Buoyancy

Buoyancy is the net upward force a fluid exerts because pressure is greater at greater depth. Archimedes' principle packages that pressure difference into one practical rule: the buoyant force equals the weight of displaced fluid.

Archimedes' principle

F_B = ρ_fluid g V_displaced

The displaced volume is the part of the object below the fluid surface. For a completely submerged object, it equals the object's external volume. The formula does not depend directly on the object's own density.

Buoyancy is a pressure effect. The bottom of a submerged object is deeper, so the upward pressure force there exceeds the downward pressure force on the top.

Floating, sinking and neutral buoyancy

An unsupported object accelerates according to the competition between weight and buoyancy. If its average density is less than the fluid's density, it can rise and float. If it is greater, it tends to sink. Equal average densities allow neutral buoyancy.

Float or sink?

Compare the object's average density with the fluid density.

Object mass (kg) Object volume (m³) Fluid density (kg/m³)

Why a steel ship can float

For a floating object, the submerged fraction reflects the density ratio. With uniform densities, V_sub/V = ρ_object/ρ_fluid.

A solid lump of steel is denser than water, but a ship is mostly hollow. What matters for floating is the average density of the entire ship, including the air-filled volume. Its hull can displace enough water that the weight of displaced water balances the ship's weight.

The same idea applies to submarines, which change their average density by moving water into or out of ballast tanks. A floating body settles until it displaces exactly enough fluid for F_B = mg; adding cargo makes it sit deeper because more fluid must be displaced.

Apparent weight

When an object is supported while submerged, buoyancy reduces the support force. If the object is fully submerged and at rest, a scale often reads T = mg − F_B. The object's true gravitational weight has not changed; the fluid supplies part of the upward support.

What changes with depth?

For a rigid object fully submerged in an incompressible liquid of constant density, the buoyant force is approximately independent of depth: both top and bottom pressures increase, but their difference remains tied to ρg and the object's vertical extent. In compressible fluids or when the object changes volume, depth can matter.

Gases also produce buoyancy. A helium balloon rises not because helium has “negative weight,” but because the surrounding air displaced by the balloon weighs more than the helium plus the balloon envelope. The net upward force is buoyancy minus the total weight.

Worked examples

1. Fully submerged block

A 0.020 m³ block is fully submerged in water. What buoyant force acts on it?

Solution

F_B = ρgV = 1000 × 9.81 × 0.020 = 196 N upward.

2. Floating fraction

A uniform piece of wood has density 650 kg m⁻³ and floats in freshwater. What fraction of its volume is submerged?

Solution

For floating equilibrium, ρ_fluidgV_sub = ρ_woodgV. Thus V_sub/V = 650/1000 = 0.65, or 65%.

3. Apparent weight

A 10 kg solid of volume 0.0040 m³ is fully submerged in water and held by a scale. What does the scale read?

Solution

Weight = 10 × 9.81 = 98.1 N. Buoyant force = 1000 × 9.81 × 0.0040 = 39.2 N.

Tension = 98.1 − 39.2 = 58.9 N.