Tautochrone: the cycloid's isochronism

What you are seeing: five frictionless beads released simultaneously from different heights along the same cycloid bowl. Despite the wide range of release positions, all five reach the bottom at exactly the same time. Huygens proved this in 1673 by showing that the equation of motion in arc-length $s$ from the bottom is $\ddot{s} = -(g / 4R) s$ , an exact simple harmonic oscillator independent of amplitude.

This isochronism is what makes the cycloid the curve of equal time, the tautochrone. It is also (separately) the brachistochrone, the curve of least time from a fixed start to a fixed end. Huygens used this to design pendulum clocks whose period would not drift with swing amplitude.

Figure 1. Tautochrone. Method: closed-form

s(t) = s_0 \cos(\omega t)

with

\omega = \sqrt{g / 4R}

, mapped through the cycloid parametrization.

speed2

WHAT TO TRY

Vary each control and watch the rail readouts respond.
Compare the diagnostic plot against the live scene.