulty can be avoided by defining the unit of force as the weight of a kilogram or pound at some definite place, say at London, or in latitude 45° at sea level. With this modification, 180.] MOMENTUM; FORCE; ENERGY 137 the gravitation system desenves the name of an absolute system as much as does the system in which mass is the thii-d fundamental unit. The general equations of mechanics are of course independent of the system of measurement adopted; they hold as well in the gravita- tion as in the scientific or absolute system. In the present work the language of the latter system is generally used in the text (not always in the exercises). This system, since its introduction by Gauss and Weber, has found general acceptance in scientific .work. In statics where we are mainly concerned with the ratios of forces and not with their absolute values it rarely makes any difference which system is used provided all forces are expressed in the same unit. And as elementary statics deals largely with the effects of gravity, the gravitation system is often used in statical problems. 178. The numerical relation between the scientific and gravitation measures of force is expressed by the equations 1 kilogram (force) = 1000 g dynes, 1 pound (force) = g poundals, where g is about 981 in metric units, and about 32.2 in British units. In most eases the more convenient values 980 and 32 may be used. 179. Exercises. (1) What is the exact meaning of "a force of 10 tons"? Express this force in poundals and in dynes. (2) Reduce 2,000,000 dynes to British gravitation measure. (3) Express a pressure of 2 lbs. per square inch in kilograms per square centimeter. (4) Show that a poundal is very nearly half an ounce, and a dyne a little over a milligram, in gravitation measure. 180. The quantity \mv'^, i. e. half the product of the mass of a particle into the square of its velocity, is called the kinetic energy of the particle. Let us consider again a particle of constant mass ?n moving with a constant acceleration, and hence with a constant force; let v be the velocity, s the space described, at the time i; y', s' the corresponding values at the time t'. Then the last 138 STATICS [182. of the three fundamental equations (see Arts. 167 and 171) gives ^v''~ - hnv- = F{s' - s); (8) hence F = -, — . (9) s — s If F be variable, we have in the limit F = — ^^-^ — - = mv ^ . (10) as as Force can therefore be defined as the rate at ivhich the kinetic energy changes with the space. (Compare the end of Art. 171.) 181. Integrating the last equation (10), we find £'Fds = hnv'^ - hnv'-. (11) The product F{s' — s) of a constant force F into the space s' — s described in the direction of the force, and in the case of a variable force, the space-integral f Fds, is called the work of the force for this space. The equations (8) and (11) show that the work of a force is equal to the corresponding change of the kinetic energy. We have here assumed that the force acts in the direction of motion of the particle. A more general definition of work including the above as a special case will be given later (Art. 261). The ideas of energy and work have attained the highest importance in mechanics and mathematical physics within comparatively recent times. Their full discussion belongs to Kinetics (see Part III). 182. According to their definitions, both momentum (Art. 168) and force (Art. 171) may be regarded mathematically 183.] MOMENTUM; FORCE; ENERGY 139 as mere numerical multiples of velocity and acceleration, respectively. They are therefore so-called vector-quantities; i. e. a momentum as well as a force can be represented geo- metrically by a segment of a straight line of definite length, direction, and sense. Moreover, as they are referred to a particular point, viz., to the point whose mass is in, the line representing a momentum or a force must be drawn through this point; the hne has therefore not only direction, but also position; i. e. a momentum as well as a force is represented geometrically hy a rotor (compare Art. 115). It follows that concurrent forces, for instance, can be com- pounded l^y geometrical addition, as will be explained more fully in Chapter X. On the other hand, kinetic energy and work are not vector- quantities. 183. The ideas of momentum, force, energy, work, with the funda- mental equations connecting them, as given in the preceding articles, form, the groundwork of the whole science of theoretical dynamics. The application of this science to the interpretation of natural phenom- ena gives results in close agreement with observation and experiment. It is therefore important to inquire what are the physical assumptions and experimental data on which this application of dynamics is based. These assumptions were formulated with remarkable clearness by Sir Isaac Newton m his Philosophioe naturalis prindpia malhematica, first published in 1687, and have since been known as Newton's laws of motion. As these three axiomata sive leges motus, as Newton terms them, are very often referred to and, at least bj^ English writers on dynamics, are usually laid down as the foundation of the science, they are given here in a literal translation: I. Every body persists in its state of rest or of uniform motion along a straight line, except in so far as it is compelled by impressed {i. e. external) forces to change that state. II. Change of motion is proportional to the impressed moving force and takes place along the straight line in which that force acts. 140 STATICS [184 III. To every action there is an equal and contrary reaction; or, the mutual actions of two bodies on one another are always equal and directed in contrary senses. 184. Some explanation is necessary to understand correctly the meaning of these laws. Indeed, Newton's laws should not be studied by themselves; they become intelUgible only if taken in connection with the definitions preceding them in the Prindpia, and with the ex- planations and corollaries that Newton himself has appended to them. The word 'body" must be taken to mean particle; the word "mo- tion" in the second law means what is now called momentum. All three laws imply the idea of force as the cause of any change of momentum in a particle. 185. With this definition of force the first law, at least in the ordi- nary form of statement, for a single particle, merely states that where there is no cause there is no effect. While this law may appear super- fluous to us, it was not so in the time of Newton. Kepler and Galileo, less than a century before Newton, were the first to insist more or less clearly on this so-called law of inertia, viz. that there is no intrinsic power or tendency in moving matter to come to rest or to change its motion in any way. 186. The second law gives as the measure of a constant force the amount of momentum generated in a given time (see Art. 171); it can be called the law of force. If force be defined as the cause of any change of momentum, the second law follows naturally by assuming, as is usually done, that the effect is proportional to the cause. The first two laws may thus be regarded from the mathematical point of view as nothing but a definition of force; but they are certainly meant to emphasize the phj'sical fact that the assumed definition of force is not arbitrary, but based on the characteristics of motion as observed in nature. In the corollaries to his laws Newton tries to show how the compo- sition and resolution of forces by the parallelogram rule follows from his definition. In deriving this result he tacitly assumes that the action of any force on a particle takes place independentlj' of the action of any other forces that may be acting on the particle at the same time, a principle that would seem to deserve explicit statement. Some writers on mechanics prefer to replace Newton's second law by this principle of the independence of the action of forces. 187.] MOMENTUM; FORCE; ENERGY 141 187. The third law expresses the physical fact that in nature all forces occur in pairs of equal and opposite forces. Two such equal and opposite forces in the same line are often said to constitute a stress. Newton's third law has therefore been called the law of stress. This law, which was first clearly conceived in Newton's time, involves what may be regarded as the second fundamental property of matter or mass (the first being its indestructibility) ; viz. that any two particles of matter determine in each other oppositely directed accelerations along the line joining them. CHAPTER X. STATICS OF THE PARTICLE. 1.88. According to the definition of force (Arts. 171, 173), a single force F acting on a particle of mass m produces ari acceleration j such that F = mj; i. e. the vector F is m times the vector j. If two forces Fi, Fo act on the same particle, it is assumed (Art. 186) that each acts as if the other were not present; / -z^ hence, if ju jo are the ac- / ^^ I celerations which Fi, Fo / * ^-^^^^ / would produce separately, •l ^"^ j I then the combined effect of J^ ^ FJ Fi and Fo will be to produce .^. an acceleration equal to the Fig. 41. , ^ . resultant, or geometric sum, i = Ji + J2, of the accelerations ji, j^; and this resultant ac- celeration j can be produced by a single force R = mj (Fig. 41). The combined effect of the two forces Fi, F2 acting on the same particle m is thus the same as that of that single force R which is the resultant, or geometric sum, of Fi and F2. The two forces Fi, Fo are said to be equivalent to the single force R; R\s called the resultant of Fi, Fo, which are called components of R. 189. Thus, the resultant R of two forces Fi, F2 acting on the same particle is found (Fig. 42) as the diagonal of the parallelogram constructed with Fi. Fo as adjacent sides. 142 190.] STATICS OF THE PARTICLE 143 Hence R = VFi- + Fa^ + 2FiF2 cos^, R sin/3 sina sin0 where 6 is the angle between Fi and i^2, « that between J? and Fi, /3 that between /^ and F-y. This proposition is known as the parallelogram of forces. It enables us to find the vector R when the vectors F], F^ are given; and conversely, to find Fi, F2 if, in addition to the '3 Fig. 42 vector R, the directions of Fi, Fo (the angles a, /3) are given. The latter operation is called resolving a force along given directions. To find 7^ when Fi, F2 are given it suffices (instead of con- structing the whole parallelogram) to lay off (Fig. 43) 1 2, equal to the vector Fi (in magnitude, direction, and sense), and 2 3, equal to the vector F2; then 1 3 is the resultant R. 123 is called the triangle of forces. 190. Let any number 71 of forces Fi, F2, • • • F„ be applied at the same point 0, i. e. act on the same particle at 0. By Art. 189, we can find the resultant Ri of Fi and Fo, next the resultant 7?2 of 7?i and F3, thvn the resultant R3 of Ro and Fi, and so on. The resultant R of Rn-- and Fn is evidently equivalent to the whole system Fi, F2, F3, • • • F„, and is 144 STATICS [191. called its resultant. It thus appears that a system consisting of any niwiber of forces acting on the same particle is equivalent to a single resultant. It may of course happen that this resultant is zero. In this case the system is said to be in equilibrium. The con- dition of equilibrium of a system of forces acting on the same particle is therefore: R = 0. The system of forces in this case produces no acceleration; notice that equilibrium of the forces does not mean that the particle is at rest. Under forces that are in equilibrium the particle, if at rest, will remain at rest; if in motion, it will continue to move uniformly in a straight line. 191. In practice, the process of finding the resultant indicated in Art. 190 is inconvenient when the number of forces is large. If the forces are given geometrically, as Fig. 44. vectors, we have only to add these vectors; and this can best be done in a separate diagram, called the force polygon. Thus, in Fig. 44, 1 2 is drawn equal and parallel to Fi, 2 3 equal and parallel to F., 3 4 to F3, 4 5 to F^, 5 6 to F^. The 194.] STATICS OF THE PARTICLE 145 closing line of the force polygon, viz. 1 6 in the figilre, is equal and parallel to the resultant R, which is therefore obtained by drawing through the point of application of the forces a line equal and parallel to 1 6. The geometrical condition of equilibrium consists in the closing of the force polygon, that is, in the coincidence of its terminal point 6 with its initial point 1. 192. Analytically, a system of concurrent forces is reduced to its rnost simple equivalent form, i. e. to its single resultant, by resolving each force F into three components A^, Y, Z, along three rectangular axes passing through the particle, or point of application of the given forces. All components lying in the direction of the same axis can then be added algebraically, and the whole system of forces is found to be equivalent to three rectangular forces SX, SF, SZ, which, by the parallelogram law, can be replaced by a single resultant 7^ = V(2Ap + (SF)2 + (2Z)2. The angles a, /S, 7 made by this resultant with the axes are given by the relations cosa _ cosjg _ C0S7 ^ 1^ SX ~ SF ~ SZ ~ R' 193. If the forces all lie in the same plane, only two axes are required and we have SF R = V(2X)2 + (SF)2, tan0 = ^, where 6 is the angle between the axis of X and R. 194. The condition of equilibrium (Art. 190) R = be- comes, by Art. 192, {^Xy + (SF)^ + (SZ)^ = 0. As all terms in the left-hand member are positive, their sum can vanish only when each term is zero. The analytical conditions 11 146 STATICS [195. of the equilibrium of any mmiber of concurrent forces are therefore : SX = 0, 2F = 0, SZ = 0. 195. The forces of nature receive various special names according to the circumstances under which they occur. Thus the weight of a mass has already been defined (Art. 176), as the force with which the mass is attracted by the mass of tlie earth. A string carrying a mass at one end and suspended from a fixed point, is subjected to a certain tension. This means that if the string were cut it would require the application of a force along the line of the string to keep the weight in equilibrium. This force, which may thus serve to replace the action of the string, is called its tension. When the surfaces of two physical bodies A, B are in contact, a pressure may exist between them; that is, if one of the bodies, say B, be removed, it may require the intro- duction of a force to keep A in the same state of rest or motion that it had before the removal of B. This force, which acts along the common normal of the surfaces at the point of contact if there is no friction, is called the resistance, or reaction, of B, and a force equal and opposite to it is called the pressure exerted by A on B. For the case of friction see Arts. 237 sq. [196. Exercises. (1) Show that the resultant of two equal forces F including an angle 6 is 2F cos^O. Observe the variation of the resultant as 9 varies from to tt; for what angle e is the resultant equal to F? (2) Show that the resultant of two forces OA, OB is twice OC, where C is the midpoint of A and B. (3) Find the magnitude and direction of the resultant of two forces of 12 and 20 lb., including an angle of 60°. 196.] STATICS OF THE PARTICLE 147 (4) Find the resultant of 6 equal concurrent forces, each inclined to the next at 45°. (5) Show that the forces OA, OB, OC are in equilibrium if is the centroid of the triangular area ABC. (6) Show (by Art. 194) that if any number of concurrent forces are in equilibrium, their point of concurrence is the centroid of their ex- tremities. (7) A mass m rests on a plane inclined to the horizon at an angle 6; it is kept in equilibrium: (a) by a force Pi parallel to the plane; (b) by a horizontal force Pi] (c) by a force P? inclined to the horizon at an angle 6 + a. Determine in each case the force P and the pres- sure R on the plane. (8) A weight W is suspended from two fixed points A, B hy means of a string AC B, C being the point of the string where the weight W is attached. If AC, BC be inclined to the vertical at angles a, §, find the tensions in AC, BC: (a) analytically; {h) graphically. (9) Show that the resultant R of three concurrent forces A, B, C in the same plane is given by P^ = ^2 _[_ 52 ^ (72 _(_ 2BC cos{B, C) + 2CA cos(C, A) + 2AB cos(^, B). (10) A weightless rod AC, hinged at one end A so as to be free to turn in a vertical plane, is held in a horizontal position by means of the chain BC, the point B lying vertically above A. If a weight W be suspended at C, find the thrust P in ^C and the tension T of the chain. Assume AC = 8 ft., AB = Q ft. (11) In Ex. (10), suppose the rod AC, instead of being hinged at A, to be set firmly into the wall in a horizontal position; and let the chain fastened at B run at C over a smooth pulley and carry the weight W. Find the tension of the chain and the magnitude and direction of the pressure on the pulley at C. (12) In "tacking against the wind," let W be the force of the wind; a, ^ the angles made by the axis of the boat with the direction in which the wind blows, and with the sail, respectively. Determine the force that drives the boat forward and find for \\'hat position of the sail it is greatest. (1.3) A cylinder of weight W rests on two inclined planes whose inter- section is horizontal and parallel to the axis of the cylinder. Find the pressures on these planes. 148 STATICS 1196. (14) Find the tensions in the string ABCD, fixed at A and D, and carrying equal weights W at B and C, ii AD = c is horizontal, AB — BC = CD, and the length of the string is 3L (15) In the toggle-joint press two equal rods CA, CB are hinged at C] a force F bisecting the angle 2a between the rods forces the ends A, B apart. If A be fixed, find the pressure exerted at B at right angles to F ii F = 100 lbs. and a = 15°, 30°, 45°, 60°, 75°, 90°. (16) A stone weighing 800 lbs. hangs from a derrick by a chain 15 ft, long. If pulled 5 ft. away from the vertical by means of a hori- zontal rope attached to it, what are the tensions of the chain and the rope? What if pulled 9 ft. away? (17) A rope 16 ft. long has its ends fastened to two points, 10 ft. apart, at the same height above the ground; a weight W is suspended from the rope by means of a ring free to slide along the rope. Find the tension of the rope. (18) A string with equal weights W attached to its ends is hung over two smooth pegs A, B fixed in a vertical wall. Find the pressure on the pegs: (o) when the line AB is horizontal; (b) when it is inclined to the horizon at an angle d. CHAPTER XI. STATICS OF THE RIGID BODY. 197. A system of forces acting on a rigid body can, in general, not be reduced to a single resultant, as is the case for concurrent forces (Art. 190) ; in other words, there does not always exist a single force having the same effect that the system of forces has in changing the motion of the body. Before discussing the general case it is best to consider certain particular kinds of systems of forces, viz. concurrent, parallel, and complanar systems. Throughout the statics of the rigid body it is assumed that the effect of a force is not changed if the force is transferred to any other position on its line of action; in other words, a body is called rigid if, and only if, it possesses this property. Thus the ''point of application" of a force acting on a rigid body is not an essential characteristic of the force; what characterizes the force is its magnitude, line of action, and sense. This is what is meant by saying that a force is a localized vector or rotor (Art. 182). 1. Concurrent forces. 1Q8. In the case of concurrent forces there exists a single resultant, viz. the geometric sum of the forces. If this resultant happens to be zero, i. e. if the force polygon (Art. 191) closes, the forces are in equilibrium. As the projection of a closed polygon on any line is zero, it follows that the projection of the resultant on any liiie is equal to the algebraic sum of the projections of its components. 149 150 STATICS [199. Thus, if the forces P, Q intersect at and have the re- sultant R we find by projecting on any line I: R cos(?, R) = P'cosil, P) -\-Q cos{l, Q). Let the hne I be drawn through 0, in the plane of P and Q, and let an arbitrary length OS = s (Fig. 45) be laid off at right angles to / in the same plane. Then, multiplying the last equation by .s we find R-scos(l,R) = P-scos(/, P) + Q-scos(l,Q); or since s cos(Z, R) = r, s cos{l, P) =p, s cos{l, Q) ^ q are the perpendiculars from S to R,P,Q: Rr = Pp + Qq. Now the product of a force into its perpendicular distance from a point is called the moment of the force about the point; the product is taken with the positive or negative sign according as the force tends to turn counterclockwise or clockwise about the point. We have therefore proved that the algebraic sum of the moments of any two intersect- ing forces about anij point in their plane is equal to the moment of their resultant about the same point. This proposition is known as the theorem of moments, or Varignon's theorem. It is readily extended to any number of concurrent forces in the same plane. As a corollary it follows that the sum of the moments of any such forces about any point of their resultant is zero. 199. As the moment of a force represents twice the area of the triangle having the force as base and the reference Fig. 45. 200.] STATICS OF THE RIGID BODY 151 point as vertex, the theorem of moments can also be proved by comparing areas. Thus, with the notation of Fig. 46 we have SOR = SOQ + SQR + QOR, I. e. or smce ->R R.r = Q-q+ P-ST+ PTU, ST + TU = SU = p: Rr = Qq + Pp. It is often convenient to think of the moment Rr of a force R about the point S as a vector drawn through S at right angles to the plane deter- mined by S and R. This is in agreement with the representa- tion of a parallelogram area by such a vector, mentioned in Art. 119. Indeed, the moment Rr is the cross-product of the radius vector drawn from S to any point of R into tlie force-vector R. This representation is of special advantage when the concurrent forces do not lie in the same plane. It can then be shown that the moment of the resultant about any point is equal to the geometric sum of the vectors representing the moments of the components. 2. Parallel forces. 200. It will be proved in the next article that any two parallel forces acting on a rigid body have a single resultant, except when the two parallel forces are of equal magnitude and opposite sense. In the latter case, the two equal and opposite parallel forces are said to constitute a couple, and no further reduction is possible. FiR. 46. 152 STATICS [201. It follows readily that any system of parallel forces acting on a rigid body can be reduced either to a single force or to a single couple. 201. Resultant of two parallel forces. In the plane of the given parallel forces P, Q, resolve P, at any point p of its line of action, into any two components, say P' and F (Fig. 47); and at the point q where F meets the line of Q, Fig. 47. resolve Q into two components F', Q', selecting for F' a force equal and opposite to, and in the same line with, F. The two equal and opposite forces F, F' in the same line pq have no effect on the rigid body so that the given forces P, Q are together equivalent to the two components P', Q' alone. The lines of P' and Q' will in general intersect at a point r and these forces can therefore be replaced by a resultant R passing through r. By placing the triangles pP'P and qF'Q together so that their equal sides PP' and qF' coincide (as is done in Fig. 47, on the right) it appears at once that the resultant of P' and 202] STATICS OF THE RIGID BODY 153 Q' , and hence the resultant R of P and Q, is parallel to P and Q and in magnitude equal to the algebraic sum of P and Q: R^ P + Q. In Fig. 47, the two given parallel forces P, Q were as- sumed of the same sense. The construction applies, how- ever, equally well to the case when they are of opposite sense. The resultant R will then be found to lie not between P and Q, but outside, on the side of the larger force. The con- struction fails only when the two given forces are equal and of opposite sense, since then the lines pP' and qQ' become parallel. This exceptional case will be considered in Art. 208. 202. The theorem of moments for parallel forces. As the forces R, P', Q' (Fig. 47) are concurrent the theorem of moments (Art. 198) can be applied to these three forces. Hence, taking moments about any point S of the plane of P' and Q' and denoting the perpendiculars from S to the forces by the corresponding small letters, we have* Rr = P'p' + Q'q'. Now P' can be regarded as the resultant of P and — F, and Q' as the resultant of Q and — F' ; hence P'p' = Pp - Ff, Q'q' =Qq- F'f; substituting these values and remembering that F and F' are equal and opposite and in the same line, we find Rr = Pp + Qq; i. e. the sum of the moments of two parallel forxes about any point in their plane is equal to the moment of their resultant about the sanUb point. If, in particular, the point of reference be taken on the resultant so that r = 0, we find Pp = - Qq; 154 • STATICS [203. i. e. the resultant of two 'parallel forces divides their distance in the inverse ratio of the forces. This proposition, well known from its application to the lever, is often referred to as the principle of the lever. 203. It has been shown that two parallel forces P, Q acting on a rigid body, provided they are not equal and of opposite sense, have a resultant R = P -\- Q, parallel to P and Q, and that its position in the rigid body can be found either ana- lytically from the fact that R divides the distance between P and Q in the inverse ratio of these forces, or geometrically by the construction of Art. 201. This geornetrical construction is best carried out in the tollowing order (Fig. 48). The parallel forces P, Q being ._->0 Fig. 48. given in position, begin by constructing the force polygon, which here consists merely of a straight line on which the forces P = 12, Q = 23 are laid off to scale ; the closing line, 1 3, gives the resultant in magnitude, direction, and sense; it only remains to find its position, and for this it suffices to find one point of its line of action. Now, to resolve P and Q each into two components (as is done in Art. 201) so that one component of P and 205.] STATICS OF THE RIGID BODY 155 one of Q are equal and opposite and in the same line, it is only necessary to draw from an arbitrary point 0, called the pole, the lines 1, 2, 3; then 1 0, 2 can be regarded as components of P = 1 2, and 2 0, 3 as components of Q = 23. Next construct the so-called funicular polygon by drawing a hne I parallel to 1, intersecting P say at p; through p a line II parallel to 2 meeting Q say at q; through q a line III parallel to 3. The intersection r of I and III is a point of the resultant R as appears by comparing Figs. 48 and 47; Fig. 48 being the same as Fig. 47, with the superfluous lines left out. 204. Analytically, the resultant of n parallel forces Fi, F2, • • • Fn, whether in the same plane or not, can be found as follows : The resultant of Fi and F2 is a force Fi + F2 situated in the plane (Fi, F2), so that F^pi = F2P2 (Art. 202), where Pi, P2 are the (perpendicular or oblique) distances of the resultant from Fi and Fo, respectively. This resultant Fi + F2 can now be combined with Fz to form a resultant Fi-\- F2 + Fz, whose distances from Fi + F2 and F3 in the plane determined l^y these two forces are as Fz is to Fi + F2. This process can be continued until all forces have been combined; the final resultant is Pi + 7^2 + • • • + Pn. Amj number of parallel forces are, therefore, equivalent to a single resultant equal to their algebraic sum, provided this sum does not vanish. 205. To find the position of this resultant analytically, let the points of application of the forces Pi, P2, • • • Fn be (xi, yi, Zi), (x2, y2, Z2), • • • (xn, Vn, Zn) ■ The point of applica- tion of the resultant Pi -f Po of Pi and P2 may be taken so as 156 STATICS [206. to divide the distance of the points of application of Fi and F<2, in the ratio F^jFi; hence, denoting its co-ordinates by x', y', z' , we have Fi{x' — Xi) = F^ix^ — x'), or (Fi + F~^x' = F,x, + F2X2, and similarly for 7/ and z\ The force Fi + F2 coml)ines with F3 to form a resultant F1 + F2+F3, whose point of apphcation {x" ,y" , z") is given Iw (Fi + F2 + F3)x" = Fixi + F2X2 + Fszs, with similar expressions for y", z" . Proceeding in this way, we find for the point of application {x, y, z) of the resultant of all the given forces (Fi + F2 + • • • + Fn)x = F,xr + F,Xo + • • • + F„a'„, with corresponding equations for y and I. We may write these equations in the form: - _ ^^ - _ ^l]L - - ^ ^ ~ 2F ' ^ ~ SF ' ^ ~ ZF ' unless 2F = 0. As these expressions for x, y, z are independent of the direction of the parallel forces it follows that the same point (x, y, z) would be found if the forces were all turned in any way about their points of application, provided they remain parallel. The point {x, y, z) is for this reason called the center of the system of parallel forces. It is nothing but the centroid of the points of application if these points are re- garded as possessing masses equal to the magnitudes of the forces. 206. Conditions of equilibrium. It follows from what pre- cedes that Jor the equilibrium of a system, of 'parallel forces the condition 2F — Q, or R = Q, though always necessary, is not sufficient. 207.] STATICS OF THE RIGID BODY 157 Now, if the resultant R of the n parallel forces Fi, F2, ■ • • F„ is zero, the resultant R' of the n — 1 forces Fi, F2, • • ■ Fn-i cannot be zero, and its point of application is found (by Art. 205) from x = (F^Xi + F2X2 -\- • • • + F n-iXn-i) I {F , -\- F^ + • • • Fn-i) and similar expressions for y and z. The whole system of parallel forces is therefore equivalent to tl.e two parallel forces R' and Fn- Two such forces can be in equi- librium only when they lie in the same straight line; i. e. Fn must lie in the same line with R' and must therefore pass through the point (x, y, z), which is a point of R'. The additional condition of equilibrium is, therefore, X Xn y yn Z Zn cosa cos/3 C0S7 ' where a, jS, 7 are the angles made by the direction of the forces with the axes. For practical application it is usually best to replace the last condition l)y taking moments about a convenient point. Thus, the analytical conditions of equilibrium can be written in the form SF = 0, Si^p = 0. Graphically, to the former corresponds the closing of the force-polygon, to the latter, in the case of complanar forces, the closing of the funicular polygon. 207. Weight; center of gravity. The most important special case of parallel forces is that of the force of gravity which acts at any given place near the earth's surface in approximately parallel lines on every particle of matter. If g be the acceleration of gravity, the force of gravity on a particle of mass m is w = mg, and is called the weight of the particle or of the mass m. 158 STATICS [208. For a system of particles of masses w-i, iih, • • • nin we have W\ = '^niQ, W2 = rrhg, • • ■ Wn — mnQ- If the particles are rigidly connected, the resultant W of these parallel forces, W = Wi-\- 102+ ■ ■ ■ + Wn = (wh + //?2 + • • • + mn)g = Mg, where M is the mass of the system, is called the weight of the system. The center of the parallel forces of gravity of a system of rigidly connected particles has, by Art. 205, the co-ordinates ' _ _ Xmgx _ _ ^mgy _ _ If^nigz ~ '^mg ' ^ " :^???g ' " 2/?ig ' or since the constant g cancels, _ _ 1,mx _ _ Zmy _ _ llmz ^^^m^' ^~~^i' ^~S^' This point is called the center of gravity of the system, and is evidently identical with the center of mass, or centroid (see Art. 159). For continuous masses the same formulse hold, except that the summations become integrations. The weight TF of a physical body of mass M is therefore a vertical force passing through the centroid of its mass. 3. Theory of couples. 208. The construction given for the resultant of two par- allel forces given in Arts. 201 and 203 fails if, and only if, the given forces are equal and of opposite sense. In this case, the lines pP' and qQ' in Fig. 47, and the lines I and III of the funicular polygon (Fig. 48), become parallel, so that their intersection r lies at infinity. The magnitude of the resultant is of course zero. 208.] STATICS OF THE RIGID BODY 159 The combination of two equal and opposite parallel forces {F, — F) acting on a rigid body is called a couple. A couple is, therefore, not equivalent to a single force, although it might be said to be equivalent to the limit of a force whose mag- nitude approaches zero while its line of action is removed to infinity. The perpendicular distance AB = p (Fig. 49) of the forces of the couple is called the arm, and the product Fp of the force F into the arm p is called the moment of the couple. The moment, or B the couple itself, is also — F called a torque. Notice that the moment of a couple is simply the sum of the moments of its forces about any point in its plane. If we imagine the couple {F, p) to act upon an invariable plane figure in its plane, and if the midpoint of its arm be a fixed point of this figure, the couple will evidently tend to turn the figure about this midpoint. (It is to be observed that it is not true, in general, that a couple acting on a rigid body produces rotation about an axis at right angles to its plane.) A couple of the type {F, p) or {F' , p') (see Fig. 49) will tend to rotate counterclockwise, while a couple of the type {F" , p") tends to turn clockwise. Couples in the same plane, or in parallel planes, are therefore distinguished as to their sense and this sense is expressed by the algebraic sign attributed to the moment. Thus, the moment of the couple {F, p) in Fig. 49 is + Fp, that of the couple {F", p") is - F"p", Fis. 49. 160 STATICS [209. -F Fig. 50. 