Pendulum Calculator
Pendulum period, frequency and length, with large-swing correction.
About this tool
Enter the pendulum's length and swing angle and choose the gravity. The tool gives the period (with and without the large-angle correction), frequency, swings per minute and maximum speed, and the length needed for a target period.
The pendulum formula
For small swings, a simple pendulum's period depends only on its length and gravity: T = 2Οβ(L Γ· g). The mass of the bob doesn't matter, which is what made pendulums such good clocks. A 1 m pendulum on Earth swings back and forth in 2.006 seconds, and rearranging, L = g (T Γ· 2Ο)Β², the length for exactly 2 seconds is 0.994 m, the 'seconds pendulum' whose each swing takes one second. The formula assumes small angles; wider swings take longer, so the tool adds the series correction T β Tβ(1 + ΞΈΒ²/16 + 11ΞΈβ΄/3072), about 0.2% longer at 10Β° and 7% at 60Β°. The speed at the bottom of the swing comes from energy conservation, v = β(2gL(1 β cos ΞΈ)). Change gravity to see that the same pendulum swings about 2.5 times slower on the Moon. Honest limits: real pendulums have air drag, pivot friction and a rod with mass, so they behave as physical pendulums; measure length to the bob's centre of mass. The Hooke's law calculator and the metronome are related.
Frequently asked questions
What is the period of a 1 metre pendulum?
About 2.006 seconds on Earth for a full back-and-forth swing.
Does the mass of a pendulum affect its period?
No. For a simple pendulum the period depends only on length and gravity.
How long is a seconds pendulum?
About 0.994 m: its full period is 2 s, so each swing takes 1 s.
Why does a bigger swing take longer?
The small-angle formula is an approximation. At larger angles the restoring force grows more slowly than the angle, lengthening the period.
Is my data sent anywhere?
No. Everything is calculated in your browser, and nothing is stored or sent.