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0:00:02 --> 00:00:08
With our knowledge of torque...
calm down.

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With our knowledge of torque
and angular momentum,

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we can now attack rolling
objects which roll down a slope.

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For instance, the following...
I have here a cylinder

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or it could be a sphere,
for that matter,

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and this angle is beta.

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I prefer not to use alpha

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because that's angular
acceleration,

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and there is
this friction coefficient

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with the surface, mu,

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and this object
is going to roll down

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and you're going to get
an acceleration

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in this direction, a.

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And I will evaluate
the situation

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when we have pure roll.

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That means the object is not
skidding and is not slipping.

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What is pure roll?

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If here is an object, the
cylinder is here with radius R,

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and I'm going to rotate it
like this

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and roll it in this direction,
the center is called point Q.

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Once it has made a complete
rotation,

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if then the point Q has moved
over a distance 2pi R,

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then we call that pure roll.

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When we have pure roll,
the velocity of this point Q,

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and the velocity of
the circumference,

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if you can read that--
I'll just put a c there--

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are the same.

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In other words, vQ is then
exactly the same

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as v circumference, and v
circumference is always omega R.

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This part always holds,
but for pure roll, this holds.

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You can easily imagine that
if there is no friction here,

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that the object could be
standing still,

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rotating like crazy,
but Q would not go anywhere.

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So then we have skidding
and we have slipping

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and then we don't have
the pure roll situation.

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If the object is skidding
or slipping,

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then the friction must always
be a maximum here.

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If the object is in pure roll,

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the friction could be
substantially less

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than the maximum
friction possible.

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Now I would like to calculate
with you the acceleration

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that a cylinder would obtain.

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When it pure rolls
down that slope,

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it has mass M, it has length l
and it has radius R.

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And I would like you to use
your intuition

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and don't be afraid
that it's wrong.

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I'm going to roll down
this incline two cylinders.

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They're both solid,
they have the same mass,

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they have the same length but
they're very different in radii

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and I'm going to have a race
between these two.

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Which one will reach
the bottom first?

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So I repeat the problem.

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Two cylinders, both solid,
same length, same mass,

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but one has a larger radius
than the other.

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There's going to be a race.

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We're going to roll them down,
pure roll.

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Which one will will?

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Win win?

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Will win?

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Who thinks that the one with
the largest radius will win?

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Who thinks the large with the
smallest radius will win?

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Who thinks there will be
no winner, no loser?

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Wow, your intuition is
better than mine was.

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We'll see how it goes.

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Keep in mind what your vote was

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and you will see it come out
very shortly.

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Okay, let's put all the forces
on this object that we know.

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This one is Mg.

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And we're going to decompose
that into one a longer slope,

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which is Mg sine beta, and one
perpendicular to the slope.

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We have done that
a zill... zillion times now.

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And this one
equals Mg cosine beta.

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Then there is right here
a normal force,

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and the magnitude of that normal
force is Mg cosine beta,

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so there is no acceleration
in this direction

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and then we have a frictional
force here

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for which I will write F of f.

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There is an angular velocity
at any moment in time, omega,

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which will change
with time, no doubt.

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And then this center point Q,
which is the center of mass,

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is going to get a velocity v

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and that v will also change
with time.

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And the v of the point Q
which changes with time

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is v of the circumference,

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because that is the condition
of pure roll.

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That equals omega R.

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This is always true, but this is
only true when it is pure roll.

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I take the time derivative.

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The derivative of the velocity
of that point Q

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is, per definition,
its acceleration,

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so I get a equals omega dot
times R,

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and that equals alpha R,

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alpha being the angular
acceleration.

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So this is the condition
for pure roll.

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Now I'm going to take the torque
about point Q.

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When I take the torque
about point Q,

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N has no effect because it goes
through Q,

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and g has no effect
because it goes through Q,

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so there's only one force
that adds to the torque.

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If this radius is R,
the magnitude is RF

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and the direction is
in the blackboard.

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But I'm only interested
in the magnitude for now,

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so I get R times
the frictional force.

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This must be I alpha,

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I being the moment of inertia
for rotation about this axis

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through point Q, times alpha,
but I can replace alpha by a/R.

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So I get the moment of inertia
about Q times a/R.

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And this is my first equation,

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and I have as an unknown
the frictional force,

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and I have as an unknown a,
and so I cannot solve for both.

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I need another equation.

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The next equation that I have
is an obvious one,

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that is Newton's second law:
f = MA.

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For the center of mass,

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I can consider all the mass
right here at Q.

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We must have f = MA.

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And so M times the acceleration
of that point Q,

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which is our goal, by the way,

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equals this component,
equals Mg sine beta.

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That is the component downhill.

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And minus Ff,
the frictional force,

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which is the component uphill,

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and this is
my equation number two.

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Now I have two equations
with two unknowns.

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So I can solve now.

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I can eliminate Ff and I will
substitute for Ff in here

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this quantity divided by R.

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And so I get Ma equals
Mg sine beta

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minus moment of inertia
about point Q,

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times a divided by R squared.

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And now notice that
I've eliminated F of f,

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and so now I can solve for a.

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So I'm going to get...
I bring the a's to one side.

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So I get a times M plus moment
of inertia divided by R squared

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equals Mg sine beta,
and a now we have.

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I multiply both sides
with R squared.

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I get MR squared g sine beta
upstairs,

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and downstairs I get MR squared
plus the moment of inertia

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about that point Q.

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This is my result, and
all I have to put in now

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is the moment of inertia
of rotation about that axis.

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I want to remind you, though,

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that it is only true if we have
a situation of pure roll.

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So we can now substitute
in there

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the values that we have
for a solid cylinder.

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If we have a solid cylinder,

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then the moment of inertia
about this axis

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through the center of mass,
which I've called Q,

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equals 1/2 MR squared.

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And if I substitute that
in here,

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notice that all my M's...
MR squares go away.

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I get 1 + 1/2, which is 1 1/2.

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Upside down becomes 2/3.

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So a = 2/3 times g
times the sine of beta.

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There is no M, there is no l
and there is no R.

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So if I have two cylinders,

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solid cylinders
with totally different mass,

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totally different radii,
totally different length

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and they have a race,
neither one wins.

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Very nonintuitive.

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Every time that I see it
I find it kind of amazing.

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Notice that everything
disappears.

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M, R and l disappear.

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So those of you who said
that if I take two cylinders

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with the same mass,
different radii,

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those of you who said that there
is no winner, there is no loser,

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they were correct.

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But even more amazing is that
even the mass you can change.

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You can change anything as long
as the two cylinders are solid.

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That's what matters.

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So if we take a hollow cylinder,

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then the moment of inertia
about this axis

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through the center of mass,
through Q, if this...

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if really most of the mass
is really at the surface,

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then it's very close
to MR squared,

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and then the acceleration--

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if I substitute in here
MR squared,

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I get a 2 there-- equals
1/2 times g times sine beta.

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So this acceleration
is less than this one.

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So the hollow cylinder will lose

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in any race against a solid
cylinder regardless of mass,

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regardless of radius,
regardless of length.

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And I want to show that to you.

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We have a setup here and I'll
try to show that to you also

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on the screen there,

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but for those of you
who are sitting close,

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it's probably much better

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that you just look at the
demonstration right here.

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I have here... ooh.

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Uh-uh.

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I have here to start with a very
heavy cylinder made of brass

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and this one is made
of aluminum.

