Rotation Problem: Angular Velocity & Work Done Calculation

[neelakash]

Homework Statement

A rotating sphere contract slowly due to internal forces to (1/n)th of its original radius.What happens to its angular velocity.Show that increase in its energy equals the work done during its contraction.

Homework Equations

The Attempt at a Solution

(2/5)MR^2*w_1=(2/5)M(R-R/n)^2*w_2

From this we should find the change in w.Am I correct?

Regarding the workdone: Please help me to start with

 

[denverdoc]

Shouldn't that just be (R/n). say n is 3, the new radius is 1/3 the old one, from your eqn it would be 2/3. As for work I would assume thit would be due to change in gravitational state--say like a collapsing star, but not sure on this part.

 

[neelakash]

Yes,that is R/n.I went wrong first time.

Since w changes to w',the energy increases by (1/2)I'^2-(1/2)Iw^2---this we can compute.
The nexttask is to show this is equal to work done.We are told that internal forces are responsible for contraction...but does it do work?Self work should be zero...isn't it?

 

[denverdoc]

Not if the internal force is say gravity. I believe there is a differnce in potential energy between the two shells which is equal to the change in kinetic energy. But again not sure, it may be the work in going from I to I'

 

[neelakash]

That sounds good,but contains an inconsistency.Say,the internal force is gravity.Then,work done will be a function of G,M(not given),r1,r2 etc.Unless you are working with numerals,how can you say that will equal
(1/2)I'^2-(1/2)Iw^2?

 

[neelakash]

I think the internal force may be given by mw^2*r, where w as well as r are changing.