# Confused by parallel axis theorem

The original scenario describes an object of mass M rotating about a parallel axis d distance away from the center of mass. I wonder how this scenario differs from the rotation of a mass point of the exact same mass M in radius d about an axis, as illustrated in the following image:

The two situations produce totally different moments of inertia. But I simply cannot see how the scenarios differ! Thanks in advance for any clarification and help!

• Just to clarify, you cannot see how an extended object differs from a single point mass? Or are you asking why the moment of inertia depends on the object's shape? Jul 2, 2021 at 10:23
• Sorry about the unclarity. I can't see how an extended object differs from a single mass point.
– K-V
Jul 2, 2021 at 10:55

These two cases are essentially of the same spirit. Your right hand side is a particular case of the right hane side. In the left hand side, the inertial moment at the cernter of mass $$I_{CM}=0$$, thus $$I_d = I_{CM} + M d^2 = 0 + M d^2 = M d^2.$$
The parallel axis theorem considers two factors contributing to resistance to rotation about the new axis. You can think of all of the mass being concentrated at the center of mass. This gives the M$$d^2$$. But remember, as a ridig body goes about the new axis, it must also still rotate about its center of mass.
$$I = \int r^2 dm$$