A smooth homogeneous hollow right circular cylinder with open, flat ends stands freely on smooth horizontal ground so that its axis is vertical. Two spheres 
and 
of radii 
and 
(
) and weights 
and 
respectively rest in equilibrium inside the cylinder as shown in the diagram. Suppose that the internal and external radii of the cylinder are 
and 
respectively, where 
.
i. Show that the vertical forces acting on the spheres reduce to a couple. Determine the moment of the couple.
ii. Determine the minimum weight of the cylinder such that it will not overturn.
iii. A third sphere 
, identical to , is then placed on top of in contact with the cylinder. Determine the minimum weight of the cylinder so that it will not overturn for the two possible equilibrium positions of .
You may assume that the cylinder is tall enough to hold all the spheres.
Roughwork.
WLOG reduce the problem from three-dimensional to two. Begin with three free-body diagrams as follow:
So many unknowns I don’t know how to get set.
(to be continued)