If we go back to the hemisphere of specific gravity 1.5, the pressure on the bottom (at the top of the cylinder) will need to greater in order to support the extra weight. (The horizontal components do cancel out.) This downward force from water pressure will be the same no matter what material is in the hemisphere. (You may want to think of it as being enclosed in a thin film.) Then the upward force,U, from water pressure on the bottom must equal the weight of the water plus the downward force, D, from the pressure on the top of the hemisphere. To get the force from the water pressure on the upper surface of the hemisphere one could integrate the (variable) vertical component of force on each segment of the upper surface, or we can replace the hemisphere with water. I am going to assume that the piston rod exerts no net force on the hemisphere, that there is no friction between the hemisphere and the cylinder, and that seals on the hemisphere and the rod prevent water from leaving the cylinder.
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