Radius of gyration:

Let us consider that a rigid body consists of n number of particles, each of the same mass m. So the total mass of the rigid body is \( M=nm \). Let \( r_1 \), \( r_2 \), \( r_3 \), \( \cdots\ r_n \) be the distances of the particles from the axis of rotation. So the moment of inertia of the body about that axis of rotation is
\( I=m({r_1}^2+{r_2}^2+{r_3}^2+\cdots+{r_n}^2) \),
or, \( I=mn\frac{({r_1}^2+{r_2}^2+{r_3}^2+\cdots+{r_n}^2)}{n} \)
or, \( I=MK^2 \).
where, \( \displaystyle{K=\sqrt{\frac{({r_1}^2+{r_2}^2+{r_3}^2+\cdots+{r_n}^2)}{n}}} \), This \( K \) is called the radius of gyration of the body. So the radius gyration of the body is definesd by the root mean square distance of the particles, with which the rigid body is composed, from the axis of rotation.

If the entire mass of the rigid body is supposed to be concentrated at a point, so that the moment of inertia of the rigid body about an axis of rotation is equal to the moment of inertia of the concentrated point mass about that same axis of rotation, then the distance of that concentrated point mass from the axis of rotation is called the radius of gyration of the body.

Again we know that the moment of inertia \( I=\sum{mr^2}=MK^2 \)
or, \( K=\sqrt{\frac{I}{M}} \)
therefore, for continious mass distribution \( \displaystyle{K=\sqrt{\frac{\int{r^2}dm}{\int{dm}}}} \)