The ratio of radii of gyration of a circular disc and a circular ring of the same radii and same mass about a tangential axis in the plane is

For an elementary determination of the ratio of the radii of gyration for a circular disc and a circular ring of the same radius and mass about a tangential axis in their plane, let us begin with their moments of inertia.

Let’s consider that the moment of inertia of a circular disc, depending on its mass and radius. To find the moment of inertia about a tangential axis, we apply the parallel axis theorem, which accounts for distance from the center of the disc to the new axis. This will add a term related to mass and the square of the distance. So, the moment of inertia of the disc about the tangential axis will be derived from both its central inertia and the additional component due to the shift.

On the other hand, the moment of inertia for the circular ring is easier since all its mass is concentrated at the radius. Applying the parallel axis theorem here again, we consider the distance to the tangential axis. Thus, the computation is straightforward.

We now compute the radius of gyration from the moments of inertia. If we then take the ratio of the radii of gyration for the disc and the ring, we will get a larger value of the radius of gyration for the ring as compared to the disc. Eventually, it comes out to be the reason for getting a simple expression in the ratio of the radii of gyration and that option also gives this relation.

Show more :
https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/