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Balancing Hemispheres (Posted on 2019-11-15) Difficulty: 5 of 5
Two rigid hemispheres A and B with uniform volume density p have radii a and b, respectively. Hemisphere B has its flat face glued to a plane. Hemisphere A is then balanced on top of hemisphere B such that their curved surfaces are in contact.

Naturally, A is in equilibrium when its flat face lies parallel to the flat face of B. However, if given a small nudge, A rolls without slipping on the curved surface of B and will either oscillate about the equilibrium position or fall.

The constraint on aa such that A can oscillate is given to be kb>a, where k is some positive real number.

Find the value of k.

Assume that gravity points down, perpendicular to the plane of B's flat face.

No Solution Yet Submitted by Danish Ahmed Khan    
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re: thoughts ... | Comment 3 of 9 |
(In reply to thoughts ... by Larry)

Larry - Agree. I think the key is the "small nudge" mentioned in the problem.  As with a mass/string pendulum, the differential equations become much more complex, and the resultant solution is not simple harmonic motion if one cannot assume "small angles".  For this problem, unless one deals with extreme values of a<<b, a thought experiment will show that with sufficient angular displacement of A, it will always fall off of B.

So assuming the above is correct, I believe, one can rephrase the problem as such: Assuming small angular displacements of A from equilibrium on B (implying limit sin(angle)-->angle in radians as angle-->0), then what is the relationship between a and b such that the weight vector of A is indeed directly above the contact point of A & B in the displaced state, as you correctly point out.  Then remembering the inequality would give the desired answer to the problem.

  Posted by Kenny M on 2019-11-18 11:32:57
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