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- [Instructor] So in the previous video,

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we defined all the new
angular motion variables

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and we made an argument
that those are more useful

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in many cases to use than
the regular motion variables

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for things that are rotating in a circle.

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Since every point on the
string in tennis ball,

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let's say this is a tennis
ball you tied a string to

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and you're whirling around in a circle.

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Every point on the string
including the tennis ball

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will have the same angular displacement,

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angular velocity and angular acceleration.

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But even though using
angular motion variables

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is more convenient for these
rotational motion problems,

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it's also really important to know

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how to translate those
angular motion variables

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back into the regular motion variables.

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So that's what I wanna
show you in this video

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how to translate angular motion variables

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back into regular motion variables.

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So let's do this.

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The simplest possible
angular motion variable

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was the angular displacement

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because that just represented

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how much angle an object
has rotated through.

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So let's say it rotated through this much.

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We represented the angular
displacement with a delta theta

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and we call it the angular displacement.

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In Physics, we typically choose
to measure this in radians

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for a reason and I'll
show you in just a second.

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Now, how would we convert this

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into a regular motion variable?

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What regular motion
variable would that be?

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If I were to come at
this for the first time,

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I'd be like all right, this
is the angular displacement.

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Let's figure out how to relate it

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to the regular displacement

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but that would be weird.

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Because just think about it,

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the regular displacement for the ball

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that started over here
and made it over here

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would be from this point to that point,

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that would be the regular
displacement of the ball,

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the regular linear
displacement of the ball.

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That's a little weird.

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I don't wanna show you how to find that

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for one, you have to
use the law of cosines.

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That's a little more in depth

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than I'd wanna get to in this video.

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For two, the better reason

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this isn't all that useful in turns out.

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There's a much more useful quantity

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that would tell you how far the ball went.

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That's the arc length of the ball.

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So the ball traced out
of path through space

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around the circle.

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We call this the arc
length that it turns out

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this is much more useful
in a variety of problems.

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Good news is it's much easier to find

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than that regular displacement.

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So this is the arc length.

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People vary on what letter to use here.

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I've seen an l but most math books use s

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so we'll just use s as well.

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You might think this is
hard to find but it's not.

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In fact, if we use radians and
this is why we use radians,

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it's extremely easy to find.

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If we wanted to find the arc
length of this tennis ball,

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we're just gonna take the
radius of the circular path

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that tennis ball is tracing out.

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So in this case it'd be
the length of the string.

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We take that radius.

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If we're in radians,

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we just multiply by the
angular displacement.

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This is why the radians are so convenient.

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We just take that measurement in radians

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multiplied by the radius
of the circular path

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the object is tracing out.

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You get the arc length which
is the number of meters

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along this path that
the object has traveled.

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If that seems miraculous, it really isn't.

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I mean the reason why this works so well

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because this is how
the radian was defined.

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One radian is defined to be the angle

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through which you have to travel

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so that the arc length is equal

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to the radius of that circle.

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So this isn't a surprise.

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This was selected and
defined strategically

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so that we can use this unit

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and we get a really easy way to convert

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between the angular displacement,

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how many radians something
has rotated through,

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and how many meters it
has actually traveled

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through its arc.

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So this arc length is
gonna have units of meters

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as long as we measure
the radius in meters.

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All right, so that's one relationship

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between angular displacement,

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how much angles something
has rotated through,

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and how many meters it
has actually traveled.

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The next relationship I
wanna talk about relates

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the angular velocity to
the regular velocity.

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So remember in the previous video,

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we defined the angular velocity

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to be the angular displacement per times.

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So this is the rate at
which something is rotating

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through a certain amount of angle

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and the letter we use
to represent velocity

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was the Greek letter omega.

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So this angular velocity
represented the rate

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at which something in a circle.

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So it's rotating slowly.

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It's gonna have a small angular velocity.

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If it's rotating quickly,

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it's gonna have a large angular velocity.

