Pendulum - Compound Pendulum

Compound Pendulum

The length L of the ideal simple pendulum above, used for calculating the period, is the distance from the pivot point to the center of mass of the bob. A pendulum consisting of any swinging rigid body, which is free to rotate about a fixed horizontal axis is called a compound pendulum or physical pendulum. For these pendulums the appropriate equivalent length is the distance from the pivot point to a point in the pendulum called the center of oscillation. This is located under the center of mass, at a distance called the radius of gyration, that depends on the mass distribution along the pendulum. However, for any pendulum in which most of the mass is concentrated in the bob, the center of oscillation is close to the center of mass.

Using the parallel axis theorem, the radius of gyration L of a rigid pendulum can be shown to be

Substituting this into (1) above, the period T of a rigid-body compound pendulum for small angles is given by

where I is the moment of inertia of the pendulum about the pivot point, m is the mass of the pendulum, and R is the distance between the pivot point and the center of mass of the pendulum.

For example, for a pendulum made of a rigid uniform rod of length L pivoted at its end, I = (1/3)mL2. The center of mass is located in the center of the rod, so R = L/2. Substituting these values into the above equation gives T = 2π√2L/3g. This shows that a rigid rod pendulum has the same period as a simple pendulum of 2/3 its length.

Christiaan Huygens proved in 1673 that the pivot point and the center of oscillation are interchangeable. This means if any pendulum is turned upside down and swung from a pivot located at its previous center of oscillation, it will have the same period as before, and the new center of oscillation will be at the old pivot point. In 1817 Henry Kater used this idea to produce a type of reversible pendulum, now known as a Kater pendulum, for improved measurements of the acceleration due to gravity.