Explain the radius of gyration.

(N/A) The radius of gyration is a parameter that describes how the mass of a rotating rigid body is distributed with respect to the axis of rotation.
It is related to the moment of inertia $(I)$ and the total mass $(M)$ of the body.
Consider a rigid body rotating about a given axis,consisting of $n$ particles,each of mass $m$. The total mass of the rigid body is $M = n m$.
The moment of inertia about the given axis is:
$I = m_{1} r_{1}^{2} + m_{2} r_{2}^{2} + \ldots + m_{n} r_{n}^{2}$
Since $m_{i} = m$ for all particles:
$I = m r_{1}^{2} + m r_{2}^{2} + \ldots + m r_{n}^{2} = m (r_{1}^{2} + r_{2}^{2} + \ldots + r_{n}^{2})$
By multiplying and dividing by $n$:
$I = (m n) \left[ \frac{r_{1}^{2} + r_{2}^{2} + \ldots + r_{n}^{2}}{n} \right]$
We define the radius of gyration $k$ such that $k^2 = \frac{r_{1}^{2} + r_{2}^{2} + \ldots + r_{n}^{2}}{n}$.
Thus,the moment of inertia is given by:
$I = M k^{2}$
Here,$k$ represents the root mean square distance of the particles from the axis of rotation.