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When we analyzed a single
particle motion-- say,

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we have an object, which
is moving with momentum p--

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then we have two principles.

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Our Newton's second law told
us that the force on the object

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causes the momentum to change.

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And now, what we'd like to do
is consider another principle.

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Suppose we're looking
at some point about s.

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We'd like to show that
the torque about s

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is equal to the change
in angular momentum

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of the particle about s.

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So let's now investigate that.

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Well, recall that angular
momentum is defined

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to be about a point, a
vector s, and will indicate m

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to show that this is the
vector from s to where

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the object is located cross
the momentum of the object.

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And what we want
to now calculate

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is the time derivative
of that quantity.

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Well, when you take the time
derivative of a vector product,

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it's the product rule.

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And so there's two terms here.

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It's drs/mdt cross p
plus rsm cross dp/dt.

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Now, no matter what
the point s is,

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the derivative of
the vector that's

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measuring where s to
the mass is is always

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the velocity of the object.

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So that's the velocity.

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And momentum is just
mass times the velocity.

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And you can see that
that quantity is 0

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because a vector direct
product with its cross product

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with itself is 0.

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And this second term--

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notice that we
have dp/dt in here.

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And it's always crucial
to understanding

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these calculations where
the second law comes in.

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This is just rsm cross f.

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And recall that when we
talk about the torque

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on an object about the
point s by the action

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of, say, some force.

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And so suppose we had a
force acting on this object.

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And that's going
to cause the object

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to change its direction
by some amount delta p.

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Then the torque, by definition,
is rsm cross the force.

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And we see that that is exactly
equal to the change in angular

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momentum.

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And so for a single particle,
the torque about a point

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s causes the angular momentum
about that point to change.

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And this result will generalize
for a collection of particles

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in a calculation
that's very simple.