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We have here, going back
to rotating objects...

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I have an object here
that has a certain velocity v,

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and it's going around
with angular velocity omega,

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and a little later

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the angle has increased
by an amount theta

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and then the velocity is here.

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We may now do something
we haven't done before.

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We could give this object
in this circle an acceleration.

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So we don't have to keep
the speed constant.

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Now, v equals omega R, so
that equals theta dot times R.

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And I can take now
the first derivative of this.

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Then I get
a tangential acceleration,

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which would be
omega dot times R,

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which is
theta double dot times R,

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and we call theta double dot...

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we call this alpha, and alpha
is the angular acceleration

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which is in radians
per second squared.

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Do not confuse ever
the tangential acceleration,

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which is along
the circumference,

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with a centripetal acceleration.

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The two are both there,
of course.

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This is the one
that makes the speed change

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along the circumference.

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If we compare our knowledge
of the past of linear motion

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and we want to transfer it
now to circular motion,

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then you can use all your
equations from the past

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if you convert x to theta,
v to omega and a to alpha.

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And the well-known equations
that I'm sure you remember

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can then all be used.

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For instance, the equation

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x equals x zero plus v zero t
plus one-half at squared

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simply becomes
for circular motion

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theta equals theta zero
plus omega zero t

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plus one-half alpha t squared--
it's that simple.

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Omega zero is then the angular
velocity at time t equals zero,

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and theta zero is the angle
at time t equals zero

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relative to
some reference point.

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And the velocity was
v zero plus at.

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That now becomes
that the velocity goes

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to angular velocity omega
equals omega zero plus alpha t.

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So there's really not much added

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in terms of
remembering equations.

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If I have a rotating disk,

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I can ask myself
the question now

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which we have never done before,
what kind of kinetic energy,

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how much kinetic energy is there
in a rotating disk?

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We only dealt
with linear motions,

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with one-half mv squared,

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but we never considered
rotating objects

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and the energy
that they contain.

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So let's work on that a little.

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I have here a disk, and
the center of the disk is C,

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and this disk is rotating
with angular velocity omega

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that could change in time,

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and the disk has a mass m,
and the disk has a radius R.

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And I want to know
at this moment

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how much kinetic energy of
rotation is stored in that disk.

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I take a little
mass element here, m of i,

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and this radius equals r of i

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and the kinetic energy
of that element i alone

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equals one-half m of i
times v of i squared,

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and v of i is this velocity--
this angle is 90 degrees.

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This is v of i.

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Now, v equals omega R.

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That always holds
for these rotating objects.

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And so I prefer to write this

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as one-half m of i
omega squared r of i squared.

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The nice thing
about writing it this way

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is that omega, the angular
velocity, is the same

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for all points of the disk,
whereas the velocity is not

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because the velocity of a point
very close to the center

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is very low.

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The velocity here is very high,

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and so by going to omega, we
don't have that problem anymore.

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So, what is now the kinetic
energy of rotation of the disk,

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the entire disk?

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So we have to make a summation,

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and so that is omega squared
over two

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times the sum
of m of i r i squared

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over all these elements mi

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which each have
their individual radii, r of i.

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And this, now, is what we call
the moment of inertia, I.

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Don't confuse that with impulse;

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it has nothing to do
with impulse.

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And this is moment of inertia...

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So the moment of inertia is
the sum of mi ri squared.

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In...

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So this can also be written
as one-half I,

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I put a C there--
you will see shortly why,

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because the moment
of inertia depends

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upon which axis of rotation
I choose-- times omega squared.

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And when you see that equation

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you say, "Hey, that looks quite
similar to one-half mv squared."

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And so I add to this list now.

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If you go from linear motions
to rotational motions,

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you should change the mass
in your linear motion

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to the moment of inertia
in your rotational motion,

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and then you get back
to your one-half mv squared.

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You can see that.

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So we now have

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a way of calculating
the kinetic energy of rotation

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provided that we know how to
calculate the moment of inertia.

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Well, the moment of inertia
is a boring job.

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It's no physics, it's pure math,

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and I'm not going
to do that for you.

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It's some integral, and if
the object is nicely symmetric,

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in general you can do that.

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In this case, for the disk

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which is rotating about
an axis through the center

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and the axis--
that's important--

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is perpendicular to the disk--
that's essential-- in that case

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the moment of inertia equals
one-half m times R squared.

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And I don't even want you
to remember this.

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There are tables in books,
and you look these things up.

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I don't remember that.

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I may remember it for one day,

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but then, obviously, you forget
that very quickly again.

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Needless to say, that
the moment of inertia depends

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on what kind of object you have.

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Whether you have a disk

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or whether you have a sphere
or whether you have a rod

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makes all the difference.

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And what also makes
the difference--

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about which axis
you rotate the object.

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If we had a sphere,
a solid sphere, then...

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So here you have a solid sphere,

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and I rotate it about an axis
through its center.

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Then the moment of inertia,
I happen to remember,

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equals two-fifths mR squared

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if R is the radius
and m is the mass of the sphere.

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My research is in astrophysics.

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I deal with stars, and stars
have rotational kinetic energy.

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We'll get back to that
in a minute--

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not in a minute but today--

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and this is the one moment
of inertia that I do remember.

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If you have a rod,
and you let this rod rotate

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about an axis
through the center,

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and this axis is
perpendicular to the rod--

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the latter is important,
perpendicular to the rod--

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and it is length l
and it has mass m,

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then the moment of inertia--

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which I looked up this morning;
I would never remember that--

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equals 1/12 ml squared.

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And all these moments
of inertia

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you can find in tables
in your book on page 309.

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So the moment of inertia
for rotation about this axis

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of a solid disk
is one-half mR squared.

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But it's completely different,
the moment of inertia,

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if you rotated it
about this axis.

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So you take the plane
of the disk.

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Instead of rotating it this way,
you rotate it now this way.

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You get a totally different
moment of inertia.

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And most of those you can find
in tables, but not all of them.

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Tables only go so far,

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and that is why I want
to discuss with you

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two theorems which will help you

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to find moments of inertia
in most cases.

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Suppose we have a rotating disk,

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and I will make you see
the disk now with depth.

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So this is a disk,
and we just discussed

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the rotation
about the center of mass.

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And I call this axis l.

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And so it was rotating like this

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and was perpendicular
to the disk.

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This is the moment of inertia.

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But now I'm going
to drill a hole here,

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and I have here an axis l prime
which is parallel to that one.

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And I'm going to force
this object

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to rotate about that axis.

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I can always do that--
I can drill a hole

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have an axle,
nicely frictionless bearing

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and I can force it
to rotate about that.

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What now is
the moment of inertia?

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If I know the moment of inertia,

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then I know how much rotational
kinetic energy there is.

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That's one-half I omega squared.

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And now there is a theorem

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which I will not prove,
but it's very easy to prove,

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and that is called
the parallel axis theorem.

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And that says

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that the moment of inertia
of rotation about l prime--

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provided that l prime is
parallel to l--

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is the moment of inertia

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when the object rotates
about an axis l

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through the center of mass

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plus the mass of the disk
times the distance d squared.

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So this is the mass.

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And that's
a very easy thing to apply,

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and that allows you
now in many cases,

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to find the moment of inertia

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in situations
which are not very symmetric.

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Imagine that you had
to do this mathematically,

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that you actually had to do

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an integration of all these
elements mi from this point on.

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That would be
a complete headache.

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In fact, I wouldn't even know
how to do that.

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So it's great.

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Once you have demonstrated,
once you have proven

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that this parallel
axis theorem works,

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then, of course, you can always
use it to your advantage.

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Notice that
the moment of inertia

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for rotation about this axis--

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which is not through a center
of mass-- is always larger

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than the one through the center.

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You see, you have this md
squared; it's always larger.

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There is a second theorem
which sometimes comes in handy,

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and that only works when you
deal with very thin objects,

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and that is called
the perpendicular axis theorem.

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If you have some kind
of a crazy object--

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which of course
we will never give you;

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we'll always give you a square
or we'll give you a disk...

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But it has to be a thin plate.

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Otherwise the perpendicular
axis theorem doesn't work.

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And suppose I'm rotating it

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about an axis perpendicular
to the blackboard

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through that point.

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I call that the z axis.

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It's sticking out to you.

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That's the positive z axis.

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I can draw now any xy axis where
I please, at 90-degree angles,

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anywhere in the plane
of the blackboard.

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So I pick one here,
I call this x,

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and I pick one here
and I call that y.

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So z is pointing towards you.

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Remember, I always choose

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a positive right-handed
coordinate system.

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My x cross y is always
in the direction of z.

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I always do that.

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And so you see that here,
x cross y equals z.

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Now, you can rotate
this thin plate about this axis.

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You can also rotate it
about that axis.

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And you can also rotate it
about the z axis.

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And then the
perpendicular axis theorem,

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which your book proves
in just a few lines,

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tells you
that the moment of inertia

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for rotation
about this axis here

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is the same as the moment
of inertia for rotation about x

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plus the moment of inertia
for rotation about the axis y.

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And this allows you
to sometimes...

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in combination with
the parallel axis theorem

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to find moments of inertia in
case that you have thin plates

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which rotate about axes
perpendicular to the plate

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or sometimes
not even perpendicular.

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Sometimes you can use... if
you know this and you know this,

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then you can find that.

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So both are useful,

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and in assignment 7
I'll give you a simple problem

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so that you can apply
the perpendicular axis theorem.

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There are applications

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where energy is temporarily
stored in a rotating disk,

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and we call those disks
flywheels.

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And the rotational kinetic
energy can be consumed, then,

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at a later time,
so it's very economical.

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And this rotational
kinetic energy

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can then be, perhaps,
converted into electricity

259
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or in other forms of energy.

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And there are really remarkably
inventive and intriguing ideas

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on how this can be done.

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Of course,
whether it is practical

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depends always
on dollars and cents

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and to what extent it is
economically feasible.

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But I have always,
even when I was a small boy...

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I remember when I was seven
years, it already occurred to me

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that all this heat
that is produced

268
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when cars slam their brakes--

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all you're doing is
you produce heat;

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you lose all that kinetic energy
of your linear motion--

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whether somehow that couldn't
be used in a more effective way.

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And this is what I want
to discuss with you now

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and see where we stand.

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This is actually
being taken seriously

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by the Department of Energy.

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So I want to work out
with you an example of a car

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which is in the mountains and
which is going to go downhill.

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And the mountains are
very dangerous-- zigzag roads--

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and so he or she can
only go very slowly.

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And the maximum speed
that the person could use is

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at most ten miles per hour--
without killing him or herself--

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which is about
four meters per second.

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And so here is your car,

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and let's assume
you start out with zero speed.

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And let's assume
that the mass of the car--

286
00:15:45 --> 00:15:51
we'll give it nice numbers--
is just 1,000 kilograms.

287
00:15:48 --> 00:15:54
And so you zigzag
down this road.

288
00:15:53 --> 00:15:59
Let us assume
that the height difference h--

289
00:15:58 --> 00:16:04
let's give it a number,
500 meters...

290
00:16:01 --> 00:16:07
And you arrive here at point p.

291
00:16:04 --> 00:16:10
And you later have to go
back up again.

292
00:16:07 --> 00:16:13
What is your kinetic energy
when you reach point p?

293
00:16:11 --> 00:16:17
Well, you have a speed
of four meters per second,

294
00:16:15 --> 00:16:21
and as you went down, you've
been braking all the time.

295
00:16:18 --> 00:16:24
One way or another,
you got rid of your speed

296
00:16:21 --> 00:16:27
and that's all burned up--
heat, you heat up the universe.

297
00:16:25 --> 00:16:31
So when you reach point p, your
kinetic energy at that point p

298
00:16:29 --> 00:16:35
is simply one-half mv squared.

