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LESSONS ON RIGID DYNAMICS. 
 
LESSONS 
 
 ON 
 
 RIGID DYNAMICS: 
 
 BY THE : , ' : . , 
 
 REV. G. PIRIE, M.A., 
 
 FELLOW AND TUTOR OF QUEENS' COLLEGE, CAMBRIDGE; 
 AND LATELY EXAMINER IN THE UNIVERSITY OF ABEKDEEN. 
 
 L 
 
 Hontron : 
 MACMILLAN AND CO. 
 
 1875. 
 
 [All Rights reserved.] 
 
vr 
 
 
 
 l*ilth. O.o 
 
 Cambritigc : 
 
 PRTNITED BY C. J. CLAY, MA. 
 AT THE UNIVERSITY PRESS. 
 
PKEFACE. 
 
 It will be generally acknowledged, I think, that there is no 
 subject of Natural Philosophy, equal in importance to that 
 familiarly known as Rigid Dynamics, of w^hich the study is 
 so exclusively restricted to the more advanced students of 
 Mathematics. Yet this restriction cannot be said to be 
 necessary, for the treatment of the subject involves none 
 of the higher mathematical methods ; and it must be allowed 
 to be unfortunate, for the science of motion is the basis of 
 Mechanical Engineering, and furnishes the explanation of 
 many interesting terrestrial and cosmical phenomena. 
 
 This restriction of the study is chiefly due to the fact 
 that, while the conceptions and reasoning peculiar to the 
 subject are somewhat difficult, the explanations of its lead- 
 ing principles, given in the books commonly used by stu- 
 dents, are for the most part very brief, and often, through 
 brevity, obscure. 
 
 It is this deficiency of explanation which I have at- 
 tempted to supply in the following little book. It is not 
 my purpose to acquaint the student with the splendid gene- 
 ralizations of Lagrange and of more recent philosophers. 
 For that the books in present use leave nothing to be 
 desired. My aim is to render more general the study of 
 this interesting science, by presenting as simple a view of 
 its principles as is consistent with scientific accuracy, and 
 to give a sound foundation to the student who is to proceed 
 higher. 
 
vi PllEFACE. 
 
 It is my hope that tlie book may be useful not only to 
 students of Natural Philosophy, but also to engineers. 
 Most of them possess a knowledge of the principles of Me- 
 chanics, of the method of Co-ordinate Geometry, and of 
 the Integral Calculus ; and that is all that is here required. 
 
 The principle on which this science is based has been 
 so Ions: connected with the name of D'Alembert that it 
 would hardly be recognised under any other. Nevertheless 
 there is no doubt that Euler has more claim to its author- 
 ship, inasmuch as he first used it. D'Alembert admits this, 
 but says that Euler gave no proof. I believe D'Alembert's 
 real merit to be, that his explanation was exactly suited 
 to clear away the difficulties which Avere perplexing men's 
 minds. 
 
 The works to which I am principally indebted are : — 
 Thomson and Tait's Natural Philosophy; Routh's Rigid 
 Dynamics ; Resal's Cinematique Pure; Rankine's Machinery 
 and Millworh ; Walton's Mechanical Prohlems; Whewell's 
 History of the Inductive Sciences; Willis' Principles of Me- 
 chanism; Muller's Lehrhuch der kosmischen Physik ; Mon- 
 tucla's Histoire des Mathematiques; D'Alembert's Traite de 
 Dynamique, and Euler's Mechanik. 
 
 My thanks arc due to Dr Campion, of Queens' College, 
 for many valuable suggestions which he has made; and to 
 several of my pupils for their frank statement of their 
 difficulties. 
 
 G. PIRIE. 
 
 Queens' College, Cambridge. 
 December, 1874. 
 
CONTENTS. 
 
 Lessox page 
 
 I. Geometry of Motion 1 
 
 II. Geometry of Motion 12 
 
 III. D'Alembert's Principle 19 
 
 rV. Keduction of the erpressiona for the effective forces . . 36 
 
 V. First Applications 49 
 
 TI. Moments and Products of Inertia ..,,.. 60 
 
 VII. Moments and Products of Inertia ...... 69 
 
 "N^II. Moments and Products of Inertia 73 
 
 IX. Problems, Without Eotation 82 
 
 X. „ Fixed Centres 89 
 
 XI, „ One Kigid Body 102 
 
 Xn. ,, A System of Bodies 117 
 
 Xni. Energy 142 
 
 XIV. Precessional Motion 163 
 
 XV. Differential Equations 171 
 
 Miscellaneous Problems 177 
 
 Notes on the Examples 180 
 
GEOMETRY OF MOTION. 
 
 I. 
 
 1. A RIGID body is an assemblage of particles such 
 that the distance between each pair is unchangeable. 
 The movements of such a body are very different from 
 those of a set of independent points. Its fixed connec- 
 tions introduce a common movement. Any straight line or 
 any plane of particles in the body must remain pJways a 
 straight line or plane. If all such planes remain parallel to 
 themselves, the motion is one of translation. But if any 
 such plane makes an angle with its former position the 
 motion is rotational. And the velocity of rotation— angular 
 velocity — is measured by the rate at which the plane is de- 
 scribing angles. 
 
 Thus the connecting rod of the driving wheels of a goods' 
 locomotive has only a translational motion; — so also (ap- 
 proximately) the axis of the earth in its yearly motion round 
 the sun. In a well-thrown quoit the motions are combined. 
 
 2. From this definition of rotation it follows that a 
 point cannot rotate. It may revolve about another point, 
 but it contains no lines nor planes which can describe angles. 
 For rotation there must be an extended system. A point in 
 motion may be said to be revolving about any point whatever 
 situated in the line through it at right angles to its direc- 
 tion of motion, for it is moving at the moment in a circle 
 with the point as centre. But the body of which this is a 
 point may not be rotating. For rotation it is necessary that 
 the different points of the body should be at the moment 
 revolving about the same axis. 
 
 P. G. .. " 1 
 
2 EOTATIOX AND REVOLUTIOX. 
 
 Suppose a man to move round a column viewing its parts 
 in succession. In this case he is also rotating. Were he 
 to move without rotation, he must work round the column 
 sometimes forAvards, sometimes sidewards, sometimes back- 
 wards, but always facing the same point of the compass. 
 
 3. From the definition it follows also that rotation is di- 
 rectional, i.e. it takes place not so much about an axis or point 
 
 as about a direction or in a plane. The bod}^ AB lias rotated 
 in passing to the position AB'; but the amoimt of the rota- 
 tion is measured by the angle between the straight lines AB 
 and A'B'. It matters not to the angular velocity of a car- 
 riage-wheel whether it is rolling along a level road or up a 
 hill, or whether the wdiecl, being raised from the ground, is 
 whirled round its own axle. Or on a larger scale, whether a 
 ship rounds a promontory or swings with her anchor fixed 
 through the same angle is indifferent to the amount and 
 direction of the rotation performed. There is indeed in 
 general an axis round whicli the body may, in a stricter sense, 
 be said to rotate, for every point of the body moves in a 
 circle about it. This rotation is the more easily imagined, 
 but it ought not to be allowed to expel the idea of the other. 
 It is a pity there are not separate words to distinguish them. 
 We will in future speak of them as rotation round a point or 
 axis, and rotation round a direction or in a plane. 
 
SIMULTANEOUS MOTIONS. o 
 
 4. The method which mathematicians adopt in treating 
 of simultaneous motions is to consider them one after the 
 other. A velocity is the describing of a certain length or 
 angle in a certain time. Properties of small linear and 
 angular displacements are properties of linear and angular 
 velocities. If, therefore, a body is subject to two inde- 
 pendent motions, as rotations about two axes or a rotation 
 and a translation, it is considered to obey them in turn each 
 for a very short time. A rifle bullet moving towards the 
 target and rotating all the time is supposed to approach the 
 target without turning, through an infinitely small space, and 
 then to turn round through an infinitely small angle, much 
 like a man descending a spiral staircase. This is not the 
 actual motion, any more than a polygon is a curve, but it 
 differs as little as Ave please from the real motion, and it 
 clears our ideas and enables us to apply mathematical 
 methods to the problems. 
 
 If a body is solicited to two different motions by two 
 simultaneous causes, it will in reality follow neither ; but it 
 may be supposed to have followed both. 
 
 Thus the very extravagant idea of some of the earliest 
 writers on projectiles that a cannon-ball went straight until 
 it had exhausted the force of projection and then fell down 
 straight under gravity, had in it, notwithstanding its 
 grievous confusion of force with velocity, a germ of truth, 
 (viz.) that the causes of motion must be considered 
 separately. 
 
 A skater describing circles, the nut of a screw, a crank 
 rod one end of which moves in a straight line while the 
 other describes a circle, the arms of a common form of 
 reaping m.achine which rotate about an inclined axis while 
 carried forwards by the machine, and hundreds of other 
 familiar cases, supply examples of translation combined v/ith 
 rotation. 
 
 Examples of combined rotations are seen in the screw or 
 paddles of a steamer, which are rotating about a horizontal 
 axis while the steamer may bo moving round a curve 
 
 1—2 
 
4 PARALLEL AXES. 
 
 and thus rotating about the vertical; or in the common 
 gyroscopic toy, where a metal ring rotates about a diameter 
 of a circle, and is borne along also by the rotation of this 
 circle about the vertical ; or in the sails of a windmill, which 
 may have rotations about their own axis, about the vertical 
 (if the wind veers) and with the whole body of the windmill 
 round the polar axis of the earth. 
 
 5. Motions in one plane and in ixiralUl planes. 
 
 Let A be a point of a rigid body which is moved to A'. 
 Bisect AA' perpendi ularly by the straight line -^^A^ On 
 
 NX take any point B. We may suppose that the point A 
 has been moved to A! by the body having been caused to ro- 
 tate about an axis through B at right angles to the plane of 
 the paper. In this case the line of particles AB has taken 
 up the position A'B. Now cause the body to rotate about an 
 axis through A' perpendicular to the plane of the paper 
 through an equal and opposite angle. The line A'B takes 
 up the position AB' parallel to AB, whence we infer that a 
 displacement of translation is equivalent to two equal and 
 opposite displacements of rotation about two parallel axes. 
 If these displacements are small, yl^' is at right angles to 
 AB, and the proposition becomes that two equal and opposite 
 angular velocities about two parallel axes are equivalent to a 
 translational velocity in a direction at right angles to the 
 plane of the axes. 
 
PARALLEL AXES. O 
 
 It will be convenient to denote sucli axes perpendicular 
 to the plane of the paper by the point where they cut this 
 plane. Thus we might have spoken of A' and B as axes. 
 
 This important proposition may be reduced to the paral- 
 lelosfram of linear velocities. 
 
 Let P be any point of a body which has simultaneous equal 
 and opposite rotations round A and B. The velocities of P 
 
 due to these are represented by Pa and Ph perpendicular and 
 proportional to PA and PB respectively. The resultant 
 velocity is represented by the diagonal PB of the parallelo- 
 gram on Pa, Ph. But this parallelogram is similar to that 
 whose sides are PA, PB. Hence the velocity of Pis propor- 
 tional and perpendicular to AB. As this holds for every 
 point of the body, the wliole is being translated at right 
 angles to AB, and v/ith velocity proportional to it. 
 
 6. From the definition of rotation it is clear that two 
 equal and opposite rotations cannot produce a rotation ; for 
 they turn a straight line in the body through equal but 
 opposite angles. 
 
 For the same reason the angular velocity of a body 
 rotating about two parallel axes is the sum or difference of 
 their separate angular velocities. 
 
 Let P be any point of a body which has angular velocities 
 in the same direction round A and B. And take on PB a 
 point G such that PB and PC are proportional to the angu- 
 lar velocities round ^ and P respectively. Then the linear 
 
TARALLEL AXES. 
 
 velocities of P will be at right angles to PA and PB, and 
 proportional to PA . PB and PB . PC respectively, i. e. to PA 
 
 and PC. Hence, joining AC and bisecting it in N, the result- 
 ant velocity of P is at right angles to PN and is measured 
 by twice Pk. P may therefore be taken to be revolving about 
 any point in PN. Let PN produced cut AB in R ; we can 
 shew that R is the same point wherever P is, and therefore 
 the whole body is rotating about R. The angular velocity 
 has been settled independently. 
 
 Produce CP to C, making PC equal to PC. Join A C 
 PNR is parallel to A C, and therefore 
 
 AR :RB:: C'PiPB; 
 
 or R divides AB inversely as the angular velocities round A 
 andB. 
 
 7. Any displacement whatever may be given to a body 
 by a translation and a rotation about an axis. 
 
 jB 
 
 For to bring AB to A'B' it is only necessary to translate 
 the body till a point A reaches its new position A\ and then 
 to rotate the body about A\ 
 
INSTANTANEOUS AXIS. / 
 
 Thus ill general any motion of a body may be composed 
 of a rotation round an arbitrary point and a translation. 
 
 And this point may generally be so chosen that the move- 
 ment of translation shall not be required. In other words, 
 there is one point which is unaffected by the change of posi- 
 tion. To find this point. Bisect A A' and BB' perpendicu- 
 larly, and let the bisectors meet in K 
 
 Join NA, NA\ NB, NB\ 
 
 Then ^''A is equal to JS^'A' and NB to NB'. 
 
 If, then, we can shew that the angle ANA' is equal to 
 the angle BNB', we shall have proved that when the body is 
 rotated about N, so as to move A to A', B is brought to B', 
 and so for any other point. For ^ is a point of the body, 
 since the triangles ANB and A'NB' are equal in all respects. 
 
 Now the triangles ANB, A'NB', have all their sides equal 
 each to each; therefore the angles ANB and A'NB' are equal. 
 Take away the common angle A'NB, and the remainder 
 ANA' is equal to the remainder BNB'. 
 
 Hence any motion of a rigid body, except one of transla- 
 tion, is one of rotation round some axis ; and this is called 
 the instantaucous axis. 
 
 If the body be a plane one, and be moving in its plane, 
 this axis cuts the plane in the instantaneous centre. To find 
 its position it is only necessary to take two points whose 
 directions of motion are known, and to draw perpendiculars 
 from them to these directions. Their intersection gives the 
 centre required. 
 
ROLLING. 
 
 8. For the illustration of these principles consider the 
 motion of a circular hoop C, rolling on another cylinder 0, 
 which is fixed. The rolling of the hoop from position (1) to 
 
 an infinitely near position (2) may be considered as takiii! 
 place in one of three principal ways. 
 
 First, by two rotati(^us. 
 
 The whole hoop may rotate througli a small angle round 
 0, thus coming into position (o), and then a rotation round C 
 will bring the hoop into position (2). 
 
ROLLING. 
 
 Secondly : 
 
 The hoop may be translated into position (4), and then 
 brought by a rotation round C to position (2). 
 
 Thirdly : 
 
 By one rotation round the instantaneous centre. This 
 point must lie in 00; for 00 is at right angles to the 
 direction of motion of 0. To get another line on which it 
 lies, consider the motions of points veiy near to N. These 
 are moving away from or towards 0. Hence iVmust be the 
 instantaneous centre. That the body is at the moment 
 rotating round N will be better seen by looking on the 
 circles as many-sided polygons of equal sides. Each angular 
 point becomes in turn the centre of rotation. But the 
 axis is continually changing, and if the question considered 
 be one of change of velocity, the motion must not be con- 
 sidered as if it were round a fixed point at N. 
 
 If n be the angular velocity of revolution of C about 0, 
 measured by the angle at 0, the velocity of rotation of the 
 hoop round in the first method will be measured by the 
 angle at 0; and since the arcs of the tv/o circles which 
 have been in contact are equal, it wdll be 
 
 ^^ ON ' 
 
 If w^e combine these by Ai^t. 6, they give a resultant 
 rotation round JSl. 
 
10 EXAMPLES. 
 
 Tn the second method the linear velocit}^ is measured by 
 the distance the centre has moved, i.e. by OC.D., and the 
 angular velocity by the angle between CJS^ (fig. 4) and CJ^ 
 (fig. 2), i.e. by (/ (9+^C) (fig. 2). 
 
 It is therefore O + II . -.-^ 
 
 or 
 
 ^•cw 
 
 ojsr 
 
 c. 
 oc 
 
 To reduce this to either of the others, consider the linear 
 velocity OC.H as the resultant of two equal and opposite 
 angular velocities. 
 
 The single angular velocity in the third method is the 
 same as in the second, but it is about N. For the whole 
 rotation must be measured by the same angle whatever be 
 the axis. Or we may see it thus. Calling it w, the linear 
 velocity of G due to it is w ON, and this must be the same 
 as that found by the last method, viz. Q. .00. 
 
 EXAMPLES. 
 
 1. If two points are rigidly connected, their velocities in 
 the direction of the straight line joining them are equal. 
 
 2. A mirror rotates about a vertical axis with an angular 
 velocity w, and a ray of light falls on it from a distant fixed 
 point on the horizon. What is the angular velocity of the 
 reflected ray ? 
 
 3. Express a velocity of 100 revolutions a minute in 
 units of angular velocity. 
 
 4. Compare the velocity of rotation of the earth with 
 the mean angular velocity of revolution of its centre. 
 
 5. If V be the linear velocity in Art. 5, which is equiva- 
 lent to tlie angular velocities w, - w about A and B, shew 
 that v = AB,(o. 
 
 G. Where is the instantaneous centre when a ladder 
 is slipping down iu a vertical plane between a wall and the 
 ground ? 
 
EXAMPLES. 1 1 
 
 7. The paddle-wheel of a steamboat is rotating with 
 velocity co, and the vessel is moving with velocity v ; whore 
 is the instantaneous axis of the paddle-wheel ? 
 
 8. Prove that any motion of a rigid body of which the 
 points move in parallel planes may be represented by sup- 
 posing a right cylinder fixed in the body to roll on a right 
 cylinder fixed in space. 
 
 9. What are these cylinders in the case of question 7 ? 
 
 10. If a straight rod be moving in any manner in a 
 plane, the directions of motion of all its points will in general 
 touch a parabola. 
 
 11. AF, BQ are two arms moveable round the fixed 
 centres A, B; and the points P and Q are connected by a 
 link (rod) PQ ; shew that the angular velocities of the arms 
 AP, BQ are inversely proportional to the segments into 
 which the link, or its direction produced, divides AB. 
 
II. 
 
 GEOMETRY OF MOTION. 
 
 1. We now come to the case of simultaneous rotations 
 about axes inclined to one another. The motions of points are 
 no longer in one plane or in parallel planes. It will be neces- 
 sary to represent the axes themselves. The way in which 
 an angular velocity is geometrically represented is as follows : 
 take the axis xx ; on it [take a point ; let Ox be the di- 
 rection which is considered positive. Place a watch at 
 with its face towards x. A rotation whose direction coincides 
 with the direction of motion of its hands is considered positive. 
 It is measured by the line OA, which contains as many units 
 
 of length as the angular velocity contains units of angular 
 velocity. An angular velocity in the opposite direction is 
 represented by a straight line measured along Ox'. 
 
 With this convention a positive angular velocity round 
 Ox — one of a rectangular system of axes as usually drawn — 
 will tend from Oy to Oz ; one round Or/ from Oz to Ox ; 
 one round Oz from Ox to Oi/. 
 
 The basis of this subject is the proposition called the 
 parallelogram of angular velocities, which is : 
 
 // a body have simultaneous angular velocities about two 
 inclined intersecting axes, and if these he represented by the 
 adjacent sides of a j^^if^^dleloyram, then shall the resultant 
 angular velocity he about the diagonal wJiich passes through 
 their intersection and proportional to it in magnitude. 
 
INCLINED AXES. 
 
 13 
 
 Let a body be rotating simultaneously about OA and OB 
 with velocities proportional to OA and OB, 
 
 taking 
 
 Then, first, the points on OG are at rest. For, 
 such a point P and drawing PM, PN at right angles to OA 
 and OB, P's linear velocity due to its rotation round OA is 
 upwards from the plane of the paper, and proportional to 
 OA . PM\ while that due to rotation round OB is downwards 
 and proportional to OB . PN. As these products are twdce 
 the areas of the triangles OPA, OPB respectively, they are 
 equal. As they are opposite the point P is at rest. The body 
 is therefore rotating about G. 
 
 To settle its angular velocity about OC', draw a perpen- 
 
 dicular to the plane ^405 through 0, and on this take a 
 point Q. 
 
14 GEOMETRY OF MOTION. 
 
 The angular velocities of the body round A, OB wiW 
 l>e proportional to the linear velocities of Q perpendicular to 
 these, i.e. Oa, Ob, which are drawn in the plane of A OB, 
 at right angles to OA, OB and proportional to them respec- 
 tively. 
 
 The resultant angular velocity about OC will be repre- 
 sented by the resultant linear velocity of Q, i.e. by Oc, the 
 diagonal of the parallelogram Oa, Oh. But the parallelograms 
 OA, OB and Oa, Ob are similar, the latter being turned 
 round through a right angle. Therefore the diagonal Oc is 
 proportionar^nd perpendicular to 00. And the resultant 
 angular velocity of the body is proportional to OC. 
 
 2. Angular velocities, then, are quantities which obey 
 the parallefogram law, and all its consequences will hold good 
 for them. A body rotating with velocity co about any axis 
 may be considered to have a component angular velocity 
 CO cos a about any other axis inclined to the former at an 
 angle a. There will be a parallelopiped of angular velocities: 
 and in general the analogy between angular velocities and 
 forces in" Statics is complete. 
 
 "We will take for the illustration of this the pendulum 
 experiment by Foucault, by which the rotation of the earth 
 is rendered visible. 
 
 Draw a circle representing a section of the earth through 
 its polar axis NS. Let be the centre, and A any place on 
 its surface. 
 
 In tliis experiment, a pendulum is set swinging in a.ny 
 vertical plane at A. We assume that wherever the point of 
 suspension may be, the plane in wliich the pendulum swings 
 will remain parallel to itself. If the earth were rotating 
 about OA, the effect of this would be that the plane of the 
 pendulum would be left behind by the earth, and would 
 appear to an observer, unconscious of the earth's motion, to 
 follow the sun. Now this is in part what happens. The 
 earth does not indeed rotate about OA; ils rotation is 
 about NS; but this is equivalent to one about OA pro- 
 portional to cos NOA, and one about a perpendicular to OA 
 
GEOMETRY OF MOTIOX. 15 
 
 proportional to sin NOA. This latter is what carries the 
 building and the whole apparatus eastward. It does not 
 affect the present question. But the other rotation — that 
 about OA — causes the plane of the pendulum to follow that 
 of the sun with an angular velocity, which is to that of the 
 earth as cos NOA to 1. 
 
 This experiment requires the greatest care for its exhibi- 
 tion. If the pendulum move in even the most elongated 
 oval, instead of swinging in a plane, the axis of this oval will 
 rotate from a very different cause, viz. the resistance of the 
 air. When Foucault exhibited his experiment in Paris to 
 the French * savants,' he used a heavy ball hung from the 
 roof of the Obsorvatory, and set it off by burning a threa^d 
 which held the ball out of the position of rest. 
 
 3. If a rigid body has one point fixed, there is at any 
 moment a straight line of points at rest. In other words, any 
 displacement of a rigid body, one point of which is fixed, may 
 be effected by a single rotation about some axis through that 
 point. The proof is the same as that by which we shewed 
 that any displacement of a plane body in its plane can be 
 given by a rotation round one point ; if instead of a plane we 
 consider a sphere in the body with the fixed point as centre. 
 The points, then, represent straight lines through the centre; 
 the straight lines in the figure become arcs of great circles 
 and represent planes passing through the centre ; but the 
 reasoning is precisely the same. 
 
 Any displacement may therefore be given to a rigid body 
 by translating it so that a chosen point comes into its new posi- 
 tion, and then making it rotate round some axis through that 
 point. The direction of this axis and the angular displace- 
 ment remain the same whatever point be chosen. The 
 direction and amount of translation may change, but the 
 translation cannot affect the angular movement. 
 
 The point may be chosen so that the direction of transla- 
 tion is that round which the rotation takes place. For let 
 0' be the new position of G, and let C X be the axis of rota- 
 tion. Let AB represent a plane in the body perpendicular 
 to this axis. Let A"B' be its final position. This is parallel 
 
16 
 
 GEOMETRY OF MOTION. 
 
 to AB; for neither the translation nor the rotation about C'X 
 affects its direction. 
 
 If, then, we first translate the Avhole body along a parallel 
 to C'X until AB comes to A'B' in the same plane with 
 A"B", we shall be able by one rotation about a parallel to 
 C'X to brincr A'B' to A"B", i. e. the direction of translation 
 will be the axis of rotation. 
 
 Hence every small motion is reducible to that of a screw 
 in its nut. And all points of any rigid body are at the same 
 moment moving in coaxial helices. If the pitch of the screw 
 be zero the motion will be one of rotation simply, if it be 
 infinite it will be a translation. 
 
 It is of course not always equally easy to see what these axes 
 and directions are. In the case of a rifle bullet, for example, 
 the motion is already reduced. In the general case the first 
 point is to find out the series of planes w^hich remain parallel, 
 01- — what is the same thing — to find the direction of rotation. 
 Thus suppose we are considering the motion of the earth at 
 alay instant. This consists of a rotation round its polar axis 
 and a revolution of its centre in the plane of the ecliptic 
 round the sun. And suppose we wish to reduce it to the 
 screw motion. We observe that the planes which remain 
 parallel are those parallel to the ec[uator. Hence the axis 
 of the screw is perpendicular to the equator. To find the 
 actaal position of this axis we must consider all velocities 
 projected on a plane parallel to the equator. Then the 
 motion is similar to that of the hoop in Lesson I. Art. 8, 
 which rotates about its centre while the centre revolves 
 about a fixed point. 
 
GEOMETRY OF MOTION. 17 
 
 Let H be tlie velocity of rotation of the earth, V the 
 component velocity of its centre in the plane of the equator, 
 E the distance between two axes each perpendicular to the 
 equator, through the centres of the sun and earth respec- 
 tively. By Lesson i. Art. 8, V and H are equivalent to two 
 
 angular velocities, -p about the axis through the sun, and 
 
 V 
 O — 51 about the axis throus^h the centre of the earth. And 
 
 these are equivalent to an angular velocity round a parallel 
 
 . V 
 
 axis in the plane of the others, distant -^ — 7y from that 
 
 through the sun. This last is therefore the axis of rotation. 
 
 4. An extremely elegant geometrical conception of the 
 motion of a body round a fixed point was introduced by 
 Poinsot. Any such motion may be completely represented 
 by imagining a cone fixed in the moving body to roll on a 
 cone fixed in space. For every body with one point fixed is 
 rotating about a certain axis. As the motion changes, this 
 axis takes up different positions, and describes a cone whose 
 vertex is the fixed point. Now by reasoning exactly similar 
 to that of Lesson i. Art. 8, any cone rolling on another with 
 the same vertex has for its instantaneous axis its line of 
 contact with the other. This axis therefore describes a cone 
 whose vertex is the fixed point. But this is precisely the 
 motion to be represented. The ra.te of rotation will depend 
 on the dimensions of the rolling cone. 
 
 As an example of this take the case of a top spinning 
 with angular velocity co about its axis G, while that axis is 
 rotating with angular velocity 12 about the vertical V, to 
 which it is inclined at an angle a. By the parallelogram of 
 angular velocities the resultant axis is OB — between G and 
 OV — inclined to (7(7 at an angle given by 
 
 sin BOG _n 
 sin {^OL — BOG) co' 
 
 Hence the motion is the same as if an imaginary right 
 circular cone in the top, whose axis was the axis of the top, 
 and whose semivertical angle was BOG, were to roll on the 
 P. G. 2 
 
18 EXAMPLES. 
 
 cone in space whose axis was OF and semivertical angle HOV. 
 As &), the anguhxr velocity of the top about C, is in general 
 large compared with fl, the angular velocity oi 00 about V, 
 the angle EOG will be small. If a series of concentric 
 coloured circles round be drawn on the head of the top, 
 that which corresponds to R will be the only pure colour 
 seen as the top spins. 
 
 EXAMPLES. 
 
 1. Prove that the proposition (Art. 1) holds for angular 
 accelerations. 
 
 2. If a ship is rolling and pitching with equal angular 
 velocities, what is her actual motion ? 
 
 3. Two circular discs can turn about fixed perpendicular 
 intersecting axes. If the axes be so placed that the circum- 
 ference of one of the discs (which is rough) presses against 
 the plane of the other, and if the former disc be caused to 
 rotate about its axis with given angular velocity, find the 
 angular velocity of the other. 
 
 4. A heavy cylindrical crushing stone rolls on a hori- 
 zontal table round a vertical axis. Represent its motion 
 (1) as two rotations, (2) as a single rotation. 
 
 5. Two bevil-wheels, with fixed axes, roll together ; 
 prove that the ratio of their angular velocities is that of the 
 cosecants of their semivertical angles. 
 
 6. Prove tliat if a motion be reduced, as in Art. 3, to that 
 of a screw in its nut, the translational velocity will be less 
 than if it had been reduced in any other manner. 
 
 7. Prove that the rotation is not altered however the 
 motion is reduced. 
 
IIL 
 
 D ALEMBERT S PRINCIPLE. 
 
 1. The ideas of force and matter would seem to be 
 equally fundamental. One cannot be conceived except as 
 acting on or being acted on by the other. Force is 'that 
 which changes or tends to change the state of rest or motion 
 of matter ;' or as Newton's first law of motion might be ex- 
 pressed, ' without force a body can experience no change 
 either in the. quantity or direction of its velocity.' 
 
 The second law is that the force in any direction is 
 proportional to the quantity of motion it produces in that 
 direction. 
 
 Two lumps of matter (masses) are defined to be equal, 
 when the same force acting during equal times on both 
 generates in them equal velocities. Two forces are defined 
 to be equal, when acting on the same mass for equal times 
 they generate in it equal velocities. Then it is found hy 
 experiment, that double forces acting on the same mass for 
 equal times generate double velocities ; and in general that 
 the whole force in any time is proportional to the product 
 of the mass moved and the velocity generated. And this 
 product is called the quantity of motion or momentum. The 
 force at any moment is measured by the rate of change of 
 quantity of motion, i.e. by the product of the mass and the 
 rate of change of the velocity. 
 
 2. Such are the laws by which the motions of a single 
 particle are determined. Newton's third law, promulgated 
 
 ^—2 
 
20 EARLY ATTEMPTS. 
 
 m 1087, — tliat action and reaction between connected bodies 
 Avh ether at rest or in motion are equal and opposite — gives 
 the means of determining the motion of a system of particles. 
 But this was not at first appreciated. The student will find 
 in the second volume of Whewell's History of the Inductive 
 Sciences, or in Walton's Mechanical Problems^ an interesting 
 sketch of the errors into which mathematicians fell, and the 
 difiiculties they overcame before arriving at that principle 
 first correctly stated by D'Alembert in 1742. 
 
 The first Kigid Dynamics problem which \vas solved on 
 correct principles had been proposed by Mersenne in 1646. 
 It Avas 'to find the centre of oscillation/ or the length of 
 the simple pendulum which swings in the same time as any 
 given rigid body swinging about a horizontal axis. This 
 was solved by Huyghens in 1673 by the help of the correct 
 principle, that if a pendulum in the shape of a rigid rod 
 loaded with any weights make part of an oscillation, and if 
 then the weights be disengaged from the constraining rod 
 and reflected upwards with the velocities acquired, the centre 
 of gravity will rise to the same height as it came from. But 
 the main difficult}^ in the transition from a particle to a 
 system still remained, viz. what effect motion impressed 
 on one part of a rigid body has on another part. 
 
 Tn 1686 James BernoulU gave expression to this difficulty 
 by proposing to physicists the following query : '• Given m, m 
 two ecpial bodies attached to an inflexible straight rod, which 
 is capable of motion in a vertical plane about one end which 
 is fixed; let r, r denote the distances of m, iii from this 
 end ; v, v their velocities for any position of the straight 
 line in its descent from an assigned position ; u, ii! the 
 velocities they would have acquired in descending the same 
 arcs unconnectedly. Then through the connection m has 
 lost u — V, and m has gained v' - it, (Query) Whether the 
 relation (similar to that of forces and arms in a lever) 
 u — v:v—u'::r':r be the correct expression of the circum- 
 stances of the motion ?" 
 
 The nearness of this to the true expression — which is, 
 that u, V, u', V must be the velocities acquired in an infinitely 
 
d'alembert's principle. 21 
 
 small time — illustrates strikingly the groping of mathema- 
 ticians for the principle which was to be the pillar supporting 
 this science. 
 
 Bernoulli's query was shortly afterwards correctly an- 
 swered by the Marquis de I'Hopital. 
 
 In 1716 a solution of Mersenne's problem was given by 
 Hermann, founded, (says Whewell), " on the statical equi- 
 valence of the solicitations of gravity and the vicarious 
 solicitations which correspond to the actual motion of each 
 part, or, as it has been expressed by more modern writers, 
 the equilibrium of the impressed and effective forces." 
 
 In 1736 Euler published his Mechanics, in which, while 
 recosrnizincr the correctness of the solutions of individual 
 problems by other philosophers, he complains that a new 
 geometrical solution is required for each separate problem, 
 and therefore tries to reduce their methods to analysis. 
 Thereafter the impulse .^iven by D'Alembert in 1742 caused 
 this science to spring almost at once into the maturity of the 
 present day. Before the death of Euler the solution of the 
 problems of the subject was pushed as far as the knowledge 
 of differential equations would allow. 
 
 3. The following is a translation of D'Alembert's own 
 statement of his principle. It will be better understood, if 
 it be borne in mind that forces may be measured by the 
 quantities of motion they give or would give were they 
 allowed; in fact, that for purposes of reasoning force and 
 motion produced are convertible. Also that a principle 
 proved for any number of rigidly connected particles is proved 
 ibr a rigid body. 
 
 " Given a system of bodies related to one another in any 
 manner whatever ; and supj^ose that on each of these bodies 
 a particidar motion is impressed, which it cannot folloiu on 
 account of the constraint of the other bodies; to find the 
 motion which each body tuill take. 
 
22 DYNAMICS. 
 
 " Solution. Let A, B, C, &c. be the bodies which compose 
 the system, and suppose that the movements* a, h, c, &c. have 
 become impressed on them, which they are forced by their 
 mutual actions to change into the movements a , h', c\ &c. 
 It is clear that we can regard the movement a impressed on 
 A as composed of the movement a which it has taken, and 
 of another a; that in the same way we may consider the 
 movements h, c, &c. as composed of the movements 6', ^ ; 
 c, 7, &c. ; whence it follows that the movements of the 
 bodies A, B, C, &c., among one another would have been 
 the same if, instead of giving them the impulses a, h, c, &c., 
 we had given them the double impulses a, a;b',^; c, 7; &c. 
 
 "Now by the supposition the bodies^, 5, (7, &c., have of 
 themselves taken the movements a, b', c, &c., hence the 
 movements a, /3, 7, &c., must be such that they do not 
 derange anything in the movements a, h', c, &c., that is to 
 say, that had the bodies only received the movements 
 a, /§, 7, &c., these movements must have destroyed one 
 another and the system have remained at rest. 
 
 " Whence results the following principle for finding the 
 motion of several bodies which act on one another. Decom- 
 pose the movements a, h, c, &c., impressed on each body, each 
 into two others a, a ; h\ ^ ; c\ 7, &c., which are such that, 
 had the bodies only received the movements a, h', c, &c., 
 they might have kept these movements without interfering 
 with one another; and had they only been subject to the 
 movements a, 13, 7, &c., the system would have remained 
 at rest. It is clear that a, h', c\ &c. will be the motions 
 which these bodies will take." 
 
 4. Thus if the impressed forces are such as would make 
 a certain body acquire the velocity represented by Aa, while 
 
 its actual velocity is AA', A' a represents a velocity which the 
 * a on A, h on B, c ou C, &c. 
 
D ALEMBERT S PRINCIPLE. 
 
 23 
 
 body is invited to take but does not. Aa being proportional 
 to the impressed force, AA' is proportional to that part of it 
 which is effective in producing motion, while A' a is propor- 
 tional to the force of constraint which that body exercises on 
 those arouDd it, and a A' is proportional to the force of con- 
 straint which they exercise on it. 
 
 We may look on this triangle in three ways : (1) that the 
 impressed force is the resultant of the part which has gone 
 to cause motion, and of the part gone to balance the force of 
 constraint ; (2) that the impressed forces and the force of 
 constraint on the body have as their result the motion pro- 
 duced ; or (3) that the force of constraint on the body balances 
 the impressed force and gives the motion. 
 
 Any one of these forces is the resultant of the other two, 
 or any one reversed is in equilibrium with the other two. 
 
 So much for one body or element of mass. When the rigid 
 system is considered, of which this body is a part, the forces 
 of constraint are in equilibrium among themselves, and there- 
 fore the remaining two sets, the impressed and reversed 
 effective forces, are in equilibrium. 
 
 . 5. The first problem to which D'Alembert applies his 
 principle is — to find the velocity of a rod CB, fixed at C, and 
 
 loaded with any number* of bodies A, B, R ; supposing that 
 these bodies, if the rod did not hinder them, would have 
 
 * A,BfR here and elsewhere denote not only the positions but the masses. 
 
