Question

A simple pendulum can be modeled as exhibiting simple harmonic motion when$\mathit{\theta }$is small. Is the motion periodic when$\mathit{\theta }$is large?

Short answer

Even though the motion is not harmonic, at large angles, the motion will be periodic. Through the small values, the period is constant when the amplitude increases. After that the period will become very larger because $\theta$ increases.

Step-by-step solution

Simple Harmonic Motion

A simple pendulum of length L can be modeled to move in simple harmonic motion for small angular displacements from the vertical. Its period is

$T=2\pi \sqrt{\frac{L}{g}}$

A physical pendulum is an extended object that, for small angular displacements, can be modeled to move in simple harmonic motion about a pivot that does not go through the center of mass. If it will repeat and it is not harmonic at large angles. The motion will be periodic.

Find the motion is periodic

Even though the motion is not harmonic, at large angles, the motion will be periodic. Through the small values, the period is constant when the amplitude increases. After that the period will become very larger because$\theta$ increases.