Explanation of Angular Speed Change

[Ryan H]

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?

 

[Subductionzon]

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.

 

[jmatejka]

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".

 

[A.T.]

Why does the angular speed decrease when the radius is increased?

To keep the linear speed constant: linear_speed = radius * angular_speed

 

[willem2]

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.

 

[rcgldr]

If you increase the radius, the tension decreases, if you decrease the radius, the tension increases. The math is explained here:

https://www.physicsforums.com/showthread.php?t=328121