299
00:16:32 --> 00:16:38
m is the mass of the car,

300
00:16:34 --> 00:16:40
so that is 500 times 16--
v squared--

301
00:16:37 --> 00:16:43
so that is 8,000 joules.

302
00:16:41 --> 00:16:47
Now compare this
with the work that gravity did

303
00:16:45 --> 00:16:51
in bringing this car down.

304
00:16:47 --> 00:16:53
That work is mgh,
and mgh is a staggering number.

305
00:16:54 --> 00:17:00
1,000 times ten times 500--
that is five million joules!

306
00:17:02 --> 00:17:08
And all of that was converted
to heat using the brakes.

307
00:17:06 --> 00:17:12
It actually even gives you also
wear and tear on the brakes.

308
00:17:09 --> 00:17:15
So who needs it?

309
00:17:11 --> 00:17:17
Is there perhaps
a way that you can salvage it

310
00:17:14 --> 00:17:20
or maybe not all of it,
maybe part of it?

311
00:17:16 --> 00:17:22
And the answer is yes,
there are ways.

312
00:17:21 --> 00:17:27
At least in principle
there are ways.

313
00:17:23 --> 00:17:29
You can install
a disk in your car,

314
00:17:26 --> 00:17:32
which I would call,
then, a flywheel,

315
00:17:28 --> 00:17:34
And you can convert the
gravitational potential energy.

316
00:17:33 --> 00:17:39
You can convert that
to kinetic energy of rotation

317
00:17:37 --> 00:17:43
in your flywheel.

318
00:17:39 --> 00:17:45
And to show you that
it is not completely absurd,

319
00:17:42 --> 00:17:48
I will put, actually,
in some numbers.

320
00:17:44 --> 00:17:50
Suppose you had a disk
in your car

321
00:17:48 --> 00:17:54
which had a radius
of half a meter.

322
00:17:50 --> 00:17:56
That's not completely absurd.

323
00:17:52 --> 00:17:58
That's not
beyond my imagination.

324
00:17:54 --> 00:18:00
That's a sizable disk.

325
00:17:56 --> 00:18:02
And I give it a modest mass--

326
00:17:59 --> 00:18:05
so that the mass of the car
is not going to be too high--

327
00:18:03 --> 00:18:09
200 kilograms.

328
00:18:05 --> 00:18:11
That's reasonable.

329
00:18:06 --> 00:18:12
That would be a steel plate
only five centimeters thick,

330
00:18:10 --> 00:18:16
so that's quite reasonable.

331
00:18:11 --> 00:18:17
And the moment of inertia
of this disk if I rotate it

332
00:18:15 --> 00:18:21
about an axis through the center
perpendicular to the disk--

333
00:18:18 --> 00:18:24
that moment of inertia,
we know now, is one-half m...

334
00:18:23 --> 00:18:29
oh, we have a capital M--
R squared, and that equals 25.

335
00:18:27 --> 00:18:33
The units are kilograms,
if you're interested,

336
00:18:30 --> 00:18:36
kilograms/meter squared.

337
00:18:34 --> 00:18:40
So we know
the moment of inertia.

338
00:18:36 --> 00:18:42
Now, what we would like to do
is we would like to convert

339
00:18:40 --> 00:18:46
all this gravitational
potential energy

340
00:18:43 --> 00:18:49
into kinetic energy
of that disk.

341
00:18:45 --> 00:18:51
If you think of a clever way
that you can couple that--

342
00:18:48 --> 00:18:54
people have succeeded in that--

343
00:18:49 --> 00:18:55
then you really would like
one-half I omega squared...

344
00:18:55 --> 00:19:01
You would really like that

345
00:18:56 --> 00:19:02
to be five times
ten to the six joules.

346
00:18:59 --> 00:19:05
And so that
immediately tells you

347
00:19:02 --> 00:19:08
what omega should be for
that disk, and you find, then,

348
00:19:05 --> 00:19:11
if you put in your numbers,
which is trivial...

349
00:19:08 --> 00:19:14
you find that omega is
about 632 radians per second,

350
00:19:13 --> 00:19:19
so the frequency of the disk
is 100 hertz,

351
00:19:16 --> 00:19:22
100 revolutions per second.

352
00:19:19 --> 00:19:25
I don't think that that is
particularly extravagant.

353
00:19:23 --> 00:19:29
So as you would come
down the hill,

354
00:19:25 --> 00:19:31
you would not be braking
by pushing on your brake,

355
00:19:29 --> 00:19:35
you would not be heating up
your brakes,

356
00:19:32 --> 00:19:38
but you would somehow
convert this energy

357
00:19:34 --> 00:19:40
into the rotating disk
and that would slow you down.

358
00:19:38 --> 00:19:44
So the slowdown,
the "braking" is now done

359
00:19:43 --> 00:19:49
because of a conversion
from your linear speed--

360
00:19:46 --> 00:19:52
which comes from gravitational
potential energy--

361
00:19:48 --> 00:19:54
to the rotation of the disk.

362
00:19:53 --> 00:19:59
And when you need that energy,
you tap it.

363
00:19:55 --> 00:20:01
So you should also be able

364
00:19:57 --> 00:20:03
to get the rotational
kinetic energy out

365
00:20:00 --> 00:20:06
and convert that again
into forward motion.

366
00:20:03 --> 00:20:09
And if you could really do this,
then you could go back uphill

367
00:20:07 --> 00:20:13
and you wouldn't have to use
any fuel.

368
00:20:09 --> 00:20:15
All your five million joules
can be consumed, then,

369
00:20:13 --> 00:20:19
in an ideal case, and you
would not have to use any fuel.

370
00:20:18 --> 00:20:24

371
00:20:20 --> 00:20:26
Now, you can ask yourself
the question,

372
00:20:21 --> 00:20:27
is this system only useful
in the mountains

373
00:20:23 --> 00:20:29
or could you also use this
in a city?

374
00:20:25 --> 00:20:31
Well, of course
you can use it in a city.

