OSCILLATION and CENTRE OF OSCILLATION. When any system is slightly disturbed from its position of equilibrium, it either falls altogether or endeavours to recover the position which it lost.
In the latter case the equilibrium is said to be stable, and in the former unstable. A pendulum banging downwards is an instance of the stable kind, and the same pendulum, if it could be so nicely adjusted as to rest immediately over the pivot, would be of the unstable kind, [STABLE AND UNSTABLE.] When a system endeavours to recover its position, it acquires some velocity in the process ; so that, though it would rest at the position of equilibrium if that velocity were then destroyed, it is really urged through the position by the velocity acquired, and continues to depart from it on the other side until, by the forces which act to restore it to the position, all the velocity acquired has been destroyed. Repetitions of the same phenomenon then take place in succession, the body never remaining still when it has attained the position of equilibrium, since it never is in that position except when moving with the velocity acquired in its descent to that position. If then there were neither friction nor resistance of the air to help in destroying this velocity, it would be a universal law of mechanics that a system disturbed from its position of equilibrium would never recover it, but would make perpetual oscillations about it In the widest sense, the problem of oscillations includes most of those which occur in astronomy, optics, &c. The moon and planets add to their average motions small oscillations about their mean places : the tides consist of oscillations of the ocean about the uniform spheroid, which, but for the action of the heavenly bodies, would be carried round in the diurnal rotation of the earth ; the phenomena of light are produced by the oscillations which take place in an elastic tether ; those ..of sound by the oscillations of the air ; and so on.
Usually, however, the problem of oscillation refers to nothing more than the oscillations of a solid system, acted on by gravity, about a horizontal axis, the original departure from the position of equilibrium being but small ; In fact, to the purely theoretical part of the problem of the PENDULUM, to which we shall here confine ourselves, giving the investigations in a brief form, since it is impossible, within our limits, to explain the numerous points alluded to with sufficient illustration for a learner.
Let a material point, a very small body, be attached by a string or rod without weight to an immovable pivot. In the position of rest the string hangs vertically : let us now suppose it removed out of the vertical position, and let go when it makes an angle a with the vertical. When t seconds have elapsed, let it make an angle 0 with the vertical.
The material point is acted on by gravity with a force which would produce an acceleration g (or 321908) feet nearly, in one second : if then 1 be the length of the string, 1 B is the arc, through which the point must move before it arrives at the lowest point of its course, and we have by the well-known equations of motion,