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Obviously the speed and
the angular velocity

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are gonna be related because the higher

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the angular velocity,
the higher the speed.

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But, what is that relationship?

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How would we get from angular velocity

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to the regular velocity?

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Well it's not actually that hard at all

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because all we need to
do is turn this number

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of radians per second
into meters per second

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and I can do that.

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If multiply both sides of
this equation down here by R,

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I'll get R times omega is gonna equal

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R times delta theta

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and I still have to divide by delta t.

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So you just multiply both
sides of this equation by R.

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But look what I get.

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R times delta theta is
just the arc length.

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So this whole side over here is just

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how many meters that object has traveled

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around the edge of the circle

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divided by the time that it took.

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But that is just the speed.

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This arc length is just the
distance the object has traveled

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and the time is the time that it took

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and distance per time is just speed.

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So this is the speed of the object.

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I'm gonna write that as v

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even though it's not velocity.

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This is not a vector and it's not velocity

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because think about it,

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this arc length isn't displacement.

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This was the distance the object traveled.

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Distance per time is the speed.

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Displacement per time is the velocity.

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We didn't use displacement.

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Displacement was this weird one.

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We didn't wanna deal with that.

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So since we're choosing
to deal with arc length

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which is distance, what
we're gonna do is relate

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the angular velocity into the speed.

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Now we have that relationship.

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Look at this.

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This is R the radius
times the angular velocity

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equals the speed of the object.

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So this is the relationship

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between the angular
velocity and the speed.

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The speed of the object
is gonna equal the radius

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of the circular path the
object is traveling in

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times the angular velocity.

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I should box these.

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These are important.

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This arc length formula was how you relate

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the number of radians and
object has rotated through

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to how much arc length it's traveled,

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i.e. how much distance it's gone through.

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In this formula down here relates

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the angular velocity omega,

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the number of radians per second

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something has rotating with

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to how many meters per
second it's traveling.

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In other words, how many meters
per second it's tracing out

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along this arc length.

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So this is good.

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Now we know how to relate
the angular displacement

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to the distance the object has traveled

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and we know how to relate
the angular velocity

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to the speed of the object

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so you probably know what's coming next.

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We have to relate the angular acceleration

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to the regular acceleration.

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So we're called that
the angular acceleration

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which we represented
with a Greek letter alpha

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was defined to be the change

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in the angular velocity per times.

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It's the rate at which your
angular velocity was changing.

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So there's moving at the constant rate.

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You've got no angular acceleration

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because there's no change in omega.

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But if omega starts off slow

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and then it gets faster and faster,

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you do have angular acceleration.

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It's probably not a surprise

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that if you have angular acceleration,

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this ball is gonna have
regular acceleration too

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because it's speeding up
in its angular rotation.

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It's gonna be changing
its velocity as well.

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So how do we this?

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How do we relate the angular acceleration

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to the regular acceleration?

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Well the simplest thing to
try is we go work down here.

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We multiply both sides of
our equation by the radius

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and we found the relationship
the related speed

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to angular velocity.

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So let's try it again.

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Let's multiply both sides
of this equation by radius

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and see what we get.

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On the left hand side,
we can get the radius

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times the angular acceleration.

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That's gonna equal the
radius times the change

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in angular velocity
over the change in time.

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So all I've done here is multiply

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both sides of this equation.

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This definition of angular
acceleration by the radius.

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So let's see what we get
on the right hand side.

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We got R times delta omega.

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So this is really R times
the change in omega.

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Well that's just omega
final minus omega initial

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and then divide it by delta t

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so I can distribute this R and get that.

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This would equal R times omega final

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minus R times omega initial

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divided by the time that it took.

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But now look what happens.

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We've got R times omega final
and R times omega initial.

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We know what R times omega is.

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It's the speed, not the
velocity, but the speed.

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So I could rewrite this.

