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# Angular velocity and speed

## Video transcript

- [Instructor] What we're going to do in this video is look at the 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. And so 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 seven meters. We know that at time is equal to three seconds, our angle is equal to, theta is equal to pi over two radians, which we've seen in previous videos. We would measure from the positive x-axis, just like that. And let's say that at T is equal to six seconds, T is equal to six seconds, theta is equal to pi radians. And so after three seconds, the ball is now right over here. And so if we wanted to actually visualize how that happens, let me see if I can rotate this ball in three seconds. So it would look like this. One Mississippi, two Mississippi, three Mississippi. Let's do that again. It would be... One Mississippi, two Mississippi, three Mississippi. So now that we can visualize or conceptualize what's going on, see if you can pause this video and calculate two things. So the first thing that I want you to calculate is what is the angular velocity of the ball? And 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. Alright, well let's go 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. And we are going to divide that by our change in time. So delta T. And so what is this going to be? Well this is going to be our angular displacement. Our final angle is pi. Pi radians minus our initial angle, pi over two radians. And then all of that's going to be over our change in time, which is six seconds, which is our final time minus our initial time, minus three seconds. And so we are going to get in the numerator, we have been rotated in the positive direction, Pi over two radians. Because it's positive, we know it's counterclockwise. And that happened over three seconds. And so we could rewrite this as this is going to be equal to pi over six. And let's remind ourselves about the units. Our change in angle, that's going to be in radians, and then that is going to be per second. So we're going pi over six radians per second, and if you do that over three seconds, well then you're going to go pi over two 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, and 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. 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, of our angular displacement, times the radius. Times the radius. And in this case, our radius would be seven 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 don't want people to get confused. Our speed is going to be equal to the distance we travel, which we just wrote is the magnitude of our angular displacement. And 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. Let me write that in a different color. So times the radius. All of that over our change in time. 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 two. You take the absolute value of that. It's still going to be pi over two. We know that our radius in this case is the length of that string. It is seven meters. And we know that our change in time here, we know that this over here is going to be three seconds, and we can calculate everything. But what's even more interesting is to recognize that what is this right over here? 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. Absolute value of our angular velocity, times our radius. Times our radius. And now so this is super useful. Our speed in this case, is going to be pi over six radians per second. So pi over, pi over six. Times the radius. Times seven meters. Times seven meters. And so what do we get? We are going to get seven pi over six meters per, meters per second, which will be our units for speed here. And the reason why we're doing the absolute value is 'cause 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's 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. And what's nice is there's a nice, simple formula for it. And 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 touched on in the video relating angular displacement to arc length, or distance traveled.
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