209. The effect of a couple is not changed by translation, i. c. by moving its plane parallel to itself without rotating it. Let AB = p (Fig. 50) be the arm of the couple {F, p) in its original position, and A'B' the same arm in a new position i:)arallel to the original one in the same plane, or in any par- allel plane. By introducing at each end of the new arm A'B' two oppo- site forces F, — F, each equal and parallel to the original forces F, the given S3'stem is not changed. But the two equal and parallel forces F at A and B' form a resultant 2F at the midpoint of the diagonal AB' of the parallelogram ABB' A'. Similarly, the two forces — F at B and A' are together equivalent to a resultant — 2F at the same point O. These two resultant forces, being equal and opposite and acting in the same line, are together equivalent to zero. Hence the whole sys- tem reduces to the force F at A' and the force — i^ at B' , which form, therefore, a couple equivalent to the orig- inal couple at AB. 210. The effect of a couple is not changed by rotation in its plane. Let AB (Fig 51) be the arm of the couple in the orig- inal position, C its midpoint, and let the arm be turned about C into the position A'B'. Applying again at A', B' equal and opposite forces each equal to F, the forces - F aAA' and F at A will form a resultant acting along CD, while F at B' and - F &i B give an equal and oppo- site resultant along CE. These two resultant forces destroy each other and leave nothing but the couple formed by Fat yl'and - F at B' which is therefore equivalent to the original couple, 212.] STATICS OF THE RIGID BODY 161 Any other displacement of the couple m its plane, or to a parallel plane, can be effected by a translation combined with a rotation in its plane about the midpoint of its arm. The effect of a couple is therefore not changed by any displacement in its plane or to a parallel plane. 211. The effect of a couple is not changed if its force F and its arm j) he changed simultaneously in any ivay, p' provided their product Fp remain the same. ^ Let AB = p) be the original arm ' (Fig. 52), F the original force of the B C -F F A' -F' couple; and let A'B' — p' be the new arm. The introduction of two equal ' -p- rn and opposite forces F' at A' , and also at B', will not change the given system F, — F. Now, selecting for F' a magnitude such that F'jj' = Fp, the force F at A and the force - F' at A' combine (Arts. 201, 203) to form a parallel resultant through C, the midpoint of the arm, since for this point F ■ ^p + {- F') ■ Ip' = 0. Similarly, - F at B and F' at B' give a resultant of the same magnitude, in the same line through C, but of opposite sense. These two resultant forces thus destroying each other, there remains only the couple formed by F' at A' and — F' at B', for which Fp = F'p'. 212. It results from the last three articles that the only essen- tial characteristics of a couple are: (a) the numerical value of the moment; (6) the sense, or direction of rotation; and (c) what has been called the "aspect" of its plane, i. e. the direction of any normal to this plane. It is to be noticed that the plane of the two forces forming the couple is not an essential characteristic of the couple; just as the point of application of a force is not an essential characteristic of the force (see Art. 197); provided, of course, that the couple (or force) is acting on a rigid body. Now the three characteristics enumerated above can all be indi- cated by a vector which can therefore serve as the geometrical repre- sentative of the couple. Thus, the couple formed by the forces F, — F (Fig. 53), whose perpendicular distance is p, is represented by the vector AB = Fp laid off on any normal to the plane of the couple, 12 162 STATICS [213. The sense is indicated by drawing the vector toward that side of the plane from which the couple is seen to rotate counterclockwise. We shall call this geometrical representative AB oi the couple simply the vector of the couple. It is sometimes called its niomcnl, or its axis, or its axial moment. 213. As was pointed out in Art. 208, a couple can be regarded as the limit of a force whose magnitude approaches zero while its line of action is removed to infinity. Similarly, in kinematics an angular velocity whose magnitude tends to zero while its axis is removed in- definitely becomes in the limit a velocity of translation. Fig. .53. Just as, in kinematics (see Art. 122), two equal and opposite angular velocities about parallel axes produce a velocity of translation, so in statics two equal and opposite forces along parallel lines form a new kind of quantity called a couple. It should, however, be noticed that while angular velocities and forces are represented by rotors, i. e. by vectors confined to definite lines, velocities of translation and couples have for their geometrical representatives vectors not confined to particular lines. It is due to this analogy between the two fundamental conceptions that a certain dualism exists between the theories of statics and kine- matics, so that a large portion of the theory of kinematics of a rigid body might be made directly available for statics by simply substituting for angular velocity and velocity of translation the corresponding ideas of force and couple. 215.] STATICS OF THE RIGID BODY 163 214. It is easily seen how, by means of Arts. 209-211, any number of couples acting on a rigid body can be reduced to a single resultant couple. It can also be proved without much difficulty that the vector of the resultant couple is the geo- metric sum of the vectors of the given couples; in other words, vectors reyreseiiting couples acting on the same rigid body are combined by the parallelograin law. In the particular case when the couples all lie in parallel planes, or in the same plane, their vectors may be taken in the same line and can, therefore, be added algebraically. Generally, the resultant of any number of couples is a single couple whose vector is the geometric sum of the vectors of the given couples. Conversely, a couple can be resolved into components by resolving its vector into components. 215. To combine a single force P with a couple {F,p) lying in the same plane it is only necessary to place the couple in its plane in such a position (Fig. 54) that one of its forces, say — F, shall lie in the same line and in opposite sense with the single force P, and to transform the couple {F, p) into a couple (P, p'), by Art. 211, so that Fp = Pp'. The original force P and the force — P of the transformed couple destroying each other at A, there remains only the other force P, at A', of the transformed couple, that is, a force parallel and equal to the original single force P, at the distance i i , p P , A -F ^ v' > < ' ~V~' K > > , -P 1 f Fis. 54. 1G4 STATICS [216. from it. Hence, a couple and a single force in the same plane are together equivalent to a single force equal and parallel to, and of the same sense with, the given force, but at a distance from it which is found by dividing the moment of the couple by the single force. Conversely, a single force P applied at a point A of a rigid body can always be replaced by an equal and parallel force P of the same sense, applied at any other point A' of the same body, in combination with the couple formed by P at A and — P at A'. This follows at once by applying at A' two equal and opposite forces each equal and parallel to P. 216. The proposition of Art. 215 applies even when the force lies in a plane parallel to that of the couple, since the couple can be transferred to any parallel plane without chang- ing its effect. If the single force intersects the plane of the couple, it can be resolved into two components, one lying in the plane of the couple, while the other is at right angles to this plane. On the former component the couple has, according to Art. 215, the effect of transferring it to a parallel line. We thus obtain two non-intersecting, or skew, forces at right angles to each other. Let P be the given force, and let it make the angle a with the plane of the given couple, whose force is F and whose arm is p. Then P sina is the component at right angles to the plane of the couple, while P cosa combined with the couple whose moment is Fp is equivalent to a force P cosa in the plane of the couple; this force P cosa is parallel to the pro- 218.] STATICS OF THE RIGID BODY 165 jection of P on the plane, and has the distance Fp/P cosa from this projection. Hence, in the most general case, the combination of a single force and a couple can be replaced by the combination of two single forces crossing each other {without meeting) at right angles; it can be reduced to a single force only when the force is parallel to the plane of the couple. 4. Complanar forces. 217. If the forces acting on a rigid body all lie in the same plane, i. e. if the forces are complanar, the system can be reduced to a single force and a single couple by applying the last proposition of Art. 215. For, selecting an arbitrary point of the plane as point of reference, we can replace each force F of the system by an equal force F applied at 0, together with a couple Fp, whose arm p is the perpen- dicular from to the line of action of the given force F at P. We thus obtain, in the plane, a number of concurrent forces at which are equivalent to a single resultant R, passing through and equal to the geometric sum of the given forces ; and in addition a numl)er of couples in the same plane which give a single resultant couple, say H = llFp. Notice that the moment H of the resultant couple is simply the sum of the moments about of all the given forces. It follows that the conditions of equilibrium are: 7^ = 0, // = 0; i. e. a system of complanar forces is in equilibrium if, and only if, (a) its resultant is zero, and (h) the algebraic sum of the moments of all its forces is zero about any point in its plane. 218. By Art. 217, a system of complanar forces reduces, for any point of reference in its plane, to a force R and a 166 STATICS I219. couple H. But as these lie in the same plane, it follows from the first proposition of Art. 215 that they can be reduced to a single resultant R (unless R = 0). The distance r of this smgle resultant from is such that Rr = — H; i. e. r = — H/R. The line of action of this single resultant is called the central axis of the system. Thus, a system of complanar forces can always be reduced either to a single force i2 or to a single couple H. 219. For a purelj^ analytical reduction of a plane system of forces the system is referred to rectangular axes Ox, Oy, arbitrarily assumed in the plane (Fig. 55). Every force F is ).Y -X ■^X X / -Y Fig. resolved at its point of application P (x, y) into two com- ponents X, Y , parallel to the axes, so that X = F cosa, Y = F sina, a being the angle made by F with the axis Ox. At the origin two equal and opposite forces X, — X are applied along Ox, and two equal and opposite forces Y, — Y along Oy. Thus, X at P is equivalent to X at together with the couple formed by X at P and — X at 0; the moment of 220.] STATICS OF THE RIGID BODY 167 this couple is evidently — yX. Similarly, F at P is re- placed by F at together with a couple whose moment is xY. The force P at P is therefore equivalent to the two forces X, F at together with a couple whose moment is xY — yX. Proceeding in the same way with every given force, we obtain a number of forces X along Ox whose algebraic sum we call 2X, and a number of forces F along Oy which give 2F. These two rectangular forces form the resultant n = -V(SZ)2 + (2F)2 whose direction is given by SF tana=^^^, where a is the angle between Ox and R. In addition to this, we obtain a number of couples xY — yX whose algebraic sum forms the resulting couple H = SCtF - yX). The whole system is thus found equivalent to a resultant force R together with a resultant couple H in the same plane with R The conditions of equilihrimn R = 0, H = (Art. 217) can therefore be expressed analytically by the three equations SX = 0, SF = 0, Z(xY - yX) = 0. 220. If R be not zero, R and H can be reduced to a single resultant R' equal and parallel to R at the distance — H/R from it (see Art. 218). The equation of the line of this single resultant R', i. e. the central axis of the system of forces, is found by considering that it makes the angle a with the axis of X and that its distance from the origin is H/R = ^{xY - 7/X)/V(:CX)2+ (SF)2. 168 STATICS [221 Hence its equation is ^SF - Tj-SZ - 2(xY - yX) = 0. If i2 = 0, the system is equivalent to the couple H = i:(xY - 7jX). If H itself be also zero, the system is in equilibrium. 221. Exercises. (1) A homogeneous straight rod AB = 21 (Fig. 56) of iveight W rests with one end A on a smooth horizontal plane AH, and with the point E{AE = e) on a cylindrical support, the axis of the cijlinder being at right angles to the vertical plane coritaining the rod. Determine what horizontal force F must be applied at a given point F of the rod {AF = f > e) to keep the rod in equilibrium irhen inclined to the horizon at an angle 6. The rod exerts a certain unknown pressure on each of the supports at A and E, in the direction of the normals to the surfaces of contact, provided there be no friction, as is here assumed. The supports may therefore be imagined removed if forces A, E, equal and opposite to these pressures, be introduced; these forces A, E are called the reactions of the supports. The rod itself is here regarded as a straight line; its weight W is applied at its middle point C. Taking A as origin and AH as axis of x, the resolution of the forces gives •Z.X = F - E sine = 0, (1) sy = A -w + E cose = 0. Taking moments about A, we find E e -W -l cos(9 - F ■ f sine = 0. (2) (3) 221.] . STATICS OF THE RIGID BODY 169 Eliminating F from (1) and (3), we have hence from (2), and finally from (1), _lcosd_ ^ e -f sin20 ^ ' \ e — f sm-0 / jj, _ I sin^ cos9 ^ ~ e -f sm20 (2) A weightless rod AB of length I can tm-n freely about one end A in a vertical plane. A weight W is suspended from a point C of the rod; AC = c. A cord BD attached to the end B of the rod holds it in equilibrium in a horizontal position, the angle ABD being a = 150°. Find the tension T of the cord and the resulting pressure A on the hinge at A. (3) A cylinder of length 21 and radius r rests with the point A of the circumference of its lower base on a horizontal plane and with the point B of the circumference of its upper base against a vertical wall. The vertical plane through the axis of the cylinder contains the points A, B and is perpendicular to the intersection of the vertical wall with the horizontal plane. If there be no friction at A, B, what horizontal force F applied at A will keep the cylinder in equilibrium? When is this force /^ = 0? (4) A weightless rod AB rests without friction on two planes inclined to the horizon at angles a, p, and carries a weight W at the point D. The intersection C of these planes is horizontal and normal to the vertical plane through AB. Find the inclination e of AB to the horizon and the pressures at A and B. (5) A weightless rod AB = I can revolve in a vertical plane about a hinge at A; its other end B leans against a smooth vertical wall whose distance from A \s AD = a. At the distance AC == c from A a weight W is suspended. Find the horizontal thrust A^ at A and the normal pressures Ay and B aX A and B. (6) The same as (5) except that at B the rod rests on a smooth hori- zontal cylinder whose axis is at right angles to the vertical plane through AB. In which of the two problems is the horizontal thrust A at A least? 170 STATICS . [222. 5. The general system of forces. 222. To reduce any system of forces acting on a rigid body to its most simple form the same methods are used as for complanar forces (comp. Art. 217) Selecting as origin any point rigidly connected with the body, let two equal and opposite forces F , — i^ be applied at 0, for every one of the given forces F. The effect of the given system of forces on the body is not changed by the introduction of these forces at 0. But we may now regard the given force F acting at its point of application P as replaced by the equal and parallel force F at 0, in combination with the couple formed by the original force F at P and the fcroe — i^ at 0. All the forces of the given system are thus transferred to a common point of application 0, and these forces at can be replaced by a single resultant* R, passing through and represented in magnitude and direction by the geometric sum of the forces. In addition to this resultant R, we obtain as many couples {F, — F) as there were forces given; and their resultant is found by geometrically adding the vectors of the couples (Art. 214). Thus the given system of forces is seen to be equivalent to a resultant R in combination with a couple whose vector we shall call H; in other words, it has been proved that any system of forces acting on a rigid body can be reduced to a single resultant force in combincdion with a single resultant couple. It follows at once that the geometrical conditions of equilibrium are* R = 0, H = 223. Of the two geometrical elements representing a general system of forces, viz. the rotor R and the vector H, the for- 224.1 STATICS OF THE RIGID BODY 171 mer being merely the geometric sum of the forces, is inde- pendent of the point of reference 0, while the vector H is in general different for different points of reference. If the elements R, H for a point are given, those for any other point 0' can readily be found. It suffices to apply at 0' equal and opposite forces R and — R. We then have R at 0' , and two couples, viz. the couple whose vector is H and the couple formed by 7^ at and — RaXO'; the resultant of the vectors of these two couples is the vector H' corre- sponding to 0'. Here, as well as in the following articles, it is assumed that R ^ 0; when R = Q the system reduces to a couple, the same whatever the point of reference. If the new point of reference 0' had been selected on the line I of the original resultant, no new couple would have been introduced, and H would not have been changed. But whenever the new point of reference 0' is taken on a line V different from I, the vector of the resultant couple H is changed. By increasing the distance r between I and V the moment Rr of the additional couple is increased. The effect of com- bining this additional couple Rr with H is, in general, to vary both the magnitude of the resulting vector H' and the angle ^ it makes with the direction of the resultant R. It can be shown that the line V of the new resultant can always be selected so as to reduce the angle to zero. The line ?o for which = 0, i. e. for which the vector H of the resultant couple is parallel to the resultant force R, is called the central axis of the given system of forces. We proceed to show how it can l)c found (comp. Art. 123). 224. Let the vector H be resolved at into a component Ho = H coscf) along I, and a component Hi = H sin<^, at 172 STATICS (224. right angles to I (Fig. 57). In the plane passing through I at right angles to Hi, it is always possible to find a line lo parallel to Z at a distance ro from I, such as to make Rro = -Hi. The line Zo so determined is the central axis. For, if this line be taken as the line cf the resultant R, the additional Ht- Fig. 57. Fig. 58. couple Rro destroys the component Hi, so that the resulting couple ^0 has its vector parallel to R. As the direction of the vector H is always changed in passing from line to line, there can be but one central axis for a given system of forces. It appears from the construction of the central axis given above, that the vector of the resulting couple for this axis Zo is Ho = H cos0; it is, therefore, less than for any other line. It is mstructive to observe how the vector H increases and changes its direction as we pass from the central axis Zo to any parallel line I. 226.J STATICS OF THE RIGID BODY 173 The transformation from lo to I requires the introduction of a couple whose vector Rro (Fig. 58) is at right angles to the plane {U, I) and combines with Hq to form the resulting couple H for I. As the distance ro of I from Zo is increased, both the magnitude of H and the angle it makes with I increase, the angle 4> approaching ^-tt as ro becomes infinite. 225. It is evident that since Ho = H cos0, the product RH COS0 is a constant quantity for a given system of forces. It may be called an invariant of the system. If the elements of reduction for the central axis R, Ho be given, those for any parallel line I at the distance ro from the central axis are determined by the equations H^ = Ho' + RW, tancp = ■-- . no To sum up the results of the preceding articles, it has been shown that any system of forces acting on a rigid body can be reduced, in an infinite number of ways, to a resultant R in combination with a couple H. For all these reductions the magnitude, direction, and sense of the resultant R are the same, but the vector H of the couple changes according to the position assumed for the line of R. There is one, and only one, position of R, called the central axis of the system, for which the vector H is parallel to R and has at the same time its least value, Ho] this value Ho is equal to the projec- tion of any other vector H on the direction of the resultant R. 226. While, in general, a system of forces cannot be reduced to a single resultant, it can always be reduced to two non- intersecting forces. This easily follows by considering the system reduced to its resultant R and resulting couple H for any point (Fig. 59). Let F, — F be the forces, p the arm of the couple H, and place this couple so that one of the 174 STATICS [227. forces, say — F, intersects R at 0. Then, if R and — F be replaced by their resultant F' , the given system of forces is evidently equivalent to the two non-intersecting forces F, F' (compare Art. 216). The two forces F, F' determine a tetrahedron OABC; and it can be shown that the volume of this tetrahedron is constant and equal to one sixth of the invariant of the system (Art. 225). The proof readily appears from Fig. 59. The volume of the tetrahedron OABC is evidently one half of the volume of the quadrangular pyramid whose vertex is C and whose base is the parallelogram ODAB. The area of this parallelogram is Fp = // ; and the altitude of the pyramid is = 7^ cos0, being equal to the perpendicular let fall from the extremity of R on the plane of the couple; hence the volume of the tetra- hedron = IRH COS0 = iRHo. 227. To effect the reduction of a given system of forces analyt- ically, it is usually best to refer the forces F and their points of application P to a rectangular system of co-ordinates Ox, Oy, Oz (Fig. 60). Let x, y, z be the co-ordinates of P and X, Y, Z the components of F parallel to the axes. 227.1 STATICS OF THE RIGID BODY 175 To transfer these components to and at the same time to introduce only couples whose vectors are parallel to the axes, we proceed in two steps. Thus to transfer, say X, we introduce at P' , the foot of the perpendicular let fall from P on the plane zx, two equal and opposite forces X, — X; and we do the same thing at 0. Then the single force X at P is replaced by the force A' at in combination with the two couples formed by X at P, — X at P' , and X at P', — X Fig. 60. at 0. The vector of the former couple is parallel to Oz, its moment is — yX; the negative sign being used because for a person looking on the plane of the couple from the positive side of the axis Oz the couple rotates clockwise. The vector of the latter couple is parallel to Oij, and its moment is zX. The transfer of Y to the origin requires the introduction of two couples, — zY having its vector parallel to Ox and xY having its vector parallel to Oz. Finally, transferring Z to O, we have to introduce the couples — xZ with a vector parallel to Oij, and yZ with a vector parallel to Ox. 176 STATICS [228. Thus each force F is replaced by three forces, X, Y, Z along the axes of co-ordinates and applied at 0, in combination with three couples whose vectors are yZ — zY parallel to Ox, zX — xZ parallel to Otj, xY — yX parallel to Oz. 228. If this be done for every force of the given system and the components having the same direction be added, the system will be found equivalent to the three rectangular forces SX, ZY, ZZ, applied at 0, together with the three couples Z{yZ-zY), i:{zX-xZ), Z{xY-yX), whose vectors are at right angles. The three forces can now be replaced by a single resultant R = V(2ZF+l2F)2TT2Z)2, whose direction is determined by the angles a, /3, 7 which it makes with the axes Ox, Oy, Oz: SX ^27 SZ COSa = -^ , COS/S = —^, COS7 = -^. In the same way the three couples can be replaced by a single resulting couple whose moment is H = V[S(?/Z - zY)Y + [S(2X - xZ)Y + [Z{xY - yX)]\ 229. Since R~, as well as H"-, is thus found as the sum of three squares, each of these quantities can vanish only if the three squares composing it vanish separately. The con- ditions of equilibrium of a rigid body (Art. 222) are therefore expressed analytically by the following six equations: 2X = 0, 27 = 0, 2Z = 0, 2(2/Z - zY) = 0, 2(zX - xZ) = 0, 2(x7 - yX) = 0. 230.] STATICS OP THE RIGID BODY 177 As the system of co-ordinates can be selected arbitrarily, the meaning of the first three equations is that the sum of the components of all the forces along any three lines not parallel to the same plane must vanish. The last three equations express that the sum of the moments of all the forces about any three axes not parallel to the same plane must also vanish. The moment of a force about an axis must be understood as meaning the moment of its projection on a plane at right angles to the axis with respect to the point of intersection of the axis with the plane. This definition is in accordance with the somewhat vague notion of the moment of a force as representing its " turning effect." For, if we regard the force as acting on a rigid body with a fixed axis, the force can be re- solved into two components, one parallel, the other perpen- dicular, to the axis; the former component evidently does not contribute to the turning effect which is, therefore, measured l^y the moment of the latter alone. 230. The equations of the central axis (Art. 223) can be found by a transformation of co-ordinates. Let the system be reduced for any point to its resultant R, whose rectangular components we denote by A = 2X, B = ^Y, C = 2Z, and to the vector H of its resulting couple with the com- ponents L = 2(7/Z - zY), M = 2(2X - xZ), N = 2(a;7 - yX). If a point 0' whose co-ordinates are ^, rj, f be taken as new point of reference and the co-ordinates of any point with respect to parallel axes through 0' be denoted by x', y', z', we have x = ^-\-x',y = 'n-[-y',z = ^-\-z'. Substituting these values, we find 13 178 STATICS [231. L = 2[(7, + y')Z - (r + z') Y] = v^Z - ^ZY + :c(^'Z - z'Y) = r,C - ^B^ L', where L' is the .r-component of the couple H' resulting for 0' as point of reference. Similar expressions hold for M and A''. The components of H' are therefore U = L-7iC + ^B, M' = M - f A + ^C, N' =^N-^B+7]A; and its direction cosines are H" ^ ~ ^" " 7 //' ■ The central axis being defined (Art. 223) as that line for which the vector of the resulting couple is parallel to the direction of the resultant, the point 0'{^, t], ^) will lie on the central axis if the direction cosines of H' are equal to those of R, viz. to a = A/R, 13 =^ B/R, y = C/R. Hence the equations of the central axis are L' ^ ilf ' ^ N^ A ~ B C ' that is, L-r]C -\-^B ^ M - rA + ^C _ N - ^B -{-yA A B C 6. Constraints; friction. 231. It has been shown in Art. 229 that the number of the conditions of equilibrium is six, for a rigid body that is perfectly free. This number will be diminished whenever the body is subject to conditions restricting its possible motions. Such conditions, or constraints, may be of various kinds; the body may have a fixed point, or a fixed axis, or one of its points may be constrained to move along a given curve or to remain on a given surface, etc. 232.] STATICS OF THE RIGID BODY 179 Now a free -point is said to have three degrees of freedom because its position is determined by three co-ordinates. One conditional equation between its co-ordinates restricts the point to the surface represented by that equation; it has then one constraint and two degrees of freedom. Two con- ditions restrict the point to a curve, viz. the intersection of the two surfaces represented by the two equations of con- dition; the point then has two constraints and one degree of freedom. The position of a rigid body is determined by the position of any three of its points, not in a Hne, i. e. by nine co-ordi- nates between which, however, there exist three conditions, expressing the constancy of the distances of the three points. A free rigid body has therefore six degrees of freedom, since six independent quantities determine its position. The most general instantaneous state of motion that a free rigid body can have is a twist, or screw-motion (Art 123), consisting of an angular velocity about a certain axis and a linear velocity along this axis; each of these velocities has three components along the rectangular axes, and these six components can be regarded as the six independent possible motions of the body, on account of which it is said to have six degrees of freedom. Equilibrium will exist only when these six possible motions are prevented; hence there must be six conditions of equi- librium. 232. We proceed to consider some forms of constraint and the corresponding changes in the equations of equi- Hbrium. It is often convenient in dynamics to replace such re- straining conditions by forces, usually called reactions. Whenever it is possible to introduce such forces having the 180 STATICS [233. same effect as the given conditions, the body may be re- garded as free, and the general equations of equihbrium can be apphed. 233. Rigid Body with a Fixed Point. A body that is free to turn about a fixed point A can be regarded as free if tlie reaction A of this point be introduced and combined with the other forces acting on the body. Let Ax, Ay, Azhe the components of A; then, taking the fixed point A as origin, the six equations of equihbrium (Art. 229) are ZX +Ax = 0, 27 + Ay =0, ^Z +A, =0, ^{yZ - zY) = 0, S(2Z - xZ) = 0, Z{xY - yX) = 0. B, >fA, The first three of these equations serve to determine the reaction of the fixed point; tlie last three are the actual conditions of equilibrium corresponding to the three degrees of freedom of a body with a fixed point. Hence, a rigid body having a fixed point is in equilibrium if the sum of the moments of all the forces vanishes for any three non-com-planar axes passing through the fixed point. 234. Rigid Body with a Fixed Axis. A body with a fixed axis has but one degree of freedom; indeed, the only possible mo- tion consists in rotation about this axis. An axis is fixed as soon as two of its points, say A, B, are fixed. Hence, after intro- ducing the reactions Ax, Ay, Az, Bx, By, Bz, of these points, the body can be regarded as free. If the point B be taken as origin, the line BA as axis of z (Fig. 61), the equations of equilibrium become SZ + Ax + fix = 0, 27 + A,i + By = 0, 2Z + Az + Bz = 0, -LiyZ - zY) - Aya = 0, 2(2X - xZ) -f Am = 0, 2 (a: 7 - yX) = 0, where a = BA. The last of the six equations is the only independent condition of A.- y/B. Fig. 61. 236.] STATICS OF THE RIGID BODY 181 equilibrium of the constrained body; the first five determine Ax, Bx, Ay, By, Az + Bz. The two 2-components cannot be found separately, since they act in the same straight line. Hence, a rigid body having a fixed axis is in equilibrium if the sum of the moments of all the forces vanishes for the fixed axis. 235. If, in the preceding article, the axis be not absolutely fixed, but only fixed in direction so that the body can rotate about the axis and also slide along it, we have evidently Az = 0, Bz = 0; hence, by the third equation of equilibrium. 2Z = 0, as an additional condition of equilibrium. The body has in this case two degrees of freedom. 236. Rigid Body with a Fixed Plane. A body constrained to slide on a fixed plane (that is, to move so that the paths of all its points lie in parallel planes) has three degrees of freedom. At every point of contact between the body and the plane, the latter exerts a reaction. As all these reactions are parallel, they are equivalent to a single resultant N. Taking the fixed plane as the plane xy, N will be parallel to the axis of z; hence, if o, b, be the co-ordinates of its point of application, the six equations of equilibrium are 2X = 0, Si' = 0, XZ + N = 0, •ZiyZ - zY) +bN = 0, i:(zX - xZ) - aN = 0, Z{xY - ijX) = 0. The third, fourth, and fifth equations determine the reaction N and the co-ordinates a, b of its point of apjjlication. The three other equations are the actual conditions of equilibrium; they agree, of course, with the three conditions of equilibrium of a plane system as found in Art. 219. If there be not more than three points of contact (or supports) between the body and the fixed plane, the reactions of these points can be found separately. Let Ai, A2, A3 be the three points of contact; Ni, N2, Ns the required reactions; ai, b^, «•>, b^, az, bs the co-ordinates of Ai, Ai, As; then A^" must be resolved into three parallel forces passing through these points, and the conditions are 182 STATICS [237. Ni + N2 + N3 = N, UiNi + a^Ni + a3A'"3 = aN , biNi + b2N2 + hsNs = bN. These three equations always determine A'^i, N^, N3. For if the determinant of the coefficients of Ni, N2, Ns vanished, 1 1 1 cti a2 as bi bi bs = 0, the three points Ai, A2, A3 would lie in a straight line, and hence the body would not be properly constrained. The reactions become indeterminate whenever there are more than three points of contact. 237. Friction. The reaction between two surfaces in contact has so far been regarded as directed along the -com- mon normal of the surfaces (Art. 195). If this is true the surfaces are said to be perfectly smooth. The surfaces of physical bodies are rough, i. e. they pre- sent small elevations and depressions; when two such sur- faces are " in contact " the projections of one will more or less enter into depressions of the other; the greater the normal pressure between the surfaces, the more will this be the case. Hence when a tangential force acting on one of the bodies tends to slide its surface over that of the other body, a resistance will be developed whose magnitude must depend on the roughness of the surfaces and on the normal pressure between them. This resistance is called the force of sliding friction, or simply the friction. The study of friction belongs properly to applied mechan- ics, and will here only be touched upon very briefly. 