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Distance or arc length from angular displacement

Relating angular displacement to distance traveled (or arc length) for a ball traveling in a circle. Derivation of formula for arc length.

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Video transcript

- [Instructor] What we're going to do in this video is try to draw connections between angular displacement and notions of arc length or distance traveled. Right over here, let's imagine I have some type of a tennis ball or something, and it is tethered with a rope to some type of a nail. If you were to try to move this tennis ball, it would just rotate around that nail. It would go along this blue circular path. Let's just say for the sake of argument the radius of this blue circle right over here, let's say it is six meters. You could view that as the length of the string. We know what's going on here. Our initial angle, theta initial, the convention is to measure it relative to the positive x-axis. Our theta initial is pi over two. Let's say we were to then rotate it by two pi radians, positive two pi radians. We would rotate in the counterclockwise direction two pi. Then the ball would end up where it started before. Theta final would just be this plus two pi, so that would be five pi over two. Of course, we just said that we rotated in the positive counterclockwise direction. We said we did it by two pi radians, so I kind of gave you the answer of what the angular displacement is. The angular displacement in this situation, delta theta, is going to be equal to two pi radians. The next question I'm going to ask you is what is the distance the ball would have traveled? Remember, distance where we actually care about the path. The distance traveled would be essentially the circumference of this circle. Think about what that is, and actually while you pause the video and think about what the distance the ball traveled is, also think about what would be the displacement that that ball travels. Well the easier answer, I'm assuming you've had a go at it, is the displacement. The ball ends up where it starts, so the displacement in this situation is as. We're not talking about angular displacement. We're just talking about regular displacement would be zero. The angular displacement was two pi radians, but what about the distance. We'll denote the distance by S. As we can see, as we'll see, we can also view that as arc length. Here the arc is the entire circle. What is that going to be equal to? Well, we know from earlier geometry classes that this is just going to be the circumference of the circle which is going to be equal to two pi times the radius. It's going to be equal to two pi times six meters which in this case is going to be 12 pi. Our units, in this case, is going to be 12 pi meters would be the distance that we've traveled. Now what's interesting right over here is at least for this particular case to figure out the distance traveled to figure out that arc length, it looks like what we did is we took our angular displacement, in fact we took the magnitude of our angular displacement, you could just view that as the absolute value of it, and we multiplied it times the radius of our circle. If you view that as the length of that string as R, we just multiply that times R. We said our arc length, in this case, was equal to the magnitude of our change in displacement times our radius. Let's see if that is always true. In this situation, I have a ball, and let's say this is a shorter string. Let's say this string is only three meters, and its initial angle, theta initial, is pi radians. We see that as measured from the positive x-axis. Let's say we were to rotate it clockwise. Let's say our theta final is pi over two radians. Theta final is equal to pi over two radians. Pause this video, and see if you can figure out the angular displacement. The angular displacement in this situation is going to be equal to theta final which is pi over two minus theta initial which is pi which is going to be equal to negative pi over two. Does this make sense? Yes, because we went clockwise. Clockwise rotations by convention are going to be negative. That makes sense. We have a clockwise rotation of pi over two radians. Now based on the information we've just figured out, see if you can figure out the arc length or the distance that this tennis ball at then end of the string actually travels. This tennis ball at the end of the three meter string. What is this distance actually going to be? Well, there's a couple of ways you could think about it. You could say, hey look, this is one fourth of the circumference of the circle. You could just say, hey, this arc length, S, is just gonna be one fourth times two pi times the radius of three meters, times three. This indeed would give you the right answer. This would be, what, two over four is one half, so you get three pi over two, and we're dealing with meters, so this would be three pi over two. But let's see if this is consistent with this formula we just had over here. Is this the same thing if we were to take the absolute value of our displacement. Let's do that. If we were to take the absolute value of our, I'd should say our angular displacement, so we take the absolute value of angular displacement, and we were to multiply it by our radius. Well, our radius is three meters. These would indeed be equal because this is just going to be positive pi over two times three which is indeed three pi over two. It looks like this formula is holding up pretty well. It makes sense because what you're really doing is you're saying, look, you have your traditional circumference of a circle. Then, you're thinking about, well, what proportion of the circle is this arc length? If you were to say, well, the proportion is going to be the magnitude of your angular displacement. This is the proportion of the circumference of the circle that you're going over. If this was two pi, you'd be the entire circle. If it's pi, then you're going half of the circle. Notice, these two things cancel out. This would actually give you your arc length. Let's do one more example. Let's say in this situation, the string that is tethering our ball is five meters long, and let's say our initial angle is pi over six radians right over here. Let's say that our final angle is, we end up right over there. Our theta final is equal to pi over three. Based on everything we've just talked about, what is going to be the distance that the ball travels? What is going to be the arc length? Well, we could, first, figure out our angular displacement. Our angular displacement is going to be equal to theta final which is pi over three minus theta initial which is pi over six which is going to be equal to pi over six, and that makes sense. This angle right over here, we just went through a positive pi over six radian rotation. We went in the counterclockwise direction. Then, we just want to figure out the arc length. We just multiply that times the radius. Our arc length is going to be equal to pi over six times our radius times five meters. This would get us to five pi over six meters, and we are done. Once again, no magic here. It comes straight out of the idea of the circumference of a circle.