24! d'alembert's ppjnciple. 
 
 described in equal times the infinitely short lines AO, BQ, 
 itj([^j)erpendicular to tlie rod. 
 
 The Avliole difficulty consists in finding the length RS 
 traversed by one of the bodies R in the time in which it 
 would, unconstrained, have traversed RT \ for then the 
 velocities EG, AM oi the other bodies will be known. Now 
 consider the impressed velocities RT, BQ, AO i\^ composed 
 of the velocities RS, ST; BG, - QG ; AM,-OM. By our 
 principle the lever CAR would have remained at rest if the 
 bodies R, B, A had only received the velocities ST, — QG, 
 - OM. Hence 
 
 A,MO.AC-^B. GQ.BC=R.ST. OR 
 
 (since A . MO is proportional to the force which produces the 
 motion MO in the body A) ; that is 
 
 A.ACiAM-AO)+B.BC{BG-BQ) = R.CR {RT-RS). 
 
 Now AM : RS :: AC : CR, 
 
 and BGiRS r.BCiCR. 
 
 Substituting these values of AM and BG, there results a 
 simple equation for RS. 
 
 D'Alembert's solution, given above, is geometrical. The 
 analytical expressions and methods (introduced by Euler) are 
 so much more convenient and powerful, that they have been 
 universally adopted. Our future proceedings will consist in 
 finding convenient expressions for the effective forces, and 
 then solving problems by any statical method. 
 
 The reasoning by which D'Alembert's principle is esta- 
 blished is obviously applicable to any system of bodies how- 
 ever connected ; as, for instance, to fluids. The science of 
 fluid motion does not branch off until we come to introduce 
 the condition of rigidity in finding expressions for the re- 
 sultant of the effective forces. 
 
 C. The impressed or external forces are the cause of the 
 motion and of all the other forces. Which are the impressed 
 forces will appear by the particular system which happens to 
 
IMPEESSED FORCES. Zo 
 
 be under consideration. The same force may be external to 
 one system and internal to another. 
 
 The constraining force on A (Art. 5) is internal when the 
 whole system GABB is under consideration; but did we 
 wish to find this force of constraint, CA would be considered 
 as a system in motion, and the action of BR on A would be 
 one of the external forces. The pressure between the foot 
 of a man and the deck of a ship on which he is, is external 
 to the ship and also to him, and is the cause of his own 
 forward motion and of a shght backward motion of the ship; 
 but if the man and ship be looked on as parts of one system 
 the pressure is internal, and the man may dash himself as 
 violently as he pleases against any part of the vessel without 
 quickening the voyage for himself and his fellow-passengers. 
 
 It is most important that the student should have in 
 every problem a clear idea as to the system which he is con- 
 sidering. 
 
 7. Before the time of Poinsot (early in the present 
 century), mechanicians had no better idea of the effect of a 
 system of forces than that it could be reduced to two or 
 more forces acting at separate points. The action of a door 
 on its hinges was taken as consisting of forces acting at the 
 different hinges. This has disadvantages. When the hinges 
 are more than two, the forces cannot be found by the 
 formulse for rigid bodies. And when problems of motion are 
 considered, it becomes very inconvenient to have no cause 
 for rotation other than the moment of a force, which is 
 different round different axes. 
 
 This defect is supplied by Poinsot's theory of couples. 
 He was led to it by considering what could be the resultant 
 of two parallel and equal forces acting in opposite directions. 
 In this theory the force P acting at A is equivalent to the 
 parallel force P at any point and to the couple (PP), 
 which is reducible to no single resultant. The effect of the 
 force P Sit A on a given body must be the same whatever 
 point we may choose, for it will not be altered by our 
 looking at it ; but in certain cases, the most conve- 
 nient position of will be suggested. Thus might be 
 
26 
 
 COUPLES. 
 
 a fixed point round which the body could turn. In that 
 case, P at will be a pressure, and the couple will make 
 
 o 
 
 ■-A. 
 
 Y 
 IP 
 
 the body rotate. If it be objected that the couj^le, to 
 
 /P P\ 
 
 cause rotation about 0, should be f -^ , -^ j , with an arm 
 
 double OA ; the answer is, that rotation is not about a line 
 or point, but about a direction or in a plane; and that these 
 two couples are in fact exactly equivalent. Of course a single 
 force need not always be resolved into a force and a couple. 
 If A were such a point that a force acting there trans- 
 lated the whole body, it would not simplify our conceptions, 
 but the reverse, to look on it as a force causing partly trans- 
 lation, partly rotation. In this theory the united actions of 
 all the hinges of a door would be a single force and a 
 single couple. The couple is the same with respect to all 
 parallel axes, but varies in magnitude as the line of action 
 of the force changes. 
 
 Couples exist uncombincd in nature in the case of mag- 
 nets. There is no pressure on the point of support of a 
 magnet due to the earth's magnetic action ; for that consists 
 of two equal and opposite forces acting on the North and 
 South poles respectively. 
 
 8. Forces are also divided in Kinetics into impulsive and 
 
FORCES AND BLOWS. 27 
 
 accelerating forces, i. e. into blows and finite forces. The 
 impulse, generating a finite momentum instantaneously, is the 
 simpler in theory. Time is no element in the calculation. 
 The other forces require time to develop a finite quantity of 
 motion, and their effect in an infinitely small time is as 
 nothing compared with that of the class of blows. They 
 are called finite forces. Attractions, tensions of elastic 
 strings, pressures of gases, are examples of this class. 
 There is probably in nature no perfect impulse which takes 
 absolutely no time in its action ; but it is usual to consider 
 as such all forces which produce an appreciable change of 
 motion in an inappreciable time, as explosions and impacts. 
 
 A blow is measured by the momentum or quantity of 
 
 motion it generates. A constant finite force is measured by 
 
 the momentum it generates in a unit of time ; a variable one 
 
 by the quantity of motion it would generate in the unit of 
 
 time if it had throusfhout that time the same mao^nitude as 
 
 . . . 
 
 at the moment of consideration, or, in other words, by the 
 
 rate of generation of momentum, or the momentum generated 
 
 in an infinitely short time (during which it may be supposed 
 
 constant) divided by this time. The total force daring a 
 
 finite time, or the force at a moment, is comparable in effect 
 
 with a blow ; but not the force during an infinitely small time. 
 
 The same laws of motion apply to both classes. Mo- 
 mentum o^enerated is the measure of both. Hence the fio^ures 
 and reasoning of DAlembert are applicable to both. 
 
 The dynamics of impulses introduces only algebraical 
 equations ; that of finite forces depends on differential equa- 
 tions. The equations of either can be deduced from those of 
 the other. A finite force may be looked on as the limit of a 
 series of small impulses ; an impulse as the limit of the total 
 of a very large finite force acting during a very short time. 
 
 The student must be cautioned ag-ainst reofardino^ a sudden 
 change or annihilation of a finite force as an impulse. If 
 a cricket-ball is struck by a bat, it moves off with a 
 measurable velocity. If it is let fall, it begins to move 
 with an infinitely small velocity. What is finite is the ac- 
 
28 d'alemceet's principle. 
 
 celeration. Ao-aio, if it is movino: in one direction and is 
 struck by a bat, it is sent off at once in a different direction ; 
 but a ball rolling off a table moves at first horizontally. 
 But although the direction of motion experiences no imme- 
 diate change, the curvature of the path does ; for there is 
 a sudden accession of downward force, and therefore of 
 downward acceleration. If a body be supported by two 
 strings, and if one be cut, the tension of the other will be 
 instantaneously but not impulsively changed. No finite 
 change of a finite force can convert it into an impulse. 
 
 9. The effective force is that component of the impressed 
 force which is effective in causing motion. It is found, not 
 from the impressed force, but from its being the force neces- 
 sary to produce the actual motion, for the same motion must 
 alwa3^s be caused by the same force. It is the force which is 
 required for producing the deviation from the straight line 
 and the change of velocity. If a particle is revolving with 
 constant velocity round a fixed axis, the effective force is the 
 centipetal force. If a heavy body falls without rotation, the 
 whole force of gravity is effective. But if it is swinging 
 about a horizontal axis, the weight goes partly to balance the 
 pressure on the axis. 
 
 If the motion is known, the force requisite to produce 
 it is easily found. But the problem of this science is in- 
 verse. It is : " Given the forces, find the motion." Now the 
 method of treating any inverse problem is to solve it as if 
 it were a direct problem, and thus get equations for the un- 
 known quantities. If the question is one of an impulse, 
 we suppose an element hm to have the components of its 
 velocity along the axes changed from ti, v, w to it, v\ lu. 
 The change of momentum is then hm{ii —u), Bm{v—v)y 
 Em(w' — w), and these are the measures of the components 
 of the effective force. 
 
 If the motion is accelerated, the rate of increase of 
 
 momentum is 8?h -, , ^m-f-, Bm-. ; and those are the 
 dt at at 
 
 measures of the comjoonents. 
 
 If the co-ordinates of the element hn be x, y, z, the velo- 
 
EFFECTIVE FORCES. 29 
 
 O T (til UZ 
 
 cities u, V, w are -^ , -j^- , -,,-, and the accelerations are 
 
 d^x d'^2f d^z 
 
 — r , -vi , —r; . We will in s^eneral work with the velocities, 
 df ' df ' di' * 
 
 because the connection between impulsive and accelerating 
 
 forces is brought out ; but if in any problem the position 
 
 of the system be required, it will be necessary to use the 
 
 co-ordinates. (It will be as well to mention here that for 
 
 purposes of abbreviation we will use the fluxional nota- 
 
 dx d'x 
 
 tion. Thus --. =x, -.-^ ='x\ so that %i = x and ii = x. A.^ t is 
 
 (tt Ctu 
 
 always the independent variable, this can cause no am- 
 biguity.) 
 
 There may be effective forces when there are no im- 
 pressed forces, as in the case of a circular ring set rotating 
 about a vertical axis and then left to itself. The effectives 
 are then in equilibrium among themselves. They are in 
 that case the centripetal forces. 
 
 It is convenient to suppose the velocity and acceleration 
 to increase, and therefore to suppose that the effective forces 
 act in the direction towards which the co-ordinates or angles 
 are positive. The result shews by its sign whether the 
 velocity is in that direction or the other, and whether the 
 acceleration is a retardation or not. 
 
 Example. A carriage-iuheel, whose radms is a, is rolling 
 ivith constant velocity v along a road. What is the force 
 which gives its motion to a particle (Sm) of mud on it just 
 passing the top of the luheel ? 
 
 The question is : What is the acceleration of this particle ? 
 Now this in any direction is equal to its value relatively to 
 any point, (say) the centre, together with the absolute ac- 
 celeration of this point. The acceleration of the centre is 
 zero. And the only acceleration of Bm relatively to it is 
 the centripetal one (o^a (co being the angular velocity). 
 
 V 
 
 Since the point of contact with the road is at rest, co=- . 
 
 Therefore the effective force acts towards the centre and is 
 
 Bm . — . 
 a 
 
30 LOST FORCES. 
 
 If the particle were now to leave the wheel and to move 
 freely with an acceleration P which would be vertically down- 
 wards, the force necessary to give this would be hn.^. 
 Whence we may infer that the force of adhesion to the 
 
 wheel is 3m [ - - /3j downwards. 
 
 10. The internal forces, or forces of constraint, or the 
 lost forces, — as some have called them, in contrast to the 
 effective forces — which are in equilibrium among them- 
 selves, can exist only in a system of particles. They are 
 stresses, or couples -which we call tendencies to break, or any 
 molecular forces in a rigid body ; or they may be pressures, 
 tensions, attractions between the bodies of a system. They 
 vanish not collectively only, but separately, when the im- 
 pressed forces on the particles of the system are entirely 
 effective in producing motion. Thus there are no internal 
 pressures between a number of bricks falling in a block with- 
 out rotation. 
 
 To find these forces at any point, we must consider a 
 system to which they are external, i. e. we must reduce the 
 system to one bounded by the point at which they act. 
 
 11. If then a system be acted on by impressed forces 
 XYZ^, X^Y.^Z.,, &c. acting at various points, and if S^^^ 
 3.7??, &c. be elements of mass at points whose co-ordinates i 
 x^y[z^, xij/..^, &c. and whose velocities are u^v^io^, '^^^.^^^ ^ 
 the system of forces X^Y^Z^, X^Y^Z^, &c., and the system 
 
 are 
 
 5, du^ 
 
 5. diL 
 
 _ dv, ^ dw. 
 dil, ^ dio.^ 
 
 acting at x^y^z^, ^^^A' 
 librium. 
 
 ... all taken together i 
 
d'alembeet's principle. 81 
 
 If the impressed forces are impulsive, they are in equi- 
 librium with 
 
 - \m [u; - w J, - ^^m {v; - v^), - 8^m (w^ - w,). 
 
 12. It will illustrate the application of D'Alembert's 
 principle if we solve the following problem. 
 
 A and B are Uuo masses connected hy a rigid massless 
 frameiuoi'h. They receive impidses tuhich luoidd, if they were 
 unconnected, gener^ate in them given velocities Aa, Bb. What 
 velocities and directions will they take ? 
 
 The force of constraint must act along AB, for an impulse 
 along that line is that which neither body can obey. If then 
 we measure from the points a and b, aA' and IB', to 
 represent the velocities caused by the constraint, they will 
 be parallel to AB and opposite in direction, and 
 
 A,aA:=B.hB' (1), 
 
 for these represent the forces of constraint. If, further, 
 
 AB' = AB (2), 
 
 AA' and BB represent the velocities of A and B. 
 
 The process above indicated is a reverse one, and although 
 in the above case the positions of A' and B' can be found by 
 a geometrical construction, in general an equation is required. 
 
 Such is the geometrical solution. 
 
 The analytical one is derived from the condition — which 
 is equivalent to (1) — that the forces 
 
 A.Aa, B.Bh, -A.AA', -B.BB 
 
 are in equilibrium. 
 
 Writing down the statical equations to which this gives 
 rise, and the geometrical equation that the components of 
 A A and BB' along AB are equal — equivalent to (2) — we 
 can determine AA' , BB', and their directions. 
 
32 EXAMPLES. 
 
 13. Let us suppose that the lever in D'Alembert's example 
 (Art. 5) is supported at an angle a with the vertical and then 
 let go, and that it is required to find the initial accelerations 
 
 The accelerations will afterwards be partly along the rod 
 and partly at right angles to it, but as each body begins _ to 
 move in a perpendicular to the rod, the initial acceleratioii 
 will be in that direction. If the angular velocity be called 
 &j, the accelerations will be 
 
 ^Ot- BC'^^, BC'^. 
 
 dt .dt dt 
 
 The forces causing tliese must be 
 
 A.AC^, B.BO^, B.Bcf^, 
 
 aciing at A,B,R perpendicularly to CABR. These reversed 
 are in equilibrium with the weights Ag, Bg, Rg, and the 
 pressure on the axis C. 
 
 Taking moments about (7, 
 
 {A.AC'-VB.BC'-VR.RC')^^^ 
 
 = g. sma. {A. AC +B.BG+R.RC), 
 which determines the initial angular acceleration. 
 
 This solution should be carefully compared with that in 
 Art. 5. 
 
 14. To illustrate this subject farther we will answer the 
 following question. 
 
 In the case of the compound pendulum^ (a rigid body 
 
 swinging about a horizontal axis), find the force which acts 
 
 on any infinitesimal part of the whole mass to balance iif^ 
 
 weight and to give it its acceleration. Hence shew that if 
 
 an infinitely small mass be hung by an infinitely short cord 
 
 from a point (x, y) of the pendulum, the inclination of the 
 
 cord to X will he 
 
 , - r/ sin ^ — d.x + w^y 
 
 tan"' — ^ 3- ; — r^ , 
 
 ^ cos ^ + w^/ + w iC 
 
FORCE OF CONSTRAINT. 33 
 
 the axis of x hein(/ the line through the point of support 
 and a fixed point in the pendulum, and co, (o heing used to 
 
 denote jr , j.^ respectively. 
 
 It will save repetition if we mention that we shall in 
 future adhere to these meanings of the symbols and also use 
 hm for an element of mass, so that %hn is the mass of the 
 whole body, r for the distance of hn from an axis of rotation, 
 6 for the angle between a fixed line in the body and a fixed 
 
 line in space. 
 
 That -jT is the same for all lines fixed in the body may 
 
 be seen by considering one whose vectorial angle is ^ + a. 
 This angle a is quite independent of the time or motion, 
 being the angle between tv,^o lines or planes of particles. 
 
 And -,- IS thereiore equal to — ^^ —- . -77 is m fact the 
 dt ^ dt dt 
 
 angular velocity of the body. 
 
 We are here asked to find the constraining force on any 
 
 element Sm situated at a point x, y, for that is the resultant 
 
 of the reversed impressed forces and the effective forces ; in 
 
 other words, it balances the weight and gives the acceleration. 
 
 Let be the point of intersection of the plane of the 
 
 P. G. 
 
84 d'alembert's principle. 
 
 paper by tlie axis of suspension, OV the vertical, Ox a 
 convenient line fixed in the pendulum ; P the position oi 
 8m on the positive side of Ox from V; PN a perpendicular 
 on Ox. Then ONisx; PN, y. 
 
 The impressed force is hm . g, parallel to V, Eeversing 
 this it is equivalent to 
 
 — hn g cos 6 parallel to Ox, 
 BrngsinO „ „ Oy. 
 
 The accelerations of P are «"?' from P to ; and cor at 
 right angles to OP and away from V. These are resolvable 
 into — co'^x along and — co'^y perpendicular to Ox, 
 
 -0)1/ „ „ (i)X „ „ „ Ox. 
 
 H>enc-e the effective forces on P are 
 
 hn (— oi^x — (oy) along and ] ^ 
 
 hn (— &>"?/ 4- wx) perpendicular toj 
 
 The required constraining force is then tlie resultant of 
 hyi (— gcosO — co^x — coy) along and hm {g sin 6 — cory + cjx) 
 perpendicular to Ox. 
 
 In answer to the second part, we observe that if the ele- 
 ment be connected with its neighbours only by a short cord, 
 the tension of that cord is the constraining force whose value 
 we have just found. The direction of the cord will therefore 
 be the direction of the resultant found above, or wdll be 
 inclined to Ox at an angle whose tangent is 
 
 — g sin + (o'y — cox 
 g cos 6 + co'^x + (oy 
 
 EXAMPLES. 
 
 1. A and B are two masses connected rigidly. If A 
 receives a blow wliicli is fitted to impress a velocity AC, 
 and if it actually takes the velocity A C' ; shew that CC is 
 parallel to A B. 
 
EXAMPLES. 35 
 
 2. A ball A is in motion. A blow is given which would 
 (were it at rest) impress a velocity AB on it. It moves 
 instead with velocity AB', With what velocity and in what 
 direction was it moving before ? 
 
 S. A cricket-ball is rotating with velocity co round the 
 direction of the line of wickets ; on touching the ground 
 it is acted on by an impulsive couple, which would have 
 given it if at rest an angular velocity co' about a horizontal 
 line perpendicular to this direction. Round what direction 
 will it actually rotate ? 
 
 4. Does the grooving of a rifle increase or diminish the 
 force of recoil ? 
 
 5. A blow and a constant finite force acting for half 
 a second produce in the same mass the same velocity. Prove 
 that the measure of the force is double that of the blow. 
 
 6. If i^ be a finite force which acting during a time t 
 causes the same momentum as a blow P, prove that 
 
 p= rFdt, 
 
 J 
 
 7. How must a particle move that the effective forces 
 may vanish 1 
 
 8. A carriage-wheel is rolling with given velocity and 
 acceleration along a road. Find the force which gives its 
 motion to a particle of mud passing the top of the wheel, 
 
 9. A circular ring of mass m and radius r is rotating 
 with constant velocity about its centre. It can bear without 
 breaking a stretching force T; prove that the angular velocity 
 
 must not exceed •! [ • 
 
 [ mr J 
 
 3—2 
 
IV. 
 
 EEDUCTION OF THE EXPEESSIONS FOPw THE EFFECTIVE 
 FORCES. 
 
 1. Any system of forces is equivalent to a single re- 
 sultant force acting at any point and to a resultant couple. 
 This must be the case with the effective forces of any moving- 
 system. These consist, if the change of motion be sudden, 
 of a force on each element of mass 8m whose components are 
 
 8m {u — ti), Sm (y' ~ v), 8m (lu — w) ; 
 
 if the change be gradual, of 
 
 ^ du -, (Iv J dw 
 cm —=- , dm J- , dm ,- • 
 at dt dt 
 
 If X, y, z be the co-ordinates of S??z, referred to any origin, 
 and S denote summation over the whole system, the resul- 
 tant force will be the resultant of 
 
 S 8m {iL —u)y S 8m {y —v), S 8m {lu — w) 
 
 acting at the origin. If the motion be accelerated, of 
 
 ^ ^ du ^^ dv ^ , dto 
 
 and the couple will be the resultant of 
 
 1cm [y [iv - xo) - z {u - v)}, 18m [r. [u -ii)-x (^w' - w)], 
 18m [x {v -v)— y {ii - w)l J 
 
CENTRE OF MASS. 87 
 
 or, if the motion be accelerated, of 
 
 ^^ f dw dv\ ^^ f du dw\ ^^ f dv du\ 
 
 round the directions Ox, Oij, Oz. 
 
 The symbol S introduced above has clearly the following 
 properties : 
 
 (1) that if a be a quantity which is the same for every 
 point, 
 
 2 i^m . a) = a .^Bm; 
 
 (2) that 2 (Sm U)±^ {Sm V) = S Sm ( U± F), 
 
 in which U and V are an}^ functions of x, y, z or their differ- 
 ential coefficients; 
 
 (3)tliat |.2(S«F) = S(S»»^). 
 
 We proceed to put the above expressions into forms more 
 suitable for practical purposes. 
 
 2. Let m^, ini^... be any masses; x^y x^... their distances 
 from a fixed plane. 
 
 If m^cCj + 77i^a?2 + . . . = (^^1 + m^ + • . ■)'x, x may be said to be 
 the average distance from the plane of all the masses. The 
 point determined by this and similar ecpiations for y and i is 
 called the Centre of Mass; a name which was first used 
 by Daniel Bernoulli. As masses are proportional to their 
 w^eights at the same locality, this point coincides with the 
 centre of gravity. 
 
 If we differentiate the above equation with respect to the 
 time, we have m^u^ + m^u^ + . . . = {m^ + m^+ ...)u, whence the 
 proposition ; that the sum of the linear momenta of any 
 masses in a given direction is equal to the momentum in 
 that direction of their united masses moving with the velocity 
 of the centre of mass. From this property the point is called 
 the Centre of Inertia. 
 
88 CENTRE OF INERTIA. 
 
 Differentiating once more we have 
 
 and so for the other component accelerations. 
 
 Hence the single resultant of the effective forces is that 
 force which would be necessary to move the whole mass 
 collected at the centre of inertia with the motion of the 
 centre of inertia. 
 
 These equations may be written 
 S i^nx) — X ^m, 
 S (mu) = u Xrrif 
 ^ ( du\ du ^ 
 
 ^ rdt) = dt -'"• 
 
 This point will in future be denoted by the letter G, 
 It is obvious that if G lie on the plane yz, 
 
 m^x^ + m,^x^+ = 0. 
 
 If it be moving in or parallel to the plane of yz, 
 
 m^u^ + m^u.^ + = 0, 
 
 and so for higher differentials. 
 
 3. Let any fixed point be taken as origin of coordinates. 
 Let the co-ordinates of any point P referred to be x, ?/, z ; 
 while those of P referred to parallel axes through G are 
 f , 7], f, and tliose of G with respect to are x, y, z. 
 
 Then x = x-\-^, y = y + 'ny z = z-\-^. 
 
 If 7?ij + m^ + = ^^, 
 
 we have by definition 
 
 QH^x^ + m.jr^ + = Mx, 
 
 and m^^^ + 7n^^., + =0, 
 
 whence w , - ,/ + m„ , / + = 0, 
 
 ^ dt ^ dt 
 
 and so for higher differentials and for the other co-ordinates. 
 
CENTRE OF INERTIA. 39 
 
 dw . d'^z 
 
 Now consider terms of tlie form my -j- , i.e. my -^ 
 d\ , d\ ^ r- , ^ (d"^ . dX\ 
 
 + 
 
 d'~~ 
 
 dx d% 
 
 - / d% d% \ 
 
 The last two expressions vanish and there remains 
 d^z. d?z,, , ^ d'z 
 
 d't dX 
 + ™.'7.^ + mA^; + 
 
 ^^ d"z -.jr-d'^z , ^-, d^K 
 By similar reasoning 
 
 Hence 
 
 .a..g=.wf.sa»rS 
 
 
40 PROPERTIES OF THE 
 
 The left-hand expression is the moment of all the effec- 
 tive forces about Ox, whence we have the important result : 
 the moment of the effective forces on a system about any 
 axis is equal to their moment about a parallel axis through 
 the centre of inertia taken as if this axis were fixed, together 
 with the moment of the effective force on the whole mass 
 supposed collected at the centre of inertia and moving witli 
 it, about the original axis. 
 
 Now suppose that G is passing through at the moment 
 whose circumstances Vv^e are considering. 
 
 The interpretation of this is that the' moment of the effec- 
 tive forces about an axis through the centre of inertia is the 
 same as if that point were fixed. 
 
 Precisely similar results hold in the case of the following 
 functions ; (all of them of the greatest importance). 
 
 ^hmyz, %hnzx, l.Smxij, 
 thus l,hnx?/=Mx7/-{-tSm^r}; 
 
 l.Bm (/ + ^'), ^^'n (r + x~), ^hn (x' -\- /), 
 thus 
 
 or their sums, 
 
CENTRE OF INERTIA. 41 
 
 «- -^'4S-^S)=^^(^i-^ 
 
 dy ^^ dx\ _^ ^j. f- dy _ _ dx 
 ~^'dt 
 
 %Sm^. tSm^, tSm^, 
 
 thus SS,.5=iff + SS„.f . 
 
 We leave the working out of these to the student. 
 
 These equalities must be carefully distinguished from the 
 somewhat similar equations which arise from the properties 
 of relative velocities and accelerations. If G were anij point 
 iuhatever it would be true that 
 
 dx dx d^ 
 di^'di'^'dt' 
 d^^d^ d^^ 
 
 de ~ de ^ de ' 
 
 But these are purely geometrical. There is in them no 
 mention of mass. Now the equations which we have proved 
 above are physical. They are true only when G is the centre 
 of inertia. 
 
 4. It will be noticed that the effective forces would 
 reduce to a force acting at any point and a couple ; but the 
 centre of inertia is that point at which the resultant effec- 
 tive force would produce the actual motion of the point on 
 the whole mass collected there ; and it is also the point 
 which may be taken as fixed while we consider the ro- 
 tation. 
 
 It will, therefore, in general be convenient to choose our 
 origin, so that the centre of inertia is just passing through 
 it. But — it may be objected — our co-ordinate axes are fixed ; 
 velocities and accelerations must be measured by reference 
 to fixed points and lines. How then can we choose our 
 origin to be coincident with a moving point ? The answer 
 is : Our origin is fixed, and the centre of inertia is passing 
 
42 FURTHER REDUCTION. 
 
 throiigli it at the moment under consideration. The equa- 
 tions are found from the consideration of the motion at an 
 instant chosen to embrace its most general conditions : and 
 provided we find the acceleration or velocity rightly at the 
 moment we consider, it matters not where the origin may be, 
 after or before. If, however, the mind of the student insists 
 on contemplating the motion in successive intervals, the pro- 
 ceedings of the origin (as we must imagine them) can easily 
 be represented. Suppose the centre of mass a material visi- 
 ble point, and suppose it illuminated by a series of electric 
 sparks (which do not remain on the eye, and therefore shew 
 a moving body as if it were fixed), then the centre of mass 
 will be seen by the light of the successive sparks standing 
 still in its different positions. The student has already come 
 across the same difficulty in the case of the accelerations 
 of a point measured along and perpendicular to a revolving 
 line. 
 
 5. So far our conclusions are true for any system of con- 
 stant mass; for systems of free particles, for strings, for 
 fluids. There remains to introduce the condition of rigidity, 
 i. e. to reduce for one rigid body the couples 
 
 'S.Sm (f (y' — v) — r}{u— t/-)}, 
 and 28,«(^?^-,-^- 
 
 to a form directly connected with rotation. 
 
 In the general case this is beyond the limits proposed in 
 these introductory Lessons. Whatever problems involving 
 three dimensions we attempt shall be considered with the 
 help of the unreduced expressions. 
 
 We suppose then henceforth that the rotation is alto- 
 gether about the direction of Oz, 
 
 The following Lemma will be of service. If x, y be the 
 rectangular, and r, 6 the polar co-ordinates of a point, 
 
 dy dx _ ^ dd 
 
ONE RIGID BODY. 43 
 
 For 
 
 dii dx „ d 
 dt -^ dt dt 
 
 © 
 
 .de 
 
 Now let r and 6 be the radius vector and the vectorial 
 angle of an element of mass of a rigid body, which can onlv 
 
 rotate round a fixed axis, -y- is the rate of chano^e of the 
 
 dt ^ 
 
 angle between a line of particles in the body and a line fixed 
 
 in space. Therefore -j- and -j-^ are the same for every element, 
 
 being in fact the angular velocity and angular acceleration of 
 the body. Denote tlie former by w. Also during the motion 
 and for the same element Sm, r does not change. 
 
 Hence 
 
 dif dx 
 
 dt'^^di 
 
 differentiating ^"^"^^ = ^^^7 
 
 dv du\ day ^^ „ 
 dt 
 
 Hence 2Sm^^^^^-y^J 
 
 If the change of motion be sudden, 
 
 ^Sm [x {v —v)—y {ii — ii)} = {ay — w) ^hm r^. 
 
 These are the moments of the effective forces round the 
 fixed axis. The single resultant through the axis is the same 
 as through any other point, and is therefore the resultant 
 effective force of the whole mass collected at the centre of 
 inertia. 
 
 If there is no fixed axis the centre of inertia is taken 
 as the point, the motion of which and the motion round 
 which determine the circumstances. The resultant effective 
 
44 ANGULAR MOMENTUM. 
 
 force is that of the Avhole mass collected there and moving 
 with its motion. And the motion round it is the same as 
 if it were a fixed point. The forces therefore reduce to 
 
 Mi^iL-ii), MXv'-v) 
 
 acting at the centre of inertia, and a couple {w - co) Shnr"^', 
 or, if the motion be accelerated, to 
 
 ■n.du -nrdv 
 
 dt ' dt ' 
 
 and a couple -^ ^dm r~, (r being the distance of Bm from the 
 
 centre of inertia). ^ 
 
 If the system consists of rigid bodies not rigidly connected 
 these forces may be reduced to one at the common centre of 
 mass, but the couples must be taken for the separate bodies. 
 
 G. The expression ^Bm (xv — yu) or l.hn ( ?•" -y- ) is called 
 
 the angidar momentum about the origin. It has sometimes 
 been called the moment of the momentum. These names 
 might with advantage be kept separate. 
 
 If there is a fixed axis of rotation, 
 
 If not, 
 
 ^Sni {xv — 1/u) = M {xv — yu) + co SBm 7'^, 
 
 in which r is the distance of the element Sm, from the axis 
 through the centre of inertia. Let the term " moment of 
 momentum" be kept for the whole mass collected at the 
 centre of inertia, i.e. for the expression 31 (xv — yu), and let 
 the term coXSmr^ be called the ''quantity of rotational motion." 
 ]iound a fixed axis or round the centre of inertia, angular 
 momentum is tlien the same as quantity of rotation. 
 
 It is important to notice that the quantity of rotational 
 7notion of an element is measured by the square of the 
 distance from the axis. Wlieu Newton attacked tlie problem 
 of precession, he proved that if a rotating ring commuui- 
 
ANGULAR MOMENTUM. 45 
 
 cated motion to a mass attached to it, the whole quantity 
 of motion would remain the same. This is right; but 
 Newton measured the quantity of motion by the sum of 
 the linear motions of the elements, which is wrong. In 
 Rigid Dynamics we introduce a new kind of motion — 
 rotation — caused by another kind of force (viz. a couple). 
 Now a couple is measured by the moment of a simple 
 force ; quantity of rotation therefore is measured by the 
 moment of a momentum. 
 
 Imagine a fly-wheel whose mass is m and radius a, rotating 
 about its centre with velocity o). Every point is moving 
 with linear velocity aw. And therefore in one sense (ex- 
 cluding the idea of direction) the whole quantity of linear 
 momentum is maco. The impulse which applied at a point 
 on the wheel will stop the motion is mao). There is some- 
 times an advantage in considering the motion thus ; but our 
 knowledge of the Geometry of motion indicates distinctly that 
 the only complete way of treating problems of motion will be 
 to consider a body as animated by a directional translation 
 and a rotation. The whole linear momentum, in any direc- 
 tion, of the wheel mentioned above is zero; the angular 
 momentum is moJ^co. In this view the force which stops 
 the motion is a couple, and there is also a single force, — a 
 pressure on the axle. 
 
 7. Our present results applied to those of Lesson III. 
 enable us to assert that the impressed forces are in equi- 
 
 librum with the forces M—r and M-^ actino^ at the centre 
 
 at at ^ 
 
 of inertia, reversed, and the couple — XSjnr^ reversed. If 
 
 ■^ at 
 
 the forces are impulsive they are in equilibrium witli 
 
 M (u' — u), M (jj' — v) and («' — co) 1,hm r^ all reversed. 
 
 8. The reduction of the expressions for the effective 
 forces is now theoretically complete. They have been shewn 
 to be equivalent, when the motion is continuous, to a resultant 
 
 force M -^ , M -rr acting at the centre of inertia and to a 
 
46 THE KESULTANT FORCE. 
 
 resultant couple. But practically the solutions of problems 
 may be much simplified by a proper choice of co-ordinates. 
 Thus if we use the above expressions or those in x, y when 
 the motion is referable to a fixed point, it is obvious that we 
 shall have for each problem to work through those differenti- 
 
 ations which shew that ^ and -^^, parallel to the axes are 
 equivalent to .,. - W ^J along and - ^-^ (,- ^) at nght 
 
 angles to the radius vector. 
 
 It might be convenient to employ the expressions for the 
 accelerations along the tangent and normal to the path of the 
 centre of inertia. Or, yet again, if the motion of G is best 
 defined by reference to a point A which is itself moving, we 
 can use the proposition that the acceleration (or velocity) 
 of G in any direction is equal to that of G relatively to A 
 (supposing A fixed) in that direction, together with the 
 acceleration (or velocity) resolved in the same direction. 
 
 In o-eneral in every analytical solution of a motion of a 
 rio-id system equations are required connecting the velocities 
 of^ different parts. These are called the geometrical equa- 
 tions, and may often be simplified or reduced in number by 
 a proper choice of variables. 
 
 9. A system consisting of two masses A and B rigidly 
 connected by a straight massless luire is moving without rota- 
 tion with velocity Y. A j^oint of the luire hetiveen A and 
 the centre of inertia suddenly becomes fiived, and the system 
 m-oceeds to rotate about O with angular velocity w. It is 
 required to find the resultant impidsive forces which must 
 have caused this change of motion. 
 
 The momentum of translation has suffered a change, 
 
 B.OJJ.co-A.OA.co- {A + i?) V. 
 
 This therefore is the measure of the force which acting at (7 
 at ri^ht angles to AB would cause the change. 
 
EXAMPLES. 
 
 47 
 
 The angular momentum about G has been changed from 
 zero to 
 
 [A.AG' + B.BG') (o. 
 
 This expression is therefore the measure of the couple 
 which must have caused the change. 
 
 10. Given that a circular hoop of radius a is rolling with 
 uniform velocity/ v aloug a road. What are the resultant 
 effective forces (1) on the ichole, (2) on a given part 9 
 
 (1) As the centre of inertia is moving uniformly in a 
 straight line, the resultant force is zero. 
 
 To find the angular velocity (w), consider the motion of 
 the point in contact with the ground. 
 
 This is carried forwards with velocity v by the motion of 
 translation. It is carried backwards by the rotation round 
 the centre with velocity aco, and its resultant velocity is 
 zero ; for it is the instantaneous centre. Hence v = a(o, or 
 the rotation is also uniform. Hence the couple effective in 
 producing rotation is zero also. 
 
 (2) Let 771 be the mass and G the centre of inertia of 
 the part AB, the effective forces on which are to be found. 
 
 The acceleration of G in any direction is equal to that of 
 C in that direction, together with that of G relatively to C 
 measured in the same direction. Now G moves uniformly 
 and G moves uniformly about G. Hence the only accelera- 
 tion of G is towards G and is co^. CG. The resultant effec- 
 
48 EXAMPLES. 
 
 tive force on AB is therefore mco'OG, and acts at G to- 
 wards 6'. 
 
 The moment of the effective couple is -, '^Smr^, where r 
 is the distance of Bm from G. As w is constant this vanishes. 
 