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They have very different masses,
same radii, same length.

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Should make no difference.

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There should be no winner,
there should be no loser.

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I'm going to start them off
at the same time.

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I hope you can see that there.

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This is... this is
the starting point.

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Can lower it a little.

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I will count down three to zero,

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and then you can see
that they reach the bottom

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almost at the same time.

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So very different in mass.

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The mass difference is
at least a factor of three.

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All other dimensions
are the same.

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Three, two, one, zero.

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00:12:00 --> 00:12:06

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Completely in unison.

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Not intuitive for me.

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Now I have one that has
a very small radius

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compared to this one.

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This is a sm... small
aluminum rod.

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Maybe you can see it here,
television.

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This is way more heavy,
almost 30 times heavier.

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Should make no difference.

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As long as it's solid,
should make no difference.

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No winner, no loser.

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Radii are different,
masses are different.

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Should make no difference.

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00:12:28 --> 00:12:34

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Okay?

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Here we start the race.

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Three, two, one, zero.

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And they hit the bottom
at the same time.

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But now here I have
a hollow one,

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and you better believe it,
that it's hollow.

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So now all the mass is
at the circumference,

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and now it takes more time.

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Now the acceleration as you w...
as you will see,

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is half times g sine beta; 
in the other case it was 2/3.

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And you may want to think
about it tonight,

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why this one takes more.

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It has to do, of course,
with the moment of inertia,

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but again, it's independent
of mass, radius and length.

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So it's purely a matter
of geometry.

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This one is going to be
the loser,

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and this one, regardless
of mass or length,

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is going to be the winner.

242
00:13:20 --> 00:13:26
So you see them.

243
00:13:21 --> 00:13:27
One is hollow, one is not.

244
00:13:22 --> 00:13:28
This is very light; 
this is very heavy.

245
00:13:25 --> 00:13:31
I'll put the hollow one
on your side.

246
00:13:28 --> 00:13:34

247
00:13:32 --> 00:13:38
Three, two, one, zero.

248
00:13:34 --> 00:13:40

249
00:13:37 --> 00:13:43
The hollow one lost
and even fell on the floor.

250
00:13:41 --> 00:13:47

251
00:13:44 --> 00:13:50
Yeah, I find these things
always quite amazing,

252
00:13:47 --> 00:13:53
that nature works this way,

253
00:13:51 --> 00:13:57
and I'm impressed
that most of you

254
00:13:53 --> 00:13:59
or many of you had
the right intuition

255
00:13:55 --> 00:14:01
when they said it would make
no difference

256
00:13:58 --> 00:14:04
for the two solid cylinders.

257
00:14:01 --> 00:14:07

258
00:14:05 --> 00:14:11
We now come to the most
nonintuitive part of all of 8.01

259
00:14:10 --> 00:14:16
and arguably perhaps the most
difficult part in all of physics

260
00:14:14 --> 00:14:20
and that has to do
with gyroscopes.

261
00:14:18 --> 00:14:24
And I really urge you to pay
a lot of attention

262
00:14:21 --> 00:14:27
and not even
to miss ten seconds,

263
00:14:23 --> 00:14:29
because you're going to see some
mind-boggling demonstrations

264
00:14:26 --> 00:14:32
which are so incredibly
nonintuitive

265
00:14:29 --> 00:14:35
that unless you have followed
the steps that lead up to it,

266
00:14:32 --> 00:14:38
you won't have any idea
what you're looking at.

267
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It will be fun, it will be cute,

268
00:14:36 --> 00:14:42
but it won't do anything
for you.

269
00:14:42 --> 00:14:48
Imagine that you and I
go in outer space.

270
00:14:45 --> 00:14:51
No gravity.

271
00:14:46 --> 00:14:52
We're somewhere in outer space

272
00:14:47 --> 00:14:53
and we have
this bicycle wheel there.

273
00:14:50 --> 00:14:56
And I'm going to put a torque
on this bicycle wheel

274
00:14:54 --> 00:15:00
in this direction, so I'm going
to put my right hand towards you

275
00:14:57 --> 00:15:03
and my left hand away from you.

276
00:14:59 --> 00:15:05
And I'll do that
for a short amount of time

277
00:15:01 --> 00:15:07
and I'll let it go.

278
00:15:03 --> 00:15:09
It's obvious
what's going to happen.

279
00:15:05 --> 00:15:11
This wheel is going to spin like
this forever and ever and ever.

280
00:15:10 --> 00:15:16
I've given it a little torque.

281
00:15:12 --> 00:15:18
That means if there's torque,

282
00:15:14 --> 00:15:20
there's a change
of angular momentum.

283
00:15:16 --> 00:15:22
The angular momentum change
must be torque times delta t.

284
00:15:19 --> 00:15:25
And so I do this and let it go

285
00:15:21 --> 00:15:27
and it will rotate about this
axis forever and ever and ever.

286
00:15:24 --> 00:15:30
Simple, right?

287
00:15:25 --> 00:15:31
Okay.

288
00:15:27 --> 00:15:33
Now I'm going to torque it
in this direction.

289
00:15:30 --> 00:15:36
So we're in outer space,
wheel is standing still

290
00:15:33 --> 00:15:39
and all I do is do this
and I let it go.

291
00:15:35 --> 00:15:41
Then it will rotate forever
and ever and ever and ever

292
00:15:39 --> 00:15:45
in this direction.

293
00:15:41 --> 00:15:47
That's clear.

294
00:15:42 --> 00:15:48
Now comes
the very nonintuitive part.

295
00:15:45 --> 00:15:51
Now I'm going to give it
a spin in your direction

296
00:15:51 --> 00:15:57
and now again I'm going
to torque like this.

297
00:15:54 --> 00:16:00
What will now happen?

298
00:15:56 --> 00:16:02
You will say...
or you might say.

299
00:15:58 --> 00:16:04
I'm not accusing you
of anything.

300
00:16:00 --> 00:16:06
You might say, well, you give
the wheel a spin

301
00:16:05 --> 00:16:11
so this... the wheel will
probably continue to spin

302
00:16:09 --> 00:16:15
and you do this, so maybe
what you're going to see is

303
00:16:12 --> 00:16:18
that it will rotate like this
as it did before

304
00:16:16 --> 00:16:22
and in the same time
the wheel will be spinning.

305
00:16:20 --> 00:16:26
But that cannot be

306
00:16:21 --> 00:16:27
because if the wheel would be
spinning like this,

307
00:16:25 --> 00:16:31
then the angular momentum
of the spinning wheel is

308
00:16:27 --> 00:16:33
in this direction.

309
00:16:29 --> 00:16:35
And if I would give it a twist

310
00:16:30 --> 00:16:36
and if it would continue to spin
and it would rotate like this,

311
00:16:34 --> 00:16:40
then this spin angular momentum
would go around like this

312
00:16:38 --> 00:16:44
and that cannot be because
there's no torque on the system,

313
00:16:41 --> 00:16:47
because once I let go,
there is no longer any torque.

314
00:16:43 --> 00:16:49
So it is not possible
for the wheel to keep rotating,

315
00:16:47 --> 00:16:53
and as a result of this torque
that I give it,

316
00:16:49 --> 00:16:55
that it simply goes around.

317
00:16:51 --> 00:16:57
That is not possible.

318
00:16:52 --> 00:16:58
How is nature going to deal
with that?

