Tautochrone Problem

Find the curve down which a bead placed anywhere will fall to the bottom in the same amount of time. The solution is a Cycloid, a fact first discovered and published by Huygens in Horologium oscillatorium (1673). Huygens also constructed the first pendulum clock with a device to ensure that the pendulum was isochronous by forcing the pendulum to swing in an arc of a Cycloid.

The parametric equations of the Cycloid are

$$x = a(\theta-\sin\theta)$$ (1)

$$y = a(1-\cos\theta).$$ (2)

To see that the Cycloid satisfies the tautochrone property, consider the derivatives

$$x' = a(1-\cos\theta)$$ (3)

$$y' = a\sin\theta,$$ (4)

and

$$x'^2+y'^2 = a^2[(1-2\cos\theta+\cos^2\theta)+\sin^2\theta]$$

$$= 2a^2(1-\cos\theta).$$ (5)

Now

$${\textstyle{1\over 2}}mv^2=mgy$$ (6)

$$v={ds\over dt}=\sqrt{2gy}$$ (7)

$$dt = {ds\over \sqrt{2gy}} = {\sqrt{dx^2+dy^2}\over \sqrt{2gy}}$$

$$= {a\sqrt{2(1-\cos\theta)}\,d\theta\over\sqrt{2ga(1-\cos\theta)}} = \sqrt{a\over g}\,d\theta,$$ (8)

so the time required to travel from the top of the Cycloid to the bottom is

$$T=\int_0^\pi dt=\sqrt{a\over g} \,\pi.$$ (9)

However, from an intermediate point $\theta_0$,

$$v={ds\over dt}=\sqrt{2g(y-y_0)},$$ (10)

so

$$T = \int_{\theta_0}^\pi {\sqrt{2a^2(1-\cos\theta)}\over 2ag(\cos\theta_0-\cos\theta)}\,d\theta$$

$$= \sqrt{a\over g} \int_{\theta_0}^\pi\sqrt{1-\cos\theta\over\cos\theta_0-\cos\theta}\,d\theta$$

$$= \sqrt{a\over g} \int_{\theta_0}^\pi {\sin({\textstyle{1\over 2}}\... ...{\cos^2({\textstyle{1\over 2}}\theta_0)-\cos^2({\textstyle{1\over 2}}\theta)}}.$$ (11)

Now let

$$u = {\cos({\textstyle{1\over 2}}\theta)\over\cos({\textstyle{1\over 2}}\theta_0)}$$ (12)

$$du = -{\sin({\textstyle{1\over 2}}\theta)d\theta\over 2\cos(\theta_0)},$$ (13)

so

$$T=-2\sqrt{a\over g}\int_1^0 {du\over\sqrt{1-u^2}} = 2\sqrt{a\over g}\, [\sin^{-1}u]^1_0 = \pi\sqrt{a\over g}\,,$$ (14)

and the amount of time is the same from any point!

See also Brachistochrone Problem, Cycloid

References

Muterspaugh, J.; Driver, T.; and Dick, J. E. ``The Cycloid and Tautochronism.'' http://php.indiana.edu/~jedick/project/intro.html.

Muterspaugh, J.; Driver, T.; and Dick, J. E. ``P221 Tautochrone Problem.'' http://php.indiana.edu/~jedick/project/project.html.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 54-60 and 384-385, 1991.