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 mass
and radius
at either end of a rod of length
and mass
shown below:
The moment of inertia of the rod about the centre is
The moment of inertia of each mass about its centre is
but the centre of each mass is distance
from 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