Rotation Problem: Angular Velocity & Work Done Calculation

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Homework Help Overview

The problem involves a rotating sphere that contracts to a fraction of its original radius, prompting questions about the effects on angular velocity and the relationship between energy increase and work done during contraction.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the relationship between the sphere's initial and final angular velocities, with attempts to derive equations relating these variables. Questions arise regarding the nature of work done by internal forces during contraction, particularly in the context of gravitational forces.

Discussion Status

The discussion is active, with participants exploring various interpretations of the problem. Some have offered insights into the energy changes involved, while others question the assumptions about internal forces and their contributions to work done. No consensus has been reached yet.

Contextual Notes

Participants note the lack of specific values for mass and gravitational constants, which complicates the discussion of work done and energy changes. The problem is framed within the constraints of a homework assignment, emphasizing the need for careful consideration of the underlying physics.

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

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