Intuitive explanation for parallel axis theorem

Question

The parallel axis theorem states that you can relate the moments of inertia defined with the center of mass as the origin to the moments of inertia defined with respect to some other origin. It is summarized in this equation:

$I=I_c+Mh^2$

For a cone, the center of mass is $1/4$ of the height from the base so we can define $I_{base}$ using the parallel axis theorem. However, since the parallel axis theorem does not care for the sign of $h$, the moments of inertia defined at $h/4$ above the center of mass are equivalent to the base.

The moment of inertia can be thought of as the opposition that a body exhibits to having its speed of rotation about an axis altered by the application of a torque. This means that turning the cone about $I_1$ in the base will require the same force as it takes to rotate it about $I_1$ which is at the center of the cone. Intuitively, I would say that is harder to rotate the cone about its base than it is to spin it around its center so the fact that they have the same moments of inertia is confusing to me. Can someone please point out the flaw in my logic? Also, as a bonus question, is the parallel axis theorem still valid when you extend the origin outside of the shape?

Answer 1

Consider the following conceptual steps

1. The MMOI of a point particle of mass $m$ that is orbiting at a distance $d$ is $$I = m\, d^2$$
2. The motion of a rigid body is decomposed into the motion of the center of mass, and the motion about the center of mass.
3. The MMOI of a rigid body about the center of mass $I_c$ is derived from the angular momentum of the body whilst rotating about the center of mass only.
4. The MMOI of a rigid body about any other point $I$ is the superposition of the MMOI due to rotation $I_c$ and due to the motion of the center of mass as if the body was a concentrated point mass $m d^2$ $$ I = I_c + m d^2$$