Simple Pendulum Calculator

The simple pendulum calculator finds the period and frequency of a pendulum.

Period2.006409 sfrom ω = √(g/L) with L = 1 m

Frequency0.498403 Hz

Angular frequency3.131557 rad/s

Oscillations per minute29.904

Maximum speed0.156578ωA, reached at the centre of the swing

Maximum acceleration0.490333ω²A, at the extremes where speed is zero

Maximum kinetic energy6.1279 mJmgL(1 − cos θ₀) — all of it at the lowest point, where the bob has fallen 1.25 mm

Speed at the lowest point0.156562 m/s√(2g × rise) — the free-fall result again

Amplitude as an angle2.865°the amplitude field is read as radians for a pendulum, not metres

Small-angle estimate of the energy6.1292 mJ½mgLθ₀², from cos θ ≈ 1 − θ²/2 — 0.021% out at this amplitude

Assumes a small swingerror grows with amplitudeat 10° the true period is about 0.2% longer, at 30° about 1.7%, at 90° about 18% — the simple formula is a small-angle result

Length for a one-second period0.248405 m

Length for a two-second period0.993621 mthe seconds pendulum, about 0.994 m — it ticks once per swing

Mass does not appearperiod is independent of massa heavy and a light pendulum of the same length keep the same time

Damping ratio ζ0.063246underdamped — it oscillates, with the amplitude decaying

Critical damping coefficient6.324555 N·s/m

Damped angular frequency3.125288 rad/sslightly slower than the undamped value

Damped period2.010434 s

Q factor7.9057roughly the number of radians of oscillation before the energy falls by a factor of e

The formula

T = 2π√(m/k) for a spring, 2π√(L/g) for a pendulum

Why the period ignores amplitude

Simple harmonic motion arises whenever the restoring force is proportional to the displacement. That proportionality is what makes the period independent of amplitude: a larger swing has further to travel but a proportionally stronger force pulling it back, and the two effects cancel exactly. Galileo noticed this with pendulums and it is the reason they could be used as clocks.

For a pendulum the proportionality is only approximate. The restoring force goes as sin θ, not θ, and those agree only for small angles — so the period does creep up with amplitude, by about 0.2% at 10° and 18% at 90°. A pendulum clock therefore needs a small, constant swing.

A pendulum's period does not depend on its mass, while a spring's does. The difference is that gravity supplies a restoring force proportional to mass, so mass cancels, whereas a spring's stiffness is fixed regardless of what hangs from it.

Damping is measured by the ratio ζ. Below 1 the system oscillates with a decaying amplitude; above 1 it creeps back without overshooting; at exactly 1 it returns as fast as possible with no overshoot, which is what a car suspension or a door closer aims for.