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Bowl Balanced on a Bowl

  1. Oct 22, 2015 #1
    1. The problem statement, all variables and given/known data
    A solid hemisphere of radius b has its flat surface glued to a horizontal table. A second solid hemisphere of different radius a rests on top of the first one so that the curved surfaces are in contact. The surfaces of the hemispheres are rough (meaning that no slipping occurs between them) and both hemispheres have uniform mass distributions. The two objects are in equilibrium when the top one is "upside down", i.e. with its flat surface parallel to the table but above it. Show that this equilibrium position is stable if a < 3b / 5 .

    2. Relevant equations
    U=mgh
    v=ωr
    CM of a solid hemisphere is 3r/8

    3. The attempt at a solution
    I know I am supposed to do this problem using the energy and it's derivative to analyze the equilibrium points but I honestly have no idea how to go about setting up the problem, a push in the right direction would be much appreciated.
     
  2. jcsd
  3. Oct 22, 2015 #2

    mfb

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    2016 Award

    Staff: Mentor

    Well, start with the energy... did you draw a sketch?
    Where is the center of mass initially? What does that mean for its energy?
    Where is the center of mass after the hemispheres rolled a tiny bit? Small-angle approximations are fine as you are interested in the limit anyway.
     
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