Rongzhou: /* See also */

2026-10-07T13:16:40Z

Rongzhou at 09:18, 7 October 2026

2026-10-07T09:18:17Z

← Older revision	Revision as of 09:18, 7 October 2026
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		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,

		{{#content:Q2049}}

		[[File:Moment-of-inertia.svg\|center\|500px]]

		<blockquote>
		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 $\Delta$ $d(M)$.
		</blockquote>

		== Orthogonal base ==

		In an [[Orthonormality#Orthonormal_basis\|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 tensor]]

2026-10-06T14:26:29Z

Created the wiki page for moment of inertia

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