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.
THE CENTRE OF PERCUSSION.
Fig. 90.