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Physical Pendulum Calculator

The physical pendulum calculator helps you compute the period and frequency of a physical pendulum.

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Period1.794587 sfrom ω = √(mgd/I)
Frequency0.557231 Hz
Angular frequency3.501187 rad/s
Oscillations per minute33.434
Maximum speed0.175059ωA, reached at the centre of the swing
Maximum acceleration0.612916ω²A, at the extremes where speed is zero
Equivalent simple pendulum length0.8 mI/(md) — a simple pendulum this long swings at the same rate
Damping ratio ζ0.063246underdamped — it oscillates, with the amplitude decaying
Critical damping coefficient6.324555 N·s/m
Damped angular frequency3.494178 rad/sslightly slower than the undamped value
Damped period1.798187 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.