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Moment of Inertia of Composite Bodies

The moment of inertia of a body made up of simpler shapes can be found because the moment of inertia of a body about an axis is the sum of individual moments of inertia about that axis. By using this linearity property, along with the parallel axis theorem, we can find the moment of inertia of a great many composite bodies.

For example, to find the moment of inertia of a dumbbell consisting of two spherical masses each of massand radiusat either end of a rod of lengthand massshown below:

The moment of inertia of the rod about the centre is

The moment of inertia of each mass about its centre isbut the centre of each mass is distancefrom the centre of the rod, so the parallel axis theorem gives the moment of inertia of each spherical mass about the centre of the dumbbell shape as

The moment of inertia of the dumbbell is then