WEBVTT

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- [Instructor] What we're
going to do in this video

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is look at the tangible example

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where we calculate angular velocity,

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but then we're going to
see if we can connect that

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to the notion of speed.

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So let's start with this
example where once again

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we have some type of a ball tethered

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to some type of center of
rotation right over here.

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Let's say this is connected with a string.

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And so if you were to
move the ball around,

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it would move along this blue
circle in either direction.

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Let's just say for the sake of argument,

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the length of the string is seven meters.

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We know that at time is
equal to three seconds,

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our angle is equal to,

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theta is equal to pi over two radians,

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which we've seen in previous videos.

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We would measure from the
positive x-axis, just like that.

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And let's say that at T
is equal to six seconds,

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T is equal to six seconds,
theta is equal to pi radians.

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And so after three seconds, the
ball is now right over here.

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And so if we wanted to actually
visualize how that happens,

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let me see if I can rotate
this ball in three seconds.

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So it would look like this.

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One Mississippi, two
Mississippi, three Mississippi.

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Let's do that again.

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It would be...

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One Mississippi, two
Mississippi, three Mississippi.

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So now that we can visualize

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or conceptualize what's going on,

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see if you can pause this
video and calculate two things.

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So the first thing that I
want you to calculate is

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what is the angular velocity of the ball?

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And actually, it would be the ball

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and every point on that string.

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What is that angular velocity
which we denote with omega?

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And then I want you to figure out

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what is the speed of the ball?

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So what is the speed?

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See if you can figure
out both of those things,

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and for extra points,
see if you can figure out

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a relationship between the two.

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Alright, well let's go
angular velocity first.

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I'm assuming you've had a go at it.

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Angular velocity, you might remember

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is just going to be equal
to our angular displacement,

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which we could say is delta theta,

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and it is a vector quantity.

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And we are going to divide
that by our change in time.

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So delta T.

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And so what is this going to be?

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Well this is going to be
our angular displacement.

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Our final angle is pi.

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Pi radians minus our initial
angle, pi over two radians.

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And then all of that's going
to be over our change in time,

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which is six seconds,
which is our final time

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minus our initial time,
minus three seconds.

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And so we are going to
get in the numerator,

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we have been rotated in
the positive direction,

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Pi over two radians.

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Because it's positive, we
know it's counterclockwise.

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And that happened over three seconds.

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And so we could rewrite this as

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this is going to be equal to pi over six.

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And let's remind
ourselves about the units.

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Our change in angle, that's
going to be in radians,

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and then that is going to be per second.

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So we're going pi over
six radians per second,

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and if you do that over three seconds,

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well then you're going to
go pi over two radians.

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Now with that out of the way,

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let's see if we can calculate speed.

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If you haven't done so
already, pause this video

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and see if you can calculate it.

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Well speed is going to
be equal to the distance

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the ball travels, and we've
touched on that in other videos.

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I encourage you to watch
those if you haven't already.

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The distance that we travel
we could denote with S.

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S is sometimes used to denote arc length,

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or the distance traveled right over here.

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So the speed is going to be our arc length

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divided by our change in time.

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Divided by our change in time.

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But what is our arc length going to be?

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Well we saw in a previous video

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where we related angular
displacement to arc length,

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or distance, that our arc
length is nothing more

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than the absolute value of
our angular displacement,

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of our angular displacement,
times the radius.

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Times the radius.

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And in this case, our radius
would be seven meters.

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So if we substitute all of this up here,

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what are we going to get?

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We are going to get that our speed,

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I'm writing it out because
I don't want to overuse,

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well I am overusing S,

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but I don't want people to get confused.

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Our speed is going to be equal
to the distance we travel,

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which we just wrote is the magnitude

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of our angular displacement.

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And this is all fancy notation,

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but when you actually apply it,
it's pretty straightforward.

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Times the radius of the circle.

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I guess you can say we
are traveling along.

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Let me write that in a different color.

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So times the radius.

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All of that over our change in time.

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All of that over our change in time.

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Well we could put in the
numbers right over here.

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We know that this is
going to be pi over two.

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You take the absolute value of that.

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It's still going to be pi over two.

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We know that our radius in this case

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is the length of that string.

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It is seven meters.

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And we know that our change in time here,

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we know that this over here
is going to be three seconds,

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and we can calculate everything.

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But what's even more
interesting is to recognize

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that what is this right over here?

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What is the absolute value
of our angular displacement

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over change in time?

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Well, this is just the absolute value

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of our angular velocity.

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So we could say that speed

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is equal to the absolute
value of our angular velocity.

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Absolute value of our angular
velocity, times our radius.

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Times our radius.

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And now so this is super useful.

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Our speed in this case,
is going to be pi over six

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radians per second.

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So pi over,

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pi over six.

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Times the radius.

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Times seven meters.

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Times seven meters.

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And so what do we get?

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We are going to get seven
pi over six meters per,

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meters per second, which will
be our units for speed here.

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And the reason why we're
doing the absolute value

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is 'cause remember, speed
is a scalar quantity.

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We're not specifying the direction.

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In fact, the whole time we're traveling,

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our direction is constantly changing.

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So there you have it.

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There's multiple ways to approach
these types of questions,

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but the big takeaway here is one,

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how we calculated angular velocity,

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and then how we can relate
angular velocity to speed.

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And what's nice is there's a
nice, simple formula for it.

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And all of this just came out of something

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that relates to what we
learned in seventh grade

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around the circumference of the circle,

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which we touched on in the video

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relating angular
displacement to arc length,

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or distance traveled.