Theory

In Fig. 1, O is the point of suspension of the compound pendulum and G is its centre of mass; we consider the force of gravity to be acting at G. If h is the distance from O to G, the equation of motion of the compound pendulum is

Where I₀ is the moment of inertia of the compound pendulum about the point O.

Comparing to the equation of motion for a simple pendulum

We see that the two equations of motion are the same if we take

It is convenient to define the radius of gyration k₀ of the compound pendulum such that if all the mass M were at a distance k₀ from O,

the moment of inertia about O would be I₀, which we do by writing I₀ = Mk₀² Substituting this into (1) gives us

The point O², a distance l from O along a line through G, is called the center of oscillation. Let h² be the distance from G to O², so that l=h+h'. Substituting this into (2), we have

If I_G is the moment of inertia of the compound pendulum about its centre of mass, we can also define the radius of gyration k_G about the centre of mass by writing IG = Mk_G². The parallel axis theorem gives us

Comparing to (3), we have,

If we switch h with h², equation (4) does not change, so we could have derived it by suspending the pendulum from O². In that case, the center of oscillation would be at O and the equivalent simple pendulum would have the same length l. Therefore the period would be the same as when suspended from O. Thus if we know the location of G, by measuring the period T with suspension at O and at various points along the extended line from O to G, we can find O² and thus h².

Then using equation (4), we can calculate k_G and I_G = Mk_G².

Knowing h² gives us l = h + h², and since for small angle oscillations the period

We can calculate g using

The minimum period T_min , corresponds to the minimum value of l. Recall that l = h + h² and that k_G² = hh' is a constant, depending only on the physical characteristics of the pendulum.

Thus, l=h+k_G²/h, and the minimum I occurs when,

i.e, when h²=k_G², h=h' and l=2h=2k_G.

Thus, at Tmin, l=2k_G.