238. Imagine a body resting with a plane surface on a horizontal plane. Let a small horizontal force P be applied at its centroid (which is supposed to be situated so low that 239.] STATICS OF THE RIGID BODY 183 the body is not overturned), and let the force P be gradu- ally increased until motion ensues. At any instant before motion sets in, the friction is equal to the value of P at that instant. The value of P at the moment when motion just begins is equal and opposite to the frictional resistance F between the surfaces at this moment, and this resist- ance is called the limiting static friction. Careful experiments with dry solids in contact have shown this force to be subject to the following laws: (1) The magnitude of the limiting friction F hears a con- stant ratio to the normal pressure N between the surfaces in contact; that is F = fj^N, where /i is a constant depending on the condition and nature of the surfaces in contact. This constant which must be determined experimentally for different substances and surface conditions is called the coeflEicient of static friction. It is in general a proper fraction; for jDerfectly smooth sur- faces ;u = 0. (2) For a given normal pressure the limiting static friction, and hence the coefficient of static friction, is independent of the area of contact, provided the pressure be not so great as to produce cutting or crushing. 239. The frictional resistance between two surfaces in relative motion is called kinetic friction. It is subject, in addition to the two laws just mentioned, to the third law: (3) For moderate velocities, kinetic friction is nearly inde- pendent of the velocities of the bodies in contact. The coefficient of static friction is somewhat greater than that of kinetic friction. A slight jarring will often reduce the coefficient from its static to its kinetic value. 184 STATICS 1240. It must not be forgotten that these so-called laws of friction are experimental laws, and therefore true only ap- proximately and within the limits of the experiments from which they were deduced. When the relative velocity of the surfaces in contact is high, or when, as is usually the case in machinery, a lubricating material is introduced between the two surfaces, the frictional resistance is found to depend on a number of other circumstances, such as the temperature, the form of the surfaces, the velocity, the nature of the lubricator, etc. Indeed, when the supply of the lubricant is sufficient, the two solid surfaces are kept by it out of actual contact; the coefficient of friction in this case varies w4th the pressure, area of contact, velocity, and temperature. 240. Consider again a body resting on a horizontal plane (Fig. 62) and acted upon by a horizontal force P just large enough to equal the limiting friction F. The normal reaction N of the plane is equal and opposite to the weight W. The body is thus in equilibrium under the action of the two pairs of equal and opposite forces; but motion will ensue as soon as P is increased. If P be decreased, F will decrease at the same rate, so that the equilib- rium remains undisturbed. The force of friction F can be combined with the normal reaction A'' to form a resultant, R VF^ + m = VP^ + TF2, which represents the total reaction of the horizontal plane. If 4, be the angle between N and R when F has its limiting value F = fxN (Art. 238), we have, since tan^ = F/N, tan0 = M- 242. STATICS OF THE RIGID BODY 185 The angle ^ thus furnishes a graphical representation for the coefficient of friction /x; it is called the angle of friction. 241. If the plane be not horizontal, but incHned to the horizon at an angle d, the weight W of the body (regarded as a particle) resting on the plane can be resolved into a component W sin9 along the plane, and a component W cos9 perpendicular to it (Fig 63). Hence, if no other forces act on the body it will be in equilibrium, provided the component W sin9 be not greater than the hmiting friction F = nW cos9. The limit- ing condition of equilibrium is therefore. ixW cosO = W sin0, or n = tan9; Fig. 63. in other words, if the angle be gradually increased, the body will not sUde down the plane until d > . This furnishes an experimental method of determining the angle of friction . Determine the required tractive force P if it is to act at an inclination a to the horizon, and show that this force is least when a. = 4>. (2) A particle ot weight W is in equilibrium on a rough plane in- clined to the horizon at an angle d, under the action of a force /-* parallel to the plane along its greatest slope. Determine P: (a) when > 0, (6) when B = 4>, (c) when < (j>, = tan'^/u being the angle of friction. (3) Solve Ex. (2) (o) graphically by means of the friction angle and determine what part of P is required to overcome friction. (4) A body weighing 240 pounds is pulled up a plane inclined at 45°, by means of a rope. If m = li, find the tension of the rope. What portion of it is due to friction? (5) A homogeneous straight rod AB = 21 of weight W rests with one end A on a horizontal floor, with the other end B against a vertical wall whose plane is at right angles to the vertical plane of the rod. If there be friction of angle 4> at both ends, determine the limiting position of equilibrium. (6) A straight homogeneous rod AB =21, of weight W, rests with the lower end A on a rough horizontal plane and with the point C {AC = c) on a smooth cylindrical support. The rod is in equilibrium when inclined at a given angle 6 to the horizon ; determine the coefficient of friction at A and the reactions at A and C. (7) If in Ex. (6) there be friction both at A and C, the friction angle <^ being the same, find the position of equilibrium and the reactions at A and C. (8) A solid homogeneous hemisphere is placed with its curved surface on a rough inclined plane; investigate the conditions of equilibrium. CHAPTER XII. THEORY OF ATTRACTIVE FORCES. 1. Attraction. 244. Among the various kinds of forces introduced in physics for describing and interpreting natural phenomena, forces of attraction and repulsion occupy a most prominent place. According to Newton's law (the law of universal or cosmical gravitation, the " law of nature ") every particle of matter attracts every other such particle with a force proportional to the masses and inversely proportional to the square of the distance of the particles, and this force acts along the line joining the particles. Thus, if m, m' are the masses of the particles, r their dis- tance, and K a constant, the force with which m attracts m' and fn' attracts in is ^ 7nm' F = K- 2 . Each particle is here regarded as a mass concentrated at a point; otherwise we could not speak of the distance of the particles and of the line joining them (comp. Art. 156). As the distance r approaches zero, the magnitude of the force F becomes infinite and its direction indeterminate. 245. In the theory of gravitation, the masses m, m' are essentially positive. The constant k, called the constant of gravitation, evidently represents the force with which two particles, each of mass 1 , attract each other when at the 187 188 STATICS [246. distance 1. It is a physical constant to be determined by experiment, and its numerical value depends on the units of measurement adopted for mass, length, and time. What can be directly observed is of course not the force itself, but the acceleration it produces. Dividing the force F (Art. 244) by the mass m of the attracted particle on which it acts we have the acceleration j produced by the force with which m' attracts m at the distance r from rn: m' 246. It will be shown later (Art. 253) that the attraction of a homo- geneous sphere at any external point is the same as if the mass of the sphere were concentrated at its center. Hence if m' be the mass of the earth (here regarded as a homogeneous sphere) the acceleration it produces in any mass m concentrated at a point P above its surface, at the distance OP = r from the center 0, is j = Km'/r^. Now for points near the earth's surface this acceleration is known from experi- ments; it is the acceleration ^ of a body falling in vacuo (apart from the slight effect due to the earth's rotation, see Arts. 334, 461). Hence, taking the radius of the earth as r = 6.37 X 10« cm., its mean density as p = 5K, and g = 980 cm./sec.^, we find in C.G.S. units K = 6.7 X 10-8. This, then, is the force in dynes with which two masses, of 1 gram each, would attract each other if concentrated at two points 1 cm. apart. Conversely, the mean density of the earth can be found with con- siderable accuracy by a direct experimental determination of the attrac- tion of gravitation between two given masses at a given distance. 247. Exercises. (1) With r = 3960 miles, g = 32 ft. /sec. 2, p = 5H, show that the attraction between two masses of 1 lb. each, at a distance of 1 ft., is equal to the weight of 0.33 X lO-i" lb. (2) In astronomy and in the general theory of attraction it is con- venient to take the unit of mass so that k = 1. Show that this astro- nomical unit of mass, i. e. the mass which acting on an equal mass at unit distance would produce unit acceleration, is = l//c. 248.] THEORY OF ATTRACTIVE FORCES 189 (3) Show that k = 1 if, with the ordinary unit of mass, the unit of time be taken as 3862 sec. This has been called the "natural hour." 248. If more than two particles are given the forces of attraction exerted on any one of the particles, m, being con- current are equivalent to a single resultant. This resultant, divided by the mass m of the attracted particle, is called the attraction at the point P where m is situated. If, instead of a finite number of particles, any continuously distributed masses of one, two, or three dimensions (Art. 155) are given they can be resolved into elements which in the limit can be regarded as particles. The first problem in the theory of attraction consists in determining the attraction at any point, due to any given masses. Notice that the ''attraction at any point," as thus defined, has the dimensions of an acceleration and not of a force. Let P(x, y, z) be the attracted point of mass 1, dm' an element of the attracting masses at Q{x' , y' , z'), PQ = r the distance of these points; then the attraction at P due to dm' is Kdm'/r^, and if a, |3, y are its direction cosines, its components are Kadm'/r^, K^dm'/r^, Kydm'/r^. Hence the attraction A at P, due to all the given masses, has the rec- tangular components: X /adm' ,, rfidm' „ rydm' with r^ = {x' — xy -\- {y' — y)~ + {z' — z)-, the integrations extending over all the masses. The attraction A itself and its direction cosines I, m, n are: X Y Z A^ ^X'- + Y^ + Z\ I =~, m==-r, n= , . ' A ' A ' A It is in general most convenient to take the attracted point P as origin so that r"^ = x'"^ + y'- + z'- 190 STATICS [249. 249. If the point P were situated within the attracting masses, l/r^ would become infinite within the hmits of integra- tion; hence a special investigation would be necessary to determine whether the integrals representing A", Y, Z have a meaning. It can be shown without difficulty in the case of three-dimensional masses that the integrals have a meaning and represent the attraction even at an internal point P. But for the sake of simplicity, we here confine ourselves to the external field. In other words, we assume, when nothing is said to the contrary, that P is an external 'point, i. e. a point such that a sphere can be described about it such as not to contain within it any portion of the attracting matter (except the unit mass at P itself). 250. The problem of attraction can be generalized in various ways. Thus, in electricity and magnetism, we have to consider both positive and negative masses, and the force may be a repulsion as well as an attraction. The force be- tween two electric charges as well as that between two magnetic poles follows Newton's law (Art. 244); i. e. the force is directly proportional to the charges, or pole-strengths, and inversely proportional to the square of the distance. But the constant k has a very different value. It is cus- tomary to select the units of electric charge and magnetic pole-strength so that /c = 1. It is sometimes necessary to consider forces that do not fol- low Newton's law of the distance. Indeed, Newtonian attrac- tion is merely a particular case of the more general type of force F = Kmm'fir), viz. the case when/(r) = 1/r^. 251. Spherical shell. The attraction due to a mass spread uniformly over a sphere is zero at any point within the sphere, while at any outside point it is the same as if the mass were concentrated at the center. 251.] THEORY OF ATTRACTIVE FORCES 191 Geometrical method, (a) Attraction at an inside point. Let C be the center, a the radius of the sphere (Fig. 65). A cone of vertex P and solid angle dw {i. e. cutting out an area element dw on the sphere of radius 1 about P as cen- ter) cuts out on the given sphere a surface element da at Q and a surface element da' at Q'. It wUl be shown that the mass elements on these surface elements produce equal and opposite attractions at P. As the whole sphere can thus be divided into pairs of elements whose attractions at P balance it follows that the attraction at P is zero. Put PQ = r, PQ' = r'\ on the sphere of radius r about P the cone cuts out an element r-c/co at Q, and we have evidently da = r^dw/cosCQP; hence if the surface density is p', the mass on da is p'r'^dw/cosCQP, and the attraction at P due to this mass is up'dw/cosCQP. In the same way we find that the mass on da' at Q' produces at P the attraction Fig. 66. Kp'du/cosCQ'P. As for the sphere the angles CQP and CQ'P are equal, the attractions are equal. (6) Attraction at an outside point. Let P' (Fig. 66) be the point 192 STATICS [251. inverse to P with respect to the given sphere, i. e. the point P' on CP such that if CP = p, CP' = p', we have pp' = a^. The extremities Q, Q' of any cliord tlirough P' determine with C, P, P' two pairs of similar triangles: CQP' and CPQ, CQ'P' and CPQ'; for, each pair has the angle at C in common and the sides including the equal angles proportional owing to the relations pp' = a^, CQ = CQ' = a. It follows that 2^ CQP' = CPQ, 2^ CQ'P' = CPQ'; hence, as the triangle QCQ' is isosceles, the line CP bisects the angle QPQ'. With the aid of these geometrical properties it can be shown that equal attractions are produced at P by the masses on the elements da at Q and da' at Q', cut out by a cone of solid angle dw with vertex at the point P' inverse to P. For the mass elements at Q, Q' we have as in the case (a) : , r-dci , , , , , r'-dw dm=pda=p ^^^^Q^, dm =pdcr =p -^^^jq^, , where r = P'Q, r' = P'Q'. Hence the corresponding attractions at P are: Kp'rHoi Kp'r"^do} PQ^ cosCQP' ' PQ'2 cosCQ'P" and these are equal, since for the sphere ^ CQP' = CQ'P', and the similar triangles give r a ^' _ i? PQ~ p' W ~ P' As shown above; these attractions make equal angles with PC', hence their components along this line are equal while their components at right angles to CP are equal and opposite. The two elements da at Q and da' at Q' produce therefore together at P an attraction along PC equal to 2/cp'rt^dw V' The coefficient of dw is constant; the summation over the unit sphere gives J"dw = 2ir, since a double cone was used. Hence the total attraction at P is A A lO^ "*' P' p^ 252.] THEORY OF ATTRACTIVE FORCES 19.3 where m' = 47rp'a- is the whole mass on the sphere. This shows that the attraction is the same as if this mass were concentrated at the center of the sphere. (c) AttracHon at a poinl on the sphere. If the point P approaches the surface from within the attrac- tion remains constantly zero; if P approaches the surface from without the attraction approaches the limit Km'/a'^ = iwKp'. At a point P on the sphere (Fig. 67) the attraction can be shown to be A = 27r/cp'. For, the mass on da at Q is p'da = Fig. 67 p'r^du/cosCQP; its attraction at P is = Kp'dcj/cosCQP, and as the angles at P and Q are equal, the projection of tliis attraction on PC is Kp'doi. As P lies on the surface, f doi = 2-k; hence the total attraction is = I-kkp . The attraction exerted by the whole mass on the mass element p'da situated at P is of course = 2i^Kp''^d, Fig. 6S. Q being any point of the sphere (Fig. 68), and as before CP = p, CQ = a. As mass element take the mass p' ■ 2x0 sin<^ • ad(t>, contained between the plane through Q at right angles to PC and an infinitely near parallel plane. The attraction produced at P by this element is directed along PC and 14 194 STATICS 1254. , „ . , cosCPQ „ , „ . J p — a coscf) = 2Tr Kp a- sma ■ ^ — = ^ttkt) a- sin<^a0 • ^ , where J.2 = q2 _j_ p'i _ 2ap co.s their expressions from these relations we find for the attraction of the ring element at P: , a p^ — a- + r'^ J TTKp' — • dr. (a) For an inside point P we have p < a, and the limits of integration for r are from a — p to a + p. Hence the resultant attraction at P is A = TTKp' - I I — , + 1 I f/r = TT/v'p' , I +r) =0. p2 J„-p \ r- J p' \ r J.,-p For an outside point P we have p > a and the limits are from p — a to p + a; hence , a / a- — p- , \?'+« , , d? vn' A = TTKp' , — +r) = iTTKp' - , = K - Y ■ p^ \ r Jp-a p^ p^ 253. From the results of Arts. 251, 252, it readily follows that the attraction due to a homogeneous solid shell (mass between twoconcentric spheres) is zero within the hollow of the shell, while at an outside point it is the same as if the mass were concentrated at the center. It suffices to resolve the shell into concentric shells of infinitesimal thickness da and put p'da = p, the volume density. In particular, for a homogeneous solid sphere of radius a and volume density p the attraction at the distance p > a from the center is . ?n' 4 , o3 p^ 6 p^ 254. Exercises, (1) Show that the results of Art. 253 hold for a solid shell whose density is any function of the distance from the center. (2) By Art. 252, the attraction due to a mass distributed uniformly over a sphere when considered as a fimction of p has a point of dis- continuity; illustrate this by a sketch. (3) Prove that the attraction at the center due to a mass distributed uniformly along a circular arc of angle la and radius a is = 2/cp" sina/a; show that a mass equal to that of the chord, if it had the same density 254.1 THEORY OF ATTRACTIVE FORCES 195 p", placed at the midpoint of the arc, would produce the same attraction at the center. (4) Prove that the attraction of a homogeneous rectilinear segment A1A2, at a point P whose perpendicular distance PO from A1A2 makes the angles di, 62 with PAi, PA2, bisects the angle A1PA2 and has the value 2kp" sins (^2 — 6i)/p. Show that the arc of the circle of radius PO = p about P, bounded by PAi and PA2, if of the same density, produces at P the same attraction. (5) Show that in any plane through A1A2 the confocal hyperbolas having A1A2 as foci are the lines of force in the field of the rectilinear segment; i. e. they have the property that the attraction at any point P is tangent to the hyperbola through P. (6) Show that for a homogeneous rod of infinite length the attraction at any point is normal to the rod and inversely proportional to the distance from the rod. Hence show that the attraction due to a homogeneous circular cylinder, of radius a and infinite length, at any point P at the distance PC = p > a from the axis, is = 2irKpa-/p. (7) Prove that the attraction due to a mass spread uniformly over the area of a circle of radius a, at a point P on the axis of the circle, at the distance PC = p from the center C, is = 2irKp'il — p/Va^ + P')- (8) Two parallel homogeneous straight rods of equal density p" are placed so that the line joining their midpoints is at right angles to each; if their lengths are 2a, 25, and their distance is c, find their mutual attrac- tion, i. e. the force required to hold them apart. (9) Show that the attraction exerted by a homogeneous right circular cone of vertical angle 2a and height h, at the vertex, is = 2irKph{l — cosa). Show that the same expression holds for a frustum of height h and angle 2a. (10) Two equal circular disks, of radius a, are placed at right angles to the line joining their centers whose distance is c. If one attracts while the other repels, determine the resultant force at a point P on the line of the centers, at a distance p from the nearer center. Wliat becomes of this force when c is indefinitely diminished? (11) Show that the attraction of a homogeneous solid hemisphere at a point on its edge is = f^Kpa i/tt^ + 4, and that it is inclined to the base at an angle of about 32J^°. 196 STATICS [255. 2. The potential. 255. As shown in Art. 248, the determination of the at- traction, due to given masses, at any particular point P is a mere problem of integration. The next problem that pre- sents itself in the theory of attraction is to express the attraction A as a function of the point P, or rather the com- ponents X, Y,Z oi A as functions of the co-ordinates x, y, z of P, and to study the nature of these functions. The solu- tion of this problem is greatly facilitated by observing that there exists a function C/, known as the potential of the given masses, which has the property that the comyonents of A are its first partial derivatives : dU dU dy' Z = dz A function having this property may exist for forces that are not Newtonian attractions; it is then called a force- function. Forces for which a force-function exists are called conservative forces. 256. Let us consider the most simple case of Newtonian attraction, viz. the field generated by a single particle m', situated at Q (Fig. 69). The attraction at P(x, y, z), due 257.] THEORY OF ATTRACTIVE FORCES 197 to m' at Q,{x\ y', z') is A = kvi' Ir"^, where r'^ = {x — x'Y + {y — y'Y -{- iz — z'y. As this attraction has the sense from P toward Q, its direction cosines are — {x — x')lr, — {y — y')l'>'j — (2 — z')lr; hence the rectangular components of the attraction are : X =- Kill' ^^', Y =- Km' '^^, Z =- km' ^^. It is easily verified that these expressions are the partial derivatives with respect to x, y, z of one and the same func- tion, viz. r this then is the potential of a single particle m'. 257. Notice that this function is one-valued and con- tinuous throughout the whole of space, except at Q where it has a simple pole {i. e. U becomes infinite like 1/r for r = 0), and that it vanishes at infinity. The same properties hold for all derivatives of U except that Q becomes a pole of higher order. For the projection of the attraction A on any direction s we have . _ ydx ydy ydz __ dU dx dU dy dU dz _ dU _ ds ds ds dx ds dy ds dz ds ds ' i. e. the s-component of A is the s-dcrivative of U. For the second x-derivative of U we have since dr/dx = (x — x')/r: §^ _dX _ _ , / 1 X - x' dr\ dx^ dx \r^ r* dx/ ,ri 3(x - x'y i and similarly: 198 STATICS [258. STATICS d'~U dY dy' dy — Km' 1 3(?/ - y'Y dz"- dz — Km' "1 3(z- z'y~\ ^6 ■]■ Adding and observing that (x — x')~ + (?/ — ?yO^ + (^ — 2')^ = r' we find dnj SHI dHj ^ dx' dy- dz' This equation, satisfied l^y the potential at every point excepting the point Q where the attracting mass m' is situated, is known as Laplace's equation, or the potential equation. 258. These results are readily generalized. If the field is due to any finite number of particles m/, rih', • • • at the distances ri, r^, • • • from the attracted point P, their potential is defined as ^, /CW?i' , KfUo' , ^ Km' * U = 1 ■ + • • • = -S . ri 7-2 r If the field is due to continuous masses their potential is \bn' U rcbv/ For, as the limits of integration are constant the derivatives of U with respect to x, y, z can be found ])y differentiating under the integral sign; we have therefore, at any rate at any external point P\ dx J r^ dy J r^ du r^-^'j . -dz=-\l ^^^^^' where the right-hand members are evidently the components X, Y, Z of the attraction at P. 259.] THEORY OF ATTRACTIVE FORCES 199 For masses of finite density and not extending to infinity it is not diflacult to show that the function U has a single definite finite value at every point P external (and even internal) to the given masses and that it is a continuous function of x, y, z. As in Art. 257 it can be shown that, at any external point, U satisfies Laplace's equation d^U , d^U dHJ dx^ dy- dz~ 259. The potential is a scalar point-function; i. e. it is not a vector, but its value at any point is given by a single real number. The locus of those points at which the potential U has a constant value c, i. e. the surface U = c, is called an equipotential surface (level, or equilibrium, sur- face) . As the first derivatives of U with respect to x, y, z are on the one hand equal to the components of the attraction while, on the other, they are proportional to the direction cosines of the normal to the surface U = c, it follows that the attraction A at any external point P is normal to the equi- potential surface passing through P. In the language of vector analysis, the attraction A is the gradient of the potential U. The orthogonal trajectories of the family of equipotential surfaces U = c are called lines of force since each of these curves has the property that the tangent at any one of its points has the direction of the attraction at that point. The differential equations of the lines of force are evidently 200 STATICS [260. 1 lid dx _ dy _ dz dx dy dz 260. Exercises. (1) For a mass spread uniformly over the surface of a sphere prove that, witliin the sphere, the potential is zero while, outside the sphere, it is the same as if the mass were concentrated at the center. Hence deduce the corresponding results for a homogeneous solid spherical shell. (2) A mass is distributed uniformly along the arc of a parabola bounded by the latus rectum 4a; show that at the focus the potential is = 3.5245 Kp" and the attraction is = 1.8856 Kp"la. (3) Find the potential due to a homogeneous circular plate, of radius a, at a point P of its axis, at the distance x from the plate. (4) Determine the equipotential surfaces for a straight homogeneous rod; comp. Art. 254, Ex. 4 and 5. (5) For a mass distributed uniformly along the circumference, of a circle, determine the potential at any point in the plane of the circle, and show that at a distance from the center equal to % the radius it is = 7.2418 Kp". (6) Show that a force-function exists when the resultant force is con- stant in magnitude and direction. (7) Find the force-function in the case of a free particle moving under the action of the constant force of gravity (projectile in vacuo); determine the equipotential surfaces. (8) Show the existence of a force-function when the direction of the resultant force is constantly perpendicular to a fixed plane, say the a;y-plane, and its magnitude is a given function /(z) of the distance z from the plane. (9) Find the force-function, the equipotential surfaces, and the kinetic energy when the force is a function /(r) of the perpendicular distance r from a fixed line, and is directed towards this line at right angles to it. ■ (10) Show the existence of a force-function for a central force, i. e. a force passing through a fixed point (.xo, yo, zo), if the force is a function of the distance r from this point. What are the level surfaces? (11) Show that a force-function exists when a particle moves under the action of any number of such central forces as in Ex. (10). 262.1 THEORY OF ATTRACTIVE FORCES 201 3. Virtual work. 261. The importance of the potential in the theory of attraction and of the force-function for any conservative forces (Art. 255) is largely due to their connection with the idea of work. The work W of a constant force F in a rectilinear displaces^ ment s of its point of application is defined as the product of the projection oi F on s into s: W = Fs cosi/', "'""^ where \f/ is the angle l^etween the vectors F and s. In other words, work is the dot-product (Art. 141) of force and dis- placement : W = F-s. Thus, e. g. when a body of weight F = mg slides down the greatest slope of a smooth plane inclined at the angle 6 to the horizon, through a distance s, the work of the vertical force F is Fs cos(^7r - e) = Fs sin0 = Fh, where h = s smd is the vertical height through which the body has descended. It follows from the theory of projection (Art. 198) that the work of a force is the sum of the works of its components. Hence, if X, Y, Z are the rectangular components of F, X, y, z those of s, we have (comp. Art. 141) W = Xx-^ Yxj + Zz. 262. Work is not a vector, but a scalar quantity (Art. 259). If, in the definition of Art. 261, we take for ^p the lesser of the two angles made by the vectors F and s, the work is positive or negative according as i/' is < or > ^r The dimensions of work are evidently ML'^T~^. 202 STATICS [263. The unit of work is the work of a unit force (poundal, dyne) through a unit distance (foot, centimeter). Tlie unit of work in the F.P.S. system is called the f oot-poundal ; in the C.G.S. system, the erg. Thus, the erg is the amount of work done by a force of one dyne acting through a distance of one centimeter. These are the scientific units. In the gravitation system where the pound, or the kilogram is taken as unit of force, the British unit of work is the foot- pound, while in the metric system it is customary to use the kilogram-meter as unit. 263. The numerical relations between these units are obtained as follows. Let X be the number of ergs in the foot-poundal, then (comp. Art. 175), em. cm.2 ^ lb. ft.^ sec.^ sec.2 hence lb / ft V a; = i^ . ( '^- ) = 4.2141 X 10^; gm. Vcm./ i. e. 1 foot-poundal = 4.2141 X 10^ ergs, and 1 erg = 2.3730 X IQ-^ foot-poundals. Again, let x be the number of kilogram-meters in 1 foot-pound, then X kg. m. = 1 ft. lb., hence ^ ^ Ik ft. ^oi3g257 kg. m. i. e. 1 foot-pound = 0.138 257 kilogram-meters. Finally, 1 foot-pound = g foot-poundals (Art. 179); hence 1 foot- pound = 1.356 X 10^ ergs, and 1 erg = 7.3730 X 10~^ foot-pounds, if g = 981. 264. Exercises. (1) A joule being defined as 10^ ergs, show that 1 foot-pound = 1.356 joules, and that 1 joule is about 3/4 foot-pound. (2) Show that a kilogram-meter is nearly 10^ ergs. (3) "WTiat is the work done against gravity in raising 300 lbs. through a height of 25 ft. : (a) in foot-pounds, (5) in ergs? 265.] THEORY OF ATTRACTIVE FORCES 203 (4) Find the work done against friction in moving a car weighing 3 tons through a distance of 50 yards on a level road, the coefficient of friction being 0.02. (5) A mass of 12 lbs. slides down a smooth plane inclined at an angle of 30° to the horizon, through a distance of 25 ft.; what is the work done by gravity? 265. The work of a variable force i^ in a very small dis- placement PP' = 8s is defined (like that of a constant force in any displacement, Art. 261) as the product of 5s into the projection F cosi^ of F (at P) on 8s: 8W ^ F cos^ 8s = F-8s = X8x + Y8y + Z8z. This expression is often called the virtual work of F in the virtual displacement 5s, the term virtual and the letter 5 meaning that the displacement is arbitrary and not neces- sarily the actual displacement along the path of the particle. But it should be carefully observed that even if the dis- placement 5s were taken along the actual path we do not in general have in the limit dW „ , —^ = /< cos;/'; as i. e. the s-component of the force is not necessarily an exact derivative. The work done by the variable force F as the particle on which it acts is moved along an arbitrary curve from Po to any position P is written W = lim SP cosiA 5s = ^F cos;/' ds = f V • ds hs=Q 'JPo '^Po = S^{Xdx + Ydy + Zdz). This integral can in general not be evaluated unless the path of the particle from Po to P is known; and it has in general 204 STATICS [266. different values for different paths between these points. But we have seen (Arts. 257, 258) that for a particle m in a field of Newtonian attraction the component of the resultant attraction in any direction s is the s-derivative of the potential: As = dU/ds. Hence, multiplying by m, we have in this case for the virtual work: ""''"■' 8W = mAs8s = mdU. It follows that the work done on the particle m by the New- tonian attraction, as it is moved from Pq to P along any path, is ui H "^ = mfl'sU = m{U - Uo), where Uo is the value of U at Pq. Hence the work of attrac- tion is independent of the -path; it is m times the difference of 'potential at P and Pq] it is zero in any closed- path. • More generally, whenever the force F is conservative (Art. 255) so that it possesses a one-valued force-function, i. e. a function U{x, y, z) such that dU/dx, dll/dy, dU/dz are the rectangular components of F, the projection of F on any direction s will be the s-derivative of U, and hence the work of F is independent of the path. 266. For a particle in equilibrium, since the resultant force F is zero, it follows that the virtual work bW = F cos;/' bs is zero whatever the displacement 5s. And conversely, if the virtual work is zero whatever 5s, or more exactly, if the virtual work is small of an order higher than that of 5s for every sufficiently small 5s, the resultant force F must be zero, i. e. the particle is in equilibrium. The virtual work is zero for every 5s if it is zero for any three non-complanar displacements. . 'Using rectangular co-ordinates we have 8W = X8x -{- 268.] THEORY OF ATTRACTIVE FORCES 205 Ydy + Z8z; hence 8W = when X = 0, F = 0, Z = 0; and conversely, if 8W = for a virtual, i. e. arbitrary, dis- placement, we have owing to the independence of 8x, 8y, 8z'. m = 0. The proposition that the vanishing of the virtual work (apart from terms of a higher order) is a necessary and sufficient condition of equilibrium for a particle is known as the principle of virtual work for the particle. 267. In the particular case of a particle in a field of con- servative forces whose force-furiction is U, the condition of equilibrium assumes the form ds for any ds • or, with reference to rectangular axes : dx dy dz Now these are necessary conditions for a maximum or minimum of U. Hence the positions of equilibrium of a particle under conservative forces are found by determining the maxima and minima of the force-function or potential. It can be shown that a minimum of U corresponds to stable, a maximum to unstable, equilibrium. 268. The principle of virtual work, proved above only for the single free particle, has a far wider field of application. It can be shown that for any system of particles or rigid bodies, subject to any constraints, expressible by equations (not in- equalities) a7ul not involving friction, the vanishing of the virtual work (apart from terms of higher order) for any displacement compatible with the constraints is a necessary and sufficient condition of equilibrium. 206 STATICS [268. If in the expression of the virtual work 8W = F cosi/' 8s we replace 8s by {8s/8t)8t we can regard 8s/8t as a velocity. This is the reason why the principle of virtual work is often called the principle of virtual velocities. PART III: KINETICS. CHAPTER XIII. MOTION OF A FREE PARTICLE. 1. The equations of motion. 