A square plate $ABCD$ of mass $m$ and side $a$ is suspended from point $A$ in vertical plane. A disc of the same mass and $\sqrt2a$ diameter is attached at point $C$ as shown in the figure. The disc can freely rotate about point $C$.
Now my doubt is when they mention that a body is free to rotate about an axis, why do we consider the moment of inertia of that body about that axis as $0$? I thought probably this would make the body non-rigid about the axis, hence it's moment of inertia is $0$.
But this seems counter intuitive to me that a free to rotate about actually "doesn't rotate" and vice versa.
Is this true? How do I convince myself this?
In this particular example, the moment of inertia of the system about a horizontal axis passing through $A$ is
$(ma^2/6 + m(a/\sqrt{2})^2) + (0 + m(\sqrt{2}a)^2) = \frac{8ma^2}{3}$,
by parallel axis theorem. (My doubt is regarding the $0$ in the second term, which is the moment of inertia of the disc).