Buoyant Force (Archimedes)

Multiply the density of water (1000 kg/m³) by the submerged volume in m³ and by g = 9.81 m/s². A fully submerged body of 2 litres displaces V = 0.002 m³, so F_A = 1000 × 0.002 × 9.81 = 19.62 N. That equals the weight of about 2 kg of water, completely independent of what the body is made of or how heavy it is. Two points matter: convert the volume from litres to cubic metres (1 L = 0.001 m³), and for partially submerged bodies count only the part below the water surface.

What counts is not the density of the material but the average density of the whole body including the enclosed air. A hull is hollow inside: steel walls plus huge air spaces together give an average density well below 1000 kg/m³. The ship therefore sinks in only until the displaced water weighs as much as the whole ship; then F_A = F_G and it floats stably. A solid steel block, by contrast, has ρ ≈ 7850 kg/m³ and sinks. If the hull floods, the air spaces vanish, the average density rises above that of water, and the ship goes down.

Divide the buoyant force by density and gravitational acceleration: V = F_A/(ρ_Fl·g). This measures the volume of irregular bodies elegantly: weigh the body once in air and once fully submerged. The difference between the two readings is the buoyant force. If the scale shows 49.05 N in air and 29.43 N under water, then F_A = 19.62 N and V = 19.62/(1000 × 9.81) = 0.002 m³ = 2 L. According to tradition, Archimedes checked King Hiero golden crown with exactly this idea: weight and volume give the density, and the density reveals the material.

Compare the average density of the body with the density of the fluid. If ρ_body is smaller, buoyancy wins when fully submerged, the body rises and finally floats with only part of it submerged. If both densities are equal, weight and buoyancy cancel exactly and the body hovers at any depth (this is how submarines work with their ballast tanks). If ρ_body is larger, weight wins and it sinks. For a floating body the submerged fraction can be read off directly: V_sub/V = ρ_body/ρ_fluid. Ice at 917 kg/m³ therefore floats about 92 % submerged, only the tip of the iceberg shows.

Yes, the formula F_A = ρ·V·g holds in any fluid, including gases. However, the density of air, about 1.2 kg/m³, is roughly 800 times smaller than that of water, so we hardly notice air buoyancy in everyday life. A person with about 75 L of body volume still experiences about 0.9 N of buoyancy in air, equivalent to roughly 90 grams. Hot-air balloons use the principle deliberately: hot air inside the balloon is less dense than the cold outside air, the displaced cold air weighs more than the filling, and the difference carries basket and envelope. Precision weighing must also correct for air buoyancy.