What is radius of gyration ? please explain ith examples.

Radius of gyration of a body about a given axis is the perpendicular distance of a point from the axis, where if whole mass of the body were concentrated, then the body shall have the same moment of inertia as it has with the actual distribution of mass. This distance is represented by K. The momentum of inertia of a body of mass M and radius of gyration K is given by,

Since, the radius of gyration of a body is defined about its axis of rotation it will change if we change the axis of rotation of the object.

For example:-

1) The moment of inertia of a ring about an axis passing through its center of mass and perpendicular to plane of ring is given by, I = MR²

Where, M = mass of the ring, R = radius of the ring

So, the radius of gyration 'K' can be found as,

MR² = MK²

=> R = K

2) The moment of inertia of a ring about its diameter is given by,

I = (1/2) MR²

So here the radius of gyration is,

MK² = (1/2) MR²