Question

Consider a cube of side "H" totally immersed in a fluid, with the top of the cube at depth "h." DERIVE Archimedes' Principle by first calculating the difference in fluid pressure between the bottom and top of the cube. SHOW your steps.

Answer 1

Starting with the fact that water at the top surface will push the cube down (with force F_top say) and water at the bottom will push the cube up (with Force F_bottom say), then the total upward force which is called the buoyant force is the difference as,

F_buoyant = F_bottom - F_top

we can now substitute the values of force by the definition of pressure(F = P*A)

A is the area top and bottom (A = H²)

F_buoyant = P_bottom*A - P_topA

From the formula of hydrostatic pressure(P = gh)

F_buoyant = gh_bottom*A - gh_topA

F_buoyant = gA(h_bottom - h_top)

according to given conditions, h_top= h; h_bottom= H+h

F_buoyant = gA*(H+h - h)

F_buoyant = gAH

where AH is volume of the cube, say V

F_buoyant = gV   

F_buoyant = gA*(H)