Pendulum Period Calculator
Calculate the period of a simple pendulum: T = 2π√(L/g). Enter pendulum length and gravity. Includes a large-angle correction for amplitudes above 10°.
How to use this tool
- Enter pendulum length l, gravity g and amplitude θ in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your period (small-angle approx.) and the full breakdown beneath it.
The simple pendulum period for small angles is T = 2π√(L/g), independent of mass and (approximately) amplitude. For swings above ~10° the correction T ≈ T₀(1 + sin²(θ/2)/4 + …) increases the period noticeably.
Formula
Small-angle period: T0 = 2π √(L / g)
Large-angle correction (Lindstedt series): T = T0 × (1 + sin2(θ/2)/4 + 9 sin4(θ/2)/64)
How it works
The small-angle formula assumes sin(θ) ≈ θ (valid below roughly 10°), yielding a period that depends only on length and gravity, not on amplitude. For larger amplitudes the period grows; this calculator applies the first two terms of the power-series correction using sin(θ/2), giving accuracy to better than 0.1% up to about 60°. Frequency is reported as f = 1/T.
Worked example
Worked example
- Inputs: L = 1.0 m, g = 9.81 m/s², amplitude θ = 5°.
- Small-angle period: T₀ = 2π × √(1.0 / 9.81) = 2π × 0.31929 = 2.00617 s.
- Correction factor at 5°: sin(2.5°) ≈ 0.04362; 1 + 0.04362²/4 + 9×0.04362⁴/64 ≈ 1.000476, giving T ≈ 2.00712 s.
Period (small-angle approximation) = 2.00617 s.
Key terms
- Simple pendulum
- An idealised pendulum consisting of a point mass on a massless, inextensible string swinging in a vertical plane.
- Small-angle approximation
- The assumption that sin(θ) ≈ θ (in radians), valid for amplitudes below about 10°, which makes the restoring force linear.
- Period (T)
- The time for one complete oscillation (swing and return). For a simple pendulum it depends on length and gravity, not the bob's mass.
- Isochronism
- The property whereby the period of a pendulum is (approximately) independent of amplitude for small angles, discovered by Galileo.
- Frequency (f)
- The number of complete oscillations per second, equal to 1/T and measured in Hz.
Frequently asked questions
- What is the period of a 1-metre pendulum on Earth?
- Approximately 2.006 seconds, which is why 1-metre pendulums were historically used in grandfather clocks (half-period ≈ 1 s per swing).
- Does mass affect the period?
- No — the small-angle period T = 2π√(L/g) is entirely independent of mass.