Moment of inertia

Similar to how inertia describes the tendency of an object or system to resist changes to its motion, moment of inertia $I_{\Delta}(S)$ describes the tendency of an object or system $(S)$ to resist changes specifically to its rotational motion about axis $\Delta$.

By definition,

$$I_{\Delta}(S)=\int_{S}\left\| \vec r_m \wedge \hat \delta \right\|^2 dm$$

where $dm$ is a unit mass of solid/system $(S)$, $\vec r_m$ is the position vector of $dm$, and $\delta$ is a unit vector along the axis of rotation $\Delta$.

Denote the position of the unit mass $dm$ as $M$, then $\vec r_m =\overrightarrow{OM} \Rightarrow \vec r_m \wedge \Delta=\overrightarrow{HM}$, where $H \in \Delta \land \overrightarrow{HM}\bot \Delta$. i.e., $\left\| \vec r_m \wedge \Delta \right\|$ measures the vertical distance from $M$ to $\Delta$ $d(M)$.

Orthogonal base

In an orthonormal basis, it can be derived that:

$$ \begin{cases} I_{Ox}(S)=\int_S(y^2+z^2)\,dm\\ I_{Oy}(S)=\int_S(x^2+z^2)\,dm\\ I_{Oz}(S)=\int_S(x^2+y^2)\,dm \end{cases} $$

And it is thereby abstracted a combined moment of inertia according to the origin $O$:

$$ I_O=\int_S(x^2+y^2+z^2)\,dm $$

which estimates the general tendency of $(S)$ to resist an arbitrary rotation around $O$.

See also

Inertia matrix