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We already showed that
the torque about a point

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can also be thought
of as a decomposition.

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We take the vector from the
point P to the center of mass

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and apply all the forces
acting on the particle

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at the center of mass.

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And we can calculate the
torque about the center

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of mass due to the
action of some forces

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where we're having forces
acting about the center of mass.

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Now if we choose the point P
to equal the center of mass,

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then we know that the
vector r center of mass

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to the center of mass is 0.

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So the torque about
the center of mass

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is just equal to the
forces about that point.

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We know that torque
is always just Lcm/dt.

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Now, again, how could we
justify that statement

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that because we're
only calculating

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the torque about
the center of mass,

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it's only the rotational angular
momentum about the center

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of mass that's changing.

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We saw before that if
we thought of Lp, again,

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as a translational and a
rotational angular momentum--

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I'm sorry, rotational
angular momentum, omega--

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and the point p was equal
to the center of mass,

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then this first
piece would be 0.

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And L about p is only
Icm omega and dLp/dt

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is equal to Icm alpha
for a fixed axis.

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And that's exactly
equal to dLcm/dt.

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And so the point
here is that when

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we're applying problems
involving rotation

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and translation, we can
just analyze the torque

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about the center of
mass and only consider

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how the angular momentum about
the center of mass is changing.

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And then for the
center of mass motion--

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so this gives us our
rotational dynamics.

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And for our linear
dynamics, we will still

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apply F equals the
total mass times Acm.

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So that's our linear dynamics.

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And this is our
overall decomposition

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of rotational motion.

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To analyze it, we study
the rotational dynamics

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and the linear dynamics.