Buoyancy and Archimedes’ Principle — Why Giant Ships Don’t Sink 🚢

Buoyancy and Archimedes’ Principle — Why Giant Ships Don’t Sink 🚢

A puzzle 🤔: a small steel nail sinks in water. But a 100,000-tonne steel ship stays afloat 🤯. How? Archimedes cracked this 2200 years ago in his bathtub — and reportedly ran naked into the streets shouting “Eureka!”. The answer is captured completely by two simple formulas.

The core idea in one paragraph 📌

Any submerged object experiences an upward force (the buoyant force) that comes from the pressure difference between its top and bottom. Archimedes’ principle states this force exactly equals the weight of the fluid displaced: \( F_b = \rho_\text{fluid} \cdot V_\text{submerged} \cdot g \). Whether the object floats, hovers, or sinks depends on the ratio of the average density of the object to the density of the fluid — not on total weight. That’s why a hollow ship floats while a solid nail sinks.

Where it comes from — a pressure difference 🎯

Recall that \( P = P_0 + \rho g h \). The pressure on the bottom face of a submerged object is greater than on the top:

\[ F_b = P_\text{bottom} A – P_\text{top} A = \rho_\text{fluid} \cdot g \cdot h \cdot A = \rho_\text{fluid} \cdot V \cdot g \]

\( \rho \cdot V \) is exactly the mass of the fluid displaced. Multiply by \( g \) to get its weight. That’s Archimedes’ principle in a single derivation.

The master formula 📐

\[ \boxed{F_b = \rho_\text{fluid} \cdot V_\text{submerged} \cdot g} \]

Three scenarios — float, hover, sink ⚖️

Compare the object’s weight (\( W = \rho_\text{obj} V g \)) to the maximum buoyant force (\( F_b^\text{max} = \rho_\text{fluid} V g \)):

Ratio Outcome Example
\( \rho_\text{obj} > \rho_\text{fluid} \) Sinks 🪨 Nail in water
\( \rho_\text{obj} < \rho_\text{fluid} \) Floats on surface 🛟 Ice in water
\( \rho_\text{obj} = \rho_\text{fluid} \) Hovers at any depth Fish with swim bladder

Submerged fraction for a floating object — a beautiful rule 🧊

For a floating object in equilibrium: \( F_b = W \), which gives:

\[ \rho_\text{fluid} \cdot V_\text{submerged} \cdot g = \rho_\text{obj} \cdot V_\text{total} \cdot g \]

\[ \Longrightarrow \boxed{\frac{V_\text{submerged}}{V_\text{total}} = \frac{\rho_\text{obj}}{\rho_\text{fluid}}} \]

Why ships stay afloat 🚢

Steel has density ~7870 kg/m³ — about 7.8× denser than water. A solid-steel ship would sink instantly. The trick: a ship is hollow. Its average density = total mass / (total volume, including the air inside) — which is less than water.

Kitchen experiment 🧪: float a sheet of foil on water → it floats. Crumple it into a ball → it sinks. Same mass, different average density.

Submarines — tuning your own density 🤿

A submarine has ballast tanks:

Archimedes’ principle turned into engineering ⚙️.

Python analysis 🐍

1) Submerged fraction for various objects

rho_water, rho_sea, rho_mercury = 1000, 1025, 13595
rho_dead_sea = 1240

objects = {
    "Ice":                917,
    "Pine wood":          420,
    "Ebony wood":        1200,   # denser than water!
    "Human body":        1010,
    "Aluminum":          2700,
    "Iron":              7870,
    "Gold":             19300,
}

def submerged_frac(rho_obj, rho_fluid):
    """Fraction of the object's volume below the fluid surface (if it floats)."""
    if rho_obj > rho_fluid:
        return None   # it sinks
    return rho_obj / rho_fluid

for name, rho in objects.items():
    f = submerged_frac(rho, rho_water)
    if f is None:
        print(f"{name:12s} → sinks")
    else:
        print(f"{name:12s} → {f*100:5.1f}% under freshwater")

# Would gold float on mercury? (gold 19300, mercury 13595)
print(f"\nGold on mercury: {submerged_frac(19300, rho_mercury)*100:.1f}%")
# → 141.9% → sinks (gold is denser than mercury)

2) A ship’s cargo capacity

# A steel ship 200 × 30 m (footprint) and 20 m tall
L, W, H = 200, 30, 20    # meters
V_total = L * W * H       # 120,000 m³

# Maximum total mass (ship + cargo) before it sinks?
rho_water = 1000
max_displacement = rho_water * V_total   # kg
print(f"Max total mass: {max_displacement/1000:,.0f} tonnes")
# 120,000 tonnes — comparable to a midsize supertanker

# If the empty ship is 20,000 tonnes:
empty_mass = 20_000_000   # kg
max_cargo = max_displacement - empty_mass
print(f"Max cargo capacity: {max_cargo/1000:,.0f} tonnes")
# 100,000 tonnes

Take-home summary 🎁

\( F_b = \rho V g \) — three letters that describe the entire physics of floating. Archimedes’ principle: the buoyant force equals the weight of the fluid displaced. What matters is the density ratio, not total weight. Ice floats ~92% submerged; a human floats ~98% submerged. Ships float because they’re hollow. Submarines tune their density with ballast tanks. One equation, endless applications 😎.

“Nice to know” box: “Eureka!” 💡

King Hiero of Syracuse asked Archimedes to determine whether his crown was pure gold or had been diluted with silver by a dishonest jeweler — without damaging the crown. Archimedes puzzled over it for days. Then, stepping into his bathtub, he watched the water spill over the edge and it hit him: measure the water displaced, get the crown’s volume, then compute its density and compare to pure gold’s! Legend has it he was so excited he ran naked through the streets shouting “Eureka! Eureka!” (I found it!) 🏃‍♂️. Twenty-two centuries later, “eureka” still means that flash of discovery.

Test yourself 📝

References and further exploration 📚

Articles and reference

Videos (YouTube)

External simulators

On this site 🔗

In the final section of this chapter, we hit Bernoulli’s principle 🛩️ — why airplanes fly, why roofs blow off in storms, and how spinning balls curve their trajectories ⚽. See you there 👋

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