Pendulum Period & Length Calculator

Pendulum Physics Guide

Simple Pendulum Formula

T = 2π√(L/g), where T is the period in seconds, L is the length in metres, and g is gravitational acceleration (9.81 m/s² on Earth). The period is independent of mass and (for small angles under 15°) independent of amplitude — this is the isochrone property discovered by Galileo. A pendulum of length 0.994m has a period of almost exactly 2 seconds (one second each way), which was the basis of the original metre definition.

The Seconds Pendulum

A pendulum with a period of 2 seconds (one beat per second) has a length of approximately 0.994m at sea level in London (g = 9.812 m/s²). This was historically used to define the metre — the French Academy proposed defining the metre as the length of a seconds pendulum, before settling on the one ten-millionth of the Earth's circumference definition. Pendulum clocks use escapement mechanisms to count swings precisely, with pendulum length adjusted by moving the bob to regulate accuracy.

Effect of Gravity

The pendulum period depends on local gravity, which varies slightly with latitude and altitude. At the equator (g = 9.780 m/s²), a pendulum runs slightly slower than at the poles (g = 9.832 m/s²). A seconds pendulum calibrated in London would run slow on the equator by approximately 16 seconds per day. On the Moon (g = 1.62 m/s²), the same pendulum would have a period of √(9.81/1.62) ≈ 2.46× longer — beating once every 4.9 seconds instead of 2.

Limitations of the Simple Model

The simple pendulum formula assumes: small amplitude (under 15° for under 0.5% error), massless rigid rod, point mass bob, no air resistance or friction, and uniform gravity. Real pendulums deviate from the formula at large amplitudes, and physical pendulums (distributed mass) require the moment of inertia in the calculation. For large amplitude corrections, the period is: T ≈ T₀(1 + θ²/16 + ...) where θ is in radians.