MECHANICS 735 Vying c ." lint " n ? nf Till OI aody. f the point of the sheet whose motion is assigned. Let a, 6 be the relative polar coordinates of the centre of inertia, then M[ + rt(cos 0)] = X, M[7/ + a(sm0)} = Y, MF0 = - Yrt cos 6 + Xa sin ; where X and Y are the forces requisite to produce the motion. Eliminating them, we find (k" + a")0 = - (T;COS 6 - i si with which we can do no more until further data are specified. Suppose |, f to move with uniform acceleration^ in a direction assigned by o, then Jl=pcosa, rj=p sin a, and ( 2 + rt 2 )0 = -ffXsinacos0 ~ cos a sin 0) op sin (5- a) . The centre of inertia of the mass therefore moves, relatively to the constrained point, precisely as docs a simple pendulum ; but the direction of p is reversed. Again suppose the constrained point to move uniformly in a circle of radius b, with angular velocity ca. We have = bcostat, 77 = Ssinw<, and (k- + a-}0= + 2 a&(sin2cos0- cos cat sin 0) ,
at
This is, again, the equation of motion of a simple pendulum, but the angle of displacement cat is no longer measured from a fixed line but from the uniformly rotating radius of the guide circle. Hence the mass oscillates, pendulum-wise, about this uniformly revolving line. Gilistic 246. Let us take, as an instance of impulse, the case of jidu- Robins s " ballistic pendulum," a massive block of wood movable about a horizontal axis at a considerable distance above it, employed to measure the velocity of a cannon or musket shot. The shot is usually fired into the block in a horizontal direction perpendicular to the axis. The impulsive penetration is so nearly instantaneous, and the mass of the block so large compared with that of the shot, that the ball and pendulum are moving on as one mass before the pendulum has been sensibly deflected from the position of equilibrium. This is the essential peculiarity of the ballistic method, which is used also extensively in electromagnetic researches and in practical electric testing, when the integral quantity of the electricity which has passed in a current of short duration is to be measured. The line of motion of the bullet at impact may be in any direction whatever, but the only part which is effective is the component in a plane perpendicular to the axis. We may therefore, for simplicity, consider the motion to be in a line perpendicular to the axis, though not necessarily horizontal. Let m be the mass of the bullet, v its velocity, and p the distance of its line of motion from the axis. Let M be the mass of the pendulum with the bullet lodged in it, and k its radius of gyration. Then, if w be the angular velocity of the pendulum when the impact is complete, mvp = M 2 w , from which the solution of the question is easily determined. For the kinetic energy after impact is changed into its equivalent in potential energy when the pendulum reaches its position of greatest deflexion. Let this be given by the angle 6 ; then the height to which the centre of inertia is raised is 7i(l - cos 0), if h be its distance from the axis. Thus ttgh (1 - cos 0) = JMfcV = i -^f , . m p v or 2sm = ^4 --- 7=-, 2 M k J g h an expression for the chord of the angle of deflexion. In practice the chord of the angle 6 is measured by means of a light tape or cord attached to a point of the pendulum, and slipping with small friction through a clip fixed close to the position occupied by that point when the pendulum hangs at rest. 247. As another example of impulse let us consider the case of a body moving in any way in a plane perpendicular to one of its principal axes. It is required to fiid what point of the body must be suddenly fixed in order that the whole may be brought to rest ; also, what will be the consequent impulsive pressure at this point. It is easy to see that impulse this is exactly the same question as to find the impulse, required and its point of application, so that it may produce a for a given motion of a body in a plane perpendicular to one ^llioi of its principal axes. The impulse must obviously act in a plane passing through the centre of inertia. And the physical conditions are that the change of momentum of translation is equal to, and in the direction of, the impulse, while the change of moment of momentum about the centre of inertia is equal to the moment of the impulse. Let the impulse acting at the point , rj have components R, S parallel to rectangular coordinates in the plane of motion, and let w be the angular velocity, u, v the linear velocities, generated by it. Then the physical conditions are M = R, Mv = S, Mi 2 w = S|-Rij. When u, v, w are given, E and S are found from the first two equations, and the third is then the equation of the line in which the impulse must act. Similarly, when the impulse and its line of action are given, we have in terms of these data the quantities it, v, w. 248. As a simple practical example, suppose one strikes Centre of a hard object with a stick in such a way that his hand is at percus- rest at the instantof the impact ; with what part of the S10D> stick must he strike so that there may be no jar on his hand] Let be measured along the stick from its centre of inertia in the direction which it has at the instant of impact. Then the kinematical condition is v = aca , where a, is the distance from the hand to the centre of inertia of the stick. Thus, as the impulse S is the sole cause of the stick s being brought to rest, we have so that = k"/a . Hence if the stick be uniform and be held by one end, so that its length is 2a, and therefore 3& 2 = 2 , we have and a + , the distance of the point of impact from the hand, is a+ ^ = f . 2a , ie., it is at two-thirds of the length of the stick. If, however,the hand be moving, at the instant of impact, perpendicularly to the stick with velocity V, the kinematical condition is v V = w, which introduces a corresponding change in the result. 249. The reaction of the axis is easily calculated. If Reaction the axis about which the body is constrained to rotate on axis. be perpendicular to a plane though the centre of inertia about which the body is symmetrical, and if the applied forces act in that plane, it is clear that the reaction of the axis is a single force in that plane. Let its components be H and H. Then y - yx) Let a, 0, be the polar coordinates of the centre of inertia, then 6, are the angular velocity and the angular accelera tion for all particles of the mass, and we have From the third we find Q, and the others give 5 and H. When there is no plane of symmetry perpendicular to the axis, there must be two points of it at which reactions are exerted on the revolving body. Let the coordinates of these bearings be c, c, and
the reactions there H, H, Z, H , H , Z respectively.