Angular PENDULUM

Angular PENDULUM, is formed of two pieces of legs Angular Pendulum like a sector, and is suspended by the angular point. This pendulum was invented with a view to diminish the length of the common pendulum, but at the same time to preserve or even increase the time of vibration. In this pendulum, the time of vibration depends on the length of the legs, and on the angle contained between them conjointly, the duration of the time of vibration increasing with the angle. Hence a pendulum of this construction may be made to oscillate in any given time. At the lower extremity of each leg of the pendulum is a ball or bob as usual. It may be easily shown, that in this kind of a pendulum, the squares of the times of vibration are as the secants of half the angle contained by the legs: hence if a pendulum of this construction vibrates half seconds when its legs are close, it will vibrate whole seconds when the legs are opened, so as to contain an angle equal to $15^\{\backslash circ\}\; 2\backslash frac\{1\}\{2\}\text{'}$.

The Conical or Circular PENDULUM, is so called Conical or Circular Pendulum from the figure described by the string or ball of the pendulum. This pendulum was invented by Mr Huygens, and is also claimed by Dr Hooke.

In order to understand the principles of this pendulum, it will be necessary to premise the following lemma, viz. the times of all the circular revolutions of a heavy globular body, revolving within an inverted hollow paraboloid, will be equal, whatever be the radii of the circle, described by that body.

In order therefore, to construct the pendulum so that its ball may always describe its revolutions in a paraboloid surface, it will be necessary that the rod of the pendulum be flexible, and that it be suspended in such

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Pendulum such a manner as to form the evolute of the given parabola. Hence, let KH (fig. 9.) be an axis perpendicular to the horizon, having a pinion at K moved by the last wheel in the train of the clock; and a hardened Reel point at H moving in an agate pivot, to render the motion as free as possible. Now, let it be required that the pendulum shall perform each revolution in a second, then the paraboloid surface it moves in must be such whole latus rectum is double the length of the common half second pendulum. Let O be the focus of the parabola MEC, and MC the latus rectum; and make $AE\; =\; MO\; =\; \backslash frac\{1\}\{2\}MC$ the length of a common half second pendulum. At the point A of the verge, let a thin plate AB be fixed at one end, and at the other end B let it be fastened to a bar or arm BD perpendicular to DH, and to which it is fixed at the point D. The figure of the plate AB is that of the evolute of the given parabola MEC.

The equation of this evolute, being also that of the femicubical parabola, is $\backslash frac\{27\}\{16\}x^3\; =\; y^3$.—Let $\backslash frac\{27\}\{16\}P\; =\; P$; then $Px^3\; =\; y^3$, and in the focus $P\; =\; 2y$. In this case $2x^3\; =\; y^3\; =\; \backslash frac\{1\}\{2\}P^3$; hence $x^3\; =\; \backslash frac\{1\}\{4\}P^3$, and $x\; =\; P\backslash sqrt[3]\{\backslash frac\{1\}\{4\}\}\; =\; \backslash frac\{27\}\{16\}P\backslash sqrt[3]\{\backslash frac\{1\}\{4\}\}$ = the distance of the focus from the vertex A.—By assuming the value of $x$, the ordinates of the curve may be found; and hence it may be easily drawn.

The string of the pendulum must be of such a length that when one end is fixed at B, it may lie over the plate AB, and then hang perpendicular from it, so that the centre of the bob may be at E when at rest. Now, the verge KH being put into motion, the ball of the pendulum will begin to gyrate, and thereby conceive a centrifugal force which will carry it out from the axis to some point F, where it will circulate seconds or half seconds, according as the line AE is 9.8 inches, or $2\backslash frac\{1\}\{2\}$ inches, and AB answerable to it.

One advantage possessed by a clock having a pendulum of this construction is, that the second hand moves in a regular and uniform manner, without being subject to those jerks or starts as in common clocks; and the pendulum is entirely silent.

Theory has pointed out several other pendulums, known by the name of Elliptic, Horizontal, Rotulary, &c. pendulums. These, however, have not as yet attained that degree of perfection as to supplant the common pendulum.

Observing that both the gridiron and mercurial pendulums are subject to many inconveniences and errors, Mr Kater has attempted to construct one possessing such properties in respect of cheapness and accuracy as he thinks might justly give it the preference to any other. As wood possesses a less degree of expansibility by means of heat than any other substance; on this account, if it could be rendered quite impervious to moisture, it would be the best of all substances for the rod of a pendulum; and as it also appears that zinc, above all other metals, possesses the greatest degree of expansibility by means of heat, he considered it the best substance which could be employed for a compensation. His next object was to institute a set of delicate experiments, in order to ascertain the precise degree of the expansibility of wood by the application of heat, and he discovered by the use of a pyrometer, that a rod of very

dry, well seasoned white wood, four feet long, three-fourths of an inch broad, and one-fourth of an inch thick, when exposed in an oven to the temperature of $235^\backslash circ$, had contracted. Being again put into the oven, where it was permitted to remain for a long time, till it became a little discoloured, with a view to dissipate the whole of the moisture, it was placed in the pyrometer, and allowed to remain till it reached the temperature of the room, or $49^\backslash circ$, when it was found to have contracted $0.0205$ of an inch with $186^\backslash circ$ of Fahrenheit, from which we obtain by proportion $0.0049$ of an inch for the expansion of one foot with $186^\backslash circ$ difference of temperature. Thus,

But for a general description of this pendulum, and a full account of the manner in which it is constructed, we must refer our readers to the inventor's own paper, Nichol. Jour. vol. xx. p. 214.

Besides the use of the pendulum in measuring time, it has also been suggested as a proper standard for measures of length. See MEASURE.