Buoyancy is the upward force that a fluid exerts on any object immersed in it, and it is the reason a 100,000-tonne aircraft carrier made of steel floats while a small iron nail sinks to the bottom of a bathtub. The principle governing buoyancy was discovered by the Greek mathematician Archimedes of Syracuse around 250 BC and remains one of the most elegant and practically important laws in all of physics: any object submerged in a fluid experiences an upward force equal to the weight of the fluid it displaces. Whether an object floats or sinks depends not on what it is made of but on its average density -- the total mass divided by the total volume, including any hollow spaces inside. A solid steel cube sinks because its density (7,900 kg/m3) far exceeds water's (1,000 kg/m3). A steel ship floats because its vast hollow interior gives it an average density of only 200-300 kg/m3.

A US Navy Gerald R. Ford-class aircraft carrier displaces approximately 100,000 tonnes of water when fully loaded. Its hull is made of high-strength steel -- a material roughly eight times denser than water. It carries 75 aircraft, over 4,500 personnel, millions of gallons of jet fuel, ammunition, food, and equipment. It floats effortlessly.

Drop a small iron nail into a glass of water. It 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 has nothing to do with the material objects are made of and everything to do with their shape.

"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 -- the earliest account comes from Vitruvius writing two centuries later. 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.

The Physics: Why Objects Float or Sink

Pressure and the Origin of Buoyancy

To understand buoyancy at its deepest level, start with fluid pressure. In any fluid -- liquid or gas -- pressure increases with depth because deeper fluid bears the weight of all the fluid above it. At any given depth, fluid exerts equal pressure in all directions, a relationship formalized by Blaise Pascal in 1653 and known as 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 mathematical expression is straightforward:

The buoyant force equals the fluid density multiplied by the gravitational acceleration multiplied by the volume of fluid displaced (F = rho x g x V). For water at sea level, this means every cubic meter of submerged volume generates approximately 9,800 newtons (about 1,000 kg-force) of upward push.

The size of this force depends entirely on how much fluid the object displaces. A larger object displaces more fluid, producing a larger buoyant force. The shape of the object determines how much fluid it displaces relative to its weight -- and this ratio is what determines whether it floats or sinks.

Archimedes' Principle: The Formal Statement

Archimedes' principle states: 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. Archimedes published this in his treatise On Floating Bodies (c. 250 BC), making it one of the oldest physical laws still taught in its original form.

The principle has a crucial corollary for floating objects. When an object floats, it displaces exactly enough fluid to equal its own weight. A 10,000-tonne ship floating in the ocean displaces exactly 10,000 tonnes of seawater. If you add 500 tonnes of cargo, it sinks a little deeper, displacing an additional 500 tonnes of water, until the buoyant force once again matches the total weight.

Archimedes' Principle in Numbers

Consider a solid cube of steel, 10 cm on each side, submerged in water:

• Volume: 1,000 cm3 = 0.001 m3
• Water displaced: 0.001 m3 x 1,000 kg/m3 = 1 kg
• Buoyant force: approximately 9.8 newtons upward
• Mass of steel cube: 0.001 m3 x 7,900 kg/m3 = 7.9 kg
• Weight of steel cube: approximately 77.4 newtons downward

The buoyant force (9.8 N) is far less than the weight (77.4 N). Net force is downward. The steel cube sinks.

Now consider a hollow steel cube with the same outer dimensions (10 cm per side), but with walls only 2 mm thick:

• Outer volume: 1,000 cm3

• Steel volume (shell only): approximately 118 cm3

• Air volume inside: approximately 882 cm3

• Mass of steel: 118 cm3 x 7.9 g/cm3 = 932 g

• Mass of air inside: approximately 1 g

• Total mass: approximately 933 g

• Buoyant force (displacing 1,000 cm3 of water): 9.8 N upward

• Weight of hollow box: 9.1 N downward

Buoyant force exceeds weight. 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

Ships in Practice: Naval Architecture

Designing for Displacement

Displacement in naval engineering refers to the weight of water a ship pushes aside -- which, by Archimedes' principle, equals the ship's total weight when floating. 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 science of hull design has progressed enormously since the pioneering work of William Froude in the 1860s and 1870s. Froude, a British engineer, developed the first systematic method for predicting ship resistance from scale model tests conducted in a purpose-built tank at Torquay, England. His work established the dimensionless Froude number -- the ratio of ship speed to a function of waterline length -- which remains the fundamental parameter in hull design. The testing methodology he developed is still used: ship designers build scale models and test them in towing tanks before committing to full-scale construction.

The Plimsoll Line: A Reform Born of Tragedy

The Plimsoll line, marked on the hull of commercial ships since 1876, indicates the maximum safe waterline in different water conditions. Its existence is the result of one of the most effective safety campaigns in maritime history.

