Solve Disk Rotation Problem: Conservation of Energy vs Momentum

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Discussion Overview

The discussion revolves around a physics problem involving the conservation of angular momentum and energy in the context of two disks, one rotating and the other dropped onto it. Participants explore the implications of the problem's wording regarding energy loss due to friction and the nature of the collision between the disks.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant proposes using conservation of angular momentum to find the new angular velocity ω after a second disk is dropped onto a rotating disk, yielding ω=((R**2)ω0)/(R**2+r**2).
  • Another participant questions the assumption of negligible energy loss to friction, arguing that friction must act for the disks to rotate together, indicating that kinetic energy is not conserved and the collision is inelastic.
  • Participants discuss the wording of the problem, particularly the phrase regarding negligible energy loss, and its implications for the validity of using energy conservation in this scenario.
  • One participant references the problem's source, indicating it is from a GRE preparation book, and expresses concern over the clarity of the problem statement.
  • Another participant mentions an errata list that acknowledges an error in the problem's wording.

Areas of Agreement / Disagreement

Participants generally disagree on the implications of the problem's wording regarding energy conservation and friction. There is no consensus on whether the problem can be solved using energy conservation due to the inelastic nature of the collision.

Contextual Notes

The discussion highlights potential ambiguities in the problem statement, particularly concerning the assumptions about energy loss and the nature of the collision. Participants express uncertainty about the implications of these assumptions on the validity of their approaches.

Teachme
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Ok here is the question that is being asked. Note that I know how to solve it one way, but when I go about trying to solve this problem another way that I think should give the the same result, I don't end up with the same result.

Question:
A disk of mass M and radius R rotates at angular velocity ω0. Another disk of mass M and radius r is dropped on top of the rotating disk, in its center, causing both disks to spin at a new angular velocity ω. Assuming a negligible loss of energy to friction what is ω?

Ok so this problem is pretty easy just set initial angular momentum to final angular momentum

I(large disk)(ω0)=I(large disk)(ω)+I(small disk)(ω) and just solve for ω.

which gives me an answer of ω=((R**2)ω0)/(R**2+r**2)

So I know that is the correct answer. But then I also thought that I should be able to solve this using conservation of energy. I set this up like so

(1/2)I(large disk)(ω0**2)=(1/2)I(large disk)(ω**2)+(1/2)I(small disk)(w**2)

however when I solve I get

ω=R(ω0)/sqrt(R**2+r**2)

I checked my work multiple times and cannot find an error. I just don't know where my logic is going wrong. I mean if there is conservation of angular momentum doesn't that imply conservation of energy? Or since you are adding mass to the system can I not set up my energy conservation equation that way because I am not accounting for the rest energy of the second disk? I guess it is an inelastic collision when you drop the second disk on the top of the first so maybe that is why as well. But they say assuming a negligible loss to friction which throws me off. I mean isn't there significant loss of friction in inelastic collisions or do they just mean a small loss compared to the total rotational energy.

Thanks for reading
 
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Teachme said:
Question:
A disk of mass M and radius R rotates at angular velocity ω0. Another disk of mass M and radius r is dropped on top of the rotating disk, in its center, causing both disks to spin at a new angular velocity ω. Assuming a negligible loss of energy to friction what is ω?
That last sentence is problematic. The only way for both disks to end up rotating together is for friction to act, and that means kinetic energy is not conserved. It's an inelastic collision.

Was that sentence really part of the problem?

I checked my work multiple times and cannot find an error. I just don't know where my logic is going wrong. I mean if there is conservation of angular momentum doesn't that imply conservation of energy?
No, not at all!
 
Yeah that's exactly what I was just thinking. Yeah it is a problem in a conquering the physics GRE book by Yoni Kahn and Adam Anderson. Well I guess not word for word. The last sentence is actually. "Assuming negligible a negligible loss of energy to friction, what is w?" I figured the first negligible was a typo.
 
Teachme said:
Yeah it is a problem in a conquering the physics GRE book by Yoni Kahn and Adam Anderson. Word for word.
Yikes. They messed up!
 
Wait sorry I read wrong. It says "Assuming negligible a negligible loss of energy to friction, what is w?" I figured the first negligible was a typo but maybe my english reading is wrong and that changes the sentence.
 
AHH THANK you sooo much. You just solved another problem I had with the book as well that I thought was weird! Just solved two problems that I was stuck on. Saved me from posting for the other. Thanks so much!
 

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