Large amplitudes: correction to the period

The isochronism of the pendulum holds only as long as $\sin\theta \approx \theta$. When the amplitude is no longer “small”, this approximation breaks down and the period starts to depend on the amplitude $\theta_0$. The correction can be written as a series expansion:

$T \approx 2\pi\sqrt{\frac{\ell}{g}}\left(1 + \frac{\theta_0^2}{16} + \frac{11\,\theta_0^4}{3072} + \cdots\right)$

The dominant term $\theta_0^2/16$ tells us that the period increases with the amplitude. Some figures: at $\theta_0 = 30^\circ \approx 0{,}52$ rad the correction is already about $1{,}7\%$; at $90^\circ$ it reaches almost $20\%$.

Why the period increases

At large amplitude the true restoring force ($-mg\sin\theta$) is weaker than the linear one ($-mg\theta$), because $\sin\theta < \theta$. A spring “softer” than expected holds the body back less, so it takes longer to complete the oscillation.

This is why pendulum clocks, before Huygens’s escapement mechanisms that keep the amplitude constant, had to swing with a very small amplitude: only there is the period genuinely independent of the amplitude, so the clock does not run slow when the drive weakens.

Collegamenti

Argomenti: Oscillations and harmonic motion Concetti: Simple pendulum Competenze: Limiting-case analysis and sci-fi physics Oggetti: Simple pendulum

Esercizi collegati: Problem — Length of a pendulum that beats seconds · Problem — Lead and cork pendulums · Problem — Pendulum clock and temperature