319
00:16:57 --> 00:17:03
I'll show you that
on a view graph.

320
00:17:01 --> 00:17:07
It's very nonintuitive
what will happen.

321
00:17:04 --> 00:17:10
And we will then also... I
will also demonstrate it to you.

322
00:17:10 --> 00:17:16
So here is the situation

323
00:17:11 --> 00:17:17
precisely as I described it
to you.

324
00:17:14 --> 00:17:20
You are, as an observer,
in this direction.

325
00:17:18 --> 00:17:24
This is the direction of 26.100.

326
00:17:20 --> 00:17:26
So you are viewing the wheels
like this.

327
00:17:23 --> 00:17:29
They're spinning towards you.

328
00:17:25 --> 00:17:31
That's what I will do shortly
and that's what I implied now.

329
00:17:29 --> 00:17:35
This is my right hand
and this is my left hand.

330
00:17:31 --> 00:17:37
The separation between
my height... right and left hand

331
00:17:34 --> 00:17:40
is little "b."

332
00:17:35 --> 00:17:41
So the torque that I apply
is bF,

333
00:17:40 --> 00:17:46
is this arm, so to speak,
times this force.

334
00:17:43 --> 00:17:49
And the force is perpendicular
to the arm.

335
00:17:45 --> 00:17:51
I apply a torque for a certain
amount of time-- delta t.

336
00:17:49 --> 00:17:55
When I do that, I apply,
I add angular momentum

337
00:17:51 --> 00:17:57
in this direction.

338
00:17:54 --> 00:18:00
But the wheel was spinning in...
in... in this direction.

339
00:17:56 --> 00:18:02
You see it.

340
00:17:57 --> 00:18:03
And so the angular momentum
of spin of the wheel

341
00:18:00 --> 00:18:06
is in this direction.

342
00:18:02 --> 00:18:08
I add angular momentum in this
direction and you see that here.

343
00:18:07 --> 00:18:13
So this was originally the spin
angular momentum of the wheel.

344
00:18:11 --> 00:18:17
I torque for time delta t,

345
00:18:14 --> 00:18:20
and so I add angular momentum
like this.

346
00:18:16 --> 00:18:22
And then I stop.

347
00:18:18 --> 00:18:24
I only torque for a short amount
of time and I stop.

348
00:18:21 --> 00:18:27
That means after I have stopped,

349
00:18:23 --> 00:18:29
the angular momentum
of the system as a whole

350
00:18:26 --> 00:18:32
can no longer change

351
00:18:27 --> 00:18:33
because there's no torque
on the system,

352
00:18:29 --> 00:18:35
and the only way that nature now
can solve that problem is

353
00:18:32 --> 00:18:38
to tilt this wheel in the way
I've indicated here,

354
00:18:36 --> 00:18:42
and to make it spin
in this direction

355
00:18:38 --> 00:18:44
and it will stand still.

356
00:18:40 --> 00:18:46
In other words, I hold it
in my hand, the wheel--

357
00:18:43 --> 00:18:49
I will get it--
I give it a spin.

358
00:18:50 --> 00:18:56
I hold it in my hand,
I spin it towards you...

359
00:18:56 --> 00:19:02
and I'm going to put
my right hand towards you

360
00:18:58 --> 00:19:04
and my left hand away from you.

361
00:19:01 --> 00:19:07
The spin angular momentum
is now in this direction,

362
00:19:04 --> 00:19:10
so I'm going to give
it a torque like so.

363
00:19:06 --> 00:19:12
That means up.

364
00:19:07 --> 00:19:13
And what will the wheel do?

365
00:19:08 --> 00:19:14
The wheel will do this.

366
00:19:10 --> 00:19:16
Very nonintuitive.

367
00:19:11 --> 00:19:17
Watch it.

368
00:19:13 --> 00:19:19
Isn't that strange?

369
00:19:15 --> 00:19:21
You wouldn't expect that.

370
00:19:16 --> 00:19:22
I will do it again.

371
00:19:17 --> 00:19:23
I'm going to torque by pushing
my hand towards you,

372
00:19:21 --> 00:19:27
and the wheel does something
completely unexpected.

373
00:19:23 --> 00:19:29
It simply tilts.

374
00:19:25 --> 00:19:31
If I torque
the other way around,

375
00:19:26 --> 00:19:32
then the torque, of course,
will make it flip like this.

376
00:19:29 --> 00:19:35
I'll give it a little bit more
spin angular momentum,

377
00:19:33 --> 00:19:39
so now I move my left hand
towards me and my... towards you

378
00:19:38 --> 00:19:44
and my right hand towards me,

379
00:19:40 --> 00:19:46
and then I expect that the wheel
will do this.

380
00:19:42 --> 00:19:48
And that's what it does.

381
00:19:45 --> 00:19:51
Extremely nonintuitive.

382
00:19:48 --> 00:19:54
These torques applied
to a spinning wheel

383
00:19:51 --> 00:19:57
always do something
that you don't expect.

384
00:19:54 --> 00:20:00
However, there is one thing
that always helps me

385
00:19:58 --> 00:20:04
in terms of guiding me

386
00:20:00 --> 00:20:06
and that is
you can always predict

387
00:20:03 --> 00:20:09
that the spin angular momentum
will always move

388
00:20:10 --> 00:20:16
in the direction of the torque,

389
00:20:17 --> 00:20:23
which is this external torque
that I applied.

390
00:20:20 --> 00:20:26
Let's go over that again.

391
00:20:22 --> 00:20:28
We have here the spin angular
momentum that you saw.

392
00:20:27 --> 00:20:33
It was pointing
in this direction.

393
00:20:29 --> 00:20:35
And I applied the torque
in this direction, the vector.

394
00:20:34 --> 00:20:40
And what does
the spin angular momentum do?

395
00:20:36 --> 00:20:42
It goes in the direction
of the torque.

396
00:20:40 --> 00:20:46
And then when I stop
with my torque,

397
00:20:42 --> 00:20:48
then of course nothing
changes anymore,

398
00:20:44 --> 00:20:50
and so what happens is
this wheel tilts.

399
00:20:47 --> 00:20:53
But notice that L, the spin
angular momentum, has moved

400
00:20:52 --> 00:20:58
from here to here.

401
00:20:54 --> 00:21:00
And I was torqueing
in this direction,

402
00:20:56 --> 00:21:02
so it moved towards the torque.

403
00:20:59 --> 00:21:05
If you have digested this,
then you can test yourself now.

404
00:21:05 --> 00:21:11
Now we have the same wheel.

405
00:21:07 --> 00:21:13
I'm going to rotate it
in exactly the same direction,

406
00:21:11 --> 00:21:17
but now I'm not going to torque
like this, in the Z direction,

407
00:21:15 --> 00:21:21
or like this,
in the minus-Z direction.

408
00:21:17 --> 00:21:23
Now I'm going to do this.

409
00:21:20 --> 00:21:26
Or I'm going to do this.

410
00:21:23 --> 00:21:29
Try now to really concentrate
on what I just taught you.

411
00:21:27 --> 00:21:33
And try to give an answer
to the following question.

412
00:21:30 --> 00:21:36
The wheel is rotating.

413
00:21:33 --> 00:21:39
I hold it in my hand and I'm
going to torque it like this

414
00:21:36 --> 00:21:42
so that the torque factor is
in your direction.

