Experimental Mechanics / A Course of Lectures Delivered at the Royal College of Science for Ireland

by Robert S. Ball

THE COMPOUND PENDULUM.

Fig. 87.

623. Pendulous motion must now be studied in other forms besides that of the simple pendulum, which consists of a weight and a cord. Any body rotating about an axis may be made to oscillate by gravity. A body thus vibrating is called a compound pendulum. The ideal form, which consists of an indefinitely small weight attached to a perfectly flexible and imponderable string, is an abstraction which can only be approximately imitated in nature. It follows that every pendulum used in our experiments is strictly speaking compound.

624. The first pendulum of this class which we shall notice is that used in the common clock ([Fig. 87]). This consists of a wooden or steel rod a e, to which a brass or leaden bob b is attached. This pendulum is suspended by means of a steel spring c a, which being very flexible, allows the vibration to be performed with considerable freedom. The use of the screw at the end will be explained in [Art. 664]. A pendulum like this vibrates isochronously, when the amplitude is small, but it is not easy to see precisely what is the length of the simple pendulum which would oscillate in the same time. In the first place, we are uncertain as to the virtual position of the point of suspension, for the spring, though flexible, will not yield at the point c to the same extent as a string; thus the effective point of suspension must be somewhat lower than c. The other extremity is still more uncertain, for the weight, so far from being a single point, is not exclusively in the neighbourhood of the bob, inasmuch as the rod of the pendulum has a mass that is appreciable. This form of pendulum cannot therefore be used where it is necessary to determine the length with accuracy.

625. When the length of a pendulum is to be measured, we must adopt other means of supporting it than that of suspension by a spring, as otherwise we cannot have a definite point from which to measure. To illustrate the mode that is to be adopted, I take here an iron bar 6' long and 1" square, which weighs 19 lbs. I wish to support this at one end so that it can vibrate freely, and at the same time have a definite point of suspension. I have here two small prisms of steel e ([Fig. 88]) fastened to a brass frame; the faces of the prisms meet at about an angle of 60° and form the edges about which the oscillation takes place: this frame and the edges can be placed on the end of the bar, and can be fixed there by tightening two nuts. The object of having the edges on a sliding frame is that they may be applicable to different parts of the bar with facility. In some instruments used in experiments requiring extreme delicacy, the edges which are attached to the pendulum are supported upon plates of agate; they are to be adjusted on the same horizontal line, and the pendulum really vibrates about this line, as about an axis. For our purpose it will be sufficient to support the edges upon small pieces of steel. a b, [Fig. 88], represents one side of the top of the iron bar; e is the edge projecting from it, with its edge perpendicular to the bar. c d is a steel plate bearing a knife edge on its upper surface; this piece of steel is firmly secured to the framework. There is of course a similar piece on the other side, supporting the other edge. The bar, thus delicately poised, will, when once started, vibrate backwards and forwards for an hour, as there is very little friction between the edges and the pieces which support them.

Fig. 88.

626. The general appearance of the apparatus, when mounted, is shown in [Fig. 89]. a b is the bar: at a the two edges are shown, and also the pieces of steel which support them. The whole is carried by a horizontal beam bolted to two uprights; and a glance at the figure will explain the arrangements made to secure the steadiness of the apparatus; the second pair of edges shown at b will be referred to presently ([Art. 635]).

627. This bar, as you see, vibrates to and fro; and we shall determine the length of a simple pendulum which would vibrate in the same period of time. The length might be deduced by finding the time of vibration, and then calculating from [Art. 606]. This would be the most accurate mode of proceeding, but I have preferred to adopt a direct method which does not require calculation. A simple pendulum, consisting of a fine cord and a small iron sphere c, is mounted behind the edge, [Fig. 89]. The point from which the cord is suspended lies exactly in the line of the two edges, and there is an adjustment for lengthening or shortening the cord at pleasure.

Fig. 89.

628. We first try with 6' of cord, so that the simple pendulum shall have the same length as the bar. Taking the ball in one hand and the bar in the other, I draw them aside, and you see, when they are released, that the bar performs two vibrations and returns to my hand before the ball. Hence the length of the isochronous simple pendulum is certainly less than the length of the bar; for a pendulum of that length is too slow.

629. I now shorten the cord until it is only half the length of the bar; and, repeating the experiment, we find that the ball returns before the bar, and therefore the simple pendulum is too short. Hence we learn that the isochronous pendulum is greater than half the length of the bar, and less than the whole length.

630. Let us finally try a simple pendulum two-thirds of the length of the bar. I make the experiment, and find that the ball and the bar return to my hand precisely at the same instant. Therefore two-thirds of the length of the bar is the length of the isochronous simple pendulum.

We may state generally that the time of vibration of a uniform bar about one end equals that of a simple pendulum whose length is two-thirds of the bar; no doubt the bar we have used is not strictly uniform, because of the edges; but in the positions they occupy, their influence on the time of vibrations is imperceptible.

632. For this rule to be verified, it is essentially necessary that the edges be properly situated on the bar; to illustrate this we may examine the oscillations of the small rod, shown at d ([Fig. 89]). This rod is also of iron 24" × 0"·5 × 0"·5, and it is suspended from a point near the centre by a pair of edges; if the edges could be placed so that the centre of gravity of the whole lay in the line of the edges, it is evident that the bar would rest indifferently however it were placed, and would not oscillate. If then the edges be very near the centre of gravity, we can easily understand that the oscillations may be very slow, and this is actually the case in the bar d. By the aid of the stop-watch, I find that one hundred vibrations are performed in 248 seconds, and that therefore each vibration occupies 2·48 seconds. The length of the simple pendulum which has 2·48 seconds for its period of oscillation, is about 20'. Had the edges been at one end, the length of the simple pendulum would have been

24" × ⅔ = 16".

A bar 72" long will vibrate in a shorter time when the edge is 15"·2 from one end than when it has any other position. The length of the corresponding simple pendulum is 41"·6.