Well, we've done a lot of work with how fast something moves,
let's see if we can work with how fast something spins.
Let's see what we can do.
Since we're going to be working with things spinning,
let me draw a circle.
Since things that spin go in circles.
And let me just draw the positive x-axis because it'll
come in handy in a second.
That's the positive x-axis.
And let's say that I have an object, and the circle is the
object's path.
So let's say this is the object.
And it's going around in a circle in a
counterclockwise direction.
Not squiggly counterclockwise, it's just
going around this way.
Let's say I wanted to figure out, or I wanted to quantify
how much, or how fast this thing is spinning.
So one thing that you're probably familiar with is
revolutions per second, or rotations per second.
So let's write that down, let's just say for the sake of
argument that this was moving at, I don't know, 1 revolution
per second.
So after 1 second it goes back, then another second.
So that's how fast it's spinning, 1
revolution per second.
1 revolution-- I'll just put rev-- per second.
So let's see if we can quantify that in angles, and
we'll do it in radians, but you could always convert it
back to degrees, if you want.
I don't know if you can see that line.
Let's just say that theta is the angle between the radius
from the center to that object, and
the positive x-axis.
So if this object is travelling at 1 revolution per
second, how many radians per second is it traveling?
Well, how many radians are there in a revolution?
Well there's 2 pi radians in a revolution, right?
1 go-around in a circle is 2 pi radians.
So we could say, so this equals 1 rev per second, times
2 pi radians per rev, right?
And then the revolutions will cancel out.
And you have 1 times 2 pi, so you have 2
pi radians per second.
So this equals 2 pi radians per second.
So that's interesting, we now know exactly after 5 seconds
how many radians it has gone.
Or after half a second, how many radians has it gone.
But that might be vaguely useful.
Let's see if we can somehow convert from this notion of
how fast something is spinning to its actual speed.
I was tempted to say velocity, but its velocity is always
changing, because the direction is always changing.
But the magnitude of the velocity is staying the same,
so its speed is staying the same.
But we'll say v for speed, because that's what they tend
to do in most formulas that you'll see.
So let's think about it this way, in 1 revolution-- so
there's a couple ways you can think about this, but as we go
1 revolution, how far has this object traveled?
Well, it's traveled the circumference of this circle.
And in order to know the circumference, we have to know
the radius of the circle.
So let's say that the radius is r-- let's say it's in
meters, r meters.
So how many meters will I travel in 1 second, then?
Well, you could do the same thing up here.
1 revolution per second, times 2 pi r, where r is the
radius-- whoops, 2 pi r, you can ignore that line-- meters
per revolution, that's just the circumference of the
thing, of the circle.
And that equals-- the revolutions cancel out-- 2 pi
r meters per second.
So it's interesting, given the radius and how many
revolutions per second, we can now figure out its velocity.
So this right here is how fast it's spinning, and this is the
object's actual speed, right?
And this term of how fast something's spinning, that's
called angular velocity.
And of course you know that the term for how fast
something is actually moving is velocity.
And just so you know, the term for angular velocity is this
curvy w, I think that's lower case omega,
that's angular velocity.
So in this case, angular velocity is equal to 2 pi
radians per second.
And what's the velocity equal to, or at least the magnitude
of the velocity-- I know the direction's always changing.
Well, we know that the velocity is equal to 2 pi r
meters per second.
So if we just ignore the units for a second, where do you see
the difference between the angular
velocity and the velocity?
The angular velocity in this case is 2 pi, and the
velocity is 2 pi r.
So in general, if you just multiply the angular velocity
times r, you get the velocity.
So angular velocity times the radius is equal to velocity.
Or you can divide both sides of that by r, and you get the
angular velocity is equal to the velocity
divided by the radius.
And this is a formula that you should know by heart, although
it's good to know where it came from.
I guess I did it this way to maybe give you an intuition,
because I always have to work with numbers.
Especially when I'm new to a concept-- so that's why I said
1 revolution per second, instead of just putting
everything as a variable-- but another way to think about it
is, what is the definition of a radian?
By definition, a radian-- if this angle is x radians, it's
an angle, and it also tells us that the arc that is kind of
projected by this angle, is equal to x radiuses.
So if each radius is 2 meters, it would be x times 2 meters.
So if this is x radians, then this is going to
be x times r meters.
And that actually comes from the definition of the radian.
And that might be more intuitive to you, than the
original explanation, or less, so hopefully one
of those two works.
But as you can see, if this angle is x, and this distance
is x times r, and if omega is change in that angle, over
change in time.
Then we know this is true too, that velocity is just change
in this, over change in time, right.
Velocity is change in-- the radius doesn't change-- change
in x times r, divided by change in time.
And we know once again that this is omega.
So another way we just showed again, that omega times the
radius is equal to the velocity.
Or the angular velocity times the radius is
equal to the velocity.
And this is a useful thing to learn, we'll see it in a
couple of things, when I do the proof for centripetal
acceleration in calculus, I'm going to use this fact.
And when I-- and actually I'm probably going to record that
video now-- I'm actually going to show you the law of
conservation of angular momentum, which is very
similar to the law of conservation of momentum, but
it deals with things spinning.
And this notion of angular velocity is
going to come in useful.
So this is the important takeaway, that w
equals v over r.
And hopefully my video has not confused you, and has shown
you that w, the rate at which the angle is changing, is
equal to the velocity of the object, or the magnitude of
the velocity, divided by the radius of the
circle that it's spinning.
I'll see you in the next video.