A US Navy aircraft carrier weighs approximately 100,000 tonnes fully loaded. Its hull is made of steel — a material roughly 8 times denser than water. It floats effortlessly, carrying 75 aircraft, 6,000 personnel, jet fuel, ammunition, food, and equipment.
A small iron nail dropped into a bathtub sinks immediately.
This apparent paradox — that a 100,000-tonne steel structure floats while a few-gram nail sinks — is one of the most counterintuitive demonstrations in everyday physics. The answer involves a principle discovered 2,250 years ago by a mathematician in a bathtub in Syracuse, and it has nothing to do with the material objects are made of and everything to do with their shape.
The key insight: floating depends not on what an object is made of, but on its average density — the total mass divided by the total volume, including any hollow spaces inside.
"Eureka! Eureka!" — Archimedes, upon realizing the principle of displacement, circa 250 BC, according to Vitruvius (De Architectura, Book 9)
Whether the legend of Archimedes leaping from his bath is historically accurate is uncertain. What is accurate is the principle he is credited with discovering, which explains everything from why ships float to why hot air balloons rise to why your body feels lighter in a swimming pool.
Key Definitions
Buoyancy — The upward force exerted by a fluid on any object immersed in it. Buoyancy results from the pressure difference between the bottom of a submerged object (higher pressure) and its top (lower pressure). The net upward pressure force is the buoyant force. Buoyancy acts on every object in every fluid — air, water, oil — though its magnitude varies with fluid density.
Archimedes' principle — The physical law stating that any object fully or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object. Discovered by Archimedes of Syracuse (circa 287–212 BC). The most precise and complete statement: the buoyant force on an object equals the weight of the fluid the object's volume displaces.
Displacement — The volume (and correspondingly the mass) of fluid that an object pushes aside to occupy its submerged space. In naval engineering, displacement refers specifically to the weight of water a ship displaces — equal to the ship's total weight when floating. A ship displaces water exactly equal to its own weight.
Density — Mass per unit volume (typically expressed in kg/m³ or g/cm³). The density of liquid water is approximately 1,000 kg/m³. Seawater is slightly denser — approximately 1,025 kg/m³ — due to dissolved salts. Steel has a density of approximately 7,900 kg/m³. Whether an object floats or sinks in a fluid depends on its average density compared to the fluid's density.
Average density — For a composite object like a ship (solid hull + hollow air-filled interior), the average density is calculated by dividing the total mass by the total volume including all interior spaces. A ship is essentially a hollow steel shell; its total volume vastly exceeds the volume of steel in its hull. The air inside is approximately 1.2 kg/m³ — far lighter than water. The average density of the entire ship (steel + air) falls well below 1,000 kg/m³.
Center of buoyancy — The geometric center of the submerged volume of an object. The buoyant force acts upward through this point. When a ship tilts, the shape of the submerged volume changes, moving the center of buoyancy. The relationship between the center of buoyancy and the center of gravity determines whether the ship is stable or unstable.
Center of gravity — The point at which all the mass of an object appears to be concentrated for the purpose of gravitational calculations. The gravitational force (weight) acts downward through this point. A ship's stability depends critically on the relative positions of its center of gravity and center of buoyancy.
Metacenter — The point at which a vertical line through the tilted ship's center of buoyancy intersects the original vertical line through the upright ship's center of gravity. If the metacenter is above the center of gravity, the ship is stable and will return upright when tilted. If the center of gravity rises above the metacenter (from cargo loading or flooding), the ship is unstable and will capsize.
Waterline — The line where the hull of a floating vessel meets the water surface. The waterline moves up or down as the ship is loaded or unloaded. Naval architects mark the designed maximum load waterline (the Plimsoll line) on ship hulls to prevent dangerous overloading.
Ballast — Weight added to a ship (traditionally water, gravel, or iron) to lower the center of gravity and improve stability. Modern ships use water ballast tanks — compartments that can be filled with seawater to lower the center of gravity or redistribute weight. Submarines use ballast tanks to control their buoyancy, diving when tanks flood and surfacing when tanks are blown empty with compressed air.