In the mid-nineteenth century, unscrupulous ship owners in Britain routinely overloaded vessels and sent them to sea in dangerous conditions, profiting from insurance payouts when they sank. The ships were called "coffin ships." Samuel Plimsoll, a Member of Parliament from Derby, campaigned for legislation requiring load lines after documenting hundreds of preventable deaths. His 1873 book Our Seamen: An Appeal caused a public sensation. After years of fierce opposition from shipping interests, the Merchant Shipping Act of 1876 mandated load line markings.

Different Plimsoll marks indicate 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 (1,025 vs. 1,000 kg/m3), so a ship rides higher in saltwater and lower in freshwater -- the Plimsoll marks account for this. A ship entering a freshwater river from the ocean sinks measurably lower in the water.

Stability and the Metacenter

The most dangerous thing that can happen to a ship other than collision or fire is capsizing -- rolling over due to instability. Understanding why ships are normally stable, and what causes them to capsize, requires three concepts: the center of gravity, the center of buoyancy, and the metacenter.

The center of gravity is the point at which the ship's entire weight effectively acts. The center of buoyancy is the geometric center of the submerged volume -- the point at which the buoyant force effectively acts. When a ship is upright, both points lie on the centerline, with buoyancy acting upward and gravity acting downward.

When a floating ship tilts, the shape of the submerged hull changes. The center of buoyancy shifts toward the side that has tilted into the water (more hull is submerged on that side). The metacenter is defined as the point where a vertical line through the new center of buoyancy intersects the ship's original centerline. The critical stability rule:

• Metacenter above center of gravity: The ship is stable. When tilted, the offset between buoyancy (acting upward through the shifted center of buoyancy) and gravity (acting downward through the center of gravity) creates a righting moment that pushes the ship back upright.
• Metacenter below center of gravity: The ship is unstable. The same offset creates a capsizing moment that pushes the ship further over.

This is why cargo placement matters enormously. 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, producing strong stability.

Disasters That Illustrate the Physics

The capsizing of the MS Herald of Free Enterprise on March 6, 1987 -- which killed 193 people off the coast of Zeebrugge, Belgium -- is one of the most studied maritime disasters in stability engineering. The roll-on/roll-off ferry departed port with its bow doors open. Water flooded the vehicle deck, a large flat space high in the ship. This raised the center of gravity above the metacenter. The ship rolled over within 90 seconds of first listing. The Marine Accident Investigation Branch report identified the open bow doors as the immediate cause and systemic management failures as the root cause.

The sinking of the MV Sewol in April 2014 off the coast of South Korea, which killed 304 people (most of them high school students), involved a combination of factors that all relate to buoyancy and stability: the ship had been modified to add extra passenger decks (raising the center of gravity), cargo had been inadequately secured (allowing weight to shift when the ship turned), and ballast water had been reduced to increase cargo capacity (further raising the center of gravity). When the ship made a sharp turn, the shifting cargo and high center of gravity produced a capsizing moment the ship could not recover from.

"Every naval architect knows the fundamental rule: the metacenter must always stay above the center of gravity. When it doesn't, no amount of speed or skill can save the vessel." -- adapted from C. B. Barras, Ship Stability for Masters and Mates (2004)

Submarines: Buoyancy Under Deliberate Control

Submarines demonstrate Archimedes' principle applied deliberately and actively. A submarine controls its own buoyancy by adjusting its average density:

Neutral buoyancy: The submarine adjusts its ballast to make its total weight exactly equal to the weight of displaced water. It neither floats up nor sinks -- it stays at depth. Achieving precise neutral buoyancy is a continuous process, since seawater density varies with temperature, salinity, and depth.

Negative buoyancy (diving): Ballast tanks flood with seawater through valves called Kingston valves (or their modern equivalents). The submarine's total mass increases while its outer volume stays the same. Average density exceeds seawater density. The submarine sinks.

Positive buoyancy (surfacing): Compressed air stored in high-pressure flasks is blown into ballast tanks, forcing out seawater through flood ports. The submarine's total mass decreases while its volume stays the same. Average density falls below seawater density. The submarine rises.

Modern nuclear submarines like the US Navy's Virginia class carry approximately 35,000 tonnes of displacement when submerged and use a sophisticated trim system -- smaller tanks distributed throughout the hull -- to maintain precise neutral buoyancy and level attitude at depth. They also use dive planes -- horizontal fins -- like aircraft wings, generating hydrodynamic lift forces that can push the boat up or down as it moves through water, providing faster depth control at speed than ballast manipulation alone.

Buoyancy Beyond Ships

The same principle explains many other everyday and extraordinary phenomena:

Hot air balloons float because heated air is less dense than the surrounding cooler air. The air inside a typical hot air balloon is heated to approximately 100 degrees Celsius, reducing its density from about 1.2 kg/m3 to about 0.9 kg/m3. The balloon envelope, gondola, fuel, and passengers together with the hot air have a combined average density slightly less than the ambient atmosphere. The principle is identical to a ship floating on water -- just applied to a different fluid.

