Pendulum Period Calculator
Enter the pendulum length and, if you want, the swing angle. You get the period, the frequency and the correction for wide swings.
Worked examples
One metre
Length 1 m, small swing
2.006 s
2 x pi x sqrt(1 / 9.80665).
Wide swing
Length 1 m, 90 degree swing
2.368 s, 18% longer
Exact period with the arithmetic-geometric mean.
About Pendulum Period Calculator
A simple pendulum swings back and forth with a period that depends only on its length and the strength of gravity, not on its mass. For small swings the period is 2 pi times the square root of the length divided by gravity. A one metre pendulum has a period of just over 2 seconds, and a pendulum of about 0.994 metres has a period of exactly 2 seconds, one second in each direction, as used in old clocks.
The simple formula is accurate only for swings up to about 10 degrees. Wider swings take slightly longer, because the restoring force is not exactly proportional to the angle. This calculator uses an exact method based on the arithmetic-geometric mean, so you can enter any angle below 180 degrees and see how much longer the true period is. For a swing of 90 degrees the period is about 18% longer than the small angle value.
How to use Pendulum Period Calculator
- Enter the length from the pivot to the centre of the weight.
- Enter the swing angle from the vertical in degrees, or leave it at 0 for a small swing.
- Change gravity if you are not on Earth.
- Read the period, frequency and the effect of the wide swing.
Common uses
- Physics practicals measuring gravity with a pendulum
- Designing a clock or a metronome
- Finding the length for a given swing time
- Seeing how a swing differs on the Moon
Frequently asked questions
What is the pendulum period formula?
For small angles T = 2 x pi x the square root of (L / g), with L in metres and g in metres per second squared.
Does the mass of the bob matter?
No. The period of a simple pendulum does not depend on the mass, only on length and gravity.
How accurate is the small angle formula?
Within 0.2% at 10 degrees, about 1% at 23 degrees and 4% at 45 degrees. For wider swings use the corrected value this page gives.
How long is a pendulum with a one second beat?
About 0.994 metres on Earth, which has a period of two seconds. It is called a seconds pendulum.