What is the formula for a physical pendulum period?

T = 2π√(I / (m·g·d)), where I is the moment of inertia about the pivot in kg·m², m is the total mass in kg, g is gravitational acceleration in m/s², and d is the pivot-to-center-of-mass distance in meters. The approximation holds for small amplitudes (below ~15°).

How is a physical pendulum different from a simple pendulum?

A simple pendulum treats all mass as concentrated at a single point (the bob). A physical (compound) pendulum accounts for mass distributed throughout the rigid body, requiring the full moment of inertia rather than just m·L².

How do I find the moment of inertia for common shapes?

Use the parallel-axis theorem: I_pivot = I_cm + m·d², where I_cm is the moment of inertia about the center of mass and d is the pivot-to-cm distance. For a uniform rod pivoting at one end: I = (1/3)m·L². For a disk pivoting at its rim: I = (3/2)m·R². Use the shape presets in this calculator to load these automatically.

What value of g should I use?

Earth standard is 9.81 m/s². It varies from 9.78 m/s² at the equator to 9.83 m/s² at the poles. The Moon is roughly 1.62 m/s² and Mars about 3.72 m/s². Use the gravity presets to switch quickly.

What is the equivalent simple pendulum length?

The equivalent (reduced) pendulum length is L_eq = I / (m·d). A simple pendulum of this length would have the same period as the physical pendulum. This tool displays L_eq in the results.

Why does the formula only work for small angles?

For large amplitudes the restoring torque is proportional to sin(θ), not θ, so the motion is no longer simple harmonic. The true period is longer and requires an elliptic integral correction. The small-angle approximation is accurate within 1% for amplitudes below about 15°.

Physical Pendulum Calculator

Enter moment of inertia, mass, and pivot distance to instantly compute the period and frequency of a physical pendulum.

Pendulum Parameters

Moment of Inertia (I) — kg·m²

Moment of inertia about the pivot axis

Mass (m) — kg

Pivot-to-CM Distance (d) — m

Distance from pivot point to center of mass

Gravitational Acceleration (g) — m/s²

Gravity:

Shape Presets

Load a common rigid-body shape. Set mass and relevant dimension, then click a preset.

Mass (kg)

Length / Radius (m)

Results

Enter values and click Calculate

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Summary

A physical (compound) pendulum is any rigid body that swings about a fixed pivot point — unlike a simple pendulum, the mass is distributed throughout the object. Its period depends on the moment of inertia about the pivot, the total mass, the distance from pivot to center of mass, and local gravity. This calculator applies the exact formula T = 2π√(I / (m·g·d)) to give period, frequency, and angular frequency in one step.

How it works

Enter the moment of inertia (I) of the rigid body about the pivot axis in kg·m².
Enter the total mass (m) of the body in kilograms.
Enter the pivot-to-center-of-mass distance (d) in meters.
Optionally adjust gravitational acceleration — Earth standard is 9.81 m/s².
Click Calculate to apply T = 2π√(I / (m·g·d)) and see period, frequency, and angular frequency.
Use the presets panel to load common rigid-body shapes and see how their geometry affects oscillation.

Use cases

Analyze the oscillation of a rigid rod, disk, or ring pivoting about an edge.
Verify pendulum clock designs that use non-point-mass bobs.
Compute the natural frequency of a mechanical linkage or robotic arm.
Compare oscillation rates of different rigid bodies in a physics lab.
Assist students with rotational dynamics and simple harmonic motion coursework.
Estimate the period of a swinging structural member during seismic analysis.
Validate finite-element predictions of rotational natural frequency.
Design tuned-mass dampers using a calibrated physical pendulum.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu