Pendulum CalculatorSimple pendulum · T = 2π√(L ÷ g)
The period of a simple pendulum is T = 2π√(L ÷ g): it depends only on the length and gravity, not on the mass. Enter a length to get the period and frequency, enter a period to find the length you need, or measure both to work out your local g. Large swings get a built-in amplitude correction, and the pendulum on the right swings at your real period.
Formula verified against OpenStax — University Physics, Volume 1 · HyperPhysics, Georgia State University
By Manzoor Ahmad · Pharmacist & calculator developerReviewed & updated
Pendulum setup
Length is measured from the pivot to the centre of the bob.
Step-by-step working
What is a simple pendulum?
A simple pendulum is a mass on a light string or rod that swings back and forth under gravity. Its period, the time for one complete swing, is set only by its length and the local gravitational acceleration.
Galileo noticed that a pendulum’s period barely depends on how wide it swings. Huygens turned that into the pendulum clock in 1656, the most accurate timekeeper for nearly three centuries.
Quick answers
How long should a pendulum be for a 1-second swing?
For a period of 1 s (a full back-and-forth swing), L = 24.8 cm. For a clock that ticks once per second on each pass, the period is 2 s and L = 99.4 cm, the classic seconds pendulum.
How do I find the length of a pendulum from its period?
Rearrange to L = gT² ÷ 4π². A swing that takes 3 s needs 2.24 m. Choose Find length above and enter your period and amplitude.
How do I measure gravity with a pendulum at home?
Hang a small heavy nut on 1 m of thread, time 20 swings at under 10°, divide by 20, then use g = 4π²L ÷ T². Careful measurements land within 1% of 9.81 m/s². The Measure g tab grades your result.
Why does amplitude change the pendulum period?
The restoring force grows more slowly than the angle at wide swings, so the bob takes longer. At 30° the period is 1.7% longer than T = 2π√(L/g). The swinging bob is also a neat way to see energy trading between potential and kinetic energy.
Pendulum period formula
The first formula is the small-angle approximation. The series on the third line (θ₀ in radians) corrects it for larger swings.
| Symbol | Meaning | SI unit |
|---|---|---|
| T | Period: time for one full swing (there and back) | s |
| L | Length from pivot to centre of mass | m |
| g | Gravitational acceleration | m/s² |
| f | Frequency | Hz |
| θ₀ | Amplitude (maximum angle) | rad |
How to calculate the period of a pendulum
- Measure L from the pivot to the centre of the bob, in metres.
- Divide L by g and take the square root.
- Multiply by 2π (≈ 6.2832) to get the period in seconds.
- For swings above about 15°, multiply by the amplitude correction factor shown in the results.
Worked example: the 1 m pendulum and the “seconds pendulum”
What is the period of a 1.000 m pendulum on Earth?
f = 1 ÷ 2.0064 = 0.4984 Hz
Reverse it: for T = exactly 2 s (one second per swing in each direction), L = 9.80665 × 2² ÷ 4π² = 0.9936 m. That is the classic “seconds pendulum” used in longcase clocks.
Period vs length on Earth (small swings)
Quadrupling the length doubles the period.
| Length | Period | Frequency | Swings per minute |
|---|---|---|---|
| 0.25 m | 1.003 s | 0.997 Hz | 59.8 |
| 0.50 m | 1.419 s | 0.705 Hz | 42.3 |
| 1.00 m | 2.006 s | 0.498 Hz | 29.9 |
| 2.00 m | 2.837 s | 0.352 Hz | 21.1 |
| 10.0 m | 6.345 s | 0.158 Hz | 9.5 |
How much does amplitude matter?
The simple formula assumes small swings. Here is the true period relative to T₀:
| Amplitude | Correction factor | Error of simple formula |
|---|---|---|
| 5° | 1.0005 | 0.05 % |
| 10° | 1.0019 | 0.19 % |
| 20° | 1.0077 | 0.77 % |
| 30° | 1.0174 | 1.7 % |
| 45° | 1.0400 | 3.8 % |
| 60° | 1.0732 | 6.8 % |
Common mistakes to avoid
- Measuring to the top of the bob instead of its centre, which shortens L.
- Timing a half swing. The period is a full there-and-back cycle.
- Expecting a heavier bob to swing faster. Mass cancels out of the equation.
Manzoor Ahmad
Pharmacist turned Web Developer & Calculator Engineer · 13+ years in web developmentI’m a pharmacist by training who turned into a full-time tech geek, and for 13+ years I’ve designed and coded web tools. My pharmacy degree included physical pharmacy and pharmaceutics — states of matter, heat, flow and reaction kinetics — so the physics behind these tools is familiar ground, not a new hobby.
I’m not a physicist, so I don’t ask you to take my word for it: every formula on this page is traced to an open university textbook or NIST reference data, and the calculator is tested against published worked examples before it goes live. Spot an error? Tell me and it will be fixed and dated on the next update.
Pendulum Calculator: frequently asked questions
For a simple pendulum with small swings, T = 2π√(L ÷ g), where L is the length and g the gravitational acceleration.
No. Mass cancels out in the equation of motion, so a heavy and a light bob on the same length swing with the same period, apart from small air-drag effects.
L = gT² ÷ 4π² = 9.80665 × 1 ÷ 39.478 ≈ 0.2484 m (24.8 cm).
Heat expands the rod, increasing L, so the period gets longer and the clock loses time. Compensated pendulums use materials that cancel the expansion.
Measure L, time 20 full swings and divide by 20 to get T, then use g = 4π²L ÷ T². With care, school experiments reach better than 1% accuracy.
It replaces sin θ with θ (in radians), which makes the motion simple harmonic. It is within 0.2% for swings up to 10°.
Frequency is 1 ÷ T. A 1 m pendulum has f ≈ 0.498 Hz, or about 30 full swings per minute.
References & sources
- OpenStax — University Physics, Volume 1 — Ch. 15.4, Pendulums
- HyperPhysics, Georgia State University — large-amplitude pendulum period
- NASA NSSDC — Planetary Fact Sheet — surface gravity values
Disclaimer: This calculator is an educational tool built on idealised textbook models. It is not a substitute for professional engineering or safety calculations.