Question

Show that the square of radius of gyration of a hollow cylinder is twice the square of
radius of gyration of a solid cylinder, having same radius when rotating about their
natural axis of symmetry

Answer 1

The answer to that is because the moment of inertia is not the same for the solid cylinder than for the hollow one.

As you write the formula for the moment of inertia, it depends on the distribution of the mass. The further away the mass is from the rotation axis, the more contributes to the moment of inertia (as in distance squared r2).

So, since the hollow cylinder has all it's mass at the border in comparison with the solid one which distributes all it's mass from the center (with very small contribution) to the border, it has a higher moment of inertia and thus more rotational energy.