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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(2): thoughts ... | Comment 5 of 9 |
(In reply to re: thoughts ... by Kenny M)

But hold on a second.  The way I'm looking at it, we just need to find 'k'.  We only need to determine where the tipping point is, we don't need to figure out either simple harmonic motion or difficult harmonic motion. We don't need to be concerned with any motion; just the static state where the CG is over the contact point.

My view of friction in the context of this problem is, we are told the upper object "rolls without slipping" whatever the friction is, we know there is enough to prevent slipping.  Beyond that, we don't need to know the friction since the solution to our problem would be to tilt the object until it just balances without moving.  No motion, no friction.

For example, if instead of lower sphere B, we just had a flat table, we'd be able to rotate the half sphere (or half circle) a bit beyond 90 degrees until the CG was directly over the 'corner' of the half circle where it rests on the table.

  Posted by Larry on 2019-11-18 14:28:13
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