Pendulum Calculator (period and length)

The simple pendulum

A weight on a string swings back and forth at a rate set almost entirely by its length. The period — the time for one complete there-and-back swing — follows a strikingly simple formula.

T = 2π √(L / g)  ·  L = g (T / 2π)²

What doesn't matter

Neither the mass of the bob nor (for small swings) the size of the swing appears in the formula. A heavy pendulum and a light one of the same length keep identical time, and as a pendulum clock's swing dies down it keeps ticking at the same rate — the property that made pendulums the basis of accurate timekeeping for nearly 300 years.

The square-root relationship

Because length sits under a square root, you must quadruplethe length to double the period. That's why a grandfather clock's one-second-per-swing pendulum is very nearly 1 metre long, and why making it slightly shorter speeds the clock up only slightly — the classic way to regulate one.

Worked example

A 1 m pendulum on Earth has a period of 2π√(1/9.807) ≈ 2.006 s, so each individual swing takes about one second. Take the same pendulum to the Moon, where gravity is six times weaker, and the period stretches to about 4.9 s — a pendulum clock would run drastically slow.

The small-angle caveat

This formula assumes small swings, below roughly 15°. Beyond that the real period grows a little longer than the formula predicts, and at large amplitudes the error becomes significant.