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I could say that this is
really the final speed

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minus the initial speed over the time

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that it took to change by that much speed.

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So this is, if I just
rewrite the left hand side,

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this is what R times alpha is equal to.

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Now if I were you, I'd be
tempted to just be like,

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oh look, we did it.

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That's the acceleration which
change in speed over time.

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But you gotta be careful.

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Acceleration, the true acceleration vector

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is the change in velocity per time,

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but these are not velocity vectors.

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These were speeds.

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So this isn't the true
acceleration vector.

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This is something different.

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This is the change in speed per time.

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So that's still an acceleration

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but it's not necessarily
the entire acceleration

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because there's two ways to accelerate.

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You can change your speed
or change your direction.

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Basically this acceleration we just found

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doesn't take into account any acceleration

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that's coming from
changing your direction.

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This is only the acceleration

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that's gonna be changing your speed.

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If I were you, I'd probably
be confused at this point.

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So let me try to show you what this means.

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If this ball is rotating in a circle

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just by the mere fact that the
ball is rotating in a circle,

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it has to be accelerating

263
00:09:39,684 --> 00:09:42,738
even if the ball isn't
speeding up or slowing down.

264
00:09:42,738 --> 00:09:44,430
There's got to be an acceleration

265
00:09:44,430 --> 00:09:45,603
because this ball is changing

266
00:09:45,603 --> 00:09:47,608
the direction of its velocity.

267
00:09:47,608 --> 00:09:48,659
These are gonna be a force.

268
00:09:48,659 --> 00:09:51,420
It's centripetal force, in this
case it would be detention.

269
00:09:51,420 --> 00:09:53,808
There's gotta be a
centripetal acceleration

270
00:09:53,808 --> 00:09:56,701
that's changing the
direction of the velocity.

271
00:09:56,701 --> 00:09:59,229
That is not this acceleration over here.

272
00:09:59,229 --> 00:10:01,111
This is a different acceleration.

273
00:10:01,111 --> 00:10:04,262
We know the centripetal
acceleration is directed inward.

274
00:10:04,262 --> 00:10:06,671
We already know how to find
this centripetal acceleration.

275
00:10:06,671 --> 00:10:09,011
Remember the formula for
centripetal acceleration

276
00:10:09,011 --> 00:10:12,252
is the speed squared
divided by the radius.

277
00:10:12,252 --> 00:10:14,798
This component, this
centripetal acceleration is

278
00:10:14,798 --> 00:10:16,459
the component of the acceleration

279
00:10:16,459 --> 00:10:19,756
that changes the
direction of the velocity.

280
00:10:19,756 --> 00:10:21,429
So I'm gonna say that again
because this is important.

281
00:10:21,429 --> 00:10:23,893
The centripetal acceleration
which you can find

282
00:10:23,893 --> 00:10:27,565
with v squared over R is the
component of the acceleration

283
00:10:27,565 --> 00:10:30,951
that changes the
direction of the velocity.

284
00:10:30,951 --> 00:10:32,631
If something is going in a circle,

285
00:10:32,631 --> 00:10:34,826
it must have centripetal acceleration.

286
00:10:34,826 --> 00:10:36,414
But this acceleration component

287
00:10:36,414 --> 00:10:37,814
that we found down here is different.

288
00:10:37,814 --> 00:10:39,688
This is what's changing the speed.

289
00:10:39,688 --> 00:10:42,197
You don't have to have this
if you're going in a circle.

290
00:10:42,197 --> 00:10:44,212
You could imagine
something going in a circle

291
00:10:44,212 --> 00:10:45,636
at a constant rate.

292
00:10:45,636 --> 00:10:48,436
If that's happening, it's
got centripetal acceleration

293
00:10:48,436 --> 00:10:51,028
but it doesn't have this thing down here

294
00:10:51,028 --> 00:10:53,544
because this thing we found R times alpha

295
00:10:53,544 --> 00:10:56,214
is the change in the speed
of the object per time.

