Dynamics of extended bodies
Torque = force \(\times\) lever arm, $\tau=F \, l$
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sign of $\tau$ = sense of rotation (+ve c.c.w.)
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lever arm = shortest distance to the line of force
Equilibrium: $\sum \vec{F} = 0$ and $\sum \tau = 0$
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provides another source of equations
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the point about which $\sum \tau = 0$ is arbitrary
Center of gravity
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c.o.g. = the point about which total \(\tau_{\rm weight}=0\)
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In problems involving weight: $\vec{W}$ acts on the body at c.o.g., and thus causes no torque about c.o.g.
N2L for rotations
\[
\tau = I \, \alpha, \quad \mbox{or} \quad \sum \tau = I_{\rm body} \alpha
\]
where the moment of inertia \(I\) is a property of the extended body, determined by the
spatial distribution of mass:
\[
I=\sum_i m_i r_i^2
\]
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