269. Let a particle of mass m be acted upon by any number of forces; as these forces are concurrent they are equivalent to a single resultant R (Art. 190). The definition of force (Art. 171) then gives for the acceleration j the fundamental equation of motion mi = R. (1) The mass m being regarded as a positive constant the equa- tion shows that the vectors j and R have the same direction and sense. The vector equation (1) assumes various forms according to the method selected for resolving j and R into components. If the motion be referred to fixed rectangular axes, (1) is replaced by the three equations (Art. 53) : mx = X, my = Y, mz = Z, (2) X, Y, Z being the components of R along Ox, Oy, Oz. If polar co-ordinates r, 6, cp are used we have (Art. 56, Ex. 9) : m(r — rd^ — r mi'^d (f"^) = Rr, m(rB + 2fd - r sin0 cos0 are the components of R along the radius vector, at right angles to the radius vector in the meridian plane, and at right angles to this plane. Finally, resolving along the tangent, normal, and bi- normal to the path we have (Art. 51) : mv = ms = Rt, m — = i?„, — Rb. (4) P In the case of plane motion the equations (2), (3), (4) reduce to the first two, with ^ = in (3); in the case of rectilinear motion the first equation of (2) or (4) suffices. 270. If the components X, Y, Z were given as functions of the time t alone, each of the three equations (2) could be integrated separately. In general, however, these com- ponents will be functions of the co-ordinates, and perhaps also of the velocity and of the time. No general rules can be given for integrating the equations in this case. By com- bining the equations (2) in such a way as to produce exact derivatives in the resulting equation, it is sometimes possible to effect an integration. Two methods of this kind have been indicated for the case of two dimensions in a particular example in Kinematics, Arts. 102-104. We now proceed to study these principles from a more general point of view, and to point out the physical meaning of the expressions involved. 271. The Principle of Kinetic Energy and Work. Let us combine the equations of motion (2) by multiplying them by X, y, z, respectively, and then adding. As xx is the time derivative of ^x-, the left-hand member of the resulting equation will be the ^-derivative of ^m{x- -\- ij^ -\- z^) = ^ww^, i. e. of the kinetic energy of the particle (Art. 181). We find therefore j^^mv'- = Xx+ Yy + Zz. 272.] MOTION OF A FREE PARTICLE 209 Hence, integrating from any point Po of the path where z; = i^o to any point P we obtain : ^mv^ - ^vo' = fl^iXdx + Ydy + Zdz). (5) The left-hand member represents the increase in the kinetic energy of the particle; the right-hand member represents the work done by the resultant force R, since its work is equal to the sum of the works of its components X, Y, Z (Art. 261). Equation (5) states, therefore, that the amount hij which the kinetic energy increases, as the particle passes from Po to P, is equal to the work done by the resultant force R on the particlt. 272. The principle of kinetic energy and work can also be deduced from the former of the two equations (4). Multiply- ing this equation by w = dsjdt, we have dihrnv"^) T^ ds „ ,ds hence, integrating as in Art. 271: ^iv^ — ^Vo^ = J R cosi/' ds, (5') where \p is the angle made by the force R with the tangent to the path. The integrand in (5) or (5'), i- e. the expression R cofi\p ds = R-ds = Xdx + Ydy + Zdz, is called the elementary work. It is the value of the virtual work (Art. 265) when the displacement 5s is taken infini- tesimal and along the actual path. As explained in Art. 265, the evaluation of the work integral in general requires a knowledge of the path. As in many problems the path is not known ])eforehand, but is 15 210 KINETICS [273. one of the things to be determined, it is very important to notice that in the case of conservative forces (Art. 255) the work integral has a value independent of the path (Art. 265). In this case, denoting the force-function, or potential, by U, we have f^iXdx + Ydy + Zdz) = f^^dU = U - Uo, so that the equation (5) or (5') becomes ^nv" — ^Vq^ = U — Uo- (6) Hence in the case of conservative forces the principle of kinetic energy and work at once gives a first integral of the equations of motion. 273. The negative of the force-function, say V= -U, is called the potential energy. If this quantity be intro- duced and the kinetic energy be denoted by T, the equation (6) assumes the form T+V=To+ Vo, (6') which expresses the principle of the conservation of energy for a particle: the total energy, i. e. the sum of the kinetic and potential energies, remains constant throughout the motion ij the forces are conservative. In other words, whatever is gained in kinetic energy is lost in potential energy, and vice versa. 274. The physical idea to which the term potential energy is due can perhaps best be explained by considering the Newtonian attraction between two particles m, m'. We think of the attracting particle to' as generating a field. ^\Tierever in this field a particle m be placed (say, Nvath zero velocity), it will become subject to the attraction A of m' and move toward m' with increasing velocity, thus acquiring kinetic energy; at the same time the force A does an amount of work on ni which is exactly equivalent to the kinetic energy gained by m. It follows that, 276.] MOTION OF A FREE PARTICLE 211 the farther away from m' the particle m is placed, initially, the greater will be the amount of work that m' can do upon it. It is this "poten- tiality" for doing work, due to the distance of m from m', which is denoted as energy of -position, or potential energy. The equation (6), or the equation (6') which differs from (6) merely in notation, shows that what the particle m in moving toward m' gains in kinetic energy it loses in potential energy so that the sum of kinetic and potential energy always remains constant. 275. The conditions for the existence of a force-function are (Art. 255) : X = ^J1 Y = ^^ Z ^ — dx ' djj ' dz ' Differentiating the second equation with respect to z, the third with respect to y we find dY _ dnj_ dZ _ dHJ_ dz dzdy ' dy dydz ' whence dY/dz = dZ/dy. Proceeding in the same way with the other two pairs of equations we find : dY _dZ dZ _ dX dX _ dY_ dz dy' dx dz ' dy dx ' These relations which are necessary and sufficient for the existence of a force-function U furnish a simple criterion for recognizing whether the given forces are conservative. 276. The principle of the conservation of energy, i. e. of the constancy of the sum of kinetic and potential energy, has been proved mathematically in the preceding articles for a very particular case, viz. for the motion of a particle under conservative forces. By a generalization as bold and far-reaching as was New- ton's extension of the property of mutual attraction to all matter (Art. 244), modern physics has been led to the assumption that .work and energy are quantities which can 212 KINETICS [277. never he destroyed, but can be transformed in a variety of ways. This assumption, the general principle of the con- servation of energy, while fully borne out so far by the results deduced from it, is of course not capable of math- ematical proof. Indeed, it may be said that in defining the various forms of energy, such as heat, chemical energy, radio-activity, etc., the definitions are so formulated as to conform to this principle; it has always been found possible to do this. The general principle of the conservation of energy cannot be fully discussed here, since this would require a study of all the forms of energy known to physics. 277. In its application to machines, the principle states that the total work W suppUed to a machine in a given time by the agent, or motor, driving it (such as animal force, the expansive force of steam, the pressure of the wind, the impact of water, etc.) is equal to the sum of the useful work Wv, done by the machine in the same time and the so-called lost, or wasteful, work Wu- spent in overcoming friction and other passive resistances of the machine: W = Wu + Wu: While W and Wu can be determined with considerable accuracy, it is difficult to determine Ww directly with equal precision; but it is found that the more accurately in any given machine Ww is determined, the more nearly will the above equation be found satisfied. This serves as a verification of the principle of the conservation of energy in its application to machines. The ratio W„/W of the useful work to the total work is called the efficiency of the machine. The term modulus is sometimes used for efficiency. 278. The time-rate at ivhich irork is performed by a force has received a special name, power or activity. The source from which the force for doing useful work is derived is commonly called the agent, or motor; and it is customary to speak of the power of an agent, this meaning the rate at which the agent is capable of supphang work. The dimensions of power are evidently ML-T~^. Tlie unit of power is the power of an agent that does unit work in unit time. Hence, 279.] MOTION OF A FREE PARTICLE 213 in the scientific system, it is the power of an agent doing one erg per second in the C.G.S. system, and one foot-poundal per second in the F.P.S. system. As, however, the idea of power is of importance mainly in engineering practice, power is usually measm'ed in gravitation units. In this case, the unit of power is the power of an agent doing one foot- pound per second in the F.P.S. system, and one kilogram-meter in the metric system. A larger unit is frequently found more convenient. For this reason, the name horse-power (H.P.) is given to the power of doing .550 foot- pounds of work per second, or 550 X 60 = 33,000 foot-pounds per minute. 279. The principle of angular momentum or of areas. By multiplying the first of the equations of motion (2), Art. 269, by y, the second by x, and then subtracting the first from tiie second we obtain the equation m{xij — yx) = xY — yX, or since the left-hand member is the time-derivative of m{xy — yx): jm^xy — yx) = xY — yX. Here the right-hand member is the moment of the resultant force R about the axis Oz (Art. 229) while, on the left, the quantity x • my — y • mx is the moment about the same axis of the inomentum mv whose components are mx, my, mz (Art. 168). This moment of momentum m{xy — yx) is also called angular momentum. As any line might have been chosen as axis Oz, our equa- tion expresses the proposition: In the motion of a ^article, the time-rate of change of the angular momentum about any line is equal to the moment of the resultant force about the same line. Applying this result to each of the axes of reference we find : 214 KINETICS [280. -^^miyz - zy) = yZ - zY, m{zx — xz) = zX — xZf m(xy — yx) = xY — yX. (8) These equations express the principle of angular momentum or of areas. 280. To interpret these equations geometrically consider first the right-hand members which are the moments of the resultant force R about the axes. The vector PA = R (Fig. 70) forms with the origin a triangle whose area is Fig. 70. ome-half the moment of R about 0; let us represent this moment, which is the cross-product of the radius vector OP = r and the force-vector PA = R, hj a vector H per- pendicular to the plane of the triangle OPA (comp. Arts. 199 and 119): H = rXR, 281.] MOTION OF A FREE PARTICLE 215 the length of this vector H being equal to twice the area OP A. The projection of the triangle OP A on the a;y-plane has the area ^{xY — yX) since the vertices of this projection have the co-ordinates (0, 0), {x, y), and {x-\- X, y -{- Y)- hence the right-hand members of (8) are the components Hjc, Hy, Hz of the vector H. Next consider in the same way the momentum-vector mv = PB; it forms with a triangle OPB whose area is one-half the moment of momentum about 0. We can represent this moment of momentum, or angular momentum, by a vector h, perpendicular to the plane OPB, and of a length equal to twice the area of the triangle OPB; the vector h is then the cross-product of r = OP and mv = PB : h = r X mv. The components of angular momentum m(yz — zy), m(zx — xz), m(xy — yx) are the components hx, hy, hz of the vector h. The equations (8) can therefore be written in the form dT-^- rfT-^- df^^" ^^^ and these equations can be combined into the single vector equation dh „ which means that the geometrical increment of the vector h, divided by A^, gives in the limit the vector H; i. e. the (geometrical) time-rate of change of the angular-momentum vector is equal to the moment-vector of the residtant force. 281. If instead of the momentum- vector mv we consider the velocity-vector v, its moment about would be repre- sented by the vector {l/m)h, whose components are yz — zy, 216 laNETICS [282. zx — xz, xy — yx. These quantities are (Art. 47) equal to twice the- sectorial velocities about the axes while the vector {\/ni)h represents twice the sectorial velocity of the particle about 0. This explains the name principle of areas. 282. If, in particular, the resultant force R is central, i. e. such as to pass always through a fixed point, then, for this point as origin, the right-hand members of the equations (8) are zero, and we find at once the first integrals of the equa- tions of motion (2) : m{yz — zy) = hi, m{zx — xz) =h2, m(xy — yx) = h, (9) where hi, ho, hs are constants. Thus, in the motion of a particle in the field of a central force, the angular momentum, and hence the sectorial veloc- ity, about any axis through the center is constant. If the resultant force always intersects a fixed line, the angular momentum, and hence the sectorial velocity, about this line as axis remains constant. These propositions are often referred to as the principle of the conservation of angular momentum or of areas. It may be noted that the equations (9), multiplied by X, y, z and added give, hix + h^iy + hzz = 0; this shows that the particle moves in a plane passing through the center of force, as is otherwise evident. 283. Exercise. In the case of plane motion, if the plane be taken as the xy-plane, the principle of areas is expressed by the third of the equations (8). If the perpendicular from the origin to the tangent at P be denoted by p (comp. Art. 100), this equation can be written in the form d{mpv)/dt = xY ~ yX. Show that the two terms mpdv/dt and mvdp/dt of the left- hand member represent the moments of the tangential and normal components of the resultant force R, respectively. 285.1 MOTION OF A FREE PARTICLE 217 2. Examples of rectilinear motion. 284. Free Oscillations. As an example of rectilinear mo- tion consider the motion of a particle of mass m under a force directly proportional to the distance OP = s of the particle from a fixed point 0. If the force is attractive, i. e. directed toward the point and if the initial velocity passes through or is zero so that the motion is rectilinear, the single equation of motion is m's = — MK^s, (10) and the motion (see Arts. 26, 27, 71) is a simple harmonic oscillation or vibration about the point as center. This point 0, at which the force R ^ — ynn^s is zero, is therefore a position of ecjuilibrium for the particle. The potential energy V due to the force R = — mKrs is, by Art. 273, F = — I Rds = mK~ I sds = ^vikts'^ + C. Hence the principle of the conservation of energy gives y2 _j_ ^2^2 = const. If the initial velocity be zero for s = So, we have y = =f: /C V.S'o" — s^. 285. As in the applications the moving particle m is generally subject to the constant force of gravity, it is important to notice that the intro- duction of a constant force F along the line of motion does not essentially change the character of the motion. For, the equation of motion ms = — m.K^s -\- F = — niK^ { s — „] reduces, with s — FlmK^ = x, to mx = — niK^x, which agrees in form with (10). The only change in the results is that 218 KINETICS [286. the center of the oscillations, i. e. the position of equilibrium of the particle m, is not the point 0, but a point at the distance e = F/mK^ from 0. 286. Forces proportional to a distance, or length, are directly ob- served in the stretching of so-called elastic materials. Thus, a homo- geneous straight steel wire when suspended vertically from one end and weighted at the other end is found to stretch; and careful measure- ments have shown that the extension, or change of length, is directly proportional to the weight apphed (the weight of the wire itself being assumed, for the sake of simplicity, as very small in comparison with the load applied). Conversely, the tension, or elastic stress, of the wire is proportional to the extension produced. INIoreover, when the weight is removed the wire is found to contract to its original length. This physical law, known as Hooke's law of clastic stress, holds only within certain limits. If the weight exceeds a certain limiting value, the extension is no longer proportional to the weight, and after removing the weight, the wire does not regain its original length, but is found to have acquired a permanent set, or lengthening; it is said in this case that the elastic limits ha,ve been exceeded. Materials for which Hooke's law holds exactly witliin certain limits of tension and extension are called perfectly elastic. Strictly speaking, such materials probably do not exist; but many materials follow Hooke's law very closely within proper limits. Thus, elastic strings, such as rubber bands, and spiral steel springs show these phenomena very clearly on account of the large extensions allowable within the elastic limits. 287. The elastic constant mn.^. Let an elastic string whose natural length is I assume the length I + x when the tension is F, so that accord- ing to Hooke's law, F = - mn^x. To determine the factor of proportionaHty ?nK^ for a given string, we may observe the length h assumed by the string under a known ten- sion, e. g. the tension— mig produced by suspending a given mass nii from the string (the weight of the string itself being neglected). We then have — mig = — rnK^ili — l), whence 289.] MOTION OF A FREE PARTICLE 219 and 288. Let the same string be placed on a smooth horizontal table, one end being fixed at a point (Fig. 71), while a particle of mass m is |<- j^-aji ^ oU 1 W/MZ/ymMSm Fig. 7L attached to the other end. Stretch the string to a length OPo = i + Xo (within the limits of elasticity) and let go; the particle m. will move under the action of the tension F alone, its weight being balanced by the reaction of the table. The equation of motion is vix = — ^ X, the distance QP = x being counted from the point Q at the distance OQ = I from the fixed point 0, Putting again (Art. 287) \mih - ' and integrating, we find X = Ci coskI + C2 smd, whence V = X = — KCi sind + kCz coskL As X = Xo and v = ior t = 0, we have Ci = xo, ct = 0; hence X = Xo cosd, V = —KXo sind. It should be noticed that these equations hold only as long as the string is actually stretched, i. e. as long as x > 0. The subsequent motion is, however, easily determined from the velocity for x = 0. 289. It was assumed, in the preceding article, that the particle m is let go from its initial position Po with zero velocity. This can be brought about by pulling the particle from Q to Po with a gradually increasing force which at any point P is just equal and opposite to the 220 laNETICS 1290. corresponding elastic tension, or stress, P = friK^x. The work thus done against the tension, i. e. in stretching or straining the string, is stored in the particle m as potential energy, or strain energy, V. To find ita amount, observe that, as the particle 7n is pulled through the short dis- tance A.r, the work of the force is = 7«AAx; this being the potential energy AV gained in the distance Ax, we have AT = wk^xAx; hence Vo = I niK-xdz = Ijuk-xo'. fJ Thus, in the initial position Po the particle m possesses this potential energy, but no kinetic energy. During its motion from Po to Q, the particle gains kinetic energy and loses potential energy. At any inter- mediate point P, for which QP = x, the kinetic energy is T = Imv^, while the potential energy is V = im/c^x^. By the principle of the con- servation of energy (Art. 273), the sum of these two quantities, the so-called total energy, E, remains constant as long as no other forces besides the elastic stress act on the particle: ^mv- + huK-x^ = const. The value of the constant is = hnK-xo', since this is the total energy at Po; hence, I'- + K-X- = K-Xlf. (Comp. Art. 284). This relation also follows from the values of x and V given in Art. 2SS, upon eliminating /. When the particle arrives at the position of equilibrium Q, the potential, or strain, energy has been consumed, having been converted completely into kinetic energy. 290. Exercises. (1) In the problem of Art. 2SS let the string be a rubber band whose natural length of 1 ft. is increased 3 in. when a weight of 4 oz. is suspended from it; determine the motion of a 1-oz. particle attached to one end, the band being initially stretched to a length of 1)^ ft.; find (a) the greatest tension of the band, (b) the greatest velocitj^ of the particle, (c) the period, (d) the work done by the tension in a quarter oscillation. (2) Discuss the effect of friction, of coefficient fi, in the problem of Art. 28 e, the tension vanishes for x = — e, i. e. at Q; the velocity at this point is vi = — kV xa' — e^, and the particle rises to the height h = (xo^ — e2)/2e above Q. The total time of one up pj^. 72 and down motion is 2V^g[hTr + sin-Ke/xo) + Vjxo/c)^ - 1]. (4) How is the motion of Ex. (3) modified if the elastic string be replaced by a spiral spring suspended vertically from one end? Assume the resistance of the spring to compression equal to its resistance to extension. (5) The particle in Ex. (3) is let fall from a height h above Q; deter- mine the greatest extension of the string. (6) An elastic string whose natural length is I is suspended from a fixed point. A mass nii attached to its lower end stretches it to a length h; another mass m2 stretches it to a length h. If both these masses be attached and then the mass ???2 be cut off, what will be the motion of nil? (7) If a straight smooth hole be bored through the earth, connecting any two points A, B on the surface, in what time would a particle slide from A to B? The attraction in the interior is directly proportional to the distance from the center of the earth. 291. Resistance of a Medium. It is known from obser- vation that the velocity y of a rigid body moving in a liquid or gas is continually diminished, the medium apparently exert- 222 KINETICS [291. ing on the body a retarding force which is called the resistance of the medium. This force F is found to be roughly propor- tional to the density p of the medium, the greatest cross- section A of the body (at right angles to the velocity v), and generally, at least for large'*velocities, to the square of the velocity v. F = kpAv"", where A; is a coefficient depending on the shape and physical condition of the surface of the body. This expression for the resistance F can be made plausible by the following consideration. As the body moves through the medium, say with constant velocity v, it imparts this velocity to the particles of the medium it meets. The portion of the medium so affected in the unit of time can be regarded as a cylinder of cross-section A and length V, and hence of mass pAv. To increase the velocity of this mass from to f in the unit of time requires, by equation (5) of Art. 171, a force pAv ■ V . „ ^— = pAi^. The retarding force of the medium must be equal and opposite to this force multiplied by a coefficient k to take into account various disturbing influences. For small velocities, however, the resistance can be assumed pro- portional to the velocity, F = kv, the coefficient k to be determined by experiment. The above consideration is only a very rough approximation. Thus the particles of the medium are not simply given the velocity v in the direction of motion; they are partly pushed aside and move in curves backwards, causing often whirls or eddies alongside and behind the body. If the medium is a gas, it is compressed in front, and rarefied behind the body; indeed, when the velocity is great (greater than that of sound in the gas), a vacuum will be formed behind the body. More- over, a layer of the medium adheres to and moves with the body, thus increasing the cross-section. It is therefore often found necessary to assume a more general expression for the resistance; and this isj in ballistics, generally written in the form F = KpAv'^fiv). 292. MOTION OF A FREE PARTICLE 223 The careful experiments that have been made to determine the re- sistance offered by the air to the motion of projectiles have shown that for velocities up to about 250 meters per second, as well as for velocities above 420 m./sec, J{v) can be regarded as constant, i. e. the resistance is proportional to the square of the velocity. But for velocities between 250 and 420 m./sec, i. e. in the vicinity of the velocity of sound in air (330-340 m./sec), the law of resistance is more complicated. 292. Falling Body in Resisting Medium. Assuming the resistance proportional to the square of the velocity, the equation of motion for a body falling (without rotating) in a medium of constant density is d^s dv , „ at at where A; is a positive constant. To simplify the resulting formulae, put g then the separation of the variables v and t gives gdv g^ — jjL^v^ whence 2m g - tJ-v the constant of integration being zero if the initial velocity is zero. Solving for v, we have V = - of*' n ^^ = — tanhjui. Writing dsjdt for v and integrating again, we find, since s = for t = 0, s = ~ log h (e*^' + e"*^') = ■, log coshyuf. 224 KINETICS [293. The relation between v and s can be obtained by eliminating t between the expressions for v and s, or more conveniently by eliminating / from the original differential equation by means of the relation dv _ dvds _ dv dt ds dt ds This gives whence, with y = for s = 0, s = -% log 9 9 2 • 2m- g- - fi-v^ 293. Exercises. (1) Show that, as t increases, the motion considered in Art. 292 approaches more and more a state of uniform motion without ever reaching it. (2) Determine the motion of a body projected vertically upward in the air with given initial velocity vo, the resistance of the air being pro- portional to the square of the velocity. (3) In iTx. (2) find the whole time of ascent and the height reached by the particle. (4) Show that, owing to the resistance of the air, a body projected vertically upward returns to the starting point with a velocity less than the initial velocity of projection. (5) A ball, 6 in. in diameter, falls from a height of 300 ft.; find how much its final velocity is diminished by the resistance of the air, if k = 0.0C090. (6) Determine the rectilinear motion of a body in a medium whose resistance is proportional to the velocity, when no other forces act on it. (7) A body falls from rest in a medium whose resistance is propor- tional to the velocity; find v and s in terms of t, v in terms of s. 294. Damped Oscillations. Let a particle of mass m be attracted by a fixed center 0, with a force proportional to the distance from 0, and move in a medium w^hose resistance 294.] MOTION OF A FREE PARTICLE 225 is proportional to the velocity. If the initial velocity be directed through (or be zero) , the motion will be rectilinear, and the equation of motion is (P'S m-r;;, = — mK}s — mkv, or, putting k = 2X, f + 2x*; + .=. = o. rii) This is a homogeneous linear differential equation of the second order with constant coefficients, which can be in- tegrated by a well-known process. The roots of the auxiliary equation, - X± VX2 - k\ are real or imaginary according as X > k, or X < k. The limiting cases X = k, X = 0, /c = 0, also deserve special men- tion. (a) If X > K, the roots are real and different, and as X is positive, both roots are negative; denoting them by — a and — 6, so that a and h are positive constants, and b > a, the general solution is S = CiC""' + €26"''^. As the force has a finite value at the center 0, we can take s = 0, V ^ vo ior t = as initial conditions. This gives s = T-^°- -- (e-°' - e-^'). V = T^^^^ (he-"" - ae""'). b — a b — a The velocity reduces to zero at the time h = r log-- b — a a 16 226 KINETICS (294. As a and b are positive and b > a, s has always the sign of i'o, i' e. the particle remains always on the same side of 0; it reaches its elongation at the time ti, for which v vanishes, and then approaches the point asymptotically. Hence, in this case, the damping effect of the medium is sufficiently great to prevent actual oscillations. Such mo- tions are sometimes called aperiodic. (6) If X = K, the roots are real and equal, viz. = — X, and the general solution is s = (ci + C2t)e~^. With s = 0,v =^ voiov t = 0, we find s = wote"^', V = i'o(l - X^e-'^'. The velocity vanishes for ti = 1/X, and then only. The nature of the motion is essentially the same as in the previous case. (c) If X < K, the roots are complex, say = — « ± ^i, where a and /3 are positive constants. The general solution s = e~"'(ci coS(8i + Co sinjS^) gives with s = 0, v = Vq for ^ = 0: s = ^e-«' sin/3^, v = -^ e-"'(/3 cos^t - a sin^t). Here v vanishes whenever tan/3^ = (3 /a = V(k/X)^ ~ 1; s vanishes (i. e. the particle passes through 0) whenever / is an integral multiple of 7r//3; s has an infinite number of maxima and minima whose absolute values rapidly diminish. The resistance of the medium, while not sufficient to ex- tinguish the oscillations, continuall}^ shortens their amplitude ; this is the typical case of damped oscillations. (d) If X = 0, the roots are purely imaginary, viz. = ± ki. In this case, the second term in equation (11) is zero; there 296.] MOTION OF A FREE PARTICLE 227 is no damping effect, and we have the case of free oscillations (see Arts. 284-290). (e) If K = 0, one of the roots is zero, the other is = — 2 X. The attracting (or elastic) force being zero, we have the case of Ex. (6), Art. 293. 295. As shown in Arts. 273, 274, the principle of the conservation of energy holds for the free oscillations of a particle (under a force pro- portional to the distance). In the case of damped oscillations (Art. 294), this principle, in the restricted sense in which it has been proved so far, is not applicable, the resistance of the medium not being given as a function of the distance s. The total energy E = T + V oi the particle, or rather the energy stored in the system formed by the spring with the particle attached (in the example used above), diminishes in the course of time because the spring has to do work against the resistance of the medium, thus transferring part of its energy to the medium (setting it in motion, heating it, etc.). Thus, in a generalized meaning, the principle of the conservation of energy can be said to hold for the larger system, formed by the spring, together with the medium (see Art. 276). The rate at which the total energy E diminishes with the time is here proportional to the square of the velocity : d^ o ^ 2. -^ = - 2?ttXz;2; for, substituting for E its value E = T + V = imv^ + Imk'^s'^ (Art. 289) and reducing, we find the equation of motion (11). The space-rate of change of the total energy E is proportional to the velocity, and is nothing else but the resistance of the medium : -r- = — 2 m\v, ds for we have dE ^ dE^ds ^ dE dt ds dt ds 296. Forced Oscillations. In the case of free simple har- monic oscillations, while the force regarded as a function of 228 KINETICS [297. the distance s is directly proportional to s, the same force regarded as a function of the time is of the form R = — mK~So coskI, since s = So cosk^. Conversely, a particle acted upon by a single force R = mk cosjit, or R = mk shifxt, directed toward a fixed center 0, will, if the initial velocity passes through 0, have a simple harmonic motion. Suppose that such a force in the line of motion be super- imposed in the case of Art. 294 so that the equation of motion becomes cPs ''i ^7^ = ~ mii~s — 2m\v -\- mk cos/jLt, dr or g + 2X 'J^ + K^s = A; sin/zf. (12) The particle is then said to be subject io forced oscillations. For a particle suspended from a spiral spring this could be realized by subjecting the point of suspension to a vertical simple harmonic motion of amplitude k and period 27r/ju. The non-homogeneous linear differential equation (12) with constant coefficients can be integrated by well-known methods. 297. Exercises. (1) With/x = 2, ^0 = 4, sketch the curves representing s as a function of t in the five cases of Art. 294; take (a) X = 3, (6) X = 2, (c) \ = H, (e) X = 2. (2) Compare the cases (c) and (d) of Art. 294; show that the os- cillations in a resisting medium are isochronous, but of greater period than in vacuo. The ratio of the amplitude at any time to the initial amplitude is called the dam-ping ratio; show that the logarithm of this ratio, the so-called logarithmic decrement, is proportional to the time. (3) Derive the equation of motion in the case of free oscillations from the principle of the conservation of energy. 299.] MOTION OF A FREE PARTICLE 229 (4) Integrate and discuss the equation s -\- k^s = a sinjui; show that the amphtude of the forced oscillation becomes very large if the periods of the free and forced oscillations are nearly equal. Discuss the limiting case when fx = k. (5) Integrate (12), assuming a particular integral of the form c cosul + c' siufjil and determining the constants c, c' by substituting this expression in (12). Discuss the result, 3. Examples of curvilinear motion. 298. Central Forces. The motion of a particle in the field of a central force has been studied in Kinematics, under central motion, Arts. 96-113. It will here suffice to add certain further developments that are best expressed in dynamical terms. 299. Force Proportional to the Distance : f(r) = /cV. The equations of motion (2) are in this case the upper sign holding for attraction, the lower for repulsion. Their solution is very simple, because each equation can be integrated sepa- rately. We find, in the case of attraction, X = Ci cosK< + a-z sind, y = bi cosk< + 62 siuKt, and in the case of repulsion, X = aiC' + 026-"^', y = bie'^' + hiC-"*; di, Oi, 61, hi, being the constants of integration. To find the equation of the orbit, it is only necessary to eliminate t in each case. In the case of attraction, this elimination can be performed by solving for cosd, sinx/, squaring and adding. The result is {aiy — bixy + {aiy — bnx)"^ = (aJh — aihi)-, and this represents an ellipse, since (fli^ + a22)(6,2 + 62=) - {aA + aMY = (aA - aA)' is always positive. The center of the ellipse is at the origin, and the lines aiy = b\x, aiy = box are a pair of conjugate diameters. 