375
00:20:27 --> 00:20:33
You wouldn't be braking
like this, then,

376
00:20:30 --> 00:20:36
but again, you would slow down

377
00:20:32 --> 00:20:38
by taking out kinetic energy
of linear forward motion,

378
00:20:35 --> 00:20:41
dump that into kinetic energy
of rotation of your flywheel

379
00:20:39 --> 00:20:45
and that would slow you down.

380
00:20:41 --> 00:20:47
And when the traffic light turns
green, you convert it back--

381
00:20:45 --> 00:20:51
rotational kinetic energy
into linear kinetic energy--

382
00:20:49 --> 00:20:55
and you keep going again.

383
00:20:51 --> 00:20:57
Now, of course, this is
all easier said than done,

384
00:20:54 --> 00:21:00
but it is not complete fantasy.

385
00:20:56 --> 00:21:02
People have actually made
some interesting studies,

386
00:21:00 --> 00:21:06
and I would like to show you
at least one case

387
00:21:03 --> 00:21:09
that I am aware of,
that I found on the Web,

388
00:21:06 --> 00:21:12
that shows you that United
States Energy Department

389
00:21:10 --> 00:21:16
is taking this quite seriously.

390
00:21:14 --> 00:21:20
This view graph is also
on the 801 home page.

391
00:21:19 --> 00:21:25
And so you see here

392
00:21:21 --> 00:21:27
the idea of mounting such
a flywheel under the car here.

393
00:21:25 --> 00:21:31
And it has the location

394
00:21:27 --> 00:21:33
of the "flywheel energy
management power plant."

395
00:21:31 --> 00:21:37
Wonderful word, isn't it?

396
00:21:32 --> 00:21:38
And here you see
a close-up of this flywheel.

397
00:21:36 --> 00:21:42
I didn't get any numbers on it.

398
00:21:38 --> 00:21:44
I don't know
which fraction of the energy

399
00:21:41 --> 00:21:47
can be stored in your flywheel,
but it's an attempt.

400
00:21:44 --> 00:21:50
People are seriously thinking
about it.

401
00:21:47 --> 00:21:53
And it may happen
in the next decade

402
00:21:50 --> 00:21:56
that cars may come on the market

403
00:21:53 --> 00:21:59
whereby some of your energy,
at least, can be salvaged.

404
00:21:58 --> 00:22:04
Instead of heating up
the universe, use it yourself,

405
00:22:02 --> 00:22:08
which could be very economical.

406
00:22:05 --> 00:22:11

407
00:22:11 --> 00:22:17
I have here a toy car--
I'll show it on TV first.

408
00:22:18 --> 00:22:24

409
00:22:21 --> 00:22:27
And this toy car has a flywheel.

410
00:22:26 --> 00:22:32
Do you see it?

411
00:22:29 --> 00:22:35
That the flywheel itself is
the wheel of the car,

412
00:22:32 --> 00:22:38
but the idea is there.

413
00:22:34 --> 00:22:40
In this case, I cannot convert
linear motion into the flywheel.

414
00:22:38 --> 00:22:44
I could do that, but I'm going
to do it in a reverse way.

415
00:22:42 --> 00:22:48
I'm going to give this flywheel

416
00:22:45 --> 00:22:51
a lot of kinetic energy
of rotation,

417
00:22:47 --> 00:22:53
and you will see shortly
how I do that.

418
00:22:49 --> 00:22:55
And then I will show you

419
00:22:51 --> 00:22:57
that that can be converted back
into forward motion--

420
00:22:54 --> 00:23:00
in this case, it's very easy

421
00:22:57 --> 00:23:03
because the flywheel itself
is the wheel.

422
00:23:00 --> 00:23:06
So let me try
to... power this car.

423
00:23:06 --> 00:23:12

424
00:23:09 --> 00:23:15
I do that
with this plastic... okay.

425
00:23:14 --> 00:23:20

426
00:23:15 --> 00:23:21
So I'm going to put
some energy into this wheel,

427
00:23:19 --> 00:23:25
into this flywheel,
and then we'll see

428
00:23:22 --> 00:23:28
whether the car can use that
to start moving.

429
00:23:25 --> 00:23:31

430
00:23:27 --> 00:23:33
Great that
my lecture notes were there.

431
00:23:29 --> 00:23:35
So, you see, it works.

432
00:23:31 --> 00:23:37
And, of course, if
you could reverse that idea,

433
00:23:34 --> 00:23:40
that when the car...

434
00:23:36 --> 00:23:42
before it stops,
get it back into the flywheel,

435
00:23:39 --> 00:23:45
then you have the idea
that I was trying to get across.

436
00:23:43 --> 00:23:49
Very economical,

437
00:23:45 --> 00:23:51
and definitely that will happen
sometime in the future.

438
00:23:50 --> 00:23:56
Flywheels are used more often
than you may think.

439
00:23:55 --> 00:24:01
MIT, at the Magnet Lab, has
two flywheels which are amazing.

440
00:24:05 --> 00:24:11
They have a radius,
I think, of 2.4 meters--

441
00:24:09 --> 00:24:15
that is correct--

442
00:24:12 --> 00:24:18
and each one of those flywheels
has a stunning mass

443
00:24:16 --> 00:24:22
of 85 tons, 85,000 kilograms...

444
00:24:20 --> 00:24:26

445
00:24:23 --> 00:24:29
and they rotate
at about six hertz.

446
00:24:28 --> 00:24:34
You can calculate
the moment of inertia.

447
00:24:31 --> 00:24:37
They rotate about their center
axis perpendicular to the plane.

448
00:24:37 --> 00:24:43
You know now what
one-half I omega square is,

449
00:24:41 --> 00:24:47
and so you can calculate
the kinetic energy of rotation.

450
00:24:46 --> 00:24:52
And that kinetic energy
of rotation is, then,

451
00:24:49 --> 00:24:55
a whopping 200 million joules

452
00:24:53 --> 00:24:59
in each of
those rotating flywheels.