 EXAMPLES. 
 
 1. Prove that the centres of mass and inertia necessarily 
 coincide; but that the centre of gravity does not coincide 
 with these unless tlie weights of the different parts of the 
 body may be supposed to act in parallel lines. 
 
 2. A grindstone is rotating about its axle. Shew that 
 its angular momentum is the same round all axes parallel 
 to this. 
 
 8. A cannon-ball whose mass is 30 lbs. is fired with a 
 velocity of 1000 feet per second. What is its momentum at 
 the moment of discharge ? 
 
 What is the moment of its momentum about a point 
 10 feet immediately above the mouth of the gun ? 
 
 4. How is the centre of inertia of a rigid body moving 
 when the resultant effective force is zero ? 
 
 5. Find a general expression for all functions of co-ordi- 
 nates and differential coefficients for which the properties of 
 Art. (3) are true. 
 
 G. Find the effective forces for the systems in Art. 10, 
 supposing the hoop to be rolling with given acceleration v. 
 
 7. Two uniform rods OA, AB, of lengths 2a, 2h are 
 hinged at A — being fixed — and they rotate in one plane 
 with angular velocities o), co'. Prove that the force which 
 causes the motion of translation of AB is the resultant of the 
 following : 
 
 mass AB . (o)'\ h + 2a)'a cos ^-2 -^ a sin <^j along BA, 
 
 and mass AB T ^ . 6 + 2 -,^ a cos + 2a)- a sin (/>! , at right 
 
 angles to BA ; (/> being the angle between AB and OA pro- 
 duced. 
 
V. 
 
 FIRST APPLICATIONS. 
 
 1. We have now proved that the forces on each element 
 of a rigid body, which are effective in causing the motion, 
 are together equivalent to that force which, acting on the 
 whole mass collected at the centre of inertia, would cause the 
 actual motion of that point ; and, to a couple which, were 
 the centre of inertia fixed, would cause the actual motion of 
 rotation of the body. The impressed forces are also reducible 
 to a single force at a point and to a couple. The reversed 
 effective forces are in equilibrium with the impressed forces. 
 Putting these together, we infer (1) that the motion of the 
 centre of inertia is the same as if the whole mass were col- 
 lected there, and the impressed forces acted at that point each 
 in its own direction ; (2) that the eifect of the impressed 
 forces to cause rotation is the same as if the centre of inertia 
 were fixed. 
 
 These principles are fruitful of important consequences. 
 From (1): 
 
 If there be no impressed force or no resultant impressed 
 force, the centre of inertia must either remain at rest or move 
 on with velocity unchanged in magnitude or direction. 
 When a shell explodes in the air before striking, the forces 
 of explosion are all internal ; and the centre of inertia of all 
 the fragments moves on in the same curve as if nothing had 
 occurred. A plank sliding on smooth ice will move so that 
 its centre of inertia will describe a straight line with con- 
 stant velocity. Supposing the solar system to be so far from 
 the stars that their attraction may be neglected ; the centre 
 
 P. G. 4 
 
50 MOTION OF AND ROUND 
 
 of inertia of the sun and planets must be at rest or moving 
 uniformly in a straight line. If there are no external forces 
 in a particular direction, or if their resultant is at right 
 angles to that direction, then the velocity of the centre of 
 inertia or the linear momentum of the system in that direc- 
 tion remains constant. The impressed forces acting on a 
 chain- shot are vertical. Hence the horizontal velocity of its 
 centre of inertia is constant. 
 
 In an old and instructive problem it is supposed that a 
 man is placed upon a perfectly smooth hard sheet of ice, so 
 that skates avail him nothing. How is he to get off ? The 
 only external forces are his weight and the support of the 
 surface. These acting down and up cannot help him along. 
 His centre of mass will not move by any action internal to 
 his person. He must get external force. Let him throw 
 away something that he has about him. This becomes a 
 body external to himself, and its reaction gives him an im- 
 pulse backwards. Or the projected body may be looked on 
 as still part of his system ; in which view the common centre 
 of mass remains at rest, but as the body moves in one 
 direction, his centre of mass must move in the other. 
 
 The resistance of water is very small to a boat moving 
 slowly through it ; and, accordingly, every one has noticed 
 that on his moving to the end of a boat, in order to get out, 
 the boat, if not previously fastened to the shore, moves back. 
 The common centre of mass, however, of himself and the 
 boat has not moved. Nor will it move although he springs 
 out, until by the pressure of his feet on the land he intro- 
 duces an external force. 
 
 Again, whatever the impressed forces may be, or vrlier- 
 ever they may act, they produce the same effect on the 
 motion of tlie centre of inertia as if they acted there. 
 
 If a ship be pulled to shore by a rope of given tension, 
 it matters notliing to the motion of the centre of inertia at 
 what point the rope is attached. A billiard-ball struck 
 horizontally will move off ecpially quickly wherever it is struck, 
 provided the force of the blow be the same. Every one has 
 noticed how a table-napkin ring, squirred out between the 
 
THE CENTRE OF INERTIA. 5l 
 
 finger and the table, keeps rotating, but soon stops moving 
 away and comes back. The only impressed force is the 
 friction of the table. And this, though acting at the rim, 
 first stops the velocity of the centre and then causes it 
 to acquire a velocity in the opposite direction. And this 
 effect of the force is the same as if it acted at the centre. 
 
 2. From (2) : 
 
 If a body at rest, or moving \Yithout rotation, be acted on by 
 a blow or force whose line of action passes through the centre 
 of inertia, it will move on without rotation : if in such a case 
 the motion of the centre of inertia be stopped, the whole 
 motion vv^ill be stopped. 
 
 To pull a boat to land without making it rotate, pull it 
 by this point. A sportsman once told the author that a certain 
 salmon he hooked gave him much trouble. He tried as 
 usual to get the fish's head down stream, but could not. 
 When at last the fish was landed, he found that he had 
 hooked it by the belly. The scientific expression of this is, 
 that he had been pulling at the centre of inertia, and con- 
 sequently had been unable to turn the fish round. 
 
 Example. 
 
 AB is a smooth fixed inclinedj straight wire. CO is a 
 rod furnished with a ring O at one end by which it hangs 
 
bZ CONSERVATION OF 
 
 from the wire. CO is taken, and being held at right angles 
 to AB is then let go. What will he the inotion ? 
 
 There are only two impressed forces, the weight and the 
 pressure. Both these act through the centre of inertia. 
 Hence they have no moment about it, and there will be 
 no rotation. Thus the rod GO will slide down, keeping 
 always perpendicular to AB. 
 
 8. The resultant effective couple 
 ^^ / dv du\ 
 
 is the rate of change of the angular momentum ^Sm {xv—yu). 
 This couple reversed is equal to the moment of the impressed 
 forces about the point taken as origin. Hence angular mo- 
 mentum plays the same part in rotation that linear momentum 
 does in translation. If there are no impressed forces, the angu- 
 lar momentum about any axis is constant. If the resultant 
 of the impressed forces has no moment about a certain axis, as 
 by passing through it or acting parallel to it, the whole 
 angular momentum of the system about that axis remains 
 constant. 
 
 In the general case, the angular momentum of any body 
 about any axis is equal to the moment of the momentum of 
 the mass collected at the centre of inertia about that axis, viz. 
 m (^^ — 2/ w), together with the angular momentum about a 
 parallel axis through the centre of inertia, w .^Bmr\ 
 
 In the solar system — there being no external force — the 
 sum of the moments of the momenta of the sun and planets 
 about any axis, together with the sum of their angular 
 momenta about parallel axes through their centres,is constant. 
 When an iceberg becomes detached from near the jDole 
 where its motion round the polar axis of the earth is small, 
 and floats to near the equator where its motion is large, the 
 angular momentum of the earth must have diminished as 
 much as the moment of the momentum of the iceberg has 
 increased. (The angular momentum of the iceberg about an 
 axis in itself is neglected.) When Don Quixote was lifted up 
 by the windmill and became a part of its system, the only 
 impressed force external to both being an impulse on the 
 
ANGULAR MOMENTUM. 5S 
 
 axle of the mill, the angular momentum of the sails about 
 the axle must have diminished as much as his increased. 
 
 If a number of particles moving freely becomes rigidly 
 connected, this does not change their united angular mo- 
 mentum. If a rotating body contains liquid, and this liquid 
 solidifies, the angular momentum remains unchanged. 
 
 It has been mentioned that rotation takes place not so 
 much about a point as about a direction. If a watch that is 
 going be finely enough poised horizontally on a point, it 
 will be seen to make small oscillations. These are due to 
 the oscillations of the balance-wheel inside it. The angular 
 momentum of the whole watch is at every moment zero. 
 Hence the angular momentum of the balance-wheel, together 
 with its moment of momentum, is equal and opposite to the 
 angular momentum of the rest of the w^atch. Spin a top on 
 a plate, and float the plate in water. When the top has come 
 to rest, its angular momentum must have been communi- 
 cated to the floating system. 
 
 The angular momentum about a fixed axis or one through 
 the centre of inertia of a rigid body is 
 
 Hence ^ve are enabled to assert that if a body is rotating 
 about a fixed direction under the action of no impressed 
 forces, or of impressed forces whose resultant passes through 
 the fixed axis or through the centre of inertia, the angular 
 velocity remains constant. 
 
 A grindstone set rotating and left to itself, or any heavy 
 body thrown up into the air and rotating in the plane of 
 projection, or an awkward man upon a slide, are examples of 
 this. 
 
 4. It has been stated before, that the tendencies of forces 
 and couples are to cause translation and rotation respectively. 
 We now see that it would be more correct to say that they 
 cause translatioa of the centre of inertia and rotation round 
 it. A couple does not necessarily cause rotation about the 
 direction of its own axis. We shall soon see when this is the 
 case. All we can at present say is that the impressed couple 
 
54< 
 
 EXAMPLE. 
 
 is in equilibrium with the reversed effective couples neces- 
 sary to produce the motion. We have previously seen the 
 complete analogies which subsist between the theory of 
 forces and that of rotations. This analogy must not be con- 
 fused with the physical connection mentioned above, which 
 is between couples and rotations. 
 
 In 1777, shortly after our science had been established 
 on its present basis, an English mathematician of some emi- 
 nence, called Land en, wrote a treatise, in which he professed 
 himself dissatisfied Avith the principles and conclusions of 
 Euler and D'Alembert ; and in v^hich he gave a new theory 
 of motion of his own. This theory gave rise to a good deal 
 of discussion in England. When carefully examined it 
 is found to depend on the principle, that any force whose 
 effect in causing rotation in a body is the same as that of a 
 given force may be substituted for it. This is fallacious ; for 
 it takes no account of the action of a force in causing trans- 
 latioD. Since that time there has been no doubt thrown on 
 the accuracy of our principles. 
 
 5. A body is rotating ivith no impressed forces and with 
 constant angular velocity (o about the axis Oz, which is fixed. 
 
PKESSURES ON A FIXED AXIS. 55 
 
 What pressures has the axis to sustain in consequence of the 
 motion ? 
 
 Suppose tlie axis acted on by forces X, F at (there 
 will evidently be none along Oz), and by a couple whose 
 components are G^, Gy about the directions Ox, Oy. 
 
 Consider the motion of an element of mass hn at P. 
 PMQL is a plane parallel to xy. PM, PL, PN are the co- 
 ordinates of P. 
 
 The acceleration of P is ay\PQ along PQ. This is 
 equivalent to — w\ PM along Ox-, - w\ PL along Oy, 
 
 The effective force at P is therefore 
 — hn oy^PM along Ox, 
 and - Bm co'^ PL along Oy, 
 
 These reversed and taken all over the body are in equi- 
 librium with X, Y and G^, Gy. 
 
 Hence X-\-o>'SBmPM= 0, 
 
 . Y+oy^l.BmPL = 0, 
 
 G,-co'XBmPL.PN=0, 
 
 Gy-Vco'tBniPM.PX=:0. 
 
 It must be remembered that rotation is positive from x 
 to y, from y to z, and from z to x. 
 
 These equations give the pressures on the axis. 
 
 Suppose the whole of Oz set free except the point 0, 
 and that we wished to know whether the forces of the 
 motion would permit the body to continue rotating about O2. 
 This will be the case if G^ and Gy are zero. Hence 
 
 tBmPL.PM and tBmPM.PN 
 
 must both be zero. 
 
 If these conditions hold, Oz is called an axis of permanent 
 rotation, or a principal axis of the body. 
 
56 FIRST APPLICATIONS. 
 
 If the "whole axis be set free, and the body still continiie 
 to rotate about it, X and Y must be zero. 
 
 Hence 
 
 S^mPil/and tBmPL 
 
 must be zero. This means that the axis O2; must pass 
 through the centre of mass. 
 
 It is easy to see how the axis may suffer a breaking couple, 
 but no single pressure. Take a rod and attach it rigidly to a 
 fixed axis at its centre of mass. Let this axis be inclined to 
 it. If it be now made to rotate it will endeavour to get into 
 a position perpendicular to the axis, but if it were set free 
 the centre of mass would not move. 
 
 6. A raft of any form, whose Qiiass is M and lulwse 
 centre of inertia is C, is at rest on the surface of still luater. 
 A man whose mass is m, who is standing at a given j^oint 
 P of the raft, commences to move with velocity v relatively 
 to the raft in a direction at right angles to CP. WJiat motion 
 of the raft will this cause ? 
 
 We take the man to be a moving point, and the water 
 not to hinder the motion of the raft. 
 
 The common centre of mass of the raft and man remains 
 at rest : in other words, there is no resultant linear mo- 
 mentum. 
 
 The velocity of C must then bo in a direction opposite 
 to that of the man. Let V be this velocity of C. Let O 
 be the angular velocity with which the raft begins to move. 
 Then the absolute velocity of the man is v — V— CF.D.; 
 the momenta are ilf Fand m [v — V— CP .fl), and 
 
 - MV+ m {v - V- cr.n) = 0. 
 
 Secondly ; as there are no horizontal external forces, 
 the wliole angular momentum about any vertical axis re- 
 mains unchanged. 
 
. EXAMPLES. . 57 
 
 Consider tlie angular momentum round C. The angular 
 momentum of the raft is 
 
 in which r is the distance of an element Silf from C. 
 
 The moment of momentum of the man round C is 
 miy+^.GP) CP due to the motion of the raft, and 
 — mv . CP due to his own on the raft. 
 
 Hence 
 
 ,n ( F_ V + n . CP) CP+ I12S ilfr^ = 0, 
 
 which gives H in terms of known quantities. 
 
 These two equations give V and O in terms of v and 
 known quantities. 
 
 It is indifferent whether the unknown quantities V and 
 H are assumed to be in the direction in which from previous 
 considerations we know them to be, or in some assumed 
 positive direction. 
 
 7. We can now write down the equations of motion for 
 any system moving under any forces in one plane or in 
 parallel planes. 
 
 Let there be one rigid body acted on by impressed forces 
 reducible to forces X, Y along the axes, and whose moment 
 is L round the centre of inertia ; then (iv. 6) if the forces are 
 impulsive, 
 
 M {a' -u)= X, M [v - r) = 7, 
 (o)'- o})XBmr^= L. 
 
 If they are finite, 
 
 dt 
 
58 EXAMPLES. 
 
 If vv^e desire to take moments about another point than 
 the centre of inertia; let the co-ordinates of the centre of 
 inertia Avith respect to it be x, y, and let the moment of the 
 forces about it be L' ; then, the other (quantities remaining 
 as before, the third equation will become 
 
 (o)' — (d) ^hmr^ + M[x{v —v)—y {u -u)]= L\ 
 
 If there are a number of bodies in the system similar 
 equations hold for each or for any number taken together. 
 
 EXAMPLES. 
 
 1. A person, who has been standing on smooth ice, falls 
 down. In what line does his ceutre of inertia move ? 
 
 2. The earth in cooling contracts. Does this make the 
 day longer or shorter? 
 
 3. A man walks across the deck of a small yacht. Has 
 the yacht rotated ? Has its centre moved ? 
 
 4. Two smooth spheres rest one above another on a 
 smooth horizontal plane. If the equilibrium be disturbed, 
 in what line will their common centre of gravity move ? 
 
 5. A straight uniform rod can slide with its ends on two 
 smooth fixed straiglit wires placed at right angles to one 
 another in a horizontal plane. It is set moving. Prove 
 that its angular velocity will remain constant. 
 
 6. How does it appear that linear and angular momenta 
 obey the parallelogram law ? 
 
 7. A rigid body attached to a string is allowed to fall 
 until the string becomes tight. Shew that if it f^iU so that 
 there is no immediate rotation, there will be no subsequent 
 rotation. 
 
EXAMPLES. 59 
 
 8. A man is being weighed in one scale of a large balance. 
 If he jump straight up, what will be the effect on the 
 machine? and what will be the result when he meets the 
 scale as^ain ? 
 
 9. Explain how it is that a boy in a swing can increase 
 his arc of swing by crouching Avhen at the lowest point. 
 
 10. If a rigid body previously at rest be set in motion 
 by a single blow ; prove that after moving for any time it can 
 be again reduced to rest by an equal and opposite blow acting 
 in the same line. 
 
 11. The centre of gyration of a rigid body capable of 
 revolving about a fixed axis is the point at which the whole 
 mass must be collected, that the angular velocity communi- 
 cated by a given couple may be the same as before. If h be 
 the distance of this point from the axis, prove that 
 
 12. A man is placed in a canoe without a paddle or any 
 means of touching the water. Can he work round the head 
 of the canoe ? 
 
YI. 
 
 MOMENTS AND PRODUCTS OF IXEUTIA. 
 
 1. We have had occasion to remark that tlie quantity 
 '^07)1 r", in which r is the distance of an element hn from an 
 axis, and the summation extends over the whole of a rigid 
 body, will be of constant occurrence in problems of rotation. 
 It is called the moment of inertia of the body about the axis. 
 We have also found that expressions of the form 
 
 ^hnyz^ Xhnzx, XBmxi/, 
 
 are of common though less frequent occurrence. These have 
 been called products of inertia. 
 
 It wdll save us much repetition if we investigate once for 
 all some of the properties of these moments and products, 
 and their values for the most common axes pf the most com- 
 mon bodies. 
 
 The values of moments and products of inertia must de- 
 pend ultimately on summation or integration for the various 
 elements of the body ; but after this has been accomplished 
 for the simplest axes possible, they can be found without 
 summation for any other axes. 
 
 2. The moment of inertia of a iinifonn rod of mass 
 m and length 2a about an auis thvoufjh its middle point at 
 I'lgJtt angles to it. 
 
 Suppose tlie rod a line of particles. Let the distance of 
 one of those from the middle j^oint be r, and its mass jihr, so 
 that fi2a is m. 
 
MOMENTS OF INERTIA. 61 
 
 The moment of inertia is X/jiSr . 7^^ Supposing Sr to dimi- 
 nish indefinitely this becomes //-/r^ dr, the limits of r being —a 
 
 and + a. The value of this integral is —^— . But fjL.2a = m, 
 
 3 
 
 2 
 
 hence the required moment of inertia is m.-z. 
 
 o 
 
 It is clear from the nature of the expression SSmr^ that 
 every moment of inertia will be the product of the mass and a 
 square. It is usually written mF. A; is called the "radius 
 of gyration." It is the distance of the centre of gyration 
 from the axis. (See Less. V. Ex. 11.) The value of k in the 
 
 above example is --= . 
 
 The product of inertia of the rod about two axes x, y 
 through the middle point and in the same plane with it, of 
 which X makes an angle a with the rod, is 
 
 ^fiSr.xT/, i.e. S/aS?^ . r sin a . r cos a. 
 
 In the limit /z, sin a cos a I r^ dry 
 
 or /^ si^ ^ <^^s a . -^ , 
 
 o 
 
 but /jb.2a = m, therefore the product of inertia is 
 
 a' 
 m . sm a . cos a . -- . 
 o 
 
 3. The moment of inertia of a uniform circular j^lctte of 
 mass m and radius a about an axis through its centre 2)er- 
 pendicular to its j^lctne. 
 
 Imagine the circle composed of narrow concentric rings. 
 Let the radius of one be r, its breadth 8r, its mass fi . 27rr8r, 
 so ihut /jL.Tra^ = 7n. 
 
 The moment of inertia of this ring is fi 27rrSr . r^, for all 
 points are equidistant from the centre. Therefore the mo- 
 
62 SUMMATION. 
 
 ment of the wliole is fi27rr^ dr, i.e. /xtt . — . Now/x7ra-=77?; 
 
 therefore the moment of inertia is ??i — , and tlie radius of 
 
 . a 
 Of y ration is --. . 
 
 V2 
 
 The product of inertia of the circle about this perpen- 
 dicular and a diameter is plainly zero ; for the co-ordiuato 
 perpendicular to the circle of every element vanishes. 
 
 The product of inertia about any two axes in its plane is 
 also zero. For each of the axes divides the body symmetri- 
 cally. Hence for every element Sm at w, y Avhose product is 
 hm.xy, there is an element hm dX—cc,y for which the pro- 
 duct is — hm xy ; and the sum will consequently vanish. 
 
 4. The moments of inertia of any body about the axes 
 of co-ordinates are 
 
 23m (2/^ -{- z"), ISm {^ + x') , tSm {x' + /). 
 
 In the case of a body altogether in one plane (x, y), one 
 co-ordinate {z) is ahvays zero. Hence the moment of inertia 
 of such a body about an axis perpendicular to its plane is 
 equal to the sum of the moments of inertia about two axes 
 at right angles to one another in its plane. Hence Ave can 
 
 infer that the moment of inertia of a circular plate about a 
 
 diameter is m -r . 
 4 
 
 The products of inertia of such a body about axes, one of 
 ■which is perpendicular to it, are zero. 
 
 5. TJie moment of inertia of a uniform rigid circular 
 cone hounded by a iilane i:^erpendicular to the axis about its 
 axis. 
 
 Imagine the cone made up of a great number of equally 
 thin circular disks. 
 
 Let the distance of one of these fi-om the vertex be x, its 
 thickness hx. If the semivertical angle be «, its radius will be 
 
MOMENTS OF INERTIA. 
 
 G^ 
 
 X tan a. Let its mass be /llttx^ tair a . Sx. Its moment of in- 
 
 trtia about tlie axis of the cone is mass x - tan" a, 
 
 x^ 
 or fjix^ tan" airSx . — tan" a. 
 
 The moment of inertia of the Avhole cone will be the sum 
 of all these, each indefinitely diminished, or 
 
 fjL tan^ ttTT r '' 4 , _ fJbiT tan* a. , 5 
 where li is the cone's height. 
 
 '^ o « 1 o 
 
 Nov*" the mass of the cone is | ^i.i:x^ tan' adx — .^ ^l tan" octtA". 
 
 Whence 
 
 fJUlX 
 
 -J 
 7j = 7i tan a -y 
 
 10 
 
 6. T7ie product of inertia of an isosceles triangular plate 
 ABC about the base BC, a?zcZ a line through B a^ right angles to 
 BC. 
 
 Take BC as the axis of ^ and the perpendicular as that 
 of y. Draw AD to the middle point of BC. Let AD = h. 
 Imagine the triangle made up of infinitely narrow strips 
 parallel to BC, of thickness Sg, one of which is FQ, cutting 
 AD in L. From P, Q draw FM, QM at right angles to BC. 
 
04 SUMMATION. 
 
 Consider an element of mass ^^x .hy d<X R ^ point on 
 PQ. Its product of inertia is fihx . hy .x.y. That of PQ is 
 fxSy.yfx dx, the limits being BM and BN. 
 
 It is therefore 
 
 ^'^fx^y.y.BD.PL. 
 
 Now FL={h-y)taiij, BD = htan^^; 
 
 therefore the whole product of inertia 
 
 4 f' 
 = 2/z,.Atan'^ y(Ji-y)cy 
 
 But the mass of the triangular plate = ^Ji" tan — 
 
 therefore the product of inertia 
 
 h tan -^ 
 
 = m. 
 
 3 
 
 This is easily seen to be the same as the product of inertia 
 about the same axes of two masses, each equal to ^ , placed 
 at the middle points of AB, A C. 
 
 There will be found in the first chapter of Kouth's Eiyid 
 Dynamics an instructive proof that the moments and pro- 
 
.'principal axes. 65 
 
 ducts of iuertia of a triangular plate about any axes whatever 
 are the same as those of three particles, each of mass equal 
 to one-third of the mass of the plate, placed at the middle 
 points of the sides. 
 
 7. When '^Smxz = and X^m yz = 0, the axis of ^ is a 
 principal axis (Lesson V. Art. o). When all three products of 
 inertia vanish we have a set of three principal axes. But 
 XBmxz and "XSm yz may vanish without 'Xhinxy vanishing; 
 and any one, as Xhmxy, may vanish without either x ov y 
 being a principal axis. If indeed the body is altogether in 
 the plane of xy, two axes for which Xhnxy vanislies are 
 principal axes, for we know that Xhm xz = and XBm yz = 0. 
 
 In many cases the position of a principal axis can be seen 
 at once. Thus at every point in a plane body one principal 
 axis is the perpendicular to the plane. Again, if in any uni- 
 form body a straight line can be drawn with respect to which 
 the body is exactly symmetrical, this must be a principal 
 axis at every point in its length. Any diameter of a uniform 
 circle or sphere is a principal axis at any point in its line ; 
 but the diagonal of a rectangular plate is not for this 
 reason a principal axis at its middle point ; for every straight 
 line drawn perpendicular to it is not equally divided by it. 
 
 8. Table of the squares of the radii of gyration for seve- 
 ral bodies (supposed uniform) of frequent occurrence. 
 
 1. For a rod of length 2a about a perpendicular through 
 
 the centre of mass, F= -^ . 
 o 
 
 2. For a rectangular plate, one of whose sides is of 
 length 2a, about a line through the centre parallel to the 
 
 other, k^ = — , 
 o 
 
 S. For an elliptic plate about either axis, 
 
 72 _ square of other semiaxis 
 
 " 4 • 
 
 P. o. 5 
 
6G MOMENTS OF INERTIA. 
 
 4. For an ellipsoid about any of the three axes, 
 
 2 sum of s quares of the other semiaxes 
 k = ^ . 
 
 These include as particular cases squares, circles, and 
 spheres. 
 
 EXAMPLES. 
 1. Prove that 
 
 (a) For a straight rod of length 2a about an axis passing 
 through one end and making an angle (3 with the rod 
 
 Jc= -j^. a Bin p. 
 
 t\JO 
 
 (/3) For any arc of a circle of radius a about an axis 
 through the centre perpendicular to its plane lc = a. 
 
 (7) For a right circular cylinder of radius a about its 
 
 . , a 
 
 axis Ic == -7-, - 
 
 (S) For a straight rod about an axis parallel to itself 
 at distance c, h — c. 
 
 2. The moments of inertia of a uniform elliptic plate of 
 semiaxes a, h about its major and minor axes are 
 
 That about a perpendicular to its plane through its centre 
 a' + l/ 
 
 IS Qn : . 
 
 3. The moment of inertia of a sphere of radius a about 
 
 2 
 a diameter is ^ 7?ia'. 
 o • 
 
EXAMPLES. 67 
 
 4 The moment of inertia of a uniform rectangular 
 plate whose sides are a and h about the side a is m ^ . 
 
 5. For what bodies and for what axes is the moment of 
 inertia zero ? 
 
 6. The moment of inertia of a uniform cube whose 
 
 leno-th of edge is 2a about a line through the centre parallel 
 
 ° , . 2r/ 
 
 to an edge is m . -^ . 
 
 7. What are the principal axes of a uniform rectangular 
 plate at the middle point of one side ? 
 
 8. Investigate the product of inertia of a right-angled 
 triangle about the sides containing the right angle. 
 
 9. What is the moment of inertia of a shell, whose mass 
 is m and which is bounded by concentric spherical surfaces 
 whose radii are a and A, about an axis through its centre ? 
 
 10. The centre of a uniform parallelepiped is fixed. 
 About what axes will it rotate permanently when no forces 
 act on it but those whose resultant passes through the 
 centre ? 
 
 11. Prove that the moment of inertia of a uniform 
 ellipsoid whose semiaxes are a, h, c about the axis a is, 
 
 h^ + c^ 
 m. — = — . 
 
 12. The density of a cylinder of mass m and radius a 
 varies as the n"-^ power of the distance from the axis. Prove 
 that the moment of inertia about the axis is 
 
 2 71 + 2 
 
 ma . — — ; 
 72 + 4 
 
 5—2 
 
C)H EXAMPLES. 
 
 13. Prove that the quantity of rotation in a cylinder 
 of radius a rotating with velocity co about its axis is 
 
 (0,7)1 -r, 
 
 14. Prove that the quantity of rotation in a rotating 
 
 2 
 sphere of radius a is w . m . -^ a^ 
 
 15. A uniform rectangular plate is rotating with con- 
 stant velocity about an axis in its plane through the centre. 
 If the centre be set free and the plate keep rotating about 
 the same axis, which is that axis ? 
 
 16. A grindstone of mass m and radius a is rotating 
 with velocity co, and is stopped at a constant rate in t seconds. 
 What is the rate of loss of ansfular momentum ? 
 
VII. 
 
 MOMENTS AND PRODUCTS OF INERTIA. 
 
 1. When the moments and products have been found 
 about any axes through the centre of mass of a body, we can 
 find them at once about any other parallel axes by the theo- 
 rem of Lesson iv. Art. 2. 
 
 The following is a geometrical proof that the moment of 
 inertia about any axis is equal to that about a parallel axis 
 through the centre of mass, together with the moment of 
 inertia of the whole mass at the centre of mass about the 
 former axis. 
 
 Let Ax and Gx' be parallel axes, the latter through the 
 centre of mass. And let F denote any element of the body. 
 
 ivr 
 
 Let a plane through P at right angles to the axes cut 
 them in iVand 31; join these and draw FL perpendicular to 
 
 NM. 
 
70 PARALLEL AXES. 
 
 Tlie moment of inertia about Ax is tBm PN\ Now by 
 Euc. II. 12, 
 
 FN^ = PM' + MN'' + 2 JfiY. ML, 
 
 Hence 23m FN^ = tSm FM' + m . il/^« 
 
 ■{■^MN.XhmML, 
 
 since il/jV is the same for all elements, being the distance be- 
 tween the parallel axes. 
 
 Also 23m ML is zero, since Gx passes through the 
 centre of mass ; 
 
 .-. ^^mFN'=l.^mFM'^m.MN\ 
 
 2. If then h is the radius of gyration about an axis 
 through the centre of mass, and a the distance of a parallel 
 axis, the moment of inertia about this latter is m {k^ + a^). 
 Call this A. If there be another axis in the same plane at a 
 distance h from the centre of mass, the moment of inertia 
 about it is m (Jc" + 6'). Call this B. Then 
 
 A — ma^ = B — ml/, 
 
 and thus knowing the positions of two parallel axes relatively 
 to the centre of mass, and the moment of inertia about either, 
 we can find that about the other. 
 
 3. If a, h, c are the co-ordinates of the centre of mass, and 
 f , 7j, f the co-ordinates of any element 3m relatively to it, 
 
 23m xy = 23m f 7 -f 7nah, 
 
 and SBmxz — XSm ^^+ mac. 
 
 If the centre of mass lies on the axis of x, 
 5 = 0, c = ; .*. 23??i xi/ = 23m ^rj, and 23??i xz = 23??z ^f. 
 If this line is a principal axis at the centre of mass, 
 XSm ^rj = 0, and 23m f f = ; 
 therefore 23m X9/ = 0, and 23m xz = 0, 
 
 or it is a principal axis at every point in its length. 
 
EXAMPLES. 71 
 
 4, The moment of inertia of a sphere of radius a about 
 
 2 
 a diameter has been found to be ^ m . c^, 
 
 5 
 
 Hence about a tangent it is 
 
 2 7 
 
 o o 
 
 5. To find the product of inertia of a rectangular plate 
 about two sides. 
 
 The straight lines drawn through the centre parallel to 
 the sides are clearly principal axes. Hence the required pro- 
 duct is equal to the product of inertia of the whole mass 
 collected at the centre about the sides. It is the product of 
 the mass and one-fourth the area of the rectangle. 
 
 EXAMPLES. 
 
 1, The radius of gyration of a circular arc about a 
 line through its middle point at right angles to its plane is 
 radius x V2. 
 
 2. That of a circular plate about a tangent is 
 
 1 
 
 radius x ^ , . 
 
 3. Find the moment of inertia of a right cone about a 
 line through the vertex perpendicular to the axis, 
 
 4. The moment of inertia of a parallelogram whose sides 
 are 2a and 26 in length about an axis through the centre at 
 
 right angles to its plane is ^ {c^-k-Jf), 
 
 5. What are the principal axes at a corner of a square ? 
 
EXAMPLES. 
 
 6. Prove that for an elliptic area about a line through a 
 focus parallel to the minor axis 
 
 P = : . 
 
 7. Prove that for a circular arc of radius r, which sub- 
 tends an angle 2a at the centre about an axis through its 
 centre of gravity perpendicular to its plane. 
 
 \ ay 
 
VIII. 
 
 MOMENTS AND PRODUCTS OF INERTIA. 
 
 1. Having now found the moments and products of in- 
 ertia of a body about all axes parallel to a given set about 
 which they are known, we proceed to complete the subject by 
 finding them about sets of axes inclined to these. We will 
 take first the case of axes in one plane. 
 
 2. At any point of a given rigid lody and in any plane 
 there are akuays two axes at right angles, such that the pro- 
 duct of inertia about them is zero. 
 
 Let be the point, Ox, Oy any perpendicular axes in the 
 plane. Let the moments of inertia about Ox, Oy be denoted 
 by a, h, and the product SSm xy by /. Let Ox, Oy be an- 
 other set of axes in the plane inclined to the others at an 
 angle a. i. e. z xOx = i yOy = a. 
 
 ' ' We will endeavour to choose a so that ^hn xy — 0. 
 
 If the polar co-ordinates of Bm be r, 6, 
 
 x = r cos Q, y — r sin Q, 
 
 X =r cos (0 — a), y =r sin {6 — ol), 
 
 XBmxy=ltBmr'sm2(^d-a) 
 
 — cos 2a 25/71 r^ sin cos 
 
 - ^^^5;am r^ (cos^ ^ - sin^ ^). 
 
74 MOMENTS OF INERTIA. 
 
 If this vanishes 
 
 ^ _ 2 SB m r" sine cos e 
 tan Za - ^^^^ ^^, cos^ l9"-7Qn*^ 6) 
 
 2 IB 
 
 tn xif 
 
 If the distance of Bm from the plane xOy be z, 
 a = ^Bm {if 4- 2^), 
 b=^tBm(z'-\-x''), 
 
 2f 
 
 and tan 2a =-r^^^ — . 
 
 b — a 
 
 As this equation has always a solution, there is always a 
 set of axes having the required property. 
 
 3. Farther, for each axis of that set, the moment of inertia 
 is a maximum or minimum. 
 
 For let a be variable and find the value for which 
 
 2Sm(y^ + s^), or 'XBmy'^ is greatest or least, 
 
 ^Bm y"" = tBm r' sin'^ {0-ol). 
 
 Differentiating with respect to a ; 
 
 2tBm / . sin (6> - a) cos (l9 - a) = 0, 
 
 but we have seen that this is equivalent to 
 
 %Bm xy — 0. 
 
 4. Farther, the moment of inertia about any line Ox' in- 
 clined at an angle a to Ox {5 
 
 a cos^ a + 6 sin^ 0L — 2f sin a cos a. 
 
 YoxXBm{y"-\-z') 
 
 = XBm r' sin' (<9 - a) + XBm z" 
 
 = cos'^ a XBm r^ sin'^ 0—2 cos a sin a SSm r' sin 6 cos ^ 
 
 + sin* a %Bm r^ cos" ^ + (cos^ a + sin^ a) tBm ^ 
 
 = cos'' a tBm (/ + ^') + sin* a SSwi (a;' + s*) 
 
 — 2 cos a sin a ^Bni xy 
 
 — a cos* a + Z* sin* a — 2/ sin a cos a. 
 
INCLINED AXES. 75 
 
 If y = 0, or if Ox, Oy are principal axes and the moments 
 of inertia about them are A and B, this becomes 
 
 A cos^ 0L + B sin^ a. 
 
 5. The moment of inertia of a rectangular plate about a 
 diagonal. 
 
 If the lengths of the sides are 2a, 2h, the moments of inertia 
 about two lines through the centre parallel to the sides are 
 
 ' 
 
 
 b' a'' 
 
 And 
 
 
 h 
 tan a = - , 
 a 
 
 whence 
 
 
 
 
 ^°" ^- a'-vh'' 
 
 therefore the moment 
 
 required is 
 
 
 
 a' a' h^ 
 2m a%^ 
 
 = — r^ J^ . 
 
 3 o:' + F' 
 
 6. To find the position of the principal axes of a uni- 
 form rectayigidar plate ABCD at the point A. 
 
 The moment of inertia about AB is m — ^r— , that about 
 
 AD w> m — ^ . 
 
