Archimedes' Principle: Buoyancy & Density¶

The Story Behind the Science¶

The King's Crown¶

Syracuse, Sicily, 250 BCE. King Hiero II had a problem. He'd given a goldsmith a chunk of pure gold to make a crown. The crown came back beautiful, weighing exactly as much as the gold he'd provided. But Hiero was suspicious. Had the goldsmith stolen some gold and mixed in cheaper silver instead?

He called Archimedes, the greatest mathematician and engineer in Syracuse. "Find out if my crown is pure gold," the king said. "But don't damage it."

This was impossible. To test purity, you'd normally melt it down or cut into it. Archimedes couldn't do either. He was stuck.

The Bath That Changed Science¶

According to legend (probably embellished, but the core is true), Archimedes went to the public baths, still obsessing over the problem. As he lowered himself into the water, he noticed something: water spilled over the edge of the tub.

More importantly: he felt lighter. The water was holding him up.

And suddenly, the pieces clicked. Different materials have different densities. Gold is denser than silver. If the crown had silver mixed in, it would be less dense than pure gold - which means for the same weight, it would take up more space. More volume means more water displaced.

"Eureka!" he allegedly shouted - Greek for "I have found it!" - and ran home naked through the streets, too excited to dress.

The Test¶

Here's what Archimedes did (probably not naked this time):

He took the crown and a piece of pure gold with the exact same weight. He submerged each one in water and measured how much water each displaced.

If the crown was pure gold: same weight, same density, same volume → same water displaced
If silver was mixed in: same weight, lower density, bigger volume → more water displaced

The crown displaced more water than the pure gold. The goldsmith had cheated. Whether he lost his head or just his reputation, history doesn't record. But Archimedes had discovered something fundamental about how fluids work.

Why It Matters¶

Before Archimedes, people knew things floated or sank, but they didn't know why. Ships floated, rocks sank, and nobody could explain the pattern.

Archimedes figured it out: an object submerged in a fluid experiences an upward force (buoyancy) equal to the weight of the fluid it displaces. That's why huge steel ships float - they displace massive amounts of water, more than enough to counteract their weight. That's why a stone sinks - it's denser than water, so it can't displace its own weight.

This principle became the foundation of naval architecture, submarine design, hot air balloons, and understanding how fish control their depth. It's why you feel lighter in a pool. It's why icebergs float with most of their mass underwater.

The Principles¶

Density¶

Mass per unit volume. How much "stuff" is packed into a given space.

\[ \rho = \frac{m}{V} \]

Where: - ρ (rho) = density (kg/m³) - m = mass (kg) - V = volume (m³)

Why this matters: Two objects with the same mass can have different volumes if they have different densities. Gold is about 19 times denser than water. Silver is about 10 times denser.

Archimedes' Principle (Buoyancy)¶

An object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced.

\[ F_b = \rho_{fluid} \cdot V_{displaced} \cdot g \]

Where: - F_b = buoyant force (N) - ρ_fluid = density of the fluid (kg/m³) - V_displaced = volume of fluid displaced (m³) - g = gravitational acceleration (≈9.8 m/s²)

Derivation: Why Does Water Push Up?¶

Step 1: Water Pressure Increases with Depth¶

Imagine a column of water. The water at the bottom has to support all the water above it. More depth = more weight = more pressure.

Pressure at depth h:

\[ P = \rho_{fluid} \cdot g \cdot h \]

This is why your ears hurt when diving - more water above you means more pressure.

Step 2: Pressure Pushes in All Directions¶

Water pressure doesn't just push down. It pushes in every direction with equal force. That's why a submerged ball doesn't get crushed flat - pressure from the sides balances pressure from above.

Step 3: More Pressure on the Bottom Than the Top¶

Now imagine a cube submerged in water: - The top face is at depth h₁ - The bottom face is at depth h₂ (deeper, so h₂ > h₁)

Pressure on top face: P₁ = ρ·g·h₁
Pressure on bottom face: P₂ = ρ·g·h₂

The bottom experiences more pressure. This creates a net upward force.

Step 4: Calculate the Net Upward Force¶

Force = Pressure × Area

Each face has area A (the cross-section of the cube).

Force pushing down on top: F₁ = P₁·A = ρ·g·h₁·A
Force pushing up on bottom: F₂ = P₂·A = ρ·g·h₂·A

Net upward force:

\[ F_b = F_2 - F_1 = \rho \cdot g \cdot (h_2 - h_1) \cdot A \]

But (h₂ - h₁) is the height of the cube, and height × area = volume:

\[ F_b = \rho_{fluid} \cdot g \cdot V_{object} \]

And ρ_fluid · V_object is the mass of the water that would fit in that volume - the mass of water displaced. Multiply by g and you get the weight of water displaced.

That's Archimedes' Principle: The buoyant force equals the weight of fluid displaced.

Why Things Float or Sink¶

An object floats if the buoyant force ≥ its weight.

Weight of object: W = m_object · g = ρ_object · V_object · g
Buoyant force: F_b = ρ_fluid · V_object · g

For floating:

\[ \rho_{fluid} \cdot V_{object} \cdot g \geq \rho_{object} \cdot V_{object} \cdot g \]

Cancel V_object and g:

\[ \rho_{fluid} \geq \rho_{object} \]

An object floats if its density is less than the fluid's density.

That's why ice floats in water (ice is less dense), why oil floats on water, why helium balloons rise (helium is less dense than air).

The Crown Test Explained¶

Archimedes had: - Crown: mass m, unknown density ρ_crown - Pure gold: mass m, density ρ_gold ≈ 19,300 kg/m³

If the crown was pure gold, its volume would be:

\[ V_{gold} = \frac{m}{\rho_{gold}} \]

If silver (ρ_silver ≈ 10,500 kg/m³) was mixed in, the crown would be less dense, so same mass = bigger volume.

By measuring water displacement (which gives volume), Archimedes found:

\[ V_{crown} > V_{gold} \]

Therefore:

\[ \rho_{crown} < \rho_{gold} \]

The crown was not pure gold. Busted.

Why This Matters Today¶

Ships: Designed to displace more water weight than the ship weighs
Submarines: Control buoyancy by flooding/emptying ballast tanks
Hot Air Balloons: Heat makes air less dense than surroundings
Hydrometers: Measure liquid density by how deep they float
Swimming: You float because you're mostly water with air in your lungs

Archimedes figured this out 2,200 years ago, sitting in a bathtub. One moment of insight that explained why the world works the way it does.

Pressure in fluids
Specific gravity
Center of buoyancy vs center of gravity (stability)
Terminal velocity in fluids (drag + buoyancy)