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Pendulum Period Calculator
Work out how long a simple pendulum takes to swing from its length and the local gravity, or go backwards to the length for a given period, or to g from a classroom measurement. Large release angles use the exact period, not just the small-angle approximation.
Last updated:
Solve for
s
m/s²
°
0 = small-angle formula; larger angles use the exact period
Enter your values to see the result.
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How to use the Pendulum Period Calculator
- Choose what to solve for: period, length or gravity.
- Enter the length from the pivot to the centre of the bob, or the period.
- Pick the gravity (Earth by default) and optionally the release angle.
- Read the result with frequency and the step-by-step formula.
Pendulum formulas
| Find | Formula | Example |
|---|---|---|
| Period (small angle) | T = 2π √(L / g) | L = 1 m, g = 9.80665 → T = 2.0064 s |
| Length | L = g (T / 2π)² | T = 2 s → L = 0.9936 m |
| Gravity | g = 4π² L / T² | L = 0.75 m, T = 1.74 s → g = 9.78 m/s² |
| Frequency | f = 1 / T = (1 / 2π) √(g / L) | T = 2.0064 s → f = 0.4984 Hz |
| Exact period, amplitude θ₀ | T = 2π √(L / g) ÷ AGM(1, cos(θ₀/2)) | θ₀ = 60° → 7.3% longer |
Period of a 1 m pendulum by release angle
| Amplitude | Period (Earth) | Longer than small-angle by |
|---|---|---|
| 5° | 2.0074 s | 0.05% |
| 10° | 2.0102 s | 0.19% |
| 20° | 2.0218 s | 0.77% |
| 30° | 2.0413 s | 1.74% |
| 45° | 2.0866 s | 4.00% |
| 60° | 2.1532 s | 7.32% |
| 90° | 2.3682 s | 18.03% |
These assume an ideal simple pendulum: a point mass on a light, rigid string with no air resistance. Real pendulums with a large bob or heavy rod swing a little slower.
Frequently asked questions
What is the formula for the period of a pendulum?
For small swings T = 2π √(L / g), where L is the length from the pivot to the centre of mass of the bob and g is the gravitational acceleration. A 1 m pendulum on Earth has a period of about 2.006 s.
Does the mass of the bob change the period?
No. For an ideal simple pendulum the period depends only on length and gravity, not on the mass. A heavy and a light bob on the same string swing at the same rate.
How long is a seconds pendulum?
A seconds pendulum ticks once per second, so its full period is 2 s. With standard gravity that needs L = g (T / 2π)² ≈ 0.994 m, a little under a metre.
Does a bigger swing take longer?
Yes, slightly. The small-angle formula is accurate to about 0.2% at 10°. At 30° the true period is about 1.7% longer and at 60° about 7.3% longer. This calculator uses the exact formula T = 2π√(L/g) ÷ AGM(1, cos(θ₀/2)).
How can I measure g with a pendulum?
Measure the length, time 10 or 20 full swings with small amplitude and divide to get T, then use g = 4π² L / T². A 0.75 m pendulum with T = 1.74 s gives g ≈ 9.78 m/s².
How would a pendulum clock behave on the Moon?
With g = 1.62 m/s², about one sixth of Earth's, a pendulum swings √(9.81/1.62) ≈ 2.46 times slower, so a clock tuned for Earth would lose more than half of each hour.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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