453
00:24:57 --> 00:25:03
Now, they use this
rotational kinetic energy

454
00:25:00 --> 00:25:06
to create very strong
magnetic fields

455
00:25:03 --> 00:25:09
on a time scale
as short as five seconds.

456
00:25:06 --> 00:25:12
So they convert
mechanical energy of rotation

457
00:25:11 --> 00:25:17
to magnetic energy,
which is not part of 801

458
00:25:13 --> 00:25:19
so I will not go
into how they do that.

459
00:25:15 --> 00:25:21
This is part of 802,

460
00:25:17 --> 00:25:23
and I'm sure all of you
are looking forward to 802,

461
00:25:19 --> 00:25:25
and that's when you will see

462
00:25:21 --> 00:25:27
how you can convert mechanical
energy into magnetic energy.

463
00:25:24 --> 00:25:30
We have already seen
a demonstration in class

464
00:25:27 --> 00:25:33
whereby we converted
mechanical energy

465
00:25:29 --> 00:25:35
when someone was rotating,
into electric energy.

466
00:25:32 --> 00:25:38
I think that was you, wasn't it?

467
00:25:34 --> 00:25:40
And we got these light bulbs on.

468
00:25:36 --> 00:25:42
Well, you can also convert it
into magnetic energy.

469
00:25:39 --> 00:25:45
And then when they have created

470
00:25:41 --> 00:25:47
these strong magnetic fields
that they do their research with

471
00:25:44 --> 00:25:50
and when they want
to get rid of them,

472
00:25:46 --> 00:25:52
they go the other way around
and they dump that energy,

473
00:25:48 --> 00:25:54
that magnetic energy,
back into the flywheels,

474
00:25:51 --> 00:25:57
who then start spinning again
at six hertz.

475
00:25:55 --> 00:26:01

476
00:25:56 --> 00:26:02
Needless to say

477
00:25:58 --> 00:26:04
that huge amount
of rotational kinetic energy

478
00:26:03 --> 00:26:09
must be stored in planets
and in stars,

479
00:26:08 --> 00:26:14
and I would like to spend
quite some time on that.

480
00:26:12 --> 00:26:18
It's a very interesting subject.

481
00:26:15 --> 00:26:21
I will first discuss with you
the sun and the earth

482
00:26:21 --> 00:26:27
and see how much rotational
kinetic energy is stored

483
00:26:25 --> 00:26:31
in the earth and in the sun.

484
00:26:28 --> 00:26:34
This is also on the 801
home page, so don't copy this.

485
00:26:32 --> 00:26:38
Let's first look at the sun.

486
00:26:35 --> 00:26:41
We have the mass of the sun,
we have the radius of the sun

487
00:26:38 --> 00:26:44
so you can calculate the moment
of inertia of the sun.

488
00:26:42 --> 00:26:48
I have used
my two-fifths mR squared,

489
00:26:44 --> 00:26:50
which is really
a crude approximation,

490
00:26:47 --> 00:26:53
because the two-fifths m
R squared for a solid sphere

491
00:26:50 --> 00:26:56
only holds if the mass
is uniformly distributed

492
00:26:53 --> 00:26:59
throughout that sphere.

493
00:26:54 --> 00:27:00
With a star, that's not the
case; not with the earth either,

494
00:26:58 --> 00:27:04
because the density is higher
at the center.

495
00:27:01 --> 00:27:07
But this sort of gives
you a crude idea.

496
00:27:03 --> 00:27:09
So we have there
the moment of inertia,

497
00:27:05 --> 00:27:11
which is easy to calculate

498
00:27:07 --> 00:27:13
with that
two-fifths mR squared,

499
00:27:09 --> 00:27:15
and I get the same
for the earth.

500
00:27:11 --> 00:27:17
This is the radius of the earth,

501
00:27:14 --> 00:27:20
and you see the moment
of inertia of the earth.

502
00:27:17 --> 00:27:23
Now I want to know

503
00:27:18 --> 00:27:24
how much kinetic energy
of rotation these objects have.

504
00:27:21 --> 00:27:27
Well, the sun rotates
about its axis in 26 days,

505
00:27:24 --> 00:27:30
the earth in one day,

506
00:27:25 --> 00:27:31
and so I finally convert
everything to MKS units

507
00:27:30 --> 00:27:36
and I find these numbers for
the rotational kinetic energy.

508
00:27:35 --> 00:27:41
Now, look at the number
of the sun--

509
00:27:37 --> 00:27:43
1½ times ten to the 36th joules.

510
00:27:42 --> 00:27:48
Our great-grandfathers
must have been puzzled

511
00:27:45 --> 00:27:51
about where
the solar energy came from--

512
00:27:47 --> 00:27:53
the heat and the light,
where it came from.

513
00:27:50 --> 00:27:56
And conceivably
it came from rotation.

514
00:27:53 --> 00:27:59
Maybe the sun is spinning down,
is slowing down

515
00:27:56 --> 00:28:02
and maybe the energy that we get

516
00:27:58 --> 00:28:04
is nothing but
rotational kinetic energy.

517
00:28:01 --> 00:28:07
If that were the case, however,

518
00:28:04 --> 00:28:10
since the sun produces four
times ten to the 26th watts--

519
00:28:08 --> 00:28:14
four times ten to the 26th
joules-- per second,

520
00:28:11 --> 00:28:17
it would only last 125 years.

521
00:28:13 --> 00:28:19
So you can completely
forget the idea

522
00:28:16 --> 00:28:22
that the energy from the sun

523
00:28:17 --> 00:28:23
that we now know,
of course, is nuclear,

524
00:28:20 --> 00:28:26
but our great-grandparents
didn't know that--

525
00:28:22 --> 00:28:28
that the energy would be tapped
from kinetic energy of rotation.

526
00:28:26 --> 00:28:32
Let's look at the earth.

527
00:28:28 --> 00:28:34
2½ times
ten to the 29th joules.

528
00:28:32 --> 00:28:38
Well, let me try something...

