Simple Pendulum Period

Galileo noticed — reputedly while timing a swinging cathedral lamp against his own pulse — that a pendulum's period depends on neither its mass nor, for small swings, its amplitude: only on its length and the local pull of gravity. Christiaan Huygens turned that insight into the pendulum clock in 1656, and for nearly three centuries it remained the world's best timekeeper. The solver uses standard gravity, g = 9.80665 m/s², a value exact by definition.

A worked example: a "seconds pendulum" beating once per second has a full period of 2 s, so L = g(T/2π)² = 9.80665 × (2/6.2832)² ≈ 0.994 m — the reason grandfather clocks stand about a metre tall inside. The formula is a small-angle approximation, accurate to about 1% for swings under 15°. Surveyors once ran it in reverse, timing precision pendulums to map tiny local variations in g across the Earth's surface.