For a thin planar ring of radius ‘r’ mm and thickness ‘t’ mm, its radius of gyration about the polar axis in mm is:

Explanation:

Radius of Gyration

The radius of gyration (k) of a body is a measure that describes how the mass of the body is distributed with respect to a given axis. For a thin planar ring of radius 'r' mm and thickness 't' mm, the radius of gyration about the polar axis can be calculated using the principles of rotational dynamics and the geometry of the ring.

Calculation:

For a thin planar ring, the mass distribution is essentially concentrated at a distance 'r' from the center. The radius of gyration is derived from the moment of inertia (I) of the ring about the polar axis (which is perpendicular to the plane of the ring and passes through the center).

The moment of inertia (I) for a thin ring of radius 'r' and mass 'm' about its polar axis is given by:

I = m × r²

The radius of gyration (k) is related to the moment of inertia and the mass by the following equation:

I = m × k²

By equating the two expressions for I, we get:

m × k² = m × r²

k² = r²

k = r