Distance or arc length from angular displacement | AP Physics 1 | Khan Academy

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

is try to draw connections
between angular displacement

and notions of arc length
or distance traveled.

Right over here, let's
imagine I have some type

of a tennis ball or something,

and it is tethered with a
rope to some type of a nail.

If you were to try to
move this tennis ball,

it would just rotate around that nail.

It would go along this blue circular path.

Let's just say for the sake of argument

the radius of this blue
circle right over here,

let's say it is six meters.

You could view that as
the length of the string.

We know what's going on here.

Our initial angle, theta
initial, the convention is

to measure it relative
to the positive x-axis.

Our theta initial is pi over two.

Let's say we were to then
rotate it by two pi radians,

positive two pi radians.

We would rotate in the
counterclockwise direction two pi.

Then the ball would end up
where it started before.

Theta final would just
be this plus two pi,

so that would be five pi over two.

Of course, we just said that we rotated

in the positive
counterclockwise direction.

We said we did it by two pi radians,

so I kind of gave you the answer

of what the angular displacement is.

The angular displacement
in this situation,

delta theta, is going to
be equal to two pi radians.

The next question I'm going to ask you is

what is the distance the
ball would have traveled?

Remember, distance where we
actually care about the path.

The distance traveled would be essentially

the circumference of this circle.

Think about what that is,

and actually while you pause
the video and think about

what the distance the ball traveled is,

also think about what
would be the displacement

that that ball travels.

Well the easier answer, I'm
assuming you've had a go at it,

is the displacement.

The ball ends up where it
starts, so the displacement

in this situation is as.

We're not talking about
angular displacement.

We're just talking about regular
displacement would be zero.

The angular displacement
was two pi radians,

but what about the distance.

We'll denote the distance by S.

As we can see, as we'll see,

we can also view that as arc length.

Here the arc is the entire circle.

What is that going to be equal to?

Well, we know from
earlier geometry classes

that this is just going to be

the circumference of the circle

which is going to be equal
to two pi times the radius.

It's going to be equal to
two pi times six meters

which in this case is going to be 12 pi.

Our units, in this case,
is going to be 12 pi meters

would be the distance that we've traveled.

Now what's interesting right over here is

at least for this particular
case to figure out

the distance traveled to
figure out that arc length,

it looks like what we did is

we took our angular displacement,

in fact we took the magnitude
of our angular displacement,

you could just view that as
the absolute value of it,

and we multiplied it times
the radius of our circle.

If you view that as the
length of that string as R,

we just multiply that times R.

We said our arc length,
in this case, was equal

to the magnitude of our
change in displacement

times our radius.

Let's see if that is always true.

In this situation, I have
a ball, and let's say

this is a shorter string.

Let's say this string
is only three meters,

and its initial angle, theta
initial, is pi radians.

We see that as measured
from the positive x-axis.

Let's say we were to rotate it clockwise.

Let's say our theta final
is pi over two radians.

Theta final is equal
to pi over two radians.

Pause this video, and see if you can

figure out the angular displacement.

The angular displacement
in this situation is going

to be equal to theta
final which is pi over two

minus theta initial which
is pi which is going

to be equal to negative pi over two.

Does this make sense?

Yes, because we went clockwise.

Clockwise rotations by convention
are going to be negative.

That makes sense.

We have a clockwise rotation
of pi over two radians.

Now based on the information
we've just figured out,

see if you can figure out the arc length

or the distance that this tennis ball

at then end of the
string actually travels.

This tennis ball at the end
of the three meter string.

What is this distance
actually going to be?

Well, there's a couple of
ways you could think about it.

You could say, hey
look, this is one fourth

of the circumference of the circle.

You could just say,
hey, this arc length, S,

is just gonna be one fourth times two pi

times the radius of three
meters, times three.

This indeed would give
you the right answer.

This would be, what, two
over four is one half,

so you get three pi over two,
and we're dealing with meters,

so this would be three pi over two.

But let's see if this is
consistent with this formula

we just had over here.

Is this the same thing if we were to take

the absolute value of our displacement.

Let's do that.

If we were to take the
absolute value of our,

I'd should say our angular displacement,

so we take the absolute value
of angular displacement,

and we were to multiply it by our radius.

Well, our radius is three meters.

These would indeed be equal because this

is just going to be positive
pi over two times three

which is indeed three pi over two.

It looks like this formula
is holding up pretty well.

It makes sense because
what you're really doing

is you're saying, look,

you have your traditional
circumference of a circle.

Then, you're thinking
about, well, what proportion

of the circle is this arc length?

If you were to say, well,
the proportion is going to be

the magnitude of your
angular displacement.

This is the proportion of the
circumference of the circle

that you're going over.

If this was two pi, you'd
be the entire circle.

If it's pi, then you're
going half of the circle.

Notice, these two things cancel out.

This would actually give
you your arc length.

Let's do one more example.

Let's say in this situation, the string

that is tethering our
ball is five meters long,

and let's say our initial
angle is pi over six radians

right over here.

Let's say that our final angle is,

we end up right over there.

Our theta final is equal to pi over three.

Based on everything
we've just talked about,

what is going to be the
distance that the ball travels?

What is going to be the arc length?

Well, we could, first, figure
out our angular displacement.

Our angular displacement is
going to be equal to theta final

which is pi over three minus theta initial

which is pi over six which is going to be

equal to pi over six,
and that makes sense.

This angle right over
here, we just went through

a positive pi over six radian rotation.

We went in the counterclockwise direction.

Then, we just want to
figure out the arc length.

We just multiply that times the radius.

Our arc length is going
to be equal to pi over six

times our radius times five meters.

This would get us to
five pi over six meters,

and we are done.

Once again, no magic here.

It comes straight out of the idea

of the circumference of a circle.