The compound pendulum

A simple pendulum theoretically has the mass of the bob concentrated at one point, but this is impossible to achieve exactly in practice. Most pendulums are compound, with an oscillating mass spread out over a definite volume of space.

Let G be the centre of gravity of a compound pendulum of mass m that oscillates about a point O with OG = h If the pendulum is moved so that the line OG is displaced through an angle q (Figure 1), the restoring couple is:

- mghsinq = - mghq = if q is small.

Therefore:

Ia= = - mghq = and so q = - mgq h /I

Since the angular acceleration is directly proportional to the angular displacement the motion is simple harmonic of period T where:








But I is the moment of inertia about an axis through 0, and therefore

I = I_G + mh² = mk² + mh²
where k is the radius of gyration about a parallel axis through G. The period can therefore be written as:





If a uniform rod is used as a compound pendulum and the period of oscillation T measured for different values of h on either side of the centre of gravity then a graph like the one in Figure 2 may be obtained.

Since the formula for a simple pendulum is T = 2p(L/g)^1/2 we can define a quantity L called the length of the simple equivalent pendulum.

This is given by L = [k² + h²]/h


For two distances h₁ and h₂ on either side of the centre, L = h₁ + h₂ (as can be seen from the graph in figure 2) and h₁h₂ = k². At the minimum h₁ = h₂ and h = k. A value of g can be determined by measuring L from the graph.