# Cycloidal Pendulum

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This Demonstration illustrates the isochronous movement of the cycloidal (tautochrone) pendulum.

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Contributed by: Erik Mahieu (November 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Considered here is a simple pendulum consisting of a bob, idealized as a point mass, attached to a pivot point by a massless string having fixed length . The pendulum is assumed to be subject only to gravity.

The path of the bob is the cycloid formed by a circle of radius 1 running from to , passing through the point . With as the angle through which the circle has rolled by time , this cycloid is given by the parametric equation

.

Using Lagrangian dynamics, we have

and the resulting equation of motion is .

The period of a cycloidal pendulum is for any amplitude. With , the period is 4.01213. In this Demonstration, the function period[amplitude] verifies this fact by escaping NDSolve with the "EventLocator" method at the point where the pendulum passes the vertical.

Much more information about the Huygens clock can be found in the following reference.

Reference

[1] A. Emmerson. "Things Are Seldom What They Seem — Christiaan Huygens, the Pendulum and the Cycloid." *The Horological Foundation*. (2010) www.antique-horology.org/Piggott/RH/Images/81V_Cycloid.pdf.

## Permanent Citation