Problem — Amplitude and period of the swing

Problem

A child on a swing oscillates with period $T$. Pushed harder, they reach double the amplitude. Does the period $T$ stay the same, increase or decrease? Explain in three lines why, using the formula for the period of the pendulum.

Solution

The period of the pendulum for small oscillations is $T = 2\pi\sqrt{\frac{\ell}{g}}$ and it does not contain the amplitude: so $\ev{T \text{ stays the same}}$ This is Galilean isochronism. At double amplitude the child travels more distance, but also passes through the lowest point faster: the two effects compensate exactly. This holds as long as the amplitude is small; at large angles the period increases slightly, according to the correction $T \approx T_0\left(1 + \theta_0^2/16\right)$.

Links

Topics: Oscillazioni e moto armonico Concepts: Pendolo semplice · Periodo e frequenza Skills: Analisi di casi limite e fantafisica Objects: Pendolo semplice