529
00:28:36 --> 00:28:42
some fantasy on you,
some crazy, some ridiculous idea

530
00:28:39 --> 00:28:45
and I'm telling you first,
it is ridiculous.

531
00:28:42 --> 00:28:48
Remember that
the world consumption...

532
00:28:44 --> 00:28:50
Six billion people
on earth consume

533
00:28:46 --> 00:28:52
about four times ten
to the 20th joules every year.

534
00:28:49 --> 00:28:55
So if somehow...
I thought if you could tap

535
00:28:52 --> 00:28:58
the rotational energy
of the earth

536
00:28:54 --> 00:29:00
by slowing the earth down,

537
00:28:57 --> 00:29:03
maybe we could use it to satisfy
the world energy consumption.

538
00:29:02 --> 00:29:08
Um, I wouldn't know
how to do it,

539
00:29:05 --> 00:29:11
and it is, of course,
complete fantasy.

540
00:29:07 --> 00:29:13
All you would have to do
is slow the earth down

541
00:29:11 --> 00:29:17
by about... 2.4 seconds.

542
00:29:16 --> 00:29:22
After one year...
So you slow it down.

543
00:29:18 --> 00:29:24
After one year,
the day wouldn't last...

544
00:29:21 --> 00:29:27
Day and night wouldn't last
24 hours

545
00:29:24 --> 00:29:30
but only 2.4 seconds longer.

546
00:29:26 --> 00:29:32
But, of course,
after a billion years, then,

547
00:29:28 --> 00:29:34
you would have consumed up all
the rotational kinetic energy

548
00:29:31 --> 00:29:37
and then the earth would
no longer be rotating.

549
00:29:33 --> 00:29:39
It is, of course, a crazy idea

550
00:29:35 --> 00:29:41
but sometimes it's cute
to speculate about crazy ideas.

551
00:29:41 --> 00:29:47
There is an object
which we call the Crab Pulsar.

552
00:29:45 --> 00:29:51
It is a neutron star and it is
located in the Crab Nebula.

553
00:29:51 --> 00:29:57
The Crab Nebula is the result
of a supernova explosion

554
00:29:54 --> 00:30:00
that went off in the year 1054,

555
00:29:56 --> 00:30:02
and during my next lecture I
will talk a lot more about that.

556
00:30:00 --> 00:30:06
For now, I just want
to concentrate

557
00:30:02 --> 00:30:08
on this neutron star alone.

558
00:30:05 --> 00:30:11
And so here you have
the data on the Crab Pulsar.

559
00:30:09 --> 00:30:15
The mass of the Crab Pulsar
is not too different

560
00:30:11 --> 00:30:17
from that of the sun.

561
00:30:12 --> 00:30:18
It's about 1½ times more.

562
00:30:14 --> 00:30:20
The radius is
ridiculously small--

563
00:30:16 --> 00:30:22
it's only ten kilometers.

564
00:30:17 --> 00:30:23
All that mass is compact in
a ten-kilometer-radius sphere.

565
00:30:21 --> 00:30:27
It has ahorrendous density

566
00:30:23 --> 00:30:29
of ten to the 14th grams
per cubic centimeter.

567
00:30:26 --> 00:30:32
So, of course, the moment
of inertia is extremely modest

568
00:30:30 --> 00:30:36
compared to the sun,
because the radius is so small

569
00:30:33 --> 00:30:39
and the moment of inertia goes
with the radius squared.

570
00:30:37 --> 00:30:43
However, if you look
at rotational kinetic energy,

571
00:30:41 --> 00:30:47
the situation is very different,

572
00:30:43 --> 00:30:49
because this
neutron star rotates

573
00:30:46 --> 00:30:52
in 33 milliseconds
about its axis.

574
00:30:48 --> 00:30:54
So it has a phenomenal
angular velocity.

575
00:30:51 --> 00:30:57
And so if now you calculate
one-half I omega squared,

576
00:30:56 --> 00:31:02
you get a fantastic amount
of rotational kinetic energy.

577
00:31:00 --> 00:31:06
You get an amount which is
more than a million times more

578
00:31:04 --> 00:31:10
than you have in the sun.

579
00:31:08 --> 00:31:14

580
00:31:10 --> 00:31:16
And this object,
this pulsar in the Crab Nebula

581
00:31:15 --> 00:31:21
is radiating copious amounts
of x-rays, of gamma rays.

582
00:31:21 --> 00:31:27
There are jets coming out
of ionized gas,

583
00:31:26 --> 00:31:32
and we are certain

584
00:31:28 --> 00:31:34
that all that energy
that this object is producing

585
00:31:32 --> 00:31:38
comes from
rotational kinetic energy.

586
00:31:36 --> 00:31:42
And I will give you
convincing arguments

587
00:31:38 --> 00:31:44
why there is no doubt
about that.

588
00:31:41 --> 00:31:47
If you take the Crab Pulsar
and you calculate

589
00:31:49 --> 00:31:55
how much energy comes out
in x-rays and gamma rays

590
00:31:51 --> 00:31:57
and everything that
you can observe in astronomy,

591
00:31:54 --> 00:32:00
then you find
that it has a power

592
00:31:56 --> 00:32:02
roughly of about six times
ten to the 31st watts.

593
00:32:01 --> 00:32:07
It's a phenomenal amount

594
00:32:02 --> 00:32:08
if you compare that
with the sun, by the way.

595
00:32:04 --> 00:32:10
The sun is only four times
ten to the 26th watts.

596
00:32:08 --> 00:32:14
So the Crab Pulsar
alone generates

597
00:32:11 --> 00:32:17
about 150,000 times
more power than the sun.

598
00:32:17 --> 00:32:23
We know the period of the pulsar

599
00:32:20 --> 00:32:26
to a very high degree
of accuracy.

600
00:32:22 --> 00:32:28
The period of rotation
of the neutron star

601
00:32:26 --> 00:32:32
is 0.0335028583 seconds.

602
00:32:41 --> 00:32:47
That's what it is today.

