602. We have seen that the time of vibration of a pendulum depends neither upon its amplitude, material, nor weight; we have now to learn on what the time does depend. It depends upon the length of the pendulum. The shorter a pendulum the less is its time of vibration. We shall find by experiment the relation between the time and the length of the cord by which the weight is suspended.
Fig. 84.
603. I have here ([Fig. 84]) two pendulums a d, b c, one of which is 12' long and the other 3'; they are mounted side by side, and the weights are at the same distance from the floor. I take one of the weights in each hand, and withdraw them to the same distance from the position of rest. I release the balls simultaneously; c moves off rapidly, arrives at the end c´ while d has only reached d´, and returns to my hand just as d has completed one oscillation. I do not seize c: it goes off again, only to return at the same moment when d reaches my hand. Thus c has performed four oscillations while d has made no more than two. This proves that when one of two pendulums is a quarter the length of the other, the time of vibration of the shorter one is half that of the other.
604. We shall repeat the experiment with another pendulum 27' long, suspended from the ceiling, and compare its vibrations with those of a pendulum 3' long. I draw the weights to one side and release them as before; and you see that the small pendulum returns twice to my hand while the long pendulum is still absent; but that, keeping my hands steadily in the same place throughout the experiment, the long pendulum at last returns simultaneously with the third arrival of the short one. Hence we learn that a pendulum 27' long takes three times as much time for a single vibration as a 3' pendulum.
605. The lengths of the three pendulums on which we have experimented (27', 12', 3'), are in the proportions of the numbers 9, 4, 1; and the times of the oscillations are proportional to 3, 2, 1: hence we learn that the period of oscillation of a pendulum is proportional to the square root of its length.
606. But the time of vibration must also depend upon gravity; for it is only owing to gravity that the pendulum vibrates at all. It is evident that, if gravity were increased, all bodies would fall to the earth more than 16' in the first second. The effect on the pendulum would be to draw the ball more quickly from d to d´ ([Fig. 84]), and thus the time of vibration would be diminished.
It is found by calculation, and the result is confirmed by experiment, that the time of vibration is represented by the expression,
3·1416 √ ( Length / Force of gravity).
607. The accurate value of the force of gravity in London is 32·1908, so that the time of vibration of a pendulum there is 0·5537√ length: the length of the seconds pendulum is 3'·2616.