Page:Collier's New Encyclopedia v. 02.djvu/465

CENTER  at its center. Such a body has a true center of gravity. When such a point exists, it necessarily coincides with the center of inertia.  

 Center of Oscillation.—A heavy particle suspended from a point by a light, inextensible string constitutes what is called a simple or mathematical pendulum. For such a pendulum it is easily proved that the time of an oscillation from side to side of the vertical is proportional to the square root of its length for any small arc of vibration. A simple pendulum is, however, a thing of theory, as in all physical problems we have to deal with a rigid mass, and not a particle, oscillating about a horizontal axis. In a pendulum of this kind the time of oscillation will not vary as the square root of the length of the string, for it is obvious that those particles of the body which are nearest the point of suspension will have a tendency to vibrate more rapidly than those remote. The former are, therefore, retarded by the latter, while the latter are accelerated by the

former. There is thus one particle which will be accelerated and retarded to an equal amount, and which will, therefore, move as if it were a simple pendulum unconnected with the rest of the body. The point in the body occupied by this particle is called the center of oscillation.

As all the particles of the body are rigidly connected, they all vibrate in the same time. Hence it follows that the time of vibration of the rigid body will be the same as that of a simple pendulum, called the equivalent or isochronous simple pendulum, whose length is equal to the distance between the centers of suspension and oscillation.

The determination of the center of oscillation of a body requires the aid of the calculus. It may be stated, however, that it is always farther from the axis of suspension than the center of inertia, and is always in the line joining the centers of suspension and oscillation. Let A be the center of suspension, B the center of inertia, and C the center of oscillation, and let AB be equal to h, and k to the