Does ω0 = ωf when radius of circular motion changes?

[Vitani11]

Homework Statement

A particle of mass m is moving on a frictionless horizontal table and is attached to a massless string that passes through a tiny hole of negligible radius in the table, and I am holding the other end of the string underneath the table. Initially the particle is moving in a circle of radius r0 with angular velocity ω0, but I now pull the string down until the radius reaches r. How much kinetic energy did the particle gain?

Homework Equations

The Attempt at a Solution

I integrated to find the potential energy and I know the force is conservative for the kinetic energy = -potential energy. Is that right? I also figured that the final angular velocity is the same as the initial angular velocity. Is that correct?

 

[phinds]

I also figured that the final angular velocity is the same as the initial angular velocity. Is that correct?

Have you ever seen an ice skater start to spin with arms and one leg extended and then pull in the arms and leg. Does the skater's shoulders stay at the same rate before and after? How might this apply to your current problem?

 

[TomHart]

Have you ever watched figure skaters - how they start spinning with their arms stretched outward. But as they bring their arms in toward their body, their angular velocity increases significantly?

Edit: phinds, you beat me to the punch. :)

 

[phinds]

Edit: phinds, you beat me to the punch. :)

Yeah, I'm feeling punchy today

EDIT: and by the way, you really should not have given him the conclusion. Notice that I was asking him to think it through himself. That's really more appropriate for PF.

 

[rude man]

Are you applying a torque to the mass about the center of rotation by pulling on the string?

 

[haruspex]

the final angular velocity is the same as the initial angular velocity

If that were true, how could any KE have been gained?

 

[Khashishi]

Fill in the relevant equations.
You mentioned potential energy. What potential energy are you thinking of?
How much work did you do by pulling the string?

 

[haruspex]

How much work did you do by pulling the string?

Reading between the lines, I would guess Vitani tried to calculate that by multiplying the centripetal force by the displacement. The problem with that is knowing the velocity at all points. Vitani wrongly took the angular velocity to be constant for this purpose.
In short, energy is not the easiest way to solve this.

 

[Khashishi]

I can't see how to solve this without using energy. The trick is that you have to write the angular velocity in terms of the energy, and integrate energy as a function of radius.

 

[haruspex]

I can't see how to solve this without using energy. The trick is that you have to write the angular velocity in terms of the energy, and integrate energy as a function of radius.

See the hint in post #5.

 

[Khashishi]

Right, of course. I had a brain malfunction.

 

[Vitani11]

I used the fact that angular momentum is conserved (Torque is 0 since the direction of the force lies along the same line as the distance to the mass) to find the final angular velocity in terms of known variables and then used conservation of energy to solve. Thank you for the help.