In a certain liquid, the acceleration due to gravity varies with depth z below the free surface according to g(z)=g0(1−αz), where g0 and α are positive constants and αz≪1. A cubical block of side a is completely immersed so that its top face is just below the surface.
(a) Write an expression for the pressure at a depth z.
(b) Obtain the pressure on the top and bottom faces of the cube.
(c) Hence derive, correct to first order in αa, an expression for the buoyant force on the cube.
(d) Comment on how the result reduces to the usual expression when α→0.