Much of what we know about the distribution of mass within
planets comes from their moments of inertia. The moment of
inertia controls the rate at which a planet spins
and the location of the spin axis depend on the
distribution of mass in each body. We illustrate this concept
by "racing" a metal ring and disk, which have the same mass
but different moments of inertia, down an incline.
The
moment of inertia, and can be found from 
where mi is the mass and pi is the
distance from the axis of rotation of the ith particle. If
the particles are sufficiently small, we can replace the first equation
with the integral 
.
What do these equations mean? First, objects with more mass at the center have lower moments of inertia. Because the relationship of the moment of inertia to rotation is exactly analogous to that for inertia and motion, a body with a lower moment of inertia will rotate faster than a body with a higher moment of inertia, even if they have the same masses and sizes. Second, if the mass is distributed differently with respect to different axes, the body will have a different moment of inertia. This is important not just for planetary studies, but for things such as satellites; more than one satellite has been lost because of an incorrectly calculated moment of inertia!
We illustrate these ideas by considering disks, which we take as
two-dimensional. For this experiment, you will need:
, most
students predict that the object with the larger moment of inertia will
travel further. However, because the total energy of the system is
constant, both objects should travel approximately the same
distance.