Page:Elementary Text-book of Physics (Anthony, 1897).djvu/56

through the centre of mass. The kinetic energy of the body rotating about this centre is ${\frac {\omega ^{2}}{2}}\sum ml^{2}$, and the kinetic energy of the whole body moving with the velocity of the centre of mass is ${\tfrac {1}{2}}\omega ^{2}R^{2}\sum m.$. By § 31 we have

${\tfrac {1}{2}}\omega ^{2}\sum mr^{2}={\tfrac {1}{2}}\omega ^{2}R^{2}\sum m+{\tfrac {1}{2}}\omega ^{2}\sum ml^{2}.$

(29)

When a rigid body is so small that its kinetic energy due to its rotation about its centre of mass is negligible in comparison with that due to its translation, it is called a particle. This definition supplements that of § 35.

38. Moment of Inertia.— The expression $\sum mr^{2}$ is called the moment of inertia of the body about the axis from which $r$ is measured. The formula just obtained shows that the moment of inertia about any axis is equal to the moment of inertia about a parallel axis passing through the centre of mass plus the moment of inertia of a particle of which the mass is equal to the mass of the body placed at the centre of mass.

The moment of inertia depends entirely upon the magnitude of the masses making up the body and their respective distances from the axis. If the mass of the body be distributed so that each element of volume contains a mass proportional to the volume of the element, the moment of inertia then becomes a purely geometrical magnitude, and may be found by integration.

It is evident that it is always possible to find a length $k$ such that $k^{2}\sum m=\sum mr^{2}.$ This length $k$ is called the radius of gyration of the body about its axis.

The moment of inertia of any body, however irregular in form or density, may be found experimentally by the aid of another body of which the moment of inertia can be computed from its dimensions. We will anticipate the law of the pendulum—which has not been proved—for the sake of clearness. The body of which the moment of inertia is desired is set oscillating about an axis under the action of a constant force. Its time of oscillation is, then, $t=\pi {\sqrt {\frac {I}{f}}},$