This pendulum calculator works the simple-pendulum equations both ways: period and frequency from length, or the length required for a target period — the physics that ran the world’s clocks for three centuries.
Galileo’s Beautiful Simplification
For small swings, a pendulum’s period depends on exactly two things: T = 2π√(L/g) — its length and gravity. Not its mass. Not (to first order) how wide it swings. The bob’s weight cancels out of the equation entirely, which stunned seventeenth-century natural philosophers and delighted clockmakers: a length of pendulum is a unit of time you can build. Part of our Physics Calculators collection.
Worked Examples (Verified)
- The 1-meter pendulum: T = 2π√(1/9.81) = 2.006 seconds, f ≈ 0.498 Hz — almost exactly a second each way, which is no coincidence: the “seconds pendulum” (T = 2.000 s) needs L = 0.994 m and was once proposed as the definition of the meter itself
- Grandfather clocks use that ~1 m pendulum; their stately tick is 2 seconds per full cycle, escapement advancing on each pass
- On the Moon (g = 1.62): the same 1 m pendulum takes 4.99 s per swing — clocks would crawl; on Mars (3.71): 3.26 s
- A playground swing is a pendulum of ~2–3 m effective length: period ~3 s, which is why pushing “in rhythm” means pushing every 3 seconds, not whenever
Where the Simple Formula Bends
- Large amplitudes: the small-angle approximation (sin θ ≈ θ) underpins the formula; at a 30° swing the true period runs ~1.7% long, at 60° about 7% long. Precision horology fights this with cycloidal cheeks and tiny arcs.
- Length is measured to the bob’s center of mass, and rods expand with temperature — the gridiron and mercury pendulums of the 1700s existed precisely to cancel thermal length drift.
- g varies by ±0.3% over Earth’s surface (altitude, latitude, geology) — enough to make a good pendulum clock a gravity meter, an actual scientific instrument of the era.
- Physical (rigid-body) pendulums — swinging rods, planks, bodies — use their moment of inertia instead of a point-mass length; the formula generalizes, the charm remains.
Related Tools
- Gravitational Force Calculator — where g comes from
- Wavelength Calculator — frequency’s other home
- Physics Calculators — the full collection
Frequently Asked Questions
What is the formula for a pendulum’s period?
T = 2π√(L/g): length in meters, g = 9.81 m/s² on Earth. A 1 m pendulum has a period of 2.006 s.
Does the bob’s mass affect the period?
No — mass cancels out of the derivation entirely. Doubling the bob’s weight doubles both the restoring force and the inertia, leaving the timing untouched (air resistance aside).
How long is a “seconds pendulum”?
0.994 m at Earth’s surface gravity — defined by T = 2.000 s (one second per swing direction), the length that beat time in regulators and grandfather clocks.
Why do pendulum clocks lose time in summer?
Heat expands the rod; longer L means longer T. Uncorrected clocks drift seconds per day seasonally — the problem temperature-compensated pendulums were invented to solve.
Does swing amplitude matter?
Slightly: the formula assumes small angles. Real period grows with amplitude — about +1.7% at 30°, +7% at 60° — negligible for a porch swing, decisive for a chronometer.
Is this pendulum calculator free?
Yes — free, instant, no sign-up.