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

 * Brachistochrone.png

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:

Thus the time taken for the bead to slide down the wire is given by:
 * $\ds T_1 = \int \sqrt {\dfrac {\d x^2 + \d y^2} {2 g y} }$

From Equation of Cycloid, we have:
 * $x = a \paren {\theta - \sin \theta}$
 * $y = a \paren {1 - \cos \theta}$

Substituting these in the above integral:


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

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:

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:

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

Hence the result.

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

Also see

 * Tautochrone Problem


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