Understanding Buoyancy: Archimedes Principle Explained
Buoyancy is the upward force exerted by a fluid on any object immersed in it. This fundamental principle of fluid mechanics explains why ships float, hot air balloons rise, and why you feel lighter when swimming. The principle was discovered over 2,200 years ago by the Greek mathematician Archimedes, reportedly while taking a bath.
Archimedes famously shouted "Eureka!" (I found it!) when he realized that the buoyant force on an object equals the weight of fluid displaced by that object. This elegant principle connects the abstract concept of force to the easily measurable quantity of displaced volume, making it one of the most practical discoveries in physics.
Archimedes' Principle: The Fundamental Equation
The buoyant force acting on an object is given by:
Buoyant force equals fluid density times displaced volume times gravitational acceleration
Where:
- Fb = Buoyant force (Newtons)
- ρ = Density of the fluid (kg/m³)
- V = Volume of fluid displaced (m³)
- g = Gravitational acceleration (9.81 m/s²)
Notice that the buoyant force depends only on the fluid's density and the volume displaced—not on the object's material, shape, or mass. A hollow steel ship and a solid wooden ball displacing the same volume of water experience identical buoyant forces.
Common Fluid Densities
| Fluid | Density (kg/m³) | Notes |
|---|---|---|
| Fresh water | 1000 | At 4°C (maximum density) |
| Seawater | 1025 | Average ocean salinity |
| Dead Sea water | 1240 | Very high salt content |
| Mercury | 13,600 | Very dense liquid metal |
| Gasoline | 680-720 | Floats on water |
| Air (sea level) | 1.225 | At 15°C, 101.3 kPa |
| Helium | 0.164 | At STP; 7.5× lighter than air |
Floating, Sinking, and Neutral Buoyancy
Whether an object floats or sinks depends on the balance between its weight and the buoyant force:
Floats: Object density < Fluid density (buoyant force > weight)
Sinks: Object density > Fluid density (buoyant force < weight)
Neutral: Object density = Fluid density (buoyant force = weight)
For a floating object, only part of its volume is submerged. The fraction submerged equals the ratio of object density to fluid density. A piece of wood with density 600 kg/m³ floating in water (1000 kg/m³) will have 60% of its volume below the waterline.
Material Densities for Comparison
| Material | Density (kg/m³) | Floats in Water? |
|---|---|---|
| Styrofoam | 25-200 | Yes |
| Cork | 120-240 | Yes |
| Pine wood | 500-600 | Yes |
| Oak wood | 600-900 | Usually yes |
| Ice | 917 | Yes (barely) |
| Bone | 1800-2000 | No |
| Aluminum | 2700 | No |
| Steel | 7850 | No (but ships float!) |
| Gold | 19,300 | No |
Worked Examples
Example 1: Buoyant Force on a Swimmer
Problem: A swimmer with volume 0.07 m³ is completely submerged in fresh water. What buoyant force do they experience?
Solution:
ρ = 1000 kg/m³ (fresh water)
V = 0.07 m³
g = 9.81 m/s²
Fb = ρVg = 1000 × 0.07 × 9.81 = 686.7 N
This equals the weight of about 70 kg of water.
Example 2: Will a Wooden Block Float?
Problem: A block of wood has mass 400 g and volume 500 cm³. Will it float in water? What fraction will be submerged?
Solution:
Object density = mass/volume = 0.4 kg / 0.0005 m³ = 800 kg/m³
Water density = 1000 kg/m³
Since 800 < 1000, it floats
Fraction submerged = 800/1000 = 80%
Example 3: How Steel Ships Float
Problem: A steel cargo ship weighs 10,000 tonnes. How much water must it displace to float?
Solution:
Weight = 10,000,000 kg × 9.81 = 98,100,000 N
For floating: Fb = Weight
V = Fb/(ρg) = 98,100,000/(1025 × 9.81)
V = 9756 m³ of seawater
The ship must displace nearly 10,000 cubic meters of water.
Example 4: Apparent Weight Underwater
Problem: A scuba diver weighs 80 kg and has volume 0.08 m³. What is their apparent weight underwater in seawater?