415
00:21:39 --> 00:21:45
Angular momentum goes like this,
torque is like this.

416
00:21:43 --> 00:21:49
What will the angular momentum
vector do?

417
00:21:46 --> 00:21:52
Move in the direction
of... of the torque.

418
00:21:53 --> 00:21:59
What will the angular...

419
00:21:54 --> 00:22:00
what will the spin angular
momentum vector then do

420
00:21:56 --> 00:22:02
if the torque is
in this direction?

421
00:21:59 --> 00:22:05
It will do this.

422
00:22:01 --> 00:22:07
It's going to move
in the horizontal plane.

423
00:22:04 --> 00:22:10
Very nonintuitive,
but that's what it will do.

424
00:22:07 --> 00:22:13
And I will show that to you.

425
00:22:09 --> 00:22:15
I'm going to spin this wheel.

426
00:22:15 --> 00:22:21
I'm going to spin it with
a high angular momentum,

427
00:22:17 --> 00:22:23
high spin angular momentum.

428
00:22:19 --> 00:22:25
And then I'm going to sit
on this stool

429
00:22:22 --> 00:22:28
and I'm going to torque
exactly as you see

430
00:22:24 --> 00:22:30
on the picture on the right.

431
00:22:25 --> 00:22:31
I'm going to torque like this.

432
00:22:27 --> 00:22:33
And as long as I torque it
like this,

433
00:22:29 --> 00:22:35
that spin angular momentum
wants to go around

434
00:22:32 --> 00:22:38
in the horizontal plane.

435
00:22:34 --> 00:22:40
And when I torque
the other way around,

436
00:22:36 --> 00:22:42
it will go back
in the horizontal plane.

437
00:22:39 --> 00:22:45

438
00:23:02 --> 00:23:08
I'm going to torque exactly
as you see on the picture there.

439
00:23:05 --> 00:23:11
Are you ready?

440
00:23:09 --> 00:23:15
I stop the torque; 
nothing happens.

441
00:23:10 --> 00:23:16
I torque backwards; 
I keep torqueing.

442
00:23:13 --> 00:23:19
I keep torqueing.

443
00:23:14 --> 00:23:20
I feel it in my hand.

444
00:23:15 --> 00:23:21
I really have to push.

445
00:23:16 --> 00:23:22
I keep torqueing.

446
00:23:17 --> 00:23:23
And I stop torqueing
and it stops.

447
00:23:18 --> 00:23:24
The angular momentum vector

448
00:23:20 --> 00:23:26
is chasing, so to speak,
the torque.

449
00:23:23 --> 00:23:29
Is that nonintuitive?

450
00:23:25 --> 00:23:31
Very nonintuitive?

451
00:23:29 --> 00:23:35
It's also dangerous sometimes.

452
00:23:32 --> 00:23:38

453
00:23:36 --> 00:23:42
We call this motion of the
stool, and in this case,

454
00:23:39 --> 00:23:45
the motion of the spinning
wheel, we call that precession.

455
00:23:45 --> 00:23:51
So you apply a torque
to a spinning wheel.

456
00:23:48 --> 00:23:54
Then what you obtain
is a precession.

457
00:23:55 --> 00:24:01
I can show you the precession
in another way

458
00:23:59 --> 00:24:05
which is, in fact,
very intriguing.

459
00:24:04 --> 00:24:10
Suppose I have here a string,
a rope, like we have there.

460
00:24:11 --> 00:24:17
And I stick into that rope,
I attach to the rope this wheel,

461
00:24:17 --> 00:24:23
just like so.

462
00:24:23 --> 00:24:29
And I let it go.

463
00:24:24 --> 00:24:30
Well, we all know
what will happen.

464
00:24:26 --> 00:24:32
Clunk.

465
00:24:27 --> 00:24:33
It's clear.

466
00:24:29 --> 00:24:35
All right.

467
00:24:30 --> 00:24:36
But now I'm going to spin it
before I let it go.

468
00:24:33 --> 00:24:39
So here at the bottom, at this
point P, there is a loop.

469
00:24:39 --> 00:24:45
And here is the... the axis of
rotation of the bicycle wheel

470
00:24:44 --> 00:24:50
which is solid brass,
it's a solid piece,

471
00:24:48 --> 00:24:54
and I give that a length
little "r,"

472
00:24:50 --> 00:24:56
not to be confused
with capital "R,"

473
00:24:53 --> 00:24:59
which is the radius
of the bicycle wheel.

474
00:24:59 --> 00:25:05
So this is capital R.

475
00:25:05 --> 00:25:11
And it can rotate about this
here reasonably freely.

476
00:25:08 --> 00:25:14
I call that center point Q

477
00:25:12 --> 00:25:18
and let this be the part of
the wheel that is on your side.

478
00:25:18 --> 00:25:24
I'm trying to make you see it a
little bit three-dimensionally.

479
00:25:22 --> 00:25:28
Suppose now I give it a spin
in this direction.

480
00:25:26 --> 00:25:32
Omega s.

481
00:25:27 --> 00:25:33
"S" stands for "spin."

482
00:25:30 --> 00:25:36
In what direction is now
the spin angular momentum?

483
00:25:33 --> 00:25:39
Use your hands, your thumbs.

484
00:25:36 --> 00:25:42
Spinning in this direction.

485
00:25:37 --> 00:25:43
Yeah.

486
00:25:39 --> 00:25:45
Spinning this direction, angular
momentum is in this direction.

487
00:25:42 --> 00:25:48
That's a spin angular momentum.

488
00:25:46 --> 00:25:52
L spin.

489
00:25:50 --> 00:25:56
Well, there is a force
on this system, Mg,

490
00:25:56 --> 00:26:02
and that force is
in this direction.

491
00:25:59 --> 00:26:05
It has a mass M,
the bicycle wheel,

492
00:26:03 --> 00:26:09
and it has a radius,
capital "R,"

493
00:26:05 --> 00:26:11
and this part is little "r."

494
00:26:09 --> 00:26:15
So relative to point P,
there is a torque

495
00:26:16 --> 00:26:22
and the torque is R times Mg.

496
00:26:18 --> 00:26:24
This is 90 degrees
so the cross product is nice.

497
00:26:21 --> 00:26:27
The sine of the angle is one.

498
00:26:23 --> 00:26:29
So the torque relative to point
P is r times M times g.

499
00:26:29 --> 00:26:35
In what direction
is that torque?

500
00:26:33 --> 00:26:39
R cross F.

501
00:26:35 --> 00:26:41
In what direction
is that torque?

502
00:26:38 --> 00:26:44
Use your hands, thumbs,
whatever you want.

503
00:26:41 --> 00:26:47
You think in this direction?

504
00:26:43 --> 00:26:49
I disagree.

505
00:26:45 --> 00:26:51
I disagree.

506
00:26:46 --> 00:26:52
R cross F is...
you must be kidding.

507
00:26:52 --> 00:26:58
In the blackboard.

508
00:26:53 --> 00:26:59
It's not out of the blackboard; 
it's in the blackboard.

509
00:26:56 --> 00:27:02
R cross F is in the blackboard.

510
00:27:00 --> 00:27:06
There is a torque
in this direction.

511
00:27:03 --> 00:27:09
Nature, gravity
provides that torque.

512
00:27:07 --> 00:27:13
What will the spin angular
momentum do?

513
00:27:10 --> 00:27:16
It's going to move in the
direction of the torque.

