Subsection 7.1.10 Radius of Gyration
The radius of gyration, \(k\) is a measure used to describe the distribution of mass or the shape of an object in rotational motion. It is the distance from the axis of rotation where the entire mass of the body is assumed to be concentrated that give the same moment of inertia of that body when it is rotating about an axis of rotation. It is an equivalent distance of the mass from the axis of rotation. It is helpfull to obtain the moment of inertia of a complex object. The radius of gyration of mass \(m\) is given by
\begin{equation*}
I =mk^{2} \Rightarrow k=\sqrt{\frac{I}{m}}
\end{equation*}
Here \(k\) refers to the radius of gyration of the object. The radius of gyration of a spherical body about its diameter can be obtained by considering the entire body is concentrated as a point mass at a distance \(k\) from the axis of rotation and hence
\begin{equation*}
I=\frac{2}{5}mr^{2} = mk^{2}\Rightarrow\quad k=\sqrt{\frac{2}{5}} \,r
\end{equation*}
The radius of gyration provides a way to quantify how the mass of an object is distributed relative to its axis of rotation. Objects with a larger radius of gyration have their mass distributed farther from the axis of rotation, which typically means they are harder to rotate. Conversely, objects with a smaller radius of gyration have their mass distributed closer to the axis of rotation, making them easier to rotate.