 The product about AB, ADis'^^.AB. AD, 
 
7G 
 
 MOMENTS OF INERTIA. 
 
 Hence, substituting in Art. % if a be the angle wbidi one 
 of the required axes makes with AB, 
 
 tan 2a = 
 
 %,AB,AJ) 
 
 Li 
 
 AB"" AD' 
 
 _ S AB.AD 
 ~2AB'-AB'' 
 
 7. There are at every j^oint of a rigid body three axes 
 at right angles to one another, for which the products of inertia 
 vanish. 
 
 We might apply the same method as before, but an 
 indirect method is here more simple. 
 
 Given the moments of inertia a, h, c about three axes 
 Ox, Oy, O^at right angles; and the products d,e,f\ [^hnyz, 
 l^Sni zx, '%hn xy respectively) let us find the moment of inertia 
 about a line Ox inclined at angles a, /?, 7 to Ox, Oy, Oz. 
 
 Let OL, LM, MP be the co-ordinates x, y, z of an ele- 
 
. INCLINED AXES. 77 
 
 ment of mass Sni at a point P. From Plet fall P^ perpen- 
 dicular to Ox\ 
 
 Projecting 0L3IP on Oi/, 
 
 OJS^ = OX cos a + />i/ cos y5 + 2IF cos 7, 
 and FN' == OP' -OF'-, 
 
 /. PjV'^ = ix^^ + y + £^ — (:c cos a + ?/ cos /3 + 2 cos y)'^ 
 
 = x\l- cos' a) + ?/' (1 - cos' ^) + 5' (1 - cos' 7) 
 — 2j/z cos /3 cos 7 — 22X cos 7 cos a — 2;cy cos a cos /?. 
 But cos' a + cos^ /3 + cos' 7 = 1; 
 
 .-. FN' = ^' (cos' /3 + cos' 7) + 7/ (cos' 7 + cos' a) 
 + ;s' (cos' a + cos""/8) 
 — 2j/;<; cos /8 cos 7 — 2zx cos 7 cos a — 2x2/ cos a cos l3 
 = cos' a (?/' + s') + cos'/3 (^' + x') + cos' 7 (x' -f- ?/') 
 ~ 2^^ cos l3 cos 7 — 2zx cos 7 cos a — 2x2/ cos a cos /3 ; 
 .% SS//i PiV = a cos' a + Z> cos' /3 + c cos' 7 
 — 2(f cos /3 cos 7 — 2e cos 7 cos a — 2y cos a cos /5. 
 To represent this geometrically take a point Q on ON. 
 Let its distance from be p^ and its co-ordinates ^, rj, f ; 
 then f = p cos a, 7j = pcos jS, ^ = p cos 7, 
 
 and 
 
 S8..P^-==-g^-^^-^'^-^-^-f^^-^^^^-^^^^ 
 
 Now the equation 
 
 af + hrf + or - 2c^;?r- 2erf - 2f^rj = 1 
 
 denotes an ellipsoid whose centre is at ; for a, h^ c are 
 necessarily positive. If then Q is a point on this 
 
 tBmFN' = \, 
 
 P 1 
 
78 MOMENTAL ELLIPSOID. 
 
 or the moment of inertia about any line through 0, is mea- 
 sured by the square of the reciprocal of the radius vector of 
 this ellipsoid which coincides with the line. 
 
 This is called the momental ellipsoid. It has no physical 
 existence, but is an artifice to bring under the methods of 
 geometry the properties of moments of inertia. The mo- 
 mental ellipsoid has a definite form for every point of a 
 rigid body. 
 
 If this ellipsoid be referred to another set of axes, and its 
 equation become 
 
 a'P + 6V + cV - ^d'v^- 2e'r? - 2/f77 = 1, 
 
 the coefficients a\ b', c will be the moments of inertia about 
 the new axes, and d', e, f will be the products. 
 
 Now every ellipsoid has three axes, to which if it is 
 referred its equation takes the form. 
 
 With respect to these axes, the products of inertia vanish. 
 
 8. Hence we see that the moment of inertia about one of 
 the principal axes is the greatest, and about another the 
 least possible. It was from this property that Euler, who 
 first thoroughly investigated the subject, gave them the 
 name. 
 
 It is now clear, that for all questions depending only on 
 moments and products of inertia, any body may be' replaced 
 by its momental ellipsoid. And farther, that any two 
 systems which have the sfime momental ellipsoid at a point, 
 are about that point kinetically identical. 
 
 If the moments of inertia of a body about three axes at 
 right angles through a point are equal, the ellipsoid becomes 
 a sphere. They are therefore equal about all axes, and 
 every axis is a principal axis. The body is then said to be 
 kinetically symmetrical with respect to that point. Thus a 
 cube is kinetically synnnetrical about its centre. 
 
KINETIC CENTRE. 79 
 
 • The following question is of some interest. 
 
 9. Under what circumstances is there a j)oint in a hody 
 such that the moments of inertia about all axes through it are 
 equal ? 
 
 If there is such a point, all sets of axes through it are 
 principal axes. 
 
 Let the co-ordinates of the point referred to the principal 
 axes at the centre of mass be a, h, c. Then the products of 
 inertia of the body about the parallel axes through the 
 point are 
 
 m .he, m . ca, m . ah^ 
 
 for those about the axes through the centre of mass are 
 zero. 
 
 If all axes at the point are to be principal axes, these 
 must be so ; 
 
 .'. he = 0, ca— 0, ah = 0, 
 equations which require that two of a, h, c should be zero. 
 
 Let & = 0, c = 0, then the point required lies on the axis 
 of X, — one of the set of principal axes at the centre of mass. 
 
 But further, it is necessary that the moments of inertia 
 about these axes should be equal. Let A, B, G be the 
 moments of inertia about the axes through the centre of 
 mass. Then those about the parallel axes through the point 
 required are 
 
 A, B+ ma^, C + ma^ 
 
 If thfese are to be equal, we have 
 
 B=G and A-B = ma\ 
 
 Hence our condition is, that two of the principal mo- 
 ments at the centre of mass should be equal. In other 
 words, the momental ellipsoid at the centre of mass must be 
 a spheroid. And then the point lies on the unequal axis at 
 a distance from the centre equal to 
 
80 
 
 EXAMPLES. 
 
 1. Given A, B the moments of inertia of a body about 
 two principal axes Ox, Oy, prove that the product of inertia 
 about axes Ox , Oy in the same plane, inclined to the 
 former set at an angle a, is 
 
 2. Prove that any two of the principal moments of 
 inertia are together greater than the third. 
 
 3. No ellipsoid except a sphere can be its own mo- 
 mental ellipsoid at its centre. 
 
 4. Every elliptic plate is similar to the section of its 
 momental ellipsoid made by its own plane. 
 
 5. If a, h, c are the semiaxes of the momental ellipsoid 
 of a rigid body in order of magnitude, shew that 
 
 ah 
 
 c is greater than 
 
 Va-' + Z/^ 
 
 6. Given the angular velocity of a body which is ro- 
 tating about a fixed point; about what axis must it be 
 rotating, so that the angular momentum shall be greatest 1 
 
 7. Two systems of equal mass have the same principal 
 axes, and the same moments of inertia about them at 
 some one point; prove that they have the same principal 
 axes at any point, and the same moments of inertia about 
 any axis. 
 
 8. Prove that there can always be foimd three points, 
 one on each of the three principal axes of any system at any 
 point, such that the momenta and products of inertia of 
 
EXAMPLES. 81 
 
 three suitable equal masses collected at them, are equal to 
 the moments and products of inertia of the system about 
 any axes whatever through that point. 
 
 9. If these equal masses be each one-third of the mass 
 of the system {m), shew that the distances of the three points 
 along the axes are 
 
 (3 (B+C-A) [^ r3 {Cj-A-BY i^ j 3 (A -{- B - C) ]^^ 
 [2 m J ' [2 m } ' -[2 m J * 
 
 10. In a triangular plate ABC, D is the middle point 
 of BG, and E the foot of the perpendicular let fall from A on 
 BG. Shew that the middle point of DE is the point at 
 which i?(7 is a principal axis. 
 
 11. Shew that the difference of the moments of inertia 
 of a body round two axes in a given plane which are equally 
 inclined to a fixed line in the same plane, is proportional to 
 the sine of the ande between those axes. 
 
 P. G. 
 
IX. 
 
 €ASES OF MOTION WITHOUT EOTATIOX. 
 
 1. The complete solution of a problem of motion would 
 involve the finding of the position of the system at a given 
 time, of the velocities at a given time or in a given position, 
 and of the values of any previously unknown forces, such as 
 pressures or frictions which may act on the system. If the 
 forces are impulsive, only velocities and forces can be re- 
 quired ; for the position is unaltered during the impulse, and 
 to follow the subsequent changes belongs to a separate pro- 
 blem of the other kind. Questions of impulsive motion can 
 then always be solved, for the changes of velocity and the 
 forces appear as unknown quantities in equations which are 
 in general simple algebraical equations. But if the forces are 
 of the kind called finite, the equations of motion are differen- 
 tial equations of the second order as regards co-ordinates of 
 position. In some simple cases these can, be completely 
 solved and the requirements of the above solution satisfied ; 
 but in more complicated cases we can get no farther than a 
 first integral, that is, an algebraical equation giving the velo- 
 cities. In such a case to find the position at a given time is 
 impossible. Our demands must be limited by what we can 
 get; and the words "to find the motion" have come to 
 mean, "to find the velocities of the system in any position." 
 
 We will therefore in general use velocities and their first 
 differential coefficients in the expressions for the effective 
 forces; but if, in any case, the co-ordinates of position must 
 ■enter, we can use their first and second differential coefii- 
 -oients to express velocities and accelerations. 
 
NATURE OF PEOBLEMS. 80 
 
 The imknown external forces can nsually be found, for 
 they depend on accelerations and velocities which by the 
 solution have been made to depend on the co-ordinates of 
 position. 
 
 An important class of problems has to do with finding 
 the stresses or internal forces at a given point of a body. 
 The action of one part of a body on another is threefold. It 
 may consist of a longitudinal stress, normal to the plane of 
 separation, a transverse or shearing stress, tangential to the 
 plane of separation, and a bending or breaking couple. These 
 are found by considering them as forces external to one part 
 of the body. As in the case of the external pressures, the 
 motion of the whole must first have been investigated. 
 
 The first difficulty of a problem in Eigid Dynamics is 
 overcome when by the reversing of the efiective forces the 
 whole is reduced to a system in equilibrium. Any method 
 which is available to find the forces or the position of equili- 
 brium of a system is equally available here to find the un- 
 known forces or the state of motion of a system. 
 
 In a problem of any complexity it is an assistance to 
 draw two diagrams ; one kinematical, representing the 
 changes in the velocities or the accelerations; and the other 
 dynamical, representing the resultant reversed effective forces, 
 and the impressed forces. This is done in Art. 4 of the pre- 
 sent Lesson. 
 
 2. A uniform rod AB of mass m and length 2a is let 
 fall in a horizontal position. After falling through a height 
 h it is brought to rest by its ends striking two fixed sup- 
 ports at the same level. What will he the breaking couple 
 at a point P ? 
 
 The stress couple or, as we may call it, the bending or 
 breaking couple in a system in equilibrium is equal and 
 opposite' to the moment of all the other forces acting on 
 either of the parts of the body which are separated at the 
 point imder consideration. 
 
 The rod in falling through h acquires a velocity ^l2gh. 
 
 G— 2 
 
84f PROBLEMS. 
 
 This is stopped suddenly by two equal pressures. Heiiee each 
 
 . 711 \/'2(j/l 
 
 01 these is — ^ . 
 
 Consider the part AP. It is in equilibrium under the 
 
 012/ r ■ 
 
 jiction of BP, the stopping blow -~ \l:Lgh acting upwards at A, 
 
 and the reversed effective force acting at the middle point of 
 AP. This is the force which has changed the momentum 
 from mass AP x \l''lgh to zero. It is therefore 
 
 — m -^ . ^'Igh, acting downwards. 
 
 j[p 
 
 Reversed it becomes on — — . x ^/2qh, actin^^ downwards. 
 AB ^ ^ 
 
 The action of BP on AP may be composed of a force and 
 a couple. Taking moments about P we shall avoid the force, 
 
 .111 i-T~T A-ry ^P /^V~r ^P 
 
 and the couple = — . \/:igli . AP - m --r . V Zgli . — ^ 
 
 m ,-—^ AP.PB 
 2 ^ Ah 
 
 8. A siraiglit rod AB of mass m hangs from a fiived 
 point O bg an elastic string {natural length, a, modulus of 
 elasticity A.), n.vhich is fastened to the end A. It is pulled 
 down and then let go. Find the longitudinal stress at any 
 point P, ivhen the j^oint A is at a distance x heloiu O. 
 
 The part BP is in equilibrium under the stress of AP 
 upwards, its own weight downwards and the revei'sed effec- 
 tive force at its centre of inertia. To know the effective 
 force we must first know the acceleration. Thus we must 
 first consider the whole rod AB. 
 
 This is in equilibrium under the tension, its own weight 
 and its own resultant effective force reversed. 
 
 Let the downward velocity of the centre of inertia be v. 
 
NO ROTATIOX. 85 
 
 The effective force is then m -r- or mv -r- actin^^ down- 
 
 dt ax ° 
 
 wards. Then the forces mg downwards and niv j- and T 
 
 (the tension) acting upwards are in equilibrium ; 
 
 dv rrr , 1 ^ ^ ^ — <^ 
 
 ,*. mv -1- =— I +mq, and 1 =X : 
 
 dx ^ a 
 
 dv ^ x — a 
 
 .'. mv ~j- = — \ h mq. 
 
 dx a '^ 
 
 Were the velocity required we should integrate this. As 
 it is we can return to the consideration of BP. The accele- 
 ration of every point on the rod is the same. Hence the 
 effective force on BP is 
 
 mass BP x v . -r- downwards, 
 dx 
 
 BP dv ^. , , 
 
 or m . -T-h • '^ ~r actmg downwards. 
 
 BP is in equilibrium under this reversed and the stress, 
 
 and m . -r^ . q acting^ downwards. 
 AB ^ ^ 
 
 Hence the stress is 
 
 BP BP dv 
 
 ''''AB^-'''AB''dx' 
 
 but onv -r- has been found above. Substitutino- • 
 dx 
 
 ^ p ^ x-a BP 
 
 stress at P = \. . —rj^ . 
 
 a AB 
 
 4. A wedge B whose angle is a and whose faces are 
 smooth, rests with one of them in contact luith a horizontal 
 table. A rigid body A with a pla7ie face is placed on the 
 other with the plane face in contact luith it. What luill he the 
 velocity of each at any subsequent time ? 
 
86 PROBLEMS. 
 
 Taking both bodies as one system, the impressed forces 
 are all vertical. Hence the common centre of inertia will 
 have no horizontal velocity; or — what is equivalent — there 
 will be no horizontal momentum of the system. 
 
 If u be the velocity of the wedge, and v be that of A 
 relatively to the wedge, i.e. down the incline, the horizontal 
 velocitv of ^ is u — v cos a. 
 
 Hence A {u — v cos a) + Bu = . .. . 
 
 Next consider A, Its effective forces are 
 
 (1). 
 
 du 
 
 dv 
 
 A -77 horizontally, A -y- down the incline, 
 
 dt 
 
 dt 
 
 and its impressed forces are its Aveight and the pressure 
 which acts at right angles to the incline. Therefore re- 
 versing the effectives and resolving along the incline, 
 
 at 
 
 A -J- cos a = Ag . sin a (2] 
 
 A-i^ /A; 
 
 These two equations suffice to determine u and v at any 
 time. 
 
 The investigation may bo continued by finding the pres- 
 sure between B and A, also by finding the actual positions 
 of A and B in terms of the time. 
 
87 
 
 EXAMPLES. 
 
 1. A railway train is going along a level with constant 
 velocity. The friction of the rails is for each carriage one- 
 hundredth part of the pressure. What is the tension of the 
 couplings of the last carriage if its mass is 2 tons ? 
 
 2. Find this tension ; supposing the mass of the carriage 
 to be m, the coefficient of friction ^la, and the train to be 
 running down an incline of one in h, with an acceleration /3. 
 
 3. Prove that the transverse stress at P, in the system 
 of Art 2, is 
 
 -.J'lgh -j^ . 
 
 4. A man is placed on a long boat which rests on the 
 surface of still water. Shew that if he could walk with 
 absolutely constant velocity along it, there would be no 
 horizontal force between his feet and the boat except when 
 starting and stopping. 
 
 5. A system consisting of two uniform rods A C, CB, 
 rigidly connected at right angles at C, falls without rotation 
 in°a vertical plane, and strikes a smooth horizontal plane at 
 B ; if there is no rotation produced by the impact, shew that 
 the inclination of BG to the horizon is 
 
 , _,BC{2.AC+BC) 
 tan -jjf, . 
 
 Find in that case the in^pulsive breaking couple at C. 
 
 6. Two small balls each of mass m, are placed at the 
 ends of a diameter within a circular tube of mass 4^m lymg 
 on a smooth horizontal table, and the balls are connected by 
 elastic strings within the tube which are stretched to twice 
 their natural length. One of the strings suddenly breaks. 
 Prove that when the other resumes its natural length, the 
 
88 EXAMPXES. 
 
 centre of the tube is moving with a velocity (-^^Y, where a 
 
 denotes the radius of the circle, and e the original tension of 
 the string. 
 
 Find all the circumstances of the problem when the balls 
 meet. 
 
 7. A wedge B, whose angle is ^, is laid on an inclined 
 plane whose inclination is a + /3, with its edge toward the 
 upper part of the plane. The wedge being at rest on the 
 l^lane, a body M is projected along its upper surface from 
 the base with a velocity due to a height a tan /S, where a is 
 the length of the upper side of the wedge ; the coefficient of 
 friction for either surface of the wedge is tan jS, and the 
 motion takes place in one vertical plane. Shew that during 
 the motion of the body on the wedge, the wedge will not 
 slip provided the ratio of M to B is greater than that of 
 tan a to tan /5. 
 
 Shew also that the body comes to rest before reach ino- 
 the edge of the wedge, and that the wedge will immediately 
 begin to slide down the incline, but that the body will not 
 .slide on the wed ere. 
 
X. 
 
 FIXED CENTRES. 
 
 1. It will be remembered that the effective forces on a 
 rigid body of mass m, whose centre of inertia is (7, rotating 
 about a fixed axis 0, and having a radius of gyration h about 
 an axis through G parallel to the fixed axis, were reduced, if 
 the change of motion was sudden, to a force mOG [m - «) 
 acting at G at right angles to G, and a couple mlc^ («' - «), 
 or, which is equivalent, to a force m 0G{(o' — (o) at and a 
 couple m (Jc^ + OG'^) (ft)' — ft)) ; if the motion was accelerated 
 they were reduced to forces at G,mOGo)^^ along GO and 
 
 mOG~p at right angles to GO, and to a couple '^^^-j^ > or, 
 
 what is the same thing, to a force at G, mOGw^ along (rO, 
 
 a force mOG -y- at 0, and a couple m ik'^ -i-OG^) -j,. 
 
 In our diagrams we will represent couples by arcs of 
 circles. The positions of these may be any whatever, for the 
 effect of a couple is absolute, mic' will always be employed 
 to denote the moment of inertia about the centre of inertia. 
 
 2. The problem of determining the law under which 
 a heavy body swings about a horizontal axis is one of the 
 most important in the history of science. 
 
 A simple pendulum is a thing of theory ; our accurate 
 knowledge of the acceleration of gravity depends therefore 
 on our understanding the rigid (or compound) pendulum. 
 We have seen that it was the first problem to wdiich D'Alem- 
 bert applied his principle. 
 
90 
 
 PROBLEMS. 
 
 The name of the problem in those days was the ^centre 
 of oscillation.' It was required to find if there were a point 
 at which the whole mass of the body might be concentrated, 
 so as to form a simple pendulum whose law of oscillation 
 was the same. 
 
 Let the plane of the paper be that in which G the centre 
 of inertia swings. Let be the intersection of the axis with 
 that plane. Take G as the line in the pendulum by which 
 its rotation is measured. Let 6 be the angle which OG 
 makes with the vertical. Let k be the radius of gyration 
 about an axis through G parallel to the fixed axis ; let OG 
 be called h, and let m be the mass of the body. 
 
 771 Q» 
 
 The only impressed force which has a moment about is 
 mfj, the weight acting at G. The anovular acceleration is 
 
 ./ , and the rate of mcreasc of the angular momentum 
 
 (It 
 
 (T-e 
 
 m (/j^ + 7r) is the measure of the effective couple. 
 
COMPOUND PENDULUM. 91 
 
 Reversing this, it must be equal and opposite to the mo- 
 ment of mg round 0. 
 
 Therefore m (F + ¥) -^ + mgh sin ^ = 0, 
 
 d^O ah . Q cw 
 
 W'=-vvl?-''''^ ^''- 
 
 Multiplying hy 2 -7- and integrating 
 
 If the pendulum began to move when 6 was equal to a. 
 It + /J 
 
 ^^"'i fiy=/ii('^°^^-'^°'"^ ^'^- 
 
 This equation cannot in general be integrated farther. 
 It is therefore not possible to- find the position in terms of the 
 time. Equation (2) enables us, what is very important, to 
 find how far the pendulum will go with a given initial angu- 
 lar velocity ; for a gives the position of instantaneous rest. 
 If then the angular velocity is O when 6 is zero, 
 
 If a = .r, O - ' ^'' 
 
 kr^lt" 
 
 this is the least angular velocity at the lowest point which 
 will send the body right round. 
 
 If VL is less than this, the motion will be one of oscillation 
 about the lowest position. 
 
o'^ COMPOUND PENDULUM. 
 
 3. The most important case is when the angle of oscillation 
 is very small. Then — ^— differs infinitely little from unity, 
 and the equation (1) becomes 
 
 cr0_ gh 
 
 df A•^ + ^•'' 
 
 The solution of this (Lesson XV.) is 
 
 and indicates an oscillation, called a simple harmonic 
 motion, whose complete period is lir a/ — -.— . 
 
 If I be the length of a simple pendulum its equation of 
 motion is 
 
 df I • ^' 
 If the times of oscillation of these are equal, 
 , ¥ + 1=' , // 
 
 If a length 00' equal to this be measured along OG, 0' 
 is the centre of oscillation. It is clear that OG. 0'G=]c\ 
 Hence and 0' are convertible. If the pendulum be hung 
 up by 0', will be the centre of oscillation. 
 
 In the above work the assumption has been made tliat 
 the solution of 
 
APPLICATIONS. 93 
 
 differs infinitely little from that of 
 
 when 6 becomes infinitely small. 
 
 The time of oscillation of a compound pendulum depends 
 on li -{- Y' Ii^ calculating the value of g from pendulum 
 
 experiments, the main advantage is that the time of ope 
 oscillation can be very accurately measured. The difficulties 
 are the determination of h and k. The point G cannot be 
 got at, and as every body is more or less irregular and 
 variable in density, k cannot be calculated with sufficient 
 accuracy. These quantities must therefore be determined 
 from experiments. Bessel observed the times of oscillation 
 about different axes, the distances between which were very 
 accurately known. Captain Kater employed the property of 
 convertibility. 
 
 4. Another interesting application of the present problem 
 is the old way of measuring the velocity of a bullet or cannon- 
 ball. The ball was fired into a mass called Eobins' ballistic 
 pendulum ; which was thereby set off with a certain angular 
 velocity about its axis. The angle through which the mass 
 ascended was found by the length of a piece of tape which 
 w^as fastened to a point in the pendulum, and came through 
 a slit immediately below the axis. 
 
 Hence by equation (2) the initial angular velocity was 
 calculated. But, the mass of the cannon-ball being m, its 
 velocity F, the common initial angular velocity of it and the 
 mass (if) o), the distance of its passage below the axis p, the 
 angular momentum of the bullet about the axis before im- 
 pact was m Vp, and after it sticks in the mass, mp^co. Now 
 the angular momentum gained by the mass must be equal to 
 the moment of the momentum about the axis lost by the 
 bullet. For, considering the two as parts of one system,^ there 
 is no external force that has any moment round the axis. 
 
 Hence mVp- mp'^co = M {k^ + h^)o). 
 
 From this V can be found. 
 
94 CENTRE OF PERCUSSION. 
 
 Hutton used to suspend his cannon as a pendulum, and 
 measure the angle through which it was raised by the dis- 
 charge. 
 
 5. A plane hodi/ at rest has a fixed point in it. It is 
 struck hy a bloiv in its oiun plane. Hoiu must this act that 
 there may he no pressure on the fixed p)oint ? 
 
 Let be the fixed point, G the centre of inertia, mJc'^ the 
 moment of inertia about G, co the angular velocity produced, 
 P the blow, p the distance of its line of action from 0. 
 
 The velocity of G is changed from zero to OG . w, which 
 requires a force ')nOG . « at G. And the angular momentum 
 about G has been changed from zero to mFo), which requires 
 a corresponding couple. 
 
 If there is no action on the fixed point, P is in equilibrium 
 with a force — m . OG . co, and the couple — mk'-co. Hence P 
 must act at right angles to G, and 
 
 P = vi.OG.(o, P (p - OG) = mJrco, 
 
 yr.2 
 
 wdience p= 0G+ -^-^ . 
 
 The name 'centre of percussion' has been given to the 
 point of action of P when there is no pressure on the 
 axis. If there be no fixed j^oint and if the blow act at 
 this point, the point will be the centre of siDontaneous 
 rotation. 
 
 G. A body is moveable about a fixed j^oint 0. It begins 
 
 to move luith given angular acceleration t: about the line Oz. 
 
 What cojiple has p)roduced this motion "^ 
 
 Let the magnitude of the couple be G, and let its axis 
 make angles a, (3, y with the axes Ox, Oy, Oz, 
 
 The impressed forces are the couple G, and the action of 
 0. We will take moments about axes through 0, and so 
 avoid this latter. 
 
EFFECT OF A COUPLE. 95 
 
 If X, y, z are the co-ordinates of an element Zm, and r its 
 distance from Oz ; its accelerations are o)^ . r towards Oz and 
 
 — r parallel to the plane xy, and perpendicular to r. Now 
 at 
 
 in the very beginning of the motion w is infinitely small, but 
 
 — is finite. Therefore the latter is the only acceleration 
 dt 
 
 that requires an effective force. This effective force 
 
 <^^^ ^ r is equivalent to tvro, viz. ^ ^"^ -J. V parallel to Ox, 
 
 and dm -^ x parallel to Oy. (See fig. Y, 5.) 
 at 
 
 Keversing these for all the elements, and taking moments 
 about the axes, 
 
 G.cosa jT. Xhnxz = 0, 
 
 dt 
 
 G cos/3 J, ^^>ny^ = ^\ 
 
 6^ cos 7 Yj. SSm?'^ = 0. 
 
 dt 
 
 Whence the cosines of the angles which the axis of the re- 
 quired couple makes with the axes are proportional to the 
 products and moment of inertia e, d^ c (Lesson viii.) ; and 
 the moment of the couple is 
 
 The student of solid geometry will have no difficulty in 
 proving that this axis is the radius of the momental ellipsoid 
 of the body at 0, diametral to the plane of the couple. 
 
 Similar reasoning holds in the case of an impulsive 
 couple. So that in general a couple will tend to cause 
 rotation round its own axis only when this is parallel to one 
 of a set of principal axes. 
 
96 
 
 FIXED AXES. 
 
 7. A uniform triangular j^lctte ACB right-angled at C, 
 V.y rotating about CA as a fixed liorizontal axis. Find the 
 ivrench couple in the vertical j^lctne through the axis when the 
 2)late comes to he horizontal in its descent. 
 
 ■rrifj - m a}IS Q- 
 
 Let tlie angular velocity in that position be &). Let G 
 be the centre uf gravity, LtN a perpendicular to CA. Let 
 m be the mass, and let a and h be the lenoths of CB, CA. 
 
 The moment of inertia about CA is, by Lesson VJ, m 
 
 Hence the value of the angular acceleration in this 
 
 position is, by Art. 2, 
 
 2.7 
 
 The impressed forces are the Aveight and the actions of 
 the axis. We are only concerned Avith that one of them 
 which is a couple in the plane VGA, Call it L. The effec- 
 tive forces are mw^ KG along ON at G, and m - NG ver- 
 
 at 
 
 tically downwards at G, of which only the latter has a 
 
COMPONENTS OF STRESS. 97 
 
 moment about BC) and a couple round an axis through G 
 parallel to GA, which has no moment about BC, 
 
 Reversing these and taking moments about CB, 
 L + mg . Cy- m -NG"^^^. CN =^ 0, 
 
 and CN= ^ , 
 
 o 
 
 whence X = ~- . 
 
 8. A uniform rod OA, of mass m and length 2a, sivings 
 as a pendidum about O. Find the components of stress at a 
 point P distant 2b from A, when the rod has reached a position 
 inclined at an angle to the vertical. 
 
 We know from Art. 2 that the angular acceleration -^ 
 
 is equal to 
 
 % sin 
 
 ' 4<a ' 
 
 and that the square of the angular velocity f -%- j is 
 
 7,*^ (cos 6 — cos a), 
 2a ^ ■ ^ 
 
 in which a is the value of 6 when the velocity is exhausted. 
 Let us consider the part FA, The impressed forces are 
 
 m-.g acting at G, its centre of inertia ; and the required 
 
 action of OF, which we will take to be (1) a longitudinal 
 stress T along F 0, (2) a transverse stress S at right angles 
 to FA ; and a couple L. 
 
 -77) aloncr GO, and 
 
 ■©■ 
 
 G . — , 2 ^t right angles to G 0. Hence the reversed effec- 
 tive forces are 
 
 mli2a-i)(§)' (1), 
 
 4tJ 
 
 p. G. 
 
98 
 
 FIXED AXES. 
 
 and 
 
 
 (2), 
 
 aciin«- at G along and perpendicular to 0G\ and a couple 
 
 1) ¥ ere 
 
 m 
 
 a ' 3 df ' 
 
 mbg ^ 
 
 Resolving for FA along and perpendicular to FA, 
 /_,H_^cose--^-(2a-&)y =0. 
 
 S-m^g.me- ^ (2a - i) ^^, = 0. 
 Taking moments about F, 
 
 L-m -asm 6--.-- , .^ (2a -I) ,., = 0. 
 
 a*^ 'Sa df a ^ ^ 
 
 'Sa df 
 Substitutincr the values of 
 
 dC 
 
 d'd - fd6^' 
 -, and - 
 
 df 
 
 lit 
 
COMPONENTS OF STEESS. 99 
 
 we have 
 
 T= — - \co^d ■\- -^ t:: (cos 6— COS a) \ , 
 
 a { a 2^ ^ 
 
 ^ ml) . .{. i2a-h)S 
 S= — ^sm ^^1 — 
 
 a [ 4ia 
 
 mh . ^fSb \ 
 
 EXAMPLES. 
 
 1. Explain why it is easier to support a long rod in a 
 vertical position on the tip of the finger than a short one. 
 
 2. A uniform rod of length 2a can rotate about one end. 
 It is allowed to fall from its position of unstable equilibrium. 
 Prove that its angular velocity when it is horizontal is 
 
 V 2a 
 
 3. A bar magnet is suspended in such a manner that it 
 can oscillate in a horizontal plane. Shew that its moment 
 of inertia about the line of suspension can be determined by 
 observing the times of oscillation with and without small 
 pieces of lead attached to it. 
 
 4. A perfectly rough cube is placed on a horizontal 
 plate. Determine the initial motion of the cube if the plate 
 is made to begin to move with a given velocity, in a direc- 
 tion at right angles to one of the faces of the cube. 
 
 5. A rectangular uniform plate, moveable in every 
 direction about its centre which is supported, rests horizon- 
 tally ; if a heavy adhesive particle be placed at one corner, 
 prove that the rectangle will begin to move about that 
 diagonal which does not pass through that corner, 
 
 7^2 
 
100 FIXED AXES. 
 
 6. Mersenne and otliers, in seeking for the centre of 
 oscillation, assumed that it was the same as the centre of 
 percussion. Why is this guess right ? 
 
 7. A slender circular ring is cracked at one point, and 
 is made to revolve in its own plane about the opposite point, 
 with a constant angular velocity co ; shew that the tendency 
 to break when greatest is measured by a couple whose 
 moment is 
 
 27r ' cos</)' 
 where m is the mass, a the radius of the ring, and a^ is the 
 arc contained between the fixed point and the point where 
 the tendency to break is greatest. 
 
 8. A uniform semicircular arc of mass m and radius a 
 is fixed at its ends to two points in the same vertical line, 
 and is rotating with constant angular velocity w. Prove 
 that the horizontal pressure on the upper end is 
 
 TT 
 
 9. If the semicircle in the previous question had an 
 angular acceleration w, shew that there would be another 
 horizontal pressure at the upper . support at right angles 
 to its plane and of magnitude 
 
 10. A uniform rod of length 2a is suspended by a 
 point in its length distant h from the centre of gravity. 
 Prove that the time of oscillation is a minimum when 
 
 11. A uniform rod of mass r.i makes small oscillations in 
 a vertical plane inside of a smooth right circular cylinder of 
 radius r, Avhich is fixed with its axis horizontal. If the rod 
 subtend an angle 2a at the centre of the cylinder, prove that 
 the length of the simple equivalent pendulum is 
 
 1 + 2 cos'a 
 S cos a 
 
EXAMPLES. 101 
 
 12. A physical line fixed perpendicularly to an axis 
 in its own plane revolves about tliat axis with a given 
 velocity ; find the pressure on the axis. Supposing the axis 
 to be set free, about what axis will the line now move ? 
 
 13. If a uniform rod capable of turning about one fixed 
 extremity be struck by a given impulse at any point, find 
 the point at which the tendency to snap in two is greatest. 
 
 14. A straight heavy rod oscillates about one end in a 
 vertical plane, coming to rest in a horizontal position ; shew 
 that if (^ be the angle included between the direction of the 
 rod and the direction of the pressure upon the axis at any 
 time, and 6 be the inclination of the rod to the horizon at 
 
 the same time, tan (f) tan = -- , 
 
XL 
 
 MOTION OF ONE BODY. 
 
 1. When the forces have been found wbich are in 
 equilibrium with one another, we employ any statical 
 method to obtain the equations of motion. Thus we resolve 
 the forces along' two directions at right angles, and take 
 moments ahcut a point. The resolutions present no diffi- 
 culty. Taking moments round an axis means that w^e ex- 
 press the conditi.cn that the forces shall have no effect in 
 jDroducing rotation round this axis. The effect of a couple 
 is the same about whatever point in its plane the motion be 
 considered ; in other words, a couple has always a certain 
 absolute rotational effect. The effect of a force not passing 
 through the axis is the moment of the force about the axis. 
 
 Now in the problems which are about to engage our 
 attention, we shall find couples and moments mixed up. 
 There will be effective forces at the centre of inertia, and an 
 effective couple. There will be impressed forces acting at 
 various points, and occasionally an impressed couple — for 
 example, a tendency to break. Taking moments about 
 any point will be expressing the condition that the alge- 
 braical sum of the couples together with the algebraical 
 sum of the moments about the point of the unreduced forces 
 is zero. 
 
 Or, if it seems convenient, each impressed force may be 
 reduced to a parallel force acting at the point round which 
 moments are to be taken and a couple. Taking moments 
 
PROBLEMS. 
 
 103 
 
 will tlien be equating to zero the algebraical sum of all the 
 couples. 
 
 Forces may be resolved in any convenient direction, and 
 moments may be taken, not only about the centre of inertia, 
 but about any convenient point. This direction and this 
 point are fixed. There is no such thing in Dynamics as 
 resolving along a moving direction, or as taking moments 
 round a moving point. Force is absolute in its action. The 
 point round which we take moments may be defined by a 
 certain point of the moving system passing through it at 
 the time, but it is itself fixed. 
 
 (mF will always be used to denote the moment of in- 
 ertia about an axis through the centre of inertia.) 
 
 2. A sphere of mass m, and radius a, is set rotating 
 with velocity 12 about a horizontal axis. It is tlien laid gently 
 doiun on a horizontal plane whose coefficient of roughness is jll, 
 and at the same time let go. Find the motion. 
 
 The first question is. Will it roll or slide ? Let v be the 
 velocity of the centre, and co the angular velocity. 
 
104 GENERAL MOTION 
 
 The effective force is m -- parallel to the plane. The 
 couple IS m v a" -r. . 
 
 The impressed forces are the friction F liorizon-tally, the 
 weight mg downwards, and the pressure of the plane B 
 upwards. Reversing the effective forces, the equations of 
 equilibrium become 
 
 m -— =F, mg — B = 0, 
 
 2 , do) ^ 
 
 m- a-' . -, = — Fa. 
 o at 
 
 F as it tends to prevent the motion of the point of contact 
 acts onwards. Hence it increases v and diminishes w. Now 
 V is at first zero, and &> is O, therefore v is at first less than 
 aco, and there is slipping. During this motion, 
 
 F= fxR = /jLing. 
 Hence ^. = mg, v = /j.gt, 
 
 and the distance travelled in time t is —-- . 
 ^''° di—2-a' 
 
 «-<»-•' '^^^ 
 
 2 a 
 
 When V becomes equal to aw, the problem changes. 
 Complete rolling begins. F is no longer equal to /z/f, but 
 instead, we have a geometrical equation v = aco, 
 
 1 dv dco 
 
 whence -- = «-, . 
 
 at dt 
 
 The dynamical equations are the same. 
 