603
00:32:43 --> 00:32:49
I called my radio astronomy
friends yesterday

604
00:32:45 --> 00:32:51
and I asked them,
"What is the rotation period

605
00:32:49 --> 00:32:55
of the neutron star
in the Crab Nebula?"

606
00:32:51 --> 00:32:57
and this was the answer.

607
00:32:53 --> 00:32:59
Tomorrow, however, it is
longer by 36.4 nanoseconds.

608
00:33:00 --> 00:33:06
So tomorrow,
you have to add this.

609
00:33:04 --> 00:33:10
That means it's slowing down.

610
00:33:08 --> 00:33:14
The Crab Pulsar is slowing down.

611
00:33:11 --> 00:33:17
That means omega is going down.

612
00:33:13 --> 00:33:19
That means one-half I omega
square is going down.

613
00:33:16 --> 00:33:22
And when you do your homework,
which you should be able to do--

614
00:33:20 --> 00:33:26
to compare the rotational
kinetic energy today

615
00:33:23 --> 00:33:29
with the rotational
kinetic energy tomorrow--

616
00:33:28 --> 00:33:34
you will see
that the loss of energy

617
00:33:31 --> 00:33:37
is six times ten to the 31st
joules per second,

618
00:33:35 --> 00:33:41
which is exactly the power
that we record

619
00:33:39 --> 00:33:45
in terms of x-rays, gamma rays
and other forms of energy.

620
00:33:43 --> 00:33:49
So there's no question

621
00:33:44 --> 00:33:50
that in the case
of this rotating neutron star,

622
00:33:47 --> 00:33:53
all the energy that it radiates

623
00:33:50 --> 00:33:56
is at the expense
of rotational kinetic energy.

624
00:33:54 --> 00:34:00
It's a mind-boggling concept
when you think of it.

625
00:33:58 --> 00:34:04
And if the neutron star
in the Crab Nebula

626
00:34:01 --> 00:34:07
were to continue to lose
rotational kinetic energy

627
00:34:05 --> 00:34:11
at exactly this rate,

628
00:34:06 --> 00:34:12
then it would come to a halt
in about 1,000 years.

629
00:34:14 --> 00:34:20
Now I would like to show you
a few slides,

630
00:34:18 --> 00:34:24
and I might as well
cover this up

631
00:34:20 --> 00:34:26
so that we get it
very dark in this room.

632
00:34:25 --> 00:34:31
I want to show you
the Crab Nebula,

633
00:34:27 --> 00:34:33
and I think I will also show you

634
00:34:31 --> 00:34:37
the beautiful flywheels
in the Magnet Lab.

635
00:34:36 --> 00:34:42

636
00:34:39 --> 00:34:45
Now I need a flashlight.

637
00:34:42 --> 00:34:48

638
00:34:45 --> 00:34:51
I need my laser pointer.

639
00:34:49 --> 00:34:55
I need a lot of stuff.

640
00:34:52 --> 00:34:58
Okay, there we go,
so I'm going to make it dark.

641
00:34:55 --> 00:35:01

642
00:34:58 --> 00:35:04
You ready for that?

643
00:35:00 --> 00:35:06
Okay, if I can get
the first slide.

644
00:35:06 --> 00:35:12
What you see here are these
flywheels at the Magnet Lab.

645
00:35:13 --> 00:35:19
These are the wheels
that have a mass of 85 tons

646
00:35:17 --> 00:35:23
and that have a radius
of 2.5 meters--

647
00:35:21 --> 00:35:27
an incredible, ingenious device,

648
00:35:25 --> 00:35:31
and you can store in there
200 million joules,

649
00:35:30 --> 00:35:36
and you can dump it
into magnetic energy

650
00:35:32 --> 00:35:38
and in five seconds dump it back
into kinetic energy of rotation.

651
00:35:37 --> 00:35:43
It is an amazing accomplishment,
by the way.

652
00:35:41 --> 00:35:47
And here you see
the Crab Nebula.

653
00:35:44 --> 00:35:50
The Crab Nebula is
at a distance from us

654
00:35:47 --> 00:35:53
of about 5,000 light-years.

655
00:35:50 --> 00:35:56
It is the remnant of a supernova
explosion in the year 1054--

656
00:35:55 --> 00:36:01
much more about that
during my next lecture--

657
00:35:59 --> 00:36:05
and what you see here is
not stuff that is generated

658
00:36:04 --> 00:36:10
at this moment in time
by the pulsar.

659
00:36:07 --> 00:36:13
This, by the way, is the pulsar,

660
00:36:09 --> 00:36:15
and the red filaments
that you see here is material

661
00:36:13 --> 00:36:19
that was thrown off
when the explosion occurred.

662
00:36:16 --> 00:36:22
The explosion,
the supernova explosion throws

663
00:36:19 --> 00:36:25
the outer layers of the star
with a huge speed--

664
00:36:21 --> 00:36:27
some 10,000 kilometers
per second-- into space,

665
00:36:24 --> 00:36:30
and that is what you are seeing.

666
00:36:26 --> 00:36:32
From here to here is
about seven light-years

667
00:36:29 --> 00:36:35
to give you an idea
of the size of this object.

668
00:36:32 --> 00:36:38
This pulsar alone, however,

669
00:36:35 --> 00:36:41
generates the...
six times 31... watts.

670
00:36:40 --> 00:36:46
And we do know that it is
this star that is the pulsar

671
00:36:45 --> 00:36:51
and we know
that it is not that star.

672
00:36:48 --> 00:36:54
And the way that
that was observed,

673
00:36:51 --> 00:36:57
that that was measured,
is as follows.

674
00:36:54 --> 00:37:00
A stroboscopic picture,
a stroboscopic exposure was made

675
00:36:59 --> 00:37:05
of the center portion
of the Crab Nebula.

676
00:37:03 --> 00:37:09
And a stroboscopic picture means

677
00:37:06 --> 00:37:12
that you are using a shutter
which opens and closes.

