A solid sphere of radius R is completely immersed in a liquid of density ρ. The centre of the sphere is at a depth H below the free surface.
(a) Consider a thin horizontal slice of thickness dy at a distance y above the lowest point of the sphere. Express the area of the slice and the pressure acting on it in terms of y,R and H.
(b) Integrate over the entire sphere to obtain the net upward force due to the liquid on the sphere.
(c) Prove that the magnitude of this force equals ρg34πR3.