Hydrostatic pressure and buoyancy

A static fluid applies pressure perpendicular to any small surface element. In a liquid of constant density, pressure increases linearly with vertical depth: $p=p_0+\rho gh$, where $p_0$ is the pressure at the free surface. If the surface is open to the atmosphere, $p_{\mathrm{g}}=\rho gh$ measures gauge pressure and $p_{\mathrm{abs}}=p_{\mathrm{atm}}+p_{\mathrm{g}}$ gives absolute pressure. At the same horizontal level within one connected stationary liquid, pressures are equal. Do not use the sloping distance along an inclined wall in place of vertical depth.

Pressure on a submerged object's lower surface is generally greater than on its upper surface, producing a net upward pressure force. Archimedes' principle gives $F_{\mathrm{B}}=\rho_{\mathrm{f}}gV_{\mathrm{disp}}$, where $V_{\mathrm{disp}}$ is the volume of fluid displaced and $\rho_{\mathrm{f}}$ is the fluid density. For a fully submerged solid, displaced volume equals its outside volume. For a floating body, only the submerged portion displaces liquid. Buoyancy depends on displaced fluid weight rather than directly on the object's own weight.

Take upwards as positive for a vertical force balance: $F_{\mathrm{net}}=F_{\mathrm{B}}-W$ if no other vertical forces act. An unrestrained object in equilibrium floats when buoyancy equals its weight; a denser-than-water object may sink if its weight exceeds the maximum buoyancy available from its full volume. For a flat vertical gate, remember that fluid pressure changes across its height, so the pressure force must be integrated or found using the appropriate area and centroid depth. A single pressure value at a point is not automatically the total pressure force on a large surface.