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Video transcript

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 so 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 and so if you were to try to move this tennis ball it would just rotate around that nail and it would go along this blue circular path and 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 and we know what's going on here the our initial angle theta initial the convention is to measured relative to the positive x axis our theta initial is PI over 2 and let's say we were to then rotate it by 2 pi radians positive 2 pi radians so we would rotate in the counterclockwise direction 2 pi and then the ball would end up where it started before so theta final would just be this plus 2 pi so that would be 5 PI over 2 and of course we just said that we rotated in the positive in the positive counterclockwise direction we said we did it by 2 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 2 pi radians the next question I'm going to ask you is what is the distance that the ball would have traveled and remember distance we actually care about the path so 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 is the displacement the ball ends up where it started so the displacement in this situation is 0 we're not talk about angular displacement we just talk about regular displacement would be zero the angular displacement was 2 pi radians but what about the distance and we'll denote the distance by s as we can see as we'll see we can also you view that as arclength but 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 so it's going to be equal to 2 pi times 6 meters times 6 meters which in this case is going to be 12 pi and 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 we took our angular displacement in fact we took the magnitude of our angular displacement you could just view this 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 is R we just multiply that times R so 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 so in this situation I have a ball and let's say this is a shorter string let's say this string is only 3 meters and it's initial angle theta initial is PI radians we see that as measured from the positive x-axis and let's say we were to rotate it clockwise and so let's say our theta final is PI over 2 radians so theta final is equal to PI over 2 radians pause this video and see if you can figure out the angular displacement so the angular displacement in this situation is going to be equal to theta final which is PI over 2 minus theta initial which is PI which is going to be equal to negative PI over 2 does this make sense yes because we went clockwise clockwise rotations by convention are going to be negative so that makes sense we have a clockwise rotation pi over 2 radians now based on the information we've just figured out see if you can figure out the arc length or the distance that these this tennis ball at the end of the string actually travels this tennis ball at the end of the 3 meter string what is this distance actually going to be well there's a couple of ways that you could think about it you could say hey look this is 1/4 of the circumference of the circle so you could just say hey this arc length s is just going to be 1/4 times 2 pi times 2 pi times the radius of 3 meters times 3 and this indeed would give you the right answer this would be what to 2 over 4 is 1/2 so you get 3 PI over 2 and we're dealing with meters so this would be 3 PI over 2 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 so let's do that so if we were to take the absolute value of our I should say our angular displacement so we take the absolute value of our angular displacement and we were to multiply it by our radius well our radius is 3 meters and these would indeed be equal because this is just going to be positive PI over 2 times 3 which is indeed 3 PI over 2 so it looks like this formula is holding up pretty well and it makes sense because what you're really doing is you're saying look you have you have your traditional circumference of a circle and then you're thinking about well what proportion of the circle is this arc length and so 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 2 pi you'd be the entire circle if it's pi then you're going half of the circle but notice these two things cancel out and so this would actually give you your arc length let's do one more example so let's say in this situation the string that is heathering our ball is 5 meters long and let's say our initial angle is PI over 6 radians right over here and let's say that our final angle so is right we end up right over there so our theta final is equal to PI over 3 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 so our angular displacement is going to be equal to theta final which is PI over 3 minus theta initial which is PI over 6 which is going to be equal to PI over 6 and that makes sense this angle right over here we just went through a positive PI over 6 radians we went in the counterclockwise direction and then we just want to figure out the arc length we just multiply that times the radius so our arc length is going to be equal to PI over 6 times our radius times 5 meters so this would get us to 5 PI over 6 meters PI over 6 meters and we are done and once again no magic here it comes straight out of the idea of the circumference of a circle
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