The Physics of Buoyancy: Why Ships Float, Submarines Dive, and Helium Balloons Rise

The Bathtub Insight

Around 250 BCE, so the story goes, a Greek mathematician named Archimedes climbed into a full bathtub and noticed the water spilling over the sides. He supposedly leapt out and ran naked through the streets of Syracuse shouting “Eureka!” — I have found it.

The story is almost certainly embellished (Vitruvius wrote it down two centuries after the fact), but the physics is sound. What Archimedes realised, watching the water level rise as his body entered the bath, was that the volume of water displaced was equal to the volume of his body submerged. And from this observation, he extracted one of the most useful principles in all of physics: the upward force on a submerged object equals the weight of the fluid it displaces.

That’s it. That’s the whole principle. And it explains why a 100,000-tonne aircraft carrier floats while a 5-gram marble sinks, why you feel lighter in a swimming pool, why hot air balloons rise, how submarines control their depth, and why continents — yes, entire continents — float on the Earth’s mantle.

Why the Force Points Up

Before we get to ships and balloons, let’s understand why a submerged object feels an upward push at all. The answer is pressure — specifically, the fact that pressure in a fluid increases with depth.

If you’re sitting in a swimming pool, the water above you pushes down and the water below you pushes up. But the water below you is deeper, and therefore at higher pressure. The upward push from below is stronger than the downward push from above. The difference is the buoyant force.

For a simple case, consider a cube submerged in water. The pressure on the top face is ρgh₁ (where ρ is the fluid density, g is gravitational acceleration, and h₁ is the depth of the top face). The pressure on the bottom face is ρgh₂, where h₂ is the depth of the bottom face. The net upward force is the pressure difference times the area of the face: ρg(h₂ - h₁) × A. But (h₂ - h₁) × A is just the volume of the cube, so the net upward force is ρgV — the weight of fluid that would occupy that volume.

That’s Archimedes’ principle, derived purely from pressure differences. No Eureka needed — just the fact that pressure increases with depth.

The beauty of this derivation is that it works for any shape, not just cubes. You can break any irregular object into infinitesimal vertical columns, apply the pressure argument to each, and sum them up. The result is always the same: buoyant force equals the weight of displaced fluid.

Why Steel Ships Float

This is the question that trips people up. Steel is about 7.8 times denser than water. A solid block of steel sinks without hesitation. So how can a ship made of steel float?

The answer is geometry. A ship isn’t a solid block of steel — it’s a hollow shell. The hull encloses an enormous volume of air, and the average density of the ship (steel + air + cargo + everything else) is what matters, not the density of the steel alone.

Consider a crude model. A ship with a mass of 50,000 tonnes floats when it displaces 50,000 tonnes of seawater. Since seawater has a density of about 1,025 kg/m³, this requires displacing roughly 48,800 cubic metres of water. A ship hull 300 metres long, 40 metres wide, and with an average draught (submerged depth) of about 4 metres displaces about 48,000 cubic metres. The numbers work.

The ship sinks into the water until the weight of water displaced equals the ship’s weight. At that equilibrium, the buoyant force exactly balances gravity, and the ship sits at a stable waterline. Load more cargo, and the ship sits lower. Remove cargo, and it rises. The waterline is a real-time readout of the ship’s total weight.

This is also why overloading a ship is dangerous. There’s a maximum draught — marked by the Plimsoll line painted on the hull — beyond which the deck approaches the waterline. Exceed it, and waves can wash over the deck, flooding the ship and potentially capsizing it. The physics gives you exactly one chance: the weight of displaced water must always exceed the ship’s weight by a safe margin.

Stability: Why Ships Don’t Flip

Floating is necessary but not sufficient. A ship also needs to be stable — if it tilts, it should right itself rather than rolling over.

When a ship heels (tilts) to one side, the submerged shape changes. More hull is submerged on the lower side, less on the higher side. The centre of buoyancy — the geometric centre of the displaced water — shifts toward the submerged side. This creates a torque that, if the geometry is right, pushes the ship back upright.

