Pendulum (mathematics) - Compound Pendulum

A compound pendulum (or physical pendulum) is one where the rod is not massless, and may have extended size; that is, an arbitrarily shaped rigid body swinging by a pivot. In this case the pendulum's period depends on its moment of inertia I around the pivot point.

The equation of torque gives:

where:

is the angular acceleration.
is the torque

The torque is generated by gravity so:

where:

L is the distance from the pivot to the center of mass of the pendulum
θ is the angle from the vertical

Hence, under the small-angle approximation ,

This is of the same form as the conventional simple pendulum and this gives a period of:

And a frequency of:

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