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

by Robert S. Ball

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.

637. We alluded in the last lecture ([Art. 610]) to the difficulty of measuring with accuracy the length of a simple pendulum; but the reciprocity of the centres of oscillation and suspension, suggested to the ingenious Captain Kater a method by which this difficulty could be evaded. We shall explain the principle. Let one pair of edges be at a. Let the other pair of edges, b, be moved as near as possible to the centre of oscillation. We can test whether b has been placed correctly: for the time taken by the pendulum to perform 100 vibrations about a should be equal to the time taken to perform 100 vibrations about b. If the times are not quite equal, b must be moved slightly until the times are properly brought to equality. The length of the isochronous simple pendulum is then equal to the distance between the edges a and b; and this distance, from one edge to the other edge, presents none of the difficulties in its exact measurement which we had before to contend with: it can be found with precision. Hence, knowing the length of the pendulum and its time of oscillation, gravity can be found in the manner already explained ([Art. 608]).

638. I have adjusted the two edges of the 6' bar as nearly as I could at the centres of oscillation and suspension, and we shall proceed to test the correctness of the positions. Mounting the bar first by the edges at a, I set it vibrating. I take the stop-watch already referred to ([Art. 612]), and record the position of its hands. I then place my finger on the stud, and, just at the moment when the bar is at the middle of one of its vibrations, I start the watch. I count a hundred vibrations; and when the pendulum is again at the middle of its stroke, I stop the watch, and find it records an interval of 110·4 seconds. Thus the time of one vibration is 1·104 seconds. Reversing the bar, so that it vibrates about its centre of oscillation b, I now find that 110·0 is the time occupied by one hundred vibrations counted in the same manner as before; hence 1·100 seconds is the time of one vibration about b: thus, the periods of the vibrations are very nearly equal, as they differ only by ¹/₂₅₀th part of a second.

639. It would be difficult to render the times of oscillation exactly equal by merely altering the position of B. In Kater’s pendulum the two knife-edges are first placed so that the periods are as nearly equal as possible. The final adjustments are given by moving a small sliding-piece on the bar until it is found that the times of vibration about the two edges are identical. We shall not, however, use this refinement in a lecture experiment; I shall adopt the mean value of 1·102 seconds. The distance of the knife-edges is about 3'·992; hence gravity may be found from the expression ([Art. 608])

3'·992 × (3·1416 / 1.102)².

The value thus deduced is 32'·4, which is within a small fraction of the true value.

640. With suitable precautions Kater’s pendulum can be made to give a very accurate result. It is to be adjusted so that there shall be no perceptible difference in the number of vibrations in twenty-four hours, whichever edge be the axis of suspension: the distance between the edges is then to be measured with the last degree of precision by comparison with a proper standard.