If F B is made too big, then the line B F can no longer be considered nearly equal to the arc of deflection E B, and the proposition is no longer true.
Hence then, both by experiment and on theory, we find that for small distances the displacement of a pendulum bob is approximately equal to the force by which that displacement is produced.
But if so, then from what has gone before, we have an example of harmonic motion. The weight of the bob, tending to pull the bob back to E, acts just as an elastic band would act, that is to say pulls more strongly in proportion as the distance F B is bigger. In fact, if we could remove the force of gravity still leaving the mass B of the pendulum bob, the force of an elastic band acting so as to tend to pull the bob back to rest might be used to replace it. It would be all one whether the bob were brought back to rest by the downward force of its own gravity, or by the horizontal force of a properly arranged elastic band of suitable length.
Fig. 37.
But the motion of the bob, under the influence of the pull of an elastic band where the strain was always proportional to the displacement, would, as we have seen, be harmonic motion, and performed in equal times whatever the extent of the swing. Whence then we conclude that if the swings of a pendulum are not too big, say not exceeding two and a half inches each way, the motion may be considered harmonic motion, and the swings will be made in equal times whether they are large or small ones. In other words, a clock with a 39-1/7 inch pendulum and side swing on each side if not over two inches will keep time, whatever the arc of swing may be.
This may be verified experimentally. Take a pendulum of wood 39-1/7 inches long, and affix to its end a bob of 10 lbs. weight. The pendulum will swing once in each second. To pull it aside two inches we should want a weight such that its moment about the point of support was equal to the moment of the force of gravity acting on the bob, about the point of support. In other words, the weight required × 39-1/7 inches = 10 lbs. × 2 inches. Whence the weight required = 1/2 lb. (nearly).
Now fix a similar pendulum A B 39-1/7 inches long, horizontally, with a weight B of 10 lbs. on it. Fasten it to a vertical shaft C D, with a tie rod of wire or string A B so as to keep it up, and attach to each side of the rod A B elastic threads E F and E G. Let these threads be tied on at such a point that when B is pulled aside two inches the force tending to bring it back to rest is half a pound. Then if set vibrating the rod will swing backwards and forwards in equal times, no matter how big, the arc of vibration (provided the arc is kept small), and the time of oscillation will be that of a pendulum, namely, one swing in a second. In fact, whether you put A B vertically and let it swing on the pivots C and D by the force of gravity, or put it horizontally, and thus prevent gravity acting on it, but make it swing under the accelerating influence of a pair of elastic bands so arranged as to be equivalent to gravity, in each case it will swing in seconds.
Fig. 38.