296
00:10:56,214 --> 00:10:57,575
How would I draw that up here

297
00:10:57,575 --> 00:10:59,582
if I wanted to represent this a

298
00:10:59,582 --> 00:11:01,706
that we found down here visually up here?

299
00:11:01,706 --> 00:11:04,684
I'd draw tangential to
the direction of motion,

300
00:11:04,684 --> 00:11:06,871
i.e. tangential to the circle

301
00:11:06,871 --> 00:11:08,535
because components of acceleration

302
00:11:08,535 --> 00:11:11,658
that are directed
perpendicular to the velocity

303
00:11:11,658 --> 00:11:13,814
change the direction of the velocity

304
00:11:13,814 --> 00:11:15,917
but components of
acceleration that are directed

305
00:11:15,917 --> 00:11:18,757
parallel to the direction
of the velocity change

306
00:11:18,757 --> 00:11:22,167
the magnitude of the
velocity, i.e. the speed,

307
00:11:22,167 --> 00:11:24,347
to change the magnitude of the velocity,

308
00:11:24,347 --> 00:11:26,884
in other words to speed
something up or to slow it down.

309
00:11:26,884 --> 00:11:28,909
You need a component of that acceleration

310
00:11:28,909 --> 00:11:30,981
that's either in the directional motion

311
00:11:30,981 --> 00:11:33,045
or opposite of the directional motion.

312
00:11:33,045 --> 00:11:34,965
If it was opposite the directional motion,

313
00:11:34,965 --> 00:11:37,537
the acceleration would be
slowing the object down.

314
00:11:37,537 --> 00:11:39,929
If the component of
acceleration is in the direction

315
00:11:39,929 --> 00:11:42,585
of motion, then it's
speeding the object up.

316
00:11:42,585 --> 00:11:44,054
That's what we found down here.

317
00:11:44,054 --> 00:11:46,221
That's what this component
of acceleration is,

318
00:11:46,221 --> 00:11:48,551
R alpha which is why it's often called

319
00:11:48,551 --> 00:11:51,182
the tangential acceleration.

320
00:11:51,182 --> 00:11:52,078
So I'm gonna write that up here.

321
00:11:52,078 --> 00:11:56,362
The tangential acceleration
which is equal to R times alpha,

322
00:11:56,362 --> 00:11:58,987
the radius times the angular acceleration

323
00:11:58,987 --> 00:12:01,113
is the component of the acceleration

324
00:12:01,113 --> 00:12:04,125
that's changing the
magnitude of the velocity,

325
00:12:04,125 --> 00:12:06,151
i.e. it's changing the speed.

326
00:12:06,151 --> 00:12:08,195
In order to do that,
it's gotta be directed

327
00:12:08,195 --> 00:12:10,928
tangential to the direction of motion.

328
00:12:10,928 --> 00:12:13,420
That's what this R times alpha represents.

329
00:12:13,420 --> 00:12:14,294
So let me box that.

330
00:12:14,294 --> 00:12:15,127
That's important.

331
00:12:15,127 --> 00:12:18,535
This is the formula to find
the tangential acceleration.

332
00:12:18,535 --> 00:12:21,209
It doesn't give you
the total acceleration.

333
00:12:21,209 --> 00:12:23,726
We know that there's always
a component of acceleration

334
00:12:23,726 --> 00:12:25,327
that's acting centripetally

335
00:12:25,327 --> 00:12:26,976
if an object is going in a circle

336
00:12:26,976 --> 00:12:28,963
and you could that with v squared over R.

337
00:12:28,963 --> 00:12:32,489
But if the object going in
a circle is also speeding up

338
00:12:32,489 --> 00:12:35,538
not only going in a circle
but changing its speed,

339
00:12:35,538 --> 00:12:38,535
it's gonna also have this
component of acceleration

340
00:12:38,535 --> 00:12:40,561
which is the tangential acceleration.

