period

The period is the time it takes for one complete cycle of a repeating motion. It answers the everyday question: how long does one full swing back and forth take? One tick-and-tock of a clock, one full up-and-down of a bobbing spring, one complete lap of an orbit each of these lasts one period.

Precisely, the period T is the shortest time after which the system returns to the same state and starts repeating. It is measured in seconds. The period is the reciprocal of the frequency f, so T = 1 / f, and it is tied to the angular frequency omega by T = 2 pi / omega. For the two most famous oscillators, a mass on a spring has T = 2 pi sqrt(m/k), and a simple pendulum has T = 2 pi sqrt(L/g), where L is the length and g is the acceleration due to gravity.

A striking feature of simple harmonic motion is that the period does not depend on the amplitude: a big swing and a small swing take the same time. This property, called isochronism, is exactly why pendulums make good clocks. One honest note: this amplitude-independence is only exact for ideal SHM; for a pendulum swung through a large angle, the period grows slightly.