Angular velocity and speed | Uniform circular motion and gravitation | AP Physics 1 | Khan Academy

What we're going to do in this video is look at a tangible example where we calculate angular velocity. But then, we're going to see if we can connect that to the notion of speed.

So let's start with this example, where once again we have some type of a ball tethered to some type of center of rotation right over here. Let's say this is connected with a string. If you were to move the ball around, it would move along this blue circle in either direction. Let's just say for the sake of argument the length of the string is 7 meters. We know that at time t is equal to 3 seconds, our angle theta is equal to pi over 2 radians, which we've seen in previous videos. We would measure from the positive x-axis just like that.

Now, let's say that at t is equal to 6 seconds, t is equal to 6 seconds, theta is equal to pi radians. So after three seconds, the ball is now right over here. If we wanted to actually visualize how that happens, let me see if I can rotate this ball in three seconds. It will look like this: one Mississippi, two Mississippi, three Mississippi. Let's do that again. It would be one Mississippi, two Mississippi, three Mississippi.

Now that we can visualize or conceptualize what's going on, see if you can pause this video and calculate two things. The first thing that I want you to calculate is what is the angular velocity of the ball. Actually, it would be the ball and every point on that string. What is that angular velocity, which we denote with omega? And then I want you to figure out what is the speed of the ball.

So what is the speed? See if you can figure out both of those things, and for extra points, see if you can figure out a relationship between the two.

All right, well, let's do angular velocity first. I'm assuming you've had a go at it. Angular velocity, you might remember, is just going to be equal to our angular displacement, which we could say is delta theta, and it is a vector quantity. We are going to divide that by our change in time, so delta t.

So what is this going to be? Well, this is going to be our angular displacement. Our final angle is pi radians minus our initial angle, pi over 2 radians. All of that is going to be over our change in time, which is 6 seconds (which is our final time) minus our initial time (3 seconds).

We are going to get in the numerator we have been rotated in the positive direction, pi over 2 radians. Because it's positive, we know it's counter-clockwise, and that happened over 3 seconds. So we could rewrite this as this is going to be equal to pi over 6. Let's remind ourselves about the units. Our change in angle is going to be in radians, and then that is going to be per second. So we're going pi over 6 radians per second.

If you do that over 3 seconds, well, then you're going to go pi over 2 radians. Now, with that out of the way, let's see if we can calculate speed. If you haven't done so already, pause this video and see if you can calculate it.

Well, speed is going to be equal to the distance the ball travels. We've touched on that in other videos. I encourage you to watch those if you haven't already. The distance that we travel, we could denote with s. S is sometimes used to denote arc length or the distance traveled right over here. So the speed is going to be our arc length divided by our change in time.

But what is our arc length going to be? Well, we saw in a previous video where we related angular displacement to arc length or distance that our arc length is nothing more than the absolute value of our angular displacement times the radius. In this case, our radius would be 7 meters.

So if we substitute all of this up here, what are we going to get? We are going to get that our speed, I'm writing it out because I don't want to overuse... well, I am overusing s, but I want people to get confused. Our speed is going to be equal to the distance we travel, which we just wrote as the magnitude of our angular displacement.

This is all fancy notation, but when you actually apply it, it's pretty straightforward times the radius of the circle. I guess you can say we are traveling along. So let me write that in a different color: times the radius. All of that over our change in time.

Well, we could put in the numbers right over here. We know that this is going to be pi over 2. You take the absolute value of that, it's still going to be pi over 2. We know that our radius in this case is the length of that string; it is 7 meters. We know that our change in time here is going to be 3 seconds.

We can calculate everything, but what's even more interesting is to recognize that what is the absolute value of our angular displacement over change in time? Well, this is just the absolute value of our angular velocity. So we could say that speed is equal to the absolute value of our angular velocity times our radius.

Now, so this is super useful. Our speed in this case is going to be pi over 6 radians per second. So pi over pi over 6 times the radius times 7 meters. What do we get? We are going to get 7 pi over 6 meters per second, which will be our units for speed here.

The reason why we're doing the absolute values is because remember speed is a scalar quantity; we're not specifying the direction. In fact, the whole time we're traveling, our direction is constantly changing.

So there you have it. There are multiple ways to approach these types of questions, but the big takeaway here is one: how we calculated angular velocity, and then how we can relate angular velocity to speed. What's nice is there's a simple formula for it. All of this just came out of something that relates to what we learned in seventh grade around the circumference of the circle, which we touch on in the video relating angular displacement to arc length or distance traveled.