- [Instructor] Things in the universe
don't just shift around,
they also rotate.
And so what we're going to do in this video
is start to think about rotations and rotational motion.
And I'm intentionally continuing to spin this
because I find it hypnotic.
But the question is,
well, how do we start to quantify these things,
measure them, and describe them
so that we can predict what will happen?
Well, what if we introduce a notion of angular displacement?
We have displacement, which is a shift in position.
But what if you have a change in angle?
That seems like it would describe something.
For example, right here, my angle would be zero radians.
And then if I move counterclockwise
pi over two radians,
now my angle is pi over two.
So why not define
going from there to there
as an angular displacement of pi over two radians?
And if I were to displace by pi over two again,
I would go over here.
And if I were to go negative pi radians,
well, that would be just like that.
Negative pi over two from there
would get me right over there.
That seems like a fairly intuitive thing.
And you might be saying y is positive counterclockwise,
and y is negative clockwise?
Well, that just tends to be the conventions
that we use for angles.
So there you have it.
We've already built our foundation
for describing rotational motion.
We could have an idea of angular displacement,
angular displacement,
which we can define more formally
just the way we just described it.
We could say, well, look,
this is going to be our change in angle,
so change in, Greek letter delta,
and angle, we use the Greek letter theta,
which we've been using since geometry or trigonometry class.
And so our angular displacement you can define as
your final angle minus your initial angle.
And we could say that this is a vector quantity
because you can either go counterclockwise,
in which case this is going to be a positive quantity.
Or you could go clockwise,
in which case this is going to be a negative quantity.
Well, that's all nice and fair,
but I know what some of you are thinking.
It's nice to be able to see how much your angle has changed,
but isn't it much more interesting to also describe
how fast that angle is changing?
For example, here it's changing quite slowly.
Let's say it takes us three seconds
to have an angular displacement of two pi.
One Mississippi, two Mississippi, three Mississippi.
That feels very different,
a lot slower than if I did that in one second.
One Mississippi, two Mississippi.
So what if we were to have something
that would describe our rate of change of angle?
Well, we could borrow some terminology
that we've used in the past.
Instead of calling that velocity,
we could call that angular velocity,
angular
velocity.
And how would you define that?
Pause this video, think about that.
Well, angular velocity, you could just say that is our
change in angle
over a certain amount of time.
If I change two pi radians
in one second,
well, that's going to be a lot slower
than if I change two pi radians in half a second.
And the letter we use for angular velocity,
the convention, is the Greek letter omega,
which looks like a fancy lowercase w.
Or at least this right over here is the lowercase omega,
which looks like a fancy lowercase w,
which we use for angular velocity.
And this, too, is a vector quantity
because it's measuring a rate of another vector quantity.
And so just to hit the point home,
just to review it all, make it all jell in your mind,
angular displacement,
it's nothing more than our change in angle.
If our change in angle is counterclockwise, like this,
it's going to be positive.
So let's say,
so an angular displacement, we might start here.
We don't always start at zero radians.
So let's say we start at pi over two.
An angular displacement of pi over four
would take us right over there.
An angular displacement from there
of negative pi over two,
might take us right over there.
And if we cared about angular velocity,
if we said, hey, we are going to go
pi radians every second,
well, then that would be like this.
It would be one second, two seconds,
three seconds, four seconds.
If we were to say we'd do two pi radians every second,
then that would be twice as fast
in terms of angular velocity.
One Mississippi, two Mississippi, three Mississippi,
four Mississippi, five Mississippi.
So there you have it.
And in the next few videos
and in the practice exercise on Khan Academy,
we will get even more familiar with these ideas
and see how powerful they are
for describing rotational motion in the universe.