Pendulum period calculator
Pick what to solve for and fill in the rest. By default the calculator uses the small-angle formula, which is accurate for swings of a few degrees. Enter an amplitude to apply a labelled series correction for wider swings.
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How to use it
- Choose to solve for the period, the length or gravity.
- Fill in the other values. The box of the quantity you solve for is ignored.
- Leave the amplitude at 0 for the small-angle formula, or enter the swing in degrees for the correction.
How it works
The ideal simple pendulum has T₀ = 2π√(L/g). With a larger amplitude θ the period is T = T₀ × (1 + θ²/16 + 11θ⁴/3072 + 173θ⁶/737280), with θ in radians. When you enter a measured period together with an amplitude, the calculator divides out that factor before solving for length or gravity.
Worked example
A 1 m pendulum on Earth: T = 2π × √(1 ÷ 9.80665) = 2.006 s. Measuring a 2-second period with a 0.99362 m string gives g = 4π² × 0.99362 ÷ 4 = 9.80665 m/s².
Tips
- Measure the length to the centre of the bob, not to its top.
- Time 10 swings and divide by 10 to reduce reading errors.
Frequently asked questions
What is the formula for a pendulum's period?
For small swings, T = 2π × √(L ÷ g), where L is the length from the pivot to the centre of the bob and g is gravity. The mass of the bob does not appear. Standard gravity is 9.80665 m/s².
How long is a one-second pendulum?
A pendulum whose full swing (there and back) takes 2 seconds is about 0.9936 m long on Earth. Enter a period of 2 and solve for length to see it. Its half-swing then takes 1 second, which is why it is called a seconds pendulum.
When is the small-angle formula not good enough?
The exact period grows with the amplitude. At 20° it is about 0.8 percent longer than the small-angle value, and at 60° about 7 percent longer. Enter the amplitude to apply the series correction.
How accurate is the amplitude correction?
It keeps the first three correction terms of the exact series, which stays within about 0.1 percent of the exact period over the whole 0 to 90° range.
Does it handle a real, swinging object?
It models an ideal point mass on a light string with no friction. A real pendulum with an extended body needs its moment of inertia, which is not included.