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 Balancing Hemispheres (Posted on 2019-11-15)
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 No Rating

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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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