Rigid rotational motion
The broom balances at its center of mass. If you saw the broom through its center of mass and weigh the two parts, which part will weigh more?
An unconstrained object rotates about its center of mass.
A rotating object has kinetic energy due to the motion of its particles.
Note that the radius for each mass element is the distance of the mass element from the axis of rotation. The moment of inertia, I, depends on the axis of rotation.
Parallel axis theorem
If an object is made to rotate about an axis that does not go through the center of mass, the moment of inertia can be found using the parallel axis theorem.
Consider a cylinder of radius r and mass m, with a string wound around it, starting from rest.
How what is the velocity of the center of mass of the cylinder when it has fallen a distance h?