The Physics: Why Objects Float or Sink
Pressure and the Origin of Buoyancy
To understand buoyancy at the deepest level, start with fluid pressure. In any fluid, pressure increases with depth — because deeper fluid has more fluid above it pressing down. At any given depth, fluid exerts equal pressure in all directions (Pascal's law).
When an object is submerged, fluid pushes on it from all sides. But the pressure on the bottom of the object is greater than the pressure on the top — because the bottom is deeper. This pressure difference creates a net upward force: the buoyant force.
The size of this force depends entirely on how much fluid the object displaces. A larger object displaces more fluid, so more fluid is pressing upward with that pressure difference — larger buoyant force. The shape of the object determines how much fluid it displaces at a given depth.
Archimedes' Principle in Numbers
Consider a cube of steel, 10 cm on each side, submerged in water:
Volume: 1,000 cm³ = 0.001 m³
Water displaced: 0.001 m³ × 1,000 kg/m³ = 1 kg of water
Weight of displaced water: approximately 9.8 N (newtons)
This is the upward buoyant force
Mass of the steel cube: 0.001 m³ × 7,900 kg/m³ = 7.9 kg
Weight of the steel cube: approximately 77.4 N
The buoyant force (9.8 N upward) is much less than the weight (77.4 N downward). Net force is downward. The steel cube sinks.
Now consider a steel cube with hollow interior — same outer dimensions (10 cm), but the steel shell is only 2 mm thick:
Outer volume: 1,000 cm³
Steel volume: approximately 118 cm³ (shell only)
Air volume inside: approximately 882 cm³
Mass of steel: 118 cm³ × 7.9 g/cm³ ≈ 932 g
Mass of air inside: 882 cm³ × 0.0012 g/cm³ ≈ 1 g
Total mass: ≈ 933 g = 0.933 kg
Buoyant force (displacing full 1,000 cm³ of water): 1 kg × 9.8 m/s² = 9.8 N
Weight of hollow box: 0.933 kg × 9.8 m/s² = 9.1 N
Buoyant force (9.8 N) exceeds weight (9.1 N). Net force is upward. The hollow steel box floats — and continues to float until enough water enters to bring the average density above that of water.
This is exactly the physics of a ship. A ship is a hollow steel structure whose total volume — steel hull plus all the air-filled interior spaces — has an average density well below that of water.
Density Comparison: Common Materials and Water
| Material / Object | Density (kg/m³) | Floats in Water? |
|---|---|---|
| Air | ~1.2 | Yes (rises) |
| Wood (pine) | ~500 | Yes |
| Ice | ~917 | Yes (90% submerged) |
| Liquid water | 1,000 | Reference |
| Seawater | ~1,025 | Reference |
| Concrete | ~2,300 | No |
| Steel | ~7,900 | No (as solid) |
| Aircraft carrier (average) | ~200-300 (hull + air) | Yes |
| Dead Sea water | ~1,240 | Yes — with much higher buoyancy |
| Lead | ~11,300 | No |
Ships in Practice: Naval Architecture
Designing for Displacement
Naval architects design ships by calculating their displacement at various loading conditions — unloaded ("light ship"), partially loaded, and fully loaded. The ship must float at a safe waterline in all conditions.
The Plimsoll line, marked on the hull of commercial ships since 1876 (following British legislation championed by Samuel Plimsoll), indicates the maximum safe waterline in different water conditions. Different lines mark maximum loading in tropical seawater (T), summer saltwater (S), summer freshwater (F), winter saltwater (W), and winter North Atlantic (WNA). Seawater is denser than freshwater, so a ship rides higher (floats with less hull submerged) in saltwater and lower in freshwater — the Plimsoll marks account for this.
