Angular speed and tangential velocity

[lonewolf219]

i am wondering if this is correct...

The angular speed is always a smaller magnitude than the tangential velocity. This is because the tangential velocity has to travel a larger distance during the same amount of time as the angular speed. Tangential velocity is dependent on two things: the extra distance it covers (radius) compared to the central angular speed, and the magnitude of the angular speed.

Sound right or only partially?

 

[Doc Al]

Angular speed and tangential velocity are two different animals with different units. You cannot meaningfully compare their magnitudes.

Angular speed = radians/sec

Tangential velocity = meters/sec

 

[Naty1]

You are considering a rigid body in purely rotational motion...circular motion, right?

This part seems ok:

The angular speed is always a smaller magnitude than the tangential velocity..

although I think you mean:

"The angular speed is always a smaller magnitude than the MAGNITUDE of the tangential velocity..."

but from there your description makes no sense to me.

The linear speed of a particle in circular motion is the product of the angular speed and the distance (r) from the axis of rotation...v = wr. So v is greater than w except at r = o.

When these are considered as vectors, v and w are othogonal so "magnitude" comparsions are somewhat moot...for uniform circular motion, the angular velocity vector w remains fixed in direction along the axis of rotation...while the velocity vector v direction changes direction at a constant rate.

 

[lonewolf219]

Yes, they do have different units.

It is true, though, the concept that the speed on the outer edge of circular motion is faster than the center speed? Is that why there are different speeds for the Earth's rotation at various distances from the equator?

 

[Doc Al]

The linear speed of a particle in circular motion is the product of the angular speed and the distance (r) from the axis of rotation...v = wr. So v is greater than w except at r = o.

For what it's worth, the numerical value of v will be less than the value of ω whenever r < 1.

 

[Doc Al]

It is true, though, the concept that the speed on the outer edge of circular motion is faster than the center speed?

The tangential speed for larger radii is greater than the tangential speed at a smaller radii. Note that you are comparing tangential speeds, not angular versus tangential.

 

[lonewolf219]

Ah, OK. Thanks Doc Al. That is true for tangential velocities.

 

[CompuChip]

You can convert one into the other using a suitable conversion factor - from Doc's units you can see that this should be something converting radians (no unit) to meters (length).

For example, for a spinning disc with radius R, for a point on the disc at distance r from the origin the formula is
$$v = \omega r$$
where v is the tangential velocity and $\omega$ is the angular velocity.

As Doc remarked though, the numbers themselves don't mean anything.
If you decide to measure v in mph, omega in degrees/century and r in inches, you will get completely different numerical values.

[edit]Wow, you guys are fast. Never mind my post - it's a bit obsolete by now!