514
00:27:13 --> 00:27:19
It's going to chase the torque.

515
00:27:15 --> 00:27:21
So what will it do if the
angular momentum is here?

516
00:27:19 --> 00:27:25
What will it do?

517
00:27:20 --> 00:27:26
It will do this.

518
00:27:22 --> 00:27:28
And as it moves, the torque
will always be perpendicular

519
00:27:27 --> 00:27:33
to the plane
through the string and r.

520
00:27:31 --> 00:27:37
You can just see that
for yourself why that is.

521
00:27:33 --> 00:27:39
At this very moment when angular
momentum is like this,

522
00:27:36 --> 00:27:42
the torque is in the blackboard
because it's r cross F.

523
00:27:40 --> 00:27:46
But when I'm here, this r
has changed position,

524
00:27:43 --> 00:27:49
and always remains perpendicular
to the wheel.

525
00:27:47 --> 00:27:53
So the torque
will also change direction

526
00:27:50 --> 00:27:56
and so this angular momentum,

527
00:27:53 --> 00:27:59
spin angular momentum
will keep chasing the torque

528
00:27:55 --> 00:28:01
and start to rotate freely.

529
00:27:57 --> 00:28:03
That is exactly what I was doing
when I was sitting on the stool,

530
00:28:01 --> 00:28:07
except that I had to apply that
torque in my hands like this.

531
00:28:05 --> 00:28:11
It's exactly the same direction.

532
00:28:06 --> 00:28:12
I had to apply it all the time,

533
00:28:08 --> 00:28:14
and when I stopped,
the precession stopped.

534
00:28:12 --> 00:28:18
Here, however, the torque
will never stop

535
00:28:14 --> 00:28:20
because this Mg
will always be there,

536
00:28:17 --> 00:28:23
and I will show that
to you shortly.

537
00:28:21 --> 00:28:27
You may say, "You must be crazy

538
00:28:24 --> 00:28:30
"because you're violating
Newton's second law, f = MA.

539
00:28:31 --> 00:28:37
"This object got to fall.

540
00:28:33 --> 00:28:39
"There's only one force
on that object.

541
00:28:36 --> 00:28:42
"f = MA.

542
00:28:37 --> 00:28:43
"How can it not?

543
00:28:38 --> 00:28:44
The center of mass must fall
with acceleration g."

544
00:28:40 --> 00:28:46
Aha.

545
00:28:41 --> 00:28:47
There is not just one force
on that object.

546
00:28:44 --> 00:28:50
What do you think is here?

547
00:28:45 --> 00:28:51
The tension in this cable, T,
will be exactly Mg.

548
00:28:51 --> 00:28:57
And so the net, the sum of all
forces on that object is zero.

549
00:28:58 --> 00:29:04
There is no net force on that
wheel but there is a net torque,

550
00:29:02 --> 00:29:08
and that's why
it's going to precess.

551
00:29:04 --> 00:29:10
If there had been a net force,

552
00:29:06 --> 00:29:12
then indeed
it would also go down,

553
00:29:08 --> 00:29:14
if this force
were larger than this.

554
00:29:11 --> 00:29:17
So nature is very clever,

555
00:29:12 --> 00:29:18
the way that it deals with these
rather difficult problems.

556
00:29:17 --> 00:29:23
Before I will show you
this demonstration

557
00:29:19 --> 00:29:25
by spinning this wheel and then
hanging it there in that rope,

558
00:29:24 --> 00:29:30
um, I want to mention

559
00:29:25 --> 00:29:31
that the angular frequency
of the precession,

560
00:29:29 --> 00:29:35
which should never be confused

561
00:29:32 --> 00:29:38
with the angular frequency
of spinning,

562
00:29:35 --> 00:29:41
is derived for you... it's only
a three- or four-minute job,

563
00:29:39 --> 00:29:45
on page 344 in your book.

564
00:29:43 --> 00:29:49
Now, I will not derive it here
but what comes out of it,

565
00:29:47 --> 00:29:53
that it is the torque

566
00:29:49 --> 00:29:55
which is the one that we have
here in this case,

567
00:29:53 --> 00:29:59
divided by
the spin angular momentum.

568
00:29:56 --> 00:30:02
That gives you the frequency
of the precession.

569
00:30:02 --> 00:30:08
In our case, for our bicycle
wheel, it is rMg

570
00:30:08 --> 00:30:14
and the spin angular momentum
of this wheel,

571
00:30:11 --> 00:30:17
if it is rotating with angular
velocity omega of s,

572
00:30:16 --> 00:30:22
would be I times omega.

573
00:30:17 --> 00:30:23
Remember, l is I times omega
of a spinning wheel.

574
00:30:20 --> 00:30:26
So I have here I
rotating about point Q--

575
00:30:24 --> 00:30:30
this is the axis of rotation--
times omega of the spin.

576
00:30:30 --> 00:30:36
This is the spin
and this is the precession.

577
00:30:32 --> 00:30:38
And then the period
of precession

578
00:30:35 --> 00:30:41
would be 2pi divided
by omega precession.

579
00:30:40 --> 00:30:46
Let's take a look
at that equation

580
00:30:41 --> 00:30:47
and see whether that sort
of intuitively makes sense.

581
00:30:46 --> 00:30:52
First of all, if you increase
the torque, the upstairs,

582
00:30:52 --> 00:30:58
then it says that the precession
frequency will increase.

583
00:30:54 --> 00:31:00
That makes sense to me

584
00:30:56 --> 00:31:02
because the torque is persuading
the angular momentum

585
00:31:00 --> 00:31:06
to follow it.

586
00:31:02 --> 00:31:08
So the torque is persuading the
spin angular momentum to change.

587
00:31:05 --> 00:31:11
Well, if the torque is stronger
then it is more powerful,

588
00:31:09 --> 00:31:15
so you expect
that the precession frequency

589
00:31:11 --> 00:31:17
will be higher.

590
00:31:13 --> 00:31:19
However, if the spin angular
momentum is very powerful,

591
00:31:19 --> 00:31:25
then the spin angular momentum
says, "Sorry, torque,

592
00:31:23 --> 00:31:29
I'm not going to go as fast
as you want me to go."

593
00:31:26 --> 00:31:32
So when you increase that spin
angular momentum in the wheel,

594
00:31:29 --> 00:31:35
it is also intuitive

595
00:31:30 --> 00:31:36
that the precession frequency
will go down.

596
00:31:36 --> 00:31:42
As the wheel spins,
it has spin angular momentum,

597
00:31:40 --> 00:31:46
but as it precesses around
like this,

598
00:31:43 --> 00:31:49
there will also be angular
momentum in this direction

599
00:31:46 --> 00:31:52
because it's rotating like this.

600
00:31:49 --> 00:31:55
Therefore there is
a total angular momentum

601
00:31:51 --> 00:31:57
which is
the vectorial sum of the two.

602
00:31:57 --> 00:32:03
This equation will only hold

603
00:32:00 --> 00:32:06
as long as the spin angular
momentum is really dominating

604
00:32:04 --> 00:32:10
the total angular momentum

605
00:32:07 --> 00:32:13
and you can see that
immediately,

606
00:32:09 --> 00:32:15
because suppose you make
the spin angular momentum zero,

607
00:32:12 --> 00:32:18
that it is not spinning at all.