A ship that sits too deep in the water — whose waterline is above the appropriate Plimsoll mark — is in danger of swamping in rough seas, where waves can push the ship lower temporarily.
Why Ships Have the Shape They Do
The hull shape of a ship is engineered to:
- Provide sufficient volume to generate enough displacement (and therefore buoyant force) to support the ship's total weight
- Minimize drag to move through water efficiently
- Maintain stability by positioning the center of buoyancy appropriately relative to the center of gravity
The V-shaped cross-section of many ship hulls concentrates displacement toward the center of the vessel, improving wave-cutting efficiency. Wide, flat-bottomed hulls generate more buoyant force for shallower draft (useful in rivers and harbors) but are less stable in rough conditions.
Stability and Capsizing
The most dangerous thing that can happen to a ship other than collision or fire is capsizing — rolling over due to instability.
When a floating ship tilts, the shape of the submerged part of the hull changes. The center of buoyancy shifts to the side the ship has tilted toward (more hull is submerged on that side). This creates a geometry question: does the buoyant force now create a righting moment (pushing the ship back upright) or a capsizing moment (pushing it further over)?
The answer depends on whether the metacenter is above or below the center of gravity:
- Metacenter above center of gravity: The ship is stable. When tilted, the buoyant force creates an upright-pulling moment. The ship rights itself.
- Metacenter below center of gravity: The ship is unstable. When tilted, the buoyant force creates a capsizing moment. The ship capsizes.
This is why cargo placement matters. Load heavy cargo high in a ship, and the center of gravity rises — possibly above the metacenter. The ship becomes unstable and capsizes in conditions it could otherwise handle. Load heavy cargo low, and the center of gravity drops — well below the metacenter — and the ship becomes highly stable.
The capsizing of the MS Herald of Free Enterprise in 1987 — which killed 193 people — occurred because bow doors were left open and the vehicle deck flooded, flooding that raised the center of gravity above the metacenter. The ship rolled over within 90 seconds of listing.
"Every naval architect knows the fundamental rule: the metacenter must always stay above the center of gravity. When it doesn't, the ship is unstable. No amount of speed or skill can fix a vessel that wants to roll over." — Barras, C. B., Ship Stability for Masters and Mates (2004, paraphrase)
Submarines: Buoyancy Under Control
Submarines demonstrate Archimedes' principle applied deliberately and actively. A submarine controls its own buoyancy:
Neutral buoyancy: The submarine adjusts its ballast to equal exactly the weight of displaced water. It neither floats up nor sinks — it stays at depth.
Negative buoyancy (diving): Ballast tanks flood with seawater. The submarine's total mass increases while its volume stays the same. Average density exceeds seawater density. The submarine sinks.
Positive buoyancy (surfacing): Compressed air is blown into ballast tanks, forcing out seawater. The submarine's total mass decreases while its volume stays the same. Average density falls below seawater density. The submarine rises.
Submarines also use their dive planes — horizontal fins — like aircraft wings, generating hydrodynamic forces that can push the boat up or down as it moves through water. This provides faster depth control at speed than ballast manipulation alone.
Other Buoyancy Phenomena
The same principle explains many other everyday phenomena:
Hot air balloons float because heated air is less dense than the surrounding cooler air. The balloon envelope + gondola + hot air has an average density less than cold ambient air. The air version of Archimedes' principle.
Icebergs float because ice (density ≈ 917 kg/m³) is slightly less dense than liquid water (1,000 kg/m³). Ice floats with about 90% of its mass below the waterline — hence "tip of the iceberg."
Fish control depth through a swim bladder — an internal gas-filled organ whose volume can be adjusted to change the fish's average density, providing neutral buoyancy at different depths.
Humans float more easily in the Dead Sea because the Dead Sea's salt concentration makes it significantly denser than ordinary seawater — higher density means greater buoyant force per unit of submerged volume.
Lava lamps work on buoyancy: wax and water are carefully chosen to have nearly identical densities at room temperature. When heated, the wax becomes slightly less dense than the liquid and rises; when it reaches the top and cools, it becomes slightly denser and sinks.
