Buoyancy and Archimedes' Principle: Formula and Examples

Ships of steel float while pebbles sink; balloons rise while stones fall. The invisible push behind it all is buoyancy — the upward force a fluid exerts on anything immersed in it. Archimedes of Syracuse found its rule over two thousand years ago, reportedly leaping from his bath shouting "Eureka!"

Buoyancy governs ships, submarines, hot-air balloons and your own body in a swimming pool. Archimedes' principle states the rule with beautiful simplicity: the buoyant force equals the weight of the fluid displaced by the object.

Key Concepts

• Buoyant force: the net upward push, because fluid pressure is greater at the bottom of an object than at its top.
• Displaced fluid: the fluid that would fill the space taken by the object's submerged part.
• Archimedes' principle: buoyant force = weight of displaced fluid, acting upward.
• Sink or float: denser than the fluid → sinks; less dense → rises until partly above the surface.
• Apparent weight: an immersed object feels lighter by exactly the buoyant force.

Formulas

F_b = ρ_fluid × V_submerged × g, with ρ in kg/m³, V in m³ and g = 9.8 m/s², giving newtons. Compare with the object's weight W = ρ_object × V_total × g: it sinks if ρ_object > ρ_fluid and floats if ρ_object < ρ_fluid. The formula assumes the fluid is at rest.

Worked Example

An iron block of 0.01 m³ (10 litres) held fully underwater feels F_b = 1000 × 0.01 × 9.8 = 98 N — the weight of 10 litres of water. Its own weight is 7870 × 0.01 × 9.8 = 771.3 N, so its apparent weight is 771.3 − 98 = 673.3 N: about 10 kg lighter, yet it still sinks since 771.3 > 98. Now shape that same iron into a hollow boat: the average density, counting the enclosed air, drops below 1000 kg/m³ — the displaced water can now weigh as much as the boat, and steel floats.

Common Misconceptions

• "Buoyancy depends on the object's weight." Only on the displaced fluid's weight — a 1 kg beach ball and a 1 kg rock of equal volume feel the same push; the rock sinks because its own weight is larger.
• "Heavy things always sink." Ships are heavy yet float: average density including enclosed air is what counts.
• "Buoyancy works only in water." Every fluid qualifies — hot-air balloons rise on displaced cool air, and you feel a tiny push from the air around you.

Real-World Applications

• Ships: hulls displace enough water that its weight equals the ship's loaded weight.
• Submarines: ballast tanks flood or blow out water to dive, hover or surface.
• Hot-air balloons: heated inside air is less dense than the displaced cool air, which lifts the balloon.
• Hydrometers: floating instruments whose depth reveals a liquid's density — testing battery acid and milk purity.
• Swim bladders: fish adjust a gas-filled sac to hover at any depth without effort.

Buoyant force equals the weight of displaced fluid — one line that explains steel ships, rising balloons and your lightness in a pool. Compare the densities, and the lighter average wins the upward battle.