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

by Robert S. Ball

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.

THE CENTRE OF OSCILLATION.

633. It appears that corresponding to each compound pendulum we have a specific length equal to that of the isochronous simple pendulum. To take as example the 6' bar already described ([Art. 625]), this length is 4'. If I measure off from the edges a distance of 4', and mark this point upon the bar, the point is called the centre of oscillation. More generally the centre of oscillation is found by drawing a line equal to the isochronous simple pendulum from the centre of oscillation through the centre of gravity.

634. In the bar d the centre of oscillation would be at a distance of 20' below the edges; and in general the position will vary with the position of the edges.

635. In the 6' bar b is the centre of oscillation. I take another pair of edges and place them on the bar, so that the line of the edges passes through b. I now lift the bar carefully and turn it upside down, so that the edges b rest upon the steel plates. In this position one-third of the bar is above the axis of suspension, and the remaining two-thirds below. a is of course now at the bottom of the bar, and is on a level with the ball, c: the pendulum is made to oscillate about the edges b, and the time of its vibration may be approximately determined by direct comparison with c, as already explained. I find that, when I allow c and the bar to swing together, they both vibrate precisely in the same time. You will remember, that when the ball was suspended by a string 4' long, its vibrations were isochronous with those of the bar when suspended from the edges a. Without having altered c, but having made the bar to vibrate about b, I find that the time of oscillation of the bar is still equal to that of c. Therefore, the period of oscillation about a is equal to that about b. Hence, when the bar is vibrating about b, its centre of oscillation must be 4' from b, that is, it must be at a: so that when the bar is suspended from a, the centre of oscillation is b; while, when the bar is suspended from b, the centre of oscillation is a. This is an interesting dynamical theorem. It may be more concisely expressed by saying that the centre of oscillation and the centre of suspension are reciprocal.

636. Though the proof that we have given of this curious law applies only to a uniform bar, yet the law is itself true in general, whatever be the nature of the compound pendulum.