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- [Voiceover] So right over here

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like I've done in previous videos,

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we have a diagram of a mass.

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We really should conceptualize
it as a point mass

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although it doesn't look like a point,

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it looks like a circle.

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But imagine a point mass here

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and it's tethered to something.

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It's kind of it's tied
to a massless string,

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a theoretical massless
string right over here.

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Where functionally a mass of string

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and it's kind of nailed down.

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And let's say it's on
a frictionless surface

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and let's say it has some velocity.

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And right here we have the
magnitude of its velocity

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in the direction that is perpendicular

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to the wire that is holding it,

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or I guess you can say perpendicular

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to the radial direction.

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Now based on that, we've had a definition

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for angular momentum.

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The magnitude of angular momentum

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is going to be equal to the mass,

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times this velocity, times the radius.

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And you could also view that

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and this is kind of always
kind of tying the connections

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between you know, translational notions

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and rotational notions.

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We can see that angular momentum

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could be the same thing.

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Well, the mass times its velocity

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you could do that as the
translational momentum row

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in the magnitude of translational momentum

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in this direction times r.

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So once again, we took
the translational idea,

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multiplied it by r

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and we're getting the rotational idea,

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the angular momentum versus
just the translational one.

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And we can also think about it

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in terms of angular velocity.

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Now this comes straight out of the idea

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that this v is going
to be equal to omega r.

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So you do that substitution,

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you get this right over here.

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Now, in previous videos we said okay.

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Like based on this and based on the idea

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that if torque is held constant

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then this does not change.

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You can describe or you could predict

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the type of behavior, explain the behavior

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that you might see.

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The figure skating competition

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where if someone pulls their
arms in while they're spinning

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and they're not, you know,
applying anymore torque to spin,

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and if they pull their arms in,

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well this thing is going to be constant

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because there's no torque being applied.

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Well, their mass isn't going to change

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so they'll just spin faster.

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When you do the opposite,

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the opposite would be happening.

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But you might have been left

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a little bit unsatisfied
when we first talked about it

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because I just told you that.

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I said, hey look, if torque is,

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if there's no net torque

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then angular momentum is constant

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and then you have this thing happening.

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But let's dig a little bit deeper

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and look at the math of it.

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So you feel good that
that is actually the case.

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So let's go back, let's go
to the definition of torque.

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And so the magnitude of torque,

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I'll focus on magnitudes in this video.

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The magnitude of torque
is going to be equal to,

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it's going to be equal to
the magnitude of the force

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that is in this perpendicular
direction, times r.

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Times r.

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Now what is this, this force?

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Well, this is just going to be equal to

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the mass force, f equals ma.

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So this is going to be mass times

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the acceleration in this direction

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which we could view as,

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which we could view as the change

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in this velocity over time,

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and we're talking about magnitudes.

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I guess you could say, it's through

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the magnitude of velocity
in that direction.

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And then of course we have times r.

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Times r.

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Now if we multiplied both
sides of this times delta t,

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we get and actually we do
tau in a different color.

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We do torque in green.

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We get torque times delta t.

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Torque times delta t is equal to,

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is equal to mass times delta v.

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Delta v in that perpendicular
direction times r.

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Well, what's this thing going to be?

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What's this?

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Well, that's just change
in angular momentum.

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So this is just going to be
change in angular momentum.

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And there's a complete analogy

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to what you might remember

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from kind of the translational world.

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The translational world
you have this notion,

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if you take your force

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and you multiply it times

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how long you're applying the force,

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should do this in a different color.

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So we multiply it by how long
you are applying the force.

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So this quantity we often call is impulse.

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Impulse.

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That's going to be equal to

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your change in translational momentum.

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Your change in translational momentum.

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And if you have no force

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then you're not gonna have
any change in momentum

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or you're gonna have your
conversation of momentum,

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or it's not gonna change,

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it's just going to be conserved.

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And then you can do all sorts of neat,

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you know, predicting
where your ball might go

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or whatever else.

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We have the same analogy here.

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The analog for force
in the rotational world

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is torque, it's obviously this force times

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the kind of radial, times
that radial distance.

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But if you take torque times

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how long you're applying that torque,

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that's going to be your
change in angular momentum.

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So if you're not applying any torque,

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if your torque is zero,

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if your torque is equal to zero,

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well that means that your delta L is zero.

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Your angular momentum is not changing

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or you could say your
angular momentum is constant.

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So if you don't apply any torque,

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like once this figure
skater is already spinning

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and he or she, she's
not pushing to get more,

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to spin so she's not applying

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or he's not applying more torque,

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well then, they're angular
momentum's going to be constant

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but they can change their rate of spinning

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by changing r, by changing how far

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in or out r is,

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or how far enough the masses are.

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And obviously I said in the last video,

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a figure skater is a much
more complicated system

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than a point mass tethered to, you know,

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tethered to a rope.

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You could view a figure skater

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as kind of being modeled
by a bunch of point masses

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but hopefully this gives you the idea.