230 KINETICS 1300. In the case of repulsion, solve for c' and e-'^', and multiply. The resulting equation, (ciy — bix)(b2X — aoy) = (0162 — a2&i)-, represents a hj'perbola whose asymptotes are the lines aiy = bix, a^y = 62X. 300. It is worthy of notice that the more general problem of the motion of a particle attracted by any ymmbcr of fixed centers, icith forces directly proportional to the distances from these centers, can be reduced to the problem of Art. 299. Let X, y, z be the co-ordinates of the particle, r, its distance from the center 0,-; x,, yi,Zi the co-ordinates of Oi] and Krri the acceleration produced by 0,. Then the x-component of the resultant acceleration is = — S/v-rr; . ^ ~— ' = - 2K-.2(.r — x.) = - x^Kr + Sk-.^x;; and similar expressions obtain for the y and z components. Hence, the equations of motion are X = — x'S.Kr + ^Ki-Xi, y = — y'^Ki^- + 'ZK^yi, z = - z'^kC- + 'Zki'^zu As the right-hand members are linear in x, y, z, there is one, and only one, point at which the resultant acceleration is zero. Denoting its co-ordinates by x, y, z, we have The form of these equations shows that this point of zero acceleration which is sometimes called the mean center is the centroid of the centers of force, if these centers be regarded as containing masses equal to kj^. It is evidently a fixed point. By introducing the co-ordinates of the mean center, we can reduce the equations of motion to the simple form X = -k2(x - x), y = - K'^iy - y), 2 = - K^iz - z), where k- = Skj-. Finally, taking the mean center as origin, we have X = — K^x, ij = — K-y, 2 = — K^z. It thus appears that the motion of the particle is the same as if there rvere only a single center of force, viz., the mean center (x, y, z), attracting loith a force proportional to the distance from this center. 302.] MOTION OF A FREE PARTICLE 231 The plane of the orbit is, of course, determined by the mean center and the initial velocity. 301. It is easy to see that most of the considerations of Art. 300 apply even when some or all of the centers repel the particle with forces proportional to the distance. It may, however, happen in this case that the mean center lies at infinity, in which case, of course, it can not be taken as origin. Simple geometrical considerations can also be used to solve such problems. Thus, in the case of two attractive centers Oi, Oi (Fig. 73) of equal intensity k^, the forces can evidently be represented by the dis- tances POi = ri, POi = Ti of the particle P from the centers. Their resultant is therefore = 2P0, if denotes the point midway between OiandO-; and this resultant always --n~-- passes through this fixed point 0, and is proporlional to the distance -p. „„ PO from this point. 302. Exercises. (1) Determine the constants of integration in Art. 299, if xo, 2/0 are the co-ordinates of the particle at the time I. = and Vi, V2 the com- ponents of its velocity Vo at the same time. The equation of the orbit will assume the form K~{xuy - yox)- + {ivj - VixY = (w2 - rjaihY for attraction, and nKxoy - VaxY - {viy - Vixy = - {xaih - yoviy for repulsion. (2) Show that the semi-diameter conjugate to the initial radius vector has the length Vu/k, where Vu- = fi^ -f- V2^. As any point of the orbit can be regarded as initial point, it follows that the velocity at any 'point is proportional to the parallel diameter of the orbit. (3) Find what the initial velocity must be to make the orbit a circle in the case of attraction, and an equilateral hyperbola in the case of repulsion. (4) The initial radius vector ro and the initial velocity Vo being given geometrically, show how to construct the axes of the orbit described 232 laNETICS 1302. under the action of a central force (of given intensity k^) proportional to the distance from the origin. (5) A particle describes an ellipse under the action of a central force proportional to the distance; show that the eccentric angle is proportional to the time, and find the corresponding relation for a hyperbolic orbit. (6) A particle of mass m describes a conic under the action of a central force F = =F mK-r. Show that the sectorial velocity is ic = },Kah, a and h being the semi-axes of the conic. (7) In Ex. (6) show that the time of revolution is' T = 2wIk, if the conic is an ellipse. (8) A particle describes a conic under the action of a force whose direction passes through the center of the conic. Show that the force is proportional to the distance from the center. (9) A particle is acted upon by two central forces of the same intensity (k^), each proportional to the distance from a fixed center. Determine the orbit: (a) when both forces are attractive; (6) when both are repulsive; (c) when one is an attraction, the other a repulsion. (10) A particle of mass m is attracted by two centers 0\, O2 of equal mass m' and repelled by a third center O3, whose mass is m" = 2m'. If the forces are all directly proportional to the respective distances, determine and construct the orbit. (11) When a particle moves in an ellipse under a force directed towards the center, find the time of moving from the end of the major axis to a point whose polar angle is d. (12) Prove that if, in the problem of Art. 301, the intensities of Oi and O2 are ki, k2, the resultant attraction F passes through the centroid G of two masses ki k2, placed at Oi, O2, and that F = {ki + k2)PG. (13) In Art. 299, in the case of attraction, the component motions are evidently simple harmonic oscillations. Show that the equation of the path can be put in the form (comp. Art. 89) x^ 2xy . ^ , y^ — , r sm5 + r, = cos^S. a' ab W (14) Show that the total energy of a particle of mass m describing an ellipse of semi-axes a, b under a force ninh directed to the center is 304.] MOTION OF A FREE PARTICLE 233 303. Force Inversely Proportional to the Square of the Distance : /(r) = ^/j"' (Newton's law). It has been shown in Kinematics (Arts. 99-108) how this law of acceleration can be deduced from Kepler's laws of planetary motion. From Kepler's first law Newton con- cluded that the acceleration of a planet (regarded as a point of mass 7n) is constantly directed towards the sun; from the second he found that this acceleration is inversely pro- portional to the square of the distance. The motion of a planet can therefore be explained on the hypothesis of an attractive force, ^ = ^^^2* issuing from the sun The value of n, which represents the acceleration at unit distance or the so-called intensity of the force, was found to be (Art. 108; or below, Art. 315) M = 4x2—; and as, according to Kepler's third law, the quantity a^/T"^ has the same value for all the planets, Newton inferred that the intensity of the attracting force is the same for all planets; in other words, that it is one and the same central force that keeps the different planets in their orbits. 304. It was further shown by Newton and Halley that the motions of the comets are due to the same attractive force. The orbits of the comets are generally ellipses of great eccen- tricity, with the sun at one of the foci. As a comet is within range of observation only while in that portion of its path which lies nearest to the sun, a portion of a parabola, with the same focus and vertex, can be substituted for this portion of the elliptic orbit, as a first approximation. 234 KINETICS [305. It is also found from observation that the motions of the moons or satelhtes around the planets follow very nearly Kepler's laws. A planet can therefore be regarded as at- tracting each of its satellites with a force proportional to the mass of the satellite and inversely proportional to the square of the distance. 305. All these facts led Newton to suspect that the force of terrestrial gravitation, as observed in the case of falling bodies on the earth's surface, might be the same as the force that keeps the moon in its orbit around the earth. This inference could easily be tested, since the acceleration g of falling bodies as well as the moon's distance and time of revolution were known. Let m be the mass of the moon, a the major semi-axis of its orbit, T the time of revolution, r the tUstance between the centers of earth and moon; then the earth's attraction on the moon is (Art. 303) a' F = Airhn ^— , or, since the eccentricity of the moon's orbit is so small that the orbit can be regarded as nearly circular, F = Air-ma/T-. On the other hand, the attraction exerted by the earth on a mass m on its surface, i. e. at the distance R = 3963 miles from the center, is F' = mg. Now, if these forces are actually in the inverse ratio of the squares of the distances, we must have F' ^ a? F ~ I^' or, since the distance of the moon is nearly = QOR, F' = GO^F. Sub- stituting the above values of F and F', we find A 2 60^-R With R = 3963 miles, T = 27'' 7" 43'", this gives g = 32.0, a value which agrees sufficiently with the observed value of g, considering the rough degree of approximation used. 308.1 MOTION OF A FREE PARTICLE 235 306. In this way Newton was finally led to his law of universal gravitation, which asserts that every particle of mass m attracts every other particle of mass m' with a force „ mm' where r is the distance of the particles and k a constant, viz. the acceleration produced by a unit of mass in a unit of mass at unit distance (see Arts. 245, 246). The best test of this hypothesis as an actual law of physical nature is found in the close agreement of the results of theoretical astronomy based on this law with the observed celestial phenomena. 307. Taking Newton's law as a basis, let us now turn to the converse prol)lem of determining the 7notion of a particle acted 7ipon by a single central force for which f{r) = fi/r'^ (problem of planetary motion). It has been shown in Kinematics (Arts. 109-112) that if the force be attractive, the particle will describe a conic section with one of the foci at the center of force, the conic being an ellipse, parabola, or hyperbola, according as Vo' = ^. (13) ^ To If the force be repulsive, the same reasoning will apply, except that fj, is then a negative quantity. The orbit is, therefore, in this case always hyperbolic; the branch of the hyperbola that forms the orbit must evidently turn its convex side towards the focus at which the center of force is situated, since the force always lies on the concave side of the path. 308. To exhibit fully the determination of tlie constants and the dependence of the nature of the orbit on the initial conditions, a solution somewhat different from that given in Kinematics will here be given for the problem of planetary motion in its simplest form. 236 KINETICS (309. With /(r) = M/r^, the equation of kinetic energy and work (5) Art. 271, gives (comp. (19), Art. 109) ' r^ r To or, if the constant of integration be denoted briefly by h and u = 1/r be introduced, 1^2 = 2u,u + /i, where /i = v^? • (14) Substituting this expression for v- in the equation (15), Art. 105, we find the differential equation of the orbit in the form ^^^y + u^ = \,{2y.u + h), (15) ( (^)'-(»-;)'+';:+' To integrate, we introduce a new variable u' by putting u , 1 1^^ h the resulting equation, ( '^ ) = 1 _ ,i'2 or dB =± , . \ dd J >/ 1 - u'2 has the general integral — a = =F cos-' a', or u' = cos (9 — a), where a is the constant of integration. The orbit has, therefore, the equation ^ =-. + ^!-,+^cos{8-a), (16) r c- y c* c^ which agrees with the equation (24) given in Kinematics, Art. 112, excepting the different notation used for the constants. 309. The equation (16) represents a conic section referred to its focus as origin. The general focal equation of a conic is ~ =\+l cos(0 - a), (17) where I is the semi-latus rectum, or parameter, e the eccentricity, and a the angle made with the polar axis by the line joining the focus to the nearest vertex. 310.1 MOTION OF A FREE PARTICLE 237 In a planetary orbit (Fig. 74), the sun S being at one of the foci, the nearest vertex A is called the perihelion, the other vertex A' the aphelion, and the angle d — a made by any radius vector SP = r with the peri- helion distance SA is called the true anomaly. Fig. 74. Comparing equations (17) and (IG), we find, for the determination of the constants: I c2 ' I ^ c*'^ c^ ' hence, l^'\ e=Jl+^, (18) or, solving for c and h, c2 - 1 C = 'SiJd, h = IX I (19) 310. The expression for the eccentricity e in (IS) determines fhe nature of the conic; the orbit is an ellipse, parabola, or hyperbola, according as e = 1; hence, by (18), according as the constant h of the equation of kinetic energy is negative, zero, or positive. Owing to the value of h given in (14), this criterion agrees with the form (13), Art. 307. It should be observed that it follows from (13) that the nature of the conic is independent of the direction of the initial velocity. The criterion (13) can be given the following interpretation. Con- 238 KINETICS [311. sider a particle attracted by a fixed center according to Newton's law. If it move in a straight line passing through the center, the principle of kinetic energy gives for its velocity, at the distance r, - 2. r-^ - V- = vo"^ — 2{i. \ '^ = ^ -{- vo' — hence, if it start from rest at an infinite distance from the center, it would acquire the velocity V 2iJ.ir at the distance r. The criterion (13) is therefore equivalent to saying that the orbit is an ellipse, a parabola, or a hyperbola, according as the velocity at any point is less than, equal to, or greater than the velocity which the particle would have acquired at that point by falling towards the center from infinity. 311. For a central conic, whose axes are 2a, 2b, we have I = b-Ja. e = V a^ =F ¥/a (the upper sign relating to the ellipse, the lower to the hyperbola), so that the equations (19) reduce to the following: c=6>, h = =f^. (20) V a a The latter relation, with the value of h from (14), gives for the major or focal semi-axis a : ±^=?--^^; (21) a ro fi while the former, with the value of c as given in Art. 100, determines the minor or transverse axis b : b = c\j = ?v'o sini/'o \/ • (22) a/ = rovo siniAo \/ ' 312. The magnitudes of the axes having thus been found, their directions can be determined by a simple construction which furnishes the second focus. In the ellipse, the focal radii have a constant sum = 2a, and lie on the same side of the tangent, making equal angles with it. In the hyperbola, they have a constant difference = 2a, and lie on opposite sides of the tangent. Hence, determining the point 0" (Fig. 75), which is symmetrical to the center of force O with respect to the initial velocity, and drawing the line PoO", we have only to lay off on this line from Po a length PoO' = ± {2a — ro) ; then 0' is the second focus, which for an elliptic 314.] MOTION OF A FREE PARTICLE 239 orbit must be taken with O on the same side of the tangent PoT", and for a hyperbolic orbit on the opposite side. Fig. 75. 313. For a parabola, since c = 1, we find, from (19), , „ , c^ Vo~ro^ sinVo (23) The axis of the parabola is redtlily found by remembering that the perpendicular let fall from the focus on the tangent bisects the tangent (i. e. the segment of the tangent between the point of contact and the axis). Hence, if Or (Fig. 76) be the perpendicular let fall from the center on the velocity vo, it is only neces- sary to make TT' = P^T, and T' will be a point of the axis. Moreover, the perpendicular let fall from T on OT' will meet the axis at the vertex A of the para- bola, so that OA = ^l. 314. The relation (21), which must evidently hold at any point of the or- bit, can be written in the form Fig. 76. ■^^Oh)' (24) the upper sign relating to the ellipse, the lower to the hyperbola, while for the parabola, the second term in the parenthesis vanishes (since a = oo). This convenient expression for the velocity in terms of the radius vector might have been derived directly from the fundamental relation (Art. 100) V = c/p, the first of the equations (19), c^ = fil, and the 240 KINETICS 1315. geometrical properties of the conic sections {r :^r' = 2a, pp' = ¥, p'r = pr', where r, r' are the focal radii, and p, p' the perpendiculars let fall from the foci on the tangent). The proof is left to the student. 315. Time. In the case of an elliptic orbit, the time 7" of a complete revolution, usually called the periodic time, is found by remembering that the sectorial velocity is constant and = ^c, whence _ 2Trab c ' or, by (20), T = 2xV'"'=?^. (25) \ /J, n The constant _ ^ = \, which evidently represents the mean angular velocity about the center in one revolution, is called the mean motion of the planet. It should be noticed that it depends not only on the intensity of the force, but also on the major axis of the orbit, while in the case of a force directly proportional to the distance the periodic time is independent of the size of the orbit (see Art. 302, Ex. 7). The periodic time T and the major axis a of a planetary orbit deter- mine the intensity n of the force: M=47r^^3, (26) whence F = mJ{r)=mt^ = Ai^^m^^, (27) where m is the mass of the planet. 316. To find generally the time t in terms of B or r, it is best to intro- duce the eccentric angle ^ of the elUpse as a new variable, and to express /, r, and 6 in terms of . In astronomy, the polar angle 6 is known as the true anomaly, and the eccentric angle as the eccentric anomaly. The relation of the eccentric angle ^ to the polar co-ordinates r, will appear from Fig. 77, in which P is the position of the planet at the time t, P' the corresponding point on the circumscribed circle, i^AOP = 6 the true anomaly, and ^ACP' = the eccentric anomaly. The focal equation of the ellipse ^ I ^ ail - e2) 1 + e cos9 1 + e COS0 317.] MOTION OF A FREE PARTICLE 241 gives r -{■ er cos^ = a — ae-; and the figure shows that r cos0 = a cos as variable by means of (28) ; this gives dl = \ ■■ a(l — c cos<^)d — e sin<^. (30) This relation is known as Kepler's equation; the quantity nt is called the mean anomaly. 318. Kepler's equation (30) can be derived directly by considering that the ellipse APA' (Fig. 77) can be regarded as the projection of the circle AP'A', after turning this circle about AA' through an angle = cos~^ (b/a). For it follows that the elliptic sector AOP is to the circular sector AOP' as b is to a. Now, for the circular sector we have AOP' = ACP' - OCP' = W- - e sm0), and this agrees with (30) since, by (25), 2Tr/T = n. 319. Kepler's equation (30) gives the time as a function of 0; by means of (28), it establishes the relation between t and r; b}' means of (29), it connects t with 0. It is, however, a transcendental equation and cannot be solved for in a finite form. For orbits with a small eccentricity e, an approximate solution can be obtained by writing the equation in the form 4> = nt -\- e sin its approximate value nt'. 320.] MOTION OF A FREE PARTICLE 243 4>=^ nt -{- e sinnl. (31) This amounts to neglecting terms containing powers of e above the first power. Substituting this vakie of 4> in (28), we have with the same approxi- mation r = a{l — e cosnt). (32) To find d in terms of t, we have from the equation of the eHipse, r = a(l — t-)(l 4- e cos0)~i = a(l — e cos9), neglecting again terms in e^; hence, r'^ =0^(1 — 2e cos9). Substituting this value in the equa- tion of areas, r^dd = cdt = V fj.a{l — e'^)dl, we find (1 - 2e cosd)dd = J-^ dl = ndt) whence, by integration, since = for i = 0, d — 2e sin& = nt, or finally, d = nt + 2c shmt. (33) Thus we have in (31), (32), (33) approximate expressions for (j), r, and 9 directly in terms of the time. The quantity 2e sinnt, by which the true anomaly d exceeds the mean anomaly nt, is called the equation of the center. 320. Exercises. (1) A particle is attracted by a fixed center according to Newton's law. What must be the initial velocity if the orbit is to be circular? (2) A number of particles are projected, from the same point in the field of a force following Newton's law, with the same velocity, but in different directions. Show that the periodic times are the same for all the particles. (3) The mean distance of Mars from the sun being 1.5237 times that of the earth, what is the time of revolution of Mars about the sun? (4) A particle describes a conic under the action of a central force following Newton's law; if the intensity ^ of the force be suddenly changed to ju', what is the effect on the orbit? (.'i) In Ex. (4), if the original orbit was a parabola and the intensity be doubled, what is the new orbit? (G) Regarding the moon's orbit about the earth as circular, what would it become: (a) if the earth's mass were suddenly doubled? (h) if it were reduced to one half? 244 KINETICS • [321. (7) In Ex. (4), determine the effect on the major semi-axis (or "mean distance") a and on the periodic time T, of a small change in the intensity m of the force. (8) If the mass M of the sun be suddenly increased by Mjn, n being very large, while the earth is at the end of the minor axis of its orbit, what would be the effect on the earth's mean distance and on the period of revolution T ? (9) Find the equation of the hodograph of planetary motion, derive from it the expression for the velocity in terms of the radius vector, and show that the velocity is a maximum in perihelion and a minimum in aphelion. (10) Show that the greatest velocity of a planet in its orbit about the sun is to its least velocity as 1 + e is to 1 — e; and find this ratio for the earth, whose orbit has the eccentricity e = 0.016 771 2. (11) Find the time exactly as a function of 0, for a parabolic orbit. (12) The latus rectum passing through the sun divides the earth's orbit into two different parts; in what time are these described if the whole time is 365 K days? (13) Show that the path of a projectile in vacuo is an ellipse, parabola, or hyperbola, according as Vts = 36,800 ft. per second ( = 7 miles per second, nearly). One of the foci lies at the center of the earth, and the ordinary assumption that the path is parabolic means that this center can be regarded as infinitely distant. Show also that the path becomes circular for v^ = 5 miles per second, nearly. 321. The Problem of Two Bodies. In the preceding discussion of the motion of a particle under the action of a central force, it has been assumed that the center of force is fixed. In the applications of the theory of central forces this assumption is in general not satisfied. Thus, in considering the motion of a planet around the sun, the force of attraction is, according to Newton's law of universal gravitation (Art. 306), regarded as due to the presence of a mass M at the center (sun), and of a mass m at the attracted point (planet); and the action between these two masses is a mutual action, being of the nature of a stress, i. e. consisting of two equal and opposite forces, each equal to „ mM F = K . Hence, the mass m of the planet attracts the mass M of the sun with 323.] MOTION OF A FREE PARTICLE 245 precisely the same force with which the mass M of the sun attracts the mass m of the planet. The attraction affects, therefore, the motions of both bodies. 322. The accelerations produced by the two forces are, of course, not equal. Indeed, the acceleration F/m = kM/t'', produced in the planet by the sun, is very much greater than the acceleration F/M = Kin/r'^, produced by the planet in the sun; for the mass of even the largest planet (Jupiter) is less than one thousandth of that of the sun. The assumption of a fixed center can therefore be regarded as a first approximation in the problem of the motion of a planet about the sun. In the case of the earth and moon, the difference of the masses is not so great, the mass of the moon being nearly one eightieth of that of the earth. It can be shown, however, that the results deduced on the assumption of a fixed center can, by a simple modification, be made available for the solution of the general -problem of the motions of two particles of masses m., M, subject to no forces besides their mutual attraction. In astronomy, this is called the problem of two bodies. In the solution below we assume the attraction to follow Newton's law of the inverse square of the distance. It will be convenient to speak of the two particles, or bodies, as planet (m) and sun (ilf). 323. With regard to any fixed system of rectangular axes, let x, y, z be the co-ordinates of the planet (m), at the time t; x', y', z' those of the sun {M), at the same time; so that for their distance r we have r2 = (x - x'Y + {y - y'Y + (2 - z')\ Then the equations of motion of the planet are mx = F •^-^— , mij = F-^-—-, mz = F - ^ ~ - , (1) while the equations of motion of the sun are Mx' = F . ^^^^- , Mij' = F y~-y' , Mz' = F ■ ^^— . (2) By adding the corresponding equations of the two sets, we find d^ d- (/'■* ^^2 (mx + Mx') = 0, ^^, {my + My') = 0, ^^piz + Mz') = 0. If it be remembered that the centroid of the two masses m, M has the co-ordinates 246 KINETICS [324. _ _ mx + Mx ' ^ _niy + My' _ _ mz + M z' it appears that these equations can be written in the form ^ = -^ = - = 0- dt^ ' dfi ' dp ' in words: the acccleralion of the common centroid of planet and sun is zero; i. e. this centroid moves with constant velocity in a straight line. 324. The integration of the equations (1) would give the absolute path of the planet. But the constants could not be determined, because the absolute initial position and velocity of the planet are, of course, not known. The same holds for the absolute path of the sun. All we can do is to determine the relative motion, and we proceed to find the motion of the planet relative to the sun. Taking the sun's center as new origin for parallel axes, we have for the co-ordinates f, i?, i' of the planet in this new system, ^ = X - x', V = y - y', t = z — z'. Now, dividing the equations (1) by m, the equations (2) by Rf, and sub- tracting the equations of set (2) from the corresponding equations of set (1), we find for the relative acceleration of the planet V M + m ^ M + m V -. il/ + w f ,„, k = — K , V = ~ K , , s' = - K ^ — • -" . (3) The form of these equations shows that the relative motion of the planet with respect to the sun is the same as if the sun ivere fixed and contained the 7nass M -\- m. Thus the problem is reduced to that of a fixed center, the only modification being that the mass of the center M should be increased by that of the attracted particle ?n. 325. This result can also be obtained by the following simple con- .sideration. The relative motion of the planet with respect to the sun would obviously not be altered if geometrically equal accelerations were applied to both. Let us, therefore, subject each body to an additional acceleration equal and opposite to the actual acceleration of the sun (whose components are obtained by dividing the equations (2) by M). Then the sun will be reduced to equilibrium, while the resulting accel- eration of the planet, which is its relative acceleration with respect to the sun, will evidently be the sum of the acceleration exerted on it by 327.] MOTION OF A FREE PARTICLE 247 the sun and the acceleration exerted on the sun by the planet. This is just the result expressed by the equations (3). 326. It can here only be mentioned in passing that, while the problem of two bodies thus leads to equations that can easily be integrated, the problem of three bodies is one of exceeding difficulty, and has been solved only in a few very special cases. Much less has it been possible to integrate the 3 n equations of the problem of n bodies. 327. According to the equations (3), the first and second laws of Kepler can be said to hold for the relative motion of a planet about the sun (or of a satellite about its primary). The third law of Kepler requires some modification, since the intensity of the center ^ should not be kM, but k{M + m). We have, by (26), Art. 315, M= k{M + ?n) = 47r2,^; in other words, the quotient a'/T^ is not independent of the mass m of the planet. Thus, if mi, ra-i be the masses of two planets, Oi, ao the major semi- axes of their orbits, and Ti, T^ their periodic times, we have OiVTii _ M + m, ^ 1 + mi/M This quotient is approximately equal to 1 if M is very large in com- parison with both nil and /«2; hence, for the orbits of the planets about the sun, Kepler's third law is very nearly true. CHAPTER XIV. CONSTRAINED MOTION OF A PARTICLE.J 1. Introduction. 328. A free particle is said to have three degrees of freedom (Art. 231) since three co-ordinates are required to determine its position, and each of these co-ordinates can vary inde- pendently of the other two. If the co-ordinates of a moving particle are subjected to one condition, say my= Y + Ny-^Nf^, (5) dz mz = Z -\- Nz — p.N -T-, ds where N"" = N,^ -\- Ny"" + A^.^ ^nd NM + N ydy + N,dz 252 KINETICS [334. = since N is normal to the path. In addition, we have of course the equations (2) of the curve. Multiplying the equations (5) by dx, dy, dz and adding we find the equation of kinetic energy and work dilmv^) = Xdx + Ydij + Zdz - fiNds. This relation might have been written down directly by con- sidering that for a displacement ds along the fixed curve the normal reaction N does no work, while the work of friction is — fxNds. If there be no friction (fi = 0) it follows from the last equation, or from the first of the equations (4), that the velocity is independent of the reaction of the curve. 334. Exercises. (1) A mass of 2 lbs. attached to a cord, 3 ft. long, is swung in a circle. Neglecting gravity, find the tension in pounds: (a) when the mass makes one revolution per second; {b) when it makes S revolutions per second, (c) If the cord cannot stand a tension of more than 300 lbs., what is the greatest allowable number of revolutions? (2) A plummet is suspended from the roof of a railroad car; how much will it be deflected from the vertical when the train is running 45 miles an hour in a curve of 300 yards radius? (3) A body on the surface of the earth partakes of the earth's daily rotation on its axis. The constraint holding it in its circular path is due to the attractive force of the earth. Taking the earth's equatorial radius as 3963 miles, show that the centripetal acceleration of a particle at the equator is about ^ ft. per second, or about j^-^ of the actually observed acceleration g = 32.09 of a body falling in vacuo. (4) If the earth were at rest, what would be the acceleration of a body falling in vacuo at the equator? (5) Show that if the velocity of the earth's rotation were over 17 times as large as it actually is, the force of gravity would not be sufficient to detain a body near the surface at the equator (comp. Ex. (13), Art. 320). (6) Show that in latitude the acceleration of a falling body, if 335. CONSTRAINED MOTION OF A PARTICLE 253 B the earth were at rest, would hegi = g + j cos-(p, where g is the observed acceleration of a falling body on the rotating earth and j the centripetal acceleration at the equator. Thus, in latitude = 45°, g = 980.6 cm. ; hence gi = 982.3. (7) Owing to the earth's rotation on its axis the direction of a plumb-line does not pass through the center of the earth, even when the earth, as here assumed, is regarded as a homogeneous sphere. Determine the angle S of the deviation in latitude I; and that it will leave the circle at the point for wliich cosO = — ^h/l, if §/i < I. (4) A particle subject to gravity moves on the outside of a vertical circle; determine where it will leave the circle: (a) if MN (Fig. 79) intersects the circle; (h) if MN touches the circle; (c) if MN does not meet the circle. (5) A particle subject to gravity is compelled to move on any vertical curve z = fix) without friction. Show that the velocity at any point is ?; = V2gz (comp. Art. 336) if the horizontal axis of x be taken at a height above the initial point equal to the "height due to the initial velocity," i. e. Vo^/2g. (6) A particle slides on the outside of a smooth vertical circle, starting from rest at the highest point of the circle. Find where it will meet the horizontal plane through the lowest point of the circle. 340. If for a particle constrained to a curve, under given forces, the time of reaching any particular point is the same from whatever point of the curve the particle starts with zero velocity, the curve is called a tautochrone for the given forces, and the point is called the point of tauto- chronism. In a vertical plane, if gravity is the only force, a cycloid with vertical axis can be shown to be a tautochrone, with the vertex as point of tautochronism. This will even be true if the curve be rough, or if the particle be subj-ect to a 250 KINETICS [341. resistance proportional to the velocity in the direction of motion; but, for the sake of simplicity, we exclude these complications. . The problem of determining a tautochrone for given forces (if such a curve exists) is rather different in nature from the ordinary problems of mechanics inasmuch as it is here required to find a curve, on which motions of a certain kind may take place. Indeed, it is a generalization of the problem of the tautochrone that led Abel to the first solution of an integral equation.* 341. With respect to a horizontal axis Ox and a vertical axis Oz through the point of tautochronism, the principle of kinetic energy and work (comp. Art. 339, Ex. 5) gives for the velocity v'^ = 2g{h - z), where h is the ordinate of the starting point P. Counting the arc s from we have dsjcit = — V2g(/i — z), whence the time of motion from P to : ^ " ~ J.=/. V2sf(/i - z) ^ ^2g X ds VT- If we put s = f{z) and hence ds = f{z)dz, the problem re- quires the determination of the function f{z) for which the integral has a value independent oi h. To make the limits independent of h let us put z = hy; we then find t = 1 r'rihy)hdy_ 1 r' ri^y This integral will be independent of h if j'Qiy) -yjhy is *See M. BocHER, Integral equations, Cambridge, University Press, 1909, p. 6. 342.] CONSTRAINED MOTION OF A PARTICLE 257 independent of h; and as this expression is symmetric in h and y, it will then be also independent of z. We can therefore put whence solving for dx we find (comp. Art. 20) : X = I -^ dz = -^z^K — z) + K- sin~i .*l- . This is the equation of a cycloid with as vertex and Oz as axis. Putting z = k sin^i^, we find the equations of the cycloid in the form X = ^K{e + sin^), z = ^k(1 - cos9), so that K is the diameter of the generating circle. For the time we find : ^ _ i-^ r dy "ViX 4y - y'' ^'2g* 342. Exercises. (1) For a heavy particle moving witliout friction on a cycloid with vertical axis, x = a(d -\- sin9), z = a(l — cos^), show tliat the equation of motion is s = — gs/4:a, s being the arc counted from the vertex. Hence, if ?> = for s = so, s = Sa cos Vgl^a t, which shows that the time of reaching the lowest point is independent of so. (2) The involute of a cycloid being an equal cycloid, with its vertex at the cusp, its cusp on the axis, of the original cycloid, the particle in Ex. (1) can be constrained to the cycloid by means of a cord of length 2a, attached to the cusp of the involute, and wrapping itself on a cylinder erected on the involute as base {cydoidal pouhdutii). Show that, if the particle starts from rest at the cusp of the original cycloid, the tension of the cord is twice the normal component of the weight of the particle. 