Conditions for Floatation

When an object is immersed in a fluid, the fluid exerts pressure on every surface of the object. Because pressure increases with depth, the upward pressure on the bottom face is greater than the downward pressure on the top face. The net upward force resulting from this pressure difference is the buoyant force, F_b. Archimedes showed that F_b = ρ_fluid × V_displaced × g, where V_displaced is the volume of fluid pushed aside by the object.

The object's weight acts downward: W = m × g = ρ_object × V_object × g. For a fully submerged object, V_displaced equals V_object. So the buoyant force equals ρ_fluid × V_object × g, while weight equals ρ_object × V_object × g. Dividing both expressions by (V_object × g) shows that the comparison reduces to simply comparing the two densities: ρ_object versus ρ_fluid.

If ρ_object > ρ_fluid, then W > F_b. The net downward force causes the object to accelerate downward — it sinks. A steel ball (ρ ≈ 7800 kg m⁻³) sinks in water (ρ ≈ 1000 kg m⁻³) because its density is about 7.8 times greater. No matter how large the ball, the ratio of forces always equals the ratio of densities, so density is the decisive factor.

If ρ_object < ρ_fluid, the object floats at the surface. It submerges only far enough so that the weight of fluid displaced equals its own weight. A wooden block of density 600 kg m⁻³ placed in water floats with 60% of its volume submerged, because it only needs to displace a mass of water equal to its own mass. The fraction submerged equals ρ_object / ρ_fluid.

If ρ_object = ρ_fluid exactly, then F_b = W at any depth and the object remains stationary wherever it is placed — this is neutral buoyancy. Fish achieve this using a swim bladder: they inflate or deflate it to adjust their overall density to match the surrounding water. Submarines pump water in or out of ballast tanks to do the same thing. This state requires precise control because even small density differences cause slow sinking or rising.

A hollow steel ship has a very large volume of enclosed air. Its total mass divided by its total volume (hull + air inside) gives an average density well below 1000 kg m⁻³ — lower than water — so it floats. This is why average density, not material density, is the physically relevant quantity. Crushing the ship into a solid block of steel would raise its average density far above water's, and it would sink. Engineering floatation is largely the art of managing average density.

• \rho_{\text{object}} — Density of object (kg m⁻³): The average density of the floating object, including any enclosed air spaces.
• \rho_{\text{fluid}} — Density of fluid (kg m⁻³): The density of the liquid or gas in which the object is placed.
• V_{\text{submerged}} — Submerged volume (m³): The volume of the object that lies below the fluid surface, equal to the volume of fluid displaced.
• V_{\text{object}} — Total volume of object (m³): The complete volume of the object, whether hollow or solid.

Problem. A wooden log has a density of 750 kg m⁻³ and a volume of 0.40 m³. It is placed in fresh water (density 1000 kg m⁻³). (a) Does it float or sink? (b) What fraction of the log is submerged when it floats at equilibrium? (c) What is the volume of water displaced at equilibrium?

Answer. The log floats with 75% of its volume submerged, displacing 0.30 m³ of water.

• Archimedes' Principle — Archimedes' principle provides the mathematical foundation for buoyant force, which is the upward force whose comparison with weight determines whether an object sinks, floats, or achieves neutral buoyancy.
• Buoyancy and Upthrust — Understanding upthrust — the actual mechanism producing the buoyant force through fluid pressure differences — is essential context for why the density comparison correctly predicts floatation behaviour.
• Density and Relative Density — Density is the single most important quantity in floatation conditions; relative density (specific gravity) directly expresses the ratio ρ_object/ρ_fluid that determines the fraction submerged.
• Pressure in Fluids — The depth-dependent pressure in fluids is the underlying cause of buoyant force, making fluid pressure the physical origin of all floatation phenomena.