CENTER of OSCILLATION. Referring to the article PENDULUM, the reader will see that the time of a pendulum's vibration increases with its length, being always propor tioned to the square root of its length. This is strictly true only of the simple pen dulum, in which the pendulous body is supposed to have no determinate magnitude, and to be connected with the point of suspension by an inflexible wire without weight. If, however, the vibrating body have a determinate magnitude, then the time of vibra tion will vary, not with the square root of its length, but with the square root of the distance from the axis of suspension of a point in the body called its center of oscillation.
If each part of the vibrating body were separately connected with the axis of sus pension by a fine thread, and entirely disconnected from the rest of the body, it would form an independent simple pendulum, and oscillate as such—the time of each vibra tion being as the square root of the length of its thread. It follows that those particles of the body which are nearest to the axis of suspension would, as simple pendulums, vibrate more rapidly than those more remote. Being connected, however, as parts of the solid body, they vibrate all in the same time. But this connection does not affect their tendencies to vibrate as simple pendulums, and the motion of the body which they compose is a compromise of these tendencies of its particles. Those nearest the axis are retarded by the more remote, while the more remote are urged on by the nearer. Among these particles there is always one to be found in which the
accelerating and retarding effects of the rest are mutually neutralized, and which vibrates in the same time as it would if it were unconnected with the other parts of the body, and simply connected by a flue thread to the axis of suspension. The point in the body occupied by this particle is its center of oscillation. By this C. of 0. the calculations respecting the vibration of a solid body are rendered as simple as those of a molecule of inconsiderable magnitude. All the properties which belong to a simple pendulum may be transferred to a vibrating body of any magnitude and figure, by considering it as equivalent to a single particle of matter vibrating at its centre of oscillation.
The determination of the position of the C. of 0. of a body usually requires the aid of the calculus. It is always further from the axis of suspension than the center of gravity is, and always in the linejoining the center of gravity and the point of sus pension, when the body is suspended from a point. The rule for finding it in such a case is: If S be the point of suspension, and 0 the C. of 0., S0=- (md=) or it is the M.Sy; quotient obtained by dividing the moment of inertia of the body by the product of its mass into the distance of its center of gravity from the point of suspension.