Pendulum Calculator
A simple pendulum, a mass swinging on a light string, is one of the most elegant systems in physics. Its most famous property is that the period, the time for one complete back-and-forth swing, depends only on the length of the string and the local strength of gravity, not on the mass of the bob or the size of the swing (for small angles). This surprising independence from mass is what made pendulums the basis of accurate timekeeping for centuries: a pendulum of fixed length swings at the same rate whether the bob is heavy or light, and small variations in how far it swings barely affect the timing. The formula is T = 2π times the square root of L divided by g, where L is the pendulum length in metres and g is gravitational acceleration in m/s2. Longer pendulums swing more slowly, but the relationship involves a square root, so to halve the frequency you must quadruple the length. This calculator takes the length and gravity as inputs and returns the period in seconds, the frequency in hertz, and the number of complete swings per minute, a useful unit for visualising the motion. It also shows the length needed for a period of exactly 1 s, which works out to about 0.248 m, a useful cross-check. The default inputs of L = 1 m and g = 9.81 m/s2 give a period of 2.006 s and a frequency of 0.498 Hz. A note on accuracy: the formula assumes small swing angles (below about 15 degrees) and a massless, rigid string. For wider swings the true period is slightly longer than this formula predicts.
T = 2π sqrt(L/g). Valid for small swing angles (below ~15 degrees). Standard gravity is 9.81 m/s². Rounded for display.
How it works
The period of a simple pendulum: T = 2π √(L/g). Frequency is 1/T in hertz. Swings per minute is 60/T. The length needed for a period of 1 s is g/(4π²), which equals about 0.248 m at standard gravity. The formula assumes the small-angle approximation (sinθ ≈ θ in radians), which is accurate to within 0.5% for angles up to about 15 degrees.
Worked example
A pendulum of length L = 1 m under gravity g = 9.81 m/s2. Period T = 2π times sqrt(1/9.81) = 2π times 0.3193 = 2.006 s. Frequency = 1/2.006 = 0.498 Hz. Swings per minute = 60/2.006 = 29.9. These match the default values pre-filled above.
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