BuoyancyWhy does ice float, a steel ship stay up, and gold sink?

Where does buoyancy come from?

A liquid presses on a submerged object from all sides, and the deeper, the harder: pressure grows with depth h as p = ρ_liq·g·h. So the liquid pushes harder on the bottom face of a cube than on the top one. The difference between these pushes is an upward force — buoyancy:

F_b=ρ_liq · g · V_sub

ρ_liq is the density of the liquid, g = 9.81 m/s² is the acceleration due to gravity, and V_sub is the volume of the submerged part of the object. The product ρ_liq·V_sub is the mass of liquid the object has pushed out of its place. Archimedes' principle: the buoyant force equals the weight of the liquid displaced by the object.

Float or sink?

The weight of an object is Q = m·g = ρ_object·g·V. If the object is less dense than the liquid, it rises and sinks in only as far as it takes for buoyancy to balance its weight. Then

V_subV=ρ_objectρ_liquid

Pine (500 kg/m³) sinks halfway into water, ice (917 kg/m³) 91.7% of the way. When the densities are equal, the object hovers anywhere in the liquid. When the object is denser than the liquid, it sinks and rests on the bottom, and the bottom supports it with a force N = Q − F_b. The liquid level in the tank rises by the displaced volume divided by the area of the bottom: Δh = V_sub / S.

Spring scale and overflow vessel

F=Q − F_b

A submerged object “loses” exactly as much weight as the liquid that spilled from the overflow vessel into the beaker. According to the Roman architect Vitruvius, Archimedes hit on this idea in the bath, when King Hiero II asked him to check whether a new crown was pure gold. A crown with silver mixed in has a larger volume than gold of the same mass, so it displaces more water. Legend has it he ran out into the street shouting “Eureka!” (“I've found it!”).

Why does a steel ship float?

Steel is almost 8 times denser than water, but the hull is hollow. What counts is the average density of the whole ship: the mass of the hull plus cargo divided by the volume the hull takes up, including the air inside:

ρ_avg=m_hull + m_cargoV_hull

As long as it is lower than the density of water, the ship floats. Once water spills over the side, it pushes the air out of the hull, the average density rises above that of water and the ship goes to the bottom. The Plimsoll line on the hull shows how deep the ship may be loaded so that it keeps a safe reserve of buoyancy for the waves.

Cartesian diver

An upside-down vial holds a bubble of air. When you squeeze the bottle, the water pressure rises and the air in the vial is compressed (at constant temperature):

p₁ · V₁=p₂ · V₂

A smaller bubble displaces less water, so the buoyancy drops and the diver sinks. When you let go, the air expands and the diver rises. Deeper down the pressure is higher (p = p₀ + Δp + ρ·g·h), which is why the balance is unstable: a diver that has started to sink sinks faster and faster.

Model simplifications. Motion is one-dimensional: objects move only up and down and don't rotate. Liquid drag and the damping of bobbing follow a simple formula, and in oil and glycerin the motion is damped more strongly. We ignore surface tension, the compressibility of liquids and temperature changes. Densities are typical (pine ranges from about 400 to 600 kg/m³, Styrofoam 15–35 kg/m³). We take g = 9.81 m/s². The tank's bottom is 50 × 16 cm, so 1 L of displaced liquid raises the level by 1.25 cm.