# Cycloid has Tautochrone Property

## Theorem

Consider a wire bent into the shape of an arc of a cycloid $C$ and inverted so that its cusps are uppermost and on the same horizontal line.

Let a bead $B$ be released from some point on the wire.

The time taken for $B$ to reach the lowest point of $C$ is:

- $T = \pi \sqrt {\dfrac a g}$

independently of the point at which $B$ is released from.

That is, a cycloid is a tautochrone.

## Proof

By the Principle of Conservation of Energy, the speed of the bead at a particular height is determined by its loss in potential energy in getting there.

Thus, at the point $\tuple {x, y}$, we have:

- $(1): \quad v = \dfrac {\d s} {\d t} = \sqrt {2 g y}$

This can be written:

\(\displaystyle \d t\) | \(=\) | \(\displaystyle \frac {\d s} {\sqrt {2 g y} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {\sqrt {\d x^2 + \d y^2} } {\sqrt {2 g y} }\) |

Thus the time taken for the bead to slide down the wire is given by:

- $\displaystyle T_1 = \int \sqrt {\dfrac {\d x^2 + \d y^2} {2 g y} }$

From Equation of Cycloid, we have:

- $x = a \paren {\theta - \sin \theta}$
- $y = a \paren {1 - \cos \theta}$

Substituting these in the above integral:

- $\displaystyle T_1 = \int_0^{\theta_1} \sqrt {\dfrac {2 a^2 \paren {1 - \cos \theta} } {2 a g \paren {1 - \cos \theta} } } \rd \theta = \theta_1 \sqrt {\dfrac a g}$

This is the time needed for the bead to reach the bottom when released when $\theta_1 = \pi$, and so:

- $T_1 = \pi \sqrt {\dfrac a g}$

Now suppose the bead is released at any intermediate point $\tuple {x_0, y_0}$.

Take equation $(1)$ and replace it with:

- $v = \dfrac {\d s} {\d t} = \sqrt {2 g \paren {y - y_0} }$

Thus the total time to reach the bottom is:

\(\displaystyle T\) | \(=\) | \(\displaystyle \int_{\theta_0}^\pi \sqrt {\dfrac {2 a^2 \paren {1 - \cos \theta} } {2 a g \paren {\cos \theta_0 - \cos \theta} } } \rd \theta\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sqrt {\dfrac a g} \int_{\theta_0}^\pi \sqrt {\dfrac {1 - \cos \theta} {\cos \theta_0 - \cos \theta} } \rd \theta\) | |||||||||||

\((2):\quad\) | \(\displaystyle \) | \(=\) | \(\displaystyle \sqrt {\dfrac a g} \int_{\theta_0}^\pi \dfrac {\sin \frac 1 2 \theta \rd \theta} {\sqrt {\cos^2 \frac 1 2 \theta_0 - \cos^2 \frac 1 2 \theta} } \rd \theta\) | Double Angle Formula for Cosine |

Setting:

- $u = \dfrac {\cos \frac 1 2 \theta} {\cos \frac 1 2 \theta_0}$

and so:

- $\d u = -\dfrac 1 2 \dfrac {\sin \frac 1 2 \theta \rd \theta} {\cos \frac 1 2 \theta_0}$

Then $(2)$ becomes:

\(\displaystyle T\) | \(=\) | \(\displaystyle -2 \sqrt {\dfrac a g} \int_1^0 \frac {\d u} {\sqrt {1 - u^2} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 \sqrt {\dfrac a g} \Big [{\arcsin u}\Big]_0^1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \pi \sqrt {\dfrac a g}\) |

That is, wherever the bead is released from, it takes that same time to reach the bottom.

Hence the result.

$\blacksquare$

## Also known as

This result is seen referred to as the **pendulum property of the cycloid**.

## Also see

- Brachistochrone is Cycloid, in which it is shown that a cycloid is also the shape for which the time is shortest.

## Historical Note

The fact that Cycloid has Tautochrone Property was discovered by Christiaan Huygens in $1658$, during his work on developing a reliable and accurate pendulum clock.

The Tautochrone Problem was also solved independently by Niels Henrik Abel in $1823$, using the technique now known as Abel's integral equation.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{V}$: "Greatness and Misery of Man" - 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VIII}$: Nature or Nurture? - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 2.11$: Problem $4$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.17$: Huygens ($1629$ – $1695$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**cycloid**