The Pendulum Is Not Isochronous — Unless It Swings on a Cycloid

For an ordinary pendulum of fixed length, a large-amplitude oscillation has a longer period than a small-amplitude one. Thus ordinary pendulums with different amplitudes slowly fall out of step. Huygens (1629–1695) fixed this by making the bob move on a cycloid instead of a circle. In the ideal model, every complete oscillation, large or small, then has exactly the same period. Below, two pendulums of each kind start together: one with a small amplitude and one with a large amplitude. Let them run for a long time and see which family stays in step.

1. Release, then let time run

speed

2. Two families, side by side

Ordinary pendulums — bob on a circle

Position over time (fraction of the swing)

Huygens' pendulums — bob on a cycloid

Position over time (fraction of the swing)

Periods and elapsed cycles

Ordinary pendulum, released from rest at angle θ0 from the downward vertical:

\[ T(\theta_0) = 4\sqrt{\frac{L}{g}}\;K\!\left(\sin\frac{\theta_0}{2}\right),\qquad K(k) = \int_0^{\pi/2} \frac{d\varphi}{\sqrt{1 - k^2 \sin^2\varphi}} = \frac{\pi}{2}\left(1 + \frac{k^2}{4} + \frac{9k^4}{64} + \cdots\right). \]

Huygens' pendulum, whose bob follows a cycloid generated by a circle of radius a, with string length L = 4a:

\[ T = 2\pi\sqrt{\frac{4a}{g}} = 2\pi\sqrt{\frac{L}{g}} = T_0 . \]

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