How this calculator works
A physical pendulum is an extended rigid body suspended from a pivot. Gravity acts through its centre of mass, while the moment of inertia is measured about the pivot axis. Linearising the restoring torque at small angles gives a harmonic period. Equivalent length is the length of a simple pendulum with the same period, not necessarily a physical dimension of the body.
Formula
T = 2 pi sqrt(I_p/(m g d)); f = 1/T; equivalent length = I_p/(m d). By the parallel-axis theorem, I_p >= m d².
Worked example
For I_p = 2 kg m², mass 2 kg, d = 0.5 m and g = 10 m/s², the equivalent length is 2 m and period is 2.809926 s. A period of 2 pi s with m = 3 kg and d = 0.5 m gives I_p = 15 kg m² at g = 10.
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Frequently asked questions
Which moment of inertia should I enter?
Use the inertia about the suspension pivot, including a parallel-axis correction if your value is about the centre of mass.
What happens if the pivot is at the centre of mass?
Gravity supplies no restoring torque, so this pendulum model has no finite gravitational period.
How large can the swing be?
The period and chart use the small-angle approximation. The chart amplitude is limited to 10 degrees.