1
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Let's say I have
some object that's

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traveling in a circular
path just like this.

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And what I've drawn here
is its velocity vector

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00:00:07,490 --> 00:00:10,140
at different points
along that path.

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And so this right over
here is going to be v1,

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velocity vector 1.

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This is going to be
velocity vector 2.

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And this right over here is
going to be velocity vector 3.

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And what we're going to
assume, in this video,

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is that the magnitude of these
velocity vectors is constant.

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Or another way to think about it
is that the speed is constant.

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So I'll just say lowercase
v without the arrow on top--

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so this is going to be a scalar
quantity-- I'll call this

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the speed.

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Or you could call this the
magnitude of these vectors.

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And this is going
to be constant.

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So this is going to be equal
to the magnitude of vector

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1, which is equal to the
magnitude of vector 2.

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The direction is
clearly changing,

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but the magnitude is going
to be the same, which

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is equal to the
magnitude of vector 3.

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And we're going to assume
that it's traveling in a path,

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in a circle with radius r.

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And what I'm going
to do is, I'm going

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to draw a position
vector at each point.

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So let's call r1-- actually I'll
just do it in pink-- let's call

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r1 that right over there.

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That's position vector r1.

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That is position vector r2.

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So the position is
clearly changing.

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That's position vector r2.

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And that is position vector r3.

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But the magnitude of
our position vectors

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are clearly the same.

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And I'm going to call the
magnitude of our position

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vectors r.

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And that's just the
radius of the circle.

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It's this distance
right over here.

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So r is equal to
the magnitude of r1,

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which is equal to the
magnitude of r2, which

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is equal to the magnitude of r3.

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Now what I want to
do, in this video,

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is prove to you visually,
that given this radius

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and given this speed,
that the magnitude

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of the centripetal
acceleration-- and I'll

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just write that as a sub c,
I don't have an arrow on top,

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so this is a scalar quantity.

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So the magnitude of the
centripetal acceleration

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is going to be equal
to our speed squared,

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our constant speed
squared, divided

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by the radius of the circle.

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I want you to feel good
that this is indeed

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the case by the
end of this video.

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And to understand
that, what I want to do

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is I want to re-plot
these velocity

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vectors on another
circle and just

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think about how the vectors
themselves are changing.

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So let's copy and paste this.

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So let me copy and paste v1.

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So copy and paste.

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So that is v-- actually I
want to do it from the center

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--so that is v1.

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Then let me do the
same thing for v2.

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So let me copy and paste it.

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That is v2.

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And then let me
do it also for v3.

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I'll just get the vector part;
I don't have to get the label.

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So copy and paste it.

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And that right over
there is vector v3.

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And let me clean
this up a little bit.

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So that's clearly v2.

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I don't think we have
to label anymore.

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We know that v2 is in orange.

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And what is the
radius of this circle

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going to be right over here?

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Well, the radius
of this circle is

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going to be the magnitude
of the velocity vectors.

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And we already know the
magnitude of the velocity

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vectors is this quantity
v, this scalar quantity.

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So the radius of this circle is
v. The radius of this circle,

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we already know, is equal to r.

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And just as the
velocity vector is

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what's giving us the
change in position

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over time, the change in
position vector over time,

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what's the vector
that's going to give us

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the change in our
velocity vector over time?

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Well, that's going to be
our acceleration vectors.

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So you will have
some acceleration.

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We'll call this a1.

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We'll call this a2.

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And I'll call this a3.

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And I want to make
sure that you get

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the analogy that's
going on here.

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As we go around in this
circle, the position

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vectors first they point out
to the left, then the upper,

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kind of in a maybe
11 o'clock position,

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or I guess the top left,
and then to the top.

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So it's pointing in these
different directions

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like a hand in a clock.

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And what's moving it
along there is the change

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in position vector
over time, which

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are these velocity vectors.

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Over here, the velocity
vectors are moving around

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like the hands of a clock.

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And what is doing
the moving around

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are these acceleration vectors.

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And over here, the
velocity vectors

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are tangential to the
path, which is a circle.

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They're perpendicular
to a radius.

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And you learned
that in geometry--

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that a line that is
tangent to a circle

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is perpendicular to a radius.

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And it's also going to be the
same thing right over here.

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And just going back
to what we learned

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when we learned
about the intuition

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of centripetal acceleration, if
you look at a1 right over here,

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and you translate this vector,
it'll be going just like that.

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It is going towards the center.

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a2, once again, is going
towards the center.

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a3, once again, if
you translate that,

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that is going
towards the center.

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So all of these are actually
center-seeking vectors.

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And you see that
right over here.

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These are actually centripetal
acceleration vectors

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right over here.

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Here we're talking about
just the magnitude of it.

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And we're going to
assume that all of these

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have the same magnitude.

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So we're going to
assume that they all

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have a magnitude of
what we'll call a sub c.

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So that's the magnitude.

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And it's equal to
the magnitude of a1.

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That vector, it's equal
to the magnitude of a2.

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And it's equal to
the magnitude of a3.

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Now what I want to think
about is how long is it

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going to take for
this thing to get

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from this point on the circle
to that point on that circle

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right over there?

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So the way to think about it
is, what's the length of the arc

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that it traveled?

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The length of this arc that
it traveled right over there.

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That's 1/4 around the circle.

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It's going to be 1/4
of the circumference.

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The circumference is 2 pi r.

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It Is going to be 1/4 of that.

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So that is the
length of the arc.

