Cycloid has Tautochrone Property

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Theorem

Let a wire $AB$ be curved into the shape of an arc of a cycloid such that:

$A$ is at the cusp

$B$ is the highest point of the arc

and inverted so that its cusps are uppermost and on the same horizontal line.

Thus $B$ is the lowest point of the arc.

Let $AB$ be embedded in a constant and uniform gravitational field where Acceleration Due to Gravity is $g$.

Let a bead $P$ be released from anywhere on the wire between $A$ and $B$ to slide down without friction to $B$.

Then the time taken for $P$ to slide to $B$ is:

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

independently of the point from which $P$ is released.

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:

$\ds \d t$

$=$

$\ds \frac {\d s} {\sqrt {2 g y} }$

$\ds $

$=$

$\ds \frac {\sqrt {\d x^2 + \d y^2} } {\sqrt {2 g y} }$

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

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

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:

$\ds T$

$=$

$\ds \int_{\theta_0}^\pi \sqrt {\dfrac {2 a^2 \paren {1 - \cos \theta} } {2 a g \paren {\cos \theta_0 - \cos \theta} } } \rd \theta$

$\ds $

$=$

$\ds \sqrt {\dfrac a g} \int_{\theta_0}^\pi \sqrt {\dfrac {1 - \cos \theta} {\cos \theta_0 - \cos \theta} } \rd \theta$

$\text {(2)}: \quad$

$\ds $

$=$

$\ds \sqrt {\dfrac a g} \int_{\theta_0}^\pi \dfrac {\sin \frac 1 2 \theta \rd \theta} {\sqrt {\cos^2 \frac 1 2 \theta_0 - \cos^2 \frac 1 2 \theta} } \rd \theta$

Half Angle Formula for Cosine and Half Angle Formula for Sine

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:

$\ds T$

$=$

$\ds -2 \sqrt {\dfrac a g} \int_1^0 \frac {\d u} {\sqrt {1 - u^2} }$

$\ds $

$=$

$\ds 2 \sqrt {\dfrac a g} \Big [{\arcsin u}\Big]_0^1$

$\ds $

$=$

$\ds \pi \sqrt {\dfrac a g}$

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

Hence the result.

$\blacksquare$

Also known as

The result can be 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.

Historical Note

The fact that 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 ($\text {1629}$ – $\text {1695}$)
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cycloid
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cycloid