OF ONE BODY. 105 
 
 „ . dv 2 doj 
 
 1 rom them y^ = — ^ a -j . 
 
 at D at 
 
 Hence -r and -j- are botli zero. F is also zero; and the 
 at at 
 
 sphere rolls on with constant velocity. The above reasoning 
 holds good for all circular bodies like wheels or barrels, for 
 the only difference would be in the moment of inertia. 
 That the constant velocity is never in practice attained, is 
 due to the imperfect rigidity of the body, and to imperfect 
 flatness and roughness of the plane. 
 
 It should be observed that, to determine the rolling 
 motion, only one dynamical equation is necessary. For 
 taking moments about N, 
 
 2 ^cZft) , dv ^ 
 m-a -J- +m-7^ ,a = 0, 
 o at at 
 
 an equation which results from the elimination of F from 
 the former two. 
 
 3. A uniform rigid circidar hoop (mass m, radius a), 
 cracked completely at one j^oint C, 'is rolling on a rough hori- 
 zontal plane. Find the breaking couple at the point A opposite 
 C, when the diameter through C is inclined to the horizon at 
 an angle a. 
 
 The angular velocity is (by the last example) constant. 
 Let this be w. Consider the upper part of the hoop from C 
 to A. Let G be its centre of inertia. Join G to the 
 centre of the circle. The part which we are considering is 
 in equilibrium under its weight at G, the action of the other 
 part at A, and the reversed effectives. The action at A is 
 composite. Let the couple be called L. The effective forces 
 
 are equivalent to a single force ^ co^ . OG from G to 0, and 
 
 a couple ^F-, J, in which k is the radius of gyration about 
 
 ^ at 
 G (Lesson iv.. Art. 10). The couple vanishes because co is 
 
106 GENERAL MOTION 
 
 constant. Eeversing the force and taking moments about A, 
 
 L+ ,y (J ' ((^coa a+ G sin a) — -^ cd^ . 6r . a = 0, 
 an equation for L. 
 
 4. A square hoard whose mass is m, and luliose length of 
 side is 2a, is rotating freely about one diagonal with angular* 
 velocity w. One end of the other diagonal is suddenly fixed. 
 What luill he the subsequent motion ? 
 
 It will clearly be a rotation about an axis through the 
 fixed point parallel to the former axis of rotation. Let the 
 new angular velocity be «'. As the only impressed force is 
 the impulse at the point which becomes fixed, the angular 
 momentum about that point remains unchanged. Now the 
 moment of inertia about a line through the centre parallel 
 
 2 
 
 to either side is m , This therefore is also the moment of 
 
 o 
 
 inertia about the diagonal. Hence m-^ (o was the quantity 
 
 o 
 
 of rotation, and was the angular momentum about any 
 axis parallel to the diagonal. The angular momentum has 
 become 
 
 m 
 
 ■a' 
 
 
 .3 + 
 
 2a' 
 
 ft) 
 
 1 
 
 = 
 
 - ~ 
 
 ft) 
 
 / 
 
 Equating these, 
 
 Consider the question another way. 
 
 Let V be the velocity of the centre after the impact. Let P 
 be the force of the blow. The momentum of translation has 
 been changed from zero to 'inv'. And the angular momen- 
 tum about the centre of inertia has been changed by 
 
 m -^ (a) — ft)j. 
 
OF ONE BODY. 107 
 
 Hence P at tlie fixed point, — mv at tlie centre, and the 
 
 couple — Qii — (ft)' — ft)), are in equilibrium ; 
 o 
 
 whence mv = P, 
 
 m^(ft)'-ft>) + PaV2 = 0. 
 
 o 
 
 Also since the centre moves round the end of a diao-onal, 
 
 V =a sj^ , ft)'. 
 Eliminatin<:j v and P we have the same result as be- 
 
 fore. 
 
 'O 
 
 5. A rough imperfectly elastic hoop is j^rojected horizon- 
 tally straight foriuards from a mans hand, an underhand 
 twist being given it so as to make it rotate about a horizontal 
 axis. Prove that on striking the ground it will rebound verti- 
 cally into the air, if the coefficient e of frictional elasticity he 
 
 1 _j_ e y 
 given by the equation = — ; a being the radius of the 
 
 J- — e aft) 
 
 hoop, and v, co the linear and angular velocities of pro- 
 jection. 
 
 Prove that if the hoop be perfectly elastic and the coefficient 
 
 of frictional elasticity be ^, the hoop tuill rebound into the 
 
 o 
 
 throiuers hand if slco = 2v. 
 
 In this problem the vertical velocity does not come under 
 consideration until the second part. The horizontal velocity 
 and the angular velocity remain constant until the ground is 
 touched. Then the point touching the ground gets stopped 
 by friction. When its velocity is zero the friction has ex- 
 hausted its force, but the friction of restitution begins, and 
 finally, when the contact with the ground ceases, the hori- 
 zontal velocity of the centre is known to be destroyed. 
 There are therefore two periods and two motions. 
 
108 PROBLEMS. 
 
 At the end of the first period v is changed into an un- 
 known velocity F, and co into an unknown velocity I>. But 
 the velocity of N' is zero ; 
 
 .-. V+an = (1). 
 
 If F be the friction, the reversed effective forces iti ( V— v) 
 and Dilc' (O — co) in equilibrium with this give 
 
 F+m{V-v)=0 (2), 
 
 Fu + mlJ" (fl-w) = 0.... (3). 
 
 In the second period the friction is Fe ; the velocity V is 
 changed to zero, and 12 to an unknown velocity a)\ 
 
 The equations will therefore be 
 
 Fe + m{O-V) = (4), 
 
 Fe . a + mk" {w - n) = (5). 
 
 No. (5) is for our purpose useless; but from (1), (2), (3), 
 (4) we deduce 
 
 1 + e V . J 
 
 = — , smce /J = a. 
 
 1 — e ao) 
 
 The rule for elimination in all such cases is : by means of 
 the geometrical relations find the value of F or other un- 
 known quantities ; then substitute these values and find the 
 final motion. 
 
 In the second part the hoop is supposed perfectly elastic ; 
 its centre will then rebound with the vertical velocity with 
 which it came down. If the horizontal velocity be exactly 
 reversed, it will bound back the same way as it came. 
 Hence equations (4) and (5) become 
 
 Fe + m{-v- V)=0 (C), 
 
 Fe. a + mk' {oy' -n) = i). 
 
 Putting e = ^, k = a, and using (1), (2), (3), (6) we have 
 the required result. 
 
ONE BODY. 109 
 
 6. A sphere of mass m and radius a, whose centre of in- 
 ertia G is distant h from its centre of figure C, is ^ilaced upon 
 a 'perfectly rough tabic. Find the time of an oscillation luhen 
 the oscillations are very small. 
 
 Here C is geometrically, G kinetically important. 
 
 Let the ande between CG and the vertical be 0. Then 
 
 o 
 
 6 measures the angular velocity, 6 the angular acceleration 
 of the body. Let v be the velocity of C. This is horizontal. 
 Let it be measured in the direction corresponding to an in- 
 crease of 0. Let mk^ be the moment of inertia of the sphere 
 about a horizontal axis through G. 
 
 The impressed forces are the weight, friction, and 
 pressure. 
 
 To c^et the effective forces. The acceleration of G is com- 
 posed (1) of that relative to G, i.e. 6'^b tov^ards (7, and 6b at 
 right angles to CG, and (2) of that of C, i. e. v horizontally. 
 
 The effective forces are therefore reducible to 
 m6rh, mOb, mv at G, 
 and a couple mk"^ 0. 
 
 Keversed they form with the impressed forces the system 
 in the figure. 
 
 Now as the motion is always very small, 6 may be 
 nesflected. 
 
110 
 
 ONE BODY. 
 
 Taking moments about JS' to avoid F and B, we liave 
 inv {a — h cos 6) + mhd (h — a cos 6) + mk 6 + mr/h sin ^ = 0. 
 
 Also since there is perfect rolling, 
 
 v = a6 and v = a6. 
 And ultimately cos 6 = 1, sin ^ = 6. 
 
 Hence (a' + A^^^ + Z>' ~ 2a?)) 6 = - hcjO, 
 
 an equation which gives the time of a small oscillation. 
 
 7. We have seen that in cases of motion whose differen- 
 tial equations do not admit of complete solution, the time of 
 oscillation can be found when the motion is small. In the 
 same way when the motion does not remain small, initial 
 circumstances of motion can be found ; such as the values of 
 unknown forces, the direction of motion of a point, or the 
 curvature of its path at the commencement of the motion. 
 Such problems are not of much physical interest. They are 
 valuable as giving examples of successive approximation. 
 
 To find the initial value of a force is usually very simple. 
 
 A uniform rod AB, of mass m, and length 2a, rests hori- 
 zontally, being partly snpjyorted by a smooth peg O. On all 
 siij^poi't but that of O being luithdratun, find the pressure at 
 and approximate to the initial motion. 
 
 Let OG = r, and let 6 be the angle AB makes with the 
 horizon. Let B be the pressure. 
 
INITIAL MOTIONS. 
 
 The general equations of motion are 
 
 111 
 
 m 
 
 d\ 
 
 dt 
 m d 
 
 — mg sin 6, 
 
 (1). 
 
 a^ d^Q 
 
 On eliminating i^, an equation presents itself whicli is 
 once integrable and leads to 
 
 ©■-©■(--^•)=%"=-«- 
 
 But no otlier equation involving velocities only can be got, 
 and therefore H cannot in the general case be found since it 
 depends on 
 
 dr , de 
 
 Putting for 6, 
 
 But the initial value of B can be found 
 -T- , -J" zero, and for r its initial value r^,, we have initially 
 
 dt 
 
 = 0, 
 
 d^ 
 df 
 
 a^d'e 
 
 'inr,-^^=mg-R, f 
 
 ^s-di^-^''^ 
 
 (2): 
 
 from which 
 
 ^-^^^ --^ df 
 
 %n 
 
 a'^ + 3r, 
 
 It is probably needless to remark that equations (2) are 
 not differential equations, and cannot be integrated. 
 
112 ONE BODY. 
 
 Let us now endeavour to find tlie initial radius of curva- 
 ture of the path of G. This involves a closer approximation 
 to the initial motion than Avas necessary in finding R. 
 Quantities which were neglected in that operation must not 
 be neglected. 
 
 G^ begins to move downwards. Hence the normal is 
 horizontal. If ON, GN are x and ?/, tlie radius of curva- 
 
 2 f\ 
 
 ture is initially — . As this assumes the form - it must be 
 evaluated. Differentiatinoj numerator and denominator and 
 
 dx , dy 
 putting X, y, -y, and -r- zero, 
 
 ^^' d£ 
 
 de 
 
 Now either by differentiating 
 
 x = r cos 6, y =^r sin Q, 
 
 or from the properties of relative accelerations, 
 
 d^y {d\ (de\-\ . ^ 1 d f ,de\ . 
 ^^" = \,„-r{-r] h sin ^ + - T I' 7 cos 6, 
 df \df \dtJ ) rdt\ dt) 
 
 whence df^"^'^' 
 
 In differentiating to find -J^^' , we may in the last differ- 
 
 ntiation neglect all vanishing terms, and in the next last 
 11 squares and products of vanishing terms, 
 
INITIAL CURVATURE. 113 
 
 d'x .(d'r _ d9 d'd\ 
 
 df = '''^[W-^'dfdl^) 
 
 d'e .de . .( d'i 
 
 dt' dt \ dtV 
 
 dt' " dt' '\dfj ' 
 
 ^''\df 
 
 
 Now 
 
 
 df d' + 3r/ ' 
 and differentiating the equation 
 
 a r 
 
 ~dt 
 
 
 zr —, — TV = Q cos o* -77 
 
 df dt dt' ^ di 
 
 :. p 
 
 d\ 
 
 at 
 
 M^h^' 
 
 dt' ' 
 
 
 8r 
 
 3r; 
 
 
 .9./ 
 
 '^df 
 
 
 -n 
 
 a"^ 
 
 To find the initial radius of curvature of a point when 
 the system is set off with a finite velocity is simple enough ; 
 
 for 
 
 sq. of velocity of point 
 
 ^ ~~ normal acceleration ' 
 
 and the normal acceleration is known when the impressed 
 forces are known. 
 
 P. G. 8 
 
114 PROBLEMS FOR SOLUTION. 
 
 If, however, as in the above problem, the system starts 
 from rest, the value of p assumes the form -, and when the 
 direction of the normal is not known the evaluation is tedious. 
 
 PROBLEMS FOR SOLUTION. 
 
 1. A uniform rod AB is whirled away on the surface of 
 smooth ice. Prove that the longitudinal stress at a given 
 point P is constant throughout the motion, and proportional 
 to AP.FB. 
 
 2. A uniform rod has a ring at one end, by which it 
 slides on a smooth straight horizontal wire. If it starts from 
 rest in any position in the vertical plane with the wire, 
 find the motion and the supporting pressure of the wire, and 
 prove that the other end of the rod moves in an ellipse. 
 
 3. A wheel in the form of a cylinder of radius B and 
 thickness A has an axle of radius r and length a cut out of 
 the same piece, the axes and centre^s of gravity being coinci- 
 dent. The whole is suspended with the axis horizontal by 
 three vertical strings, one of which is coiled round the wdieel 
 and the other two round the axle at equal distances on either 
 side of the wheel ; prove that if the first string be drawn up 
 or let down in any way the tensions of the other two will not 
 be altered provided 
 
 a _ R' (B - 27-) 
 A 7'(2B-r)'^ 
 
 4. A cylinder unrolls itself from a vertical string, the 
 other end of which is fixed. Prove that the motion is uni- 
 formly accelerated. 
 
 5. A cube is rotating with angular velocity co about a 
 diao-onal, when one of its edges which does not meet that 
 diagonal suddenly becomes fixed ; shew that the angular 
 
 velocity about this edge as axis will be — -, , . 
 
PROBLEMS FOR SOLUTION. 115 
 
 6. The ends of a rod of length 2a are constrained to 
 move on the smooth arc of a vertical circle of radius c. If 
 the rod be displaced from its position of unstable equi- 
 librium, find the breaking couple at any pomt in any 
 position. 
 
 7. A uniform rigid bar, suspended at one end by a 
 thread, rests on a perfectly smooth horizontal plane at a 
 given angle with it ; if the thread be cut, shew that the con- 
 tact with the plane will be unbroken during the motion. 
 
 8. A circular ring hangs in a vertical plane on two pegs. 
 If one be removed, prove that, P^, P^ being the instantaneous 
 pressures on the other peg calculated on tlie supposition that 
 the ring is (1) smooth, (2) rough, 
 
 tan'^ a 
 
 P' : P: :: 1 : H- 
 
 where a is the angle which the line drawn from the centre to 
 the peg makes with the vertical. 
 
 9. A uniform inelastic rod falls without rotation inclined 
 at any angle to the horizon, and hits a smooth fixed peg at a 
 distance from its upper end equal to one-third of its length. 
 Prove that the lower end begins to descend vertically. 
 
 10. A rod of length 2a has a ring at one end which 
 slides upon a smooth fixed horizontal rod. The former being 
 initially vertical, an angular velocity « is impressed upon it 
 about the ring in the vertical plane containing the fixed rod ; 
 prove that the greatest angle it will make with the vertical is 
 
 _j / ao)' 
 
 ' V 1^/ 
 
 2 sin" 
 
 - v^ 
 
 11. A uniform inelastic rod, inclined at an angle with 
 the vertical, falls without rotation and strikes a smooth hard 
 horizontal plane. Shew that its centre of gravity imme- 
 diately moves with a velocity 
 
 V. 
 
 1 -i- 3 sm' 
 V being its previous velocity. 
 
116 PROBLEMS FOR SOLUTION. 
 
 12. An inelastic ball of given radius is dropped from the 
 window of a carriage travelling uniformly along a level road 
 upon the wheel, which it hits at the highest point ; deter- 
 mine the subsequent motion of the ball relatively to the car- 
 riage, the rim of the wheel being perfectly rough. 
 
 13. The end of a uniform rod of weight W can slide by 
 a smooth riug on a vertical rod ; the other end sliding on a 
 smooth horizontal plane. The rod descends from a position 
 inclined (3 to the horizon. Shew that the rod does not leave 
 the plane during the descent, and that its minimum pressure 
 
 ., . W , ^ 
 on it IS -i-cos p. 
 4 
 
 14. A triangular lamina ABC is suspended horizontally 
 by vertical strings attached to its angular points. If the 
 strings at B and G be simultaneously cut, shew that there 
 will be no instantaneous change of tension in the string at 
 A, provided AD=GD . cos ADC, D being the middle point 
 of BC. 
 
 15. An imperfectly elastic sphere descending vertically 
 comes in contact with a fixed rough point, the impact taking 
 place at a point angularly distant a from the lowest point, 
 and the coefficient of elasticity being e. Shew that it will 
 commence moving off horizontally after the impact if 
 
 . 2 ^^ 
 
 tan a = — 
 
XII. 
 
 PROBLEMS. 
 
 A system of rigid bodies. 
 
 1. In these problems tlie expressions for the effective 
 forces are written down for each body of the system sepa- 
 rately. The equations of motion are always easily written 
 down in whatever co-ordinates the changes of velocity are 
 expressed. But their solution and the geometrical equa- 
 tions are much simplified by a judicious choice of variables. 
 
 As a general rule it is best to take co-ordinates which 
 are all independent of one another. We get by this means 
 the least possible number of variables, and so avoid having 
 to differentiate geometrical equations. 
 
 Suppose for example that two spheres, A and B, were 
 placed A above B on a plane and were disturbed, and that 
 it was required to find the motion so long as they were in 
 contact. We might take the co-ordinates of the centre of B 
 as x^, ?/j, those of A as x^, ?/,, and denote their angles of 
 rotation by other symbols. But it is clear that x^, y^ are 
 not independent of x^, y^, but connected by the relation that 
 the distance between the centres is constant. Hence it would 
 be better to denote the co-ordinates of ^'s centre relatively 
 to i5's by r, 6. Then r is constant and no other geome- 
 trical equation is needed. The properties of relative accele- 
 rations enable us to express at once the absolute acceleration 
 of A's centre in any direction. 
 
 It is not necessary nor even expedient, in drawing the 
 necessary diagrams and finding the effective forces, to con- 
 
118 A SYSTEM 
 
 sider in what direction velocities actually are or actually 
 increase. Measure them in the positive direction of the 
 co-ordinates, and the result will shew by its sign in which 
 direction they are, and in which they increase. 
 
 Forces of connection which are independent of one an- 
 other and also of the position of the system (as rolling 
 frictions and normal pressures, but not sliding frictions nor 
 tensions of elastic strings), may be avoided when the motion 
 only is required by grouping the various systems, so as to 
 make these forces internal, and by resolving and taking 
 moments in suitable ways. The equation arrived at by con- 
 sidering a whole system, is just the equation which would 
 have been arrived at, had each body been separately con- 
 sidered and the mutual actions eliminated. 
 
 2. A bullet of onass m, moving with velocity V, strikes 
 perpendicularly at the centre a uniform rectangular door of 
 mass ^I and breadth 2a. If the bullet sticks, find the angular 
 velocity of the door. 
 
 Let this be a, and let the measure of the blow on the 
 door be P. Then the blow suffered by the bullet is — P. 
 
 Consider (1) the bullet. This has its velocity changed 
 from V to aco. The force necessary to do this is m (aco — Y). 
 This reversed is in equilibrium with — P; 
 
 .*. P -f m (ao) ^ V) = 0. 
 Consider (2) the door. 
 
 The moment of P round the line of hinofes orencratcs a 
 
 4-a^ 
 quantity of angular momentum J/— cd. Hence the couples 
 
 Fa and — M ~ w are in equilibrium ; 
 o 
 
OF RIGID BODIES. 119 
 
 Eliminating P, we have 
 
 3m (ao) - F) + AaMco = ; 
 
 SmV 
 ,'. ft) = 
 
 ,{3m + 4:M)' 
 
 For example, m = 1 ounce, 31 = 200 lbs., 2a = 3 feet, 
 F= 500 feet per second, 
 
 1500 1000 „^o 1 
 
 " = |(3TiT20l)7l6)=12803 = -'''«'^^^'^'^'- 
 
 As tlie unit of angular velocity is, when the unit of 
 circular measure is described in one second, the door de- 
 scribes 
 
 •078ofo7\2^, 
 
 or about 4J*^ per second. 
 
 (a) Solve the problem by the principle that the whole 
 angular momentum about the line of hinges is not changed 
 by the impact. 
 
 (^) Find the resultant impulsive pressure on the line of 
 hinges^ by taking moments for the door about the centre, 
 and then substituting the value of co. 
 
 [ , mMV \ 
 
 Answer -z r-TTr) • 
 
 V dm + 4:MJ 
 
 (7) Find this pressure by considering the two together, 
 and resolving at right angles to the plane of the door. 
 
 3. A uniform cylinder of mass M and radius a, has a 
 hollow of any form in it filled with fluid of the same density 
 and of mass m. The cylinder being allowed to roll down a 
 perfectly rough plane inclined at an angle a. to the horizon 
 with its axis horizontal, find the motian. 
 
 Here the fluid is supposed not to rotate. As far as trans- 
 lation is concerned the whole is one, but the moment of 
 inertia must be calculated for the solid part alone. 
 
120 RIGID BODY 
 
 Take a section of the cylinder at right angles to its axis. 
 Let G be the centre, which is also the common centre of 
 inertia of the solid and fluid. Let v be the velocity of C, 
 ■and ty the angular velocity of the cylinder. Then v = a(D and 
 
 dv _ day 
 The effective forces of translation are reducible to one, 
 
 acting at C parallel to the incline. The couple is due to the 
 "motion of the solid part alone, and is therefore equivalent to 
 
 MK^ ~j, , MK" being the moment of inertia about the centre 
 
 of inertia of the solid. 
 
 To calculate this, call this point G\ and suppose the whole 
 cylinder solid. Its moment of inertia round C would be 
 
 - (ilf + m) a\ But this is made up of il/ii" round G, and of 
 
 the moment of inertia of the part which in our problem is 
 replaced by fluid. Let its centre of gravity be 6r', and its 
 moment of inertia about it ml^. Then 
 
 Suppose MK^ found from this equation. 
 
 The impressed forces are the weight (il/+ m)g at C, and 
 the friction and pressure at the point of contact. 
 
 Eevcrsing the effectives and taking moments about this 
 point ; 
 
 (il/ + m) a ^J + MK'' -^^ - (J/+ 7^0 ga . sin a = 0. 
 Since -u = ci , . the motion is uniformly accelerated. 
 
CONTAINING LIQUID. 121 
 
 (a) Prove that the cyKnder will roll down faster than if 
 it had been solid. 
 
 (/3) Calculate 3IK^, supposing the hollow a concentric 
 cylinder of radius h. 
 
 Ans. M."^. 
 
 (7) Prove that the friction acts up the plane and is 
 constant. 
 
 T . /TT- \ f • civ 
 
 It is {M + m) igsma—'-T: 
 
 (S) Prove that the normal pressure is constant. Sup- 
 posing the liquid to be in an excentric cavity, would it tend 
 to cause the cylinder to jump off the plane ? 
 
 Would it have this tendency if it were of different density 
 from the solid ? 
 
 4. A smooth circular tube (mass M) has a particle (mass 
 m) inside of it, and is set in motion in any manner luith every 
 point touching a smooth inclined j^lane. Prove that the parti- 
 cle will move with constant velocity round the centre of the 
 tube, and that if co be this angidar velocity and a the radius 
 of the tube, the pressure of the j^article on the tube is constant 
 and equal to 
 
 MnW 
 M + m' 
 
 The impressed forces on the tube are its own weight and 
 the normal pressure of the particle. These have no moment 
 round the centre, and therefore the angular velocity of the 
 tube remains constant. Let co be the angular velocity of 
 the particle round the centre of (not relatively to) the tube. 
 Let ii, V be the accelerations of the centre in and at right 
 angles to the direction of the particle at any moment. Also 
 let \g, fig be the component accelerations in these directions 
 of the force of gravity on a unit mass. 
 
1^2 A SYSTEM 
 
 Then tlie accelerations of the particle are 
 
 ii — a(xP' and v + aw. 
 
 Let R be the normal pressure, inwards on the particle, 
 outwards on the tube. Then the tube would be in equi- 
 librium under 
 
 M\g, R and Mit reversed, 
 
 and ^^f^!J> ^.nd 21 v 
 
 Tlie particle would be in equilibrium under 
 
 m\g,—R and m(u — aM^) reversed, 
 
 and 'mf^g, and 7n{v -{- aco) 
 
 Resolving p^g ^ y = 0, 
 
 fjig — V — a^ = ; 
 
 whence w = 0, or tlie particle moves round the tube with 
 constant velocity. 
 
 Kesolving again, MXg + R — 31 ii = 0, 
 
 and tnXg — R — m (d — aco') = 0. 
 
 Whence, R(-y.+ —] = aa)^ 
 
 \l\i mj 
 
 (a) Supposing 12 the angular velocity of the tube, what 
 will be the time in which the particle will come round to the 
 same point of the tube again ? 
 
 (/5) Shew from first principles that gravity has no effect 
 in altering the relative motion of the tube and particle, or 
 the mutual pressure. 
 
 (7) In wliat path does the common centre of inertia 
 move ? 
 
 (S) About what point is the whole angular momentum 
 constant ? 
 
 (e) Prove by moans of this principle that the angular 
 velocity of the particle is constant, assuming that the angular 
 velocity of the tube is constant. 
 
OF RIGID BODIES. 
 
 123 
 
 5. Tiuo uniform rods AB, BC, of masses m, m', and 
 lengths 2a, 2a', are connected by a joint at B, and are lying 
 in a straight line. A blow P is struck at C, in a dwection 
 perpendictdar to ABC. With what velocities will the system 
 begin to move ? 
 
 The immediate effect of P upon BG will be to make its 
 centre move off in a direction parallel to that of P, and to 
 make it rotate. Hence the point JS must begin to move at 
 right angles to BC. Hence the action at B between the rods 
 will at first be in this direction, and in general there will be 
 no motion nor force at first along ABC. 
 
 Let V, V be the velocities with which the centres of 
 inertia of AB, BC move off. Let w, «' be the initial angular 
 velocities measured in any but the same direction. The im- 
 pressed forces are P and the action at B. This latter we will 
 avoid. The effective forces are mv and mv at the centres 
 
 2 'a 
 
 of inertia, and couples m%Wy m' -r-co' on AB, PC respec- 
 tively. Eeversing these, as in the figure, and resolving for 
 the whole system : 
 
 7nv -\- m'v — P = (1). 
 
 yi 
 
 j3 
 
 Takinsf moments for AB round P, 
 
 mv . a — m CO = 
 
 (2). 
 
 Taking^ moments for BC round P, 
 
 P. 2a - m'v'. a — m' — w' = (3). 
 
 o 
 
124 A SYSTEM 
 
 Also, since B is common to AB and BC, 
 
 V — dw — v-V aw (4). 
 
 These are four simple equations to determine the un- 
 known quantities v, v\ co, co\ 
 
 These are probably the most convenient equations. There 
 are many others equivalent to the above, which might have 
 been chosen. Thus, taking moments for the whole about 0, 
 
 2 '2 
 
 7n~co — mv [Za + a) + m --- o) — m v a = 0. 
 
 O it 
 
 Supposing the action of BC on AC to be a force X at 
 right angles to AB ; we have, on resolving for xii? at right 
 angles to AB, 
 
 X—mv = 0. 
 
 Considering BC, and resolving, 
 
 P-X-m'v' = 0, 
 
 which two are equivalent to (1). 
 
 Taking moments for AB about its centre of inertia, 
 
 A. — m &j = I), 
 o 
 
 and so 
 
 on. 
 
 (ot) Justify, by writing down equations of motion, the 
 assumption that there is no force nor motion along ABC. 
 
 (/3) Prove that AB begins to turn about a point onc- 
 tliird of its length from A. 
 
 (7) Solve this problem by the statical method of virtual 
 velocities. (See Art. 7.) 
 
 6. A little squirrel clinr/s to a thin rough hoop, the plane 
 of which is vertical, and whicJi is rolling along a perfecthj 
 rough horizontal plane. The squirrel makes a point of keep- 
 ing at a constant height above the p)lane, and selects his place 
 
OF RIGID BODIES. 
 
 125 
 
 on the hoop so as to travel from a position of instantaneous 
 rest the greatest possible distance in a given time. Prove that, 
 m heing the mass of the squirrel and M that of the hoojy, the 
 inclination of the squirreVs distance from the centre of the 
 hoop to the vertical is equal to 
 
 m 
 
 cos 
 
 m + ^M' 
 
 Here the impressed forces are the weights, the friction, 
 and the pressure of the ground. There are also the forces by 
 which the squirrel clings to the hoop ; but we will take both 
 as one system, and so these will be internal. 
 
 The linear accelerations of the squirrel and hoop are the 
 same. Let this be v. If the angular acceleration of the 
 hoop be w, the effective forces are 
 
 m'v at >S^, Mi) at (7, 
 
 and a couple Ma^w, 
 
 [a being the radius of the hoop). 
 
 Reversing these, and taking moments for the Avliole 
 system about N, 
 
 mg a sin C — onb (a — a cos C) — Mv a — Ma^co = 0. 
 
 Also, since there is perfect rolling, 
 V = ad) I 
 .*. V [m (1 - cos G) -f 2ilf } = mj sin C, 
 
\ 
 
 126 A SYSTEM 
 
 Hence the acceleration is constant wbile . the squirrel 
 keeps to the same place. And therefore the greatest possible 
 distance Avill correspond to the greatest possible acceleration. 
 
 Now V is to be made a maximum by the variation of C. 
 Hence, differentiating 
 
 sin G 
 'IM -\-m — m cos C 
 
 with respect to C, and equating the result to zero, we have 
 
 cos C = - „,^. 
 
 We will write down some equations of motion by which 
 the problem might have been solved. Suppose F and li 
 the friction and pressure at N. Let T and Q be the tan- 
 gential and normal forces of the hoop on the squirrel, and 
 reversed of the squirrel on the hoop. 
 
 Considering the squirrel, we have, by resolving along the 
 radius CS and perpendicular to it, 
 
 mcf cos C — Q — mh sin C = 0, 
 — mg sin C + T— mv cos (7=0. 
 
 Consider the hoop. Resolving horizontally and vertically, 
 and taking moments about C, 
 
 F-v QsinC- Tcos C - Mv = 0, 
 
 Jl-Mg- QcosC-TsmC=(), 
 
 Fa — Ta + ^hrio = 0. 
 
 It is evident that on eliminating jP from the last and first 
 of this last set we shall have the equation of moments for 
 the hoop alone about iV^; and that on eliminating F, Q, T ^\c 
 shall have the single equation in the former solution. 
 
 It is also clear that F, 7?, T and Q can be found in terms 
 of the accelerations from the above equations. ^ 
 
OF RIGID BODIES. 127 
 
 In general, by writing down the equations of motion for 
 each body and the connecting equations, we obtain sutficient 
 equations to determine the motion and the forces of con- 
 nection. 
 
 7. A rectangle is formed of four uniform rods of lengths 
 2a and 2b, tuhich are connected by hinges at their ends. The 
 rectangle is revolving about its centre on a smooth horizontal 
 plane with an angular velocity n, when a jjoint in one of the 
 sides of length 2a suddenly becomes fixed. Sheiv that the 
 angidar velocity of the side of length 2b becomes immediately 
 
 3a +b 
 
 . n. 
 
 Oa-i- 4b 
 
 It was mentioned that any convenient statical method 
 might be applied to the solution of problems on motion. 
 We shall for this problem use the method of virtual velocities. 
 The virtual moment of a force is the force multiplied by the 
 displacement in its direction of the point of application. 
 The virtual moment of a couple is the sum of the virtual 
 moments of the two parallel forces which compose it. This 
 is easily seen to be the product of the measure of the 
 couple and the angular displacement of the body on which 
 it acts. 
 
 The rectangle has been revolving about the centre with 
 velocity n; hence the middle points of the rods have been 
 moving with velocities bn, an, and the rods have been 
 rotating about their centres with angular velocity n. The 
 point becomes fixed. Opposite sides, as ^5 and CD, will 
 still remain parallel; and, since they must make equal angles 
 with any direction, their angular velocities must be equal. 
 Let them be w. Let the angular velocity oi BG and A I) he 
 co\ Then the velocities may be all expressed in terms of ay 
 and ft)'. For let M, N, M' , N' be the middle points of the 
 sides, and let OM be called p. Then the velocity of B is 
 {a-\-p) ft). That of N is the same as that of B in the airec- 
 tion of jBJV, and is bwi' in the perpendicular direction. And 
 so on, as represented. 
 
128 
 
 Now subtracting from these the previous velocities, we 
 have the chang-es in the velocities. These will be : 
 
 for AB, 
 for BC, 
 for CD, 
 for DA, 
 
 29CO, — hi, 0) — n; 
 
 {a +2^) <« — «?^) ^<^'j <^' ~ '^ ) 
 
 2JW, 2h(o' — hi, (o — n; 
 
 {a — jj) CO — an, hco' , w — n. 
 
 If we multiply the changes in the linear velocities by 
 the masses, which are proportional to 2a, 26, and the changes 
 
 2 7 2 
 
 in the angular velocities by 2a. -^, 26. -^, and reverse them, 
 
 we shall have a system in equilibrium with the one im- 
 pressed force, the impulse at 0. 
 
 These reversed forces will then be (as to magnitude) : 
 
 on AB, 
 
 2ap(D, 2a6?i, at M and a couple, 2a • v- (<^ ~ ^0 5 
 
 on BG, 
 
 2 (a +p) 6a) - 2a6«, 26V; at N and a couple, 26 .- . (w - n) ; 
 
VIRTUAL VELOCITIES. 
 
 129 
 
 on CD, 
 
 'lapco, 4<ahco' — labn, at M' and a couple, 2a . ^ . (w — n) ; 
 
 on DA, 
 
 2 (a —J?) 5&) - ^dbn, ^Wa)\ at ^^' and a couple, 2& . - (w' - n) ; 
 
 and will act in the directions indicated. 
 
 C 
 
 Now the problem proposed is to find o)'. Let us suppose 
 a small angular displacement W given to the system in its 
 own plane, keeping AB fixed. The angular displacement of 
 DC will be zero. The displacements of N and B' will be 
 hW^ and that of M' will be 268^. The virtual moments of the 
 couples on AB and i) (7 vanish. 
 
 The virtual moment of the forces at J/ also vanish, and 
 that of the forces parallel to CB. And the equation of virtual 
 moments becomes 
 
 2&V . UQ + (4aZ^a)' - lain) 2hSe + 2Z/V . hSd 
 
 Ah^ AW 
 
 e7 O 
 
 whence 
 
 or 
 
 or 
 
 co' {Sh + 12a) = n {6a + 2^»), 
 
 CO = n . r. 
 
 Sa + h 
 6a + 4<b ' 
 
 P. G. 
 
130 
 
 DOUBLE PENDULUM, 
 
 (a) What displacement must be given to obtain an 
 equation involving only co ? 
 
 (/3) Write down the equation obtained by keeping the 
 point and a point on DC opposite to fixed, and giving 
 an angular displacement about these. 
 
 (ry) Prove by giving the whole a displacement parallel 
 to AB, that the component of the impulse at along AB is 
 4h (a + h) co\ 
 
 8. A pendulum consists of a uniform rod AB, of mass 
 m and length 2a, attached hy a string of length \ at A. to 
 a fixed 2^oint O. It makes small oscillations in one plane. 
 Find their law. 
 
 Let G be the centre of inertia of AB ; and denote by 6 
 and ^ the angles which OA and AB make with the vertical. 
 
 The accelerations of yl are 
 
 Z{|; along ^0, 
 
 fPfi 
 
 and I -^^ perpendicular to OA ; 
 
CO-EXISTE.VT OSCILLATIOXS. 131 
 
 therefore the accelerations of G are those in these directions, 
 together with the accelerations relatively to A, which are 
 
 ct[-j,] along GA, and ci^^ perpendicular to AB. 
 
 The angular acceleration is -^^2 . 
 
 The impressed forces are the weight mg at G, and the 
 tension of OA. The reversed effective forces will act as in 
 the figure. We can get two equations without the unknown 
 tension. 
 
 1st. Take moments about A, 
 
 m 'li-i-i) .a.sin {(p — 6) +ml^a cos (0 - 6) 
 
 + ma -j'^ . a + mk -^ + mga sm ^ = 0. 
 2nd. Resolve perpendicular to OA, 
 
 + ma -ji- . cos (^ - 6") + mg sin (9=0. 
 
