Physical Pendulum Calculator

Written by Great Medico

Updated September 25, 2026 •2 min read

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A physical pendulum is different from an ideal point-mass pendulum because the object’s mass is distributed through space. Its moment of inertia therefore matters directly to the period.

What this calculator does

The Physical Pendulum Calculator uses moment of inertia I, mass m, the distance R from the pivot to the center of mass, and gravitational acceleration g. It returns the oscillation period, frequency, and radius of oscillations.

How to use it

Enter the moment of inertia about the pivot, the pendulum mass, the pivot-to-center-of-mass distance, and gravity. The result is intended for small oscillations, where the standard physical-pendulum approximation applies.

How the calculation works

The period is T = 2π√[I/(mgR)]. The radius of oscillations is L = I/(mR), and the frequency is f = 1/T. The moment of inertia must be referenced to the same pivot used for R.

Example

For I = 0.8 kg·m², m = 2 kg, R = 0.4 m, and g = 9.80665 m/s², the radius of oscillations is 1.0 m. The period is about 2.00641 s and the frequency is about 0.498403 Hz.

How to interpret the result

A larger moment of inertia generally lengthens the period, while stronger gravity shortens it. The result also changes with the position of the center of mass because R determines the gravitational restoring torque.

Limitations and notes

The formula assumes small-angle oscillations, a fixed pivot, and negligible damping. Friction, air drag, flexible supports, large amplitudes, and an incorrectly referenced moment of inertia can all make the measured period differ from the ideal result.