PENDULUM (Neo-Lat. nen. sg. of Lat. pen (lulus, hanging down, from pendere, to hang). A pendulum is any rigid body pointed about a horizontal axis so that it can swing to and fro, making vibrations under the influence of gravity. A pendulum is the name given an imagi nary pendulum consisting of a particle of matter suspended from a point by a cord supposed to be without mass. If the length of this cord is 1 and if q is the which the suspended particle would have if allowed to fall freely to ward the earth. the period. or time taken for one complete vibration of the simple pendulum • swinging in a vertical plane through very small arcs, The period of any actual or pound' pendulum is given by the formnla 27F where I is the moment of inertia of A the pendulum about its axis of suspension, or is its mass, / is the distance from the centre of gravity to the axis, and g has it. usual value. It is seen that the period of vibration of a. pendulum is constant, being independent of the extent of the amplitude of vibration, provided only that it ' he small eompared with 1. This fact was made use of by Galileo, who also recognized the con nection between the period and the length of a pendulum.
The length of the pendulum, as will be seen from the formula, varies as the square of the • time of oscillation, and consequently a pendulum with a vibration period of one-half second has but one-quarter the length of a seconds pendu lum. It is also apparent that pendulums of dif ferent kinds of matter—but iu other• respects identical—should have the same periods of vibra tion if g is the same for the different kinds of matter. This idea was tested by Newton and • later by Bessel, who found that within experi mental limits g is the same for all forms and kinds of matter. If g is different at different places on the earth, the fact will be shown by measuring the period of the same pendulum at these places. This was shown to be the case by Huygens.
The problem of deducting the length of a sim ple pendulum which should have the same period as a given compound pendulum was first solved by Huygens. See CENTRE or (ism...I–mos and :Mren If the same pendulum is swung at various positions on the earth's surface, its time of vibra tion will vary depending on the value of g. Accordingly one of the best methods of deter mining this quantity is to cause a pendulum to swing at different stations and carefully and exactly measure its time of oscillation. In the United States this work is done by the States Coast and Geodetic Survey, and from their observations the value of the absolute force of gravity in the pendulum room of the Coast and Geodetic Survey at Washington has been found to be 9S0.165 on the C. G. S. system. The rela
tive measure between Paris and Washington gave p = 980.169. and, accordingly, the mean of these two values, 980.167, was adopted.
The pendulums used in the United States Coast and Geodetic Survey for gravity de terminations swing with a period of one half second in a ease from which the air has been exhausted. The stem is flat and carries a lenticular bob, the pendulum being supported on agate bearings. Abeam of light is thrown into the pendulum receiver every second, and when the pendulum is ver tical it is reflected in to a telescope through the coincidence of two slits. A carefully rated chronometer forms a part of the apparatus and enables the nb servsr to determine with groat the time of vibration of the pendulum. The instruments of the Survey are extremely portable, and the observers travel from station to station making observa tions.
This variation in time of oscillation is shown below in figures from three stations in North America, with considerable difference in their latitude. A pendulum whose period of vibration in Washington was .5008383 second had a. period at Key West, Fla., of .5011320 second and at limanak, Greenland, of .5002032 second.
Another application of the pendulum due to Foucault is to •show the rotation of the earth. Using a long pendulum with a heavy weight, he found that the plane of oscillation would shift in the same direction as the motion of the sun, or opposite to the rotation of the earth. '\ ere the pendulum to vibrate at the pole, it would continue in the same plane while the earth ro tated, the path of the pendulum appearing to make one complete revolution in twenty-four hours. At the equator. on the other hand, there would be no rotation of the plane of vibration.
In order to determine the length of a seconds pendulum at any particular place, the method devised by Captain hater of using a reversible pendulum is employed. Here the centre of oscil lation and the centre of suspension are inter changeable, and the apparatus will be equivalent to a simple pendulum, whose length is the dis tance between them. In practice these two points are formed by knife edges, which can be moved with respect to each other along a rod so that the vibrations of the pendulum will be syn chronous irrespective of the knife edge by which it is supported.