CENTER, a point equidistant from the extremities of an object. Among its best known applications are Center of Inertia.—If in, and 1111, be the masses of two particles placed at the points A, and and if the right line A,A, be divided in B,, so that the point B, is called the center of iner tia, or center of mass, of the two parti cles. If in, be a third mass at A,, and if B,A, be divided in B,, so that (m, x in4B,B,=m,A,B2, B, is called the center of inertia of the three particles. In general, if there be any number of particles, a continuation of the above process will enable us to find their center of inertia. Every body may be supposed to be made up of a multitude of particles connected by co hesion. From this it is obvious that the center of inertia is a definite point for every piece of matter.
Center of Gravity.—If a body be suffi ciently small, relatively to the earth, the weights of its particles may be consid ered as constituting a system of parallel forces acting on the body. Now, the magnitude of the weight of a particle is proportional to its mass. Hence, the line of action of the resultant of the parallel forces will approximately pass through the center of inertia. For this reason such bodies are said to have a center of gravity. Strictly speaking, there is no such point of necessity for every body, since the directions of the forces acting on the body are not accurately parallel. Hence, it is only approximately that we can say of a body that it has a center of gravity. On the other hand, every piece of matter has, as is shown above, a center of inertia. For all heavy bodies of moderate dimensions it is, however, sufficiently accurate to assume that the center of inertia and gravity coincide. For example, the center of gravity of a uniform homogeneous cylinder with par allel ends is the middle point of its axis, that of a uniformly thin circular lamina its center, and so on.
The center of gravity of a body of moderate dimensions may be approxi mately determined by suspending it by a single cord in two different positions, and finding the single point in the body which, in both positions, is intersected by the axis of the cord.
The term center of gravity is also used in a stricter sense than the one just explained. Thus, if a body attracts and is attracted by all other gravitating matter as if its whole mass were con centrated in one point, it is said to have a true center of gravity at that point, and the body itself is called a centro baric body. A spherical shell of uniform gravitating matter attracts an external narticle as if its whole mass were con densed 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 heavy parti cle suspended from a point by a light, in extensible string constitutes what is called a simple or mathematical pendu lum. For such a pendulum it is easily proved that the time of an oscillation from side to side of the is pro portional to the square root of its length for any small arc of vibration. A simple pendulum is, however, a thing of theory, 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 pendu lum, called the equivalent or isochronous simple pendulum, whose length is equal 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 os cillation will not vary as the square root of the length of the string, for it is ob vious that those particles of the body which are nearest the point of suspen sion 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 to the distance between the centers of suspension and oscillation.
The determination of the center of os cillation 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 cen ters 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 radius of gyration of the body about an axis through B parallel to the fixed axis, then it is easily shown that AC= h From this there follows the important proposition that the centers of oscillation and suspension are convertible, a propo sition which was taken advantage of by Kater for the practical determination of the force of gravity at any station.