Pendulum Period Calculator
Enter length and gravity to compute the small-angle pendulum period T = 2π·√(L/g).
Input Data
Results
At a glance:A simple pendulum (point mass on a massless rod) has, for small angular amplitudes (θ ≲ 15°), a period independent of amplitude: T = 2π·√(L/g), where L is the length and g gravity. This is the small-angle (linear) approximation of the exact elliptic integral. Frequency f = 1/T. For larger angles the period is longer by a factor 1 + (θ₀²/16) + ... (θ₀ in radians). The period grows with √L and falls with √g — so a pendulum runs slower on the Moon (g lower) and longer pendulums swing slower. This tool returns T and f for the small-angle case.
Formula
T = 2π·√(L/g)
f = 1/T
Large-angle correction: T ≈ T₀(1 + θ₀²/16)
$$T = 2\pi \sqrt{\dfrac{L}{g}}$$$$f = \dfrac{1}{T} = \dfrac{1}{2\pi}\sqrt{\dfrac{g}{L}}$$How to Use
- Enter the length L (m).
- Enter the gravitational acceleration g (m/s²).
- The calculator returns the period T and frequency f.
Case Studies
Grandfather clock
L = 0.994 m, g = 9.81.
T = 2π√(0.994/9.81) ≈ 2.0 s.
Half-period 1 s → 'tick-tock' seconds pendulum.
FAQ
Why is period independent of mass?
Both the restoring force (mg sinθ) and inertia (m) scale with mass, so m cancels — all masses swing with the same period (Galileo's observation).
Does amplitude matter?
Only for larger swings; the small-angle formula assumes sinθ≈θ. Big amplitudes give a longer period (corrected by the elliptic-integral series).
Related Tools
References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.