PENDULUM, in mechanics, denotes a heavy body, so sus pended that it may vibrate, or ,swing backwards and for wards about a fixed point by the action of gravity.
When a heavy body, suspended in such a manner as to be at liberty to move round a fixed axis, is drawn aside from the vertical position, and abandoned to the action of gravity, its oscillations are all sensibly equal in point of duration. It is uncertain by whom this remarkable fact was first observed ; but the celebrated Galileo is univer sally acknowledged to have been the first who pointed out the important advantages that might be derived from it, by employing the pendulum to measure small determinate intervals of time. The general properties of the pendu lum were made known by the publication of the Systema Cosmicum, in 1632, and more particularly by that of the Dialogi de Motu, in 1639, after which time the instru ment began to be generally employed by astronomers ; but, for want of some convenient method of counting the number of oscillations, and of supplying the loss of velo city occasioned by the resistance of the air, and the fric tion of the axis, no great advantage was derived from its use till these defects were obviated by Huygens. Under the article HOROLOGY, a full account has been given of the ingenious mechanical contrivance, by means of which this distinguished philosopher connected the pendulum with the wheel work of a clock ; and of the methods which have since been devised to compensate the effects of a variation of temperature, on the materials of which it may be constructed, in order to render it a perfectly ac curate regulator of time-pieces.
The theory of the pendulum forms a branch of mecha nics, and has therefore been already considered under that article. Our principal object in the present article, is to give an account of some of the more recent and accurate experiments which have been made, with a view to de termine the length of a pendulum making a certain num ber of oscillations in a given time,—an element of great importance in physical astronomy. But as the pendulum in such experiments is always made to oscillate in the arc of a circle, in which case the oscillations in arcs of differ ent lengths are not exactly isochronous, it will be requisite previously to investigate the series which expresses the time of an oscillation in a circular arc of any amplitude whatever, in order to know the value of the required cor rection. We shall then briefly notice a few formula, im
mediately derived from the series which are required in deducing the length of the seconds pendulum, from one of which the length has been determined by experiment ; referring the reader, who is desirous of information with regard to the oscillation of pendulums in cycloidal arcs, to the article on MECHANICS, already mentioned, or to the profound and elegant treatise of Huygens, entitled Horo logiunz Oscillatorium.
In order to compare with greater facility the times of the oscillations of pendulums with each other, geometri cians have imagined a simple pendulum, composed of a heavy point of matter suspended at one extremity of a wire or thread without weight, inflexible and inextensi ble ; the other extremity being attached to a fixed point. To determine the time of an oscillation of such a pendu lum in a circular arc, let BA b, Plate CCCCLVIII. Fig. 1. be the arc in which it oscillates, C the centre of the circle, or point of suspension, CA a vertical line, and P the place of the material point in any part of the arc. Draw BD b and PE pperpendiculars to CA, and put the radius, or length of the pendulum equal I, AD=b, and AE=x. Then if t, S, and V, represent the time, space, and velocity respectively, the differential relation, accord ing to the theory of dynamics, is d t= But S=arc V therefore d S = d arc AP= d x, --- where the /2 x—x2 negative sign is taken, because the arc diminishes as t in creases. Also V, or the velocity acquired by descending from B to P, is equal to that acquired by falling through DE, or equal to V2 ti) . DE, being the accelerating force of gravity. Hence, since DE= b— x, —I d 4/(2 2 0 (b—x)' or, putting this fraction under another form, d x d t =— a.