PENDULUM. Hatay heavy body, sus pended 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 reach ing the vertical position it will have ac quired a velocity equal to that which it would have acquired by falling vertically through the versed sine of the arc it has described, in consequence of which it will continue to move in the same aro until the whole velocity is destroyed ; and, if no other force than gravity acted, this would take place the body reached a height on the opposite side of the vertical equal to the 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 resist ance of the air, each successive excursion will be diminished and 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 passage from the greatest distance from the vertical on the one side to the great est distance on the other is called an os cillation.
In order to investigate the circumstan ces of the motion, the body must be re garded as a gravitating point, and the in flexible rod as devoid of weight. This is denominated the simple pendulum, and the problem to be resolved is to deter mine the motion of a point constrained to move in a circular arc in virtue of the accelerating force of terrestrial gravity.
According to the theory of falling bodies (see Guavrrv,) the time t in which a body falls through the space a, by the by the equation 2s = /; then But the time T, of the oscilla tion of a pendulum whose length is 1, is T = — ; therefore T t r: 1; con sequently 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 circum ference of a circle to its diameter.
If we suppose that the time to be ex pressed in seconds, and make T=1, we shall have g= a° 1. Now, Captain later found the length of the same pendulum at London to be 39.13929 inches, and we know that = 9.8696 ; therefore g = 9.8696 X 39.139 = 386.239 inches, or g 32.2 feet. It follows, therefore, that the space through which a body falls freely at London in a second time is 161 feet.
Compound Pendulum. The simple as above defined, is only a theoretical abstraction ; for the oseilla, ting 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 pen Sedum ; and instead of being supported by a single point it is supported an axis, or by a series of points situated in the same straight line. According to this definition, any heavy body oscilla ting about an—axis of suspension is a compound pendulum.
In every compound pendulum there is necessarily a certain point nt which if all the matter of the pendulum were collec ted the oscillations would be performed in exactly the same time. This point is the centre of oscillation. (See CENTRE OF 08. CILLILTION.) It is situated in the vertical plane passing through the centre of grav ity of the pendulum, and at a distance from the axis of suspension (the axis being always supported horizontal,) which is determined by the following formula: Let d m be the element of the mass of the compound pendulum, r its distance from the axis of rotation, and x the distance of the centre of oscillation from the same axis ; then x=f d r d na • that is, the distance of the centre of OS. ciliation from the axis of suspension is equal to the moment of inertia of the os cillating body divided by its moment of rotation. This value of x is the length of the isochronous simple pendulum, and is what is always to be understood by the term length of a pendulum ' The centre of oscillation possesses a very remarkable property, which was discovered by Huygens ; namely, that if the body be suspended from this point; or a horizontal axis passing through it parallel to the former axis of suspension, its oscillations will be performed in the same time as before ; in other words, the axis of suspension and oscillation are interchangeable. This property furnish es an easy practical method of deter mining the centre of oscillation, and thence the length of a compound pen dulum.