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If we have some mass, m,

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and it is moving with some velocity,

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let's say the magnitude of
that velocity we say is v,

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we know that this object
right over here has momentum.

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Translational momentum.

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And that momentum, and we
use the Greek letter rho

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to represent momentum.

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Translational momentum
is defined as being equal

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to the mass times the velocity.

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This is all a review, we have other videos

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where we talk about
translationial momentum,

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and one way to think about it is,

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"Well how hard is it to stop this thing?"

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Literally in everyday language you think,

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"Well how much momentum
does something have?

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"The more momentum this has
the harder it's going to,

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"the harder it is to stop it in some way."

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And so we know if we wanna get

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a little bit more mathematical,

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that if we wanna change momentum,

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we have to apply force
for some amount of time.

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And so the magnitude of our force

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times the duration of the
time that we apply it for,

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force times time and
this is called impulse.

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And this is once again, all review.

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This is equal to change in momentum.

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Let me do this in that yellow color.

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That is equal to change in momentum.

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So if you don't have any impulse,

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especially if you don't have any net force

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acting on an object,

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its momentum is going to be constant.

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You have a conservation of momentum.

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And we use that idea in all sorts of

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interesting physics
applications in the world,

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and especially a lot of
cases using billiard balls

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and whatever else.

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So now let's try to take a similar idea,

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but go into the rotational world.

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So let's imagine you have a mass,

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for the sake of this we're
gonna assume it's a point mass.

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So you have a mass there.

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And let's just say it's attached by,

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essentially a massless wire,

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to you know, it's just nailed down

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right over here.

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And so this right over here would be

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its center of rotation,

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and so you could imagine
if someone applied

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a torque to this mass,

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this mass could start
rotating in a circle.

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And you can just assume that maybe

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it's sitting on a, you know the screen,

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this is kind of a frictionless surface,

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there's no air resistance.

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And so then it will, if
you apply a torque here,

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it will start rotating.

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And so you could think about,

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"Well there might be an idea,

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"just as momentum is this idea of,

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"well how hard is it to stop something?"

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You might say, "Well how?"

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And this is, stop translating
something from moving.

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You might think, "Well
maybe there's a similar idea

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"of how hard is it for something,

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"or how hard is it to
stop rotating something?"

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And you could imagine that
that idea has been defined

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and it has been defined
as angular momentum.

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So let me make this clear,

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this right over here is momentum.

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And over here we'll talk
about angular momentum.

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And actually both momentum
and angular momentum

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are vector quantities.

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So here I just wrote
kind of the magnitudes

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of velocity and momentum.

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But momentum is a vector
and it could be defined,

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the momentum vector could be
defined as equal to the mass

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which is a scalar quantity
times the velocty.

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Times the velocity vector.

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Now the same thing is
true for angular momentum,

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but I'm gonna stay focused on

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the magnitude of angular momentum.

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Angular momentum can have
direction as you can imagine

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you could rotate in two different ways,

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but that gets a little
bit more complicated

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when you start thinking about
taking the products of vectors

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because as you may already know

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or you may see in the future,

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there's different ways of
taking products of vectors.

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But just to get the intuition
of angular momentum,

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I'll focus on the magnitudes.

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So angular momentum is defined

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and the letter used is L.

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I did a lot of research
to try to figure out

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why it is called L, and I
could not find a good reason.

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So in the message board below
if anyone has a good reason

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I would like to know

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why angular momentum is called L.

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A lot of the best arguments I saw

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is that almost everything else was,

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all the other letters were used up

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for other ideas in physics.

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But anyway, angular momentum is defined,

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and it's defined very similarly.

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Just as kind of torque
is the thing that can

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change how something rotates,

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and force is the way
that something changes

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how something translates it,

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and torque is force times distance

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from the center of rotation,

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everything in kind of the rotational world

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is defined in a similar way.

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You kind of take the analogue
in the translational world,

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and you multiply it times the distance

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from your center of rotation.

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So angular momentum is defined as

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mass times velocity

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times distance from the center of rotation

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so let's call this distance
right over here, r.

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r for radius 'cause you could imagine

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if this was traveling in a circle

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that would be the radius of the circle.

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m, v, r.

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Actually let me be a little
bit more careful here.

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It's the magnitude of the velocity

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that is perpendicular to the radius.

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Sometimes it might be called
the tangential velocity.

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So this symbol right over here,

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this is the magnitude of the velocity

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that is perpendicular to the radius.

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So it would be that
magnitude right over here.

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This is what we define
as angular momentum.

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Now what I will tell you here is,

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just as in the absence of a net force,

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momentum is constant.

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We know, and I haven't
shown it to you yet,

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I haven't proven it to
you yet mathematically,

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but in the absence of torque,

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so if torque is equal to zero,

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we'll do torque in pink.

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If torque is equal to zero,

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if there's no net torque going on here,

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if the magnitude of
torque is equal to zero,

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then we will have no change.

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No change in angular momentum.

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And we will look at that
mathematically in a few seconds.

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But just from this

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there's a very interesting
thing that arises.

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And something that you might have observed

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at even the Olympics or in other things.