The critical concept is the metacentre — the point around which the ship effectively pivots when it tilts. If the metacentre is above the ship’s centre of gravity, the ship is stable: any tilt produces a restoring torque. If the metacentre falls below the centre of gravity, the ship is unstable and will capsize.

This is why cargo must be loaded carefully. Heavy items placed high in the ship raise the centre of gravity, reducing the metacentric height and making the vessel less stable. Heavy items placed low keep the centre of gravity below the metacentre, maximising stability. Every cargo ship’s loading plan is, at its core, a buoyancy and stability calculation.

It’s also why icebreakers and sailboats have such different hull designs. An icebreaker has a broad, shallow hull with the weight concentrated low — extremely stable but not very manoeuvrable. A racing sailboat has a deep keel with a heavy bulb at the bottom — the low centre of gravity lets it heel dramatically without capsizing, but the narrow hull makes stability more sensitive to how the crew positions their weight.

Submarines: Controlling Density on Demand

A submarine is, in a sense, a ship that has solved the problem of being both denser and less dense than water — on command.

The trick is ballast tanks: large compartments in the hull that can be flooded with seawater or blown clear with compressed air. When the tanks are full of air, the submarine’s average density is less than seawater, and it floats at the surface like any other ship. To dive, valves open and seawater floods the ballast tanks, increasing the submarine’s mass without significantly changing its volume. The average density rises above 1,025 kg/m³, and the submarine sinks.

To hover at a specific depth — neutral buoyancy — the submarine adjusts its density to exactly match the surrounding seawater. This is trickier than it sounds, because seawater density varies with temperature and salinity (which change with depth and location). Fine-tuning is done with smaller trim tanks and by adjusting the angle of hydroplanes — small fins that generate lift or downward force as the submarine moves through the water, much like the wings of an aeroplane.

To surface, high-pressure air from storage tanks blows the seawater out of the ballast tanks. The submarine’s density drops below that of seawater, and it rises.

The hull must withstand the pressure of the surrounding water, which increases by about 1 atmosphere for every 10 metres of depth. At 300 metres — a typical operating depth for a military submarine — the pressure is about 30 atmospheres, or roughly 3 million pascals. At the crush depth (the depth at which the hull fails), the force on each square metre of hull can exceed 10 million newtons. This is why submarine hulls are built from high-strength steel or, in some cases, titanium — materials that combine strength with the ability to withstand repeated pressure cycling.

Hot Air and Helium: Buoyancy in Gases

Archimedes’ principle doesn’t care whether the fluid is liquid or gas. Air is a fluid. And anything less dense than air will experience a net upward force — just as anything less dense than water floats to the surface of a pool.

A hot air balloon works by heating the air inside the balloon envelope. Hot air is less dense than cool air (the gas expands, so fewer molecules occupy each cubic metre). The balloon displaces a volume of cool ambient air, and the buoyant force — equal to the weight of that displaced air — exceeds the weight of the hot air plus the balloon and basket. The balloon rises.

The numbers are instructive. Air at 20 °C and sea level has a density of about 1.2 kg/m³. Heat it to 100 °C (a typical hot air balloon operating temperature) and the density drops to about 0.95 kg/m³. The difference — about 0.25 kg/m³ — is the net lifting force per cubic metre. A balloon with a volume of 2,800 m³ (a typical sport balloon) produces a gross lift of about 700 kg — enough for the balloon envelope, basket, burner, fuel, and two to three passengers.

Helium balloons work differently: instead of heating air, they use a gas that’s inherently less dense. Helium has a density of about 0.164 kg/m³ at sea level — roughly seven times lighter than air. This gives much greater lift per cubic metre than hot air, which is why helium balloons can be much smaller than hot air balloons for the same payload.

