PENDIILIIII, in its widest scientific sense, denotes a body of any form or material which, under the action of some force, vibrates about a position of stable equilibrium. In its more usual application, however, this term is restricted, in conformity with its etymology (Lat. penden, to hang). to bodies suspended from a point, or oscillating about an axis, under the action of gravity, so that, although the laws of their motion are the same, rocking-stones (q.v.), magnetic needles, turning-forks, balance wheel of a watch, etc., are not included in the definition.
The simple pendulum consists (in theory) of a heavy point or particle. suspended by a flexible string without weight. and therefore constrained to Move as if it were always on the inner surface of a smooth spherical bowl. If such a pendulum be drawn aside into a slightly inclined position, and allowed to fall back, it evidently will oscillate from side to side of its position of equilibrium, the motion being confined to a vertical piano. If, instead of being allowed to fall back. it be projected horizontally in a direction per pendicular to that in which gravity tends to move it, the bob will revolve about its low est position; and there is a particular velocity with which, if it be projected, it describes a circle about that point, and is then called a conical pendulum. As the theory of the simple pendulum can be very easily explained by ref erence to that of the conical pendulum, we commence with the latter, which is extremely simple. 'l'o find the requisite velocity, we have only to notice that the (so-called) centrifugal force (q.v.) must balance the tendency towards the vertical. This tendency is not directly due to gravity, but to the tension of the sus pending cord. In the fig. let 0 be the point of sus pension, OA the pendulum in its lowest position, P the bob in any position in the (dotted) circle which it describes when revolving as a conical pendulum: PB, a radius of the dotted circle, is evidently perpendicu lar to OA. Now, the centrifugal force is directly as the radius PB of the circle, and inversely as the square of the time of revolution. Also the radius PB is PO sin. BOP, the length of the string multiplied by the sine of the angle it makes with the vertical; and the force towards the vertical is proportional to the earth's attraction, and to the tangent of the above angle—as may be at once seen from the consideration that the three forces acting on the bob at P are parallel, and therefore proportional, to the sides of the triangle OBP. Hence the square of the time of revolution is directly as the length of the and the sine of the angle BOP, and inversely as the earth'S attraction and the tangent of the same angle; or (what is easily seen to be equivalent) to the length of the string and the cosine of its inclination to the vertical directly, and to the earth's attraction inversely. Hence, in any given locality,
all conical pendulums revolve in equal times, whdtevcr be the lengths of their strings, so long as their heights are equal; the height being the product of the length of the string by the cosine of its inclination to the vertical. Also the squares of the dines of revolu tion of conical pendulums are as their heights directly, and as the earth's attraction inversely.
Now, so long as a conical pendulum is deflected only through a very small angle from the vertical, the motion of its bob may be considered as compounded of two equal simple pendulum oscillations in directions perpendicular to each other, such as it appears to make to an eye on a level with it; and viewing it at some distance, first from one point, say on the n., and then from another 90° round, say on the east. And these motions take place, by Newton's second law (see Moviox, LAws oF), independently. Also the time of a (double) oscillation in either of these directions is evidently the same as that of the rotation of the conical pendulum. Hence, for small arcs of vibration, the square of the time of oscillation of a simple pendulum is directly as its length, and inversely as the earth's attraction. Thus, the length of the second's pendulum at London being 39.1393 in., that of the half-second's pendulum is 9.7848 in., or one-fourth, that of the two-seconds' pendulum 156.5572 in., or four times that length. It follows from the principal now demonstrated, that so long as the arcs of vibration of a pendulum are all small relatively to the length of the string, they may differ considerably in length among themselves without differing appreciably in time. It is to this property of pen dulum oscillations, known as isochronism (q.v.), that they owe their value in measuring; time. See kThat the times of vibration of different pendulums are as the square roots of their lengths, may be demonstrated to the eye by a very simple experiment. Suspend three musket balls on double threads as in the figure, so that the heights in the dotted line may be' as 1, 4, and 9. When they are made to vibrate simultaneously, while the lowest ball makes one oscillation the highest will be found to make three, and the middle ball one and a half.