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# Relating angular and regular motion variables

In this video David shows how to relate the angular displacement to the arc length, angular velocity to the speed, and angular acceleration to the tangential acceleration. Created by David SantoPietro.

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• How is arc length a distance and not a displacement?
• Displacement is the shortest distance from the initial position to the final position, while Distance is the actual path covered by a moving object. In this case, the tennis ball covers the arc as the actual path, therefore it is the distance. Displacement, in this case, is the line joining the initial position to the tennis ball. Vote up if it helped
• At , arc length can also be found using a degree.
so why do we use radians if we can find arc length by using
(Theta/360)*2pi*r? I mean why do we have to use radians?
• radians have no units. This makes them much more convenient than degrees.
• If tangential acceleration and centripetal acceleration are both components of the total acceleration, why tangential acceleration is not a vector whereas centripetal acceleration is a vector?
• You are correct, tangential acceleration is just as much a vector as centripetal acceleration. That's why at in the video, he draws an arrow to represent it. Then, he goes on to say that you would add the tangential acceleration and the centripetal acceleration as vectors to get the total acceleration, just like we do with all vectors and their components. Did he say something somewhere that seemed to imply that tangential acceleration was not a vector?
• if v=r(omega) , then omega has units rad/s and r has units meters then what do we express v in?, (meter*rad)/s ?
• radians are unit-less because they just represent a ratio
omega has units of 1/s.
So v is m/s
• I am a bit confused about the usage of radians for angular measurements. Is pi(π) a unit of radians, or is it not? When converting different units, such as revolutions or degrees, to radians, should I solve for radians in terms of pi or terms of an integer?

For example:
[Question]: Convert 3 revolutions to radians.
To solve, one would have to do (3 rev.)(2π/1rev.)
• Radians are technically unitless since it is ratio of radius which is a distance to the circumference which is also a distance. This gives you meter/meter which cancel out leaving no units. π is not a unit it is just a constant of proportion.

How you report the answer depends on who you are reporting it to but in general an answer like 6π is more understandable that the decimal amount since using π you get a better idea of the proportion of the circle it is.
• why cant we just call angular displacement 'angular distance' or angular velocity as 'angular speed'?
• Angular displacement is the displacement of an object from the starting point to a final point. If you are talking about the distance then it's all of distance that it took us to get to a final point from an initial point. This about this in this way; if you're in a circular road driving a car and you complete one lap i.e one revolution then your displacement would be 0 because you are at the same location from where you started but you distance traveled would be the circumference of that circular road track i.e the total distance that it took you to complete one lap.
For Angular velocity, it is the change in your angular displacement per time or in other words, how fast our angle is changing for every change in time(second), and since it is velocity, it must have a direction which are counter-clockwise or clockwise but the Angular speed is simply the magnitude of the angular velocity, not specifying the direction of the motion.
• Because that's how you convert linear measures to angular.
• Is tangential acceleration equal to the angular acceleration?
• Tangential acceleration is not (numerically) equal to angular acceleration. If the motion is circular (radius is a constant) then they are related by the constant radius. alpha (angular acceleration) times the radius is equal to the tangential acceleration.