Pendulum Period Calculator
The pendulum period calculator starts with a 1 m pendulum on Earth, let go from 10° off vertical. Change the length, the swing or the gravity; the table shows how the period grows as the swing gets wider.
Period T (small-angle)
- Period T = 2π√(L/g) 2π × √(1 m / 9.80665 m/s²) = 2.006 s. The small-angle formula: exact only in the limit of a tiny swing.
- 2.006 s
- Period at a 10° swing 0.1907% longer than the small-angle value. Computed exactly as T0 / AGM(1, cos(θ₀/2)), the elliptic-integral period, so it holds at any amplitude.
- 2.01 s
- Correction series With θ₀ in radians (0.1745 here). Adding 11θ₀⁴/3072 and 173θ₀⁶/737280 gives 2.01 s. The first term alone is within 0.05% of the exact period up to 30°.
- T0 × (1 + θ₀²/16) = 2.01 s
- Frequency f 1 / T, small-angle. At this swing: 0.4975 Hz.
- 0.4984 Hz
- Angular frequency ω ω = √(g/L), small-angle. Note the mass appears nowhere: every bob on the same string keeps the same time.
- 3.132 rad/s
- Speed at the bottom √(2gL(1 − cos θ₀)), from energy conservation. Exact at any amplitude, ignoring air resistance.
- 0.5459 m/s
- Length for a 2 s period g / π² at g = 9.807 m/s²: a pendulum that ticks once a second each way, the "seconds pendulum" of a long-case clock.
- 0.9936 m
| Release angle | Exact period | Longer than 2π√(L/g) by | Series to θ₀² gives |
|---|---|---|---|
| 5° | 2.007 s | 0.04762% | 2.007 s |
| 10° | 2.01 s | 0.1907% | 2.01 s |
| 15° | 2.015 s | 0.4301% | 2.015 s |
| 20° | 2.022 s | 0.7669% | 2.022 s |
| 30° | 2.041 s | 1.741% | 2.041 s |
| 45° | 2.087 s | 3.997% | 2.084 s |
| 60° | 2.153 s | 7.318% | 2.144 s |
| 90° | 2.368 s | 18.03% | 2.316 s |
| 120° | 2.755 s | 37.29% | 2.556 s |
| 150° | 3.536 s | 76.22% | 2.866 s |
- A simple pendulum: a point mass on a light, rigid string, no air resistance. A real bob with size swings as a physical pendulum, T = 2π√(I / mgd), which is slightly slower.
- Results are shown to four significant figures. Round to the precision of your least precise measurement.
How the Simple Harmonic Motion Calculator works
T = 2π√(L/g). The period of a simple pendulum is 2π times the square root of its length over the acceleration of gravity. Length in meters and g in meters per second squared give the period in seconds. For small swings a pendulum is simple harmonic motion: the restoring force is close to proportional to how far the bob is displaced.
Worked example, the one above: L = 1 m under standard gravity, 9.80665 m/s². T = 2π × √(1 / 9.80665) = 2π × 0.3193 = 2.006 s. The frequency is 1 / 2.006 = 0.4984 Hz: a little under one full swing, out and back, every two seconds.
The mass does not appear. A heavier bob needs more force to accelerate, and gravity supplies exactly proportionally more, so the two cancel. A lead bob and a cork bob on the same string keep the same time, apart from air resistance. Only the length and g matter, and the period goes as the square root of the length: four times longer, twice as slow.
The small-angle approximation. The restoring force on the bob is mg sin θ. The formula above replaces sin θ with θ, which is accurate for small swings and increasingly wrong for big ones. The true period is longer, by a factor that depends only on the release angle θ₀:
T = T0 × (1 + θ₀²/16 + 11θ₀⁴/3072 + ...), with θ₀ in radians.
At 10° the first correction is 0.0019, so the 1 m pendulum's period is 2.010 s rather than 2.006 s, 0.19% longer. At 30° it is 1.7% longer, at 60° about 7%, and at 90° 18%. The calculator does not stop at the series: it evaluates the exact period through the arithmetic-geometric mean, T = T0 / AGM(1, cos(θ₀/2)), which is the closed form of the elliptic integral the series expands, and shows the first series term beside it so the size of the approximation is visible.
The seconds pendulum. Rearranging gives L = gT²/4π². For a 2 s period (one second each way, the tick of a long-case clock), L = g / π² = 0.9936 m under standard gravity. That near-coincidence with one meter is not an accident of nature: in the 1790s one proposal for defining the meter was the length of a seconds pendulum, rejected partly because g varies from place to place.
Other worlds. On the Moon, with g = 1.62 m/s², the same 1 m pendulum takes 4.94 s; on Mars, 3.25 s. A pendulum is a gravimeter: time it, measure the length, and g = 4π²L / T².
This is the ideal pendulum: a point mass on a light, inextensible string, with no air resistance. A real bob with size is a physical pendulum, T = 2π√(I / mgd), slightly slower than the simple formula for the same length to its center.
Simple Harmonic Motion Calculator FAQ
What is the period of a 1 m pendulum?
About 2.006 s under standard gravity: T = 2π√(1 / 9.80665). Released from 10° the exact period is 2.010 s, because the small-angle formula slightly underestimates real swings.
Does the mass of a pendulum affect its period?
No. Both the force of gravity on the bob and the inertia resisting it are proportional to the mass, so it cancels. Only the length and the strength of gravity set the period, as long as the bob is small and air resistance negligible.
When does the small-angle approximation stop working?
It is good to 0.2% at 10° and about 0.5% at 15°, which is why textbooks quote it up to around that range. At 30° the true period is 1.7% longer, and at 90° it is 18% longer. Use the release angle field to see the exact figure.
How long is a pendulum with a period of 2 seconds?
L = gT² / 4π² = 9.80665 × 4 / 39.478 = 0.9936 m under standard gravity. This is the seconds pendulum used in long-case clocks, so called because each swing one way takes one second.
More from the Simple Harmonic Motion Calculator
Sources
- NASA NSSDCA Moon Fact Sheet, bulk parameters table, row "Surface gravity (m/s2)", Moon column (page last updated January 11, 2024) (2026)
- NASA NSSDCA Mars Fact Sheet, bulk parameters table, row "Surface gravity (mean) (m/s2)", Mars column (page last updated May 19, 2025) (2026)