Fluids at rest: pressure, density and buoyancy
A fluid (any liquid or gas) cannot sustain a shear stress at rest, so its equilibrium is described entirely by pressure. Density is mass per unit volume, ρ = m/V, in kg/m³: liquid water is almost exactly 1000 kg/m³, air near sea level about 1.2 kg/m³, steel about 7800 kg/m³. Memorising those three gives an instant scale check for any density answer.
Pressure is force spread over area, P = F/A, measured in pascals (1 Pa = 1 N/m²). It acts *perpendicular* to any surface and, in a connected fluid at rest, equally in every direction — Pascal's principle. Standard atmospheric pressure is 101 325 Pa (101.325 kPa, 1 atm ≈ 760 mmHg), and gauge pressure simply subtracts that reference.
In a column of fluid, pressure grows with depth: P = P₀ + ρ g h. Every metre of water adds about 1000 × 9.81 × 1 ≈ 9810 Pa — roughly 0.1 atm. Ten metres of water doubles what a diver feels from the atmosphere alone, which is why scuba limits and dam walls are thickness-graded: the load at the base depends only on depth, not on the total volume above.
Archimedes’ principle follows: a submerged body experiences an upward buoyant force equal to the weight of fluid displaced, F_b = ρ_fluid g V_displaced. Whether something floats follows from comparing densities — ice (917 kg/m³) floats on water (1000 kg/m³) because it is lighter by volume; a submarine dives by flooding ballast until its *average* density matches seawater's. Floating itself is the equality case: ρ_object/ρ_fluid = fraction submerged.
Torricelli's result for drainage ties statics to motion: an orifice at depth h under a large reservoir exits at v = √(2 g h) — the same speed a dropped object reaches falling the same height, as energy conservation demands.