Physics and Mathematics
Radius of Gyration
It is defined as the distance from the axis of rotation at which, if the whole mass of the body were supposed to be concentrated, the moment of inertia would be the same as with the actual distribution of the mass of the body into small particles. It is denoted by $K$.
If $M$ is the mass of the body and $K$ is its radius of gyration about the axis $OZ$,
then the moment of inertia of the body about the axis $OZ$ is:
\[
I = M K^2
\]
From equation (1), we get:
$\displaystyle M{{K}^{2}}$ $\displaystyle ={{m}_{1}}r_{1}^{2}+{{m}_{2}}r_{2}^{2}+{{m}_{3}}r_{3}^{2}+$ $\displaystyle \ldots +{{m}_{n}}r_{n}^{2}$
If each of the $n$ particles constituting the rigid body is of mass $m$, then
$\displaystyle M{{K}^{2}}$ $\displaystyle =mr_{1}^{2}+mr_{2}^{2}+mr_{3}^{2}+\ldots +mr_{n}^{2}$ $\displaystyle =m\left[ {r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+\ldots +r_{n}^{2}} \right]$
Since $m \times n = M$, the mass of the body, we have:
$\displaystyle M{{K}^{2}}$ $\displaystyle =M\left( {\frac{{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+\ldots +r_{n}^{2}}}{n}} \right)$
or
$\bbox[15px, #e4e4e4, border: 2px solid #000000]{\boldsymbol {\displaystyle K=\sqrt{{\frac{{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+\ldots +r_{n}^{2}}}{n}}}}}$
i.e.,
$\displaystyle K$ = root mean square distance of the constituting particles from the axis of rotation
Thus, the radius of gyration of a body about an axis of rotation may also be defined as the root mean square average of the particles from the axis of rotation and its square when multiplied with the mass of the body gives the moment of inertia of the body about that axis.
From the study of moment of inertia and the radius of gyration, the following conclusions can be drawn:
- The moment of inertia of the body depends upon the mass of the body as well as the manner in which mass is distributed about the axis of rotation. It is because the moment of inertia of the body changes with the change in the position of the axis of rotation.
- The radius of gyration of a body is not a constant quantity. Its value changes with the change of location of the axis of rotation.
