Archimedes’ Principle

Archimedes’ principle states that any object placed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. It explains why ships float, why you feel lighter in water, why hot-air balloons rise, and why submarines can sink and resurface at will. Discovered by the Greek mathematician Archimedes around 250 BCE, the principle is still the foundation of fluid statics, naval architecture, and density measurement.

Statement and Formula

For an object fully or partially submerged in a fluid at rest, the buoyant force $ F_b $ acting upward on the object equals the weight of the displaced fluid:

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

where $ \rho_{\text{fluid}} $ is the fluid’s density, $ V_{\text{displaced}} $ is the volume of fluid pushed aside by the object, and $ g $ is gravitational acceleration (9.81 m/s²).

Why It Works

Pressure in a fluid increases with depth. The bottom of a submerged object sits in a higher-pressure region than the top, so the upward push on the bottom exceeds the downward push on the top. The net upward force is the buoyant force. A short calculation shows this difference equals exactly $ \rho g V $ — the weight of the displaced fluid. The result depends only on the fluid and the volume, not on the object’s shape or material.

Floating, Sinking, and Neutral Buoyancy

Sinks if the object’s average density exceeds the fluid’s density: its weight is greater than the maximum buoyant force.
Floats if its average density is less than the fluid’s: it settles at a depth such that the weight of displaced fluid equals its own weight.
Neutrally buoyant if the densities are equal: it stays at any depth without rising or sinking. Submarines and SCUBA divers actively adjust their average density to achieve this state.

A steel ship floats not because steel is light — it’s not — but because the ship’s hull encloses a large volume of air. The average density of (steel hull + enclosed air) is well below water’s density.

Worked Examples

Example 1: Wooden block in water. A wooden cube of side 0.10 m and density 600 kg/m³ floats in water ($ \rho = 1000 $ kg/m³). At equilibrium, weight = buoyant force, so $ 600 \cdot 0.001 \cdot g = 1000 \cdot V_{\text{sub}} \cdot g $. Solving: $ V_{\text{sub}} = 6 \times 10^{-4} $ m³, which is 60% of the cube’s volume. The cube floats with 60% submerged and 40% above water — matching the density ratio.

Example 2: Iceberg. Ice density is 917 kg/m³, seawater density 1025 kg/m³. The fraction submerged is $ 917/1025 \approx 0.895 $. Roughly 90% of an iceberg sits below the waterline — the source of the iceberg metaphor.

Example 3: Apparent weight underwater. A 5.0 kg rock of density 2500 kg/m³ has volume $ 5/2500 = 2 \times 10^{-3} $ m³. Submerged in water, the buoyant force is $ 1000 \cdot 2 \times 10^{-3} \cdot 9.81 = 19.6 $ N. Its real weight is $ 5 \cdot 9.81 = 49 $ N. Apparent weight in water: $ 49 – 19.6 \approx 29 $ N — feels 40% lighter.

Applications

Ship design. Naval architects calculate displacement tonnage from the volume of water a hull pushes aside.
Hydrometers. Floating instruments that measure liquid density (urine specific gravity, battery acid, alcohol content in spirits) work entirely by Archimedes’ principle.
Hot-air balloons. Hot air inside the envelope is less dense than the surrounding cool air; the difference produces lift equal to the weight of displaced cool air minus the weight of the heated air and the balloon.
Submarines. Ballast tanks are flooded with seawater to add weight (sink) or blown clear with compressed air to displace water (rise).
Density measurement. Archimedes legendarily used the principle to verify the gold content of King Hiero’s crown — comparing weight in air with apparent weight in water gave the density.

Related study notes: Pascal’s Law, Density, Newton’s Laws of Motion, Pressure.

Frequently Asked Questions

What does Archimedes’ principle state?

Any object fully or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. The formula is F_b = ρ × V × g, where ρ is the fluid’s density, V is the displaced volume, and g is gravitational acceleration.

Why do heavy steel ships float?

Because the average density of the ship — hull plus enclosed air — is less than water’s density. Although steel is denser than water, a hollow steel hull encloses a large volume, mostly air. The total weight is well below the weight of the water the hull can displace, so it floats.

What percentage of an iceberg is below the water?

Around 90%, sometimes called the ‘iceberg ratio.’ Ice density (~917 kg/m³) divided by seawater density (~1025 kg/m³) gives 0.895. That fraction of the iceberg’s volume must sit below the waterline to displace enough water to balance the iceberg’s weight.

Why do you feel lighter in a swimming pool?

Because the water exerts an upward buoyant force on your body. Your apparent weight equals your actual weight minus this buoyant force. Since your body’s average density (~985 kg/m³) is close to water’s, the buoyant force is close to your full weight — so you feel almost weightless when fully submerged.

How do submarines sink and float?

They adjust their average density. Flooding ballast tanks with seawater increases the sub’s total mass without changing its volume, so its average density rises above seawater’s and it sinks. Blowing the tanks empty with compressed air reduces mass and the sub rises. Modern subs make fine adjustments using trim tanks to stay neutrally buoyant.

Did Archimedes really discover this in a bathtub?

The famous ‘Eureka!’ story comes from Roman writer Vitruvius about 200 years after Archimedes. The historical claim — that he noticed his body displacing water in a bath — may or may not be accurate. What’s certain is that Archimedes (c. 287-212 BCE) wrote ‘On Floating Bodies’, a rigorous mathematical treatment of buoyancy that remains the foundation of fluid statics.