Table of Contents
Physical Pendulum
A physical (or compound) pendulum is any rigid body that oscillates about a fixed pivot point that does not pass through its center of mass. Unlike a simple pendulum (point mass on massless string), the physical pendulum's period depends on the distribution of mass, described by its moment of inertia about the pivot point.
Physical pendulums are more realistic models of real oscillating objects like swinging signs, clock pendulums with finite-size bobs, and human limbs during walking. The analysis uses the parallel axis theorem to find the moment of inertia about the pivot and relates it to the restoring torque due to gravity.
Formula
Where I is moment of inertia about pivot, m is mass, g is gravity, and d is distance from pivot to center of mass.
Moments of Inertia for Common Shapes
| Shape (pivot at end) | I about pivot |
|---|---|
| Uniform rod (end) | mL²/3 |
| Uniform disk (edge) | 3mR²/2 |
| Uniform ring (edge) | 2mR² |
| Solid sphere (surface) | 7mR²/5 |
Frequently Asked Questions
What is the center of oscillation?
The center of oscillation is the point at distance Leq from the pivot where, if all mass were concentrated, a simple pendulum would have the same period. An interesting property: if you pivot the pendulum at its center of oscillation, the original pivot becomes the new center of oscillation. This reversibility was used by Kater to measure g precisely.
How does pivot position affect the period?
Moving the pivot closer to the center of mass increases the period (approaches infinity at the COM since d approaches 0). Moving it farther away also eventually increases the period (since I grows faster than d). There is an optimal pivot distance that minimizes the period.