Stable Equilibrium of Two Hemispheres: a<3b/5

[Latao Manh]

Homework Statement

A solid hemisphere with radius $b$ has its flat surface glued to a horizontal table. Another solid hemisphere with radius $a$ rests on top of the hemisphere of radius $b$ so that the curved surfaces in contact. The surfaces of hemispheres are rough, meaning no slipping occurs between them. Both hemispheres have uniform mass distributions. Two objects are said to be in equilibrium when the top one is upside down
- that is, with its flat surface parallel to the table but above it. Show that the equilibrium position is stable if $a<3b/5$.

Variables: a,b

Homework Equations

I think it's gravitational potential energy. So $mgy =U$
and $v_cm = r\omega$ for the top hemisphere
But this does not seem to go anywhere.

The Attempt at a Solution

I am stuck at resolving gravitational potential energy and the no-slip condition into some form so that I can differentiate.

 

[BvU]

Differentiating sounds good. Need some coordinate to describe deviation form the equilibrium position. Then express height of c.o.m. in that coordinate. If the center of mass goes up, stable, if it goes down, unstable. Any idea that keeps things simple ?