18 258 KINETICS [343. (3) Prove that it is not possible to construct a tautochrone (for gravity) from P to with as point of tautochronism unless the slope of OA is in absolute value < 2/7r. 343. The cycloid (with vertical axis) has another remarkable prop- erty; it is the brachistochrone, or cui've of quickest descent, for a particle subject to gravity. More definitely: two points Pi, P2 being given we may inquire to what curve in their vertical plane must a heavy particle be constrained to reach in the shortest time the lower point P2 if it starts from Pi with a given velocity. As the time is given by a definite integral the problem requires the determination of that curve z = fix) for which this integral becomes a minimum. This problem has given rise to the invention of the calculus of variations. As the problem can hardly be solved satisfactorily without using the methods of this calculus we merely state that the required curve is the cycloid through the two points, without cusp between them and with vertical axis.* 3. Motion on a fixed surface. 344. The equations of motion of a particle constrained to a surface do not differ in form from tlie equations (5), Art. 333, for a particle constrained to a curve. The normal reaction being normal to the given surface (p(x, y, z) = 0, we have dip dip dip dx dy dz A comparatively simple problem is that of the conical or spherical pendulum, i. e. of a particle subject to gravity and constrained to the surface of a sphere. But even this problem can not be treated without introducing elliptic integrals. *See O. BoLZA, Variationsrechnung, Leipzig, Teubner, 1909, p. 207. 346.] CONSTRAINED MOTION OF A PARTICLE 259 4. The method of indetermmate multipliers. 345. The following brief discussion of the equations of motion of a constrained particle is not so much intended to furnish methods for solving particular problems, but rather as a preparation for, and an introduction to, the general methods of mechanics of systems of particles subject to conditions. For this reason we shall here assume the absence of friction on the constraining surface or curve; but, on the other hand, it is desirable to generalize by assuming that the constraints are variable, that is, that the conditional equations (1) and (2), Art. 328, contain the time t explicitly. 346. D'Alembert's Principle. The ordinary equations of motion of a free particle, mx = A^, my — Y, mz — Z, (6) where X, Y, Z are the components of the resultant R of the given forces, mejiely express the equality of this force R, as a vector, to the mass-acceleration mj, which is sometimes called the effective Jorce. It follows that // the reversed effective force — mj, or its components — mx, — my, — m'z, he combined with the given forces we have a system in equilibrium at the given instant. This is the fundamental idea of d'Alembert's principle, as it is now generally used. Owing to this idea we can apply to kinetic problems the statical conditions of equilibrium. Thus, in the case of the free particle, the conditions of equilibrium of the forces X, Y, Z, — mx, — my, — mz are X — mx = 0, Y — mij = 0, Z — mz — 0, and thus the equations of motion arc found. But the conditions of equilibrium can also be expressed 260 KINETICS [347. by means of the principle of virtual work. By Art. 266, the necessary and sufficient condition of equiUbrium of the particle under the forces — 77ix. — mij, — m'z, X, Y , Z is that (- mx + X)bx + (- vtij + r)5?/ + (- mz + Z)52 = (7) for any virtual displacement bs{bx, by, 8z). Owing to the independence of 8x, dy, 8z, their coefficients must vanish separately, and we find again the equations (6). In other words, the single equation (7) is equivalent to the three equations (6). 347. One constraint. If the particle is subject to the condition or constraint ifix, y, z, 0=0, (8) it must throughout its motion remain on the surface repre- sented by this equation. To apply d'Alembert's prin- ciple let the particle be subjected to a virtual displacement ds. If this displacement be selected along the position of the surface at the time t, the work of the reaction (which is normal to the surface (8), and hence to 8s, since we assume that there is no friction) will be zero. Hence the equation of motion is the same as for a free particle, viz. (7). But the displacement 5s must be along the surface (8), or as we shall say, compatible vrith the constraint. This requires that 8x, 8y, 8z be selected so as to satisfy the relation n — 6. For if there 268 359.] EQUATIONS OF MOTION OF A RIGID BODY 269 were but 3 particles, the number of independent conditions would evidently be 3; for every additional particle, 3 ad- ditional conditions are required. Hence, the total number of conditions is 3 + 3(n — 3) — Sn — 6. It follows that if a rigid body be subject to no other con- straining conditions, the number of its equations of motion must be Sn — (3n — 6) =6. Hence, a free rigid body has six independent equations of motion (comp. Art. 231). 'h . '< '• '''■ 359. The six equations of motion of the rigid body can be obtained as follows. Imagine the equations (1) written down for every particle, and add the corresponding equations. This gives the first 3 of the 6 equations of motion: ^mx = 2X, Zmij = 2 7, llmz = 2Z. (2) It is important to notice that the internal reactions be- tween the particles which make the body rigid occur in pairs of equal and opposite forces, and form, therefore, a system which is in equilibrium by itself. This may be regarded as an assumption which should be included in the definition of the rigid body. Hence, while these internal forces enter into the equations (1), they do not appear in the equations (2). The right-hand members of these equations (2) represent therefore the components Rx, Ry, Rz of the resultant R of all the external forces acting on the body. The left-hand members can be written in the form d{I,mx)/dt, d{'Zmij)ldt, d{'Zmz)/dt: these are the time-derivatives of the sums of the linear momenta of all the particles parallel to the axes. The equations (2) can therefore be written in the form ^Xmx^Rx, ~i:my = Ry, ^^^^^mz = R,. (2') The axes of co-ordinates are arbitrary. Hence, if we agree to 270 KINETICS [360. call linear momentum of the body in any direction the algebraic sum of the linear momenta of all the particles in that direc- tion, the equations (2') express the proposition that the rate at which the linear mofnentum of a rigid body in any direction changes with the time is equal to the sum of the comyonents of all the external forces in that direction. 360. Let us now combine the second and third of the equations (1) by multiplying the former by z, the latter by y, and subtracting the former from the latter. If this be done for each particle, and the resulting equations be added, we find I,m{yz — zij) = '^{ijZ — zY). Similarly, we can pro- ceed with the third and first, and with the first and second of the equations (1). The result is: ^m{yz - zij) = Z(yZ - zY), ^m(zx - xz) = Z{zX - xZ), ^ni{xij - yx) = Z{xY - yX). (3) Here again the internal forces disappear in the summation, so that the right-hand members are the components Hx, Hy, Hz, of the vector H of the resultant couple, found by reducing all the external forces for the origin of co-ordinates. The left-hand members are the components of the resultant couple of the effective forces for the same origin. We can also say that the right-hand members are the sums of the moments of the external forces about the co-ordinate axes (Art. 229), while the left-hand members represent the moments of the effective forces about the same axes. The latter quantities are exact derivatives, as shown in Art. 279. The equations (3) can therefore be written in the form y. 'Lm{yz - zy) = Hx, ^J^mizx - xz) = Hy, d ^^'^ -^^^mixy -yx) = H^. 362.] EQUATIONS OF MOTION OF RIGID BODY 271 As explained in Art. 279, the quantity m{yz — zy) is called the angular momentum (or the moment of momentum) of the particle m about the axis of x. We shall now agree to call the quantity I,m{yz — zy) the angular momentu7n of the body about the axis of x, just as Hmx is the linear momentum of the body along this axis; and similarly for the other axes. The meaning of the equations (3') can then be stated as follows: The rate at ivhich the angular ynomentum of a rigid body about any axis changes with the time is equal to the sum of the moments of all the external forces about this line. The equations (2) and (3), or (2') and (3'), are the six equations of motion of the rigid body. The three equations (2) or (2') may be called the equations of linear momentum, while (3) or (3') are the equations of angular momentum. 361. If, as in Art. 280, we imagine the angular momentum of each particle represented by a vector drawn from the origin of co-ordinates, the geometric sum, or resultant, of these vectors is a vector h which represents the angular momentum of the body about the origin; and its components hx, hy, hz along the axes are tlie angular momenta 1,7n{yz — zy), 'Zim{zx — xz), ^m{xy — yx) of the body about these axes. The equations (3') can then be written in the simple form afix -rj any jj anz jj /o//\ d^ == ^- 7it^ ^^- ^dt ^ ^" ^^ ^ and these equations are together equivalent to the single vector equation dh _ „ dt ~ 362. The equations of linear momentum, (2) or (2'), admit of a further simplification, owing to the fundamental property 272 KINETICS [363. of the centroid. By Art. 159, the co-ordinates x, y, z of the ^centroid satisfy the relations Mx = llmx, My = 2m2/, Mz = ^mz, where M = ^m is the whole mass of the body. Differentiat- ing these equations, we find Mx = Imx, My = limy, Mz = 2mi, and Mx = Smx, My = 2my, Mz = ^mz, where x, y, z are the components of the velocity v, and x, y, z those of the acceleration j, of the centroid. The equations (2) or (2') can therefore be reduced to the form Mx = j,Mx = R., Mi) = J. My = Ry, M2 = -Mz = Rz, at whence Mj= -^.Mv = R; i. e. if the whole mass of the body be regarded as concentrated at the centroid, the effective force of the centroid, or the time-rate of change of its momentum, is equal to the resultant of all the external forces. It follows that the centroid of a rigid body moves as if it contained the whole mass, and all the external forces were applied at this point parallel to their original directions. 363. If, in particular, the resultant R vanish (while there may be a couple H acting on the body), we have by (2") j = 0; hence v = const.; i. e. if the residtant force he zero the centroid moves uniforinly in a straight line. 364.] EQUATIONS OF MOTION OF A RIGID BODY 273 This proposition, which can also be expressed by saying that ii R = 0, the momentum Mv of the centroid remains constant, or, using the form (2') of the equations of motion, that the hnear momentum of the body in any direction is constant, is known as the principle of the conservation of linear momentum, or the principle of the conservation of the motion of the centroid. 364. Let us next consider the equations of angular momen- tum, (3) or (3'). To introduce the properties of the centroid, let us put X — X = ^, y — y = r], z — z = ^, so that ^, 77, f are the co-ordinates of the point (x, y, z) with respect to parallel axes through the centroid. The substitution of X =^ X -\- ^, y = y -\- V, z = z -{- ^ and their derivatives in the expression yz — zy gives yz - zij = yz - zy -\- ijt - zi] + -nz - ty + r]t - ^v. To form 'Zm(yz — zy) we must multiply by m and sum throughout the body; in this summation, y, z, y, z are constant and by the property of the centroid, '^mt] = 0, 2m^ = 0, Sm^ = 0, 2mf = 0. Hence we find 'Emiyz - zy) = 2m (17^ -fTJ) + M(;yz - zy). The second term in the right-hand member is the angular momentum of the centroid about the axis of x (the whole mass M of the body being regarded as concentrated at this point), while the first term is the angular momentum of the body (in its motion relatively to the centroid) about a parallel to the axis of x, drawn through the centroid. Similar relations hold for the angular momenta about the axes of y and z; and as these axes are arbitrary, we conclude that the angular momentum of a rigid body about any li7ie is equal to its angular momentum about a parallel through the 19 274 KINETICS [365. centroid plus the angular momentum of the centroid about the former line. 365. Differentiating the above expression, we find -^^m{yz - zy) = -^^^^Mvt - ^v) + M(yz - zy). The first of the equations (3') can therefore be written ^2m(7?f - In) + Miyl - ztj) = H.. Now, if at any time t the centroid were taken as origin, so that y = 0, z =0, this equation would reduce to the form J-2m(7?f -N) = H., which is entirely independent of the co-ordinates of the cen- troid. On the other hand, wherever the origin is taken, if the centroid were a fixed point, the same equation would be obtained. Similar considerations apply of course to the other two equations (3')- It follows that the motion of a rigid body relative to the centroid is the same as if the centroid were fixed. 366. If, in particular, the resultant couple H be zero for any particular origin (which will be the case not only when all external forces are zero, but whenever the directions of all the forces pass through the point 0), the equations (3') can be integrated and give i:m{yz - zy) = d, i:m{z± - xz) = Ci, ... Zm(xy - yx) = C^, where d, d, C3 are constants of integration. Hence, if the external forces pass through a fixed point, the angidar momentum of the body about any line through this point is constant; if there are no external forces, the angular momentum is constant for 368.1 EQUATIONS OF MOTION OF RIGID BODY 275 amj line whatever. This is the principle of the conservation of angular momentum. 367. Taking the equations of angular momentum in the form (3") we find when // = 0; /ix = Ci, hy = Cs, h = C^, (4') and hence the vector h (Fig. 70, Art. 280) remains constant in magnitude and direction. The term principle of the con- servation of areas which is often used instead of principle of the conservation of angular momentum is less appropriate. In the case of the single particle, where hx = m(yz — zy), etc., the vector of angular momentum h is simply 2m times the vector representing the sectorial velocity; but in the case of the rigid body, to form the vector of angular momentum h we have to multiply the sectorial velocity of each particle by twice its mass and add these " weighted" sectorial velocities geometrically. In the study of the motion of the rigid body with a fixed point where the vector h is of primary importance it has l)ecn called the impulse, or impulse-vector. Our principle then means that whenever for any point the resultant couple H is zero the impulse remains a constant vector: h = C. The direction of h is then called the invariable direction; the plane through 0, perpendicular to h, Cix + Coy + Csz = 0, is called Laplace's invariable plane. 368. Returning to the general case of the motion of a rigid body under any forces, we may say that the propositions at the end of Arts. 362 and 3G5 cstal^lish the principle of the independence of the motions of translation and rotation. Ac- 276 KINETICS [369. cording to these propositions the problem of the motion of a rigid body resolves itself into two problems; that of the mo- tion of the centroid and that of the motion of the body about its centroid. The former reduces by Art. 362 to the problem of the motion of a particle, viz. the centroid of the body, with a mass M equal to that of the body, acted upon by all the given external forces transferred parallel to themselves to the centroid. The latter problem, that of the motion of the bod}^ about its centroid, is, by Art. 365, the same as the problem of the motion of a rigid body about a fixed point. This important problem is discussed in Chap. XVIII; its solution depends on the equations (3), (3'), or (3")- 369. If the equation of motion (1), Art. 357, of the par- ticle m be multiplied by the components dx, dy, dz of the actual displacement ds of this particle, we find upon adding the equations for all the particles ^m{xdx + ijdy + zdz) = ^(Xdx + Ydy -{- Zdz), where the right-hand member represents the elementary work of the external forces since that of the internal forces is zero. The left-hand member, just as in the case of the single particle (Art. 271), is the exact differential of the kinetic energy T = 'Ehnv^ = i:hn{x^ -f 7J~ + i') of the body. Hence, integrating, say from t = to t = t, we find the relation T - To = ^hnv^ - Simvo^* = C^iXdx -f Ydy -f Zdz), where the right-hand member represents the work done by the external forces on the body during the time t. This 371.] EQUATIONS OF MOTION OF A RIGID BODY 277 equation expresses the principle of kinetic energy and work, for a free rigid body: in any motion of the body, the increase of the kinetic energy is equal to the work done by the external forces. 370. By introducing the co-ordinates of the centroid, i. e. by putting x ^ x -{- ^, y = y -\- v, z = z -\- ^, as in Art. 364, the expression for the kinetic energy assumes the form (since Sm| = 0, 2m^ = 0, I,nit = 0) : T = 7:hn(x' + y' + -z') + ^hn{t~ + ^- + f') where v is the velocity of the centroid and u the relative velocity of any particle m with respect to the centroid. Thus, it appears that the kinetic energy of a free rigid body consists of two 'parts, one of which is the kinetic energy of the centroid (the whole masss being regarded as concentrated at this point), ivhile the other may be called the relative kinetic energy with respect to the centroid. 371. By the same substitution the right-hand member of the first equation of Art. 369, i. e. the elementary work Ii(Xdx + Ydy + Zdz), resolves itself into the two parts {dx^X + dy^Y + dz^Z) + 2(Xd^ + Ydv + Zd^). The first parenthesis contains the work that would be done by all the external forces if they were applied at the centroid ; it is therefore equal to the change in the kinetic energy of the centroid, that is, to d{^Mv^). The equation of kinetic energy reduces, therefore, to the following dXiimu'') = 2(Xd^ + Ydr] + Zd^); in other words, the principle of kinetic energy holds for the relative motion with respect to the centroid. 278 KINETICS [372. 372. Impulses. The equations determining the effect of a system of impulses on a rigid body are readily obtained from the general equations of motion (2) and (3). We shall denote the impulse of a force F by F. It will be remembered that the impulse F oi a, force F is its time integral (Art. 172) ; i. e. F = j^' fdt. We confine ourselves to the case when t' — t is very small and F very large, in which case the action of the impulsive force F is measured by its impulse F. If all the forces acting on a rigid body are of this nature, and the impulses of X, Y, Z during the short interval t' — t be denoted by X, Y, Z, the integration of the equa- tions (2) from t = t to t = t' gives Sm(a;' - x) = 2X, ^m (ij' - t/) = 2 Y, ^m (i' - i) = 2Z, (5) where x, y, z denote the velocities of the particle ?n at the time t just before the impulse, and x', y' , z' those at the time t' just after the action of the impulse. Similarly the equations (3) give ^m[y{i' - i) - z{y' - ?/)] = ^{yZ - zY), 2m[2(i' - x) - x{z' - z)\ = Z{zX - xZ), (6) Xmixiy' -y)- y(x' - x)] = Z{xY - yX). 373. In detcrmiming the effect on a rigid body of a system of such impulses, any ordinary forces acting on the body at the same time are neglected because the changes of velocity produced by them during the very short time t' — t are small in comparison with the changes of velocity x' — i, y' — y, z' — z produced by the impulses. If the impulse F of an impulsive force F be defined as the limit of the integral 373.] EQUATIONS OF MOTION OF RIGID BODY 279 r'Fdt when t' — t approaches zero and F approaches infinity, it is strictly true that the effect of ordinary forces can be neglected when impulsive forces act on the body. If the rigid body be originally at rest, it will be convenient to denote by x, y, z the components of the velocity of the particle tn just after the action of the impulses. We may also denote by R the resultant of all the impulses, by H the resultant impulsive couple for the reduction to the origin of co-ordinates, and mark the components of R and H by subscripts, as in the case of forces. With these notations the effect of a system of impulses on a body at rest is given by the equations 'Zmx = Rjr, ^my = Ry, ^mz = R^, (5') I,'m(yz — zi/) = Hx, ^m{zx — xz) = Hy, Zm{xij — yx) = H,. (6') In the equations (5') we have, of course, Xmx = Mx, I,77iy — My, l^mz = Mz, where x, y, z are the components of the velocity of the centroid, and M is the mass of the body; i. e. the momentum of the centroid is equal to the resultant impulse. The meaning of the equations (6') can be stated by saying that the angular momentum of the body about any axis is equal to the moment of all the impulses about the same axis. CHAPTER XVI. MOMENTS OF INERTIA AND PRINCIPAL AXES. 1. Introduction. 374. As will be shown in Chapters XVII and XVIII, the rotation of a rigid body about any axis depends not only on the forces acting on the body, but also on the way in which the mass is distributed throughout the body. This distribu- tion of mass is characterized by the position of the centroid and by that of certain lines in the body called principal axes. It has been shown in Art. 159 that the centroid of a system of particles is found by determining the moments, or more precisely, the moinents of the first order, 'Emx, Zmy, 'Zmz, of the system with respect to the co-ordinate planes, i. e. the sums of all mass-particles m each multipUed l^y its distance from the co-ordinate plane. The principal axes of a sj'stem of particles can be found by determining the moments of the second order, 'Lmx'^, '^my^, Iimz^, llmyz, Xmzx, 'Zmxy of the system with respect to the same planes. We proceed, therefore, to study the theory of such moments. 375. If in a rigid body the mass m of each particle be multi- plied by the square of its distance r from a given point, plane, or line, the sum Zmr~ = niiri~ + nur2~ + • • • , extended over the whole body, is called the quadratic moment, or, more commonly, the moment of inertia of the body for that point, plane, or line. 280 377.] MOMENTS OF INERTIA AND PRINCIPAL AXES 281 If the body is not composed of discrete particles, but forms a continuous mass of one, two, or three dimensions, this mass can be resolved into elements of mass dm, and the sum Smr^ becomes a single, double, or triple integral J r'^dm. Expressions of the form Zmvir^, or J riVodm, where ri, r2 are the distances of m or of dm from two planes (usually at right angles), are called moments of deviation or products of inertia. 376. The determination of the moment of inertia of a con- tinuous mass is a mere prol^lem of integration; the methods are similar to those for finding the moments of mass of the first order required for determining centroids, the only dif- ference being that each element of mass must be multiplied by the square, instead of the first power, of the distance. A moment of inertia is not a directed c^uantity ; it is not a vector, but a scalar; indeed, it is a positive ciuantity, provided the masses are all positive, as we shall here assume. If the mass is homogeneous, the density appears merely as a constant factor; as the density in this case can be regarded as =1, it is customary to speak of moments of inertia of volumes, areas, and lines. The moment of inertia of any number of bodies ot masses for any given point, plane, or line is obviously the sum of the moments of inertia of the separate bodies or masses for the same point, plane, or line. 377. The moment of inertia Xmr"^ of any body whose mass is M = Sm can always be expressed in the form 2mr2 = M-ro-, where Vo is a length called the radius of inertia, arm of inertia, or radius of gyration. This length ro is evidently a kind of average value of the distances r, its value being intermediate between the greatest r' and least r" of these distances r. For 282 KINETICS [378. we have Swr'^ = Smr^ = 'Lmr"'^, or, since Swr'^ = Mr'^, Xmr^ = Mro\ ^mr"^ = Mr'"", r' ^ ro = r". 378. As an example, let us determine the moment of inertia of a homogeneous rectilinear segment (straight rod or wire of constant cross- section and density) for its middle point (or what amounts to the same thing, for a line or plane through this point at right angles to the seg- ment). Let I be the length of the rod (Fig. 80), its middle point, p" its density {i. e. the mass of unit length), x the distance OP of any element J dm = p"dx from the middle Jt i +H ^ point. Observing that the xp- QQ moment of inertia for O of the whole rod AB \s the sum of the moments of inertia of the halves AO and OB, and that the moments of inertia of these halves are equal, we have, for the moment of inertia / of AB, I = 2£^x^.p"dx = ^p'% and for the radius of inertia ro, since the whole mass is M = p" I, 379. Exercises. Determine the radius of inertia in the following cases. When nothing is said to the contrary, the masses are supposed to be homo- geneous. (1) Segment of straight line of length I, for perpendicular through one end. (2) Rectangular area of length I and width h: (a) for the side h; (6) for the side I; (c) for a line through the centroid parallel to the side h; (d) for a line through the centroid parallel to the side I. (3) Triangular area of base b and height h, for a line through the vertex parallel to the base. (4) Square of side a, for a diagonal. (5) Regular hexagon of side a, for a diagonal. (6) Right cylinder or prism of height h, for the plane bisecting the height at right angles. 381.] MOMENTS OF INERTIA AND PRINCIPAL AXES 283 (7) Segment of straight line of length I, for one end, when the density is proportional to the nth power of the distance from this end. Deduce from this: (c) the result of Ex. (1); (6) that of Ex. (3); (c) the radius of inertia of a homogeneous pyramid or cone (right or oblique) of height h, for a plane tlxrough the vertex parallel to the base. (8) Circular area (plate, disk, lamina) of radius a, for any diameter. (9) Circular line (wire) of radius a, for a diameter. (10) Solid sphere, for a diametral plane. (11) Solid ellipsoid, for the three principal planes. (12) Area of ring bounded by concentric circles of radii ai, aj, for a diameter. 380. The moment of inertia of any mass AI for a point can easily be found if the moments of inertia of the same mass are known for any hne passing through the point, and for the plane through the I point perpendicular to the line. Let (Fig. 81) be the point, / the line, tt the plane; r, q, p the perpendicular dis- tances of any particle of mass m from 0, I, T, respectively. Then we have, evidently, r^ = g^ + p~. Hence, multi- Fig. 81. plying by m, and summing over the whole mass M, ^mr"^ = Hmq^ + '^mp'^; (1) or, putting ^mr- = Mro~, '^rnq'^ = Mq^-, 1,mp^ = Mpo"^, where ^0, qo, Po are the radii of inertia for 0, I, tt, To' = qo' + Po'. (10 381. The moment of inertia of any mass M for a line is equal to the sum of the moments of inertia of the same mass for any two rectangular planes passing through the line. Thus, in particular, the moment of inertia for the axis of x in a rectangular system of co-ordinates is equal to the sum of 284 KINETICS [382. the moments of inertia for the 2a:-plane and x?/-plane. This follows at once by considering that the square of the distance of any point from the line is equal to the sum of the squares of the distances of the same point from the two planes. Thus, if q be the distance of any point {x, y, z) from the axis of x, we have g^ = y"^ -\- z-; whence Zmq- = '^my- -\- Zmz"^. 382. It follows, from the last article, that the moment of inertia I^ of a plane area, for any line perpendicular to its plane, is if ly, Iz are the moments of inertia of the area for any two rectangular lines in the plane through the foot of the perpendicular line. 383. The problem of finding the mo- ment of inertia of a given inass for a line I', when it is ktiown for a parallel line I, is of great importance. Let Smg^ be the moment of inertia of the given mass for the line I (Fig. 82), Smg'2 that for a parallel line V at the distance d from I. The distances q, q' of any particle m from I, V form with d a triangle which gives the relation g'2 = g2 _|. fp _ 2qd cos{q, d). Multiplying by m, and summing over the whole mass M, we find Zmq'~ = 2?ng2 + Md^ - 2d1mq cos(g, d). Now the figure shows that the product q cos(q, d) in the last term is the distance p of the particle ?n from a plane Fig. S2. 385.1 MOMENTS OF INERTIA AND PRINCIPAL AXES 285 through I at right angles to the plane determined by / and V. We have, therefore, Zmq'^ = i:mq^ + Aid'- - 2d'Emp, . (2) where the last term contains the moment of the first order Zm/j = Mjj of the given mass M for the plane just mentioned. If, in particular, this plane contains the centroid G of the mass M, we have I,mp = 0, so that the formula reduces to 2mg'2 = ^mq^ -j. Md\ (3) Introducing the radii of inertia 50', Qo, this can be written go'- = go- + d'\ (3') 384. Similar considerations hold for the moments of inertia Zmp"^, Zmp'^ with respect to two parallel planes tt, t' at the distance d from each other. We have, in this case, p' = p — d; hence, 2wp'2 = v^p2 _|_ ]^f^2 _ 2cZ2mp, (4) and if the plane x contain the centroid G, 385. Of special importance is the case in which one of the lines (or planes), say Z(x), contains the centroid. The for- mulae (3), (3'), and (5) hold in this case; and if we agree to designate any line (plane) passing through the centroid as a centroidal line (plane), our proposition can be expressed as follows : The moment of inertia for any line (plane) is found from the moment of inertia for the parallel centroidal line {plane) by adding to the latter the product Md^ of the whole mass into the square of the distance of the lines (planes). It will be noticed that of all parallel fines (planes) the centroidal line (plane) has the least moment of inertia. 286 KINETICS [386. 386. Exercises. Determine the radius of inertia of the following homogeneous masses : (1) Rectangular plate of length I and width h, for a centroidal line perpendicular to its plane. (2) Area of equilateral triangle of side a: (a) for a centroidal line parallel to the base; (b) for an altitude; (c) for a centroidal line per- pendicular to its plane. (3) Circular disk of radius a: (a) for a tangent; (6) for a line through the center perpendicular to the plane of the disk ; (c) for a perpendicular to its plane through a point in the circumference. (4) Solid sphere, for a diameter. (5) Area of ring bounded by concentric circles of radii ai, 02, for a line through the center perpendicular to the plane of the ring. (6) Right circular cylinder, of radius a and height h: (a) for its axis; (6) for a generating line; (c) for a centroidal line in the middle cross-section. (7) By Ex. (3) (6), the moment of inertia of the area of a circle of radius a, for its axis {i. e. the perpendicular to its plane, passing through the center), is / = ^iraK Differentiating with respect to a, we find: -7- = 2-ira^ = 2ira ■ a? ; da hence, approximately for small Aa: A7 = 27ra'Ao = 2iraAa • a^. This is the moment of inertia of the thin ring, of thickness Ao, for its axis. (Comp. Ex. (5).) If the constant surface density (Art. 155) of the circle be p', we have / = \p'Tra*\ hence A/ = 27rap'Aa • a^, where p'Aa is the linear density p" of the ring. (8) Apply the method of Ex. (7) to derive from Ex. (4) the moment of inertia of a thin spherical shell, of radius a and thickness Ao, for a diameter. (9) Area of ellipse: (a) for the major axis; (6) for the minor axis; (c) for the perpendicular to its plane through the center. (10) Solid ellipsoid, for each of the three axes. (11) Wire bent into an equilateral triangle of side a, for a centroidal line at right angles to the plane of the triangle. 388.] MOMENTS OF INERTIA AND PRINCIPAL AXES 287 (12) Paraboloid of revolution, bounded by the plane through the focus at right angles to the axis, for the axis. (13) Anchor-ring, produced by the revolution of a circle of radius a about a line in its plane at the distance h{> a) from the center, for the axis of revolution. 2. Ellipsoids of inertia. 387. The moments of inertia of a given mass for the dif- ferent hnes of space are not independent of each other. Several examples of this have already been given. It has been shown, in particular (Art. 383), that if the moment of inertia be knov/n for any line, it can be found for any parallel line. It follows that if the moments be known for all lines through any given point, the moments for all lines of space can be found. We now proceed to study the relations be- tween the moments of inertia for all the lines passing through any given point 0. Let X, y, z be the co-ordinates of any particle m of the mass; and let us denote hy A, B, C the moments of inertia of M for the axes of x, y, z; by A', B', C those for the planes ijz, zx, xy; by D, E, F the products of inertia (Art. 375) for the co- ordinate planes; i. e. let us put A = Zm(?/2 + z^), A' = 2mx% D = 2myz, B = -Emiz^ + a;2), B' = l^imf, E = Zmzx, (6) C = 2m(a;2 + ij^), C = Sms^, F = ^mxy. 388. These nine quantities are not independent of each other. We have evidently A = B' + C\ B = C + A', C = A' + B'; hence, solving for A', B' , C, A'^UB-\-C-A), B'=UC+A-B), C'=^A-^B-C). The moment of inertia for the origin is Swr2=Sm(a;2+ 1/2+22) = A'-j- B'-\- C = i{A -\- B-\-C). (7) 288 KINETICS [389. 389. The moment of inertia I for any line through can be expressed by means of the six quantities A, B,C, D, E, F; and the moment of inertia /' for any plane through can be expressed by means of A', B', C, D, E, F. Let TT (Fig. 83) be any plane passing through 0; Z its nor- mal; a, /3, 7 the direction cosines of I; and, as before (Art. 380), p, q, r the distances of any point {x, y, z) of the given mass from tt, I, and 0, re- spectively. Then, projecting the closed polygon formed by r, X, \j, z on the line I, we have p = ax + /3?/ + 72; hence, squaring, multiplying by w, and summing over the Fig. 83. whole mass, we find + 2l3y'^myz + 2yaZmzx + 2a^'Emxy, or, with the notations (6), r = A' a'- + B'I3- + CY- + 2Z)/37 + 2jE'7a + 2Fa/3. (8) Thus the moment of inertia for any j)lane through the origin is expressed as a homogeneous quadratic function of the direction cosines of the normal of the plane. 