Statement of Archimedes’ Principle

When you lower yourself into a full bath, water spills over the edge. That displaced water is the key to one of the most powerful principles in physics. Archimedes' Principle states that any object wholly or partially submerged in a fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid it displaces. This principle applies universally — to ships floating on oceans, submarines diving beneath the surface, balloons rising through air, and even tiny particles suspended in blood. The principle explains why some objects float and others sink. A solid iron ball sinks because its weight exceeds the weight of the small volume of water it displaces. A hollow iron ship floats because its large hull shape displaces a volume of water whose weight equals the entire weight of the ship. The buoyant force is not a mysterious new force — it arises directly from the pressure difference between the bottom and top of a submerged object, since pressure in a fluid increases with depth. Understanding buoyancy requires careful thinking about weight versus density. An object floats when its average density is less than the fluid's density, and sinks when it is greater. At exactly equal densities, the object achieves neutral buoyancy — hovering at any depth. This is precisely how submarines control their depth, by adjusting the mass of water in ballast tanks to match the surrounding seawater density.

Key takeaways

Fluid Pressure Increases with Depth
Calculating the Net Upward Force
The Displaced Fluid Argument
Floating vs. Sinking — Applying the Principle

Fluid Pressure Increases with Depth

In any static fluid, pressure at depth h is given by P = P₀ + ρgh, where ρ is the fluid density and g is gravitational field strength. This means the pressure pushing upward on the bottom face of a submerged object is always greater than the pressure pushing downward on its top face. The net upward pressure force is what we call the buoyant force.

Calculating the Net Upward Force

Consider a rectangular block submerged with its top face at depth h₁ and bottom face at depth h₂ (so h₂ > h₁). The downward force on the top is F_top = ρ_fluid × g × h₁ × A, and the upward force on the bottom is F_bottom = ρ_fluid × g × h₂ × A. The net upward force is F_buoy = ρ_fluid × g × (h₂ − h₁) × A = ρ_fluid × g × V_displaced. This equals the weight of the fluid occupying that volume.

The Displaced Fluid Argument

Here is an elegant way to see it: before the object was placed there, that volume was occupied by fluid. The surrounding fluid was in equilibrium, meaning the net pressure force on that parcel of fluid exactly balanced its weight. Replace the fluid parcel with a solid object — the surrounding fluid exerts exactly the same pressure forces. Therefore the buoyant force on any object equals the weight of fluid displaced, regardless of the object's shape or material.

Floating vs. Sinking — Applying the Principle

Compare the object's weight W_object with the maximum buoyant force F_b_max (when fully submerged). If W_object < F_b_max, the object floats — it only submerges far enough so that F_b equals W_object. If W_object > F_b_max, the object sinks. At equilibrium for a floating object: ρ_object × V_object × g = ρ_fluid × V_submerged × g, which simplifies to V_submerged / V_object = ρ_object / ρ_fluid. An iceberg (ρ ≈ 917 kg/m³) in seawater (ρ ≈ 1025 kg/m³) has about 89% of its volume below the surface.

The Key Equation

F_b = \rho_{\text{fluid}} \, g \, V_{\text{displaced}}

F_b — Buoyant force (Newton (N)): The net upward force exerted by the fluid on the submerged or partially submerged object.
\rho_{\text{fluid}} — Fluid density (kg/m³): The mass per unit volume of the surrounding fluid (e.g. water ≈ 1000 kg/m³, air ≈ 1.2 kg/m³).
g — Gravitational field strength (m/s² (or N/kg)): The acceleration due to gravity; approximately 9.81 m/s² near Earth's surface.
V_{\text{displaced}} — Volume of fluid displaced (m³): The volume of fluid pushed aside by the object. For a fully submerged object this equals the object's own volume; for a floating object it is only the submerged portion.

Worked Example

Problem. A solid aluminium sphere of radius 0.10 m is completely submerged in water (density 1000 kg/m³). The density of aluminium is 2700 kg/m³. Calculate (a) the buoyant force acting on the sphere and (b) the apparent weight of the sphere when submerged. Take g = 9.81 m/s².

F_b = 41.1 \text{ N}, \quad W_{\text{apparent}} = 69.8 \text{ N}

Answer. Buoyant force = 41.1 N; Apparent weight underwater = 69.8 N

Figure

Diagram

Origins · 1 minArchimedes of Syracuse and how this was discovered

Archimedes of Syracuse

-287–-212 · Greek (Sicilian)

Archimedes formulated the principle of buoyancy in his treatise 'On Floating Bodies' (c. 250 BCE) — the first known work of mathematical physics applied to hydrostatics. He proved rigorously that a solid denser than a fluid sinks, one less dense floats, and derived exactly how much of a floating body lies above the surface. His geometrical proof of the buoyant force predates calculus by nearly two millennia, yet remains entirely correct. Legend holds that Archimedes discovered buoyancy whilst stepping into an overfull bath and immediately ran through the streets of Syracuse shouting 'Eureka!' ('I have found it!'). He had been tasked by King Hiero II to determine whether a crown was pure gold without melting it — buoyancy provided the answer by revealing the crown's density.

Fluid Pressure and Depth — Buoyant force arises directly from the pressure difference between the bottom and top of a submerged object — understanding pressure variation with depth is essential to deriving Archimedes' Principle.
Density and Floating — Whether an object floats or sinks depends on the ratio of its average density to the fluid density, which follows directly from applying Archimedes' Principle at equilibrium.
Gravitational Force and Weight — The buoyant force equals the weight of displaced fluid — correctly calculating this weight requires a solid understanding of gravitational force and how weight relates to mass and g.
Apparent Weight — When submerged in a fluid, an object's apparent weight is reduced by the buoyant force. This directly applies Archimedes' Principle and is commonly measured using a spring balance in experiments.