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And then how long will
it take it to go that?

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Well, you would
divide the length

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of your path divided by the
actual speed, the actual thing

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that's nudging it
along that path.

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So you want to divide that by
the magnitude of your velocity,

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or your speed.

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This is the magnitude of
velocity, not velocity.

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This is not a vector right
over here, this is a scalar.

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So this is going to be the
time to travel along that path.

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Now the time to
travel along this path

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is going to be the exact
same amount of time

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it takes to travel along this
path for the velocity vector.

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So this is for the position
vector to travel like that.

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00:07:05,310 --> 00:07:07,590
This is for the velocity
vector to travel like that.

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So it's going to be
the exact same T.

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And what is the
length of this path?

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And now think of it in the
purely geometrical sense.

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We're looking at a circle here.

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The radius of the circle
is v. So the length

167
00:07:19,030 --> 00:07:22,640
of this path right over
here is going to be 1/4.

168
00:07:22,640 --> 00:07:25,070
It is going to be-- I'll
do it in that same color

169
00:07:25,070 --> 00:07:28,530
so you see the analogy--
it's equal to 1/4 times

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00:07:28,530 --> 00:07:30,180
the circumference of the circle.

171
00:07:30,180 --> 00:07:33,010
The circumference of
this circle is 2 pi times

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00:07:33,010 --> 00:07:35,730
the radius of the
circle, which is v.

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Now what is nudging
it along this circle?

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What is nudging it
along this path?

175
00:07:40,700 --> 00:07:43,350
What is the analogy for
speed right over here?

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00:07:43,350 --> 00:07:45,710
Speed is what's nudging it
along the path over here.

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00:07:45,710 --> 00:07:48,250
It is the magnitude of
the velocity vector.

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00:07:48,250 --> 00:07:50,750
So what's nudging it along
this arc right over here

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00:07:50,750 --> 00:07:53,530
is the magnitude of the
acceleration vector.

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00:07:53,530 --> 00:07:58,110
So it is going to be a sub c.

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00:07:58,110 --> 00:08:00,400
And these times are going
to be the exact same thing.

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00:08:00,400 --> 00:08:02,191
The amount of time it
takes for this vector

183
00:08:02,191 --> 00:08:04,060
to go like that, for
the position vector,

184
00:08:04,060 --> 00:08:06,670
is the same amount of
time it takes the velocity

185
00:08:06,670 --> 00:08:08,640
vector to go like that.

186
00:08:08,640 --> 00:08:11,310
So we can set these 2
things equal each other.

187
00:08:11,310 --> 00:08:19,030
So we get, on this side,
we get 1/4 2 pi r over v

188
00:08:19,030 --> 00:08:28,730
is equal to 1/4 2 pi v over the
magnitude of our acceleration

189
00:08:28,730 --> 00:08:29,900
vector.

190
00:08:29,900 --> 00:08:31,760
And now we can simplify
it a little bit.

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00:08:31,760 --> 00:08:33,230
We can divide both sides by 1/4.

192
00:08:33,230 --> 00:08:34,150
Get rid of that.

193
00:08:34,150 --> 00:08:37,030
We can divide both sides
by 2 pi, get rid of that.

194
00:08:37,030 --> 00:08:37,960
Let me rewrite it.

195
00:08:37,960 --> 00:08:41,830
So then we get r/v
is equal to v over

196
00:08:41,830 --> 00:08:43,510
the centripetal acceleration.

197
00:08:43,510 --> 00:08:45,370
And now you can cross multiply.

198
00:08:45,370 --> 00:08:50,020
And so you get v
times v. So I'm just

199
00:08:50,020 --> 00:08:52,490
cross multiplying right
over here. v times v,

200
00:08:52,490 --> 00:08:56,147
you get v squared, is
equal to a c times r.

201
00:08:56,147 --> 00:08:58,730
And cross multiplying, remember,
is really just the same thing

202
00:08:58,730 --> 00:09:01,490
as multiplying both sides
by both denominators,

203
00:09:01,490 --> 00:09:03,620
by multiplying both
sides times v and ac.

204
00:09:03,620 --> 00:09:06,879


205
00:09:06,879 --> 00:09:08,170
So it's not some magical thing.

206
00:09:08,170 --> 00:09:11,900
If you multiply both sides times
v and ac, these v's cancel out.

207
00:09:11,900 --> 00:09:13,700
These ac's cancel out.

208
00:09:13,700 --> 00:09:18,360
You get v times v is v squared,
is equal to a sub c times r.

209
00:09:18,360 --> 00:09:22,270


210
00:09:22,270 --> 00:09:23,990
And now to solve
for the magnitude

211
00:09:23,990 --> 00:09:26,300
of our centripetal
acceleration, you

212
00:09:26,300 --> 00:09:30,290
just divide both sides by r.

213
00:09:30,290 --> 00:09:32,030
And you are left
with-- and I guess

214
00:09:32,030 --> 00:09:35,740
we've earned a drum
roll now-- the magnitude

215
00:09:35,740 --> 00:09:37,800
of our centripetal
acceleration is

216
00:09:37,800 --> 00:09:40,470
equal to the magnitude,
our constant magnitude

217
00:09:40,470 --> 00:09:41,410
of our velocity.

218
00:09:41,410 --> 00:09:43,710
So this right here
is our speed, divided

219
00:09:43,710 --> 00:09:46,370
by the radius of the circle.

220
00:09:46,370 --> 00:00:00,000
And we're done!