Pendulum Period Calculator

About the Pendulum Period Calculator

The time a simple pendulum takes to swing there and back, its period, depends on its length and on gravity: T = 2π·√(L ÷ g) for small swings. This calculator finds the period of a pendulum, the length that gives a period, or g from a period you have timed, and it does each exactly for any amplitude, not only for small swings.

Wide swings are slower than the small-angle formula says: about 0.2 % at 10°, 1.7 % at 30° and 18 % at 90°. The exact period comes from the arithmetic–geometric mean (the same as the complete elliptic integral), so it is right up to 179°. A chart and a table show how the period grows with the amplitude, and another table puts the same pendulum on the Moon and the planets.

For a lab, enter the time for 10 or 20 oscillations, the length and the amplitude, and add your uncertainties to get g with its error. With a Pro pass, copy the result or download it as a CSV, or unlock this one result. Without a pass the free preview shows the period, length or g you solve for in full, with the first rows of each table, a marked chart and the first step of the working.

Common uses

• Physics labs: g from a timed pendulum, with the amplitude correction and the uncertainty
• Homework on simple harmonic motion and the small-angle approximation
• Choosing the length of a pendulum for a clock, a metronome or a demonstration
• Seeing how much the small-angle formula misses for wide swings
• Comparing the same pendulum on the Moon, Mars or Jupiter

The formulas

• Small swings: T₀ = 2π·√(L ÷ g), f = 1 ÷ T; good to 0.5 % below about 15°, and the period does not depend on the mass
• Any amplitude θ₀: T = T₀ ÷ M(1, cos(θ₀ ÷ 2)), where M is the arithmetic–geometric mean: start with a = 1 and b = cos(θ₀ ÷ 2), and replace them by their average and their geometric mean, √(a·b), until they agree. It equals T = (2T₀ ÷ π)·K(sin(θ₀ ÷ 2)), with K the complete elliptic integral of the first kind
• The series T ≈ T₀·(1 + θ₀²/16 + 11θ₀⁴/3072 + …), θ₀ in radians: close for small swings, too low for wide ones (the chart shows both)
• Length for a period: L = g·(T·M ÷ 2π)²
• g from a period: g = 4π²·L ÷ (M·T)², with T the time measured ÷ the oscillations counted: a wide swing is slower, so the small-angle 4π²·L ÷ T² reads g too low
• Uncertainty: δg ÷ g = √((δL ÷ L)² + (2·δT ÷ T)²); the period counts twice because g goes with 1 ÷ T²
• At the bottom: v = √(2·g·L·(1 − cos θ₀)), from the drop L·(1 − cos θ₀)

Why wide swings are slower

The small-angle formula treats the pull back to the middle as if it grew in step with the angle, but it grows with sin θ, which falls behind the angle as the swing widens. A wide swing is pulled back more gently than the formula assumes, so it takes longer. The exact period is only a little longer for small swings (0.19 % at 10°, 0.43 % at 15°), and the gap grows quickly: 1.7 % at 30°, 7.3 % at 60°, 18 % at 90°, and about 2.4 times as long at 170°.

Above 90° a string goes slack on the way down, so such swings need a light, rigid rod.

Measuring g with a pendulum

• Measure the length from the pivot to the centre of the bob, and keep the string much longer than the bob.
• Release it from a small angle, 5° to 10°, or enter the amplitude so the calculator corrects for it.
• Time 10 or 20 full oscillations, there and back each, and let the calculator divide: the stopwatch’s start and stop errors then spread over all of them.
• Start timing as the bob passes the lowest point, where it moves fastest and the moment is easiest to see.
• Repeat with a few lengths: T² against L is a straight line through zero with a slope of 4π² ÷ g.

Sources

• OpenStax, University Physics Volume 1, §15.4 Pendulums: T = 2π·√(L ÷ g) for angles below about 15° and Example 15.3, measuring g (CC BY 4.0)
• Pendulum (mechanics)): the exact period with the complete elliptic integral, its arithmetic–geometric-mean form and the series in θ₀
• R. A. Nelson and M. G. Olsson, The pendulum — Rich physics from a simple system, American Journal of Physics: the corrections needed to measure g accurately with a pendulum
• NASA JPL Solar System Dynamics: planetary physical parameters: the surface gravity of the planets

Limitations

• A simple pendulum: a small heavy bob on a light string or rod. A swinging bar, a door or a large bob is a physical pendulum, whose period depends on how its mass is spread.
• Air resistance and friction at the pivot are ignored, so the amplitude never dies down here; real swings shrink slowly, which shortens the period a little.
• Gravity differs a little from place to place on Earth (about 9.78 to 9.83 m/s²); choose 9.80665 m/s² or enter your local value.
• Swings wider than 90° need a rigid rod; a string goes slack.

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