Buoyancy and Archimedes' Principle — Why Things Float

Where the upward push comes from

Push a beach ball underwater and you feel it fight back. That upward shove is buoyancy — and it acts on everything immersed in a fluid, from submarines to your own body in a swimming pool.

Buoyancy is not magic. Deep in a fluid, pressure is higher than near the surface ($P = \rho g h$). So the bottom of a submerged object gets pushed up harder than the top gets pushed down. Add up that pressure difference over the object's surface and you get a net upward force — the buoyant force.

Archimedes' principle

Over 2,200 years ago, Archimedes realized the buoyant force has a beautifully simple size: it equals the weight of the fluid the object displaces.

$$F_b = \rho_{fluid} V_{displaced} g$$

Dunk a 0.01 m³ rock in water and it shoves aside 0.01 m³ of water weighing $1000 \times 0.01 \times 9.8 = 98\ \mathrm{N}$. The water pushes back up with exactly 98 N, no matter what the rock is made of.

For a floating object, only the submerged part displaces fluid. The object settles at whatever depth makes the displaced water's weight equal its own weight — which is why floating objects bob at a fixed waterline.

Float or sink: a density contest

Whether something floats or sinks is a density contest, not a weight contest:

• Object denser than fluid ($\rho_{object} > \rho_{fluid}$): weight beats maximum buoyancy → sinks.
• Object less dense ($\rho_{object} < \rho_{fluid}$): buoyancy wins before full submersion → floats with part above the surface.
• Equal densities: hovers, neutrally buoyant — like a scuba diver perfectly trimmed, or a fish holding depth.

A steel nail ($ho \approx 7800\ \mathrm{kg/m^3}$) sinks in water. A steel ship floats because its average density — steel plus all the air inside the hull — is below $1000\ \mathrm{kg/m^3}$. Shape matters only because it changes how much air gets included in the average.

Worked mini-example: how deep a raft sits

A wooden raft has mass $200\ \mathrm{kg}$ and floats in freshwater. What volume of the raft sits below the waterline?

• Floating means: $F_b =$ weight, so $\rho_{water} V_{sub} g = m_{raft} g$
• Cancel $g$: $V_{sub} = m_{raft} / \rho_{water}$
• Substitute: $V_{sub} = 200 / 1000 = 0.20\ \mathrm{m^3}$

Answer: $0.20\ \mathrm{m^3}$ of the raft is submerged — exactly the volume of water weighing 200 kg.

Notice the raft's own size never entered the calculation. A bigger raft would simply float higher, displacing the same $0.20\ \mathrm{m^3}$.

Steering buoyancy: submarines, balloons and hydrometers

Buoyancy is a control knob, not just a curiosity. Submarines take water into ballast tanks to raise their average density and sink, then blow the water out with compressed air to rise. Hot-air balloons heat their air until the whole balloon's average density drops below the surrounding cool air. Fish adjust a gas-filled swim bladder to hover without swimming.

Engineers use the same principle in reverse: a hydrometer is a weighted float that sinks deeper in thin liquids and rides high in dense ones, letting mechanics read battery-acid or antifreeze strength straight off its stem.

Variables and units

Common mistakes

Real-world applications

• Ship designers shape hulls so the displaced water's weight always exceeds the ship's weight, with margin for cargo.

• Submarines fine-tune buoyancy with ballast tanks — flooding them to sink, blowing air in to rise.

• Hot-air balloons fly because heated air inside is less dense than the cool air outside, creating net upward buoyant force.

• Hydrometers measure liquid density (like battery acid or antifreeze) by how high they float.