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Condition for stable equlibrium

  1. May 6, 2015 #1
    1. The problem statement, all variables and given/known data

    A homogeneous wooden bar of length 10 cm, thickness 4 cm and weight 1 Kg is balanced
    on the top of a semicircular cylinder of radius R as shown below. Calculate the
    minimum radius of the semicircular cylinder if the wooden bar is at stable equilibrium.
    ?temp_hash=ff5d8793a573727d0f3d7c73d119c317.jpg

    2. Relevant equations

    Potential energy E=mgh and its derivatives.

    3. The attempt at a solution

    Stable equilibrium means the first derivative of potential energy is zero and its second derivative must be greater than zero(local minima). So, I have to express the PE of the wooden bar in terms of R and find minimum R to satisfy above conditions. But here the CM of the bar is at R+(4/2)=R+2 cm above the ground. So, second derivative of PE is always 0? Where am I going wrong? Also, how to approach problems like these in general? I had read about equilibrium a long time back and the concepts are a bit muddled up.
     

    Attached Files:

  2. jcsd
  3. May 6, 2015 #2

    BvU

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    The CM doesn't stay at R+2 if the beam is wiggled !
     
  4. May 6, 2015 #3
    Thanks BvU. Yes, I googled similar problems and understood what needs to be done. By displacing the block by θ I calculate the new CM height which comes out to be
    h=Rcosθ+Rθsinθ+2cosθ
    Now, it is just differentiating twice. So, for equilibrium, is the width of the block(10 cm) irrelevant? I didn't find it's use in the height equation or am I missing something?
     
  5. May 6, 2015 #4

    BvU

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    Ah, you mean the length :smile: !

    Didn't work out h myself, and you don't show the steps, so I can't really tell. Suppose you're right.
     
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