Angular speed and tangential velocity

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Discussion Overview

The discussion revolves around the relationship between angular speed and tangential velocity, particularly in the context of circular motion. Participants explore definitions, comparisons of magnitudes, and the implications of these concepts in various scenarios.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that angular speed is always a smaller magnitude than tangential velocity due to the distance covered in a given time.
  • Another participant argues that angular speed and tangential velocity have different units, making direct comparisons of their magnitudes meaningless.
  • A participant acknowledges the initial claim about angular speed being smaller but clarifies that the comparison should focus on the magnitudes of tangential velocity.
  • It is noted that the linear speed of a particle in circular motion is the product of angular speed and radius, leading to the conclusion that tangential velocity is greater than angular speed except at the center of rotation.
  • Discussion includes the idea that tangential speed varies with radius, with larger radii having greater tangential speeds.
  • One participant mentions the ability to convert between angular speed and tangential velocity using appropriate conversion factors, highlighting the importance of consistent units.

Areas of Agreement / Disagreement

Participants express differing views on the comparison of angular speed and tangential velocity, with some agreeing on the differences in units while others debate the implications of their magnitudes. The discussion remains unresolved regarding the initial claim about the relationship between angular speed and tangential velocity.

Contextual Notes

There are limitations in the assumptions made about the context of motion, particularly regarding the conditions under which the comparisons are valid. The discussion also highlights the importance of unit consistency in calculations.

lonewolf219
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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?
 
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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
 
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.
 
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?
 
Naty1 said:
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.
 
lonewolf219 said:
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.
 
Ah, OK. Thanks Doc Al. That is true for tangential velocities.
 
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!
 

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