Pendulum Calculator
Find the period of a simple pendulum from its length, the length for a given period, or the local gravity from a timed swing. Add the swing angle to see how much slower a wide swing really is.
How to Use
- Pick what to solve for: Period, Length or Gravity.
- Enter the other two values with their units. Gravity starts at 9.80665 m/s², standard gravity on Earth; change it for the Moon, Mars or your own measurement.
- Optionally enter the amplitude, the angle the pendulum swings out from vertical, to compare the textbook answer with the exact large-angle period.
- Read the answer in the highlighted field and the readouts; the drawing swings at the period you entered where that is watchable.
- Press a preset to load an example, and check Show Work for the formula, the conversions and the large-angle comparison.
Worked Example
A 1 m pendulum. T = 2π√(L ÷ g) = 2π × √(1 ÷ 9.80665) = 2π × 0.31933 = 2.006 s, a frequency of 1 ÷ 2.006 = 0.4984 Hz.
The same pendulum swung out to 60°. The series T₀(1 + θ₀²/16 + 11θ₀⁴/3072), with θ₀ = π/3 = 1.0472 rad, gives 2.006 × 1.07285 = 2.1526 s. The exact value, T₀ ÷ AGM(1, cos 30°), is 2.1532 s: 7.32% slower than the small-angle answer, and the two-term series is within 0.04% of it.
The common mistake: timing one swing instead of the full period. A seconds pendulum ticks once a second, but T is the time to swing out and back, 2 s. L = 9.80665 × (2 ÷ 2π)² = 0.9936 m. Putting in T = 1 s gives 0.2484 m, a quarter of the right length.
Show Work
Formulas
From Galileo’s Chandelier to Foucault
Galileo Galilei noticed that a pendulum keeps nearly the same time whether it swings wide or narrow; his biographer Vincenzo Viviani placed the moment in Pisa Cathedral, watching a swinging lamp, in 1583. Christiaan Huygens turned the idea into the first pendulum clock in 1656 and, in his Horologium Oscillatorium of 1673, gave the period of a small swing, the result written today as T = 2π√(L/g). He also showed that the period does grow for wide swings, the effect this calculator measures with the amplitude field.
Pendulums became measuring instruments. Henry Kater’s reversible pendulum of 1817 measured local gravity so precisely that reversible pendulums served gravity surveys for more than a century. In 1851 Léon Foucault hung a 67 m pendulum in the Panthéon in Paris; its swing plane slowly turned as the Earth rotated beneath it, a public demonstration of the Earth’s spin that needed no astronomy.
A real pendulum with two hinged arms behaves very differently: it is chaotic, and tiny changes in the start send it on a completely different path. See the Double Pendulum Chaos Lab.
About This Tool
This calculator solves the simple-pendulum equation for the period, the length or gravity, with a unit on every field. Most pendulum calculators stop at the small-angle formula; this one also takes the amplitude and gives the period of a real wide swing two ways, by the classic series and exactly by the arithmetic–geometric mean, so you can see when the textbook answer is good enough. The drawing swings at the period you entered.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Double Pendulum Chaos Lab, Spring (Hooke’s Law) Calculator, and Gravity & Orbit Calculator.
Frequently Asked Questions
What is the formula for the period of a pendulum?
For a simple pendulum swinging through a small angle, T = 2π√(L ÷ g), where L is the length from the pivot to the centre of the bob and g is the acceleration due to gravity. A 1 m pendulum at standard gravity (9.80665 m/s²) has T = 2π√(1 ÷ 9.80665) = 2.006 s.
Does the mass of the bob change the period?
No. The mass cancels out of the equation, so a 1 m pendulum has a 2.006 s period with a 50 g bob or a 5 kg bob, as long as the string is light and air drag is small. Only the length, gravity and the size of the swing matter.
How long is a seconds pendulum?
A seconds pendulum ticks once a second, so each swing takes 1 s and the full period is 2 s. L = g × (T ÷ 2π)² = 9.80665 × (2 ÷ 2π)² = 0.9936 m at standard gravity, which is why tall-case clocks are about a metre from pivot to bob.
How accurate is the small-angle formula?
Very, for small swings. The real period is longer than 2π√(L/g) by 0.048% at 5°, 0.19% at 10°, 0.77% at 20° and 7.32% at 60°. A 1 m pendulum swinging out 60° takes 2.153 s, not 2.006 s.
How do I measure g with a pendulum?
Time many swings and divide, to shrink the reaction-time error, then use g = 4π²L ÷ T². A 0.5 m pendulum that makes 10 full swings in 14.2 s has T = 1.42 s, so g = 4π² × 0.5 ÷ 1.42² = 9.789 m/s². Keep the swing under about 10° so the small-angle formula holds.
How do I use the Pendulum Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Clock making
A tall-case clock with a seconds pendulum needs 0.9936 m from pivot to the centre of the bob for a 2 s period at standard gravity.
School physics labs
Timing a 0.5 m pendulum at 1.42 s per swing gives g = 9.789 m/s², within 0.2% of standard gravity.
Foucault pendulums
Foucault’s 67 m pendulum in the Panthéon has a period of 16.42 s, slow enough to watch each swing.
Other worlds
On the Moon (g = 1.62 m/s²) a 1 m pendulum takes 4.937 s per swing, 2.46 times longer than on Earth.
Playground swings
A swing on 2.5 m chains has a natural period of about 3.17 s, so pushing once every 3.17 s builds it up.
Last updated: