It is this curious property of the circle that makes the vertical force of gravity on a pendulum pull it as though it were a horizontally acting elastic band; that is the reason why a pendulum is equal-time-swinging, or, as it is called, isochronous, from two Greek words that mean “the same” and “time.”
But it must be remembered that this equal swinging is only approximate, and only true when the arc of vibration is small.
Here then we have a proof which shows us that the pendulum of a clock and the balance wheel of a watch depend on exactly the same principles. They are each an example of harmonic motion.
The next question that arises is whether the weight of the pendulum has any influence upon the time of its vibration.
A little reflection will soon convince us that it has none. For we know that the time that bodies take to fall to the ground under the action of gravity is independent of the weight. A falling 2 lb. weight is only equivalent to two pound-weights falling side by side.
In the same way and by the same reasoning we might take two pendulums of equal length, and each with a bob weighing 1 lb. They would, if put side by side close together swing in equal times. But the time would be the same if they were fastened together, and made into one pendulum.
For inasmuch as the fall of a pendulum is due to gravity, and the action of gravity upon a body is proportional to its mass, it follows that in a pendulum the part of the gravitational force that acts upon each part of the mass is occupied in moving that mass, and the whole pendulum may be considered as a bundle of pendulums tied together and vibrating together.
The same would be the case with a pendulum vibrating under the influence of a spring. If you have two bobs and two springs, they will vibrate in the same time as one bob accelerated by one spring. In this case, however, the force of the one spring must be equal to the combined force of the two springs. In other words, the springs must be made proportional in strength to the masses.
Hence, then, you cannot increase the speed of the vibration of a pendulum by adding weight to the bob.
On the other hand, if you have a bob vibrating under the influence of a spring, like the balance wheel of a watch, then if you increase the bob without increasing the spring, since the mass to be moved has increased without a corresponding increase in the accelerating force acting on it, the time of swing will alter accordingly.