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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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thoughts ... | Comment 2 of 9 |
Kenny M gave a formula for the CG of the upper half sphere (half circle if reduced to a 2 dimensional problem).
It seems that the problem is to find at what amount of rotation (as a function of a and b) leads to the CG being directly above the point of contact between the 2 bodies.   I assume that is the essence of the constraint that allows A to oscillate instead of continuing to roll.


  Posted by Larry on 2019-11-18 08:45:45
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