341
00:12:40,561 --> 00:12:41,957
So at this point you might be confused

342
00:12:41,957 --> 00:12:44,087
like wait, we've got
tangential acceleration,

343
00:12:44,087 --> 00:12:45,259
we've got centripetal.

344
00:12:45,259 --> 00:12:47,264
Which one is the acceleration?

345
00:12:47,264 --> 00:12:51,079
Well, they're both just components
of the total acceleration

346
00:12:51,079 --> 00:12:53,454
which you could find if
you really wanted to.

347
00:12:53,454 --> 00:12:56,434
You could say that the
total acceleration squared.

348
00:12:56,434 --> 00:12:58,224
You could use the Pythagorean theorem

349
00:12:58,224 --> 00:13:01,285
because these are the two
perpendicular components

350
00:13:01,285 --> 00:13:03,390
of the total acceleration.

351
00:13:03,390 --> 00:13:05,038
We said the total acceleration squared

352
00:13:05,038 --> 00:13:08,284
would equal the tangential
acceleration squared

353
00:13:08,284 --> 00:13:11,012
plus the centripetal acceleration squared.

354
00:13:11,012 --> 00:13:14,108
What we would be finding is
the total acceleration squared

355
00:13:14,108 --> 00:13:15,926
which if you wanted a direction,

356
00:13:15,926 --> 00:13:18,040
the direction of that total acceleration

357
00:13:18,040 --> 00:13:19,631
would point this way somewhere.

358
00:13:19,631 --> 00:13:22,577
If you had centripetal acceleration inward

359
00:13:22,577 --> 00:13:24,606
and let's say the object was speeding up.

360
00:13:24,606 --> 00:13:26,398
So let's say it wasn't slowing down.

361
00:13:26,398 --> 00:13:27,712
So you didn't have this component.

362
00:13:27,712 --> 00:13:30,331
You've got this forwards
component and inwards component.

363
00:13:30,331 --> 00:13:33,273
The total acceleration would
be directed here somewhere.

364
00:13:33,273 --> 00:13:35,823
Since you could form a
right triangle out of this

365
00:13:35,823 --> 00:13:38,284
with these two sides,
you can imagine moving

366
00:13:38,284 --> 00:13:41,667
this centripetal acceleration
over to this side

367
00:13:41,667 --> 00:13:42,975
and you could find the hypothenuse

368
00:13:42,975 --> 00:13:45,134
which would be the total acceleration

369
00:13:45,134 --> 00:13:47,910
by just taking the tangential
acceleration squared

370
00:13:47,910 --> 00:13:50,244
plus the centripetal acceleration squared

371
00:13:50,244 --> 00:13:52,376
and then taking a square
root would give you

372
00:13:52,376 --> 00:13:55,095
the magnitude of the total acceleration.

373
00:13:55,095 --> 00:13:58,036
So recapping, there's two
components of acceleration,

374
00:13:58,036 --> 00:14:01,493
the tangential acceleration
which is R times alpha,

375
00:14:01,493 --> 00:14:04,087
either speeds an object
up or slows it down,

376
00:14:04,087 --> 00:14:06,677
the centripetal
acceleration works to change

377
00:14:06,677 --> 00:14:09,067
the direction of the motion of the object.

378
00:14:09,067 --> 00:14:11,400
You can relate the speed of an object

379
00:14:11,400 --> 00:14:14,533
to the angular velocity
by multiplying by R.

380
00:14:14,533 --> 00:14:17,265
You could relate the arc
length, i.e. the distance

381
00:14:17,265 --> 00:14:20,254
the object traveled around
this edge of the circle

382
00:14:20,254 --> 00:14:24,573
to the angular displacement
by also multiplying by R.

383
00:14:24,573 --> 00:14:27,119
So these three equations or how you relate

384
00:14:27,119 --> 00:00:00,000
the angular motion variable
to its linear counterpart.