Waves & Optics · Physics

Pendulums turn gravity into nearly harmonic motion.

Learn the simple pendulum model, why the small-angle approximation matters, and how energy methods help beyond small angles.

This topic

Pendulums

Understand assumptions, derive the period (conceptually), and extend the idea to physical pendulums.

Model

Simple pendulum model

The simple pendulum treats the bob as a point mass on a massless string, moving in a plane under gravity.

Approximation

Small-angle approximation

The pendulum becomes SHM only for small angles, where the restoring term becomes approximately linear in the angle.

Timing

Period of a pendulum

For small angles, the period depends mainly on length and gravity—not on mass and (to first order) not on amplitude.

Extension

Physical pendulum (intro)

When the swinging object has size, rotation matters. The period then depends on the moment of inertia about the pivot.

Energy

Energy methods for pendulums

Energy provides a clean way to relate speed to position and to reason about turning points—especially when angles are not small.

Practice

Practice & Exercises

Practice pendulum assumptions, small-angle reasoning, and interpreting period dependence.