608
00:32:14 --> 00:32:20
Do you really think

609
00:32:15 --> 00:32:21
that the precession frequency
will be infinitely high?

610
00:32:17 --> 00:32:23
Of course not.

611
00:32:18 --> 00:32:24
So this only holds in situations

612
00:32:20 --> 00:32:26
where the spin angular momentum
is way, way larger

613
00:32:24 --> 00:32:30
than the angular momentum that
you get due to the precession.

614
00:32:30 --> 00:32:36
So there are restrictions.

615
00:32:32 --> 00:32:38
When the... when the wheel
comes to a halt,

616
00:32:34 --> 00:32:40
when it's no longer rotating,
you better believe it,

617
00:32:37 --> 00:32:43
then the thing will go clunk.

618
00:32:39 --> 00:32:45
There's no longer
the precession mode.

619
00:32:45 --> 00:32:51
For our bicycle wheel,

620
00:32:47 --> 00:32:53
to get a feeling for how long
the precession will take,

621
00:32:51 --> 00:32:57
uh, we can substitute
the numbers in there,

622
00:32:56 --> 00:33:02
our bicycle wheel,

623
00:32:57 --> 00:33:03
the... the... the rod,
the brass rod, little r

624
00:33:00 --> 00:33:06
has a length of 17 centimeters,

625
00:33:05 --> 00:33:11
and the... the radius
of the bicycle wheel

626
00:33:09 --> 00:33:15
is about 29 centimeters.

627
00:33:15 --> 00:33:21
And let me make the assumption

628
00:33:18 --> 00:33:24
that all the mass of the bicycle
wheel is at the circumference,

629
00:33:22 --> 00:33:28
which is not very accurate,
but it's close to that.

630
00:33:24 --> 00:33:30
I mean, there are
some spokes here,

631
00:33:26 --> 00:33:32
but let's assume
that everything is here,

632
00:33:28 --> 00:33:34
so then the moment of inertia is
MR squared.

633
00:33:32 --> 00:33:38
Well, if now you take
a frequency of five hertz,

634
00:33:39 --> 00:33:45
spin frequency,

635
00:33:42 --> 00:33:48
you can calculate now omega
of the spin frequency.

636
00:33:47 --> 00:33:53
Omega equals 2pi
times the spin frequency.

637
00:33:52 --> 00:33:58
And so I know now I can
substitute that in there,

638
00:33:56 --> 00:34:02
so I get an omega precession

639
00:33:59 --> 00:34:05
now equals rMg
times the moment of inertia.

640
00:34:04 --> 00:34:10
I assume that all the mass is
at the circumference,

641
00:34:07 --> 00:34:13
an approximation,
so we get MR squared

642
00:34:10 --> 00:34:16
and then we get omega S,
which we have here.

643
00:34:15 --> 00:34:21
We lose the M, and so we get

644
00:34:19 --> 00:34:25
rg divided by omega s
times R squared.

645
00:34:22 --> 00:34:28
That is the angular frequency
of the precession,

646
00:34:29 --> 00:34:35
and the period of the precession
is 2pi divided by omega

647
00:34:35 --> 00:34:41
and you find then for the period
of the precession

648
00:34:38 --> 00:34:44
about ten seconds.

649
00:34:41 --> 00:34:47
So if I gave it a spin frequency
of five hertz

650
00:34:46 --> 00:34:52
with these dimensions
and with this approximation

651
00:34:48 --> 00:34:54
that all the mass is
at the circumference,

652
00:34:50 --> 00:34:56
you would expect that it would
precess around very gently

653
00:34:54 --> 00:35:00
in about ten seconds.

654
00:34:56 --> 00:35:02
But I have very little control
over that frequency,

655
00:34:59 --> 00:35:05
so it is possible
I gave it seven hertz,

656
00:35:01 --> 00:35:07
it's possible
I gave it three hertz.

657
00:35:04 --> 00:35:10
But I will do what I can.

658
00:35:06 --> 00:35:12
I'll actually give it
the maximum one that I can.

659
00:35:08 --> 00:35:14
That is always guaranteed
success.

660
00:35:13 --> 00:35:19
Where is the wheel?

661
00:35:14 --> 00:35:20
The wheel is here.

662
00:35:18 --> 00:35:24
So we'll spin it up
and then we'll put it in here.

663
00:35:22 --> 00:35:28

664
00:35:27 --> 00:35:33
Notice the way I'm spinning it.

665
00:35:29 --> 00:35:35
I'm holding it away from me now
and going to change it

666
00:35:31 --> 00:35:37
and do it differently next.

667
00:35:35 --> 00:35:41

668
00:35:54 --> 00:36:00
And there it goes.

669
00:35:56 --> 00:36:02
About ten seconds.

670
00:36:01 --> 00:36:07
Isn't that amazing?

671
00:36:02 --> 00:36:08
And it rotates,
seen from below, clockwise.

672
00:36:06 --> 00:36:12

673
00:36:08 --> 00:36:14
Now it's going this way and I'm
going to redo the experiment,

674
00:36:11 --> 00:36:17
changing the direction
of rotation,

675
00:36:13 --> 00:36:19
and then it will go
the other way around.

676
00:36:15 --> 00:36:21
And now the angular momentum
is rotating like this,

677
00:36:17 --> 00:36:23
is pointing here.

678
00:36:20 --> 00:36:26
Spin angular momentum
is pointing like this,

679
00:36:23 --> 00:36:29
torque is like this,

680
00:36:26 --> 00:36:32
and so the spin angular momentum
is changing that...

681
00:36:28 --> 00:36:34
chasing that torque.

682
00:36:29 --> 00:36:35
I am the spin angular momentum.

683
00:36:30 --> 00:36:36
I am the torque.

684
00:36:31 --> 00:36:37
This is the torque.

685
00:36:32 --> 00:36:38
It's chasing it.

686
00:36:34 --> 00:36:40

687
00:36:39 --> 00:36:45
All right.

688
00:36:40 --> 00:36:46
So I have this in my right hand.

689
00:36:42 --> 00:36:48
That's all right.

690
00:36:43 --> 00:36:49
And now I will... so
when I spin it up, that's right.

691
00:36:52 --> 00:36:58
So let me now change
the direction.

692
00:36:55 --> 00:37:01
I'm turning it over and
I'm going to spin it up again.

693
00:36:59 --> 00:37:05

694
00:37:23 --> 00:37:29
Angular momentum is now
in this direction.

695
00:37:27 --> 00:37:33
See, it's turning
the other way around.

696
00:37:29 --> 00:37:35
Angular momentum is
in this direction.

697
00:37:30 --> 00:37:36
Torque is now towards me.

698
00:37:34 --> 00:37:40
Angular momentum is chasing
the torque.

699
00:37:36 --> 00:37:42
I've changed the direction
of the spin angular momentum.

700
00:37:39 --> 00:37:45
I've not changed the direction
of the torque,

701
00:37:41 --> 00:37:47
and now it is rotating, as seen
from below, counterclockwise.

702
00:37:45 --> 00:37:51
Before,
it was rotating clockwise.

703
00:37:48 --> 00:37:54
If I can increase the torque

704
00:37:50 --> 00:37:56
by putting some weight here
on the axle, I have this...

705
00:37:54 --> 00:38:00
this actually extends,
in our case,

706
00:37:57 --> 00:38:03
and I can put some weight
on here,

707
00:37:59 --> 00:38:05
then I actually add
to the torque

708
00:38:02 --> 00:38:08
and then you will see
that it's... it goes faster.

