Angular motion variables

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. 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 our 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 0 radians, and then, if I move counterclockwise π/2 radians, now my angle is π/2. So why not define going from there to there as an angular displacement of π/2 radians? If I were to displace by π/2 again, I would go over here. If I go negative π radians, well, that would be just like that. Negative π/2 from there would get me right over there. That seems like a fairly intuitive thing.

You might be saying, why is positive counterclockwise, and why 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 can 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. Our angular displacement can be defined as your final angle minus your initial angle. 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 2π. One Mississippi, two Mississippi, three Mississippi— that feels very different. A lot slower than if I did that in one second: one Mississippi, two.

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 2π radians in 1 second, well, that's going to be a lot slower than if I change 2π radians in half a second. 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 we use for angular velocity. This too is a vector quantity because it's measuring a rate of another vector quantity.

So just to hit the point home, just to review it all, make it all gel in your mind: angular displacement is 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 an angular displacement—we might start here. We don't always start at 0 radians, so let's say we start at π/2. An angular displacement of π/4 would take us right over there. An angular displacement from there of negative π/2 might take us right over there.

And if we cared about angular velocity, if we said, "Hey, we are going to go π 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 would do 2π 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! In the next few videos and in the practice exercises 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.