Explanation of Angular Speed Change

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Discussion Overview

The discussion centers around the relationship between angular speed and radius in a rotating system, specifically exploring why angular speed decreases as the radius increases. Participants examine this concept through various scenarios, including a weight on a string and the analogy of an ice skater. The conversation touches on principles of angular momentum and kinetic energy, as well as the implications of maintaining constant linear speed.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant describes a scenario involving a weight and a cork to illustrate the question of angular speed and radius.
  • Another participant introduces the concept of constant energy in the system, suggesting an inverse relationship between radius and angular velocity based on rotational kinetic energy equations.
  • A simplified explanation is provided, comparing the situation to an ice skater, emphasizing that angular momentum remains constant while moment of inertia changes with radius.
  • Some participants assert that to keep linear speed constant, the relationship linear_speed = radius * angular_speed must hold, leading to different interpretations of whether linear speed remains constant or increases.
  • One participant notes that increasing the radius decreases tension in the system, linking it to the overall dynamics of the scenario.

Areas of Agreement / Disagreement

Participants express differing views on the constancy of linear speed and its relationship to angular speed and radius. There is no consensus on the interpretations of these relationships, and multiple competing views remain present throughout the discussion.

Contextual Notes

Some assumptions regarding energy conservation and the definitions of linear and angular speed are not fully explored. The discussion also highlights the complexity of balancing angular momentum and moment of inertia without resolving the mathematical details.

Who May Find This Useful

Readers interested in rotational dynamics, angular momentum, and the principles of motion in physics may find the discussion relevant and thought-provoking.

Ryan H
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You have a weight on one end of a piece of string and you run that piece of string through a tube, and then on the other end you attach a cork. You hold on to the tube and try and keep the cork spinning at a constant radius, such that the weight stays dangling at the same height. As you increase the radius of the circle, the time to complete a revolution takes longer. Why does the angular speed decrease when the radius is increased?

I guess it's sort of the same question as, if Mercury were the same size as Earth, would it's speed during revolution still be faster than Earth's? And if so, why?
 
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Your question is incomplete as stated. If you keep the energy of the system constant then it makes more sense. The equation for rotational kinetic energy is KEr=Iw^2/2 Where I is the rotational inertia and w is the angular velocity. For a very simple problem like yours I=Mr^2. So substituting for I we get KEr=M(r*w)^2/2. Going through a lot of steps and simplifying we can see that r=k/w where k is a the constant the squareroot of 2KEr/M. So you can see in the case of constant KE there is a inverse relationship between radius and rotational velocity.
 
Ryan H said:
Why does the angular speed decrease when the radius is increased?
QUOTE]

How about a simplified laymanistic answer to your question?

Suppose we put a speedometer on your cork, it would confirm your statement about the increase or decrease of velocity. Suppose we put an odometer on your cork, the odometer would show a longer "angular" path traveled associated with the longer radius.

If we put a fixed(finite) amount of energy into "each" rotation of this object, we cannot expect it to do more "work" in some rotations than others. (Term "work" is used loosely).

I believe Angular Momentum stays the same for each revolution,(as it would for an ice skater). (this statement ignores frictional losses, etc.)

Angular momentum = moment of inertia X angular velocity

As you change the radius for the cork,(or an ice skater moves their arms in/out from center of rotation) the moment of inertia is changed. For Angular momentum to stay the SAME, velocity MUST also change to keep the equation balanced. This is what "nature" does to keep the equation balanced.

The ice skater is perhaps the best example, Hopefully my more Learned Colleges chime in, if I have told you any "half truths".
 
Ryan H said:
Why does the angular speed decrease when the radius is increased?
To keep the linear speed constant: linear_speed = radius * angular_speed
 
A.T. said:
To keep the linear speed constant: linear_speed = radius * angular_speed

The linear speed doesn't remain constant, it increases. The angular momentum, which is
linear speed * radius remains constant.
 

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