The cycloidal pendulum

The cycloid is also concerned in another problem, this time of a more technical kind. Galileo again had observed how the oscillations of a pendulum would take more or less the same time, independently from the amplitude of the oscillations. The first pendulum clocks originated from this observation by Galileo.

In reality, the pendulum's oscillations are not exactly isochronous: the time it takes to complete an oscillation does depend on the oscillation's amplitude, and it is longer for wider oscillations. Only for very small oscillations we can consider the time substantially constant, and it is those small oscillations that are used for pendulum clocks.

One can then wonder: how must a pendulum be so that the all its oscillations require exactly the same time, or in one word (also derived from Greek: isos, equal, and chronos, time) are strictly isochronous?

Let us see things from a slightly different point of view. In a normal pendulum, weight swings freely attached to a point, therefore along a circumference. Its oscillations are only approximately isochronous, and require more time the larger the arc of circumference. In this context, the question becomes: along what type of curve must a body oscillate so that the oscillations are perfectly isochronous?

The answer, once again, is: the cycloid. If we put two balls in two points of the cycloid, and we make them fall at the same time, they will hit each other exactly in the lower point of the curve, even if one started very near and the other very far from this point. In other words, the ball takes the same time to travel over the large arc and the small one, a sign that the oscillations are isochronous. In a circumference this does not happen: the ball that starts nearer arrives earlier.

>So, if we want to build a perfectly isochronous pendulum clock, we must have the weight oscillate along a cycloid. But how to make the weight move along this line without making it slide, that is without using a cycloidal profile? The situation is analogous to that of the first room, when we wanted to draw a straight line without using a profile. There we used a rather complex mechanism, Peaucellier's mechanism. Here we will use another trick: instead of allowing the weight to oscillate freely (in this case it would describe a circumference), we will condition its trajectory by making the string lie on two profiles.

In mathematical terms, the profiles will have to be shaped in such a way that their involute is a cycloid. And here we have a surprise: the involute of a cycloid is a second cycloid equal to the first! Consequently, the profiles must be arcs of a cycloid. In this case (and only in this case) the pendulum will oscillate along a cycloid, and will therefore be isochronous.

We have compared two pendulums: one free, moving along a circumference; and another cycloid, obtained with profiles at the top. Two photocellules measure their periods, that are visualised on the computers. As you may see, while the period of the ordinary pendulum decreases with the amplitude of its oscillations, the one of the cycloidal pendulum is strictly constant.