Angular Velocity or Angular Speed?

[Sammy101]

Hi,

We have been studying centripetal forces and accleration 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?

 

[Sammy101]

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?

 

[khemist]

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.

 

[sophiecentaur]

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.

 

[PJC01]

I understand your confusion and I'm happy to clarify the difference between angular velocity and angular speed. Angular velocity and angular speed are two terms that are often used interchangeably, but they actually have different meanings.

Angular velocity is a vector quantity that describes the rate of change of angular displacement over time. In simpler terms, it measures how fast an object is rotating around a fixed point. It is expressed in units of radians per second (rad/s).

On the other hand, angular speed is a scalar quantity that measures the rate of rotation of an object around a fixed point. It is expressed in units of revolutions per second (rev/s) or degrees per second (deg/s). Angular speed does not take into account the direction of rotation, only the magnitude.

To answer your first question, it is correct to use the term angular velocity instead of angular speed. This is because we are looking at the change in the central angle (angular displacement) over time, not the distance traveled around the circle.

For your second question, the formula for centripetal acceleration, v^2/r, uses the magnitude of the velocity vector, not the speed. The magnitude of the velocity vector is the same as the angular speed, but it also takes into account the direction of rotation. This is important because the direction of the velocity vector changes as an object rotates, and this change in direction is what causes the centripetal acceleration.

In summary, angular velocity and angular speed are different quantities, and it is important to understand the difference between them in order to correctly apply them in equations like the one for centripetal acceleration. I hope this helps clear up any confusion you may have had. Keep up the good work in your math and science classes!