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Competitive exam Question: Work done to shorten string

  1. Oct 10, 2009 #1
    Q) A particle of mass 'm' attached to a string rotates with velocity Vo when the length of the string is Ro. How much work is done in shortening the string to R?

    One way I thought about doing this was:
    W= {(m*r*w^2) * r}dr and integrate this from R to Ro
    But I am not sure if that is correct. So if someone can help me with this question I will really appreciate it.
    Thanks in advance!
     
    Last edited: Oct 10, 2009
  2. jcsd
  3. Oct 10, 2009 #2

    Delphi51

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    Sounds good.
    What happens to the velocity as r decreases?
    I suppose angular momentum will be conserved.
     
  4. Oct 10, 2009 #3
    You think my solution is correct? Can someone else also please confirm this.

    v=rw, w is constant. r decreases so v decreases as well. Correct me if I am wrong.

    Angular momentum I believe is conserved.
     
  5. Oct 10, 2009 #4

    Delphi51

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    I don't think w is constant. The constant is k = mrv so v = k/(mr)
    and w = v/r = k/(mr^2).

    v increases as r decreases.
     
  6. Oct 10, 2009 #5
    I guess you are right. I was taking the wrong assumption. If angular momentum is conserved then velocity increases if radius decreases.
    Now when you have corrected me I think my solution to the problem was not correct either. Can you please verify that for me as well?
    Thanks
     
  7. Oct 10, 2009 #6

    Delphi51

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    I haven't seen a solution yet. I just agreed that integrating Force*dr would be the way to do it. And you have to be careful to express v or w in terms of k and r because v and w are not a constants.
     
  8. Oct 10, 2009 #7
    Solution:

    dW= F.dr
    dW= {(m*v^2)/r}dr (v=k/mr)
    dW= {(k^2/m)*(1*r^3)}dr
    Integrate this with limits Ro to R

    W = [(k^2)/(2m)]*[(1/Ro^2) - (1/R^2)] <----Answer

    Please check and let me know if this seems correct to you?
    Thanks
     
  9. Oct 10, 2009 #8

    Delphi51

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    That is precisely what I got!
    Of course I am not infallible! I'm just a retired high school teacher missing the good feeling of helping my students.
     
  10. Oct 10, 2009 #9
    I really appreciate your help. I hope we are correct but it will be good to know if someone else here can verify our solution.
     
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