Basics of Rotational Motion
5.0 Radius of gyration $(K)$
5.0 Radius of gyration $(K)$
It is defined as the distance from the axis of rotation at which if whole mass of the body were concentrated, then the moment of inertia of the body would be same as with the actual distribution of mass of the body.
It is denoted by the symbol $K$.
The radius of gyration for $n$ particles in a body about an axis of rotation may also be defined as the root mean square of the distance of the particles from an axis of rotation.
Mathematically, $$K = \sqrt {\frac{{r_1^2 + r_2^2 + r_3^2 + ... + r_n^2}}{n}} $$
Relation between moment of inertia $(I)$ and radius of gyration $(K)$
The moment of inertia of a body about an axis of rotation is equal to the product of mass of the body and square of its radius of gyration about that axis. $$\begin{equation} \begin{aligned} I = m{K^2} \\ K = \sqrt {\frac{M}{I}} \\\end{aligned} \end{equation} $$
Graph between moment of inertia $(I)$ and radius of gyration $(K)$ is parabola as shown in the figure. $$I = m{K^2}$$
The above equation is similar to that of parabola, $$y = a{x^2}$$
