Simple Pendulum Calculator
Computes the period and frequency of a simple pendulum from its length and gravitational acceleration. Use it for clock design, physics experiments, and comparing pendulum behaviour on different planets.
About this calculator
For small angles, the period of a simple pendulum is T = 2π√(L / g), where L is the pendulum length in metres and g is gravitational acceleration in m/s². This result is remarkable: the period is independent of the bob's mass and, for small angles, independent of amplitude. The frequency is simply f = 1 / T. For larger amplitudes the small-angle approximation breaks down, and a corrected period is T_exact ≈ 2π√(L/g) · (1 + θ²/16), where θ is the angular amplitude in radians (the ¼·(θ_rad)² factor shown in the formula). This calculator offers both the simple approximation and the first-order correction, so you can see how much amplitude matters. Changing the gravitational field — for example, on the Moon where g ≈ 1.62 m/s² — dramatically lengthens the period.
How to use
A pendulum is 0.5 m long, swinging on Earth (g = 9.81 m/s²) with a small amplitude. Select method 'simple'. T = 2π × √(0.5 / 9.81) = 2π × √(0.05097) = 2π × 0.2257 ≈ 1.418 s. Frequency f = 1 / 1.418 ≈ 0.705 Hz. Now switch to 'exact' with amplitude = 20°. θ_rad = 20 × π/180 ≈ 0.349 rad. Correction factor = 1 + 0.25 × 0.349² ≈ 1 + 0.030 = 1.030. T_exact ≈ 1.418 × 1.030 ≈ 1.461 s — about 3% longer than the simple estimate.
Frequently asked questions
Why does the period of a pendulum not depend on its mass?
In the restoring force for a pendulum, mg·sin(θ), the mass m cancels exactly with the m in Newton's second law F = ma. The resulting equation of motion depends only on g and L. This mass independence was first observed by Galileo and is one reason pendulum clocks are reliable: changing the bob does not alter the tick rate. However, air resistance and pivot friction, which do scale differently with mass, can introduce small practical differences.
How accurate is the small-angle approximation for a simple pendulum?
The small-angle approximation sin(θ) ≈ θ holds to within 1% for angles below about 14°. Beyond that, the true period exceeds the approximated period by an increasingly large margin. At 30° the error is roughly 1.7%; at 45° it reaches about 4%; at 90° the true period is about 18% longer than the simple formula predicts. For precision timekeeping or large-amplitude experiments, the correction term 1 + θ²/16 (first-order expansion) provides substantially better accuracy.
How does gravity on other planets affect pendulum period?
Since T = 2π√(L/g), a weaker gravitational field means a longer period. On the Moon (g ≈ 1.62 m/s²), a 1-metre pendulum has T = 2π√(1/1.62) ≈ 4.94 s, compared with 2.01 s on Earth. On Mars (g ≈ 3.72 m/s²), T ≈ 3.26 s. A pendulum clock calibrated on Earth would run far too slowly on the Moon. Conversely, at high altitudes on Earth, where g is slightly lower, precision pendulum clocks must be adjusted to keep correct time.