Archimedes' principle | Buoyant force

Let us consider an object fully immersed in fluid. The pressure acting on the top face of the object is \(p_{\mathrm{t}},\) whilst the pressure acting on the bottom face is \(p_{\mathrm{b}}.\) The pressure at the bottom face is greater by an amount equal to the hydrostatic pressure of a fluid column whose height is the same as that of the object, \(h:\) \[p_{\mathrm{b}} = p_{\mathrm{t}} + \varrho_{\mathrm{fluid}} \cdot g \cdot h.\] The downward force resulting from the pressure at the top face: \[F_{\mathrm{t}} = p_{\mathrm{t}} \cdot A,\] whilst the upward force resulting from the pressure at the bottom: \[F_{\mathrm{b}} = p_{\mathrm{b}} \cdot A = \left( p_{\mathrm{t}} + \varrho_{\mathrm{fluid}} \cdot g \cdot h\right) \cdot A.\] Since \(F_{\mathrm{b}}\) is greater, the object will experience a net upward force, which we call the buoyant force: \[F_{\mathrm{B}} = F_{\mathrm{b}} - F_{\mathrm{t}} = p_{\mathrm{b}} \cdot A - p_{\mathrm{t}} \cdot A = \left( p_{\mathrm{t}} + \varrho_{\mathrm{fluid}} \cdot g \cdot h\right) \cdot A - p_{\mathrm{t}} \cdot A = \varrho_{\mathrm{fluid}} \cdot g \cdot h \cdot A = \varrho_{\mathrm{fluid}} \cdot g \cdot V_{\mathrm{s}},\] where \(V_{\mathrm{s}}\) denotes the volume of the object is beneath the surface of the fluid (in this special case, the full volume of the fluid).

Though we assumed at the beginning that the object is fully submerged, it can be proved that this relationship is valid also when the object is only partially submerged.