 We can solve these equations if we assume that the 
 oscillations are small, so that the velocities vanish while the 
 
 accelerations do not. Making -^ , -^ zero; and putting for 
 cos ^, cos 6 the value 1, for sin ^, sin 6 the values <j6, 6, 
 
 A solution of these is clearly (Lesson XV.) 
 6=Asin{nt + B\ 
 (j) = A' sin (nt + B). 
 
 9-2 
 
132 
 
 A SYSTEM OF 
 
 To find n we have 
 
 - aht'A - (a' + F) n'A' = - agA'] 
 
 — In^A - an^A' = -gA ] ' 
 
 Whence a'h' = [ag - {a' + Jc') n'] [g - hv) . 
 
 If the roots of this be n^ and n^, and the vahies of the 
 
 A' 
 ratio -J corresponding to tliese be fx^ and yu,,, the solution 
 
 will be 
 
 e = A^ sin (ii^t + B^) + A^ sin {nj^ + B^), 
 </) = fi^A^ sin (nj) + ^J + fM^A^ sin (??./ + ^J. 
 
 B^, B^ are to be determined by the circumstances of the 
 motion. 
 
 These equations indicate a double oscillation. The two 
 parts are independent and co-existent. 
 
 (a) Obtain these differential equations by taking .r, y as 
 the vertical and horizontal co-ordinates of G^ and ic, y as 
 its accelerations, and then differentiating the geometrical 
 equations. 
 
 (yS) Obtain them by the method of virtual velocities. 
 
 9. A cylinder of mass M and radius a can rotate about 
 its aocis which is horizontal and fixed. A rough sphere of 
 mass m and radius b is j^laced on its top and disturbed from 
 
■ RIGID BODIES. 183 
 
 its position of unstable equilibrium. Find in what position 
 it will leave the cylinder. 
 
 Let 6 be the angle the line of centres makes with the 
 vertical. Let w and H denote the angular velocities of the 
 sphere and cylinder respectively. The accelerations of C are 
 
 {a^h)e\ and {a + b)0, 
 along and perpendicular to CO respectively. The angular 
 accelerations will be co and H respectively. Thus the effec- 
 tive forces reversed will form a system as in the figure. 
 
 The impressed forces are the weights at G and 0, the 
 pressure of the axle at 0, and the mutual friction and normal 
 pressure (R) at if. We will avoid the action at and the 
 friction. The pressure B is necessary ; for the instant when 
 it vanishes is the instant of separation. 
 
 Considering the sphere and resolving along 00, 
 
 m {a + b) &' + R -mrj cos = (1) ; 
 
 taking moments about N, 
 
 2 
 
 m{a-{-b) 6b + m -Ifco- mgb sin ^ = (2). 
 
 o 
 
 Considering both the sphere and cylinder, and taking 
 moments about 0, 
 
 m 
 
 {a + If + m ~h'oy+M^ il-mg [a + h) sin ^ = 0....(:3). 
 
 o 2 
 
 We have also a geometrical equation since there is no 
 sliding and the points of the two bodies at N are moving 
 together ; 
 
 (a-\-b)e-hco=^aQ. (4). 
 
 From the last three equatious 6 has to be found, and R 
 may then be found by the first. Differentiating (4), and sub- 
 stituting for O in (3), we have 
 
 ^'{m(a4-5r + ^(a + Z>)}+e;)j|m2.^-^} = m^(a+^')sin^; 
 
134 A SYSTEM OF 
 
 eliminating co between this and (2), 
 
 {a + I) (7J/+ 4.m) = r/ sin 6/ (4??2, + 5JJ), 
 
 whence multiplying by 20 and integrating, 
 
 6' {a + h) (7if + 4m) = 0- 2^ cos 6^ (5i/+ 4»0. 
 
 Now at the beginning of the motion 6 was zero, and cos 6 
 was 1 ; 
 
 .-. C=--2fj{5M+4m), 
 
 ' _ 2g (1 - cos 6) 5il/+ 4m 
 
 Substituting this in (1) and making R zero, the equation 
 which gives 6 when the bodies separate is 
 
 27 (1 — cos 6') ^^-^TT — ;r- = 7 cos 6, 
 
 . 10i¥+ 8m 
 whence cos U = - ^ , ^ , ., ^ . 
 
 l/i]i + 12m 
 
 (a) Calling i^ the friction at N, acting at right angles 
 to OG, but upwards on the sphere and downwards on the 
 cylinder; prove that 
 
 F— mg 'An 6 ■\- m . {a -{• I) = 0, 
 
 F=-^h(o. 
 
 
 
 (/S) What forces would be introduced if we resolved for 
 the whole system at right angles to 00? 
 
 (7) What forces would be introduced if we resolved for 
 the cylinder along ON 'I 
 
 10. A string luithout rueight is coiled round a rough 
 horizontal solid cylinder, of whicli the mass is M and radius 
 a, and which is capable of turning about its axis. To the 
 free end of the string is attached a chain of mass xn and 
 length 1/ if the chain he gathered vp close and then let go, p)rove 
 
RIGID BODIES. 135 
 
 that, if 6 he the angle through which the cylinder has turned 
 after a time t, before the chain is full g stretched, 
 
 This problem illustrates the fact that an infinite number 
 of infinitely small blows has the effect of a continuous force. 
 The uncoiled part of the chain and the cylinder are moving 
 with finite acceleration, and the coil is falling like a free 
 heavy body. But at every instant a link passes from the 
 coil to the straight part, and so has its velocity instantane- 
 ously changed by a finite amount. There is therefore an 
 impulsive tension on the uncoiled part as often as a link 
 is added to it. This impulsive tension is the opposite of 
 that which changes the motion of the link, and if the whole 
 be taken as one system it will be internal and disappear. 
 
 The whole increase of angular momentum about the 
 axis of the cylinder is caused by the weight mg. 
 
 But the rate of change of angular momentum is the 
 measure of the rotational effect of a force. 'J'he statement of 
 this is what is meant by taking moments about a certain 
 axis. Hence the rate of increase of angular momentum 
 about the axis is mga. 
 
 The rate of increase- of the angular momentum of the 
 cylinder is 
 
 Mrf ere 
 
 2 • di'' 
 
 Let z be the length of the uncoiled part of the chain at 
 
 rJO 
 any time. Then its velocity is « ^, and the rate of increase 
 
 of the moment of its momentum about the axis is 
 
 d hnz ^dO 
 dt\T^lt 
 
 for s is variable as well as 6, 
 
13G SERIES OF IMPULSES. 
 
 1-2 
 The coil moves with velocity gt, and its mass is m —j— 
 
 Hence the rate of chans^e of its moment of momentum is 
 
 The sum of all these is mga, 
 
 o 
 d^f l-Z 
 dt 
 
 Ma" d'd ma d ( de\ m d ri n 
 
 Integrating once 
 
 dO Ma ma dO m ,j . mn .t ,^. 
 
 di-T+T'dt+i'^-'^^'-ir w- 
 
 Now since the coil is moving with velocity gt, and the 
 
 straight part with velocity a -j, , the rate of uncoiling, —, is 
 
 dO 1 
 
 equal to gt — a~^^, and therefore z= -gf — a6, since z, 6 and 
 
 t all bes^in from zero tosfether. 
 
 1 , ^ dO dz 
 
 d^Mal^^^ dz 
 dt m ^"^ ' dt' 
 
 Til, 
 
 Integrating MaQ = -j . z"^. 
 
 i 
 
 (of) Shew that the impulsive tension caused by the un- 
 coiling of a link ^z is — ^ [gt — a -,- ) . 
 
 (/3) Shew that the finite tension due to the weight of 
 
 the strai<]rht part is ,^, £ ~. 
 ^ ^ M + 2mz 
 
 (7) Compare these. 
 
PROBLEMS. 137 
 
 PROBLEMS. 
 
 1. Two infinitely rough wheels revolving uniformly in 
 the same plane are suddenly brought into contact and their 
 axes are kept fixed ; determine what changes are made in 
 the angular velocities. 
 
 2. Two particles of any elasticities and of masses m and 
 m', joined by an elastic string, are placed in a vertical line; 
 the string is stretched and they are simultaneously let go. 
 Prove that whenever m comes to rest, m is moving with a 
 
 velocity gt , — . 
 
 3. A string has two particles m and m' attached to its 
 ends. The mass m lies on a smooth horizontal table, and m 
 is held so that the string is horizontal with a length a beyond 
 the edge of the table. If m be let drop, prove that the initial 
 
 l-\ ; 
 
 4. A little animal, the mass of which is m, is resting on 
 the middle point of a thin uniform bar, the mass of which is 
 on and the length 2a, the ends of the bar being attached by 
 small rings to two smooth fixed rods at right angles to each 
 other in a horizontal plane. Supposing the animal to start off 
 along the bar with a relative velocity V, prove that, 6 being 
 the inclination of the bar to either rod, the angular velocity 
 initially impressed upon the bar will be 
 
 3m Fsin2^ 
 
 3m 4- 4m' ' a 
 
 5. A thin hollow smooth ring (mass M and radius a), of 
 
 which the plane is vertical, and which contains a bead of 
 
 mass m, is placed upon a smooth horizontal plane. Prove 
 
 that the bead, having been placed near the lowest point of the 
 
 ring, will oscillate synchronously with a pendulum, the length 
 
 , , . , . 3Ia 
 01 which IS 
 
 m 
 
 + M' 
 
133 niOBLEMS 
 
 G. Two equal heavy spheres, one solid, tlie other hollow, 
 and the hollow filled with fluid, are rotating with the same 
 angular velocity about a horizontal axis, and are laid side by 
 side on a rough horizontal plane, the coefficient of friction for 
 both being fju ; if the interior radius of the sphere be one-half 
 of the exterior, and the density of the fluid be equal to that 
 of the solid, find the distance between them at any time, 
 supposing that they move in parallel lines. {Iiesult. Let H 
 be the initial angular velocity, a the radius of each. Then 
 
 the distance which the solid sphere has covered is -^- 
 before perfect rolling begins, and f^OK\^~i)] ^fterw^ards, 
 where t = S . - . The same results hold for the other, only 
 
 _m an 
 '^- '61 >^-^ 
 
 7. A cylinder rolls down the rough upper face of a 
 wedge which is capable of moving on a smooth horizontal 
 table ; prove that the accelerations are uniform. 
 
 8. An iceberg floats without change of volume from 
 latitude \ to latitude X^. Shew that the angular velocity of 
 the earth is diminished (very nearly) by the fraction 
 
 - -. ^ (cos \ — COS X J 
 
 of itself, m and il/ being the masses, and the earth supposed 
 spherical and homogeneous. 
 
 9. A heavy circular disk is rotating in a horizontal 
 plane about its centre, which is fixed. An inject walks from 
 the centre with constant velocity along a certain radius, and 
 then flies away. Determine the whole motion. 
 
 10. A loaded cannon is suspended from a fixed horizontal 
 axis, and rests with its axis horizontal and perpendicular to 
 the fixed axis, the supporting ropes being equally inclined to 
 the vertical ; if ^; be the initial velocity of the ball, whose 
 
FOR SOLUTION. 139 
 
 mass is -tli of the mass of the cannon, and h the distance 
 n 
 
 between the axis of the cannon and the axis of support, 
 shew that when it is fired off, the tension of each rope is im- 
 mediately changed in the ratio 
 
 v^ + T^gh : % (71 + 1) gh. 
 
 (The moment of inertia of the cannon about its centre is 
 neglected.) If a cannon be supported in a gun-boat in the 
 manner described, what would be the effect of firinsr it off ? 
 
 11. A rod whose centre is fixed is rotating uniformly in 
 a vertical plane. A perfectly elastic ball of equal mass is 
 dropped from a height equal to one-fourth of the circum- 
 ference of the circle described by the end of the rod, and 
 strikes it when horizontal at one extremity. After eight 
 revolutions of the rod the ball again strikes it ; prove that 
 the rod was horizontal when the ball was dropped. 
 
 12. An imperfectly elastic ball is let fall upon a smooth 
 
 hoop, of which the mass is equal to that of the ball, and 
 
 which is suspended from a point in its circumference about 
 
 which it is capable of moving freely in a vertical plane ; prove 
 
 that, if e be the modulus of elasticity, and a the inclination 
 
 to the vertical of the radius passing through the point at 
 
 which the ball must strike the hoop in order that it may re- 
 
 2e 
 bound horizontally, tan^ a = — . 
 
 o 
 
 13. A square formed of equal and similar uniform rods, 
 jointed freely at the ends, is revolving with constant velocity 
 about its middle point. Shew that if one of the angular 
 points suddenly becomes fixed while the four joints remain 
 free, the angular velocity of each rod will be at once di- 
 minished in the ratio 5:2. 
 
 14. Four equal uniform rods, jointed at their ends, when 
 falling freely as a square with one diagonal vertical, are 
 caught by means of a light hook at the middle jDoint of one 
 of the lower rods. Prove that that rod will be brought to 
 rest by the impact, and will remain at rest during the rest of 
 the motion. 
 
140 PHOBLEMS 
 
 15. ABCD is a uniform heavy chain whose length 
 equals ol, which is fastened to a peg at A, hangs down to a 
 distance I, and passes over a smooth peg at C, which is very 
 near A. If the chain be slightly disturbed so that its end D 
 descends, prove that the impulsive pressures at A and C at 
 the moment when it has run entirely down are m . \/2lg and 
 2m ^Itlg, m being the mass of the chain. 
 
 16. A mass M attached to the end ^ of a chain AC, is 
 placed (with the chain) on a horizontal plane in such wise 
 that a portion AB of the chain forms a straight line, the re- 
 maining portion BG being heaped up at jB : the mass M is 
 then set in motion in the direction B to A with a given 
 velocity, and so moves in a straight line, dragging the chain; 
 determine the motion. 
 
 17. Two uniform rods OA, AB, of lengths 2a, 2h, and of 
 masses proportional to their lengths, are jointed together at 
 A, and are rotating round the fixed hinge in the same 
 straight line, and with equal angular velocities, when the 
 outer AB comes against a fixed obstacle P. If the position of 
 this be such as to reduce both rods to rest, prove that 
 
 Sa + 2h 
 2a~ + bab + So" 
 
 18. Three equal particles, each repelling with a force 
 varying as the distance, are at rest at the corners of an equi- 
 lateral triangle, being connected by three fine inextensible 
 strings, which form the sides. If one of the strings be cut, 
 shew that the tension of each of the other two is instan- 
 taneously increased by one-fifth of its previous amount. 
 
 19. CP, AP are two equal uniform heavy beams, con- 
 nected by a free hinge at P. The beam CP turns freely in a 
 vertical plane about a fixed horizontal axis through C, while 
 A slides freely on a vertical groove of which C is the liighest 
 point; prove that if the system make small oscillations 
 about its position of equilibrium, the length of the simple 
 
 CP 
 
 isochronous pendulum is - . 
 
 o 
 
FOR SOLUTION. 141 
 
 20. A uniform string hangs at rest over a smooth peg. 
 Half the string on one side is cut off. Shew that the pressure 
 on the peg is instantaneously reduced by one-third. 
 
 21. A smooth sphere 11 is on a horizontal plane, and 
 another sphere m resting on it is just disturbed from its 
 position of unstable equilibrium. The spheres being sup- 
 posed homogeneous, shew that, whatever their radii or 
 weights, the upper sphere will leave the lower before the 
 
 2 
 line of centres is cos~^ ~ from the vertical. 
 o 
 
XIII. 
 
 ENERGY. 
 
 1. We have now to consider a method into -which the 
 element of 'time' does not directly enter ; in which force is 
 considered not as generating a certain acceleration, but as 
 pulling through a certain space; in which position and 
 velocity are therefore the language, but never time nor ac- 
 celeration. 
 
 The question which of these two was the proper ex- 
 pression of the effect of force caused a controversy very 
 memorable in the annals of mathematics. We can now see 
 that both sides were right. All our methods hitherto have 
 been based on the former. We now turn to the latter. 
 
 2. When a force P drags its point of application through 
 a small space, of which Sp is the measure in the direction of 
 the force, it is said to have done work FSj). And if the 
 point has been forced back Sj) against P, work PSj} has been 
 done against the force. Thus in this system the effect of 
 a force is work, in the other it was momentum. 
 
 The Avork of a couple L, which moves a body through 
 an angle B6, is LhO, 
 
 The accumulated work of a force P is clearly jPdp, the 
 limits being the values of j3 in the extreme positions. Thus 
 the work done by gravity on a stone of mass 7n, moving from 
 a height h^ to a height //^, is mg (h^ — /^J ; and this is the 
 same by whatever path, constrauicd or free, the stone has 
 reached the lower level. 
 
 The work done by a radial force P, in pulling a body 
 from a distance r^ from the centre of force to a distance r^, is 
 
 Bdr. 
 
 I 
 
woiiK. 14 3" 
 
 And in general tlie work done by any force, wliose com- 
 ponents are X, Y, Z, in bringing its point of application 
 from x^, y^, z^ to a?^, y^, ^o, is 
 
 ^{XqIx-^ Ydy^-Zdz), 
 
 the limits being given by these points. For if P be the 
 force of which X, Y, Z are components, and if QQ' be an 
 element of the arc {hs) traversed by the point of application, 
 
 I^_ Q 
 
 F,hp = P. QN'= P cos NQQ\ QQ\ 
 
 But the sum of the components of any forces in any 
 direction is equal to the component of their resultant in 
 that direction ; therefore 
 
 whence, in the limit, 
 
 Pdp = Xdx + Ydy + Zdz. 
 
 To measure work we must have some constant and easily 
 accessible force. Take the force of gravity acting on a pound 
 of matter at a given locality. Then tbe work expended in 
 raising this pound to the height of one foot will be the 
 unit of work. And any quantity of work will be measured 
 by the number of feet to which it would suffice to raise the 
 pound. 
 
 8. The work done by an impulse I, which causes the 
 velocity of its point of application to change from u to u, is 
 
 For suppose I the limit of a very great constant finite 
 force P, acting during a very short time r. Let a be the 
 acceleration of the point of action of P resolved in its own 
 direction. The space described by a point moving with 
 
14-i WORK. 
 
 finite acceleration a is „ — , where v-u is tlie increase of 
 2a 
 
 velocity. 
 
 This holds good also in the heginning of every accelerated 
 motion whatever be its law. 
 
 Hence the work done, P^p, 
 
 _ u"-u' 
 ~ 2a • 
 
 But II -u = a.T; 
 
 u' + u 
 
 therefore work done =Ft . . 
 
 And when t is infinitely small, 
 
 I=Pt, 
 whence the formula. 
 
 A very general proof of this is to be found in the 808th 
 section of the first volume of Thomson and Tait's Natural 
 Philosophy. 
 
 From this we see that if a ball strikes perpendicularly 
 a fixed hard surface and rebounds with equal velocity there 
 is no work done. 
 
 4. When a body or system possesses the power of doing 
 work it is said to have energy. Thus a moving cannon-ball 
 could force back a resisting body through a certain space 
 before exhausting its own motion ; this energy is called 
 kinetic. Or the Avater in a mill-dam could do work before 
 falling to a lower level ; it has energy which is called poten- 
 tial, and which is due to a position of advantage relatively 
 to a force. A mass of gunpowder has in it a store of the 
 energy of chemical affinity. Steam in the boiler of an engine 
 has the energy of heat ; a man not utterly prostrate possesses 
 a certain amount of another form of energy ; and so on. 
 
 It is not often that the whole of any form of energy 
 possessed by a body can be made avaiLable for work. A 
 stone at the top of a tower has energy due to its height 
 above the bottom. At the bottom its available poten- 
 
PRINCIPLES. 145 
 
 tial energy is exhausted, but it is clear that if there ^yere 
 a pit at hand the stone could do additional work before 
 getting to the bottom of that. Its total energy due to the 
 attraction of the earth would never be exhausted till it 
 had reached the centre. 
 
 The amount which cannot be made available was called 
 by Clausius the Entropy. It has been proposed to give 
 this name to the amount which can be made available. 
 As the usage in English works has come to be different from 
 that in foreign works, we will avoid the word altogether. 
 
 5. There are two general principles to which this method 
 has led. 
 
 Imagine a system A which possesses a certain store of 
 energy, and which can be completely isolated so that no 
 energy can pass out or in. Let it be connected in any way 
 with a system B possessing similar properties, and let no 
 energy pass except between these. Then the total energy of 
 all the different forms in A and B remains the same as 
 before. This is the law of the conservation of energy. 
 
 But the available energy is now less than when the 
 systems were separate. For that depends on the excess of 
 energy which one possessed over the other ; and by the 
 passage of energy between them the amount possessed by 
 the one is lessened and by the other increased. The two 
 systems are in fact brought more nearly to a level. Now, 
 as no system in nature can be completely isolated and 
 made energy-tight, every such system as A is always com- 
 municating its energy to those around it, as B, which pos- 
 sess less. And the available energy of all is being lessened. 
 This is the law of the dissipation of energy. 
 
 These laws are proved by experiment ; it is found that 
 energy which disappears in one form reappears in some other, 
 and to an equal amount. But for those forms of which we 
 can take cognizance in this science (viz.), the energy of 
 motion and of position, the conservation law can be deduced 
 from the laws of motion. 
 
 G. The potential energy of a system relatively to a force 
 is measured by the work which it can do against the force, 
 
 P. G. 10 
 
146 CONSERVATION OF 
 
 or by minus tlie work which the force can do upon it before 
 it reaches the position of zero force. It is therefore for one 
 force 
 
 the limits being given by the actual position and the position 
 of zero force ; for a number of forces it is 
 -^^{Xdx-\-Ydij-\-Zdz). 
 
 It is almost needless to remark that the origin from^ which 
 these co-ordinates are measured is a matter of convenience. 
 
 The kinetic energy of a particle of mass 7?i, moving with 
 velocity v, is the work it can do before being stopped. This 
 may be measured against any force. Let us choose gravity. 
 A particle of mass m projected upwards with velocity v will 
 
 move through a space ^ before stopping. It has thus pushed 
 
 back the point of application of the force tti^^ through a space 
 
 7/ . , , . v^ i^(v^ m • 
 
 — Hence the work it has done is mg x— or —j-. ims 
 '2g ^ , -^ - 
 
 may of course be expressed in any co-ordinates, as 
 
 m. \fdx\^ ,duV . fdz\^ 
 
 dt) ^ \dt) 
 
 'Z \\dtl \ 
 or in one plane '^ Ig^) -r r (^^-) j . 
 
 The kinetic energy of a moving rigid body is 
 
 J'-»-l(S)'*©'- 
 
 7. To establish the law of the conservation of energy, 
 we have then to shew that the gain or loss of kinetic energy 
 while the sy.stem passes from one configuration to another is 
 equal to the loss or gain of potential energy. 
 
 Let X, ;/, z be the co-ordinates of an element^ S;?i of the 
 system; ii, r, w its component velocities; A", Y, Z the com- 
 
ENERGY. 147 
 
 ponents of the impressed forces. Then these are in. equi- 
 librium with the reversed effective forces. These last are 
 
 -^'"d-f -^"'dt' -^'"df 
 
 dii _ du dx _ dit _1 du'^ 
 dt dx dt dx 2 dx ' 
 Thus the reversed effective forces may be expressed 
 Bm diu^ __ hn d{v^) _ hn d {uf) 
 "^fdx ' '2. dy ' 2 "dz ' 
 
 Hence, by the principle of virtual velocities, if the point 
 X, y, z receive a displacement whose projections parallel to 
 the axes are hx, By, oz^ 
 
 jMow suppose (1) that the displacements 3.r, %, Bz are 
 consistent with the geometrical relations of the system ; 
 for example, that they involve no breaking of connections. 
 Then in this equation X, Y, Z come to represent the im- 
 pressed forces whose virtual moments do not vanish. 
 
 Of all possible displacements one must be the displace- 
 ment due to the actual motion. If we suppose this to be the 
 one given to the system, 
 
 dx . d]! 
 
 ly. 
 
 in which Is is the element of arc described bj x, //, 
 Whence the equation 
 
 '2 \dx dij ^ Liz 
 
 becomes 
 
 = t(Xhx-hYhiji-Zh^) 
 ^^^'\d. ^ ds^'ds)—^C^<h^^ ds-^'-ds) 
 
 10—2 
 
14-8 CONSERVATION OF 
 
 Now suppose (2) that the forces A" Y, Z are functions 
 of the co-ordinates alone. Then this equation is integrable, 
 and gives rise to 
 
 \ S Sm {ic + ?;^ -}- ^r) = (7 + :S liXcU + Ydu + Zdz). 
 
 But the left-hand side of this is the kinetic energy, and 
 2/(AWx+ Ydy-vZdz) 
 
 is the work done by the forces on the system, (i.e.) the loss 
 of potential energy. The equation may therefore be ex- 
 ]3ressed : 
 
 the sum of the kinetic and potential energies is constant. 
 On the above supposition as to the forces, 
 ]{Xdx + Ydy + Zdz) 
 
 will be a function of the limiting values of the co-ordinates. 
 Hence the work done by these forces is independent of the 
 paths pursued by the points of the body : such a system of 
 forces is called " Conservative." 
 
 8. The supposition (2) which we have made as to the 
 nature of the impressed forces limits the cases of motion 
 to which the principle, as expressed by the above equation, 
 can be applied. It excludes all cases into which sliding 
 friction enters, for this, depending on the direction of motion, 
 is not a conservative force. All effects of animal force, such 
 as a constraint of one part of the system to move according 
 to some given law, and all explosions or impacts of imper- 
 fectly elastic bodies, are excluded. 
 
 The nature of the forces thus excluded is easily seen 
 from the energy stand-point. The only forms of energy 
 which our present science can deal with, are the kinetic 
 energy of motion, and the potential energy of position. 
 That energy is conserved must mean to us that the sum 
 of the energies in these two forms is constant. If then, 
 in any case of motion, energy passes into or comes from 
 any other form, as when heat is generated by sliding friction, 
 or when energy is imported into the system through an 
 auimal or an explosion ; or when sound or other vibrations 
 
ENERGY. 119 
 
 are produced ; in such cases, althougli the conservation 
 principle still holds true, the energy passes into forms which 
 do not come within the scope of Rigid Dynamics, and we 
 can only use the principle when we can make allowance 
 for the amount of the energy which has thus changed form. 
 
 The student who is familiar with the application of 
 virtual velocities to the theory of equilibrium, knows that 
 the forces which do not appear in that equation when the 
 displacement given is consistent with the geometrical re- 
 lations of the system, are (1) all internal pressures and 
 tensions between rigidly connected parts, (2) external forces 
 (as normal pressures) whose point of application experiences 
 no displacement in the direction of the force. There is to 
 be added to these rolling friction, whose point of application 
 being the instantaneous centre experiences no displacement. 
 
 9. The expression for the kinetic energy of a body 
 or system moving in one plane, viz. -Xom \l-j-j +('^ 
 
 falls under the rule of Lesson IV, Art. 3. The kinetic 
 energy of a rigid body is therefore equal to that of the 
 whole mass supposed collected at its centre of inertia, and 
 moving with it, together with the similar function relative 
 to that point; i.e. to the kinetic energy of translation to- 
 gether with that of rotation. 
 
 Now the velocity of every point relatively to the centre 
 of inertia is rw, where w is the angular velocity, and r the 
 distance of the point from the centre of inertia. If then 
 V is the velocity of the centre of inertia, the kinetic energy 
 of the body is 
 
 or ^il/F^ + ^Fcol 
 
 If the motion is round a fixed axis about which the radius 
 of gyration is K, the kinetic energy is clearly 
 
 •»- -» r-r-ro a 
 
1 5 COXSEnVATIVE 
 
 10. A TGvgh sphere of mass m and radius a, rolls in a 
 vertical plane inside of a fixed liorizontal cylinder of radius b. 
 Find the motion. 
 
 Let the velocity of the centre be v ; the angular velocity 
 CO ; the angle which the radius to the point of contact makes 
 witb the vertical 6. Then if the centre be at a height y 
 above its lowest point, its potential energy may be measured by 
 
 mgy, its kinetic energy is —\v + -^ crco- 1 ,v/ hence 
 
 2 
 V" -f- - a'6)^ + 2gy = constant. 
 o 
 
 I^ow the velocity of the point of contact is zero ; 
 .*. ?; — «w = 0, 
 
 also v= (b-a)-^, y={l^-a) (1 - cos 6) ; 
 
 therefore the equation of energy becomes 
 
 L(j)_ay [ — ] + 2[/ {h -«)(!- cos 6) = constant. 
 
 To determine the constant, suppose the sphere projected 
 from the lowest point with velocity T. Then when 6 = 0, 
 
 (6-.)f=F; 
 
 7 ... 
 
 whence the constant is ^, F^ ; therefore the motion is given by 
 
 D 
 
 ^rlO\^ _ F' _ 10^ 1 - COS ^ 
 
 dt) ~(h-ay~ 7 b-a ' 
 
 To find the time of a small oscillation, differentiate; then 
 d'e__ry sin6> 
 tlf~ 1^h-a' 
 
 whence the period of oscillation is ^tt lU — ^ -, 
 
SYSTEMS. 
 
 151 
 
 11. A uniform ladder of mass va., slips down between 
 a smooth wall and a smooth hoi^izontal plane always keeping 
 in a vertical 'plane perpendicular to their intersection. Find 
 the motion. 
 
 Let the planes be those of y and x, and let 6 be the 
 angle which the ladder makes with the vertical at any time. 
 
 Then the angular velocity is ~j- and the linear velocity of 
 
 the centre of inertia is OG . -j. , since G keeps at a constant 
 distance from 0. 
 
 Hence the kinetic energy is 
 
 o. fd6' 
 
 f(o^'.//)Q 
 
 The potential energy is due to the height of G above 
 Ox, and is therefore 
 
 mg . OG cos 0; 
 therefore 
 
 ~{0G' + F) f ^ j + ^ . 6> 6^ cos ^ = constant. 
 
 If the motion began when AB was vertical, this constant 
 isg. OG. 
 
1j2 ■ CONSERVATIVE 
 
 \dt) ~ OG' + k' 
 
 wliicli can be aGfain inteo^rated, and 6 found in terms of t. 
 
 /f7(9\^ 2g.0G,{l- cos 6) 
 Whence i \ - ^ ^ ^ 
 
 '' -eft 
 
 12. If the object of our investigation is to find, not the 
 whole motion, but the motion in some particular position 
 or the position of instantaneous rest, this method has a 
 great advantage over earlier methods. In these, integration 
 is almost always necessary. The general problem is worked 
 out and the particular case deduced. Here the particular 
 case is the easier. Thus, suppose that in the above problem 
 it were required to find the angular velocity of the ladder just 
 before it reaches the ground. The kinetic energy is zero at 
 first, and the potential energy at last. Therefore the kinetic 
 
 energy at last, -^m (OG^ -{■ If) oy^, must be equal to the po- 
 tential energy at first, (viz.) mr/ . - AB. 
 
 13. A j^ci'^^t^cle is attached to the circumference of a 
 massive cylinder, and starting from the end of a horizontal 
 diameter pidls up another jiarticle hanging at the end of a 
 string tuoiind round the cylinder, by making the cylinder 
 rotate about its axis, which is liorizontal. Prove that if the 
 former particle first readies the loiuest point the ratio of the 
 
 TT 
 
 masses of the pa^rticles is ^ . 
 
 Here, both at first and at last the kinetic energy is zero. 
 Hence the potential energy gained by one particle is equal 
 to that lost by the other. But tlie cylinder turns through 
 a right angle ; therefore the particle attached to it descends 
 a depth equal to the radius ; while the other ascends a 
 height equal to one-fourth of the circumference. And these 
 spaces arc inversely as tlie masses of the particles. Therefore 
 the masses are in the ratio of the arc of the quadrant to 
 the radius. 
 
SYSTEMS. lo3 
 
 14. A uniform tube in the form of a common helix 
 (screw) of mass M can move round a vertical axis coincident 
 with one of the generating lines of its own cylinder. ^ A imr- 
 ticle of mass m is dro2:)ped in at the top. ^ Find its velocity 
 and the angular velocity of the tube when it has reached any 
 2)osition. 
 
 There are two principles which will give two equations 
 of motion. 1st, That the angular momentum of the whole 
 about the axis of rotation must remain zero ; and, 2nd, 
 That the kinetic energy at the end must be equal to the 
 potential energy lost by the descent. Let the radius of the 
 cylinder be a. Let the inclination of the curve of the 
 screw to the vertical be a ; let « be the angular velocity 
 of the tube at any moment, v the velocity of the particle 
 relatively to the tube. Then its vertical velocity is v cos a, 
 and its horizontal velocity is compounded of the velocity of 
 the point of the helix at which it is, which is OP . co at 
 right angles to OP; and of v sin a relatively to the helix at 
 vhAit angles to CF. 
 
 o o 
 
 The figure represents a horizontal section of the cylinder 
 of the screw, through the particle. is the axis of revolu- 
 tion, C that of the screw, F the particle. 
 
 Let /POO be called 6. Then ^OFC='^--^, and 
 
 OF =■ 2a . sin - . 
 
 The moment of the momentum of F about the vertical 
 axis through is mv sin a (a - a cos 6) + m. OF^ . w. 
 
154 
 
 CONSERVATIVE 
 
 That of tlie tube is M2a^(o. 
 Hence, by the first princij)le, 
 
 d 
 
 mv sin 
 
 in a . a (1 - cos ^) + m^a^ sin'^ -.(o-\- M 2a^a) = 0, 
 
 for initially the system was at rest. 
 
 Again, the square of the velocity of P is 
 
 v^ cos" a + v" sin" a + OP V + 2y sin a . OP 
 
 e 
 
 CO . sni 
 
 The moment of inertia of the tube about the axis through 
 is 2Ma\ Its kinetic energy is therefore Md'o}\ 
 The total kinetic energy is 
 
 J/aW + -^iv'^ -\- 4a" sin^ -, co^ + 4a sin" -vsma.co 
 
 This must be equal to the loss of potential energy, which if 
 the height fallen through by the particle be It, will be mgh. , 
 These two equations sufifice to give v and o). 
 
 15. A square formed of four similar iiniform rods 
 jointed freely at their ends, is laid upon a smooth horizontal 
 table, one of its angular points being fixed ; if angular veloci- 
 ties O, II' be communicated to the two sides containing this 
 angle, shew that the greatest value of the angle (2(p) between 
 them is given by the equation 
 
 5 (a -a')' 
 
 Let be the fixed point. The angular velocities of OA 
 and BG will be equal since these must remain parallel. Let 
 this be 0). And let that oi 00 and AB be w'. 
 
SYSTEMS. 155 
 
 We will use for the solution of this the facts that the 
 whole angular momentum round remains unchanged and 
 that the kinetic energy remains constant. 
 
 The masses of the rods are equal, and will divide out of 
 each term in the equation. We may therefore take each as 
 unity. Let the length of each be 2a. 
 
 The angular momentum of OA is - - co. 
 
 The angular momentum of AB about is equal to the 
 angular momentum about its centre of inertia G, which 
 
 is ^^^ «', tosrether with the moment of momentum of its mass, 
 3 o 
 
 concentrated at G, about 0. Draw GM and GN perpen- 
 dicular to ^0 and AB. Now the velocity of G is com- 
 pounded of the velocity of A, 2aco along MG, and of its own 
 relative velocity, aco' along GN. And the moment of the 
 resultant momentum about any point is equal to the sum of 
 the moments of the separate momenta. It is therefore 
 
 2a(o . 0M+ aco' . ON, 
 
 or 2aw (^la + a cos 2(^5) + acd' (a + 2a cos 2.^). 
 
 Hence the whole angular momentum about is 
 
 4f (o) + o)') + ''^ (o) + o)') + (« + a>') 2a' (2 -i- cos 2^), 
 + (« + w')a'(l + 2cos2^). 
 
 Now this is always equal to its initial value. But initi- 
 ally (o was fl, w was Q! and 2(f> was a right angle. In the 
 final circumstances Avhich the question contemplates, tlie 
 angle 2(f> has a maximum or minimum value. Hence co and 
 (a are equal. 
 
 Equating the values of the angular momentum in these 
 states 
 
 '5a^ . o . . o ^ A /^ . ^/v /5a^ 
 
 2a) f -^- + oa' + 4a' cos 20) = (11 + ^') f -7^-+ 5a^ 
 
156 ENERGY. 
 
 Consider next the kinetic energy. 
 
 That of OA is ^ «' ; that of AB is 
 o 
 
 — 0)'^ + -ia^o)^ + a^co'^ + 4(rwa)' cos 2^. 
 o 
 
 The whole will be 
 
 '--,- + oa'^^ + Scro) &)' cos 2(^. 
 
 The initial and final values of these also are equal. 
 AVhence 
 
 2co' . --;; +Sco"- cos 2(^ - {n' + n'^) "., . 
 
 \\ hence w' = _,^. ^, ^ (2). 
 
 10 + cos 2cf) ^ ^ 
 
 Eliminating co from (1) and (2) we have the required 
 result. 
 
