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We've defined moment of inertia
of a rigid body already.

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And we often are
interested in it,

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because when we rotate
a body about an axis,

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say an axis that's passing
through the center of mass.

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Then our kinetic energy depends
in that moment of inertia

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through that axis.

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But we may also want to consider
rotation about another axis.

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So suppose we consider
an axis passing

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through close to the
end and we rotate

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the body around this axis.

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You can see from
overhead like that.

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Then we want to now consider
how the mass is distributed

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about an axis that was
parallel to the axis passing

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through the center of mass.

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So we'll now make a
little calculation.

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But first, we want
to quote the theorem.

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So let's just draw an
example of our rigid body.

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And let's take the
center of mass,

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and let's consider an axis
that's going perpendicular

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to the center of mass.

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We can think of the object
is rotating about that axis.

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And now let's consider
another axis passing

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through a different
point, but also parallel

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to this axis separated
by a distance d.

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Then the result that we want
is at the moment of inertia

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about an axis
passing perpendicular

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to the plane of the
object through the axis s

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is equal to the moment
of inertia about an axis

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passing through the
center of mass--

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notice these are parallel
axes-- plus the mass

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of the object times the distance
between the two parallel axes.

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And this is a result
that is very useful when

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calculating moments of inertia.

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Of course, we could
calculate the moment

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about any axis we wanted.

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And we'll see that in a moment.

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For example, we know that
for a rod of length L

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that the moment of inertia
through the center of mass

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was 1/12.

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And let's call the length of the
rod L. So we have a length L.

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And let's assume it's
a uniform object.

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And we'll calculate a moment
of inertia through an axis

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through the end.

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And so in this case, d
is equal to L over 2.

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And so I about the
end axis is 1/12 mL

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squared plus the mass
times L over 2 square

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and a 12 plus a quarter
is 1/3 mL squared.

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And that means that
all you need to know

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is the moment through
the center of mass,

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and you can calculate the
moment through any other axis.

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Very useful theorem called
the parallel axis theorem.

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Now as I said, we can
calculate the moment s.

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And just to show you very
quickly, if we pick s here,

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and we pick our little dm
there, and we have a distance x,

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then the moment
about s-- and let's

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say that dm is equal to
the total mass divided

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by the length times some
little distance dx--

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then the moment is dm times
x squared, where x we'll

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give this an
integration variable

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x prime goes from 0
to x prime equals l.

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And so we have for our
dm m over L dx prime.

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Remember, that's our
integration variable.

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And we have x squared.

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So that's x prime squared.

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And this is just the
integral of x cubed over 3.

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So we have 1/3 m over L x prime
cubed evaluated from 0 to L,

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and that comes out
to 1/3 mL squared.

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And that's in agreement with
the parallel axis theorem.

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And that's an example of
parallel axis theorems.

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You can do the same thing with
the disk or other objects.