Simple Pendulum Calculator
Compute the period, frequency, and angular frequency of a simple pendulum from its length and the local gravitational acceleration.
m
m/s²
Time for one complete swing (small angles)
- 1
Ratio L/g
1 ÷ 9.81 = 0.101937 - 2
√(L/g)
√(0.101937) = 0.319275 - 3
Period T = 2π × √(L/g)
2π × 0.319275 = 2.0061
How does this calculator work?
A simple pendulum swings with period T = 2π·√(L / g), where L is its length and g is gravity. Its frequency is f = 1 / T and its angular frequency is ω = √(g / L). The period depends only on length and gravity, not the bob mass, and the formula holds for small swing angles.
Formula
How this is calculated
A simple pendulum is an idealized point mass (bob) suspended from a massless, inextensible string of length L that swings under gravity g. For small swing angles (typically under about 15°), the restoring force is approximately proportional to displacement, giving simple harmonic motion with period T = 2π·√(L / g). Enter the length L in meters and the gravitational acceleration g in m/s² (Earth is about 9.81 m/s²); the period comes out in seconds.
From the period the calculator derives the frequency f = 1 / T in hertz (swings per second) and the angular frequency ω = √(g / L) in radians per second, where ω = 2π·f. Notice the period depends only on length and gravity, not on the mass of the bob or, to first order, on the amplitude. You can also rearrange the formula to size a pendulum for a target period: L = g·(T / 2π)².
Assumptions and edge cases: the small-angle approximation breaks down for large amplitudes (the true period grows slightly with angle), and air resistance, string mass, and friction are ignored. Length and gravity must both be positive, so the inputs are guarded against zero or negative values.
Frequently asked questions
No. For a simple pendulum the period depends only on the length and gravitational acceleration, T = 2π·√(L / g). The mass cancels out, so a heavy and a light bob on equal-length strings swing at the same rate.
The formula assumes the restoring force is proportional to displacement, which is only true for small swing angles (roughly under 15°). At larger amplitudes the real period is somewhat longer than T = 2π·√(L / g).
Rearranging gives L = g·(T / 2π)². For a 2-second period (one tick per second each way) with g = 9.81 m/s², L ≈ 0.994 m, the classic "seconds pendulum".
Also known as
TG we-Calculate Editorial Team. (2026). Simple Pendulum Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/simple-pendulum-calculator
TG we-Calculate Editorial Team. "Simple Pendulum Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/simple-pendulum-calculator.
TG we-Calculate Editorial Team, "Simple Pendulum Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/simple-pendulum-calculator
@misc{wecalculate_simple_pendulum_calculator, title = {Simple Pendulum Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/simple-pendulum-calculator}}, year = {2026}, note = {TG we-Calculate} }
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