Pendulum period calculator
This pendulum period calculator uses T = 2π·√(L/g): it gives the period from the pendulum length and vice versa, plus frequency, maximum speed and the exact correction for large swing angles.
Example: Known value: pendulum length · Pendulum length 1 m · Amplitude (swing angle) 10 ° · Location (gravity): Earth (9.81 m/s²) → Result: T = 2.0102 s · f = 0.4975 Hz · L = 1 m. Source: Wikipedia – Mathematisches Pendel (Periodendauer, Großwinkelkorrektur). Updated: .
How it is calculated
For small swings, T = 2π·√(L/g). A 1 m pendulum has a period of 2.006 s on Earth, and a seconds pendulum with T = 2 s is about 99.4 cm long.
What the formula says
The period depends only on length and gravity, not on the mass of the bob. Four times the length means twice the period. On the Moon, where g = 1.62 m/s², the same pendulum swings about 2.5 times slower than on Earth.
Large angles
The approximation is only exact for small angles. With a larger amplitude the swing slows down: at 10° the period is 0.2 % longer, at 30° about 1.7 %, at 90° already 18 %. The calculator therefore uses the exact solution with the elliptic integral, computed through the arithmetic-geometric mean: T = T₀ / AGM(1, cos(θ₀/2)).
- The pendulum must be a thin, massless string with a point mass; otherwise the physical pendulum formula applies.
- The maximum speed at the lowest point is v = √(2·g·L·(1 − cos θ₀)).
Frequently asked questions
How do you calculate the period of a simple pendulum?
With T = 2π·√(L/g). For L = 1 m and g = 9.81 m/s² that gives 2.006 s. The mass does not matter.
How long is a seconds pendulum?
A pendulum with a 2 s period (one second per swing) is about 0.994 m long on Earth.
When does the small-angle approximation stop working?
At 20° the error is already about 0.8 %. The calculator therefore shows the correction factor for larger angles in a table.
Sources and legal basis
- Wikipedia – Mathematisches Pendel (Periodendauer, Großwinkelkorrektur)
- Wikipedia – Pendulum (mechanics): period for large amplitudes, AGM
- NASA Planetary Fact Sheet – Oberflächenschwerebeschleunigung
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