If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

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.

Want to join the conversation?

  • blobby green style avatar for user anas.bashar89
    shouldn't the angular displacement in the first example be zero?
    (10 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user faaeahelena.poloa
    At the last problem why did you subtract pi/3-pi/6 if it went in a positive direction? Shouldn't it be pi/3+pi/6= pi/2 (work: 2pi/6+pi/6= 3pi/6= pi/2).
    (2 votes)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user Aadit Ambadkar
      One thing that Sal mentions in the early part of the video is that pi/3 and pi/6 are positions in units of radians.

      When he tries to find the angular displacement, he tries to find how much the ball was rotated in total. The ball went from pi/6 radians to pi/3 radians, so Sal took pi/3-pi/6 radians to be the total.

      To help visualize why, try this:
      Imagine trying to find the displacement if a man moved from 4 on the number line to 8 on the number line. The answer is obviously 8-4=4.

      Now let us try to solve the original problem. Remember that with angular displacement, counterclockwise is positive and clockwise is negative (just like right is positive and left is negative in the example above). The final position is pi/3. The initial position is pi/6. So, the angular displacement is pi/3-pi/6.

      Key things to remember:
      The formula for angular displacement is not P_f - P_s, where P_f is final position and P_s is starting position. That is because angular displacement is how much has the angle moved. So, the angular displacement of completing a full rotation counterclockwise (2 pi radians) may appear to be 0. After all, the position of the ball didn't change, right? But we are not looking for the change in the ball. We are looking for the change in angle. So the answer for the displacement of a complete rotation (2 pi radians) is 2 pi radians!
      (6 votes)
  • leafers seed style avatar for user Lanieta Lasaqa
    So take for a example a wheel is rotated through angles of 30degrees, 30rads and 30rev, respectively which one will be my angular displacement when im looking for path length and my radius is 4.1m?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • leafers tree style avatar for user Ayah.A
    At how did you get Δθf=5π/2?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • starky sapling style avatar for user Bom

    why is the pie final = 5 pie / 2 ?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • primosaur seedling style avatar for user minaaxelma14
    displacement is the change in r vector so why we are more interested in the length
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby yellow style avatar for user Issam armas
    great! but why we use the letter of displacement "S" to represent distance here ?!
    (1 vote)
    Default Khan Academy avatar avatar for user
  • boggle blue style avatar for user Josue
    Are Δθ = 2π counterclockwise, Δθ = π/2 clockwise, and Δθ = π/6 counterclockwise valid answers for the angular displacements Sal solved for in this video?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • leaf blue style avatar for user Matthew Smith
    If the initial position is pi/2 and there is one full rotation, or 2pi, then how can it travel a full revolution and still land on pi/2 when he says at the angular displacemnt its 5pi/2 (because if pi/2 is x=0, and they do one revolution, x=xfinal- xinital, so x= 2pi-0)? shouldn't the final angular displacement be just 2pi?

    I've re-watched this video and the 'Angular motion variables' video multiple times and still don't get it.
    Thanks in advance!
    (1 vote)
    Default Khan Academy avatar avatar for user
    • boggle blue style avatar for user Bryan
      Theta final (theta sub f) is not the angular displacement, it is the final angle. The angular displacement would be 2pi,

      angular displacement = theta final - theta initial
      = 5pi/2 - pi/2
      = 2pi
      (1 vote)
  • blobby green style avatar for user Kvon Tayco
    Why for the second problem is it multiplied by 1/4?
    (1 vote)
    Default Khan Academy avatar avatar for user

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.