Radius of Gyration

Radius of gyration is defined as the distance from the axis of rotation to a point where the total mass of the body is supposed to be concentrated, so that the moment of inertia about the axis may remain the same. Simply, gyration is the distribution of the components of an object. It is denoted by K. In terms of radius of gyration, the moment of inertia of the body of mass M is given as,

Inertia (I) = MK2

Suppose a body consists of n particles each of mass m. Let r1, r2, r3... , rn be their perpendicular distances from the axis of rotation. Then, the moment of inertia I of the body about the axis of rotation is

Formula of moment of inertia

If all the particles are of same mass m, then

Moment of inertia if all mass is same

Since mn = M, total mass of the body,

Formula of inertia in terms of total mass or body and radius

From the above equations, we have

Radius of gyration is the root mean square distance of particles from axis formula

Therefore, the radius of gyration of a body about a given axis may also be defined as the root mean square distance of the various particles of the body from the axis of rotation.

Radius of Gyration of a Thin Rod

The moment of inertia of uniform thin rod of mass M and length l about an axis through its center and perpendicular to its length is given by

Moment of inertia of thin rod

If is the radius of gyration of the rod about the axis, then we have

Relationship between moment of inertia and radius of gyration of thin rod

From the above equations, we have

Gyration is equal to the reciprocal of square root of 12

Radius of Gyration of a Solid Sphere

The moment of inertia for a solid sphere of radius R and mass M is given by

Formulae of moment of inertia of solid sphere

If is the radius of gyration of the solid sphere, then

Radius of gyration of solid sphere is equal to the produce of radius of sphere and square root of 2/5