Physical Pendulum Calculator

About physical pendulums

A physical pendulum, also called a compound pendulum, is a rigid body that oscillates about a horizontal pivot under gravity. Unlike an ideal simple pendulum, its mass is distributed throughout the body rather than concentrated at a point on a massless string. This distribution matters because rotational resistance depends on the moment of inertia about the pivot. The center of mass determines the gravitational lever arm that supplies restoring torque. For a small angular displacement, the restoring torque is approximately proportional to angle. The resulting motion is simple harmonic, with angular frequency equal to the square root of mass times gravitational acceleration times center-of-mass distance divided by pivot moment of inertia. Period is two pi divided by angular frequency, and ordinary frequency is the reciprocal of period. This calculator evaluates those relationships and displays all three oscillation measures. The small-angle assumption is important. It replaces the sine of the angle with the angle measured in radians and is usually a good approximation for amplitudes around ten degrees or less. At larger amplitudes, the real period is longer and requires a nonlinear correction or numerical solution. Air drag, pivot friction, elastic supports, and motion outside one plane can also shift measured periods or damp the oscillation. The output therefore describes an ideal rigid pendulum at small amplitude. Moment of inertia must be measured about the actual pivot axis. If a tabulated inertia is given about a parallel axis through the center of mass, the parallel-axis theorem adds mass times the square of the offset. The calculator includes convenient shape models. A uniform slender rod pivoted at one end has inertia equal to one third of mass times length squared and a center of mass halfway along the rod. A uniform disk pivoted at a point on its rim has inertia equal to three halves of mass times radius squared and a center-of-mass distance equal to its radius. Equivalent simple-pendulum length is the pivot moment of inertia divided by mass and center-of-mass distance. A simple pendulum with this length has the same small-angle period at the same gravity. The quantity is useful in metrology and in the reversible Kater pendulum method for measuring gravitational acceleration. It also makes physical and simple pendulum behavior easier to compare. Mass often cancels from the period when an object's geometry scales its moment of inertia directly with mass. It remains explicit in the general equation because custom inertia and mass may be supplied independently. Use local gravitational acceleration for precise work; the default is standard gravity. This calculator supports classroom dynamics, clock mechanisms, laboratory design, machinery, and any rigid body whose oscillation can be approximated around a fixed pivot.

Physical pendulum examples

How to calculate a physical pendulum

1. Choose a custom body, uniform rod, or uniform disk.
2. Enter mass and the geometry or custom pivot properties requested.
3. Use standard gravity or enter the gravitational acceleration at the location.
4. Select Calculate pendulum and read the period, frequencies, and equivalent length.

Frequently asked questions