390. The moment of inertia I = llinq^ for the line I can now be found from equation (1), Art. 380, by substituting for I,7nr'^ and Zmj)^ their values from (7) and (8) : I = i:mr^ - 7' - A' + B' + C - I' = A'(l-a'-)-\-B'{l-(3'-)-\rC'{l-y'-)-2D^y-2Eya-2Fa^, or, since a- + /3- + 7- = 1, 392.] MOMENTS OF INERTIA AND PRINCIPAL AXES 289 I = A'(^' + 7') + B'W + a^) + C'(a2 + iS^) = a'-iB' + C) + fi\C' + A') + 7-(A' + 5') - 2D^y - 2Eya - 2Fa^; hence, finally, applying the relations of Art. 388, I = Aa''^ 5/32 + (7^2 _ 27)^^ _ 2Eya - 2F«/3. (9) The moment of inertia for amj line through the origin is, therefore, also a homoge^ieous quadratic function of the direction cosines of the line. 391. These results suggest a geometrical interpretation. Imagine an arbitrary length OP = p laid off from the origin on the line I whose direction cosines are a, /3, 7; the co- ordinates of the extremity P of this length will be a: = pa, y = p^, z = py. Now, if equation (9) be multiplied by p^, it assumes the form Ax^ + By^ + Cz^ - 2Dyz - 2Ezx - 2Fxy = pU, which represents a quadric surface provided that p be selected for the different lines through so as to make p^/ constant, say p2/ = K^. Hence,?/ on every line I through the origin a length OP = p = kJ -^ I he laid off, i. e. a length inversely pro- portional to the square root of the moment of inertia I for this line I, the points P will lie on the quadric surface Ax^ + By^ + C22 - 2Dyz - 2Ezx - 2Fxy = k\ The constant k^ may be selected arbitrarily; to preserve the homogeneity of the equation it will l^e convenient to take it in the form k^ = Me*, where e is an arl)itrary length. 392. As moments of inertia are essentially positive quan- tities, the radii vectores of the surface Ax^ + By^- + C22 - 2Dyz - 2Ezx - 2Fxy = Me* (10) 20 290 KINETICS [393. are all real, and the surface is an ellipsoid. It is called the ellipsoid of inertia, or the momental ellipsoid, of the point 0. This point is the center; the axes of the ellipsoid are called the principal axes at the point 0; and the moments of inertia for these axes are called the principal 7noments of inertia at the point 0. Among these there will evidently be the greatest and least of all the moments of the point 0, the greatest moment corresponding to the shortest, the least to the longest axis of the ellipsoid. It may be observed that, owing to the relations of Art. 388, which show that the sum of any two of the quantities A, B,C is always greater than the third, not every ellipsoid can be regarded as the momental ellipsoid of some mass. An ellip- soid can be a momental ellipsoid only when a triangle can be constructed of the reciprocals of the squares of its semi-axes. 393. If the axes of the ellipsoid (10) be taken as axes of co-ordinates, the equation assumes the form hx^ + hy- + hz- = Me\ (11) where 7i, h, I3 are the principal moments at the point 0. By Art. 391 we have p^ = k'-/I = Me\fl; hence 7 = Me*/p\ If, therefore, equation (11) be divided by p-, the following simple expression is obtained for finding the moment of inertia, I, for a line whose direction cosines referred to the principal axes are a, /3, 7: I = /la^ + W + hy^- (12) 394. To make use of this form for 7, the principal axes at the point 0, i. e. the axes of the momental ellipsoid (10), must be known. The determination of the axes of an ellipsoid whose equation referred to the center is given is a well-known problem of analytic geometry. It can be solved by considering that the semi-axes are those radii vectores of the surface that are normal to it. The direction cosines of the normal 394.1 MOMENTS OF INERTIA AND PRINCIPAL AXES 291 of any surface F{x, y, z) =0 are proportional to the partial derivatives dFjdx, dF/dy, dF/dz. If, therefore, the radius vector p is a semi-axis, its direction-cosines a, /3, y must be proportional to the partial deriv- atives of the left-hand member of (10); i. e. we must have Ax - Fy - Ez ^ - Fx + By - Dz ^ - Ex - Dy + Cz a " fi 7 ' or dividing the numerators by p, Aa - F0 - Ey ^ - Fa + B0 - Dy ^ - Ea - D0 + Cy a ^ y ' Denoting the common value of these fractions by / we have al = Aa - Fp - Ey, /3/ = - Fa + Bfi - Dy, yl = - Ea - Dff+Cy] multiplying these equations by a, p, y and adding we find I = Aa'^ + Bff^ + Cy^ - 2D^y - 2Eya - 2Fa/3, which, compared with (9), shows that / is the moment of inertia for the axis (a, /3, y). To obtain it in terms oi A,B, C, D, E, F, we write the preceding three equations in the form (/ - A)a + F^ + Ey =0, Fa + {I - B)p + Dy =0, (13) Ea+ D^+ {I - C)y = 0, whence, eliminating a, /3, y, we find / determined by the cubic equation I - A, F, E F,I - B, D E, D,I -C = 0. (14) The roots of this cubic are the three principal moments I\, 1 2, h of the point 0. The direction cosines of the principal axes are then found by substituting successively I\, h, h in (13) and solving for a, /8, y. In general, the three principal moments of inertia I], h, h at a point O are different. If, however, two of them are equal, saj' h = h, the momental ellipsoid becomes an ellipsoid of revolution about the third, /i, as axis; and it follows that the moments of inertia for all lines through O in the plane of the two equal axes are equal. If I\ = h = I3, the ellipsoid becomes a sphere, and the moments of inertia are the same for all lines passing through 0. 292 KINETICS [395. 395. If the equation of the momental elhpsoid at a point be of the form Ax'' + B^f + Cz^ — 2Dyz = Ale*, i. e. if the two conditions E = Xmzx =0, F = i:mxy = be fulfilled, the axis of x coincides with one of the three axes of the ellipsoid, the surface being symmetrical with respect to the yz-plane. Hence, if the coriditions E = 0, F = are satisfied, the axis of x is a 'principal axis at the origin. The converse is evidently also true; i. e. if a line is a principal axis at one of its points, then, for this point as origin and the line as axis of x, the conditions 'Zmzx = 0, I^mxy = must be satisfied. It is easy to see that if a Une be a principal axis at one of its points, say 0, it will in general not be a principal axis at any other one of its points. For, taking the line as axis of x and as origin, we have Hjtizx = 0, '^mxy = 0. If now for a point 0' on this line at the dis- tance a from the line is likewise a principal axis, the conditions 2??iz(x — a) = 0, Zm{x — a)y = must be fulfilled. These reduce to 2m2 = 0, "Liny = 0, and show that the line must pass through the centroid. And as for a centroidal line these conditions are satisfied independently of the value of a, it appears that a centroidal principal axis is a principal axis at every one of its points. Hence, a line cannot be principal axis at more than one of its points unless it pass through the centroid; in the latter case it is a principal axis at every one of its points. 396. All those lines passing through a given point for which the moments of inertia have the same value I can be shown to form a cone of the second order whose principal diameters coincide with the axes of the momental ellipsoid at 0. This is called an equimomental cone. Its equation is obtained by regarding / as constant in equation (12) and introducing rectangular co-ordinates. Multiplying (12) by a"^ + /3^ + 7^ = 1, we find (7i - IW + ih - 1)0' + (h - IW = 0; and multiplying by p-, we obtain the equation of the equimomental cone in the form (7i - I)x' + {h - I)y- + (/3 - 7)2^ = 0. (15) 397.] MOMENTS OF INERTIA AND PRINCIPAL AXES 293 397. A slightly different form of the equations (11), (12), (15) is often more convenient ; it is obtained by introducing the three principal radii of inertia gi, q^, qz defined by the relations /i = Mgi2, h = Mq2^, h = Mq^"^. The equation (11) of the momental ellipsoid at the point then as- sumes the form qi^x^ + 92^ + gaV = 64. (11') The expression of the radius of inertia q for any line (a, /3, y) through becomes g2 = ^^2^2 + g,2^2 + 532^2. (12') Dividing (11') by the square of the radius vector, p^, and comparing with (12'), we find q = ~ , p = -~, (16) P 9 as is otherwise apparent from the fundamental property of the momen- tal ellipsoid (Art. 391). The equation of the equimomental cone for all whose generators the radius of inertia has the value q is obtained from (15) in the form (5,2 _ 52)3.2 + (5^2 _ g2)y + (532 _ ^2)^2 = 0. (15') This cone meets any one of the momental ellipsoids (11') in points whose radii vectores p are all equal; and if we select the arbitrary con- stant e equal to the radius of inertia q of the generators of the equi- momental cone, it follows from (16) that p = q. This particular ellipsoid has the equation 7i-x2 ^ 5,2,^2 _|_ 532^2 = qi^ and its intersection with the equimomental cone (15') lies on the sphere a;2 _[_ 2/2 _|_ 22 = qi^ In other words, the equimomental cone (15') passes through the sphero- conic in which the ellipsoid meets the sphere. Multiplying the equa- tion of the sphere by q^ and subtracting it 'from the equation of the ellipsoid wc obtain the equation (15') of the cone. The semi-axes of the ellipsoid are q^/qi, q-jq^, q^/qs. If we assume h > h > h, and hence qi > qi > qz, q must be ^q^/qs and =5V (fjqi, the axis of 2 is the axis of the cone. For q = q^jq^, i. e. q = q2, the cone (15') degenerates into the pair of planes (gi^ — q2^)x'^ — {q^^ — qz^)z^ = 0. These are the planes of the central circular (or cyclic) sections of the elUpsoid; they divide the elUpsoid into four wedges, of which one pair contains all the equimomental cones whose axes coincide with the greatest axis of the ellipsoid, while the other pair contains all those whose axes lie along the least axis of the ellipsoid. 398. There is another ellipsoid closely connected with the theory of principal axes; it is obtained from the momental elUpsoid by the process of reciprocation. About any point (Fig. 84) as center let us describe a sphere of radius e, and construct for every point P its polar plane tt with regard to the sphere. If P describe any surface, the plane -k will envelop another surface which is called the -polar reciprocal of the former surface with regard to the sphere. Let Q be the intersection of OP with TT, and put OP = p, OQ = q; then it appears from the figure that pq = t' (16) Fig. 84. the ellipsoid 399. It is easy to see that the polar reciprocal of the momental ellipsoid (11') with respect to the sphere of radius e is + q^^ 1. (17) To prove this it is only necessary to show that the relation (16) is fulfilled for p as radius vector of (11'), and q as perpendicular to the tangent plane of (17). Now this tangent plane has the equation 4x + -^7 + ;f,-Z = l; q-^ q^ qr hence we have for the direction cosines a, /3, y and for the length q of the perpendicular to the tangent plane 401.] MOMENTS OF INERTIA AND PRINCIPAL AXES 295 7 xlqi^ ylqi^ z/q^' [x'^/q.i + y'i/q^t + 22/^34]^' These relations give qia = {x/qi)q, q^^ = {ylqi)q, qzy = {z/q3)q, whence qM' + 92^/32 + 532^2 = C ^ + ^ + ^\ g2 = g2. (18) \ qr q-r q^ J For the radius vector pof (11') whose direction cosines a, /3, 7 are the same as those of q, we have by (11')- q-^oi^ + 52-/32 _|_ ^32^2 Hence p^(^ = e^; and this is what we wished to prove. 400. The surface (17) has variously been called the ellipsoid of gyra- tion, the ellipsoid of inertia, the reciprocal ellipsoid. We shall adopt the last name. The semi-ax. s of this ellipsoid are equal to the princi- pal radii of inertia at the point 0. The directions of its axes coincide with. those of the momental ellipsoid; but the greatest axis of the former coincides with the least of the latter, and vice versa. By comparing the equations (12') and (IS) it will be seen that q is the radius of inertia of the line {a, 0, 7) on which it Ues. Thus, while the radius vector OP = p of the momental ellipsoid is inversely propor- tional to the radius of inertia, i. e. p — e-/q, the reciprocal ellipsoid gives the radius of inertia q for a line as the segment cut off on this line by the perpendicrdar tangent plane. 401. We are now prepared to determine the moment of inertia for any line in space. Let us construct at the centroid G of the given mass or body both the momental ellipsoid and its polar reciprocal. The former is usually called the central ellipsoid of the body; the latter we may call the fundamental ellipsoid of the body. As soon as this fundamental ellipsoid ^ -4_ '/ j_ ii = 1 51' q2' 33^ is known, the moment of inertia of the body for any line whatever can readily be found. For, by Art. 400, the radius of inertia q for any line lo passing through the centroid is equal to the segment OQ cut off on the line Iq by the perpendicular tangent plane of the fundamental ellipsoid; and for any line I not passing through the centroid, the square of the radius of inertia can be deterininod by first finding the 296 KINETICS [402. square of the radius of inertia for the parallel centroidal line k, and then, by Art. 385, adding to it the square of the distance d of the centroid from the line I. 402. In the problem of determining the ellipsoids of inertia for a given body at any point, considerations of symmetry are of great assistance. Suppose a given mass to have a plane of symmetry; then taking this plane as the ?/z-plane, and a perpendicular to it as the axis of X, there must be, for every particle of mass m, whose co-ordinates are X, y, z, another particle of equal mass m, whose co-ordinates are — x, y, z. It follows that the two products of inertia Smzx and Smxi/ both vanish, whatever the position of the other two co-ordinate planes. Hence, any perpendicular to the plane of symmetry is a principal axis at its point of intersection with this plane. Let the mass have two planes of symmetry at right angles to each other; then taking one as ?/z-plane, the other as zx-plane, and hence their intersection as axis of x, it is evident that all three products of inertia vanish, S?n?/z = 0, 'Lmzx = 0, 'Zmxy = 0, wherever the origin be taken on the intersection of the two planes. Hence, for any point on this intersection, the principal axes are the line of intersection of the two planes of symmetry, and the two per- pendiculars to it, drawn in each plane. If there be three planes of symmetry, their point of intersection is the centroid, and their lines of intersection are the principal axes at the centroid. 403. Exercises. Determine the principal axes and radii at the centroid, the central and fundamental ellipsoids, and show how to find the moment of inertia for any line, in the following Exercises (1), (2), (3). (1) Rectangular parallelepiped, the edges being 2a, 26, 2c. Find also the moments of inertia for the edges and diagonals, and specialize for the cube. (2) Ellipsoid of semi-axes a, b, c. Determine also the radius of inertia for a parallel I to the shortest axis passing through the extremity of the longest axis. (3.) Right circular cone of height h and radius of base a. Find 404.] MOMENTS OF INERTIA AND PRINCIPAL AXES 297 first the principal moments at the vertex; then transfer to the centroid. (4) Determine the momental eUipsoid and the principal axes at a vertex of a cube whose edge is a. (5) Determine the radius of inertia of a tliin wire bent into a circle, for a line through the center incUned at an angle a to the plane of the circle. (6) A peg-top is composed of a cone of height H and radius a, and a hemispherical cap of the same radius. The pointed end, to a distance h from the vertex of the cone, is made of a material three times as heavy as the rest. Find the moment of inertia for the axis of rotation; specialize for h = a = \H. (7) Show that the principal axes at any point P, situated on one of the principal axes of a body, are parallel to the centroidal principal axes, and find their moments of inertia. (8) For a given body of mass M find the points {spherical -points of inertia) at which the momental ellipsoid reduces to a sphere. (9) Determine a homogeneous ellipsoid having the same mass as a given body, and such that its moment of inertia for every line shall be the same as that of the given body. (10) For a given body M, whose centroidal principal radii are qi, qi, qs, determine three homogeneous straight rods intersecting at right angles, of such lengths 2a, 26, 2c, and such linear density p", that they have the same mass and the same moment of inertia (for any line) as the given body. 3. Distribution of principal axes in space. 404. It has been shown in the preceding articles how the principal axes can be determined at any particular point. The distribution of the principal axes throughout space and their position at the different points is brought out very graphically by means of the theory of con- focal quadrics. It can be shown that the directions of the principal axes at any point are those of the principal diameters of the tangent cone drawn from this point as vertex to the fundamental ellipsoid; or, what amounts to the same thing, thoy are the directions of the normals of the three quadric surfaces passing through the point and confocal to the fundamental ellipsoid. In order to explain and prove these propositions it will be necessary to give a short sketch of the theory of confocal conies and quadrics. 298 KINETICS [405. 405. Two conic sections are said to be confocal when they have the same foci. The directions of the axes of all conies having the same two points S, S' as foci must evidently coincide, and the equation of such conies can be written in the form where X is an arbitrary parameter. For, whatever value may be as- signed in this equation to X, the distance of the center from either focus will always be i^a^ + X - (6^ + X) = Va^ - ¥; it is therefore constant. 406. The individual curves of the whole system of confocal conies represented by (19) are obtained by giving to X any particular value between — oo and + oo; thus we may speak of the conic X of the system. For X = we have the so-called fundamental conic x^/a^ + y'^/b- = 1 ; this is an ellipse. To fix the ideas let us assume a > b. For all values of X > — b^, i. e. as long as — 6^ < x < oo, the conies (19) are ellipses, beginning with the rectilinear segment SS' (which may be regarded as a degenerated ellipse X = — 6^ whose minor axis is 0), expanding gradu- ally, passing through the fundamental elhpse X = 0, and finally verging into a circle of infinite radius for X = oo. It is thus geometrically evident that through every point in the plane will pass one, and only one, of these ellipses. 407. Let us next consider what the equation (19) represents when X is algebraically less than — 6^. The values of X that are < — a^ give imaginary curves, and are of no importance for our purpose. But as long as — a^ < X < — 6^, the curves are hyperbolas. The curve X = — b^ may now be regarded as a degenerated hj^perbola collapsed into the two rays issuing in opposite directions from S and S' along the line SS'. The degenerated ellipse together with this degenerated hyperbola thus represents the whole axis of x. As X decreases, the hyperbola expands, and finally, for X = — a^, verges into the axis of y, which may be regarded as another degenerated hj'perbola. The system of confocal hyperbolas is thus seen to cover likewise the whole plane so that one, and only one, hjTjerbola of the system passes through every point of the plane. 409.] MOMENTS OF INERTIA AND PRINCIPAL AXES 299 408. The fact that every point of the plane has one ellipse and one hyperbola of the confocal system (19) passing through it, enables us to regard the two values of the parameter X that determine these two curves as co-ordinates of the point; they are called elliptic co-ordinates. If X, y be the rectangular cartesian co-ordinates of the point, its elliptic co-ordinates Xi, X2 are found as the roots of the equation (19) which is quadratic in X. Conversely, to transform from elliptic to cartesian co-ordinates, that is, to express x and y in terms of Xi and X2, we have only to solve for x and y the two equations "^^ _L y"^ = 1 3:^ I y'^ ^ ■, a2 + Xi 62 ^- Xi ' 02^X26^ + X2 The two confocal conies that pass through the same point P intersect at right angles. For, the tangent to the ellipse at P bisects the exterior angle at P in the triangle SPS', while the tangent to the hyperbola bisects the interior angle at the same point; in other words, the tangent to one curve is normal to the other, and vice versa. The elliptic system of co-ordinates is, therefore, an orthogonal system; the infinitesimal elements dXi ■ dX2 into which the two series of confocal conies (19) divide the plane are rectangular, though curvilinear. 409. These considerations are easily extended to space of three dimensions. An ellipsoid -, + r, + ^ = 1, where a > b > c, a^ b- c^ has six real foci in its principal planes; two, Si, Si', in the xy-plane, on the axis of x, at a distance OSi = Va^ — ¥ from the center 0; two, S2, S-i, in the yz-plane, on the axis of y, at the distance OS2 = Vb^ — (? from the center; and two, Si, S3', in the 2x-plane, on the axis of x, at the distance 0^3 = Va^ — & from the center. It should be noticed that, since 6 > c, we have OSi > OSi] i. e. Si, Si' lie between S3, &' on the axis of x. The same holds for hyperboloids. Two quadric surfaces are said to be confocal when their principal sections are confocal conies. Now this will be the case for two quadric surfaces whose semi-axes are Oi, bi, Ci, and 02, 62, C2, if the directions of their axes coincide and if Oi' — 61^ = a-r — bi^, br — cr = bi^ — d^, Oi^ — Ci^ = a^^ — c-^. 300 KINETICS [410. Writing these conditions in the form ai — ar = b^- — bi^ = c^ — c-c, say = X, we find ai = a^ + X, bi = b{' + X, ci = c^ + X. Hence the equation ^2 i;2 yl + t^ + :t^=1. (20) a'- + X 62_^ X c2 + X where X is a variable parameter, represents a system of oonfocal quadric surfaces. 410. As long as X is algebraically greater than — c-, the equation (20) represents ellipsoids. For X = — c^ the surface collapses mto the interior area of the ellipse in the x?/-plane whose vertices are the foci Si, &■>! and aSs, &z . For as X approac hes the h mit — & , the three semi- axes of (20) approach the limits V a^ — c^, VV^ — c^, 0, respectively. This limiting ellipse is called the focal ellipse. Its foci are the points Si, Si', since a? - c" - {V- - c^) = a? - b\ When X is algebraically < — c-, but > — a^, the equation (20) repre- sents hyperboloids; for values of X < — a- it is not satisfied by any real points. As long as — 6- < X < — c-, the surfaces are hyperboloids of one sheet. The limiting surface X = — c- now represents the exterior area of the focal ellipse in the rv-plane. The limiting In^perboloid of one sheet for X = — 6^ is the area in the 2x-plane bounded by the hyper- bola whose vertices are Si, Si', and whose foci are Ss, S3'. This is called the focal hyperbola. Finally, when — a^ < X < — b-, the surfaces are hj'perboloids of two sheets, the limiting hyperboloid X = — a^ collapsing into the ?/2-plane. 411. It appears from these geometrical considerations, that there are passing through every point of space three surfaces confocal to the fundamental ellipsoid i^/a^ + y'^/b^ + z^/c"^ = 1 and to each other, viz.: an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of two sheets. This can also be shown analytically, as there is no difficulty in proving that the equation (20) has three real roots, say Xi, X2, X3, for every set of real values of x, y, z, and that these roots are confined between such limits as to give the three surfaces just mentioned. The quantities Xi, X2, X3 can therefore be taken as co-ordinates of the point {x, y, z); and these elliptic co-ordinates of the point are, geomet- rically, the parameters of the three quadric surfaces passing through the point and confocal to the fundamental ellipsoid ; while, analytically, 413.] MOMENTS OF INERTIA AND PRINCIPAL AXES 301 they are the three roots of the cubic (20). To express x, y, z in terms of the elhptic co-ordinates, it is only necessary to solve for x, y, z the three equations obtained by substituting in (20) successively Xi, X2, Xs for X. 412. The geometrical meaning of the parameter X will appear by considering two parallel tangent planes tto and tta (on the same side of the origin), the former (tto) tangent to the fundamental ellipsoid x^la? + 2/^/6^ + z-jc^ = 1, the latter (tta) tangent to any confocal surface X or x^lia? + X) + y'^Hh'^ + X) + z^l{c^ + X) = 1. The perpendiculars go, ^A, let fall from the origin on these tangent planes tto, tta, are given by the relations (the proof being the same as in Art. 399) go= = o?a^ + 6^/3^ + c'y^ (21) qK2 = (^2 + x)a2 + (62 + X)/32 + (c^ + \)y\ (22) where a, /3, 7 are the direction cosines of the common normal of the planes tto, tta. Subtracting (21) from (22), we find, since a- + /S^ + y^ = 1, ^A- — go- = X; (23) i. €. the parameter X of any one of the confocal surfaces (20) is equal to the difference of the squares of the perpendiculars let fall from the common center on any tangent plane to the surface X, and on the parallel tayigenl plane to the ftmdamental ellipsoid X = 0. 413. Let us now apply these results to the question of the distribu- tion of the principal axes throughout space. We take the centroid G of the given body as origin, and select as fundamental ellipsoid of our confocal system the polar reciprocal of the central ellipsoid, i. e. the ellipsoid (17) formed for the centroid, for which the name "fundamental ellipsoid of the body" was introduced in Art. 401. Its equation is qi' ^ 92^ ^93^ ' if ?i, ?2, qs are the principal radii of inertia of the body. The radius of inertia go for any centroidal lino In can be constructed (Art. 400) by laying a tangent plane to this ellipsoid perpendicular to the line k; if this line meets the tangent plane at Qo (Fig. 85), then 302 KINETICS [414. Qo = GQo. Analytically, if a, /3, 7 be the direction cosines of lo, go is given by formula (21) or (12')- To find the radius of inertia q for a line I, parallel to lo, and passing through any point P, we lay through P a plane tta, perpendicular to I, and a parallel plane wo, tangent to the fundamental ellipsoid; let Qk, Fig. 85. Qo be the intersections of these planes with the centroidal line k. Then, putting GQo = qo, GQk = q\, GP = r, PQ\ = d, we have, by Art. 385, q^ = qa' + d?. The figure gives the relation d^ = r^ — q\^, which, in combination with (23) reduces the expression for the radius of inertia for the line I to the simple form g2 = r2 - X. (24) 414. The value of r^ — X, and hence the value of q, remains the same for the perpendiculars to all planes through P, tangent to the same quadric surface X: these perpendiculars form, therefore, an equimo- mental cone at P. By varying X we thus obtain all the equimomental cones at P. The principal diameters of all these cones coincide in direction, since they coincide with the directions of the principal axes of the momental ellipsoid at P (see Art. 396) ; but they also coincide with the principal diameters of the cones enveloped by the tangent planes tta. It thus appears that the principal axes at the point P coincide in direction 414.] MOMENTS OF INERTIA AND PRINCIPAL AXES 303 with the principal diameters of the tangent cone from P as vertex to the fundamental ellipsoid x^jq^ + y'^lq-^ + zV?3^ = 1- Instead of the fundamental ellipsoid, we might have used any quadric surface X confocal to it. In particular, we may select the con- focal surfaces Xi, X2, X3 that pa s through P. For each of these the cone of the tangent planes collapses into a plane, viz. the tangent plane to the surface at P, while the cone of the perpendiculars reduces to a single line, viz. the normal to the surface at P. Thus we find that the prin- cipal axes at amj point P coincide in direction with the normals to the three quadric sinfaces, confocal to the fundamental ellipsoid and passing through P. For the magnitudes of the principal radii qx, qy, qz, at P, we evidently have qi = 7-2 _ y^^^ q2 = j-l — X2, 5^2 = j-2 — Xj. CHAPTER XVII. RIGID BODY WITH A FIXED AXIS. 415. A rigid body with a fixed axis has but one degree of freedom. Its motion is fully determined by the motion of any one of its points (not situated on the axis), and any such point must move in a circle about the axis. Any particular position of the body is, therefore, determined by a single variable, or co-ordinate, such as the angle of rotation. Just as the equilibrium of such a body depends on a single con- dition (see Art. 234), so its motion is given by a single equation. This equation is obtained at once by " taking moments about the fixed axis." For, according to the proposition of angular momentum (Art. 360), the time-rate of change of angular momentum about any axis is equal to the moment of the external forces about this axis. Hence, denoting this moment by H and taking the fixed axis as axis of z, we have as equation of motion the last of the equations (3'), Art. 360, viz., — 2w(x?/ - yx) = H. (1) 416. The angular momentum, 1,m{xy — yx), about the fixed axis can be reduced to a more simple form. For rotation of angular velocity co about the 2-axis we have (Art. 48, Ex. 1) X = — coy, y = o)X, so that '2m{xy — yx) = (jo1,ni{x'^ + y'') = w'Smr^ = /co. 304 417.] RIGID BODY WITH A FIXED AXIS 805 where r is the distance of the particle m from the axis and / = Swr^ the moment of inertia of the body for this axis. This expression for the angular momentum can be derived without reference to any co-ordinate system. For evidently mcor is the linear momentum of the particle m, mcor^ is its moment, i. e. the angular momentum of the particle, about the axis; andZwiwr- = uZmr- = /co is the angular momentum of the body about the axis. It thus appears that, just as in translation the linear mo- mentum of a body is the product of its mass into its linear velocity, so in the case of rotation the angular momentum of the body about the axis of rotation is the product of its moment of inertia (for this axis) into the angular velocity. As regards the right-hand member of equation (1), the reactions of the axis need not be taken into account in forming the moment H; for as these reactions meet the axis, their moments about this axis are zero. 417. Substituting 7co for Xm(xy — yx) in equation (1), and observing that the moment of inertia I about a fixed axis remains constant, we find the equation of motion in the form /1 = H; (2) ^. e. for rotation about a fixed axis the product of the moment of inertia for this axis into the angular acceleration equals the moment of the external forces about this axis; just as, in the case of rectilinear translation, the product of the mass of the body into the linear acceleration equals the resultant force R along the line of motion: dv m-TT = K. dt And just as the latter equation may serve to determine 21 306 KINETICS [418. experimentally the mass of a body by observing the accelera- tion produced in it by a given force R, e. g. the force of gravity (as in the gravitation system, Art. 177), so the former equation, (2), may serve to determine experimentally the moment of inertia of a body about a line I, by observing the angular acceleration produced in the body when rotating about I under given forces. 418. For the kinetic energy of a body rotating with angular velocity co about any axis we have T = '^hnv^ = Hhnoi^r^ = i/w^, (3) an expression which is again similar in form to that for the kinetic energy of a body in translation, viz. T = ^mv-. When the axis is fixed so that I is constant, the equation of motion (2) , multiplied by co and integrated, say from t = to t = t, gives the relation i/co2 - i/coo^ = £'Ho}dt, (4) which expresses the principle of khietic energy and work. 419. As an example consider the compound pendulum, i. e. a rigid body with a fixed horizontal axis and subject to gravity alone. If OG = h is the distance of the centroid G from the fixed axis and 6 the angle made by OG with the vertical plane through the axis we have H = Mgh sin5. Denoting by q the radius of inertia about the centroidal axis through G parallel to the fixed axis so that the moment of inertia about the fixed axis is = M{q- + h^), we find the equation of motion (2) in the form q' + h sine. (5) Comparing this with the equation of motion of the simple 420.] RIGID BODY WITH A FIXED AXIS 307 pendulum (Arts. 63, 335), 6 = — (g/l) sin^, it appears that the motion of a compound pendulum is the same as that of a simple pendulum of length l-h + ^j. (6) This is called the length of the equivalent simple pendulum. The foot of the perpendicular let fall from the centroid G on the fixed axis is called the center of suspension. If on the line OG a length OC = I be laid off, the point C is called the center of oscillation. It appears, from (6), that G lies between and C. The relation (6) can be written in the form h(l - h) = cf, or OG-GC = const. As this relation is not altered by interchanging and C, it follows that the centers of oscillation and suspension are inter- changeable; i. e. the period of a compound pendulum remains the same if it be made to swing about a parallel axis through the center of oscillation. 420. Exercises. (1) A pendulum, formed of a cylindrical rod of radius a and length L, swings about a diameter of one of the bases. Find the time of a small oscillation. (2) A cube, whose edge is a, swings as a pendulum about an edge. Find the length of the equivalent simple pendulum. (3) A circular disk of radius r revolves uniformly about its axis, making 100 rev./min. What is its kinetic energy? (4) A homogeneous straight rod of length I is hinged at one end so as to turn freely in a vertical plane. If it be dropped from a horizontal position, with what angular velocity does it pass through the ^ vertical position? (Equate the kinetic energy to the work of gravity.) (5) A homogeneous plate whoso shape is that of the segment of a parabola bounded by the curve and its latus rectum swings about the 308 KINETICS 1421, latus rectum which is horizontal. Find the length of the equivalent simple pendulum. (6) When q is given while I and h vary, the equation (6) represents a hyperbola whose asymptotes are the axis of I and the bisector of the angle between the (positive) axes of h and I. Show that Imia = 2q for h = q; also that I, and hence the period of oscillation, can be made very large by taking h either very large or very small. The latter case occurs for a ship whose mclacenter (which plays the part of the point of suspen- sion) Ues very near its centroid. (7) A homogeneous circular disk, 1 ft. in diameter and weighing 25 lbs., is making 240 rev./min. when left to itseK. Determine the constant tangential force applied to its rim that would bring it to rest in 1 min. 421. While a single equation determines the motion of a body with a fixed axis, the other five equations of motion of a rigid body must be used to determine the reactions. The axis will be fixed if any two of its points A, 5 are fixed. The reac- tion of the fixed point A can be re- solved into three components Ax, Ay, Az, that of B into B^, By, B^. By introducing these reactions the body becomes free; and the system com- posed of these reactions, of the exter- nal forces, and of the reversed effec- tive forces must be in equilibrium. We take the axis of rotation as axis of z (Fig. 86) so that the z-co-ordinates of the particles are constant, and hence i = 0, S = 0; and we put OA = a, OB = h. Then the six equations of mo- tion are (see Art. 359 (2) and Art. 360 (3)): ^mx = SX+ ^. + B„ Zmij = 37 -\- Ay+ By, Fig. 86. 423.] RIGID BODY WITH A FIXED AXIS 309 = SZ -\- A, -\- B,, — 'Zmzij = i:(yZ - zY) - aAy - hBy, llmzx = l^izX — xZ) + a^^ + bB^, 'Zm{xy — yx) = 2(xF — yX). 422. It remains to introduce into these equations the values for X, y. As the motion is a pure rotation, we have (see Art. 48, Ex. 1) X = — ooy, y = oox; hence, x = — 6:y — oi^x, y = 6)x — co^y. Summing over the whole body, we find 2mf = — (jiZmy — w-^mx = — Mioy — Mw'^x, Xmij = (li^mx — w'^^my = Miox — Mco^y, where x, y are the co-ordinates of the centroid; and - ^mzij = — ooliffizx + co^'^niyz = — E6: + -Dco^, l^mzx = — ulimyz — co^lmzx = — Du — Eo:"^, Xm{xij — yx) = coSrwo:" — (xr^mxy + os'^my^ + co-Zmxy = Co), where C = 1,m(x- -{- y-), D = Zmyz, E = l^mzx are the no- tations introduced in Art. 387. With these values the equations of motion assume the form : - MxiJ" - MyCi = SX + A, -1- B^, - Myo)"- + Mx(h = 1:Y + Ay+ By, = 2Z + ^. + B,, Dco2 - E(h = i:OjZ - zY) - aAy - hBy, ^ ' - Eoi"- - Z)w - ^{zX - xZ) + aA, + hB„ C(h = Z(xY - yX). 423. The last equation is identical with equation (2), Art. 417. The components of the reactions along the axis of rotation occur only in the third equation and can therefore not be found separately. The longitudinal pressure on the axis is = - A.- B, = 2Z. 310 • KINETICS [424. The remaining four equations are sufficient to determine ■^x> -^Vl ^x, ijy The total stress to which the axis is subject, instead of being represented l)y the two forces, at A and B, can be reduced for the origin to a force and a couple. The equa- tions (7) give for the components of the force - A,- B, - SX + Mxco2 + My6i, - Ay - By ^ ^Y + Myoi'' - Ma-w, (8) - A, - B, = SZ. This force consists of the resultant of the external forces, E = V(SX)2 + (SF)2 + (SZ)^ and two forces in the a;?/-plane which form the reversed effec- tive force of the centroid; for Mxcji^ and Myw- give as re- sultant the centrifugal force Mco^V^^ + y~ = Mw-f, directed from the origin towards the projection of the centroid on the a;?/-plane, while Myic, — Mxx, coy, uz, i. e. the moments and products of inertia for the fixed axes, are not constant, while for the mo\'ing axes the coefficients of wi, 002, W3 are constant. 434. The position of the moving trihedral at any instant with respect to the fixed trihedral can be assigned bj^ three angles as follows. Let X, Y, Z (Fig. 87) be the intersections of the fixed axes, Xi, Fi, Zi those of the moving axes Avith the sphere of radius I de- scribed about the fixed point 0; and let N be. the intersection with the same sphere of the nodal line, or line of nodes, i. e. the line in which the 1 t ^x)^' \ \e 0' \""""A A ^-^ \ y xY Fig. 87. planes XOY and XiOFi intersect. Then the angles ZZi = 6, NXi = 72 > 73 this cone is real if and only if For 5" = 1/73, the polhode reduces to a point, viz. the ex- tremity of the longest axis of the momental ellipsoid. As 5^ diminishes, the polhode is first an oval about this longest axis. When 5- = 1/72, the polhoclal cone degenerates into a pair of planes and the polhode l^ecomes an ellipse. When 5^ lies between 1/72 and l/7i the polhode is an oval about the shortest axis, and it contracts to the extremity of this axis for 52 = 1//,. For values of 5^ very close to 1/72 the motion can, in a certain sense, be called unstable since a slight disturbance might change the polhodal cone from a cone about the longest to a cone about the shortest axis, or vice versa. 439. The herpolhode is a plane curve; but it is in general not closed. The radius vector OP = p (Fig. 88), if not con- stant, has a greatest and a least value in the course of the motion, and the same is true of its projection QP on the fixed plane. Hence the herpolhode lies between two con- centric circles. When p is constant these circles coincide 441.] RIGID BODY WITH A FIXED POINT 323 and the herpolhode coincides with them. It can be shown that the herpolhode has no points of inflection. 440. The invariable line describes a cone in the moving body. Its equation may be found from the reciprocal ellipsoid ^ ,y^ i^A = ^ whose radius vector in the direction 5 is 1/5 (Arts. 398, 399), and hence constant. The cone must pass through the inter- section of the reciprocal ellipsoid and the sphere 0" Hence its equation is h) ■''-+{''- h) '■"'- + {'"- b '-'- '■ 441. When H = Eider's equations (11), Art. 432, are /iCOi = (/o — l2)W2W-i, I'i<^2 = (h — /OcOgOJl, ho:-6 = (/i — /2)C01W2. Multiplying by coi, coo, ws and adding we find ~ i(/ia;:2 + /2coo2 + /3C032) = Q; hence, by (9), Art. 429, /icoi2 + 1,0,/ + I,o:./~ = 2T = const. (14) This is the integral of kinetic energy and work. Multiplying (13) by /iwi, /2CO2, ho^z and adding we find similarly Ijy (3) : /rwr + /2"a32" + L^ws^ = li^ = const., (15) which is the integral of angular momentum. 324 KINETICS [442. As, moreover, wi^ + ^-i" + ^z' = w^, (16) we can solve (14), (15), (16) for coi^, t02-, wa^. Introducing the new constants a, /3, 7 by putting 2T{l2 + 73) -h^ = hha\ 2T{h + 7i) -h^ = IzL^\ 2T{h + h) - h^ = Iihy\ we find "^' = (/i - mh - h) ^"' " "'^' ^^^1 (^2_„2)^ (17) (/2 - /3)(/l - U) o 7i/2 . 2 ON -3- = (7, _ /3)(/2 _ 73) (- - T^)- Hence, if Ji > /o > /a we have w^ > a^, co^ < iS^, co^ > 72. 442. To find the time, multiply the equations (13) by coi//i, W2//2, (jiz.Hz and add: (/l - /2)(/ l - /3 )(/2 - /s) «(-2W ) = 7—}^^: W1CO2CO3; -1 li2^3 substituting for coi, 0)9, ws their values (17) we find: V V(C02 - «2)(^2 _ ^2) (^2 _ ^2) The positive or negative sign must be used according as d{w^) is positive or negative. As t is given by an elhptic integral, co^ is a periodic function of the time. 443. If, in particular, the momental ellipsoid at is an ellipsoid of revolution, say if /i = lo, the results assume a very simple form. Euler's equations (13) reduce to Wi = XcOoWS, ^2 = — XcOsCOi, CO3 = 0, (18) 444.] RIGID BODY WITH A FIXED POINT 325 where The angular velocity cos about the third axis Ozi (which is not necessarily an axis of symmetry for the mass of the whole body) is therefore constant: C03 = n. The first two equations (18) give coiwi + C02W2 = 0, whence toi^ + co2^ = const. = rn}. It follows that CO = l/cor + 0)2^ + COs^ = Vm- + 71^ is constant although coi and C02 vary. The inclination of the instantaneous axis to the principal axes Oxi, Oyi varies, but its inclination to the third principal axis Ozi is constant, viz. cos~^(co.-i/a)). This means that the polhodal cone is a cone of revolution about Ozi and the polhode is a circle. The herpolhode is therefore Hkewise a circle (Art. 439). As the two circular cones are in contact along the instantaneous axis, this axis lies in the same plane with the impulse h and the axis Ozi. 444. To find coi, 0^2 separately, differentiate the first equa- tion (18) with respect to t and substitute for cj^ its value from the second: 0)1 + X^n^coi = 0; hence Oil = k sin(Xn^ + e), where k, e are the constants of integration. The fir^ equa- tion (18) then gives 012 = z- Oil = k cos(Xnf + e). Kn 326 KINETICS 1445. As coi^ + C02- = 7)1^ (Art. 443) it appears that k = m. Hence coi = 7nsin(\nt + e), C02 = wz cos(\nt + e), C03 = ?i. (19) 445. To determine the position of the body with respect to fixed axes through let the invarialilc direction of h be taken as axis Oz. The direction cosines of h given in Art. 435 give hi = 1 10)1 = h smO sinv?, ho= 12(^2= h sin0 cos^, hs = hooi = h cos^. It follows that COS0 = ~T- = const., tan^ = — = tan(XwY. + e): hence

(because initially u is real), /(I) < 0, /(oo) > 0, the cubic /(w) has three real roots, say Ui, U2, Us, such that — I < Ui < Uq < ih < '^ < Ua < oo. For the time we have ^ - " y^wk j du V(w — Ui){u — Uo){u — Us) the plus or minus sign being used according as du is positive or negative. As u = cos0 must lie between — 1 and + 1 it oscillates between its least value Ui and its greatest value u^; i. e. the axis Ozi oscillates between its greatest inclina- tion di and its least inclination do to Oz. 449. Suppose, in particular, that the body is initially given a spin about the third principal axis OZi so that wi = 0, C02 = for t = 0. We may take the axes of refer- ence so that (p = and ^i- = for ^ = 0. We then have since hz is constant : hz = hn COS0O, and f{u) = (wo - u)[2IiWk(l - u'-) - /3-n2(^/,o - w)]. 330 KINETICS [450. When Uo < u < 1, f{u) is clearly negative; it is therefore U2 which is equal to Uo- Hence, at the beginning of the motion u diminishes; in other words, do is the minimum inclination of the axis OZi to the vertical OZ. 450. The centroid G descril^es a spherical curve; its pro- jection on the horizontal A"F-plane lies between the circles of radii k^il — Wi^ and k-yjl — u^^ about 0. The co-ordinates X, y of the projection of the centroid on the A^F-plane are X = k^ll — u^ sim^, y = — k^l — u^ cos^. To determine the direction in which the curve approaches the bounding circles let us determine the angle /x between the radius vector p and the tangent to the curve. We have p Vl — U" 1 — u^ d^p tan u. = -,— = =t = =F J- . ^P d r 5 u du Now by Arts. 447 and 449 d\p . I371U0 — u du Ii 1 — u^ ' hence , hnuo — u tan u = =F ~r^ : — . il uu As Iiu = ± V/(w) (Art. 448) we find , ^ Uo — u IzU Vwo — u tan n = Izii u V/(m) u ^j2WkIi{u — Wi) {u — Us) This shows that tan fi becomes infinite for u = Ui and zero for ti = Uo = u^. The curve meets therefore the inner circle at right angles (with a cusp) and touches the outer bounding circle. It is in general not a closed curve. 452.) RIGID BODY WITH A FIXED POINT 331 451. The expressions for 9, ip, \p as functions of i assume a simple form if we suppose the initial angular velocity n about OZi to be very large (Art. 446). In this case the equation (Art. 449) fiu) = (mo - u)[2IiWkil - u'~) - Mi-(wo - w)] = has its root Wi nearly equal to Wo so that the angle 6 differs but little from ^o- Hence if we put 6 = do -\- v, v will be small. This gives cos9 = cos^o — v sin^o, i- e. COS0O — COS0 . . • ra \ \ • n \ a V = .--- , sm0 = sm(»o -r v) = sm^o + v cos^o- smpo Substituting these values in the equation for 6 (Art. 447) we find Zi^^^ = 2Wkhv sin^o - h-if^ l'T'^'\.^ , (sm0o + V cos^o) or neglecting the term v cos^o in comparison with sin^o: lid = ^I2WkIlV sindo - hVv^ As 6 = V we find upon integration 7i • ,v , Wkli sin^o t = -^ — versm ^ - , where a = zr-r—;: — , Un a U^n^ and hence 6 = Oo + V = 00 -^ a (l - cos -p A = eo-\-2a sin^ ^ t. The variation v in the value of 9 is called the nutation; it is periodic, of period 2TrIi/hn. 452. By Arts. 447 and 449, » _ hn cos^o — COS0 _ IsU j'_ /i sin^^ 1 1 sin^o ' where (Art. 451) 332 KINETICS [453. p = a { 1 — cos ^^ t Hence, integrating and observing that \{/ = ior t = 0: , hna , a . I^n ^ iism^o sm^o /i Thus the first term of \p increases uniformly with the time, while the second is periodic. The angular velocity \l/ is the velocity of precession (Art. 445). 453. For

e\r\ay 838 KINETICS 1459. If the particle starts from rest at (or rather from a point very near to 0) we find hence 2u)^ sin^a g cosa 1 2 - Ci — 62 — ^ Jf_C0Sa^ /„i„ smat silia-ty^ For the projection of the path on the horizontal plane we have P = sinof, 6 = cot; hence the projection of the absolute path on the horizontal plane is which represents a spiral. 459. Motion of a -particle relative to the earth, near its surface. The earth's motion of translation (which is not uniform) need not be considered since the forces affecting it act on the particle just as Fiff. 92. they do on the earth and hence do not affect the relative motion. The earth can therefore be regarded as rotating uniformly about a fixed axis; the slight variation of direction of the axis may be neglected. The angular velocity of the earth is 27r " = ^a^c^T~^ = 0.000 072 92 rad./sec, 86 164.1 ' the sidereal day having 86 164.1 sec. of mean time. 459.] RELATIVE MOTION 339 The body-force is simply the centrifugal force (Art. 458) mco^r = mco^R coS(^, where R is the earth's radius and <{> the latitude. In most problems of relative motion near the earth's surface the introduction of this centrifugal force is unnecessary. This is best seen by considering a particle at relative rest, say the bob of a pendulum hanging at rest (Fig. 92). Let P be the bob, S the point of suspension, the earth's center, OP = R the earth's radius, r = R cos^ the radius of the parallel in latitude . As Vr = 0, the complementary force is zero; hence the only forces to be considered are the centrifugal force mu^r, the tension of the rod along PS, and the earth's attraction which is directed along PO if Fig. 93. we regard the earth as composed of homogeneous spherical layers. Hence the tension of the rod must balance the resultant of the cen- trifugal force and the attraction. But this resultant is due precisely to the actually observed acceleration g of falling bodies since this in- cludes the combined effect of centrifugal force and attraction. The complementary force, — 2ma}Vr sina, where a is the angle between the relative velocity v,- and the earth's axis (northward) is at right angles to the plane of the angle a. We take the earth's center O as origin of 340 KINETICS 1460. the fixed axes and Oz toward the north (Fig. 93); the origin of the moving axes at any point P (in latitude 0) on the earth's surface, Fzi vertical, Px\ tangent to the meridian southward, and hence Py\ tangent to the parallel eastward. We then have: . cji = CO cos(7r — <^) = — CO cos<^, C02 = 0, cos = CO sin<^. Hence the components of the complementary acceleration jc are 2(co2ii — co3?/i) = — 2co2/i sin0, 2(co3.ri — coiii) = 2oo(.ri sine/) + Zi cost^), 2(a)i7/i — C02.ri) = — 2coi/i COS0. The components of the comijlementary force Fc along the moving axes are therefore: Xc = 2mcoyi sin<^, Yc = — 2mw{zi cos. 460. Relative ynotion of a heavy particle on a smooth horizontal plane. The centrifugal force being taken into account by using the observed value of g (Art. 461) the equations of the relative motion are N ii = 2co32/i, 2/1 = 2(coi2i — cos.ri), Zi = -- — g - 2coi?/i, m where N is the normal {i. e. vertical) reaction of the plane. As Zi and ii are constantly zero, the equations reduce to xi = 2co3?/i, yi = - 2CO3X1, N = mig + 2coi?/i), where coi = — co cos. The third equation determines N as soon as 2/1 has been found from the first two. The principle of kinetic energy and work gives iCii^ + 2/1^) = const. Hence the relative or apparent velocity Vr is constant. Assuming the particle to start from the origin P we find by integrating each of the two equations by itself: .fi = .To + 2co32/i, 2/1=2/0— 2aj3Xi; as -fr + 2/1" = I'r^ = I'u" = i'o" + 2/0^ we find as equation of the path: 461.1 RELATIVE MOTION 341 \ 2a)3 J V 2a;3 / VScoj / a circle tangent to the initial velocity in the horizontal plane. The center C (Fig. 94) lies on the perpendicular to vo tlirough P, to the right of an observer at P looking in the direction of vo, in the northern hemisphere, i. e. for positive (j>, to the left in the southern hemisphere. Thus the particle deidates to the right in the 7iorthern, to the left in the southern hemisphere. The radius of the circle is very large since w is very small. Thus, for (p = 30° we have for this radius Vo 2a;3 = " = 13700 Ik 461. Particle falling from rest in vacuo. The equations of motion are the same as in Art. 400 except that N = 0: .fi = 20)3^/1, 7/1 = 2cjiii — 2co3.ri, zi = — g — 2myi. If the starting point be taken as origin, the initial conditions are Xo = 0, ?/o = 0, zo = 0, i-o = 0, 2/0 = 0, 2o = 0; hence the first integrals are i-i = 2a)3?/i, 7/1 = - 20)3X1 + 2wiZi, 2i = — gt — 2wi?/i. 342 KINETICS [462. The method of successive approximations gives the first approxi- mation ±1 =0, 2/1=0, ii = — gt, whence Xi =0, 2/1=0, 2i = — hgt^. Substituting these values in the expressions for the velocities we find the second approximation i'l = 0, 7/1 = go: COS0 • /-, 2i = — gt, whence Xi =0, iji = Igw cos(j> • t^, Zi = — \gl^. The third approximation gives Xi = J^w^ COS0 sin the latitude of the place of observation. 463.1 RELATIVE MOTION 343 463. Foucault's pendulum. It will be convenient to take the point of suspension as origin, the axis Ozi vertically downward, Oxi tangent to the meridian northward, and hence Oyi tangent to the parallel eastward. The forces acting on the bob are its weight mg, the tension N of the suspending wire, and the complementary force Fc whose components are, since coi = co cosqi, C03 = — co sin^ : Xc = — 2muiyi siuij), Yc = 2??Jco(.ri sin + Zi cos4>), Zc = — 2mco?/i cos0. If I = Vxi^ + 2/1^ + 2i^ is the length of the wire the equations of motion are: Nxi ^ . . Xi = r ~ 2,mh smrf), Vi = V + 2coXi sin0 + 2coZi cos,---)=^, Kt,xi,yi,zi,xi,- ■■)=(), ••• (4) for a holonomic system of n particles. The number of inde- pendent equations of motion will be 3n — A;. For, these equations must express the equilibrium of the given forces, together with the reversed effective forces, under the given conditions; and for this equilibrium it is sufficient that the virtual work should vanish for any displacement com- patible with the conditions, the work of the reactions and con- straining forces being zero for such virtual displacement. In other words, in d'Alembert's equation (2), Art. 469, the constraining forces clue to the conditions will not appear if the displacements 5x, by, bz be so selected as to be compati- ble with the k conditions (4) . Now this will be the case if these displacements are made to satisfy the equations that result from differentiating the conditions (4), viz. l^{iP::bx+ipyby-\-n — k = m 476.] MOTION OF A SYSTEM OF PARTICLES 351 equations of motion of a system of n particles with k conditions. To do this more systematically we may, as in Arts. 348, 351, use Lagrange's method of indeterminate multipliers: adding the equations (5), multiplied by X, /x, • • • , to d'Alem- bert's equation (2), we obtain a single equation in which the k multipliers X, n, • • • can be selected so as to make the coefficients of k of the Sn displacements 8x, By, 8z vanish. The remaining 3n — k displacements being arbitrary their coefficients must likeAvise vanish. Hence the coefficients of all the displacements must be equated to zero, and this gives n sets of 3 equations of the type 7nx = X + X«^x + M'Ax + • • • 7 mi) = F + \^y + fxxlyy + • • • , (6) mz = Z -]- \ip^ + fjL\p; -\- • • ' . These, together with the equations (4) , are sufficient to deter- mine the 3n co-ordinates x, y, z and the k multipliers X, M; • • • • It is apparent from the equations (6) that the constraining force acting on the particle m has the components: X' = \(p-c + ljL\pjc + • • • , Z' = \nts of X, /i, • • • in the resulting equation, viz. '^X^i-x + ipyij -f- (pzz)dt, 'Li^iX -\- \pyy + \pzz)dt, • • • are zero as appears by differ- 352 KINETICS [477. entiating the conditions (4) with respect to t. This means that the constraining forces in this case do no work in the actual displacement of the system, as they are all perpen- dicular to the paths of the particles. If, however, the conditions (4) contain the time explicitly, their differentiation gives so that instead of (7) we find: d^imv^ = S(Xda; + Ydy + Zdz) - \ in three dimensions (Art. 269) we have T = i //?.('■•- + r-b"- + r2sin26> -p^), and we find: vi\f — r(d- + S\D?d Ar gf = ^^ To mtegrate put i9 = A cosr/, ip = \A cosrt, whence (m + 7n')(g — ar-) = m'bXr"^, \(g — br~) = ar-, mabr^ — {m + ■m')g{a + b^- + (?« + vi')g- = 0. 486.] MOTION OF A SYSTEM OF PARTICLES 359 The last equation regarded as a quadratic in r- has real (the discriminant being positive) and positive roots, say r- and rr. Corresponding to these roots we have e = A cosrt, (p = .„ cosrt, g — f)j-2 ' e = Ax cosnt,

V2,«^-,(=^=[- fc2/^ log V2yi2 L Vso + ^^2i^ + Vso + Vso(so + k'"R) - Ms + km) (5) If ?^o < -^2gR the height above the earth's surface to which the particle rises is /i = Vo^RI(2gR — vo^) and the time of rising to this height is R ( . 2gR . _, vo 2gR - Vo~ \ ^i2gR - v} ^l2gR / ' if Vo = ^2gR, the time of rising to the distance s from, the center is ANSWERS 363 SR^i2g and the particle does not return; if vo > ^2gR the time is -S^gR L Vs + Vs + fc^ J where /c'' — Wo' - 2gR ' (6) /t = 72, ^ = (1 + ^tt) V^ = 34f min. Art. 28. (2) V = ^JgR = 5 M./seC; T = 1 h. 25 min. (3) Vso^ + (volixr. Art. 36. (1) tt; 15.7 ft./sec. (3) (a) 402; (b) 25.1 sec. (2) 0.157 rad./sec.2; 5 rev. (4) (a) 0.022 rad. /sec; (6) 15.7 ft./sec; 7.85 ft./sec. Art. 42. (l)-(5) Check graphically. (7) 36M./li.; 198 ft. (6) 20". (8) Vr = Vb sine. (9) Spiral of Archimedes r = (vo/w) d. Art. 48. (4) The projection of the velocity on the radius vector and on the focal axis are in the constant ratio e of the focal radius to the distance to the directrix. It follows that the tangent meets the directrix at the same point as does the perpen- dicular to the radius vector through the focus. Art. 56. (7) j2 = a"[2(l - COS0) ^' + 2 sin0 er- + 6']. (8) (a) Straight line; (6) circle; (c) circle of radius v; (d) it is normal. (9) Tlie cylindrical components are j\ = f' - rV", h = 2r'

= ji sine + J3 cos^, je = ji cos^ - Ja sin0, j^ = > Art. 59. (3) 45°. (5) Construct a vertical circle having the given point as its highest point and touching (a) the straight line, (6) the circle. Art. 61. (9) (a) 1374- ft. from the vertical of the starting point; (6) 6i sec; (c) 201 ft./sec, at 6^° to the vertical. (10) 227.5 ft./sec. (11) 4° 22' or 86° 48'. (13) Let Oy = vo be the given initial velocity. On the vertical through lay off 0T> = H = Vo'^!2g; then the hori- zontal through jDis the chrectrix. Make 2^ VOF = 2^ DOF, and lay ofiOF = OD = H; then F is the focus. (14)" With VoV2g = H, the locus is a;^ = - 4:H(y - H), a parabola. (17) A horizontal line. (18) (a) 1.5 sec; (h) 25.1 ft. from the building; (c) 59.7 ft./sec, at 16i° to the vertical. (19) 300 ft. from tee, in 1 see. (20) At a distance of 6250 ft. Art. 68. (1) 0.99672; 86117. (4) 28.8 ft. (2) 3.26 ft. (5) 980.4. (3) 32.158. (8) The pendulum should be lengthened by y^^ of its length. (9) It will lose 67 sec. /day. (10) About a mile. Art. 70. (3) 1.0038. (5) Use the first formula of Art. 69. (6) Determining the constant from 6 = r for v = we have iw^ = 2gl cos^i^. Putting v — — Idd/dt and inte- grating gives t = -yjl/g log tanj(7r -\- 6) if = for i = 0. ANSWERS 365 Hence the bob approaches the highest point of the circle asymptotically, i. e. without reaching it in any finite time. Art. 75. (1) X = Xq cosfxt + (volfj.) smfit. {2) V = - |x^la^ - x\ Art. 80. (1) X = 10.806 cosCiTT^ + 271°). (2) X = 2a cosiS cos(cof + i5). (3) (a) X = 2acoscof; (6) a; = 0, the case known in physics as interference. (4) xi= - 5.18 coswf, xo = 14.14 cosM + 30°). Art. 113. (1) a;^ + t/^ = a- being the circle, j = — ahi^jif where Vi is the x-component of the initial velocity. (2) WoVa. (3) Let j = )uV; then, if (xo, v/o) is the initial position, I'l, ??2 the components of the initial velocity, the path is the hyperbola : (i'2--/i-?/o-).c-+2(/x2xo2/o-i'i?'2) + 0'r-AtW)|/^=(y2a;o-?^iyo)^ ,-. iJ. r. Voro sim/'o , Vo^ sin2;/'o (5) a = — , 6 = , tana = ^- ^ .rj^ , ^ t^ e t^ -\- Vo^ cos2i^o where e^ = v^"^ . To (6) Put r = 1/m and determine d^u/dd"^ in terms of u alone: fj~')i ~= - u-\- (n - 1)(1 - e'~)q-^"u-^'"+' - (n - 2)g-"w-"+i. Hence by (IG), Art. 100: /(r) = c-iin - 1)(1 - e2)5-2"ir2"+' - (/i - 2)q-"u-"+^. n = 1 gives an ellipse if e < 1, a parabola if e = 1, a hyper- bola if e > 1, all referred to focus and focal axis; n = 2 gives conies referred to their axes; n = — 1 gives pascalian lima- 366 ANSWERS gons (cardioids for e = ±1); n = — 2 gives a lemniscate if c - ± 1. (7) (a) c2(2aV-5 + r-3); (6) cV-^; (c) c-(l + n2)r-3; {d) c-[2?i"aV~^ + (1 — n^)r"^]. (8) Sa^cV-^. (9) Ellipse, parabola, or hyperbola according as ju ^ 2'2^I/o^, 2/0 being the initial distance from the plane, V2 the component of the initial velocity normal to the plane. (10)/(r) = --,^. a^ y3 Art. 137. (3) The direction of motion passes through the highest point of the wheel. (4) With the center of the given circle as origin and the perpendicular to I through as axis of x, the fixed centrode is y^ = ex =^ a -yjx^ + ?/'^ where a is the radius of the given circle, c the distance of from I. With A as origin and h as axis of X the body centrode h x^ = ay ^ c ^x'^ + y"^- The upper sign corresponds to h sliding over the first and second cjuadrants of the circle, the lower to h sliding over the third and fourth quadrants. If c > a, the complete fixed centrode has a node at with the tangents ay = ± Vc^ — a'^x. The polar equations of the centrodes are r sin-0 = c cos0 + a and r' cos'^9' = a sin0' + c. The body centrode for c > a is (apart from position) the same curve as the fixed centrode for a > c, and vice versa. (5) •//- = 2a{x + ia). (6) The fixed centrode is a circle passing through Oi, O2; the body centrode is a circle of twice the radius of the fixed centrode. The path of any point in the fixed plane is a Pascal limagon; the points of the body centrode describe cardioids. (8) Two equal parabolas; the motion is the same as that of Ex. (5). (10) With as pole and OB as polar axis the equation of the fixed centrode is r- cos-^ — 2ar cos-9 + a^ = P. With as origin and OB as axis of x the equation is (x^ -j- a^ ANSWERS 367 — P) ^lx'^ + Z/^ = 2ax'^. The rationalized equation represents the centrode of A 5 when B moves not only on the positive but also on the negative half of the axis of x. The equation of the body centrode, with A as pole and A 5 as polar axis, AC ^ r', ^ BAG = 6', is found by observing that r = r' -j- a, I sine' = OB sin0 = r cos6 smd whence (a2 - P cos^e')^-'' - 2a/V sin^e' + l-iP - a^ cos0') = 0, i. e. , A -\- a cos^' , A — a cos0' ri = I — —. T7 , r2 =1 -. -, . a + i cosO a — i com These relations can be read off directly from the figure if perpendiculars be dropped from on AB and from B on AC. For the path of any point P whose body co-ordinates, with A as origin and A 5 as axis of x' , are x' , y' , we have x = a cos^+rc' COS97+?/' sin^, y = a smd — x' sin^+?/' cos^?, where 6 and ^ are connected by the relation ?/a = sin^/sinc^. For the path of the midpoint oi AB we find x = a cos0 + •2"^ coS(p, y = a sin0 — ^l shiip, whence x = -^a^ — 4?/- + |- -sP- — 4?/^ which is of the fourth degree. To find the velocity of B when that of A is given oljserve that as the distance AB \^ invariable the projections of the velocities of A and B owAB must be equal, whence va cos^ = Va sin(0 + >p). (12) Find first the velocity Vr of Po relative to Pi as the resultant of -i^i and v^; hence w. Art. 148. (2) A Pascal lima^on. (7) (a) co-x — CO?/ = 0; (b) 6:x — ory -]- J = 0. Art. 166. (2) Distance from midpoint = \^l, 21 being the distance of 5 from 23. (3) About 1000 M. below the earth's surface. (5) X = r sina/a = rc/s, where c is the chord, s the arc; for the semi-circle x = (2/7r)r. 368 ANSWERS . ^ 31/2 - log(l + 1/2) ^ ^ ^^^^g * 1/2 + log(l + 1/2) Ot/2 — 1 y = ^ ^- '^ i — -^ a = 1.12907a. ^ l/2 + log(l + i/2) (7) Tra, |a. (8) |a, fa. (9) r sin^/9_, r(l - cose)/0, ^fcr sin^. (10) 2a(Q! sino: -f cosa — 1)/q:^, 2a(sinQ: — a cosa)/a^. (12) AV«, ?rV«. (13) Distance from hypotenuse = 0.11a. (15) (a) ix„ %y,; (b) ^t, ^t; (c) ^ a = 0.40531a, ^6; (17) frsina/«. (18) \a. (22) If a:i, X2 are the distances of the planes from the center then a; - ,(a:i + x,) ^, _ ^^^^, ^ ^^^^ _^ ^^,^ (a) |(2a - /i)V(3a - /i); (6) fa; |a(l + cosa). (23) f/i. (25) t\7/i. (24) A2/1- (26) fa, §b, fc. (27)(a) |a,|a; (6)'--p^a = 1.85374a, |a; (c) ffa; , ,. 128 + 454 _ . . . - .__, n + 3 (^^ 907r « == ^'^^^^^^ ^28) 2-(^-^) a. Art. 170. (1) 300,000 F.P.S. units. (3) 32.000 F.P.S. units. (2) 50ft./sec. Art. 179. (1) 6.4 X 10^ poundals = 8.9 X 10^ dynes. (2) 4.5 pounds. (3) 0.1406. Art. 196. (1) 9 = 120°. (3) 28, 39° 16'. (4) 2F cos22|° = 1M8F, ANSWERS 369 (7) (a) W sine, W cos0; (6) W tsmd, W seed; (c) W sin^ seca, W cos(0 + a) seca. (8) W siii)8/sin(a! + fi), W sinQ:/sin(a + /3). (10) P = iW,T = iW- (11) P = 2W cosJ(« + Itt) = 0.8945TF. (12) P = W sm{a + j8)sin|S is greatest when the sail bisects the angle between the wind and the direction of motion. (13) W sin/3/sin(Q: + /S), W sinQ;/sin(a: + /3). (14) T = Wl/d, r = W{c - I) Id, where d^ = V - i(c - ly. (15) 13.4, 28.9, 50, 86.6, 186.6, oo. (16) 848, 282; 1000, 600. (17) 0.640T^. (18) (a) 1.414Pr; (6) 2W cosi(i7r ± 6). Art. 221. (2) T = mW, A = Vm' - m + 1 W, where m = 2c//. (3) F = i(cot0 - rll)W. (4) tan^ = (a cota - h coti3)/(a + fc). (b) A,= ^^ Tf, ^.= ( 1--^ lF,fi=-^TF. Art. 243. (1) P = TT sm(plcos{a — if). („) sin(^^ri < P < sin(9i^ 3=|a2sin«); cos(p TF cosv? ' Ty (c) if P act up the plane. p = sm(g+ (p) ^ if Pact down COS(p , , „ ^ sin(v? — d) „, the plane, P < — ^^ ^ W. eos(p (4) 226i, 56*-. (5) B = ^t - 2 e the particle performs simple harmonic oscillations about (^ (5) The length I is increased to I -\- e -{- Ve(e + 2/i). (7) 42 min. 35 sec. Art. 293. (2) The equation of motion s = i* = — g — kv'^ gives with A; = /^^/jy: _ g iJiVo cos/Lt^ — g sin/^f jx/jLio smiJLt-\-g cos/zf ' ANSWERS 371 1 , g -\- kvo'^ (4) ^-i= ^^ ^0 -yjg -\- kvo^ (5) In vacuo v = 139 ft. /sec, in air v = 122 ft./sec. (6) s = ^ (1 — e~^0> ^ = Voe~''^ = Vo — ks. k (7) 11 = ^ (1 - e-"), k^¥ ^^g - kv Art. 297. (2) The logarithmic decrement is log e~^' = — Xf . (4) If fjL =^ K, s = Ci cosk/, + Co siiiKt + -^, , sin/x^; if K- — fJL'^ fjL = K, s — Ci cosd + Co sind + ^, sin/c^. (5) The term due to the forced oscillation is a a/0c^^^2)2_^ 4X2^2 cosm(^ - h); hence the oscillation lags behind the force by the phase difference ixto; the amplitude is less than for undamped oscillations. The free oscillations (if any) will rapidly die out so that the motion soon approaches the state of motion given by the above term. Art. 302. (2) The eciuation of the orbit given in Ex. (1) is satisfied not only by Xo, ?/o, but also by Vi/k, v^/k; i. e. the orbit passes not only through the initial position Po, but also through the point Q(i>i/k, Vz/k) which is the extremity of the radius 372 ANSWERS vector OQ = vqIk parallel to fo; OPq and OQ are the con- jugate semi-diameters whose equations are Xoy = y^^, V\y = VoX. (4) The problem requires the construction of the axes of a conic from a pair of conjugate diameters. (5) Referring the orbit to its axes we have x = a cosd, y = h smd for the ellipse and x = ^0(6"' + e"*') = a eoshd, y = i&Ce"' — e""') = 6 sinh/c^ for the hyperbola. (6) From the equations of Ex. (5) it follows that for the ellipse tan0 = (6/a) tauK^ whence 6 = Kab/r-. (8) Use the equations of the conic in terms of the eccentric angle (p. (9) (a) Ellipse; (6) hyperbola; (c) parabola. (10) The parabola x — Xo=(vi/v2)iy — yo) — (2kc/v2^) (?/ — ?/ o)S where 2c is the distance of O3 from the point that bisects O1O2; the midpoint between and O3 is taken as origin and OO3 as axis of x. (U) t = -tan-i^'^tan^] Art. 320. (I) Vo = ^fji/ro. (3) 687 days. (4) As the velocity is not changed instantaneously we have by (24), Art. 314: 2fjt, jj. 2iJ.' ij! r. a r a whence a' is found. (5) An ellipse, with the end of its minor axis at the point where the change takes place. (6) (a) Ellipse with a = |r; (h) parabola. (7) Differentiate (24), Art. 314, with respect to m and a. (8) The periodic time T would be diminished by (2/n)r. (9) r = Z/(l + e COS0) gives x, y as functions of 9: hence, observing that r-^ = c, x = — {c/l) sin^, y = (c'l) (cosd + e). The hodograph is therefore the circle x^ -\- {y — cejlY = {cjiy, where c ^ -^fxl. (10) 1.034 114. (II) t = V2aVM(tani0 + i tan49). (12) 178.73 and 186.52 days. ANSWERS 373 Art. 334. (1) (a) 7h lb.; (b) 480 lb.; (c) 6.4 rev./sec. (2) 8i°. (4) 32.20. (7) tanS - Rco^simpcosip/ig - T^co^cos V) ; 44°57'. (8)7ilb. Art. 339. (4) To count the angles from the highest point of the circle put t — d = (p; then, putting h — I = h', where h' is the height to which the velocity at the highest point is due, we have N = — 3mg[coS(p — f(l + h'/l)]. The par- ticle remains on the curve as long as cos^ > |(1 + h' /I) ; distinguish the cases h' = 0. (6) At the distance 1.4617a from the lowest point of the circle if a is the radius. Art. 379. (c) tV^; (d) tVi^. (8) laK (9) ha\ (10) w- (11) ia\ \h\ \c\ (12) i(fli2 + a,^) (2) (a) i ^^: (6) W (3) i/i2. (4) y>^ (5) 2\a~. (6) xW- (1) i.{lv ' + P). (2) (a) • 2^a'; (&) ■ Art. 386. (4) fa^. fa^; (c) -Jott^. (5)|(a> + a22). (3) (a) fa^; (6) Ja^; (c) fa^. (6) (a) ia^; (6) fa^; (c) ^^(/i^ + 3a2). (8) ta2. (9) («) i62. (5) 1^2. 1(^2 + ^2). (10) 1(6^ + c2), i((;2 + a^), Ka^ + ?>^). (12) fa^. (11) ia2. (13) fa^ + 62. Art. 403. (1) The centroidal principal axes are perpendicular to the faces. The moments for these axes are ^Mih''- + c~), IMic^ + a2), Pf(o2 + 62). The central ellipsoid is (62 + c2)x2 + (c2 + a2)7/2 + (a2 + 6'-)z2 = 3e4. For an edge 374 ANSWERS 2a, I = pf (62 + c2) ; for a diagonal / = 'i-M(b'-c' + c"a- + a%^)/{a^ + 6' + c2). For the cube the fundamental ellipsoid becomes a sphere of radius ^ VOa ; for an edge of the cube, q"^ = fa- ; for a diagonal, q- = fa-. (2) Central eUipsoid: (6^ + c^)^;" +(c- + a-)!]- -\-{a" + 62)22 = Se"; forZ, g2 = |(6a2 + 62). (3) Take the vertex as origin, the axis of the cone as axis of re; then I\ = y^Ma^; 7i', t. e. the moment of inertia for the ^2-plane, = ^Mh^. As for a solid of revolution about the axis oi X B' = C and B = C, we have I2 = I3 = Hi, and h = h = // + i/i. Hence, h = h = p/(/r + ^a^). At the centroid the squares of the principal radii are -y^ci^, J?_(4(j2 _[_ /j2\ ''(4) A = J5 = C = Wa^ D = E = F = IMa"; hence momental ellipsoid: 4(a;2 _|_ ^2 _j_ 2,2) _ 3(^2 + 2a; + a;|/) = 6eVc^^; squares of principal radii: ^a^, \^a'^, Wa^. (5) g2 = ia2(i + sin^a). (6) / = TVpTTflKfa + ^ + 2}i^im ; for /i = a = ii7, (7)'i' = /i, i? = /2 + M.Ti^ C = /3 + M2;i2. (8) The centroid may be such xi point; if the central ellip- soid be an oblate spheroid, the two points on the axis of revolution at the chstance ± V(/i — l2)/M from the centroid are such points. (9) The ellipsoid must have the same central ellipsoid as the given body; its equation is x^/A' + if/B' + z^/C = 5/71f , where M is the mass and A', B', C' are the moments of inertia for the principal planes of the body at the centroid. (10) p" = M/N, where N=^^l^[(q2'+q^'-ql')'+iq^'+q^'-q■y'y'+{qi'+q2'-qz')'f. M «^ = f -77 (