A simple pendulum is a small mass $m$ suspended, as shown, by a (weightless) string. Show that for small oscillations (small $\theta$ ), both $\theta$ and $x$ are sinusoidal functions of time, that is, the motion is simple harmonic. Hint: Write the differential equation $\mathbf{F}=m \mathbf{a}$ for the particle $m$. Use the approximation $\sin \theta=\theta$ for small $\theta$, and show that $\theta=A \sin \omega t$ is a solution of your equation. What are $A$ and $\omega ?$
