Archimedes’ Principle and Buoyancy
Archimedes’ principle states that the upward buoyant force on an object that is wholly or partly immersed in a fluid equals the weight of the fluid that the object displaces: Fb = ρfluidVg. In brief: an object floats if the buoyant force can equal its weight, which happens when its average density is less than the fluid’s, and it then sinks only until it displaces its own weight of fluid. The buoyant force arises because pressure increases with depth, so the push up on the bottom is larger than the push down on the top.
Ships made of steel float, submarines can rise and sink, and a stone feels lighter in water. All of these follow from one principle. This guide states Archimedes’ principle and explains its origin, discusses floating, sinking and neutral buoyancy, compares cases in a table, works through calculations and corrects common mistakes.
What is Archimedes’ principle?
Any object in a fluid (a liquid or a gas) experiences an upward force equal to the weight of the displaced fluid. In symbols, Fb = ρfVsubg, where ρf is the density of the fluid, Vsub is the volume of the submerged part of the object (the volume of fluid displaced) and g = 9.8 N/kg. The buoyant force depends on the density of the fluid and the submerged volume, but not on the depth (for incompressible liquids) or the material of the object. The apparent weight of a submerged object is its weight minus the buoyant force.
Where does the buoyant force come from?
The pressure in a fluid increases with depth. The bottom of a submerged object is deeper than its top, so the upward force on the bottom surface is greater than the downward force on the top. The net upward force is the buoyant force. For a cylinder of height h and base area A, the pressure difference is ρfgh, and the net force is ρfghA = ρfVg, which is the weight of the fluid that would fill the volume of the cylinder.
When does an object float or sink?
- Sinks: if the weight is greater than the maximum buoyant force (when fully submerged), which means the average density of the object is greater than the density of the fluid.
- Floats: if the average density is lower, the object rises until it is partly out of the fluid, and the buoyant force on the submerged part equals the weight. The fraction submerged equals ρobject ÷ ρfluid.
- Neutral buoyancy: if the densities are equal, the object stays at any depth. Submarines achieve this by taking in or pumping out water from ballast tanks, and fish by adjusting the air in their swim bladders.
A ship floats because its hollow hull displaces a large volume of water. Ships are marked with a load line (Plimsoll line) showing the safe level of floating in different waters. Salt water is denser than fresh water, so an object floats higher in the sea, and even higher in the Dead Sea, where the density is about 1,240 kg/m3.
Floating, sinking and neutral buoyancy compared
| Condition | Weight and buoyant force (fully submerged) | Result |
|---|---|---|
| ρobject > ρfluid | W > Fb | Sinks |
| ρobject = ρfluid | W = Fb | Neutral buoyancy |
| ρobject < ρfluid | W < Fb | Rises and floats |
Worked example: buoyancy calculations
- Buoyant force. A block of volume 2.0 × 10−3 m3 is fully submerged in water. Fb = 1,000 × 2.0 × 10−3 × 9.8 = 19.6 N.
- Will it float? If the block has a mass of 1.5 kg, its weight is 14.7 N, which is less than 19.6 N. It floats. The fraction submerged = 14.7 ÷ 19.6 = 0.75 (and the density is 1.5 ÷ 2.0 × 10−3 = 750 kg/m3, so 750 ÷ 1,000 = 0.75), so 75 percent is under the water.
- Apparent weight. A 0.50 kg piece of metal with a density of 8,000 kg/m3 is hung from a spring balance and lowered into water. Volume = 0.50 ÷ 8,000 = 6.25 × 10−5 m3. Fb = 1,000 × 6.25 × 10−5 × 9.8 = 0.61 N. The weight in air is 4.9 N, so the apparent weight in water is 4.9 − 0.61 = 4.29 N.
- Ship. A ship of mass 20,000 tonnes (2.0 × 107 kg) floats in sea water (1,025 kg/m3). The displaced volume is 2.0 × 107 ÷ 1,025 = 19,500 m3.
How was the principle discovered?
According to the story by Vitruvius, Archimedes was asked by King Hiero II of Syracuse to determine whether a crown was pure gold. He noticed, in the bath, that his body displaced water, and he is said to have run out shouting “Eureka!”. His treatise On Floating Bodies (third century BCE) states the principle of buoyancy. The story is probably embellished, but the principle is the basis of naval architecture, submarine design and hot-air balloons.
Where is buoyancy used?
Ships, submarines, life jackets, hot air balloons and airships (air is the fluid), hydrometers (which float lower in less dense liquids, used to test battery acid and milk), and diving equipment with buoyancy compensators.
Which mistakes are common?
Mistake 1: Thinking that buoyancy depends on the weight of the object
It depends on the weight of the displaced fluid, which is determined by the submerged volume and the density of the fluid.
Mistake 2: Saying that heavy objects always sink
An object sinks if its average density is higher than that of the fluid. A steel ship floats.
Mistake 3: Using the total volume for a floating object
Use only the submerged volume.
Mistake 4: Thinking that the buoyant force increases with depth
For a fully submerged object in an incompressible fluid, it is constant.