A pendulum of given length is a most delicate instrument for the measurement of the relative amounts of the earth's attraction at dif ferent places. Practically, it gives the kinetic measurement of grav ity,' which is not only by far the most c.mvenient, but alsb the true measure. By this application of the pendulum, the oblateness of the, earth has been determined, in terms of the law of decrease of gravity from the poles to the equator. The instrument has also been employed to determine the mean density of the earth (from which its mass is directly derivable), by the observation of its times of vi bration at the mouth and at the bottom of a cgal-pit. It was shown by Newton that the force of attraction at the bottom of a pit depends only upon the internal nucleus which remains when a shell, everywhere of thickness equal to the depth of the pit, has been sup posed to be removed from the whole surface of the earth. - The latest oli-;ervations by this method were made by Airy, the present astrondmer-royal. in the Marton coal-pit, and gave for the mean destiny of the earth a result nearly equivalent to Mdt deduced by Cavendish and Maskelyne from experiments of a totally different nature. See EARTH.
If the bob of the simple pendulum be slightly displaced in any manner, it describes an ellipse about its lowest position as center. This ellipse may, of course, become a straight line or a circle, as in the cases already considered. The bob does not accurately describe the same curve in successive revolutions: in fact, the elliptic orbit just men tioned rotates in its own plane about its center, in the same direction as the bob moves, with an angular velocity nearly proportional to the area of the ellipse.. This is an inter esting case of progression of the apse (_upsides, q.v.), which can be watched by any one who will attach a small bullet to a fine thread; or, still better, attach to the lower end of a long string fixed to the ceiling a funnel full of fine sand or ink which is allowed to escape from a small orifice.. By this process, a more or less permanent trace of the motion of the pendulum is recorded, by which the elliptic form of the path and the phen omena of progression are well shown.
According to what is stated above, there ought to be no progression if the pendulum could be made to vibrate simply in a straight line, as then the area of its elliptic orbit vanishes. It is, however, found to he almost impossible in practice to render the path absolutely straight: so that there always is from this cause a slight rate of change in the position of the line of oscillation. But as the direction of this change depends on the' direction of rotation in the ellipse, it is as likely to effect the motion in one way as in the opposite, and is thus easily separable from the very curious result obtained by Foucault, that on account of the earth's rotation, the plane of vibration of the pendulum appears to turn in the same direction as the sun, that is, in the. opposite direction to the earth's rotation about its axis. To illustrate this now well-known case, consider for a moment a simple pendulum vibrating at the pole of the earth. Here, if the pendulum vibrates in a straight line, the direction of that line remains absolutely fixed in space, while the earth turns round below it once in 24 To a spectator on the earth, it appears, of course, as if the plane Of motion of the pendulum were turning once round in 24 hours, but in the opposite direction. To find the amount of the corresponding phenomenon in ally other latitude, all that is required is to know the rate of the earth's rotation about the vertical in that latitude. This is easy, for velocities of rotation are resolved and compounded by the same process as forces, hence the rate at which the earth rotates about the vertical in latitude A. is less than that of rotation about the polar axis in the ratio of sin. A to 1. Hence the time of the apparent retation of the plane of the pendu
lum's motion is At the pole, this is simply 24 hours; at the equator, it is infin sin. A Hely great, or there is no effect of this kind: in the latitude of Edinburgh 57' 23.2'), it is 28.63 or 28 h. 37 in. 48 seconds.
We have not yet alluded to the obvious fact, that a simple pendulum, such as we have described above, exists in theory only, since we cannot procure either a single heavy particle, or a perfectly light and flexible string. But it is easily shown, although the process cannot be given here, that a rigid body of any form whatever vibrates about an axis under the action of gravity, according to the same law as the hypothetical simple pendulum. The length of the equivalent simple pendulum depends upon what is called the radius of gyration (q.v.) Of the pendulOus body. Its property is simply this, that if the whole mass of the body were collected at a point whose distance from the axis is the radius of gyration, the moment (q.v.) of inertia of this heavy point (about the axis) would be the same as that of the complex body. The square of the radius of gyration of a body about any axis, is greater than the square of the radius of gyration about a parallel axis through the center of gravity, by the square of the distance between those lines. Now, the length of the simple pendulum equivalent to a body oscillating about any axis is directly as the square of the radiva of gyration, and inversely as the distance of the cen ter of gravity from the axis. Hence, if k be the radius of gyration of a body about an axis through the center of gravity, is that about a parallel axis whose distance from the first is h; and the length, 1, of the equivalent simple pendulum is A + This expression becomes infinitely great if h be very large, and also if is be very small (that is, a body vibrates very slowly about an axis either far from, or near to, its center of gravity). It must therefore have a minimum value. By solving the equation above as a quadratic in h, we find that I cannot be less than 2k, which is, therefore, the length of the simple pendulum corresponding to the quickest vibrations which the body can exe cute about any axis parallel to the given one. In this case the value of is is equal to k. hence, if a circular cylinder be described in a body, its axis passing through the center of gravity, and its radius being the radius of gyration about the axis, the times of oscil lation about all generating lines of this cylinder are equal, and less than the times of cseillation about any other axis parallel to the given one. Also, since the formula for L above given, may be thus it is obvious that it is satisfied if be put for Is. Hence, if any value I (of course not less than 21z) be assigned as the length of the equivalent simple pendulum, there are he° values of is which will satisfy the con ditions; that ,is, there are two concentric cylinders, about a generating line of either of which the time of oscillation is that of the assigned simple pendulum. When 1=2/.•, these cylinders coincide, and form that above described. Aud, since the sum of the radii of these cylinders is 1, it is obvious that if we can find experimentally two' parallel axis about which a body oscillates in equal times, and if the center of gravity of the body lie between, these axis, and in their plane, the distance between these axis is the length ig• the equivalent simple pendulum. This result is of very great importance, because it enabled hater (who was the first to employ it) to use the complex pendulum for the determination of the length of the simple second's pendulum in any locality. The sim ple pendulum is perfect in theory, but cannot be constructed ; and thus the method which enaules us to obtain its results by the help of such a pendulum as we can construct, is especially valuable.