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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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• shouldn't the angular displacement in the first example be zero?
• OK so don't confuse angular displacement with regular displacement, that is why Sal told in the video that displacement is zero, but the angular displacement is 2pi radians and for two complete revolutions it is 4pi radians. I would suggest you to watch the video on Angular motion again to clear your doubts.
• 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).
• 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!
• 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?
• At how did you get Δθf=5π/2?
• The original position was π/2, then there is a rotation of 2π, so by adding those up, you get the final position of 5π/2.
• Why is theta intial pi/2?
(1 vote)
• To know this, a background in algebra 2 is helpful (because that is when most people learn about the unit circle). Here is the explination:

Radians (which are used here) are like degrees, where they measure angles. In terms of circles, like are in this problem, 360 degrees is all the way around a circle once, 180 degrees is half way around the circle, and 90 degrees is a quarter around the circle. Radians are similar, however, they are measured differently. 2π is all the way around a circle, π is half way around a circle, and 1/2π (or π/2) is a quarter around the circle. So basically:
In circular motion problems like this, here's another thing to keep in mind: the point of 0 degrees (0 radians), AKA the starting point, is the positive x axis. So when the ball is at its initial point (θ initial), it has rotated a quarter around the circle (since positive rotation is counter-clockwise). Since a quarter around the circle is π/2 radians, theta initial is π/2 radians.

P.S.- radians will is a very conventional unit in math and physics at high school/college levels. Here are some lessons explaining radians and the unit circle if you want to learn more in depth:

I hope that this helps!
• At , Sal says that the units for S are meters.
I thought that Δθ was measured in radians, so why are the units for S not (meters)(radians)?
(1 vote)
• One radian is the length of the radius of the same circle. So the length of 2π radians is really just the length of 2π radiuses. Therefore, if your radius was 6 meters, multiplying 2π radians with the radius would give you 12π meters.
(1 vote)

• why is the pie final = 5 pie / 2 ?
(1 vote)
• displacement is the change in r vector so why we are more interested in the length
(1 vote)
• great! but why we use the letter of displacement "S" to represent distance here ?!
(1 vote)
• Well according to me, Sal used S to denote distance to just differentiate between angular displacement and Distance so I'm pretty sure it's just a notation to avoid confusion.
(1 vote)
• 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)