MotionLab

Pendulum Calculator

A simple pendulum's period depends only on length and gravity — not mass or amplitude (for small swings).

Pendulum length (m)

Gravity (m/s²)

Period (s)

2.006

T = 2π√(L/g)

Frequency (Hz): 0.499

What you entered

Pendulum length (m): 1Gravity (m/s²): 9.81

1
Period: T = 2π√(L/g)
2π × √(1 / 9.81)= 2.0061 s
2
Frequency: f = 1/T
1 / 2.0061= 0.4985 Hz

Result

Period (s): 2.006

A pendulum 1 m long has a period of 2.006 s (0.499 Hz).

Simple pendulum period and frequency by length

For small swings the period is 2 pi times the square root of length over g. Mass and amplitude do not appear in the formula.

Length (m)	Period (s)	Frequency (Hz)
0.25	1.0030	0.9970
0.5	1.4185	0.7050
1	2.0061	0.4985
2	2.8370	0.3525
4	4.0121	0.2492
9.81	6.2832	0.1592

A one metre pendulum takes almost exactly two seconds per full swing, which is why the seconds pendulum used in clocks is about 0.994 m. Quadrupling the length only doubles the period, since it depends on the square root - the 1 m and 4 m rows show that directly. The last row is a neat coincidence: a length numerically equal to g gives a period of exactly 2 pi. Mass genuinely does not matter, which Galileo established. The formula assumes small amplitudes; beyond about 15 degrees the true period is slightly longer.

Why pendulums were the first precision timekeepers

A pendulum's period depends only on its length and local gravity — not on the mass of the bob or the amplitude of the swing (for small angles). This remarkable independence is why pendulum clocks, invented by Huygens in 1656, were the world's most accurate timekeepers for nearly 300 years.

The small-angle approximation

The simple formula T = 2π√(L/g) assumes small swings (under ~15°). For larger amplitudes the period increases slightly — a full derivation requires elliptic integrals. For most practical purposes (clocks, lab experiments), the simple formula is accurate to well within 1%.

Frequently asked questions

A grandfather clock has a 1 m pendulum. What is its period?

T = 2π√(L/g) = 2π√(1/9.81) = 2.006 seconds — almost exactly 2 seconds. This is by design: clockmakers chose a ~1 m pendulum precisely because one tick-tock (full period) equals 2 seconds, making gear ratios simple.

Why doesn't the mass of the pendulum bob affect the period?

Mass cancels out in the derivation: the restoring force (mg×sin(θ)) and inertia (m) both scale linearly with mass. Heavier bobs swing with the same period — only length and gravity matter. This is why pendulum clocks don't need precision-mass bobs.

How does gravity affect pendulum period?

T = 2π√(L/g). On the Moon (g = 1.62 m/s²), a 1 m pendulum has T = 4.93 s — about 2.5× slower than on Earth. Pendulum clocks were actually used historically to measure local gravity variations for geological surveying.

What is the maximum angle for the simple pendulum formula to be accurate?

The formula T = 2π√(L/g) assumes sin(θ) ≈ θ (small angle approximation). At 15°, the error is about 0.5%. At 30°, the error is about 2%. At 90°, the error is about 18%. For most practical purposes, angles under 15° give excellent accuracy.

How is a pendulum related to simple harmonic motion?

A pendulum approximates simple harmonic motion for small angles — the restoring force is proportional to displacement. This is the same math as a mass on a spring (T = 2π√(m/k)). See the spring energy and Hooke's law calculators for the spring equivalent.