Pendulum Period Calculator

The simple pendulum

A small mass hung on a string and pulled slightly aside swings back and forth. The time for one full swing there and back is the period. For small angles the period depends only on the length of the string and on gravity:

T = 2 × π × √(L / g)

Mass does not appear: a heavy or a light bob makes no difference. Neither does amplitude, as long as it stays small. Making the string four times longer doubles the period.

Example

For a pendulum 1 metre long, T = 2 × π × √(1 / 9.80665) = 2.006 seconds. A pendulum with a period of exactly 2 seconds is called a seconds pendulum, and its length is 9.80665 / π² = 0.9936 metres.

Measuring g with a pendulum

Turning the formula round gives g = 4 × π² × L / T². This is one of the easiest ways to measure gravitational acceleration. Measure the length from the pivot to the centre of the bob, time 20 swings and divide by 20. The "Find g" tab does the calculation.

How good is the small angle approximation?

The formula rests on sin θ ≈ θ. As the angle grows the real period gets longer. The error is about 0.2 percent at 10 degrees, 0.8 percent at 20 degrees, 4 percent at 45 degrees and 18 percent at 90 degrees. The calculator also shows the exact period and the difference for the amplitude you enter.

Limits

The calculation is for an ideal pendulum: a point mass, a massless string that does not stretch and a frictionless pivot. A swinging rod or an extended body is a physical pendulum, and its period depends on its moment of inertia.