PENDULUM

(Lat.pendulus; from pen do, 7 Aa»^.) If any heavy body, suspended by an inflexible rod from a fixed point, be drawn aside from the vertical position, and then let fall, it will descend in the arc of a circle of which the point of suspension is the centre. On reaching the vertical position it will have acquired a velocity equal to that which it would have acquired by falling vertically through the versed sine of the arc it has described, m consequence of which it will continue to move in the same arc until the whole velocity is destroyed; and, if no other force than gravity acted, this would take place when the body reached a height on the "PP"s»^.e /''|e of the vertical equal to the height from which It fell Having reached this height it would again descend, and so continue to vibrate for ever; but in consequence of the friction of the axis, and the resistance of J Ji.""v^^'^" succe ive excursion will be diminished, ajid the body soon be brought to rest in the vertical position. A body thus suspended, and caused to vibrate, is called a pendulum; and the pa age from the greatest distance from the vertical on the one side, to the greatest distance on the other, is called an oscillation. *u w^j^^"" to investigate the cir cums Unces of the motion, the bodv must be regarded as a gravitating point, and the indexible rod as devoid of weight. This is denominated the simple pendulum; and the problem to be resolved is to determine the motion of a point constrained to move in a circular arc in virtue of the accelerating force of terrestrial gravity. Let C be the fixed point, A D B the arc described by the pendulum, C D the vertical, and P the place of the pendulum at a given instant. Draw P E perpendicular to C D, and put D E=.r, E P=y, the arc D P = s, D H (the versed sine of the arc D A) = //, and the radius CD = /; also let « = the velocity at P, and i = the time of descent through A P. The equation of the circle being y' = 2lx — x'2, we have the diflferential equation di=., -!^~ -.and bv V (2 tx — j:'^) ' ' •' the doctrine of falling bodies, the velocity at P is that!)( 6 PENDULUM. which a body would acquire in falling through H E, or equal to v/ 2g H E {g being the accelerating force of gravity): therefore v mm. ^^2g{h — x). But dt——: thereV fore, on substituting the values of d s and v above given, we get the equation dt= — • rf-r V^g V{{h-x){1lx-x'^)] This equation, being developed and integrated, gives the tinie of descent from A to D, or the time of a semi-oscillation, as follows (tr being the ratio of the circumference to the diameter, or = 3-14159): — When the arc of vibration is small, it is only nece ary to take account of the first two terms of the series; whence the time of a whole oscillation from A to B, which we shall denote by T, is given by the formula — and if the arc is so small that h (its versed sine) becomes evanescent in comparison of 8 /, we have simply T = jt a/ Whence it appears that the time of an oscillation in an infinitely small arc is directly as the square root of the length of the pendulum, and Jnversely as the square root of the accelerating force of gravity. Suppose gravity to be a constant force (as it is at any given place on the earth's surface), the time of oscillation is proportional to the square root of the length of the pendulum; consequently, if T' denote the time of oscillation of a seconds' pendulum, and I' its length, we have T: T':: \/l: \/l'. Now, let N be the number of oscillations the first pendulum makes in a given time, and N' the number the second makes in the same time; then the number of oscillations being evidently in the inverse ratio of the times, we have N: N':: T': T; or N'2 N: N':: v^Z': \/l, whence / = -— ■ I' • so that if the length of one pendulum be known, we can compute the length of another pendulum by observing the number of oscillations that each makes in any given time; for example, an hour. In like manner, if we suppose the same pendulum to be transported to different parts of the earth (in which case I is constant and g variable), and a ume that the values of N and g at the first station become respectively N' andg' at the second, we shall have T: T':: —: —; T2 N'2 ^^ ^^' whence g' = Twj^gS org' = -^g- So that the force of gravity at dif To rent places may be compared by observing the number of oscillations which a pendulum makes at each place in a given time. Or, if we suppose the times of oscillation constant (that is, if we suppose two pendulums so adjusted as to beat seconds at two different stations), and / and g variable; then, since T = a- 1/-, and I' II', ^ T' = 3- V —, we shall have -= —; or/'=Z—; whence the g g g' g length of the seconds' pendulum at any place is directly proportional to the intensity of gravity at that j)lace. According to the theory of falling bodies (see Guavity), the time t in which a body falls through the space *, by the accelerating force of gravity, is given by the equation t = \/—. Let 2 s = /; then f = v' - • But the time T,of g. g the oscillation of a pendulum whose length is /, is T=t \/-. g therefore 1: t:■: -x: 1; consequently the time of the oscillation of a pendulum is to the time that a heavy body would fall freely by the force of gravity through Half its length, as the circumference of a circle to its diameter. If we suppose the time to be expre ed in seconds, and make T = 1, we shall have ^ = jr2 /. Now Captain Kater found the length of the simple pendulum at London to be 39-13929 inches, and we know that tr" = 9-8696; thejefore^ = 9-8696 x 39-139 = 386-29 inches, or ^ = 32-2 feet. It follows, therefore, that the space through which a body falls freely at London in a second of time is 16-1 feet. Compound Pendulum. — The simple pendulum, as above defined, is only a theoretical abstraction; for the oscillating body can neither be so small that it may be regarded as a mathematical point, nor can the rod be entirely devoid of weight. When the body has a sensible magnitude, and the suspending rod a 'sensible magnitude and weight, as they must have in all actual constructions, the apparatus is called a compound pendulum j [s. 919]