Icebergs float because ice (density approximately 917 kg/m3) is slightly less dense than liquid water (1,000 kg/m3). This is unusual -- most solids are denser than their liquid form. The anomalous expansion of water as it freezes is caused by the crystal structure of ice, which forces water molecules into a more open arrangement. Ice floats with about 90% of its mass below the waterline -- the origin of the phrase "tip of the iceberg." If ice were denser than water, lakes and oceans would freeze from the bottom up, with catastrophic consequences for aquatic life and global climate. As Loren Eiseley wrote, the fact that ice floats is "one of the most important accidents in the natural world."

Fish control their depth through a swim bladder -- an internal gas-filled organ whose volume can be adjusted to change the fish's average density. By inflating the swim bladder (using gas secreted from the blood), a fish decreases its density and rises. By deflating it, the fish increases its density and sinks. This provides neutral buoyancy at any chosen depth without constant swimming effort -- an elegant biological solution to the same engineering problem submarines solve with ballast tanks.

The Dead Sea allows humans to float with minimal effort because its extreme salt concentration (approximately 34% salinity, compared to about 3.5% for normal ocean water) gives it a density of roughly 1,240 kg/m3. Since the human body has an average density of approximately 985-1,050 kg/m3, the Dead Sea's density exceeds even the densest human bodies, producing strong positive buoyancy for everyone.

Lava lamps operate on buoyancy cycling: the wax and surrounding liquid are carefully chosen to have nearly identical densities at room temperature. When the base heater warms the wax, thermal expansion makes it slightly less dense than the liquid, and it rises. At the top, away from the heat source, the wax cools, contracts, becomes denser than the liquid, and sinks back down -- a continuous demonstration of density-driven buoyancy.

Historical Note: Archimedes and the Golden Crown

The most famous story about Archimedes -- and one of the most famous stories in the history of science -- involves a crown, a bathtub, and a king's suspicion.

According to the Roman architect Vitruvius, writing in De Architectura (c. 25 BC), King Hiero II of Syracuse commissioned a golden crown and suspected the goldsmith of adulterating it with cheaper silver while keeping some of the gold. He asked Archimedes to determine whether the crown was pure gold -- without melting it down.

The problem: pure gold is denser than silver (19,300 kg/m3 vs. 10,500 kg/m3), so an adulterated crown of the same mass would occupy a larger volume. But how do you measure the volume of an irregularly shaped crown without destroying it?

According to the legend, Archimedes realized the solution while lowering himself into a full bath: an object immersed in water displaces a volume of water exactly equal to its own volume. The water that overflowed could be measured. By comparing the volume of water displaced by the crown to the volume displaced by the same mass of pure gold, he could calculate the crown's density -- and determine whether it had been adulterated.

Modern historians, including Chris Rorres of Drexel University, have noted that the displacement method as described by Vitruvius would have been extremely difficult to perform with sufficient accuracy given ancient measurement tools -- the difference in displaced water between a pure gold crown and one with 10% silver would be tiny. Rorres and others have suggested that Archimedes more likely used a hydrostatic balance, comparing the crown's weight in air to its weight when submerged -- a method that is both more accurate and more consistent with the mathematical treatment in On Floating Bodies. Regardless of the exact method, the underlying insight is the same: density determines buoyancy, and submerging an object in water reveals its volume.

For related concepts, see how bridges are engineered, how flight works, how pressure works in fluids, and what is critical thinking.

References and Further Reading

1. Archimedes (c. 250 BC). On Floating Bodies (De corporibus fluitantibus). Translated by T. L. Heath in The Works of Archimedes, Cambridge University Press, 1897.
2. Vitruvius. De Architectura (c. 25 BC), Book 9, Preface. Various translations; the source of the "Eureka" account.
3. Munson, B. R., Young, D. F., & Okiishi, T. H. (2002). Fundamentals of Fluid Mechanics. John Wiley & Sons.
4. Barras, C. B. (2004). Ship Stability for Masters and Mates. Butterworth-Heinemann.
5. Molland, A. F. (Ed.) (2008). The Maritime Engineering Reference Book: A Guide to Ship Design, Construction and Operation. Butterworth-Heinemann.
6. Plimsoll, S. (1873). Our Seamen: An Appeal. Virtue & Company.
7. Marine Accident Investigation Branch. (1987). Report of the Formal Investigation into the Sinking of the MV Herald of Free Enterprise. HMSO.
8. Rorres, C. (2004). Completing Book II of Archimedes' On Floating Bodies. The Mathematical Intelligencer, 26(3), 32-42.
9. Froude, W. (1874). On experiments with HMS Greyhound. Transactions of the Institution of Naval Architects, 15, 36-73.
10. Korean Maritime Safety Tribunal. (2014). Investigation Report on the Sinking of MV Sewol. Republic of Korea.
11. Pascal, B. (1663). Traite de l'equilibre des liqueurs. (Treatise on the Equilibrium of Liquids). Published posthumously.