Chapter 7.2, Problem 13P

A simple pendulum consists of a point mass $m$ suspended by a (weightless) cord or rod of length l, as shown, and swinging in a vertical plane under the action of gravity. 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 $F=ma$ 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 $\omega ?$