709
00:38:05 --> 00:38:11
The precession frequency
goes up.

710
00:38:07 --> 00:38:13
So I will put some weight
on there.

711
00:38:10 --> 00:38:16

712
00:38:31 --> 00:38:37
So let it first go around,
which was roughly ten seconds,

713
00:38:36 --> 00:38:42
roughly calculated,

714
00:38:39 --> 00:38:45
and now I'm going to put
two kilograms here at the end.

715
00:38:43 --> 00:38:49
And now you will see
an instantaneous increase

716
00:38:46 --> 00:38:52
in the precession frequency.

717
00:38:47 --> 00:38:53
You see it goes much faster now.

718
00:38:50 --> 00:38:56
I take it off and then it goes
back to its roughly ten seconds.

719
00:38:54 --> 00:39:00
So what I have done is
I have increased this torque

720
00:38:56 --> 00:39:02
but not at the expense of M,

721
00:38:58 --> 00:39:04
because the reason why
the M cancels is

722
00:39:00 --> 00:39:06
because the moment of inertia
has an M in it,

723
00:39:02 --> 00:39:08
but if I just hang
this object on it,

724
00:39:04 --> 00:39:10
that doesn't change
the moment of inertia

725
00:39:06 --> 00:39:12
of the spinning wheel.

726
00:39:10 --> 00:39:16
None of this is intuitive.

727
00:39:13 --> 00:39:19
None of this
is intuitive.

728
00:39:17 --> 00:39:23
You can do all of this
with a $5 toy gyro.

729
00:39:26 --> 00:39:32
And I want to show this to you.

730
00:39:27 --> 00:39:33
This is my toy gyro.

731
00:39:28 --> 00:39:34
I have it in my office.

732
00:39:30 --> 00:39:36
It's great fun.

733
00:39:31 --> 00:39:37
And this toy gyro is doing
exactly the same thing

734
00:39:34 --> 00:39:40
that this is doing.

735
00:39:37 --> 00:39:43
Let me show you
the toy gyro first.

736
00:39:45 --> 00:39:51
Toy gyro.

737
00:39:48 --> 00:39:54
Oh, yeah.

738
00:39:50 --> 00:39:56
Here is a toy gyro.

739
00:39:51 --> 00:39:57
Can you see it?

740
00:39:53 --> 00:39:59
Maybe I should make it
a little darker here.

741
00:39:59 --> 00:40:05
Can you see my toy gyro?

742
00:40:02 --> 00:40:08
Yeah?

743
00:40:05 --> 00:40:11
I'm going to spin it

744
00:40:07 --> 00:40:13
and then I'm going to hang it
exactly the same way

745
00:40:09 --> 00:40:15
that that was hanging.

746
00:40:10 --> 00:40:16
I'm going to spin it,

747
00:40:11 --> 00:40:17
for those who are
sitting close-- whoosh--

748
00:40:13 --> 00:40:19
and then putting it horizontally
and hanging it in a string,

749
00:40:17 --> 00:40:23
and you'll see exactly
the same thing is happening.

750
00:40:21 --> 00:40:27

751
00:40:44 --> 00:40:50
And now through friction,
of course,

752
00:40:46 --> 00:40:52
all this fun ultimately
comes to a halt.

753
00:40:52 --> 00:40:58
I have something
very special for you,

754
00:40:54 --> 00:41:00
or I may have something
very special for you.

755
00:40:57 --> 00:41:03
That depends on my helper
who is here behind the scenes.

756
00:41:04 --> 00:41:10
I hear him.

757
00:41:06 --> 00:41:12
He's there.

758
00:41:09 --> 00:41:15

759
00:41:11 --> 00:41:17
Great.

760
00:41:12 --> 00:41:18
Did you get full speed?

761
00:41:15 --> 00:41:21

762
00:41:18 --> 00:41:24
I'm going to the airport
and get a little tired

763
00:41:22 --> 00:41:28
and ask one of my friends
to help me.

764
00:41:25 --> 00:41:31
Would you please help me

765
00:41:27 --> 00:41:33
and just carry this suitcase
for me around?

766
00:41:31 --> 00:41:37
Pick it up, walk around
a little, make some turns.

767
00:41:36 --> 00:41:42
Turn the other way around,
please.

768
00:41:40 --> 00:41:46
(laughter )

769
00:41:42 --> 00:41:48
STUDENT:
What the hell?

770
00:41:43 --> 00:41:49
LEWIN:
What the hell, yeah.

771
00:41:45 --> 00:41:51
Exactly.

772
00:41:46 --> 00:41:52
What are you doing, man?

773
00:41:48 --> 00:41:54
You're behaving so strangely.

774
00:41:51 --> 00:41:57
Make some more turns, man.

775
00:41:52 --> 00:41:58
We've got to go the...
we've got to catch the plane.

776
00:41:56 --> 00:42:02

777
00:41:59 --> 00:42:05
It doesn't quite do what you
think it will be doing, right?

778
00:42:02 --> 00:42:08
So in here, you've guessed it.

779
00:42:05 --> 00:42:11
STUDENT:
Spinning wheel or something.

780
00:42:07 --> 00:42:13
LEWIN:
Spinning wheel.

781
00:42:08 --> 00:42:14
And when you do this,
you put a torque on it

782
00:42:10 --> 00:42:16
and it does exactly what
you least expect: it flips up.

783
00:42:13 --> 00:42:19
Isn't that fun?

784
00:42:15 --> 00:42:21
Yeah.

785
00:42:16 --> 00:42:22
You may get arrested

786
00:42:17 --> 00:42:23
when you go to Logan Airport
with this suitcase.

787
00:42:20 --> 00:42:26
Thank you very much.

788
00:42:21 --> 00:42:27
It's great.

789
00:42:24 --> 00:42:30

790
00:42:31 --> 00:42:37
Spinning objects have
a stabilizing effect.

791
00:42:35 --> 00:42:41
If you take a bicycle wheel,
and we have one,

792
00:42:42 --> 00:42:48
and I put it here and I do
nothing, it will fall.

793
00:42:48 --> 00:42:54
No one is surprised.

794
00:42:50 --> 00:42:56
However, if I give it a little
spin, then it doesn't fall.

795
00:42:56 --> 00:43:02
Why?

796
00:42:57 --> 00:43:03
Because it has angular momentum.

797
00:42:59 --> 00:43:05
It has spin angular momentum.

798
00:43:03 --> 00:43:09
And so it doesn't fall.

799
00:43:04 --> 00:43:10
And it is not only
a bicycle wheel.

800
00:43:06 --> 00:43:12
Look.

801
00:43:07 --> 00:43:13
Nicely stable.

802
00:43:09 --> 00:43:15
Not only with a bicycle wheel.

803
00:43:11 --> 00:43:17
You take... you take a quarter
and you put a quarter like this

804
00:43:14 --> 00:43:20
on your desk.

805
00:43:15 --> 00:43:21
You bet your life it will fall.

806
00:43:18 --> 00:43:24
Roll it; it becomes stable.

807
00:43:20 --> 00:43:26
You give it spin angular
momentum, it becomes stable.

808
00:43:25 --> 00:43:31
Take a top.

809
00:43:28 --> 00:43:34
You put a top on the table,
falls over.

810
00:43:31 --> 00:43:37
You twist it, you give it spin,
and the top is stable.

