Speed Pertaining to Circular Motion

[Coop]

Units for Speed Pertaining to Circular Motion

Hi,

So v=ω*r

Where v = velocity in m/s
ω = angular velocity in rad/s
r = radius in m

But I am confused...the units don't match! What happens to the rad?

m/s = (rad/s)*s

My textbook doesn't explain it, it simply does calculations like (56.5 rad/s)(0.030 m) = 1.7 m/s and the rad disappears. I have a feeling I am missing something obvious . Any help?

Thanks,
Coop

 

[D H]

The measure of an angle θ, in radians, is defined as the ratio of a circular arc subtending that angle θ to the radius of the circle. Angles thus have units of length per length. In other words, angle is a unitless quantity. Using radians makes the constant of proportionality one when computing arc length, or when computing speed along a circular path.

 

[dauto]

In other words, rads is not a unit. It's just a label.

 

[Coop]

Cool, thanks a lot guys, I wish my book explained that :p

 

[PJC01]

Hi Coop,

I can understand your confusion about the units for speed in circular motion. Let me explain it for you.

When we talk about circular motion, we are talking about an object moving in a circular path around a fixed point. The speed of this object is constantly changing as it moves along the circular path. However, the magnitude of the speed at any given point can be described by the tangential speed, which is the speed of the object along a tangent to the circular path at that point. This tangential speed is what we typically refer to when we talk about the speed of an object in circular motion.

Now, let's break down the units for tangential speed. As you correctly pointed out, the units for velocity are m/s. However, the units for angular velocity are rad/s. So what exactly is a radian (rad)?

A radian is a unit of measurement for angles, just like degrees. However, instead of being based on a 360 degree circle, a radian is based on the radius of the circle. One radian is equal to the angle subtended by an arc of length equal to the radius of the circle. So, in essence, a radian is a unit of length divided by a unit of length, and therefore it is a dimensionless unit. This is why in your textbook, the rad disappears when doing calculations. It is simply a conversion factor to get from angular velocity to tangential speed.

To summarize, the units for speed in circular motion are m/s, and the units for angular velocity are rad/s. The radian is a dimensionless unit that is used as a conversion factor in these calculations. I hope this clears up your confusion. Let me know if you have any further questions.

Best,