Pendulum Period

The engineering

Mass cancels out — a cannonball and a golf ball on equal strings keep identical time, which is the deep reason pendulums could regulate clocks for three centuries. Length is the only knob, and it's a square-root knob: quadruple the length to double the period. A 'seconds pendulum' (one-second half-swing) is 0.994 m in standard gravity — suspiciously close to the meter, and not by coincidence, since that was one of the meter's proposed definitions.

The fine print is 'small swing': the isochronism Galileo admired is only approximate, and at 20° amplitude the period runs about 0.8% long — an error a clock accumulates into minutes per day. Precision horology fought that circular error with tiny amplitudes and Huygens' cycloidal cheeks; you should just keep θ small.

Where this math comes from

Legend puts Galileo in Pisa's cathedral around 1602, timing a swinging lamp against his pulse and noticing the period didn't care about the swing's size. True or embellished, his letters from that period contain the law, and his proposal to regulate clocks with it was executed by Christiaan Huygens, whose 1656 pendulum clock cut timekeeping error from a quarter-hour a day to seconds.

Huygens' 1673 'Horologium Oscillatorium' supplied the full mathematics — including the cycloid's perfect isochronism — and the pendulum became a scientific instrument in its own right: Henry Kater's 1817 reversible pendulum measured g to a part in ten thousand, and Foucault's 1851 pendulum used the swinging plane to show the Earth turning underneath.

1. 1602Galileo Galilei (circa)Isochronism of small swings observed.
2. 1656Christiaan HuygensPendulum clock — timekeeping improves a hundredfold.
3. 1817Henry KaterReversible pendulum measures g with precision.
4. 1851Léon FoucaultPendulum demonstrates the Earth's rotation.

See the full timeline of the math behind every calculator →

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