 16. The following general proposition*, which is due to 
 Clausius, is of such simple demonstration and of such value 
 ill molecular theories, that it ought to be better known in 
 England than it is. 
 
 Let X, y, z be the coordinates of an element hn of a 
 system not necessarily rigid. 
 
 Tlien — ^hnar = llhmx -r- , 
 
 dt at 
 
 and j2S.^ = 2V8..f-f2VS,H(g;. 
 
 cPx 
 and Sm , .3 is the component in the direction of the axis of x, 
 
 of the whole force, whether external or constraining, which 
 acts uiiSm. Call this X, 
 
 ■* I am iudobted to Professor ]Max\Yell for my iulroducliou to this 
 subject. 
 
VIRIAL FUNCTION". 157 
 
 JsW = 22S».X.+ 228«.gt 
 
 so ^ SSm/ = 2tSm Yy + 2tSm (j^J, 
 
 and ~ t^mz" = 2^hnZz + ^thn Q\ . 
 
 Now let A, B, C be tlie moments of inertia of the system 
 about the axes. 
 
 Then Xhn {x' + tf + z') r=:]^ {A -\- B -\- C). 
 
 Adding the above equations we have 
 
 -^«»{©'-(S)"-@)l- 
 
 Now su^Dpose the system to be such that A+B+ (7 does 
 not vary during the interval of time U. This will be the 
 case if the system be at rest: and 1st, if the system be a rigid 
 body rotating about certain fixed axes ; 2nd, if the system be 
 a homogeneous fluid whose particles are in motion but 
 which always occupies the same portion of space, for then 
 any element hm which moves away from a point is replaced 
 by another equal element ; 8rd, if the system consist of a 
 number of molecules each moving with a periodic motion 
 whose period is much smaller than ht ; for then the different 
 values of x, y, z for any molecule will recur many times in 
 the course of ht ; and in various other cases of motion. 
 
 All these kinds of motion are called "stationary." For 
 such we have 
 
 The right-hand expression is twice the kinetic energy of the 
 system ; that on the left hand Clausius proposes to call the 
 virial function. The virial equation holds whenever 
 
 1(^ + 5+0=0. 
 
158 ENERGY. 
 
 If tlie S3rstem is at rest, then the viiial fuQctlon vanishes 
 because Xy Y, ^separately vanish for every particle. 
 
 The above equation expresses that the virial function 
 is equal to twice the kinetic energy. The ptxrt of this 
 function which depends on internal forces admits of being 
 simplified. 
 
 17. Suppose the forces between any two particles at 
 a distance r to be R, Let it be considered positive when 
 it is repulsive. Then there are two terms in the above 
 sum which arise from the action between two particles at 
 j:^, y^, z^, and x^,y^_, z^ respectively. These terms are 
 
 Xx^ — Xoc,,, 
 
 >2 
 
 or -R -^ ' .x. + R -^ X. or R —^^- • 
 
 Now r^ = {x^ - rr J' + (y. - y^ + (.?, - z^. 
 Whence 2- [Xx + Yy -^ Zz) = -V ^Rr. 
 
 If then the forces in any system are partly external, sym- 
 bolized by X, Y, Z; and partly internal actions symbolized 
 by R, the equation becomes 
 
 - % {Xx + Yy + Zz) + - Siir = kinetic energy. 
 
 The proposition will hold good for any direction and 
 for any number of particles. If the action R between the 
 particles is attractive the term %Rr is negative. 
 
 18. This equation is evidently of very general applica- 
 tion. It gives, through SiiV, a measure of the internal 
 forces in bodies, whatever the nature of these forces may 
 be, and whether the particles of the body are at rest or in 
 motion. Its most important application hitherto has been 
 to the kinetic theory of gases. This is beyond our limits ; 
 but we will shew how to apply it in one or two very simple 
 cases. 
 
 Suppose a thin ring of mass m and radins «7, composed 
 of a number of particles, to be rotating with angular velocity 
 
VmiAL EQUATION. 159 
 
 0) in its plane about its centre. Let B be the tension. Let 
 8s be the arc between two adjacent particles. Then by the 
 equation 
 
 SPtSs = ma^cD^. 
 
 But R is the same for all points of the ring, 
 
 and SSs = ^air ; 
 
 ■R _ "^^^^'^^ 
 
 These equations can easily be verified by other methods. 
 
 Again, suppose a network of light cords in equilibrium 
 under any external forces A^, Y, Z acting at points x, y, z, 
 and let T be the tension along a cord of leugth r, then 
 
 EXAMPLES. 
 
 1. Prove that a flywheel of radius a rotating w^ith 
 velocity « has in it energy enough to raise a mass equal 
 
 to its own to a height — p- . 
 
 2. A cannon-ball of mass M raises by its recoil a mass M 
 to a height of h feet. If the mass of the cannon-ball is m^ 
 shew that its velocity of projection is 
 
 — .T-^gK 
 
 m' -^ J 
 
 8. A weight is attached to an elastic string which is 
 fastened to a point. Apply the principle of energy to de- 
 termine its motion when it falls from rest, the strmg being 
 initially vertical and unstretched. 
 
 4. A nut slides smoothly on its screw. If this be 
 placed in a vertical position and the nut be allowed to 
 
1(J0 EXAMPLES. 
 
 run down, prove that its angular velocity when it has 
 descended a space h, will be 
 
 (/j" + oJ^ tan'' a) 
 
 in which a is the radius of the screw-cylinder, a is the in- 
 clination of its tangent to the horizon, and k is the radius of 
 gyration of the nut about the axis. 
 
 5. A thin uniform smooth tube of length 2a is balancing 
 horizontally about its middle point which is fixed ; a uniform 
 
 rod whose mass is - th of that of the tube and whose 
 n 
 
 length is 2a, is placed end to end in a line with the tube, 
 and then shot into it with such a horizontal velocity that 
 its middle point shall only just reach that of the tube; prove 
 that if V is the velocity of projection of the rod, the an- 
 gular velocity of the tube and rod when their middle points 
 coincide is 
 
 f >V \^ 
 
 6. A circular ring is free to move on a smooth hori- 
 zontal plane on which it lies ; and a uniform rod has its 
 extremities connected with and moveable on the smooth 
 arc of the ring ; the system being set in motion on the 
 plane, shew that the angular velocity of the rod is constant ; 
 and describe the paths of the centres of the rod and ring. 
 
 7. A narrow smooth semicircular tube is fixed in a 
 vertical plane w^ith its vertex upwards, and a heavy flexible 
 string passing through it hangs at rest; shew that if the 
 string be cut at one of the ends of the tube, the velocity 
 which the longer portion of the string will have attained 
 when it is just leaving the tube will be 
 
 (nr/7)'' |27r-^(7r=-4) 
 
 I being the length of the longer portion, and a the radius 
 of the tube. 
 
EXAMPLES. 161 
 
 8. A particle is suspended so as to oscillate in a cycloid 
 wliose vertex is at the lowest point ; if it begin to move 
 from a point distant a from the lowest point measured along 
 the curve, and the medium in which it moves give a small 
 resistance kv^ to the acceleration, prove that before it next 
 conies to rest energy will have been dissipated, which is 
 
 —^ of its original value. 
 o 
 
 9. A fine circular tube carrying within it a heavy par- 
 ticle is set revolving about a vertical diameter. Shew that 
 the difference of the squares of the absolute velocities of 
 the particle at any two given points of the tube, equidistant 
 from the axis, is the same for all initial velocities of the 
 particle and the tube. 
 
 10. A rough cylinder of radius a loaded so that its 
 centre of gravity is at a distance h from its axis is placed on 
 a board of n times its mass w^hich can move on a smooth 
 horizontal plane. Find the time of a small oscillation, and 
 prove that if I be the length of the simple equivalent 
 pendulum 
 
 ?i + 1 ^ ' 
 
 where h is the radius of gyration of the cylinder about a 
 horizontal axis through its centre of gravity. 
 
 11. A mass M of fluid is running round a circular 
 channel of radius a, with velocity u ; another equal mass 
 is running round a channel of radius h, with velocity v ; the 
 radius of the one channel is made to increase and the other 
 to diminish till each has the original value of the other. 
 Shew that the work required to produce the change is 
 
 12. A smooth thin tube in the shape of a quadrant of 
 a circle, of radius a, is fixed in a vertical plane with its 
 lowest radius horizontal. A heavy uniform inextensible 
 
 P. G. 11 
 
1G2 EX^^IPLES. 
 
 string, of length -^ , is held wholly within the tube and then 
 let go. Find the velocity during the subsequent motion. 
 
 13. A uniform imperfectly elastic beam, of length 2a 
 moving parallel to itself impinges on a fixed obstacle. 
 Prove that the kinetic energy after will be to that before 
 impact as 3c^ + eV to 8c^ + «^ ; c being the distance from 
 the middle point to the point of impact, and e the modulus 
 of elasticity. 
 
 14. A plane body is struck by a blow in its own plane. 
 Prove that the work done by the bloAV will be greater if 
 the body be free than if a point of the body were fixed. 
 
 15. Which of the systems described in the Problems at 
 the end of Lessons XI. and XII. are conservative ? 
 
 16. If A^ B, G be the moments of inertia round three 
 axes at right angles of a uniform cylinder rolling with 
 constant velocity along a plane ; prove that 
 
 -^^{A + B+ C) is constant. 
 
 17. Verify the virial equation in the case of a uniform 
 chain hanging in the common catenary. 
 
xiy. 
 
 PRECESSIONAL MOTIOK", 
 
 1. We have, in these introductory Lessons, avoided the 
 more complicated phenomena of motion. But there is one 
 class of motions of such paradoxical appearance and such im- 
 portant nature that we will state the phenomena and give a 
 general explanation of their cause. 
 
 Every one knows that a rapidly spinning top not only 
 rotates about its axis but with its axis about the vertical. 
 
 The explanation of the seasons depends on the fact that 
 the earth, while rotating and revolving, keeps its axis always 
 parallel to itself. But when its directions in successive years 
 are compared with one another, they are found not to remain 
 parallel but to move in a cone, pointing to different fixed 
 stars in the course of ages, and to take nearly 26,000 years to 
 return to the same direction. This is called the precession of 
 the equinoxes. 
 
 2. The whole of these phenomena can be illustrated by 
 one piece of apparatus, called Fessel's. 
 
 Let i^ be a metal disc or ring which can rotate freely 
 about the rod BA a.t the point A, but cannot move along 
 it. is a point at which the rod is supported either by a 
 pivot or by a long cord. Let there be means for suspending 
 irom the other side a weight W, of such size that it can 
 Ijalance B or overbalance it, or be overbalanced by it. Now 
 let B be set rapidly spinning in the direction of the arrow 
 head, and the system afterwards left to itself. And (1) let 
 W be so placed that it exactly balanced B when all was at 
 
 11—2 
 
1G4. 
 
 EXPERIMENT. 
 
 rest. Then it is found that the axis COB continues to point 
 constantly in the same direction. In fact, if the workman- 
 ship is so good that R will spin for some considerable time, 
 the apparatus may be used, like Fouconlt's pendulum, to 
 give a visible proof of the earth's daily rotation. For, as 
 the earth rotates, the axis GOB wdiose direction is abso- 
 lutely fixed in space appears to rotate slowly backwards. 
 
 This is the case of a coin thrown up into the air and 
 made to rotate in its own plane. The only force is that of 
 gravity which acts through its centre of inertia. Hence it 
 will continue to rotate about the same direction unless it is 
 disturbed. (2) Let R overbalance W, as in the case of the 
 common top or the gyroscopic toy, in which indeed every- 
 thing on the left of is wanting. Then it is found that 
 the axis GOB rotates about the vertical, so that B approaches 
 and G recedes from us. (3) Let W overbalance R when 
 at rest. Then it is found that in the motion GOB rotates 
 about the vertical, so that B recedes and G approaches. 
 
 From this experiment it is clear that the motion of the 
 axis is caused by the couple due to gravity acting round 
 a horizontal axis. 
 
 In the case of the earth the couple is the attraction 
 
EXPLANATION. 165 
 
 of tlie sun on tlie protuberant parts of the earth. Were the 
 earth a sphere there would be no process ioru 
 
 The explanation of these phenomena, which we will now 
 give, is confessedly imperfect, but on its own suppositions it 
 is satisfactory. It is applied to the complete explanation of 
 precession and to accurate calculation in Airy's Mathemati- 
 cal Tracts. 
 
 3. Imagine a sphere rotating freely about a diameter 
 AOA' with angular velocity 11, and struck by an impulsive 
 
 couple which would, if the sphere were at rest, generate an 
 angular velocity co about a perpendicular axis OB. Then the 
 resultant angular velocity will be \/Q^^ + a)^ and the sphere 
 
 will rotate about OD where tan AOD = -^. 
 
 Now suppose that co is small compared with O, i. e. either 
 that the sphere is rotating rapidly or that the impulsive 
 
 couple is small. Then (^j may be neglected in comparison 
 
 Avith -^ , and thus we may say that the velocity of rotation 
 
 will not be altered by the blow, whereas the axis will. 
 
 (That OD will lie on the same side of OA as OB will be 
 seen by considering the resultant velocity of C.) 
 
 Now let another impulsive couple act on the sphere tend- 
 insr to cause rotation about an axis at ridit angles to OB, not 
 
1C6 
 
 CONE IN SPACE. 
 
 in the plane A OB, but so that the planes of both the couples 
 pass through one line OK Then the body will begin to 
 
 rotate about the direction OD', DD' being at right angles to 
 the plane ODN\ and the angular velocity will be unchanged. 
 And if a number of such impulses act one after the other, 
 their planes all containing ON, the axis will proceed to 
 describe a pyramid (not necessarily re-entering) in space with 
 as vertex. But if the couples be equal and at equal 
 intervals, the pyramid will be regular and re-entering. And 
 again, if these impulses are numerous and small, so as to 
 approach the case of an accelerating couple, the pyramid 
 becomes a riii^ht circular cone, with as vertex and OiT 
 as axis. 
 
 Suppose that the impulse takes place at intervals of 
 time ht. Then in each interval the axis describes an angle 
 
 equal to AOD, i.e. to ^ . Now to complete a revolution 
 
 it must cover the whole cone. Let the semivertical angle 
 be a. To complete a revolution the axis must describe 
 "Itt sin a. And this it will do in time 
 
 27r sin a' 
 
 ' 6) 
 
 4. Such are tlie proceedings of the axis of rotation in 
 space. If the end of each axis on the sphere were marked 
 
CONE IN THE BODY. lO*' 
 
 with chalk, how would the successive marks appear when 
 the sphere was stopped ? 
 
 Suppose the sphere to be beginning to rotate about J. 
 Let D, n be the positions in space of the ends of the next 
 
 two axes ; and let R, E' be the points of the sphere which 
 are to coiDcide with them when they become axes. Let the 
 lines in the figure represent planes through the centre of 
 the sphere. Then BAD is the angle through which the 
 sphere must turn while it rotates about A ; and R'Ba must 
 be equal to BAD and D'Da together. Now AD, DD\ &c. 
 form an equiangular pyramid. Hence, as the sphere rotates 
 with constant velocity, and the impulses take place at equal 
 intervals of time, the locus of R — the successive chalk marks 
 — will be the angular points of an equiangular polygon. 
 If the impulses be all equal the polygon will also be equi- 
 lateral. In the limiting: case the marks will trace out a 
 circle on the sphere. 
 
 Let its radius be p. Then AD = AB, 
 And AB = /3 . / B'Ba in the limit 
 
 And since y:r ultimately vanishes 
 
 AB = pnu. 
 
 Again, AD (fig. Art. 3) = ^ z AOD = i? ^^ , 
 in which B denotes the radius of the sphere ; 
 
 therefore p^U = ^Ty and p = B ^2gT • 
 
168 PHECESSIONAL 
 
 The time in which the instantaneous axis passes through 
 
 all these positions in the sphere is-^y > i*®. the same as one 
 
 complete revolution of the sphere. This latter circle is there- 
 fore a very small one, and very quickly described compared 
 with that which its centre describes in space. 
 
 5. "We have considered a sphere, in order that every axis 
 might be a permanent axis of rotation. For if an axis of 
 rotation is not a principal one the forces introduced by the 
 motion tend themselves to alter the axis. Happily in the 
 important case of motion this supposition is not far wrong. 
 The earth is itself very nearly a sphere, and 12 is so large 
 compared with co that the circle described on the surface by the 
 locus of the end of the real axis of rotation is exceedingly small 
 (a few feet in radius), and may for all purposes be neglected. 
 The momental ellipsoid of a top is not so nearly a sphere. 
 It is an oblate spheroid. But here again the rotation is so 
 fast that the true axis is never far from the axis of figure, 
 and the centrifugal or other effective forces of motion have 
 never any effect that would interfere with the above reason- 
 ing. If the rotation is not fast the wobbling which sets in 
 shews that the axis of rotation is far from the axis of figure. 
 
 We see, then, that a top or the sphere described above is 
 not in general rotating about its axis of figure, but about a 
 not principal axis very close to it. This goes through its 
 various positions in the course of a single revolution of the 
 body, and the axis which is the mean of all these describes 
 in space a cone of finite size. 
 
 EXAMPLES. 
 
 1. A perfectly balanced gyroscope is rotating Avith given 
 angular velocity ; supposing it to be acted on by a small 
 constant couple in a vertical plane through its axis, find the 
 precession. 
 
MOTION. 169 
 
 2. Prove tliat the finite couple corresponding to « in 
 Art. 3 is ^- , A being the moment of inertia of the sphere 
 about an axis through the centre. 
 
 3. Assuming that the polar axis of the earth changes 
 its direction by 20*5" every year, and that the angle (a) 
 between the poles of the ecliptic and equator is 23J degrees ; 
 find the time which the polar axis takes to complete a revo- 
 lution in space. 
 
 4. Find the radius of the circle in which the axis of 
 rotation cuts the surface of the earth. (Radius of earth is 
 nearly 4000 miles.) 
 
 5. What time does it take to describe this circle ? 
 
 6. A top of mass m, whose centre of gravity is distant h 
 from its vertex, and whose radius of gyration about an axis 
 through the vertex at right angles to its axis of figure is k, is 
 rotatinof with its axis inclined at a to the vertical. Prove 
 that 
 
 CO gh sin a 
 
 It ^ ¥ • 
 
 7. Assuming that the rotation will be always about an 
 axis very near the axis of figure, prove that if the axis of the 
 top is observed to make a complete revolution about the ver- 
 tical in time T, the angular velocity of the top is 
 
 ghT 
 
XV. 
 
 DIFFERENTIAL EQUATIONS. 
 
 1. The complete analytical solution of a physical pro- 
 blem depends in general on that of a differential equation. 
 And each physical science depends in general on equations 
 of a particular type. It is therefore necessary for the student 
 of a department like the present to have a working ac- 
 quaintance with the class of equations peculiar to it. Hap- 
 pily this is easily acquired. A little practice, without 
 systematic study of a treatise on Differential Equations, is 
 all that is necessary. The solution of Differential Equations 
 being a reverse process is to some extent guess work. A 
 knowledge of the nature of the result seldom fails to suggest 
 its form. 
 
 In the following brief notes on differential equations, 
 which have been put together chiefly for reference, the 
 student must not expect a regular exposition of the subject, 
 but must be content to assume some of the principles and 
 solutions. 
 
 2. Many of the differential equations which present 
 themselves to the student of Rigid Dynamics are not capable 
 of complete solution. Those which are soluble, are mostly 
 of the class called "linear with constant coefficients"; 
 i.e. in which the differential coefficients and the dependent 
 variable appear only in the first power, and with constant 
 coefficients. Here are some of the most common. 
 
 (1) ^^ + ax=bf{t). 
 
PROPERTIES. 171 
 
 (3) S-.4f...=o 
 
 The order of a differential equation is the order of the 
 highest differential coefficient which occurs in it. 
 
 The complete solution of a differential equation must 
 contain as many arbitrary constants as the number which 
 expresses the order of the equation. For a differential 
 equation is formed by eliminating the constants in an ordi- 
 nary equation from the results of successive differentiations. 
 
 We can see that the solution of a physical problem 
 requires such constants; for, taking the case of pendulum 
 motion, all such bodies move after one law, but the velocity 
 or position of any one at any time will depend on the velocity 
 and position at starting. Hence in physical problems the 
 arbitrary constants are determined from the circumstances 
 of the motion being known in some one position or at some 
 one time. 
 
 Thus equations of the form (1) contain one constant, of 
 (2) two constants, and of (3) two each. Linear differential 
 equations have farther these properties. 
 
 (1) If the order is n, the addition of n independent 
 values of .r, each with an arbitrary constant, gives the com- 
 plete solution. For it satisfies all the conditions of a complete 
 solution. 
 
 (2) That part of the solution which contains the arbi- 
 trary constants is the same for all equations having the 
 same terms involving the dependent variable and its dif- 
 ferential coefficients. Thus the solution of (2) is found from 
 that of 
 
 by adding any particular value of x (without arbitrary 
 constants) which makes 
 
 If'^^di^^^ equal to cf[t). 
 
1/2 DIFFERENTIAL EQUATIONS. 
 
 Lastly, every particular solution of a differential equa- 
 tion corresponds to a possible motion or other physical 
 condition. 
 
 3. First form. 
 
 j^+ax = lf{t). 
 
 Multiplying by e"^ 
 
 Integrating 
 
 cce''' = A-\-hJG'''f{t)dt (1). 
 
 The following are important cases : 
 
 / \ dx 
 
 The solution is x = Ae 
 
 ■at 
 
 dx 
 (/5) --J- + ax = h sin nt. 
 
 The part of the solution involving the constant is ic = ^e""'. 
 For the second part we remark that x = p sin nt + q cos nt 
 will satisfy the equation provided p and q be properly 
 determined. To secure this, differentiate and substitute; 
 then 
 
 dm + qa) cos nt + {im — qn) sin nt = h sin nt 
 
 If these are identical, 
 
 p7i + qa = 0] 
 pa — qn = h\ 
 whence the solution is 
 
 X = Ae'"* + -5 o (a sin nt — n cos nt). 
 
 n' + a ^ 
 
 The solution might have been obtained from (1) by 
 remembering that Jc"' sin nt dt is of the form 
 
 e"' {p sin nt + q cos nt). 
 
SOLUTIONS. 17o 
 
 The first part {Ae'""') of the solution indicates a gradually 
 diminishing or (if a be negative) increasing motion ; the 
 second indicates a motion of oscillation superimposed on 
 this. 
 
 4. Second form. 
 
 Consider first 
 
 g+„J+ J,; =,/(<). 
 
 S+4>^-« 
 
 it is clear that x = Ae"^* will satisfy it if m^ + am + Z» = 0, Let 
 7?ij, iy\ be the roots of this quadratic. Then the solution is 
 
 x^^Ae'^'^^ + Be'''^*. 
 : If the roots of the quadratic are impossible, and of the 
 form a±l3j-l, 
 
 the solution is x = e^ {A cos ^i-\-B sin ^t) . 
 
 For the particular value that must be added when cfif) is 
 not zero, we must in general depend on happy thoughts. 
 The rules are. too long to be given here for all the cases. 
 
 The following are important : 
 
 Solution. w=Ae"'^Be-'". 
 
 Solution. x^A(io^{nt^B) or A^mni^ Bo^o^rd, 
 (7) "^j^ + '^^ ~ ^ ^^^ ^® expressed 
 
 i(«-a-''(-5)-°- 
 
 C^) -^ + w^^ = a sin mi. 
 
174 DIFFERENTIAL EQUATIONS. 
 
 The solution of 
 
 at 
 And it is clear that a; =^ sin 7nt will make 
 
 -7,5, + n^x = a sin mf, 
 at 
 
 if p have the right value. To determine p, differentiate 
 J) sin mt twice and substitute. Then ~ pm^ + 7i^p = a ; 
 
 whence « = -^ :, , and the complete solution is 
 
 ^ n — m 
 
 X = .4e"' + Be-""' + ^-^72 sin mt 
 
 The equation (/3) indicates a motion of oscillation whose 
 central point is the origin, and whose period is — -; (7) indi- 
 
 n 
 
 cates a similar motion with the central point at a distance 
 a 
 
 ti 
 
 from the origin. 
 
 5. Equations of the form (3) are in general the ex- 
 pression of co-existent oscillations, and the coefiQcients v/ill be 
 such as give a solution in sines and cosines. 
 
 d^x , dy . ^ "1 
 d'y , ,dx .r ^ j 
 
 Assume x = A sin (iit + B), 
 
 9/ = A' con {nt + B), 
 
 It is evident that these will satisfy the equations, if A, A' 
 be suitably determined. To do this, dif{:erentiatc and sub- 
 stitute 
 
 - Atv" - Ana + hA = 0} 
 
 -A'n'''^cuiA-\-h'A' = 0i' 
 
SOLUTIONS. 175 
 
 A' 
 From these the ratio . is found, and also a quadratic for n\ 
 
 Let the vakies of n^ be ??^^ r?/. Suppose them positive. 
 
 And suppose A' = /ji^A. 
 
 Then x = A^ sin {nj; + B,) + A^ sin (n.jt + ^,), 
 
 y = fiAj^ cos (Wj^ 4- B^) + /i,^2 ^^s (^^-Z + -^2)- 
 
 In this ?i,, 7?2 and /i have definite values ; A^, A^, B^, B^ 
 depend on the initial circumstances. As there are here 
 four constants no extension of generality is gained by taking 
 the roots —n^, — n^ of the biquadratic for ?i. The solution 
 indicates that there are two independent oscillations going 
 on together in x, and two also in y. If the values of ri^ are 
 not positive, x and y will involve exponentials. 
 
 6. "When the Differential Equations, at which we arrive 
 by eliminating unknown forces, are not completely in- 
 tcgrable, they can frequently be integrated once. 
 
 The equation 
 
 is of constant occurrence, especially in problems of sliding 
 triction. Assume i-j,] — lz\ 
 
 ^^^"^ dt ' df ~ dt ' 
 
 d'O dz 
 
 ^^^ ^dJ^-de' 
 
 The equation may therefore be written 
 
 J+2/(6). = <^(e). 
 
 Multiplying by e-ff^^'^'^^, both sides become perfect in- 
 tegrals. 
 
 Integrating, ^e^//(^)^^ = A +/(/)(6')e^//(^>'^^ . d9. 
 
176 EXAMPLES. 
 
 EXAMPLES. 
 
 1. What kind of motion is indicated by the equation 
 x—ct sin nt ? 
 
 2. Of what differential equation is x^at-^- Ae""^ the 
 solution ? 
 
 Solve the following equations : 
 
 dx , . 
 
 3. -r- -{■ ax = oe . 
 at 
 
 . (Fx _ cZic . „ 
 
 d^x ^dx . 
 df dt 
 
 d^x „ 
 G. -TT + ri^^ = « cos 7h/. 
 
 - dx ^1 
 
 7. 5i + «^ = 0| 
 
 J 
 
 8. -^^+"^ = 
 
 dv 
 
 -^; + «a? = c sm mt 
 dt 
 
 Prove that the equations 
 represent an oscillatory motion. 
 
C 177 ) 
 
 MISCELLANEOUS PROBLEMS, 
 
 1. A rigid body in wliicli A, B, C are three points, moves 
 so that these come into the positions a, h, c. Aa, Bh, Co 
 being very small spaces given in magnitude and direction, 
 find the motion of translation and of rotation of the body. 
 
 2. A flywheel is driven by a piston acting on a crank 
 alternately up and down with a force P; find the limits 
 between which the velocity varies. 
 
 3. If a mass is animated by simultaneous velocities, its 
 moment of momentum about any axis is equal to the sum of 
 the moments of the separate momenta about that axis. How 
 does this appear ? 
 
 4. A lamina rotating in its plane about its centre of 
 inertia is suddenly brought to rest by sticking a two-pronged 
 fork into it. Shew (1) that the impulses on the prongs are 
 equal, (2) that they are of the same magnitude wherever the 
 fork is stuck in. 
 
 5. A w^heel of which an axle projecting on each side 
 forms a part, is supported in a vertical plane by having the 
 axle on each side resting on a pair of friction wheels each 
 of which is just like the first wheel and is similarly sup- 
 ported, and so on indefinitely; compare the inertia of the 
 whole system in relation to a rotation of the first wheel 
 Avith that of the first wheel alone. 
 
 6. A pendulum performs small oscillations in a medium 
 of which the resistance varies as the square of the velocity ; 
 given the number of oscillations in which the arc of oscilla- 
 tion is reduced one half, compare the original resistance 
 with the weight of the pendulum. 
 
 P. G. 12 
 
178 MISCELLANEOUS 
 
 7. Shew that when the centre of gravity of any system 
 of material particles in motion passes through a point of 
 contrary flexure, the momentum of the system is in general 
 a maximum or minimum, and the resultant of the effective 
 forces is zero. 
 
 8. Two points B, C oi s. circular ring moveable in its 
 own plane about its centre are connected with a fixed point 
 A by elastic strings the natural length of each of which 
 is equal to the shortest distance c, between A and the ring. 
 Supposing the ring turned through any angle and let go, 
 calculate the motion and shew that the time of a small 
 
 -^ J , where m is the mass of the ring, and \ 
 the modulus of elasticity of the string. 
 
 9. A uniform bar of length 2a is suspended hori- 
 zontally by two parallel strings each of length I attached at 
 distances c from the middle point. It receives a small 
 angular disturbance so that it oscillates about a vertical 
 axis. Prove that it makes small oscillations in the same 
 
 time as a simple pendulum of length -^ . 
 
 oC 
 
 10. Prove that if any straight line (taken to be the 
 
 axis of z) is a principal axis at some point (not necessarily 
 
 , ... Xhuyz ^Smxz 
 
 the orio^in) — =^— = - — ^ — . 
 "^ ' y X 
 
 11. A uniform rod of length 2a has its ends on two 
 straight lines meeting at right angles in a point 0, and 
 makes an angle with one of them. Every point of the 
 
 rod is attracted to the point with a force .. y. 
 
 Prove that if p be the distance of an element of mass v^p 
 from the middle point of the rod 
 
 p sin lOciedp 
 
 {p' + a' - 'lap 
 
 12. A uniform revolving rod the centre of gravity of 
 which is initially at rest, moves in a plane under the action 
 uf a constant force in the direction of its length ; prove that 
 
 2 KXd. + Yd, + Zd.) = 2,.. j_^ / , ^^^^ _ ^^ J^, 
 
PROBLEMS. 179 
 
 the square of the radius of curvature of the path of the rod's 
 centre of gravity varies as the versed-sine of the angle 
 through which the rod has revolved at the end of any time 
 from the beginning of the motion. 
 
 13. A particle is attached by a string to the end of 
 a rod n times as long as the string, which rotates in a given 
 manner about the other end; the whole motion taking place 
 in a horizontal plane. If 6 be the inclination of the rod 
 and string, and &> the angular velocity of the rod at the time 
 t, prove that 
 
 14. A uniform circular disc, whose upper surface is im- 
 perfectly rough, rests on a smooth horizontal table. A par- 
 ticle is tied by a stretched inextensible string to the centre 
 and then projected along the disc at right angles to the 
 string. Prove that the particle will come to rest on the 
 disc before the strinsf becomes slack. 
 
 ^& 
 
 15. A Catharine wheel is constructed by rolling a thin 
 casing of powder several times round the circumference of 
 a circular disc of radius a. If the wheel burn for a time T, 
 and the powder be fired off with relative velocity V along 
 the circumference, shew that the angle turned through by 
 the wheel will be 
 
 niii 
 
 clo-fl-f-^)|, 
 
 where 2c is the ratio of the masses of the disc and powder. 
 The casing is supposed so thin that the distance of all the 
 powder from the centre of the disc is a. 
 
( l^->0 ) 
 
 SUGGESTIONS AND RESULTS FOE, THE 
 EXAMPLES AND PROBLEMS. 
 
 I. Page 10. 
 
 (3) The unit of angular velocity is when the unit of cir- 
 cular measure is described in one second. 
 
 (4) The earth rotates once in 28 hours oG minutes 
 (nearly). 
 
 (7) At a distance — from the axis of the paddle-wheel. 
 
 (8) The reasoning is given in Lesson li. Art. 4. 
 
 (9) The cylinder fixed in space becomes a plane. 
 
 (10) The focus is at the instantaneous centre. 
 
 (11) The velocities of P and Q are proportional to their 
 distance from the intersection of ^P and JJQ. 
 
 XL Page 18. 
 
 (3) The points of contact are moving with the same 
 velocity. 
 
 (4) It has one rotation round the vertical axis and 
 another round the line of contact with the table. 
 
 (6) and (7) These appear from the identity of the laws 
 for the composition of forces, and those for the composition 
 of anovular velocities. 
 
NOTES ON THE EXAMPLES. 181 
 
 III. Page 84. 
 
 (2) With a velocity represented by BB\ 
 
 (3) Round a horizontal axis inclined to the line of 
 
 CO 
 
 wickets at tan - . 
 
 CO 
 
 (4) Diminishes. 
 
 (8) li V, (a be the acceleration of the centre and the 
 angular acceleration, the force required will be that found in 
 the Example Art. 9; and also a horizontal force hm {;o + ad)). 
 
 TV. Page 48. 
 
 (2) The centre of inertia is fixed. 
 
 (3) 30,000; 300,000. 
 
 (4) With constant velocity in a straight line. 
 
 (5) Any function of the forms 
 
 or the sum of such terms. 
 
 (6) On the whole hoop; the effective force is horizontal, 
 and is mass of hoop x ^ ; the couple is mass . d' . co. On the 
 part AB; the force acts at G, and is compounded of m.CG. co' 
 along GO, mv horizontally, and m.CG.co ^t right angles to 
 GC. The couple is c6 (moment of inertia about G) and 
 
 v = aco. 
 
 (7) Calling the centre of inertia of AB, G; the accelera- 
 tion of G relatively to A along GA is co" . b. That of A is 
 
 compounded of o)\2a along AG and ~^. 2a at right angles 
 
 to A 0. Hence the acceleration of A measured along BA is 
 
182 NOTES ON THE 
 
 6)" . 2a COS (h ,- . 2a sin 6. The whole acceleration of G 
 
 at ^ 
 
 along BA is 
 
 o)"^b + &)^2a cos <i> — ^1.' 2a sin (^. 
 
 y. Page 58. 
 
 (2) The angular momentum remains constant while the 
 moment of inertia diminishes. 
 
 (4) Vertically downwards. 
 
 (5) The angular momentum about the instantaneous 
 centre remains constant, 
 
 (7) The direction of the string must pass through the 
 centre of inertia. 
 
 (8) (Second part). There has been no external force, 
 and therefore all will come to rest again. 
 
 (9) He increases the angular momentum of projection. 
 
 (10) The angular and linear momenta do not alter in the 
 interval between the blows. 
 
 (12) Yes. By swinging his arm round in a horizontal 
 plane. 
 
 VI. PageGG. 
 (5) The only two bodies are a particle and a straight 
 
 od. 
 
 n 
 
 (7) The side, its perpendicular bisector and the normal 
 to the plane. 
 
 (8) See Art. G. 
 
EXAMPLES. 183 
 
 (10) An axis parallel to an edge. 
 
 (16) «^. 
 
 VII. Page 71. 
 
 (3) ^"^^i^ + i^Ti^a). 
 
 YIII. Page 80. 
 (6) The moment of inertia must be greatest. 
 
 IX. Page 87. 
 
 4 
 
 (1) A force equal to the weight of 44 - lbs. 
 
 o 
 
 (2) Neglecting the square of -r , the tension is 
 
 lb 
 
 (5) If V be the velocit}^ just before striking, and the 
 angle just found be called a, the couple will be 
 
 , ^ Fsin a. AC 
 mass AC X ^ . 
 
 X. Page 99. 
 
 (1) Because the long rod takes longer to fall. 
 
 (4) Let V be the velocity of the plate, and 2a the edge 
 
 of the cube. Then the cube will begin to rise up about its 
 
 3V 
 edge with angula.r velocity ^—, 
 
 oci 
 
 (5) See Art. 6. 
 
 (6) The distance of each from the fixed axis ddpends on 
 the moment of the resultant effective couple. 
 
184 NOTES ON THE 
 
 (13) If be the fixed point, OA tlie rod, P the point 
 where the blow is struck, Q the required point, G the middle 
 point of QA^ 
 
 0A' = 2.0P.AQ.0a. 
 
 XI. Page 114. 
 
 (2) The lengtli of the rod being 2a, and the angle it 
 
 makes with the vertical at first being a ; the angular velocity 
 
 when the angle with the vertical becomes 6 is the square root 
 
 „ 6g cos ^— cos a m . n •. • X- 
 
 of -^.-= ^ ■ ■■ ,, . ihe centre oi gravity moves m a verti- 
 
 cal straight line. 
 
 (5) The angular momentum about that edge is un- 
 chansred. 
 
 o 
 
 (6) First find the value of the pressure at one end. The 
 motion is the same as that of a compound pendulum. 
 
 (7) Prove that the pressure never vanishes. 
 
