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Let's consider the motion
of a symmetric object that's

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rotating about a fixed axis.

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So we could take some object.

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And we can imagine this is
sort of what sometimes people

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refer to as a spinning top.

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And it's rotating about
this symmetry axis, k.

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And we know that the angular
velocity of the object we

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can write as omega z k hat.

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And the angular momentum
about this symmetry

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axis we just showed that was
equal to the moment of inertia

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about this axis times
the angular velocity.

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And in particular, the magnitude
of the angular momentum--

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I'll just write that
L about that axis

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is i axis times the magnitude
of the angular velocity.

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Now, we've seen separately
that the kinetic energy

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about this axis,
which we'll call

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the rotational kinetic energy,
was 1/2 times the moment

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of inertia about that axis times
the angular velocity squared.

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But here, we see that omega
squared is equal to L over i.

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So if we substitute that
in, we get L squared about

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that axis over i axis squared.

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And so we have a nice expression
for the kinetic rotational

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kinetic energy is L squared
about that axis over 2 moment

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of inertia about that axis.

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And this we can describe
is the kinetic energy

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of a symmetric body
about a fixed axis.

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It is nice to
notice the angular--

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the linear analogy
to this, when we

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wrote that the momentum
of an object was mv,

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so the magnitude of the
momentum is just m times v.

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And when we write kinetic
energy translation as 1/2 and v

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squared, we can write that as
a translational kinetic energy

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as p squared over 2m.

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So once again, we see this nice
analogy for rotational motion

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where we describe it in
terms of the angular momentum

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and the moment of inertia.

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And linear motion,
we describe in terms

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of momentum and the mass.