Cycloidal pendulum

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Exhibit
Cycloidal pendulum
Pendolocicloj.jpg
On display at Il Giardino di Archimede - Un Museo per la Matematica
Type Hands-on
Creator Il Giardino di Archimede
Topics geometry, cycloid, history
License (Copyrighted, Patented, Open source, Creative Commons, etc)
[Il Giardino di Archimede Website]
[[] Video Tutorial]

This exhibit is part of the exhibition Beyond the compasses at Il Giardino di Archimede (Florence, Italy).

Description

In the exhibit two pendulums are compared. The first one i a free pendulum, moving along a circumference. The another one is a cycloidal pendulum, where the weight moves along a cycloid. Two photocellules measure their periods, that are visualised on the computers.

Activities

As the pendulums start oscillating, you may see that while the period of the ordinary pendulum decreases with the amplitude of its oscillations, the one of the cycloidal pendulum is strictly constant.


Background

Galileo 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 is the cycloid.

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?

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.


History and museology

The exhibit is part of a path on cycloid. It is particularly connected with a previous exhibit showing the fall of two balls into a cycloidal slide. The two balls arrive at the same time in the lowest point of slide, regardless of the starting point. This shows the isochronism of the cycloid. Another exhibit shows that the involute of a cycloid is a second cycloid equal to the first.


involute of a cycloid
isochron curve

Resources

Documentation page