678
00:37:13 --> 00:37:19
In this case,
you have to open and close it

679
00:37:16 --> 00:37:22
with exactly the same frequency

680
00:37:18 --> 00:37:24
as the rotation
of the neutron star.

681
00:37:20 --> 00:37:26
This neutron star-- for reasons
that is not well understood--

682
00:37:24 --> 00:37:30
is blinking at us.

683
00:37:26 --> 00:37:32
It blinks at us

684
00:37:27 --> 00:37:33
at exactly the frequency of
its rotation, 33 milliseconds.

685
00:37:32 --> 00:37:38
That means 30 hertz.

686
00:37:35 --> 00:37:41
Roughly 30 times per second

687
00:37:37 --> 00:37:43
you see the star become bright
and then go dim again.

688
00:37:41 --> 00:37:47
If now you set your frequency
of your shutter of your...

689
00:37:46 --> 00:37:52
in front of your photographic
plate at exactly that frequency

690
00:37:51 --> 00:37:57
and you expose
the photographic plate

691
00:37:53 --> 00:37:59
only when the star is bright,

692
00:37:55 --> 00:38:01
then you will see
a very bright star

693
00:37:58 --> 00:38:04
when you develop your picture.

694
00:38:00 --> 00:38:06
If now you take another picture,

695
00:38:02 --> 00:38:08
expose it
the same amount of time,

696
00:38:04 --> 00:38:10
but the shutter is open
when the star is dim

697
00:38:07 --> 00:38:13
and you develop that picture,
the star is dim.

698
00:38:10 --> 00:38:16
But the beauty is

699
00:38:12 --> 00:38:18
that all other stars
in the vicinity, of course,

700
00:38:14 --> 00:38:20
will show up on both
photographic plates

701
00:38:16 --> 00:38:22
with exactly the same strength

702
00:38:18 --> 00:38:24
because they are
not blinking at you,

703
00:38:20 --> 00:38:26
since they don't blink at us

704
00:38:22 --> 00:38:28
with a period
of 33 milliseconds.

705
00:38:25 --> 00:38:31
That is what you will see
on the next slide,

706
00:38:29 --> 00:38:35
which is
a stroboscopic exposure.

707
00:38:35 --> 00:38:41
This star is clearly
not the pulsar

708
00:38:37 --> 00:38:43
as it is about equally bright
on both exposures.

709
00:38:40 --> 00:38:46

710
00:38:44 --> 00:38:50
This is not the pulsar,
but this one is.

711
00:38:46 --> 00:38:52
You see, this one
is missing here.

712
00:38:49 --> 00:38:55
And so this is
beyond any question

713
00:38:52 --> 00:38:58
that we know exactly
which the pulsar is.

714
00:38:57 --> 00:39:03
A very new observatory
was launched only recently,

715
00:39:04 --> 00:39:10
and that is called
the Chandra X-ray Observatory.

716
00:39:07 --> 00:39:13
And Chandra made a picture
very recently

717
00:39:11 --> 00:39:17
of the Crab Nebula,
of the pulsar,

718
00:39:13 --> 00:39:19
and that's what
I want to show you now.

719
00:39:17 --> 00:39:23
It's on the Web,
and I show you a picture

720
00:39:20 --> 00:39:26
that many of you probably
haven't seen yet,

721
00:39:23 --> 00:39:29
which is the center part
of the Crab Nebula,

722
00:39:30 --> 00:39:36
and the pulsar is located here.

723
00:39:34 --> 00:39:40
And all this is x-rays, nothing
to do with optical light.

724
00:39:37 --> 00:39:43
This is all x-rays,

725
00:39:39 --> 00:39:45
and you see there is a huge
nebula here around this pulsar

726
00:39:43 --> 00:39:49
which is about
two light-years across,

727
00:39:45 --> 00:39:51
and all that energy in x-rays
is all at the expense

728
00:39:49 --> 00:39:55
of rotational kinetic
energy of the pulsar,

729
00:39:52 --> 00:39:58
which is quite amazing.

730
00:39:53 --> 00:39:59
And when this picture was made
with Chandra X-ray Observatory,

731
00:39:57 --> 00:40:03
they discovered immediately that
the pulsar also produces a jet.

732
00:40:02 --> 00:40:08
Maybe you can see that
from where you are sitting.

733
00:40:04 --> 00:40:10
There is a jet coming out here,

734
00:40:05 --> 00:40:11
and with a little bit
of imagination

735
00:40:07 --> 00:40:13
you can see
this jet going out there.

736
00:40:10 --> 00:40:16
All that energy is
at the expense

737
00:40:11 --> 00:40:17
of rotational kinetic energy.

738
00:40:13 --> 00:40:19
MIT has a big stake, by the way,
in the Chandra Observatory,

739
00:40:17 --> 00:40:23
and not only MIT
but Cambridge as a whole.

740
00:40:20 --> 00:40:26
The Center for Astrophysics
and MIT are running

741
00:40:23 --> 00:40:29
the Chandra Science Center,

742
00:40:27 --> 00:40:33
from which
all radio commands are given,

743
00:40:31 --> 00:40:37
which is here just across
the street, a few blocks away.

744
00:40:34 --> 00:40:40
And many MIT scientists
have dedicated

745
00:40:37 --> 00:40:43
the major part of their careers
in this endeavor.

746
00:40:41 --> 00:40:47
And these are one
of the wonderful results

747
00:40:44 --> 00:40:50
that have come out.

748
00:40:45 --> 00:40:51
All right, you now have
five minutes left.

749
00:40:48 --> 00:40:54
You have a little more--
you have seven minutes left.

750
00:40:50 --> 00:40:56
I would appreciate it a lot if
you fill out the questionnaire,

751
00:40:55 --> 00:41:01
because that's the only way
we can get your feedback

752
00:40:58 --> 00:41:04
and we can make changes

753
00:41:00 --> 00:41:06
if you think these changes
are necessary.

754
00:41:03 --> 00:41:09
So, see you Friday.

755
00:41:05 --> 00:41:11.000