Solution:
Actual weight = 80 × 9.81 = 784.8 N
Buoyant force = 1025 × 0.08 × 9.81 = 804.4 N
Apparent weight = 784.8 - 804.4 = -19.6 N
Negative value means they would rise! They need weights to achieve neutral buoyancy.
Example 5: Hot Air Balloon
Problem: A hot air balloon envelope has volume 2800 m³. If the hot air inside has density 0.95 kg/m³ and outside air is 1.20 kg/m³, what is the lift force?
Solution:
Buoyant force = ρoutside × V × g = 1.20 × 2800 × 9.81 = 32,962 N
Weight of hot air = ρinside × V × g = 0.95 × 2800 × 9.81 = 26,094 N
Net lift = 32,962 - 26,094 = 6,868 N (≈ 700 kg lift capacity)
Applications of Buoyancy
Ship Design
Ships float because their overall density (including the air-filled interior) is less than water. Naval architects carefully calculate displacement to ensure stability and determine how much cargo can be loaded while maintaining safe freeboard (the height of the deck above water).
Submarines
Submarines control their buoyancy using ballast tanks. Filling tanks with water increases overall density, causing the sub to sink. Blowing water out with compressed air decreases density, causing it to rise. Fine adjustments allow submarines to hover at any depth.
Hydrometers and Density Measurement
Hydrometers are instruments that float at different levels depending on fluid density. They're used to measure alcohol content in beverages, battery acid concentration, and antifreeze strength.
Hot Air Balloons and Airships
Atmospheric buoyancy allows lighter-than-air craft to fly. Hot air is less dense than cool air, and helium is much lighter than both. Blimps and dirigibles use this principle for controlled flight.
Scuba Diving
Divers use buoyancy compensator devices (BCDs) to achieve neutral buoyancy at depth. They add or release air to hover effortlessly, conserving energy and protecting delicate underwater environments.
The Iceberg Principle
Ice floats because it's less dense than water (917 vs 1000 kg/m³). This means 91.7% of an iceberg's volume is underwater—only about 8% shows above the surface. This is why icebergs are so dangerous to ships; the visible portion gives little indication of the massive underwater bulk.
Interestingly, if ice didn't float, lakes would freeze from the bottom up, killing aquatic life. The floating ice layer actually insulates the water below, allowing fish and other organisms to survive winter.
Pressure and Depth
While buoyant force doesn't depend on depth (for incompressible fluids), pressure increases with depth according to:
Where h is depth below the surface
Every 10 meters of water depth adds approximately 1 atmosphere (101,325 Pa) of pressure. At the bottom of the Mariana Trench (11,000 m), pressure exceeds 1,100 atmospheres!
Related Calculators
- Drag Force Calculator - Fluid resistance calculations
- Bernoulli Equation Calculator - Fluid flow dynamics
- Pressure Calculator - Force per unit area
- Volume Converter - Convert volume units
Frequently Asked Questions
Why do steel ships float but steel balls sink?
A solid steel ball sinks because steel (7850 kg/m³) is denser than water. But a ship made of steel is mostly hollow, filled with air. The average density of the entire ship (steel hull + air inside) is much less than water, so it floats. What matters is the total mass divided by the total volume, not the density of the construction material.
Does buoyant force change with depth?
For incompressible fluids like water, buoyant force doesn't change with depth—only the volume of displaced fluid matters. However, for compressible fluids (gases), density increases with pressure/depth, so buoyant force can vary. For most practical applications in liquids, buoyancy is depth-independent.
Why can you float more easily in the Dead Sea?
The Dead Sea has extremely high salt content, making its water density about 1240 kg/m³ compared to 1000 kg/m³ for fresh water. This 24% higher density means 24% more buoyant force, allowing you to float with less of your body submerged—you can even read a newspaper while floating!
How do fish control their buoyancy?
Most fish have a swim bladder—an internal gas-filled organ. By adjusting gas volume in this bladder, fish can fine-tune their overall density to match surrounding water, achieving neutral buoyancy at any depth without constantly swimming. Sharks lack swim bladders and must keep swimming or they'll sink.
What's the difference between buoyancy and flotation?
Buoyancy refers to the upward force exerted by fluid. Flotation is the state of floating, which occurs when buoyant force equals or exceeds weight. All objects in fluid experience buoyancy, but only some float.