11348

# Cycloidal Pendulum

This Demonstration illustrates the isochronous movement of the cycloidal (tautochrone) pendulum.
In 1656, Dutch mathematician and scientist Christiaan Huygens discovered that the simple pendulum is not isochronous, that is, its period depends on the amplitude of the swing.
He realized that the pendulum would be isochronous if the bob of a pendulum swung along a cycloidal arc rather than the circular arc of the classical pendulum. He proved that the cycloid is a tautochrone curve.
To construct this cycloidal pendulum, he used a bob attached to a flexible rod. The movement of the pendulum was restricted on both sides by plates forming a cycloidal arc. When the rod unwraps from these plates, the bob will follow a path that is the involute of the shape of the plates. Since the cycloid is its own evolute, this is a congruent cycloid.

### 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

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.
 © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS
 Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX Download or upgrade to Mathematica Player 7EX I already have Mathematica Player or Mathematica 7+