Angular Velocity or Angular Speed?

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Discussion Overview

The discussion revolves around the concepts of angular velocity and angular speed, particularly in the context of centripetal acceleration and circular motion. Participants explore the definitions, differences, and implications of these terms, as well as their relationships to linear velocity and speed.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether the term "angular speed" should be used instead of "angular velocity," suggesting that angular velocity relates to displacement over time rather than distance traveled.
  • Another participant proposes that angular velocity is indeed the correct term, emphasizing that it refers to the change in the central angle over time, not arc length.
  • A participant clarifies that in the formula for centripetal acceleration (v^2/r), the term "velocity" refers to displacement over time, not arc length.
  • There is a discussion about the distinction between speed and velocity, noting that speed is a scalar quantity while velocity is a vector, with angular speed being a scalar and angular velocity being a vector.
  • One participant mentions that the displacement in circular motion is not simply linear but involves the radius and the change in angle (rdθ).
  • Another participant highlights that the direction of motion in circular motion implies that angular velocity is a vector quantity, reinforcing the idea that it is related to the rate of change of angle.

Areas of Agreement / Disagreement

Participants express differing views on the terminology of angular velocity versus angular speed, with some supporting the use of "angular velocity" and others advocating for "angular speed." The discussion remains unresolved regarding the implications of these terms in the context of centripetal acceleration.

Contextual Notes

Participants have not fully resolved the definitions and implications of angular velocity and angular speed, particularly in relation to centripetal acceleration and the distinction between linear and angular measures.

Sammy101
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Hi,

We have been studying centripetal forces and acceleration in my class, and my teacher has shown me how to derive centripetal acceleration as v^2/r using change in velocity over time. I recently got into angular velocity vs. linear velocity in my math class, and I know that angular velocity is the change in the central angle(in radians) over time. But is this not SPEED? Speed is distance traveled over time, which is like the change in the central angle. Velocity is change in displacement over time.

With this lack of clarity, I became confused with the formula for centripetal acceleration as change in velocity over time. The "velocities" we seemed to be using actually appear to me now as speeds.

So here are my two questions:
1) Should it be angular speed not velocity since it is really distance traveled around a circle over time and not displacement over time?
and
2) In the formula for centripetal acceleration, v^2/r, is the centripetal velocity suppossed to be termed centripetal speed since it is also the distance traveled around the circle over time? If not, then the displacement over time for an object traveling around a circle (velocity) is much less than the distance over time (speed), right?
 
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I think I had a slight enlightenment right when I posted this and I would like to clarify some things.
1) Angular velocity is the correct term because it is not the distance around the circle traveled over time (s/t) but rather the displacement over time which is not arc length over timer but theta over time. Right?
2)So in centripetal acceleration = v^2/r, they are using displacement over time for velocity right?, not arc length or "s" over time?
3)Is angular velocity the change in the central angle over time and angular speed the change in the arc length over time?
 
What is the difference in velocity and speed? Well, the most obvious is the fact that speed is a scalar and velocity is a vector. In order to find the speed, we must take the square root of the sum of the squares of the components of the velocity (I think that's right). So angular velocity would be the speed at which an object is rotating, but it would have a direction, while the angular speed would simply be a number. Note that the speed is always positive, while velocity can be positive or negative depending on the coordinate axis.

In circular motion, there is certainly a displacement, but it is not simply dx. The displacement would actually be rdθ. It would not be m/s, but rather rad/s, or radians per second.
 
The way an object moves on the end of a string will be due to the force which is 'towards' the centre of the string. This is in a particular direction i.e a vector quantity is involved so the rate of change of angle is also a vector quantity - a velocity.
 

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