Inertia Moment and Turning Radius
Moment of Inertia
$$ \begin{align*} I &= \sum_{i} m_{i} {r_{i}}^2 \\ I &= \int r^2 dm \end{align*} $$
The moment of inertia is defined as the (mass of a particle)$\times$(distance from the rotation axis to the particle) and represents the physical quantity that indicates the characteristic of a body to continue rotating. Its symbol is $I$, which seems to be derived from the initial letter of the English word Inertia. The unit is $[kg \cdot m^2]$. It can be considered to play a similar role to mass in translational motion. That is, when the angular momentum $L=I \omega $ is constant, the greater the moment of inertia, the smaller the angular velocity becomes. When there are multiple particles, the moment of inertia of the particle system is calculated by adding up the moments of inertia of each particle. For a body with continuously distributed mass points, it is calculated using integration.
Radius of Gyration
When the moment of inertia is divided by the total mass, the average value of the squared distance from the rotation axis is obtained. This is referred to as the radius of gyration and is denoted by $k$.
$$ k=\sqrt{\frac{I}{m}} $$