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.
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