811
00:43:36 --> 00:43:42
So spin angular momentum has the
property of stabilizing things.

812
00:43:44 --> 00:43:50
And you will see that addressed
in one of your assignments,

813
00:43:46 --> 00:43:52
when I want you to address that
quantitatively.

814
00:43:52 --> 00:43:58
This is the basic idea behind
inertial guidance systems.

815
00:43:56 --> 00:44:02
In inertial guidance systems,
you have a spinning wheel,

816
00:44:00 --> 00:44:06
at least in the days

817
00:44:01 --> 00:44:07
that the guidance systems
had mechanical wheels.

818
00:44:05 --> 00:44:11
You have a spinning wheel,

819
00:44:07 --> 00:44:13
but that spinning wheel is
mounted in such a way

820
00:44:12 --> 00:44:18
that you cannot put a torque
on the axis of rotation

821
00:44:17 --> 00:44:23
of the spinning wheel.

822
00:44:19 --> 00:44:25
That's the way it's mounted.

823
00:44:20 --> 00:44:26
We call that
three-axle-gimbaled gyros.

824
00:44:26 --> 00:44:32
So the moment that you put
a torque on it,

825
00:44:29 --> 00:44:35
the housings-- in this case, the
yellow and the black housing--

826
00:44:33 --> 00:44:39
will start to rotate,

827
00:44:34 --> 00:44:40
and you never managed
to get that torque

828
00:44:36 --> 00:44:42
on the spinning wheel.

829
00:44:38 --> 00:44:44
You never get it
on... on this axis.

830
00:44:41 --> 00:44:47
And therefore,
if now you put it on your boat

831
00:44:45 --> 00:44:51
or you put it in a plane,
or a missile for that matter,

832
00:44:49 --> 00:44:55
if you can never put a torque
on the spinning wheel

833
00:44:51 --> 00:44:57
and if the angular momentum
for spin is in this direction,

834
00:44:55 --> 00:45:01
it will stay there forever
and ever,

835
00:44:57 --> 00:45:03
assuming that we have
no frictional losses.

836
00:45:01 --> 00:45:07
And if then the plane turns,

837
00:45:04 --> 00:45:10
the direction of the spin
angular momentum

838
00:45:06 --> 00:45:12
will not change,

839
00:45:08 --> 00:45:14
but what will happen
of course is

840
00:45:10 --> 00:45:16
that this yellow frame
will rotate

841
00:45:13 --> 00:45:19
or this black frame will rotate.

842
00:45:16 --> 00:45:22
And in these bearings here
are shaft encoders,

843
00:45:19 --> 00:45:25
and they sense that the rotation
that the outer housing makes

844
00:45:23 --> 00:45:29
in order to keep this thing
pointing at the same direction.

845
00:45:27 --> 00:45:33
And that signal is being fed
back to the automatic pilot

846
00:45:30 --> 00:45:36
and that keeps the plane flying

847
00:45:32 --> 00:45:38
in the direction
that you want to.

848
00:45:34 --> 00:45:40
So you use, as a reference
all the time,

849
00:45:36 --> 00:45:42
the spin angular momentum
of your gyro,

850
00:45:40 --> 00:45:46
which is now mounted
in such a way

851
00:45:42 --> 00:45:48
that you cannot put
a torque on it,

852
00:45:44 --> 00:45:50
even when the plane changes
direction,

853
00:45:46 --> 00:45:52
and I want to show that to you.

854
00:45:49 --> 00:45:55

855
00:46:27 --> 00:46:33
Okay, this is the direction
of my spin angular momentum

856
00:46:32 --> 00:46:38
and I'm the airplane
and I'm going to fly.

857
00:46:37 --> 00:46:43
Look at that
spin angular momentum.

858
00:46:41 --> 00:46:47
It has no respect for me.

859
00:46:44 --> 00:46:50
It stays in the same direction
no matter how I fly.

860
00:46:49 --> 00:46:55

861
00:46:53 --> 00:46:59
And the arrow signals
that come from the bearings

862
00:46:56 --> 00:47:02
of the yellow housing
and the black housing,

863
00:46:59 --> 00:47:05
those arrow signals are fed back
to the automatic pilot

864
00:47:03 --> 00:47:09
and so the plane will stay
on course.

865
00:47:08 --> 00:47:14

866
00:47:10 --> 00:47:16
Now what I can do for you

867
00:47:12 --> 00:47:18
to come to a final test
on your thinking,

868
00:47:17 --> 00:47:23
this wheel is suspended
in such a way

869
00:47:20 --> 00:47:26
that there is
no gravitational torque on it

870
00:47:24 --> 00:47:30
like there was here.

871
00:47:26 --> 00:47:32
But I can put a torque on it

872
00:47:30 --> 00:47:36
by simply putting some weights
on the axis.

873
00:47:34 --> 00:47:40
And what do you think
will happen now

874
00:47:36 --> 00:47:42
if I put some weight here
on the axis?

875
00:47:37 --> 00:47:43
So the wheel is spinning,

876
00:47:41 --> 00:47:47
but now I'm going to put
a torque on it here.

877
00:47:43 --> 00:47:49
It is spinning
in this direction.

878
00:47:45 --> 00:47:51
Angular momentum is pointing
straight at me, away from you.

879
00:47:50 --> 00:47:56
I'm going to put a torque
on like this,

880
00:47:53 --> 00:47:59
put a little weight there.

881
00:47:54 --> 00:48:00
Torque will be
in this direction.

882
00:47:55 --> 00:48:01
What will the spin
angular momentum do?

883
00:47:58 --> 00:48:04
Torque is in this direction; 

884
00:48:00 --> 00:48:06
spin angular momentum's
in this direction.

885
00:48:02 --> 00:48:08
Spin angular momentum
will start to chase the torque.

886
00:48:06 --> 00:48:12
Watch this.

887
00:48:10 --> 00:48:16
There it goes.

888
00:48:12 --> 00:48:18
The spin angular momentum
is chasing the torque.

889
00:48:16 --> 00:48:22
You see exactly the same thing
that I've shown you before.

890
00:48:19 --> 00:48:25
And if I make the torque higher,

891
00:48:25 --> 00:48:31
then the precession frequency
will go up.

892
00:48:28 --> 00:48:34
See, it stops now immediately
when I take it off.

893
00:48:31 --> 00:48:37
Put it back on again.

894
00:48:34 --> 00:48:40
Continues.

895
00:48:36 --> 00:48:42
Put more on it.

896
00:48:37 --> 00:48:43
Goes way faster.

897
00:48:41 --> 00:48:47
What happens now if I put
the weight on this side?

898
00:48:45 --> 00:48:51
So I change the direction
of the torque.

899
00:48:48 --> 00:48:54
If I put it on this side,
torque is now in this direction,

900
00:48:51 --> 00:48:57
spin angular momentum is
in this direction.

901
00:48:54 --> 00:49:00
It's going to reverse direction.

902
00:48:56 --> 00:49:02
There we go.

903
00:48:59 --> 00:49:05
And you see it does.

904
00:49:00 --> 00:49:06
Amazingly nonintuitive.

905
00:49:04 --> 00:49:10
If you have problems with this,
you're not alone.

906
00:49:07 --> 00:49:13
See you Wednesday.

907
00:49:09 --> 00:49:15

908
00:50:00 --> 00:50:06.000