Pendulum Clock Length Calculator

The period of a simple pendulum is governed by one of the most elegant equations in physics, first derived by Christiaan Huygens in 1673 in the Netherlands.

The Pendulum Period Formula

T = 2 x pi x sqrt(L / g)

Where:

• T = period of one complete swing (seconds)
• L = length from pivot to center of mass of the bob (meters)
• g = gravitational acceleration (9.80665 m/s^2 at sea level)

Solving for length: L = g x (T / (2 x pi))^2

Beat vs. Period

A “beat” in horology is one swing (half a period). A clock that “beats seconds” swings once per second, meaning the full period T = 2 seconds.

The physics in the middle three columns is exact. The last column is not, and it is worth saying so plainly.
Clock type names are trade conventions, not specifications: plenty of clocks sold as Vienna regulators beat half-seconds, and a cuckoo pendulum has a sliding leaf precisely so it can be tuned to whatever movement is behind it. Measure your own pendulum and read the rate off this table, rather than picking a row by the name on the dial. The escapement beat rate calculator carries the same figures worked the other way round, from length to beats per hour.

Two kinds of clock are deliberately absent. A 400-day anniversary clock uses a torsion pendulum, a disc twisting on a suspension spring, and its period comes from the spring rather than from a length. A French carriage clock beats faster than anything here, around 240 a minute, but it does that with a balance wheel on a platform escapement, which is why it can be carried about at all. Neither one obeys the formula on this page.

Worked Example - Grandfather Clock (Beats Seconds)

Beat time = 1.0 second, so period T = 2.0 seconds. L = 9.80665 x (2.0 / (2 x 3.14159))^2 L = 9.80665 x (0.31831)^2 L = 9.80665 x 0.10132 L = 0.9936 m = 99.36 cm

This is why the classic “seconds pendulum” is almost exactly one meter long.

Altitude and Latitude Corrections

Gravity varies slightly by location. At higher altitudes, g is smaller, so the pendulum must be shorter. At the equator, g = ~9.780 m/s^2. At the poles, g = ~9.832 m/s^2. Rate follows the square root of g, so a 1% fall in gravity slows the clock by half a percent.

Move a clock regulated in London (g = 9.812) to Mexico City (latitude 19°, elevation 2,240 m, g = 9.779) and gravity drops by 0.33%. The clock then loses about 145 seconds a day, near enough two and a half minutes. That is a real regulation job on arrival, not a rounding error, and it is why a longcase clock shipped across the world always needs its bob resetting.

Temperature Effects

A steel pendulum rod expands about 0.0012% per degree Celsius, which is 12 parts per million. For a 1-meter pendulum, a 10°C temperature increase lengthens the rod by about 0.12 mm. At roughly 43 seconds per day per millimetre (see below), that costs about 5 seconds a day. This is why precision clocks use compensating pendulums made of invar alloy or wood-and-brass composite rods.

Practical Adjustment

Most pendulum clocks have a rating nut at the bottom of the bob. Turning the nut clockwise (raising the bob) shortens the effective length and speeds up the clock.

How much a given movement is worth follows one rule, and it is the single most useful number on this page:

Rate change (seconds per day) = 43,200 ÷ pendulum length in millimetres, per millimetre moved

A seconds pendulum near 994 mm therefore shifts about 43 seconds a day for every millimetre. A typical rating nut moves the bob roughly 0.4 mm per full turn, so one turn is worth around 17 seconds a day on a longcase clock. Short pendulums are far touchier: a 250 mm regulator moves about 173 seconds a day per millimetre, so work in quarter turns there. The clock rate error calculator turns an observed gain or loss straight into the millimetres to move.