 (12) The ball starts forwards with four-sevenths of the 
 velocity of the centre of the wheel. It remains in contact 
 with the rim and rolls off backwards, leaving the wheel when 
 its direction from the centre makes an angle with the vertical 
 
 whose cosine is ^V- -^10 + -= — ; ^i^r , V heincr the velocity 
 
 17 [ 7 (j{r + B)) ' * -^ 
 
 of the wheel and r, R the radii of the ball and wheel. 
 
 XIL Page 187. 
 
 (1) Consider each as acted on by an impulsive friction, 
 and take moments for each about its centre. 
 
 (3) Ths external force is {m + ni)g, 
 
 (4) The angular momentum about the instantaneous 
 centre of the bar remains zero. 
 
 (12) At the moment of greatest compression the velocity 
 of the ball and of the point it touches resolved along the nor- 
 mal are equal. 
 
EXAMPLES. 185 
 
 (17) Displace the system, keeping and P fixed. Or 
 take moments for AB about A and for the whole system 
 about 0. 
 
 (19) Use the method of virtual velocities and give the 
 natural displacement. 
 
 (22) There is no rotation. 
 
 XIII. Page 159. 
 
 (3) Supposing m the mass, a the unstretched length 
 of the string, x the distance from the point of suspension, 
 and taking the point of suspension as the origin, the potential 
 
 energy due to the tension is I Tdoc, and that due to gravity is 
 
 — mgx. 
 
 Whence ^ mv^ -i-XJ- — ^ dx - mgx = - mga, X being the 
 
 modulus of elasticity. 
 
 (6) The common centre of inertia will move in a straight 
 line with uniform velocity ; and the centres of the ring and 
 rod will describe circles relatively to it. 
 
 (11) The angular momentum of each remains constant, 
 and the work goes to increase or diminish the kinetic energy. 
 
 (12) Observe that the velocities of all points of the 
 string are at any moment equal. 
 
 XIY. Page 168. 
 (1) If L be the couple and A the moment of inertia ; 
 the axis will rotate about the vertical in time ^ . 
 
 (3) The true time is 25,868 years. Take sin 23J° to 
 be •41.' 
 
 (4) Nearly four feet. 
 
 (5) One day. 
 
 P. G. 13 
 
186 NOTES ON THE 
 
 XY. Page 17G. 
 
 (1) Oscillations of an increasing magnitude and constant 
 period about the same central point. 
 
 (2) -^-mx = a(l — mt). 
 
 (3) x = Ae-'' + (b-a)e\ 
 
 (4) x = Ae' + Be'' 
 
 (5) The additional terms must be of the form p + qt 
 
 (6) X = A sin 7it +5 cos nt + —. ^ cos mt 
 
 (8) Differentiating the second and substituting for -^ 
 from the first, we find 
 
 -tX — a Y^ = c sm mt, 
 dt dt 
 
 the solution of which is y = A + Be"'"'^ + p sm 7nt + q cos mt, 
 in which A and B are arbitrary while p and q can be found 
 by differentiating and substituting. 
 
 (9) Solve as in Art. 5. 
 
 MISCELLANEOUS PROBLEMS. 
 
 (1) If a direction can be found along which the com- 
 ponents of Au, Bb, Cc are all equal this will be the direction 
 of translation. To this end, from any point draw Oa, Oh\ 
 Oc parallel and proportional to them, and on these as diame- 
 ters describe spheres. If these intersect in p, Op is the re- 
 quired direction. 
 
 (2) The variation of kinetic energy is equal to the 
 variation of the potential energy. 
 
 (3) It follows from the parallelogram law. 
 
EXAMPLES. 187 
 
 (5) The angular velocities diminish in geometrical pro- 
 gression. 
 
 (10) Suppose it is a principal axis at a distance c from 
 the origin. Using the symbols of Lesson viil. Art. 2, we 
 must have 
 
 Xdmzr cos {0 — a) = 0, Xdmzr sin (6 — a) = 0. 
 
 (11) X^x + YSy -\-Ztz for one element is ^ . Sr ; and the 
 
 variation of r is due to the displacement of the whole body. 
 
 (13) Resolve at right angles to the string, and introduce 
 the expressions for relative acceleration. 
 
 (15) The whole angular momentum is constant. 
 
 THE END. 
 
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14 EDUCATIONAL BOOK'S. 
 
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SCIENCE, 17 
 
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SCIENCE. 19 
 
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 THE TRANSIT OF VENUS. By G. Forbes, M.A., Professor of 
 Natural Philosophy in the Andersonian University, Glasgow. 
 Illustrated. Crown Svo. 3J. 6^. 
 
 Other volumes to follcnv. 
 
 Ball (R. S., A.M,)~EXPERIMENTAL MECHANICS. 
 A Course of Lectures delivered at the Royal College of Science 
 for Ireland. By R. S. Ball, A.M., Professor of Applied 
 Mathematics and Mechanics in the ^Royal College of Science 
 for Ireland. Royal Svo. i6j. 
 
 Clodd.— THE CHILDHOOD OF THE WORLD: a Simple 
 
 Account of Man in Early Times. By Edward Clodd, F.R.A.S. 
 
 Third Edition. Globe Svo. 3^. Also a Special Edition for 
 
 Schools. iSmo. \s. 
 
 Professor Max Muller, in a letter to the Author, says: ''I read 
 
 yoicr book with great pleasure. I have no doubt it imll do f^ood, and I 
 
 hope you will continue your zvork. Nothing spoils our temper so mjich as 
 
 having to tmlearn in youth, manhood, and even old age, so many things 
 
 which we luere taught as children. A book like yours will prepare a far 
 
 better soil in the child's miftd, and I zms delighted to have it to read to 
 
 my children.''^ 
 
 Cooke (Josiah P., Jun.)— FIRST PRINCIPLES OF 
 CHEMICAL PHILOSOPHY. liy Josiah P. Cooke, Jun., 
 Ervine Professor of Chemistry and Mineralogy in Harvard College. 
 Third Edition, Revised and Corrected. Crown Svo. \2s. 
 
 Thorpe (T. E.)_A series OF CHEMICAL PROBLEMS, 
 for use in Colleges and Schools. Adapted for the preparation of 
 Students for the Government, Science, and Society of Arts Ex- 
 aminations. With a Preface by Professor Ro.'^coE. iSmo. 
 cloth. \s. Key. \s. 
 
 B 2 
 
20 EDUCATIONAL BOOKS. 
 
 SCIENCE PRIMERS FOR ELEMENTARY 
 SCHOOLS. 
 
 The necessity of commencing the teaching of Science in Schools at an 
 early stage of the pupil's course has now become generally recog- 
 nized, and is entorced in all Schools under Government inspection. 
 For the purpose of facilitating the introduction of Science 
 Teaching into Elementary Schools, Messrs. Macmillan are now 
 publishing a New Series of Science Primers, under the joint 
 Editorship of Professors Huxley, Roscoe, and Balfour 
 Stewart. The object of these Primers is to convey information 
 in such a manner as to make it both intelligible and interesting to 
 pupils in the most elementary classes. They are clearly printed on 
 good paper, and illustrations are given whenever they are necessary 
 to the proper understanding of the text. The following are just 
 published : — 
 PRIMER OF CHEMISTRY. By H. E. Roscoe, Professor of 
 Chemistry in Owens College, Manchester. iSmo. is. Third 
 Edition. 
 PRIMER OF PHYSICS. By Balfour Stewart, Professor of 
 Natural Philosophy in Owens College, Manchester. i8mo. \s. 
 Third Edition. 
 PRIMER OF PHYSICAL GEOGRAPHY. By Archibald 
 Gkikie, F.R. S., Murchison-Professor of Geology and Mineralogy 
 at Edinburgh. Second Edition. i8mo, is. 
 Everyone ought to knew something about the air we breathe and the 
 earth we live 7tpon, and about the relations between them; and in this 
 little work the author tvishes to shoiu what sort of questions may be put 
 about some of the chief parts of the book ofnattcre, and especially about tvjo 
 op them — the Air and the Earth. The divisions of the book are as 
 follozvs :—The Shape of the Earth— Day and Night— The Air— The 
 Circulation of Water on the Land — The Sea — The^ Inside of the Earth. 
 PRIMER OF GEOLOGY. By Professor Geikie, F.R.S. With 
 numerous Illustrations. Second Edition. iSmo. cloth, is. 
 In these Primers the authors have aimed , not so much to give inforina- 
 tioHy as to endeavour to discipline the mind in a xuay which has not 
 hithei'to been custojuary, by b}-itiging it into immediate contact ivith 
 Nature herself. For this purpose a series of sitnple experiments (to be 
 performed by the teacher) has been devised, leading up to the chief truths 
 of each Science. Thus the power of observation in the pupils will be 
 awakened and strengthe?ied. Each Man^ial is copiously illustrated, and 
 appended are lists of all the necessary apparatus, zoith prices, and 
 directions as to hon.v they may be obtained. Professor Huxley's introduc- 
 tory volume has been delayed through the illness of the author, but it is 
 now expected to appear very shortly. " They are wonderfully clear atui 
 lucid in their instruction, simple in style, and admirable in plan.''' — 
 Educational Times. 
 
MISCELLANEOUS. 21 
 
 Science Primers — co7itinucd. 
 
 PRIMER OF PHYSIOLOGY. By Michael Foster, M.D., 
 F.R. S. With numerous Illustrations. i8mo. \s. 
 
 In preparatio7i : — 
 
 INTRODUCTORY. By Professor Huxley. 
 PRIMER OF BOTANY. By Dr. Hooker, C.B., F.R.S. 
 PRIMER OF ASTRONOMY. By J. Norman Lockyer, F.R.S. 
 
 MISCELLANEOUS. 
 
 Abbott.— A SHAKESPEARIAN GRAMMAR. An Attempt to 
 
 illustrate some of the Differences between Elizabethan and Modern 
 
 English. By the Rev. E. A. Abbott, M. A., Head Master of the 
 
 City of London School. For the Use of Schools. New and 
 
 Enlarged Edition. Extra fcap. 8vo. 6s. 
 
 "A critical inquiry, conducted ivith great skill and knowledge^ and 
 
 zuith all the appliances of modern philology .... We venture to believe 
 
 that those who consider themselves most proficient as Shakespearians 
 
 will find something to learn from its pages.'" — Pall Mall Gazette. 
 
 " Valuable not 07tly as an aid to the critical study of Shakespeare, but 
 
 as tending to familiarize tlu reader with Elizabethan English in 
 
 gener-al.^'' — Athen^um. 
 
 Barker. — FIRST LESSONS IN THE PRINCIPLES OF 
 COOKING. By Lady Barker. i8mo. \s. 
 ^' An unpretending but invaluable little work .... The plan is 
 admirable in its completeness and simplicity ; it is hardly possible that 
 anyone who can read at all can fail to understand the practical lessons on 
 bread and beef, fish and vegetables ; while the explanation of the chemical 
 composition of our food must be intelligible to all who possess sufficient 
 education to follow the argtcment, in which thefeivest possible technical 
 terms are used.'''' — SPECTATOR. 
 
 Berners.— FIRST LESSONS ON HEALTH. By J. Ber- 
 NERS. iSmo. IS. Third Edition, 
 
 Besant. — STUDIES IN EARLY FRENCH POETRY. By 
 Walter Besant, M.A. Crown 8vo. Sj. dd. 
 '* In one moderately sized volume he has contrived to introdttce us to the 
 very best, if not to all of the early French poets.''— K-YYiY^iimjUL. ''In- 
 dustry, the insight of a scholar, and a gemmie enthusiasm for his subject, 
 combine to make it of very considerable value." — Spectator. 
 Breymann.— A FRENCH GRAMMAR BASED ON PHI- 
 LOLOGICAL PRINCIPLES. By Hermann Breymann, 
 Ph.D., Lecturer on French Language and Literature at Owens 
 College, Manchester. Extra fcap. 8vo. 4^. dd. 
 *' We dismiss the work with every expression of satisfaction. It can- 
 not fail to be taken into use by all schools which endeavour to make the 
 
2 2 EDUCA2I0NAL BOOKS, 
 
 study of French a means towards the higher cuUiirey — Educational 
 Times. ^^ A good, sound, valuable philological grajnmar. The author 
 presents the pupil by his method and by detail, with an enormous amount 
 of informatio7i about French not usually to be found in grammars, and 
 the information is all of it of real practical value to the student who 
 really wants to know French well, and to understand its spirit . . . At 
 tJu end a long chapter called ' Reasons and Illustrations ' forms an 
 exceedingly interesting and valuable dissertation 7ipo?i French philo- 
 Z^^/."— School Board Chronicle. 
 
 Calderwood.— HANDBOOK of moral PHILOSOniY. 
 
 By the Rev. Henry Calderwood, LL.l)., Professor of Moral 
 
 Philosophy, University of Edinburgh. Second Edition. Crown 
 
 8vo. 6s. 
 
 " A compact and useful work .... will be an assistance to many 
 
 students outside the author'' s own University.'^ — Guardian. 
 
 Deiamotte.— A BEGINNER'S drawing book. By p. H. 
 
 Delamotte, F.S.A. Progressively arranged. New Edition, 
 
 improved. Crown 8vo. 3^-. 6d. 
 '* We have seen and exa?nined a great many drazving-books, but the one 
 now before us strikes us as being the best of them all." — Illustrated 
 Times. ^* A concise, simple, and thoroughly practical zvork. The 
 letterpress is throughout intelligible and to the poitit." — Guardian. 
 
 Goldsmith. — the traveller, or a Prospect of Society; 
 and THE DESERTED VILLAGE. By Oliver Goldsmith. 
 With Notes Philological and Explanatory, by J. W. Hales, M. A. 
 Crown 8vo. (>d. 
 
 Green.— A history of the English people. By the 
 
 Rev. J. R. Green, M.A. For the use of Colleges and Schools. 
 Crown 8vo. 
 Hales. — longer English poems, with Notes, Philological 
 and Explanatory, and an Introduction on the T*eaching of English. 
 Chiefly for use in Schools. Edited by J. W. Hales, M.A., late 
 Fellow and Assistant Tutor of Christ's College, Cambridge, 
 Lecturer in English Literature and Classical Composition at King's 
 College School, London, &c. &c. Extra fcap. 8vo. ^. 6d. 
 
 Helfenstein (James).— a COMPARATIVE GRAMMAR 
 
 OF THE TEUTONIC LANGUAGES. Being at the same 
 time a Historical Grammar of the English Language, and comprising 
 Gothic, Anglo-Saxon, Early English, Modern English, Icelandic 
 (Old Norse), Danish, Swedish, Old High German, Middle High 
 German, Modern German, Old Saxon, Old Frisian, and Dutch. 
 By James Helfenstein, Ph.D. 8vo. i8j. 
 Hole.— A GENEALOGICAL STEMMA OF THE KINGS OF 
 ENGLAND AND FRANCE. By the Rev. C. Hole. On 
 Sheet. IS. 
 
MISCELLANEOUS. 23 
 
 Jephson. — SHAKESPEARE'S "TEMPEST." With Glossarial 
 and Explanatory Notes. By the Rev. J. M. Jephson. Second 
 Edition. i8mo. is, 
 
 Kington-Oliphant.— THE SOURCES OF STANDARD 
 ENGLISH. By J. Kington-Oliphant. Extra fcap. 8vo. ds. 
 " Mr.piipkant^s look is, to our 7?iind, one of the ablest and most 
 scholarly contributions to our standard English we have seen for many 
 years. . . . The arrangei)ient of the ivork and its indices make it in- 
 valuable as a work of reference, and easy alike to study and to store, when 
 studied, in the memory:'— "^QWQox. Board Chronicle, " Comes 
 nrarer to a history of the English language than anything that we have 
 seen since such a history could be written without confusion and con- 
 tradictionsy — Saturday Revi ew. 
 
 Martin. — the POET'S HOUR: Poetry Selected and Arranged 
 for Children. By Frances Martin. Second Edition. i8mo. 
 2j. (>d. 
 Nearly 200 Poems selected frovi the best Poets, ancient and modern^ 
 and intended mainly for children betzveen the ages of eight and twelve, 
 
 SPRING-TIME WITH THE POETS. Poetry selected by Frances 
 Martin. Second Edition. iSmo. y. 6d. 
 Intended mainly j or girls and boys bettveen the ages of twelve and seven- 
 teen. 
 
 Masson (Gustave).— a COMPENDIOUS DICTIONARY 
 OF THE FRENCH LANGUAGE (French-English and English- 
 French). Followed by a List of the Principal Diverging Deriva- 
 tions, and preceded by Chronological and Ili.-torical Tables. By 
 Gustave Masson, Assistant-Master and Librarian, Harrow 
 School. Square half-bound, 6s. 
 This volutne, though cast in the same form as other dictionaries, has 
 several distinctive features which increase its value for the student. In the 
 Erench- English fart, etymologies, founded on the researches of Messrs. 
 Littrc, Scheler, and Bracket, are given. The list of diverging deriva- 
 tions, at the end of the volmne, zvill be very useficl ^ to those ^ who are 
 uitcrested in tracing the various dcvtlofments of original Latin words. 
 But that zvhich makes it almost indispensable to students of the political 
 and literary history of France, is to be found at the beginning oj the work, 
 where M. Masson has drawn up clear and complete tables of historical 
 events., vienjed in connection with the developments of literature and lan- 
 guage, between the death of Charlemagne, 814 A.D., and that of Louis 
 Philippe, 1850. These tables are illustrated l)y remarks on the various 
 social moods, of which the works produced were the expression. Appended 
 also is a list of the principal Chronicles and Memoirs on the History of 
 Erance zvhich have appeared up to the present time; the French Re- 
 publican Calendar y compared zoith the Gregorian; and a Chronological 
 list of the principal Erench iVezvspapers published during the Revolution 
 and the Eirst Empire, 
 
2 4 EDUCATIONAL BOOKS. 
 
 Morris. — Works by the Rev. R. Morris, LL.D., Lecturer on 
 English Language and Literature in King's College School. 
 
 HISTORICAL OUTLINES OF ENGLISH ACCIDENCE, 
 
 comprising Chapters on the History and Development of the 
 
 Language, and on Word-formation. Third Ediiion. Extra fcap. 
 
 8vo. 6s. 
 
 " It makes an era in the study o/ the English tongue^ — SATURDAY 
 
 Review. ^^ He has done his work unth a ftdness and completeness 
 
 that leave nothing to be </^j-?>^./."— Nonconformist. ^' A genuine 
 
 and sound book. " — Athen^um. 
 
 ELEMENTARY LESSONS IN HISTORICAL ENGLISH 
 GRAMMAR, Containing Accidence and Word- formation. i8mo. 
 2s. 6d. 
 
 Oppen.— FRENCH READER. For the Use of Colleges and 
 Schools, Containing a graduated Selection from modern Authors 
 in Prose and Verse ; and copious Notes, chiefly Etymological. By 
 Edward A. Oppen. Fcap. 8vo. cloth. 4^. 6d. 
 
 Pylodet. — NEW GUIDE TO GERMAN CONVERSATION: 
 containing an Alphabetical List of nearly 800 Familiar Words 
 similar in Orthography or Sound and the same Meaning in both 
 Languages, followed by Exercises, Vocabulary of Words in 
 frequent use, Familiar Phrases and Dialogues; a Sketch of German 
 Literature, Idiomatic Expressions, &c. ; and a Synopsis of German 
 Grammar. By L. Pylodet. i8mo. cloth limp. 2s. 6d. 
 
 Sonnenschein and Meiklejohn. — the ENGLISH 
 METHOD OF TEACHING TO READ. By A. Sonnenschein 
 and J. M. D. Meiklejohn, M.A. Fcap. 8vo. 
 
 COMPRISING : 
 
 The Nursery Book, containing all the Two-Letter Words in 
 the Language. id. (Also in Large Type on Sheets for 
 School Walls. 5J-.) 
 The First Course, consisting of Short Vowels with Single 
 
 Consonants, ^d. 
 The Second Course, with Combinations and Bridges, con- 
 sisting of Short Vowels with Double Consonants, ^d. 
 The Third and Fourth Courses, consisting of Long 
 Vowels, and all the Double Vowels in the Language. 6d. 
 " These are admirable books, because they ateconstrtcctcdon a principle^ 
 and that the \simplest principle on which it is possible to learn to read 
 English. " — S ipECTATOR. 
 
 Taylor.— WORDS AND PLACES ; or, Etymological Illus- 
 trations of History, Ethnology, and Geography. By the Rev. 
 Isaac Taylor, M.A. Third and cheaper Edition, revised and 
 compressed. With Maps. Globe 8vo. 6j-. 
 Already been adopted by many teachers, and prescribed as a text-book in 
 the Cambridge Higher Exa7ninations for IVoincti. 
 
HISTORY. 25 
 
 Thring. — Works by Edward Thring, M.A., Head Master of 
 
 Uppingham. 
 THE ELEMENTS OF GRAMMAR TAUGHT IN ENGLISH, 
 
 with Questions. Fourth Edition. i8mo. 2s. 
 THE CHILD'S GRAMMAR. Being the Substance of "The 
 
 Elements of Grammar taught in English," adapted for the Use of 
 
 Junior Classes. A New Edition. i8mo. is. 
 SCHOOL SONGS. A Collection of Songs for Schools. With the 
 
 Music arranged for four Voices. Edited by the Rev. E. Thring 
 
 and H. Riccius. Folio. 7^-. 6d. 
 
 Trench (Archbishop). — HOUSEHOLD BOOK OF ENG- 
 LISH POETRY. Selected and Arranged, with Notes, by 
 R. C. Trench, D.D., Archbishop of Dublin. Extra fcap. 8vo. 
 ^s. 6d. Second Edition. 
 " The Archbishop has cottferred in this delightful vohime an impor- 
 tant gift on the whole Ejtglish- speaking population of the world.'''' — Pall 
 
 Mall Gazette. 
 
 ON THE STUDY OF WORDS, Lectures addressed (originally) 
 to the Pupils at the Diocesan Training School, Winchester. 
 Fom-teenth Edition, revised. Fcap. 8vo. 4^. ^d. 
 
 ENGLISH, PAST AND PRESENT. Eighth Edition, revised 
 and improved. Fcap. 8vo. 4.^. 6<:/. 
 
 A SELECT GLOSSARY OF ENGLISPI WORDS, used formerly 
 in Senses Different from their Present. Fourth Edition, enlarged. 
 Fcap, 8vo. 4J-. 6^/. 
 
 Vaughan (C. M.) — A SPIILLING BOOK OF WORDS 
 FROM THE POETS. By C. M. Vaughan. i8mo. cloth. 
 
 Whitney. — Works by William D. Whitney, Professor of San- 
 skrit and Instructor in Modern Languages in Yale College ; first 
 President of the American Philological Association, and hon. 
 member of the Royal Asiatic Society of Great Britain and Ireland ; 
 and Correspondent of the Berlin Academy of Sciences. 
 
 A COMPENDIOUS GERMAN GRAMMAR. Crown 8vo. 6j-. 
 
 A GERMAN READER IN PROSE AND VERSE, with Notes and 
 Vocabulaiy. Crown 8vo. 7^, 6^/. 
 
 Yonge (Charlotte M.)— the abridged book of 
 
 GOLDEN DEEDS. A Reading Book for Schools and General 
 Readers. By the Author of "The Heir of Redclyffe." i8mo. 
 cloth. IS. 
 
 HISTORY. 
 
 Freeman (Edward A.)— old -ENGLISH HISTORY. 
 By Edward A. Freeman, D.C.L., late Fellow of Trinity 
 College, Oxford. With Five Coloured Maps. Third Edition. 
 Extra fcap. Svo. half-bound. 6x. 
 
26 EDUCA770NAL BOOKS. 
 
 **Ihave, I hope,'' the author says, *' shown that it is perfectly easy to 
 teach children, frojn the very first, to distinguish trtie history alike from 
 legend and from wilful invention, and also to understand the nature of 
 historical authorities and to lueiiih one state77ient against another. I have 
 throughout striven to cofinect the history of Englaiid with the general 
 history of civilized Europe, and I have especially tried to make the 
 book serzie as an incentive to a more accurate study of historical 
 geography.''' In the present edition the whole has been carefully revised, 
 and such improraements as suggested themselves have been introduced. 
 ' • The book indeed is full of instruction and interest to students of all 
 ages, and he must be a well-ififormed man indeed xvho will not rise from 
 its pe}-usal with clearer and more accurate ideas of a too much neglected 
 portion of English History." — SPEcrATOR. 
 
 Historical Course for Schools. — Edited by Edward 
 
 A. Freeman, D.C.L., late Fellow of Trinity College, Oxford. 
 The object of the present series is to put forth clear and correct views 
 of history in simple language, and in the smallest space and cheapest 
 form in which it could be done. It is meant in the first place for 
 Schools ; but it is often found that a book for schools proves useful 
 for other readers as well, and it is hoped that this may be the case 
 with the little books the first instalment of wliich is now given to 
 the world. The General Sketch will be followed by a series ot 
 special histories of particular countries, which will take fc^r granted 
 the main principles laid down in the General Sketch. In every case 
 the results of the latest historical research will be given in as simple 
 a form as may be, and the several numbers of the series will all be 
 so far under the supervision of the Editor as to secure general ac- 
 curacy of statement and a general harmony of plan and sentiment ; 
 but each book will be the original work of its author, who will 
 be responsible for his own treatment of smaller details. 
 The first volume is meant to be introductory to the whole course. It 
 is intended to give, as its name implies, a general sketch of the history of 
 the civilized world, that is, of Europe, and of the lan'ds which have drawn 
 their civilization fnvn Europe. Its object is to trace out the general rela- 
 tions of difjei-ent periods and diffej'Oit countries to one another, tvithoul 
 going minutely into the a ff'airs of any particular country. TJiis is an 
 object of the first importance, for ivithout clear notio)is of genei'al history, 
 the history of particular countries can fiever be rightly undei^stood. The 
 narrative extends from the earliest movements of the Aryan peoples, down 
 to the latest events both on the Eastern and Western Continents. The 
 book consists of seventeen moderately sized chapters, each chapter bein<i 
 divided into a number of short numbered paragraphs, each with a title 
 prefixed clearly indicative of the subject of the paragraph. 
 
 I. GENERAL SKETCH OF EUROFEAN HISTORY. Py 
 Edward A. Freeman, D.C.L. Third Edition. i8mo. cloih. 
 
 3^. ed. 
 
 ^^ It supplies the great zuant of a good foundation for historical teach- 
 ing. The scheme is an excellent one, and this instalment has been 
 
HISTORY. 27 
 
 Historical Course for Schools — contmued. 
 
 executed in a ivay thai promises much for the vohwies that are yet to 
 appear."" — Educational Times. 
 
 II. HISTORY OF ENGLAND. By Edith Thompson. Fourth 
 Edition. i8mo. is. 6d. 
 
 ** Freedom Jrom preptdice, simplicity 0/ style, and accuracy of statement, 
 are the characteristics of this little volume. It is a. trustworthy text-hook 
 and likely to be generally serviceable in schools." — Pall Mall Gazette. 
 " Up07i the whole, this manual is the best sketch of English history for the 
 use op young people we have yet met with.''^ — ATHEN^TiUM. 
 
 III. HISTORY OF SCOTLAND. By Margaret Macartiiur. 
 l8mo. 2s. 
 
 "An excellent summary, unimpeachable as to facts, and putting them in 
 the clearest and most impartial light attainable.^'' — Guardian. ''^ Miss 
 Ma-carthjir has performed her task with admij-able care, clearness, and 
 fulness, and we have now for the first time a really good School History 
 of Scotland.'' — Educational Times. 
 
 IV. HISTORY OF ITALY. By the Rev. W. Hunt, M.A. i8mo. 
 
 ** Jt possesses the same solid merit as its predecessors . ... the same 
 scrup7clous care about fidelity i)i details. . . . It is distinguished, too, by 
 information on art, architectttre, and social politics, in ivhich the writer's 
 ^rasp is seen by the firmness and clearness of his touch." — Educational 
 Times. 
 
 V. HISTORY OF GERMANY. By J. Sime, M.A. iSmo. 3^. 
 *'^ A remarkably clear and impressive IIisto7y of Germany. Its great 
 
 events are xvisely kept as central figures, and the smaller clients are car ef idly 
 kept, not only subordinate and subservient, but most skilfully tvovcn into 
 the texture of the historical tapestry presented to the eye." — Standard. 
 The following will shortly be issued : — • 
 
 FRANCE. By the Rev. J. R. Green, M.A. 
 
 GREECE. By J, Annan Bryce, B.A. 
 
 AMERICA. By John A. Doyle. 
 
 Yonge (Charlotte M.)-~a barallel history of 
 
 FRANCE AND ENGLAND : consisting of Outlines and Dates. 
 
 By Charlotte M. Yonge, Author of "The Heir of Redclyffe," 
 
 *' Cameos of English History," &c, &c. Oblong 4to. 3^-. ^d. 
 ** We can imagine feiv more really advantageous courses of historical 
 study for a young mind than going carefully and steadily through Miss 
 Yonge'' s excellent little book."" — Educational Times. 
 CAMEOS FROM ENGLISH HISTORY. From Rollo to Edward 
 
 II. By the Author of "The Heir of Redclyffe." Extra fcap. 
 
 8vo. Second Edition, enlarged. 3J-. dd. 
 A book for young people ;ust beyond the elanentary histories of England, 
 and able to enter in some degree into the real spirit of events, and to be 
 struck tvith characters and scenes presented in some relief. " Instead of 
 
28 EDUCATIONAL BOOKS. 
 
 Yonge (Charlotte M.) — co7itinued. 
 
 dry details, we have living pictures, faithful, vivid,' and striking.'''' — 
 
 Nonconformist. 
 
 A Second Series of CAMEOS FROM ENGLISH HISTORY. 
 
 The Wars in France. Second Edition. Extra fcap. 8vo. 55-. 
 " Though mainly intended for young readers, they will, if we mistake 
 vot, be found very acceptable to those of more mature years, and the 
 life and reality imparted to the dry bones of history cannot fail to be 
 attractive to readers of eveiy age." — John Bull. 
 EUROPEAN HISTORY. Narrated in a Series of Historical Selec- 
 
 tions from the Best Authorities. Edited and arranged by E. M. 
 
 Sewell and C. M. Yonge. First Series, 1003— 11 54. Third 
 
 Edition. Crown 8vo. 6s. Second Series, 1088— 1228. Crown 
 
 8vo. 6^. Second Edition. 
 " We know of scarcely anything which is so likely to raise to a higher 
 level the average standard of English education.'^ — GUARDIAN. 
 
 DIVINITY. 
 
 *J^ For other Works by these Authors, see Theological Catalogue. 
 
 Abbott (Rev. E. A.)— BIBLE LESSONS. By the Rev. 
 
 E. A. A15R0TT, M.A., Head Master of the City of London School. 
 
 Second Edition. Crown 8vo. 4^. 6d. 
 
 *' Wise, suggestive, and really profound initiation into religious thought.'' 
 
 — Guardian. *^ I think nobody cmild read the?n without being both the 
 
 better for the^n himself, and being also able to see how this difficult duty op 
 
 imparting a sound religimis education may be efected." —BlSKOY OF St. 
 
 David's at Abergwilly. 
 
 Arnold.— A BIBLE-READING FOR SCHOOLS. The 
 Great Prophecy of Israel's Restoration (Isaiah, Chapters 
 40 — 66). Arranged and Edited for Young Learners. By Mat- 
 thew Arnold, D.C.L., formerly Professor of Poetry in the 
 University of Oxford, and Fellow of Oriel. Third Edition. iSmo. 
 cloth. IS. 
 ** There can be no doubt that it idll be found excellently caliulattxi to 
 further instruction in Biblical literature in any school into which it may 
 be introduced ; and we can safely say that whatn'er school uses the book, 
 it will enable its pupils to understand Isaiah, a great advantage compared 
 zvith other establishments which do not avail themselves of it.'' — TiMES. 
 ''Mr. Arnold has done the greatest possible service to the public. We tie7'er 
 read any translation of Isaiah tvhich interfered so little w'lth the musical 
 rhythtn and associations of our English Bible translation, while doing 
 so much to display the missing links in the connection of the parts.'" — 
 Spectator. 
 Golden Treasury Psalter. — students' Edition. Being an 
 
 Edition of "The Psalms Chronologically Arranged, by Four 
 Friends," with briefer Notes. i8mo. 3^. 6d. 
 
DIVINITY. 29 
 
 Hardwick. — a HISTORY OF THE CHRISTIAN CHURCH. 
 
 Middle Age. Yxova Gregory the Great to* the Excommunication 
 
 of Luther. Edited by William Stubbs, M.A., Regius Professor 
 
 of Modern History in the University of Oxford. With Four Maps 
 
 constructed for this work by A, Keith Johnston. Third Edition. 
 
 Crown 8vo. ioj-. ()d. 
 
 For this edition Professor Stubbs has carefully revised both text and 
 
 notes, making such correctio7ts of facts, dates, and the like as the results 
 
 of recent research warrant. The doctrinal, historical, and generally 
 
 speculative views of the late aicthor have been preserved intact. ^^ As a 
 
 manual for the student of ecclesiastical history in the Middle Ages,%ve 
 
 know no English work which can be compared to Mr. Hardwick's 
 
 book. " — Guardian, 
 
 A HISTORY OF THE CHRISTIAN CHURCH DURING THE 
 REFORMATION. By Archdeacon Hardwick. Third 
 Edition. Edited by Professor Stubbs. Crown 8vo. los. 6d. 
 
 Maclear. — Works by the Rev. G. F. MACLEAR, D.D., Head 
 
 Master of King's College School. 
 A CLASS-BOOK OF OLD TESTAMENT HISTORY. Eighth 
 
 Edition, with Four Maps. i8mo. cloth. 4J-. 6d. 
 " A careful and elaborate though brief compendium of all that modern 
 reseaj'ch has done for the illustration of the Old Testament. We knoiv 
 of no work which contains so much important information in so small 
 a compass." — BRITISH QUARTERLY Review. 
 A CLASS-BOOK OF NEW TESTAMENT HISTORY, including 
 
 the Connexion of the Old and New Testament. With Four Maps. 
 
 Fifth Edition. i8mo. cloth. 5^-, 6d. 
 "A singularly clear a?id orderly arrangcfjient of tlie Sacred Story. 
 Ills 7vork is solidly and co7)ipletely done," — AtheN/EUM. 
 A SHILLING BOOK OF OLD TESTAMENT HISTORY, 
 
 for National and Elementary Schools. With Map. l8mo. 
 
 cloth. New Edition, 
 A SHILLING BOOK OF NEW TESTAMENT HISTORY, 
 
 for National and Elementary Schools, With Map. i8mo, 
 
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 These works have been carefully abridged from the author's larger 
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 CLASS-BOOK OF THE CATECHISM OF THE CHURCPI OF 
 
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 ' ' It is indeed tfie work of a scfiolar and divine, and as such, tJiougfi 
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 mation ; and tha-e are not a fezo wJio would find it useful to tJiemselves 
 as well" — Literary Churchman. 
 A FIRST CLASS-BOOK OF THE CATECHISM OF THE 
 
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30 EDUCATIONAL BOOKS. 
 
 Maclear — continued. 
 
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 — ATHEN/EUM. 
 
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 M.A. Second Edition. iSmo. is. 6d. 
 
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DIVINITY. 31 
 
 Trench. — synonyms OF the new TESTAMENT. By 
 
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 Seventh Edition, car^fllIly revised, and with a considerable number of 
 new synonyms added. Appended is an Index to the Synonyms, and an 
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 INTRODUCTION TO THE STUDY OF THE FOUR GOSPELS. 
 
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32 EDUCATIONAL BOOKS. 
 
 Yonge (Charlotte M.)~scripture readings for 
 
 SCHOOLS AND FAMILIES. By Charlotte M. Yonge, 
 Author of " The Heir of Redely ffe." First Series. Genesis 
 to Deuteronomy. Globe 8vo. is. 6d. With Comments. Second 
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 Second Series. From Joshua to Solomon. Extra fcap. 
 
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 Actual ficed has led the author to endeavou?' to prepare a reading book con- 
 venient for study tuith children, containifig the very words of the Bible, xvith 
 only a frji) expedient omissions, and arranged in Lessons of such length as by 
 experience she has found to suit zuith children's ordinary power of accurate 
 attentive interest. The verse form has been retained, because of its con- 
 venience for children reading in class, and as more resembling their Bibles ; 
 hut the poetical poitions have been given in their lines. When Psahns or 
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 given in their chronological sequence. The Scripture portion, zuith a very 
 fav notes explanatory of mere words, is bound up apart, to be used by 
 children, while the same is also supplied with a brief comment, the purpose 
 of lijhich is either to assist the teacher in explaining the lesson, or to be 
 used by more advanced yotiug people to zuhom it may not be possible to give 
 access to the authorities whence it has been taken. Professor Huxley, at a 
 meeting of the London School Board, particularly mentioned the selection 
 made by Miss Yonge as an example of hoiv selections might be made from 
 the Bible for School Reading. .S"^^ Times, March 30, 187 1, 
 
 London: r. crAV. sons, and t.wlok, rKiNiKics. 
 
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