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Quantum mechanics, balancing problem with uncertainty

  1. Sep 24, 2009 #1
    1. The problem statement, all variables and given/known data
    Try to balance an ice pick of mass m and length l, on its point. Under ideal conditions, what is the maximum time T it can balance on its point?

    2. Relevant equations
    Potential Energy: U=mgh
    Series Expansion for Cosine: cos(x)=1-1/2(x^2)+...

    3. The attempt at a solution
    By conservation of energy,
    [tex]mgl=\frac{1}{2}ml^2\dot{\theta}^2+mglcos \theta[/tex]
    [tex]g=\frac{1}{2}l\dot{\theta}^2+gcos \theta[/tex]
    [tex]\frac{2g}{l}(1-cos \theta)=\dot{\theta}^2[/tex]

    Assume a small angle displacement, so
    [tex]cos \theta = 1-\frac{1}{2}\theta^2[/tex]
    and
    [tex]\dot{\theta}^2=\frac{g}{2l}\theta^2 ; \dot{\theta}=\theta \sqrt{\frac{g}{2l}}[/tex]

    Integrating from t=0 to T and theta from some small delta theta to pi/2 gives
    [tex]T=\sqrt{\frac{2l}{g}}ln\theta\right|^{\pi/2}_{\Delta \theta}[/tex]

    The only thing I now further is
    [tex]\Delta x\Delta p=\frac{\bar{h}}{2} ; \Delta \theta = \frac{\Delta x}{l}[/tex]

    At this point, I'm not sure how to proceed, or even if the work so far is correct. I've seen an actual solution to this, but I can't follow it and some of the math seems wrong, or is not properly explained.
     
  2. jcsd
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