Pendulum Period Calculator
Period T (s)2.0061
Frequency (Hz)0.4985
Angular Frequency ω (rad/s)3.1321
The Pendulum Period Calculator finds how long a simple pendulum takes to complete one full swing using the small-angle relationship T = 2π√(L/g). Enter the arm length and the local gravitational acceleration to get the period in seconds, the frequency in hertz, and the angular frequency in radians per second. Because the formula assumes small swing angles, the period depends only on length and gravity, not on the mass of the bob.
Formula
T = 2π√(L / g); f = 1 / T; ω = 2π / T
- T
- Period — time for one complete swing (seconds)
- L
- Length from pivot to centre of mass (metres)
- g
- Gravitational acceleration (m/s²)
- f
- Frequency — swings per second (hertz)
- ω
- Angular frequency (radians per second)
How it works
- Enter the pendulum length L in metres, measured from the pivot to the centre of the bob.
- Set the gravitational acceleration g (9.81 m/s² on Earth; lower it to model the Moon or another planet).
- The calculator computes the period T = 2π√(L/g), then derives frequency f = 1/T and angular frequency ω = 2π/T.
Worked example
A 1-metre pendulum swinging on Earth.
- L = 1 m, g = 9.81 m/s².
- T = 2π√(1 / 9.81) = 2π × 0.31928 = 2.0061 s.
- f = 1 / 2.0061 = 0.4985 Hz.
Period ≈ 2.0061 s, frequency ≈ 0.4985 Hz.
Frequently asked questions
- Does the mass of the bob affect the period?
- No. For a simple pendulum the period depends only on the length and the gravitational acceleration. Doubling the mass leaves the swing time unchanged because both the restoring force and the inertia scale with mass.
- Why does the calculator assume small angles?
- The formula T = 2π√(L/g) is the small-angle approximation, accurate to within about 1% for swings up to roughly 20 degrees. At large amplitudes the true period grows slightly and the simple formula underestimates it.
- How does gravity change the period?
- A weaker gravitational field lengthens the period. On the Moon (g ≈ 1.62 m/s²) the same pendulum swings about 2.5 times slower than on Earth, which is why you can set g to model other planets.
- How do I make a pendulum tick once per second?
- A clock that beats once per second needs a half-period of 1 s, so the full period is 2 s. Solving 2 = 2π√(L/9.81) gives a length of about 0.994 m, the classic "seconds pendulum".