The centre of mass of the combined solid will be on the axis of symmetry at distance
_
x
 

from the vertex of the cone. The total mass is
1
3
 pa2hw + 2
3
 pa3 w = 1
3
 p a2w(h+2a).
By taking moments about the vertex of the cone:
1
3
 pa2w(h+2a) _
x
 
 = 2
3
 pa3w (h+ 3a
8
 ) + 1
3
 pa2 hw 3h
4
 .
This gives
_
x
 
 = h + 3a2-h2
4(h+2a)
from the vertex of the cone.

Label the vertex of the cone as V, a point on the rim of the face where the two solids are joined as R, the centre of this circular face (on the axis of symmetry) as O and the point where the axis of symmetry cuts the curved surface of the hemisphere as B.

(i) If h=a then the centre of mass is inside the hemisphere at a distance a/8 from the centre of the circular base. For this position of the centroid (and all points inside the hemisphere) the body rolls to rest with the lowest point of the hemisphere B in contact with the table and the equilibrium is stable.

(ii) For h=2a the centre of mass of the solid is at a distance
a
16

from O and inside the cone. For this position of the centre of mass (and all positions inside the cone) the body will roll to a position of equilibrium with a generator of the cone in contact with the table. The only exceptions are when it is placed on V or B with the axis of symmetry vertical which are unstable positions of equilibrium.

(iii) If the centre of mass is at O then h=aÖ3 and in this case, with any point on the curved surface of the hemisphere in contact with the table, the centre of mass is always vertically above the point of contact, so the body will be in equilibrium.

In this case the angle of the cone is 60o and the slant height is equal to the diameter of the hemisphere.

The equilibrium is unstable when the point of contact is on the rim between the hemisphere and the cone, and stable for all other points of contact between the hemisphere and the table.

(iv) In summary
* for h < aÖ3, the angle of the cone greater than 60o, the body comes to rest with B in contact with the table and equilibrium is stable;

* for h = aÖ3, the angle of the cone 60o, the body will rest in stable equilibrium if it is placed on the table either with a point of the hemisphere in contact with the table or with a generator of the cone in contact with the table. If it is placed with R (on the rim between the cone and the hemisphere) in contact with the table then the equilibrium is semi-stable. if tipped towards the cone it will fall to rest on a generator, if tipped the other way it will rest wherever it is moved to with the hemisphere in contact with the table.

* for h > aÖ3, the angle of the cone less than 60o, the body will roll to a stable equilibrium position with a generator in contact with the table. The exceptional cases are unstable equilibrium with V or B in contact with the table.