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And this is the idea that you can,

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by changing your radius,

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you could actually change
your tangential velocity.

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And as we've seen in previous videos,

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tangential velocity is closely related to

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angular velocity.

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So let's explore that a little bit.

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So when we write it in the world where,

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well actually you see
it straight out of this,

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if L is constant, if r went down,

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so let me write this down.

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So let me rewrite it over here.

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So L, whoops.

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L is equal to mass times
tangential velocity,

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or actually well yeah,
tangential velocity,

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or the velocity that's
perpendicular to the radius,

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times the radius.

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Now what happens,

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if we assume that this is constant,

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if we assume that there's no torque,

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so we're in this world.

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So this over here is going to be constant.

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So what happens if we were to reduce r?

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Somehow this wire started
to reel in a little bit

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or started to wrap around here,

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and that's actually a reasonable thing,

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you could imagine as it rotates
it starts to wrap around

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this thing so the wire gets shorter.

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So if r goes down,

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and this is constant, the
mass isn't going to change,

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Well if L is constant,
mass isn't changing,

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r is going down, tangential velocity,

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or the velocity that's
perpendicular to the radius

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is going to go up.

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And if we wanna think about it,

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we can think about it in
terms of angular velocity,

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we know that angular velocity,

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which we would measure
in radians per second,

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we would use the symbol omega,

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omega is defined,

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and we go into much more
depth in this in other videos,

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as tangential velocity,

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the magnitude of the velocity that is

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perpendicular to the radius,

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divided by the radius.

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Or if you solve for tangential velocity,

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you get v is equal to

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is equal to omega r.

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And so if you substitute back into this,

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really this definition
for angular momentum,

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you get angular momentum

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00:09:07,108 --> 00:09:11,725
is equal to mass times this times r.

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So mass times, I'm just
substituting for velocity here,

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times omega r,

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times r.

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00:09:22,087 --> 00:09:26,621
Which of course is just omega r squared.

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So once again we do the same exercise.

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If our radius goes down,

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what happens to our angular velocity?

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Remember we could measure this in

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00:09:34,153 --> 00:09:36,449
angles per second or radians per second,

212
00:09:36,449 --> 00:09:38,688
well if this is constant,

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00:09:38,688 --> 00:09:40,896
remember we're assuming
that there's no net torque

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being applied to the system,

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00:09:42,194 --> 00:09:45,521
so we're still in this
world right over here.

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If we assume that this
thing isn't changing,

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00:09:47,864 --> 00:09:49,777
but the radius were to change,

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00:09:49,777 --> 00:09:52,022
what's going to happen to omega?

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Well omega is going to go,

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omega is going to go up.

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Now likewise, if the radius got longer,

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So the radius got longer,

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00:10:03,025 --> 00:10:04,589
what's going to happen to omega?

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00:10:04,589 --> 00:10:06,217
Omega is going to go,

225
00:10:06,217 --> 00:10:08,169
omega is going to go down.

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00:10:08,169 --> 00:10:10,305
So if you reduce the radius

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you're going to start spinning faster

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00:10:12,313 --> 00:10:13,929
if you increase that radius,

229
00:10:13,929 --> 00:10:15,903
you're gonna start spinning slower.

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And you have seen this,

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or I think there's a high likelihood

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that you have seen this

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00:10:19,878 --> 00:10:23,160
probably at the Olympics when
you have seen figure skaters.

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Where they might start spinning

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00:10:24,866 --> 00:10:26,377
and they have their arms out.

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00:10:26,377 --> 00:10:27,498
So when their arms are out

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you could say that their
radius is further out.

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00:10:29,218 --> 00:10:30,413
And obviously a figure skater is

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00:10:30,413 --> 00:10:33,514
a much more complex
system than a point mass.

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00:10:33,514 --> 00:10:34,686
You could imagine a figure skater is

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a bunch of point masses.

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Well you could just model a figure skater

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00:10:39,303 --> 00:10:41,105
as a huge number of point masses

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00:10:41,105 --> 00:10:41,895
at different radii,

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00:10:41,895 --> 00:10:44,311
and you would wanna sum
up their angular momentum,

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00:10:44,311 --> 00:10:47,428
but the essence of what is happening is

247
00:10:47,428 --> 00:10:49,605
is when her arm is out

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00:10:49,605 --> 00:10:52,509
the average radius when you're calculating

249
00:10:52,509 --> 00:10:55,874
all of the point masses in
her arms and all the rest,

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00:10:55,874 --> 00:10:58,066
the average radius is higher.

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00:10:58,066 --> 00:10:59,938
And then when she pulls them in,

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00:10:59,938 --> 00:11:02,238
when she pulls them in
that radius goes down,

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00:11:02,238 --> 00:11:05,730
and her angular velocity goes up.

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And you see that. They start spinning,

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and then without applying any torque,

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00:11:08,957 --> 00:11:10,277
when they pull their arms in

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00:11:10,277 --> 00:11:11,805
they start spinning faster.

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00:11:11,805 --> 00:11:13,070
And then if they push their arms out,

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00:11:13,070 --> 00:11:14,837
once again without applying any torque,

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00:11:14,837 --> 00:00:00,000
they start spinning slower.