the simple pendulum

The simple pendulum is a weight on a string, swinging back and forth. Think of the pendulum in a grandfather clock, a child on a playground swing, or a plumb bob nudged gently to one side. It answers a very old question that helped launch modern physics: what decides how long each swing takes?

Precisely, it is idealized as a point mass, called the bob, hanging from a massless, unstretchable string of length L. Gravity supplies the restoring force. For small swings, where the angle is small enough that sin(theta) is nearly theta, the motion is simple harmonic with period T = 2 pi sqrt(L/g) and angular frequency omega = sqrt(g/L), where g is the acceleration due to gravity. Remarkably, this period depends only on the length L and on g, not on the mass of the bob, and not on the amplitude for small swings.

This is exactly why pendulums kept accurate time for centuries, and why timing a pendulum is a classic way to measure g. One honest caveat that matters: the tidy formula relies on the small-angle approximation. Swing the pendulum through a large angle and the period grows a little, and the motion is no longer exactly simple harmonic.