734 In the case of a simple pendulum of length ?, we saw that the angular acceleration is gsind T Hence the motion of the compound pendulum will be identical with that of the simple pendulum when, and only when, As h and k are necessarily positive (or rather signless) quantities, the smallest value of I is evidently when k h. Hence the shortest time in which the mass can vibrate about any axis parallel to the original one corresponds to that of a simple pendulum of length 2k. When h is made either less or greater than , the length of the equivalent simple pendulum increases, and for any assigned value of I greater than 2k there are two corresponding values of h, one less and the other greater than &. Their sum, however, as we see by the coefficient of the second term in the equation is always equal to I. If then we can find two parallel axes in a rigid body, lying in one plane with the centre of inertia, and on opposite sides of that point, such that the time of oscilla tion is the same for each, the distance between them is the length of the equivalent simple pendulum. Kater made use of this proposition in his determination of the length of the second s pendulum, under the circumstances in which it was defined by Act of Parliament as a datum for restoring, in case of loss, the standard yard. Complex 242. Suppose now that a second body is attached to the first by an axis parallel to that about which the first is con strained to move ; and, for simplicity, suppose the centre of inertia of the first body to be in the plane containing the two axes. Here we have a complex compoimd pendulum, and it is interesting to compare the motion with that of the complex simple pendulum of 177, 178. Let m , h , (f> correspond, for the second body, to m, h, 6 for the first, and let a be the distance between the axes. For variety we will adopt Lagrange s method. We have clearly T = (mtfP + mh 2 $ 2 + m & 2 < 2 + m (a 2 ()~ + /t 2 2 + 2ah cos (0 - 6} <pd) , V = C - mgh cos 6 - m g(a cos + h cos </>) . These would enable us at once to write down the equations of motion, however large be the disturbance, but they are too complex for our present work. Let us then assume <f> and 6 to be very small, and we have com pound pendu him. m (lc 2 + h 2 )<p + m ah 8 = - m gh <f> . Combining, as before, by means of an undetermined multiplier, we have (m(& 2 + 7t 2 ) + m a 2 + m ah )& Thus the two values of A. are given by the equation m ah + m (k 2 + h 2 ) m h m(k^ + h 2 ) + m a 2 + m ah ~ mh + m a, This may be written in the form A where B is greater than A; and A, B, C are all essentially positive, if the bodies have been only slightly displaced from the position of stable equilibrium. The equation gives A 2 +(B-C)A-AC = 0, HO that the values of A are essentially real and of opposite signs. If we write p - B for A, this equation becomes so that the values of A + B are both positive, and therefore the motion of either mass is the resultant of two simple harmonic motions. 243. A well-known puzzle in connexion with this subject used to be "How to distinguish between two hollow shells, one of gold the other of silver, if their diameters and masses of holl be alike, and both be painted." If we observe that the sliells volumes of equal masses are inversely as the densities, the volume of the gold shell is seen to be less than that of the silver one, and therefore, on the whole, its mass is farther from the centre, and its moment of inertia greater. Hence any form of experiment in which the moment of inertia comes in will suffice to decide the question. Thus they might be alternately clamped tight to the end of a rod, and the system swung as a pendulum, when the gold sphere would vibrate more slowly than the silver one. Or they might be allowed to roll, not slide, down a rough plane. In this case the work done by gravity on each is the same when they have fallen through equal spaces. But its equivalent is in the form of kinetic energy, partly transla- tional and partly rotational. The relative amounts of these two depend on the moments of inertia of the spheres, for the ratio of the translational velocity to the angular velocity is the same for each. Hence the gold sphere, having the greater moment of inertia, will have the smaller velocity of translation. Another form of this question was to have a shell with a spherical mass inside, which might be either free to rotate on gimbals, or else be keyed to the outer skin. The keying would of course retard the motion of the whole down a rough plane, for part of the energy due to gravity would then be shared by the internal mass in the form of energy of rotation, from which it would other wise have been free. Another very instructive form is that of a spherical shell full of fluid. If the fluid be perfect, the moment of inertia io that of the shell alone ; if it be infinitely viscous, the moment of inertia is that of shell and fluid as if they constituted one rigid solid ; and we may have every intermediate amount. If we suppose the rotation of the outer shell to be suddenly stopped, the infinitely viscous contents would be reduced to rest also. But if they be not infinitely viscous they will not at once be brought to rest, but will be able to put the shell in rotation again if it be at once set free. Thus, in practice, we can tell a raw egg from a hard-boiled egg. The first is with difficulty made to rotate, and sets itself in motion again if it be stopped and at once let go. The second behaves, practically, like a rigid solid. 244. The problem of the rolling of a sphere down a rough inclined plane is solved at once, as above, by applying the conservation of energy. For, if x be the coordinate of its centre parallel to the plane, the angle through which it has turned, and a its radius, we have the kinematical condition x = ad (due to the perfect roughness of the plane). Also the potential energy lost is Mr/a; sin a, where a is the inclination of the plane to the horizon ; and the kinetic energy gained is made up of the two parts, -|M# 2 translational, and M& 2 2 rotational. Hence M(& 2 + a 2 )^ 2 = or x 2 = 2y^ k 2 + a? This shows that the motion is the same as that of a particle sliding down a smooth plane of the same inclina tion, under gravity diminished in the ratio a 2 : 2 + a 2 . And it shows how friction may retard motion without pro ducing any dissipation of energy. 245. Suppose one point of a rigid plane sheet be made to i.iove in any manner in the plane of the sheet, what will be the consequent rotation 1
Let M be the mass, and , 77, given in terms of t, the coordinates