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What is the physical meaning of a mass times a distance squared? Why is such a product of interest in the applications?
For straightline motion, the mass $m$ of a particle is the constant of proportionality between the applied force F, and the resulting acceleration a, a relationship captured in Newton's second law $\mathbf{F}\=m\mathbf{a}$.
This section will show that $m{r}^{2}$ is the constant of proportionality between an applied torque, and a resulting angular acceleration. (A force F applied at a distance $r$ from a pivot point generates a torque (or twist) defined as $\mathbf{\τ}\=\mathbf{R}\times \mathbf{F}$, where R is a vector from the pivot point to the point where the force is applied.)
Let an element of mass $\mathrm{dm}$ rotate about the $x$axis at a distance $r$ from the axis. Measure the angle through which the element has turned by $\mathrm{\θ}\left(t\right)$. Then the quantities in Table 6.6.2 are relevant.
Angular position

$\mathrm{\θ}\left(t\right)$

Angular velocity

$\stackrel{\.}{\mathrm{\θ}}\left(t\right)$

Angular acceleration

$\stackrel{..}{\mathrm{\θ}}\left(t\right)$

Arc length: $s\left(t\right)$

$s\left(t\right)\=r\mathrm{theta;}\left(t\right)$

Rim (or linear) speed: $\stackrel{\.}{s}\left(t\right)$

$\stackrel{\.}{s}\left(t\right)\=r\stackrel{period;}{\mathrm{theta;}}\left(t\right)$

Linear acceleration on rim: $a\=\stackrel{..}{s}\left(t\right)$

$\stackrel{..}{s}\left(t\right)\=r\stackrel{..}{\mathrm{theta;}}\left(t\right)$

Table 6.6.2 Rotation of $\mathrm{dm}$ about $x$axis



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If $r\=\u2225\mathbf{R}\u2225$ and $F\=\u2225\mathbf{F}\u2225$, then for a force F tangential to the circle through which $\mathrm{dm}$ rotates,
$\u2225\mathbf{\τ}\u2225$ = $\u2225\mathbf{R}\times \mathbf{F}\u2225$ = $\u2225\mathbf{R}\u2225$ $\u2225\mathbf{F}\u2225$ $\mathrm{sin}\left(\mathrm{\π}\/2\right)$ = $\u2225\mathbf{R}\u2225$ $\u2225\mathbf{F}\u2225$ = $rF$
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Multiply the scalar form of Newton's second law by $r$ to obtain
$F$

$\=ma$

$\mathrm{\τ}\=rF$

$\=r\left(ma\right)$

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$\=rm\left(\stackrel{..}{s}\left(t\right)\right)$

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$\=rm\left(r\stackrel{..}{\mathrm{theta;}}\left(t\right)\right)$

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$\=m{r}^{2}\stackrel{..}{\mathrm{theta;}}\left(t\right)$



The constant of proportionality between $\mathrm{\τ}$, the magnitude of the torque, and the resulting angular acceleration $\stackrel{..}{\mathrm{\θ}}\left(t\right)$ is the scalar $m{r}^{2}$. This quantity measures the "rotational inertia" and is the content of the integrals that define ${I}_{x}$ and ${I}_{y}$, the moments of inertia about the $x$ and $y$axes, respectively.