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Secondary Magazine - Issue 26: Focus on

This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 15 January 2009 by ncetm_administrator
Updated on 27 April 2009 by ncetm_administrator

Focus on...the cycloid

One arch of a cycloid of radius r is given by the parametric equations

$x = r (t-sin t)$

$y = r(1-cos t)$

What happens if the radius of the cycloid changes?
Make your prediction then test it using this animation:

The cycloid has generated fierce competition among even the most distinguished of mathematicians. It has been called 'Helen of Geometers' – the most beautiful curve in the world.

The name Cycloid is attributed to Galileo who attempted to find its area by weighing various pieces of metal slices representing the rolling disc. Many other very famous mathematicians worked on ways to calculate the exact area under the curve – those who were successful include Fermat and Descartes. Roberval and Sir Christopher Wren succeeded in calculating the length of the arc.

The brachistochrone problem, posed by Johann Bernoulli in 1696, which asks
Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time?
can be solved using a cycloid. It is thought that Bernoulli knew the solution but challenged others to solve it. He received four further solutions from Newton (who is said to have solved it in an evening after returning home from the Royal Mint), Jacob Bernoulli, Leibniz and de L'Hôpital.

If the cycloid is turned upside down it forms a tautochrone curve, that is a bead dropped anywhere on the curve will always take the same time to reach the lowest point of the curve, $T=\pi \sqrt{\frac{r}{g} }$ as illustrated here.

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