Pendulums — Question 1

Question 1

The small-angle approximation is an approximation to the torque

A point mass on a massless inextensible string of length $\ell$ swings in gravity. Angle $\theta$ is measured from downward vertical. Ignore air drag and use radians in expansions.

1. Derive the exact angular equation and its SHM approximation. Express the corresponding equation for arc displacement $s=\ell\theta$.

2. At amplitude $30^\circ$, find the ratio of exact restoring-torque magnitude to the linear approximation. Using $\sin\theta/\theta\approx1-\theta^2/6$, estimate an amplitude limit for a $1\%$ torque error relative to the linear value.

Question 1 - Solution

The small-angle approximation is an approximation to the torque

(a) Derivation and calculation

$$m\ell^2\ddot\theta=-mg\ell\sin\theta\Rightarrow\boxed{\ddot\theta=-\frac g\ell\sin\theta}.$$$$|\theta|\ll1:\quad\sin\theta\simeq\theta,\quad\ddot\theta+\frac g\ell\theta=0.$$$$s=\ell\theta\Rightarrow\boxed{\ddot s+\frac g\ell s=0},\quad\omega=\sqrt{g/\ell}.$$

(b) Result and interpretation

At $30^\circ=\pi/6$,$$\frac{\tau_{\mathrm{exact}}}{\tau_{\mathrm{linear}}}=\frac{\sin\theta}{\theta}=\frac3\pi=\boxed{.95493}.$$$$\frac{\theta-\sin\theta}{\theta}\simeq\frac{\theta^2}{6}\le.01\Rightarrow\boxed{|\theta|\lesssim.24495\,\mathrm{rad}=14.03^\circ}.$$This is the leading-order torque error, not the period error.

Physical check

The linear torque slightly overestimates the restoring torque because $\sin\theta<\theta$ for small positive angles.