Historical Note: Archimedes and the Crown
The legend says Archimedes was asked by King Hiero II of Syracuse to determine whether a newly made crown was pure gold or had been adulterated with silver by the goldsmith — without melting it down. Pure gold is denser than silver, so an adulterated crown would have greater volume for the same mass.
The problem: how to measure the volume of an irregular crown without destroying it?
According to the legend, Archimedes realized the solution while lowering himself into a bath: an object immersed in water displaces a volume of water exactly equal to its own volume. By measuring the volume of water displaced by the crown (compared to the same mass of pure gold), he could calculate its density — and determine whether it was adulterated.
Whether the story is precisely accurate, the underlying principle is: density determines buoyancy, and volume can be measured by displacement. Everything else in floating and sinking follows from that.
For related concepts, see how bridges are engineered, how flight works, and how pressure works in fluids.
References
- Archimedes (circa 250 BC). On Floating Bodies (De corporibus fluitantibus). (Surviving fragments; translated by T. L. Heath in The Works of Archimedes, Cambridge University Press, 1897.)
- Vitruvius. De Architectura (circa 25 BC), Book 9, Preface. (Various translations; the source of the "Eureka" account.)
- Munson, B. R., Young, D. F., & Okiishi, T. H. (2002). Fundamentals of Fluid Mechanics. John Wiley & Sons.
- Barras, C. B. (2004). Ship Stability for Masters and Mates. Butterworth-Heinemann.
- Molland, A. F. (Ed.) (2008). The Maritime Engineering Reference Book: A Guide to Ship Design, Construction and Operation. Butterworth-Heinemann.
- Plimsoll, S. (1873). Our Seamen: An Appeal. Virtue & Company.
- Marine Accident Investigation Branch. (1987). Report of the Formal Investigation into the Sinking of the MV Herald of Free Enterprise. HMSO.
Frequently Asked Questions
Why do ships float if steel is denser than water?
A ship floats because buoyancy depends on the average density of the entire ship — including all the empty air space inside — not on the density of the hull material. A ship is mostly hollow. When you average the mass of the steel hull with the mass of the vast air-filled interior, the overall density falls below that of water. Steel sinks; an air-filled steel box floats.
What is Archimedes' principle?
Archimedes' principle states that any object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. A ship displaces a volume of water equal to its hull's submerged volume. If that displaced water weighs more than the ship, the ship floats. If it weighs less, the ship sinks. Archimedes discovered this around 250 BC.
Why does a pebble sink but a ship floats?
A pebble sinks because it is solid — its density exceeds water. The weight of water displaced is less than the pebble's weight, so the net force is downward. A ship's total volume includes enormous air-filled spaces, making its average density well below water's, so it floats.
How does a submarine dive and surface?
A submarine uses ballast tanks — chambers filled with seawater to sink, or blown empty with compressed air to rise. Flooding the tanks increases total mass while volume stays the same, increasing average density above seawater — the submarine sinks. Blowing the tanks reduces mass, dropping average density below seawater — it rises. This is Archimedes' principle applied deliberately.
What is displacement in naval engineering?
Displacement is the weight of water a ship pushes aside. A fully loaded aircraft carrier displaces approximately 100,000 tonnes of water — meaning the water it displaced weighs exactly as much as the carrier. Naval architects design ships to ensure their displacement at full load keeps them at the desired waterline.
What makes a ship capsize?
A ship capsizes when its center of gravity rises above its metacenter. When a ship tilts, the center of buoyancy shifts. If the resulting buoyant force creates a righting moment — pushing back upright — the ship is stable. If the ship is too top-heavy or has flooded compartments, the buoyant force creates a capsizing moment instead. The Herald of Free Enterprise disaster in 1987 illustrated this: an open bow door flooded the deck, raising the center of gravity above the metacenter, and the ship rolled over in 90 seconds.