PENDULUM, in mechanics, denotes any heavy body, so suspended as that it may vibrate or swing backwards and for wards, about some fixed point, by the force of gravity. The vibrations of a pendulum are called its oscillations. See OsciLLA•ION. A pendulum, therefore, is any body, B, (Plate XII. Miscell. fig. 8), suspended upon, and moving about, a fixed point, A, as a centre. The na ture of a pendulum consists in the follow ing particulars : 1. The times of the vi brations of a pendulum, in very small arches, are all equal. 2. The velocity of the bob, in the lowest point, will be near ly as the length the chord of the arch which it describes in the descent. 3. The times of vibration in difibrent pendulums, A B, A C, are as the square roots of the times of their vibrations. 4. The time of one vibration is to the time of the de scent, through half the length of the pen dulum, as the circumference of a circle to its diameter. 5. Whence the length of a pendulum, vibrating seconds, will be found 39.2 inches nearly ; and that of an half second pendulum 9.8 inches. 6. An uniform homogeneous body B G (fig. 9) has a rod, staff &c. which is one-third part longer than a pendulum A 1), will vibrate in the same time with it.

From these properties of the pendu lum we may discern its use as an univer sal chronometer, or regulator of time, as it is used in clocks, and such-like ma chines. See CHRONOMETER, HOHOLOGY, &c.

By this instrument also we can mea sure the distance of a ship, by measuring the interval of time between the fire and the sound of the gun; also the distance of a cloud, by numbering the seconds, or half-seconds, between the lightning and thunder. Thus, suppose between the lightning and thunder, we number 10 seconds ; then, because sound passes through 1142 feet in one second, we have the distance of the cloud equal to 11420 feet. Again, the height of any room, or other object, may be measured by a pen dulum vibrating from the top thereof. Thus, suppose a pendulum from the height of a room vibrates once in three seconds ; then say, as 1 is to the square of 3, viz. 9, so is 39.2 to 352.8 feet, the height required. Lastly, by the pen

dulum we discover the different force of gravity on different parts of the earth's surface, and thence the true figure of the earth.

When pendulums were first applied to docks, they were made very short ; and„ the arches of the circle being large, the time of vibration through different arches could not in that case be equal ; to effect which, the pendulum was contrived to vibrate in the arch of a cycloid, by mak-. ing it play between two semi-cycloids G B, C D (fig. 10), whereby it describes the cycloid BEAD; the property of which curve is, that a body vibrating in it will describe all its arches, great or small, in equal times. These are, how ever, which concur in rendering the ap plication of this curve to the vibration of pendulums designed for the measures of time, the source of errors even greater than those which by its peculiar proper ty it is intended to obviate, and it is now not used.

Although the times of vibration of a pendulum in different arches be nearly equal, yet if the arches differ very con siderably, the vibrations will be perform ed in different times, and the dilIerence, though very small, will become sensible in the course of one day or more. In clocks for astronomical purposes, the arc of vibration must be accurately ascer tained, and if it be different from that described by the pendulum, when the clock keeps time, a correction must be applied to the time shown by the clock. This correction, expressed in seconds of time, will be equal to the half of three times the difference of the square of the given arc, and of that of the arc de scribed by the pendulum when the clock keeps time, these arcs being expressed in degrees ; and so much will the clock gain or lose, according as the first of these arches is less or greater than the second. Thus, if a clock keeps true time when the pendulum vibrates in an arch of 3°, it will lose 101 seconds daily in an arch of 4°, and 24 seconds in an arc of 5°, for 4 2-3 2 xi=7 Xi= 104 and generally B2 — A2 X 2 gives the time lost or gained. See Simpson's Flux ions, vol. ii. prob. xxviii.