Prove that upthrust (U) = (h2 - h1)Adg = Vdg,  where the symbols having their usual meanings.

Let us consider a cylinder of height 'h' and uniform CSA 'A' is immersed in a liquid of density 'd'. If h1 and h2 are the depths of the upper and lower surface of the immersed cylinder then,

Force on the upper surface, F1 = P1 . A = h1dgA

Force on the lower surface, F2 = P2 . A = h2dgA

Here, g is the acceleration due to gravity.

Now,    upthrust (U) = F2 - F1
                                   = h2dgA - h1dgA
                                   = (h2 - h1)dgA
                                   = hdgA    Since h2 - h1 = h
                                   = Vdg    Since Ah = V
                                   = mg    Since Vd = m