Pendulum period calculator
Set the string length and the gravity, and the simulator animates the pendulum at its real period and counts how many full swings it would make in ten minutes. It is one of the simplest experiments for measuring gravity anywhere.
Key formula
T = 2π · √(L / g) · g = 4π² · L / T²
Try it
How to use the simulator
A simple pendulum always takes the same time to swing there and back, and that period depends on only two things: string length and gravity (T = 2π√(L/g)). The simulator animates the pendulum with its real period and counts how many full swings it would make in ten minutes, which is exactly what you can measure with a watch.
Controls
- Scenarios: buttons that set Earth gravity (9.8 m/s²), Hail Mary gravity (15 m/s²) or Moon gravity (1.6 m/s²).
- String length: 0.1 to 3 m.
- Gravity: 0.5 to 30 m/s².
Things to try
- The starting values recreate Grace's experiment: with Hail Mary gravity and a 1.14 m string you get 346 cycles in 10 minutes. Now find the length that gives the same count under Earth gravity (about 0.74 m).
- Make the string four times longer and notice the period only doubles: it scales with the square root.
- Switch from Earth to the Moon with the same string. The swing is about 2.5 times slower, because √(9.8/1.6) ≈ 2.5.
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What the period depends on
A simple pendulum is a small mass hanging from a light string. Pull it slightly from vertical and let go, and it swings back and forth with a very regular rhythm. The time for one full swing (there and back) is the period, T.
For small angles the period depends only on string length and gravity. It does not depend on the mass you hang or on how far you pull it, as long as the angle stays below about 10–15°. At 10° the formula is off by only 0.2%.
Measuring gravity with a pendulum
If you measure the length and the period, you can solve for gravity: g = 4π² · L / T². To keep the error down, time many swings and divide. If you count 50 and your reaction on the stopwatch is off by 0.2 s, the error per swing is only 0.004 s.
This was the standard way to measure gravity for over two centuries. Galileo noticed how regular pendulums are, Huygens built the first pendulum clock in 1656, and surveyors used pendulums to discover that gravity varies slightly from place to place on Earth.
Worked example
Example: the seconds pendulum
- With a 1 m string on Earth (g = 9.81 m/s²), T = 2π · √(1 / 9.81) ≈ 2.006 s.
- A pendulum with a period of exactly 2 s (one second each way) needs L = g · T² / (4π²) ≈ 0.994 m. That is the classic "seconds pendulum".
- On the Moon (g = 1.62 m/s²) the same 1 m string would take 2π · √(1 / 1.62) ≈ 4.9 s per swing.
- Make the length four times longer in the simulator and check that the period only doubles: it goes with the square root.
In Project Hail Mary
Early in the novel, the main character measures the local gravity with a makeshift pendulum and a stopwatch, counting swings. The simulator's starting values recreate that experiment.
Frequently asked questions
Does the pendulum's mass affect the period?
No. Gravity pulls harder on a bigger mass, but that mass is also harder to accelerate, and the two effects cancel. That is why all pendulums of the same length swing alike.
What about large angles?
The period gets a little longer. At 30° it is 1.7% longer than the formula says, and at 90° about 18% longer. For measuring gravity, small angles are best.
What is a Foucault pendulum?
A very long, heavy pendulum that swings for hours. Its plane of swing seems to turn slowly because Earth rotates beneath it. Léon Foucault used one in 1851 to demonstrate Earth's rotation.