A pendulum period calculator that solves the classic simple-pendulum relation in any direction — period, string length or gravitational acceleration — with full working and unit checks shown at every step. Whether you are writing a physics lab report, sizing a clock pendulum, or measuring local gravity from a timing experiment, the calculator gives you the correct answer in under a second.
How it works
The simple pendulum models a point mass (the bob) suspended from a fixed pivot by a massless, inextensible string of length L. When displaced slightly and released, it executes simple harmonic motion. The governing formula, derived by linearising the equation of motion at small angles, is:
T = 2π √(L ÷ g)
where:
- T is the period (one complete back-and-forth oscillation) in seconds
- L is the effective string length from pivot to centre of mass, in metres
- g is the local gravitational acceleration, in m/s² (standard Earth value: 9.80665 m/s²)
The calculator rearranges this for all three unknowns:
| Want to find | Formula used |
|---|---|
| Period T | T = 2π √(L ÷ g) |
| Length L | L = g · (T ÷ 2π)² |
| Gravity g | g = L · (2π ÷ T)² |
It also reports the oscillation frequency f = 1/T in hertz and the half-period (time for one one-way swing), which is useful in clock design and in timing gate experiments.
Worked example
A grandfather clock uses a pendulum that must beat once per second — i.e. one half-period of 1 s, so the full period T = 2 s. What length of string is needed at standard gravity?
- Formula: L = g · (T ÷ 2π)²
- L = 9.80665 × (2 ÷ 6.28318)²
- L = 9.80665 × (0.31831)²
- L = 9.80665 × 0.10132
- L ≈ 0.9933 m (about 99.3 cm)
This is the famous seconds pendulum — just under one metre, giving the classic tick-tock of a pendulum clock. The precise length differs slightly from place to place because g varies by about 0.5% between equator and poles.
| Pendulum length (m) | Period on Earth (s) | Period on Moon (s) | Period on Mars (s) |
|---|---|---|---|
| 0.25 | 1.003 | 2.466 | 1.633 |
| 1.00 | 2.007 | 4.933 | 3.266 |
| 2.00 | 2.838 | 6.977 | 4.618 |
| 4.00 | 4.014 | 9.866 | 6.532 |
Formula note and accuracy
The formula T = 2π√(L/g) is exact only in the small-angle limit (swing angle ≤ 15° from vertical). For larger amplitudes the true period is given by a complete elliptic integral of the first kind, which adds a correction term. At 15° the error in using the simple formula is about 0.46%; at 30° it rises to 1.74%; at 45° to about 7.3%. For school and university experiments, and for clock pendulums that intentionally use small amplitudes, the simple formula is the correct one to use.
The value of g used in the calculator’s presets is the internationally adopted standard (9.80665 m/s² at sea level, 45° latitude). For a precision measurement you should look up the actual value for your location — it varies from about 9.764 m/s² at the equator to 9.832 m/s² near the poles, and decreases with altitude at roughly 3 μm/s² per metre of elevation. Use the “Custom g” option to enter your location’s value.
Everything is calculated client-side in your browser — no data is ever sent to a server.