Hydrogen is even lighter (density about 0.082 kg/m³ — nearly twice the lift of helium), which is why early airships like the Hindenburg used hydrogen. The trade-off, of course, was flammability. The Hindenburg disaster of 1937 effectively ended the hydrogen airship era, and modern lighter-than-air craft use non-flammable helium exclusively.

As any balloon rises, the atmospheric pressure decreases and the gas inside expands. A weather balloon launched at ground level — about 1.5 metres in diameter — expands to roughly 6–8 metres in diameter by the time it reaches 30 km altitude, where the pressure is less than 1% of sea level. Eventually the balloon material can’t stretch further, and it bursts.

The Dead Sea: Buoyancy You Can Feel

If you’ve ever swum in the Dead Sea — or seen photos of people reclining on its surface reading newspapers — you’ve seen extreme buoyancy in action.

The Dead Sea has a salinity of about 34%, roughly ten times that of normal ocean water. This gives it a density of approximately 1,240 kg/m³. Since the human body has an average density of about 1,010 kg/m³ (slightly above fresh water, which is why you can barely float in a pool), you’re dramatically less dense than Dead Sea water. You displace far less water than your full body volume before the buoyant force matches your weight. The result: you float high, with a significant portion of your body above the surface.

The same principle, less dramatically, is why you float more easily in the ocean than in a swimming pool. Seawater (1,025 kg/m³) provides about 2.5% more buoyancy than fresh water (1,000 kg/m³). It’s subtle, but swimmers notice it.

This also matters for ships. A ship moving from salt water to fresh water — entering a river from the ocean, for example — sits lower in the water because fresh water provides less buoyant force per unit volume. Ship designers account for this: the Plimsoll line on a hull actually includes multiple marks for different water types (tropical saltwater, summer saltwater, fresh water), each indicating a safe maximum draught for that condition.

Isostasy: Floating Continents

Here’s something that might surprise you: continents float.

Not on water, obviously. But the Earth’s crust — relatively light rock with a density of about 2,700 kg/m³ — sits on the mantle, a layer of much denser rock (about 3,300 kg/m³) that behaves as a very viscous fluid over geological timescales. The crust floats on the mantle for exactly the same reason a block of wood floats on water: it’s less dense than the fluid beneath it.

This is the principle of isostasy, and it explains why mountain ranges have deep roots. Just as an iceberg has most of its mass below the waterline, a mountain range extends a “root” of crustal rock deep into the mantle. The Himalayas — with peaks reaching nearly 9 km above sea level — are estimated to have a crustal root extending about 70 km below the surface. The total crustal thickness under the Himalayas is roughly 80 km, compared to about 35 km under flat continental regions.

When glaciers covered large parts of North America and Scandinavia during the last ice age, the weight of the ice sheet — up to 3 km thick — pushed the crust down into the mantle. When the ice melted (starting about 11,000 years ago), the crust began rebounding upward. Scandinavia is still rising at about 1 centimetre per year. The Gulf of Bothnia has risen by roughly 300 metres since the ice retreated. This post-glacial rebound is direct evidence of the Earth’s crust floating on the mantle — pushed down by added weight, rising back up when the weight is removed.

Archimedes’ principle, applied at the scale of a planet. Same physics, different medium, incomprehensibly different scale.

What Buoyancy Teaches Us

I think buoyancy is one of the most satisfying topics in physics because it connects so many scales of experience. The same principle — an object in a fluid is pushed up by a force equal to the weight of fluid it displaces — explains why you feel lighter in a bath, why supertankers float, why weather balloons rise, and why Scandinavia is still recovering from the ice age.

The physics is simple. The applications are vast. And the intuition, once you have it, never fails: if the average density of the object is less than the fluid, it floats. If it’s greater, it sinks. If it’s equal, it hovers.

Archimedes worked this out 2,200 years ago in a bathtub. We’ve built submarines, airships, cargo fleets, and models of continental dynamics on the same insight. That one moment of noticing — water rises when you get in, and the amount it rises tells you something — turned out to be one of the most productive observations anyone has ever made.

Eureka indeed.