Demonstration of the cause of the oscillation of a pendulum.

Let a pendulum AB hang from a nail A (Table 16.​1, Fig. 70). Spontaneously, it remains immobile vertically since the natural effort of the heavy body B in propelling itself towards the centre of the earth by the shortest way is directed along the perpendicular AB. Thus the pendulum is closer to the centre of the earth to which it strives instinctively. Since its fall is hindered and prevented by the opposite virtue of the strength of a string, the pendulum is immobile in this perpendicular position. The pendulum B being balanced and immobile in this position, a percussive force occurs which impels it horizontally towards C. The balanced body B obviously must be propelled by the virtue of the impulse in an uniform and even movement. The tenacity of the string forces its course to occur, not along a horizontal line, but upwards over an arc BC of the circle described by the radius AB. In this upwards movement, its normal gravity pulls it downwards and continuously weakens the impressed impetus until complete extinction at the upper point of the ascension C. The impressed impetus being deleted, the ball let to itself remains at C. Then pushed by its natural force it falls through the only possible way over the arc CB since the strength of the string prevents a vertical fall. The descent from C towards B occurs in an accelerated movement as required by the nature of gravity. Consequently, it continuously acquires new degrees of impetus. Since all these persist, the pendulum is led to the lowest point B where spontaneously it would remain immobile if it was deprived of any transverse impetus. Since it is provided with an impetus acquired in its descent, it does not stay in the position B but is pushed further over an equal arc BD in the same time. Then again, for the same reason, it falls from D towards B and from B rises towards C. The journey there and return repeats itself until external obstacles progressively slow down the excursions and extinguish the impetus completely.

Here is now another property of the pendulum which must be noticed in our instance.

If two different pendulums are unequal, their periods are proportional to the square roots of their lengths.

Two pendulums AB and EF are unequal (Table 23.​1, Fig. 76). It must be shown that the ratio of the time of oscillation of AB over the arc CD to the time of oscillation of the pendulum EF over the arc GH similar to CD is equal to the ratio of the square roots of the radii AB and EF. AI is the proportional average between AB and EF. AB is supposed to be the time of oscillation over the arc CD and AI the time of oscillation over the arc GH. The ratio of the similar arcs CD/GH or of their halves CB/GF is equal to the ratio of the radii AB/EF. CB and GF are the distances travelled by the pendulum bodies B and F in their displacements. Consequently, the ratio of the distances travelled by the pendulums B and F is equal to the ratio of the radii AB/EF. But the ratio of distances travelled in an uniformly accelerated displacement is equal to the ratio of the squares of the times in which these distances are travelled. Consequently, the ratio of the times in which the arcs CB and GF are travelled is equal to the ratio of the square roots of the radii AB and EF. They are thus in the ratio AB/AI. AB is supposed to be the time of the oscillation of the pendulum B over the arc CB. Thus, AI will be the time of the oscillation of the pendulum EF over the arc GF. Q.E.D.