Buoyancy is the force that enables boats and beach balls to float on water. The term **buoyant force** refers to the upward-directed force that a fluid (either a liquid or a gas) exerts on an object that is partially or completely immersed in the fluid. Buoyant force also explains why we can lift objects underwater more easily than on land.

### Key Takeaways: Buoyant Force

- The term buoyant force refers to the upward-directed force that a fluid exerts on an object that is partially or completely immersed in the fluid.
- The buoyant force arises from differences in
- The Archimedes principle states that the buoyant force exerted on an object that is submerged partially or completely in a fluid is equal to the weight of the fluid that is displaced by the object.

## The Eureka Moment: The First Observation of Buoyancy

According to the Roman architect Vitruvius, the Greek mathematician and philosopher Archimedes first discovered buoyancy in the 3rd century B.C. while puzzling over a problem posed to him by King Hiero II of Syracuse. King Hiero suspected that his gold crown, made in the shape of a wreath, was not actually made of pure gold, but rather a mixture of gold and silver.

Allegedly, while taking a bath, Archimedes noticed that the more he sank into the tub, the more water flowed out of it. He realized this was the answer to his predicament, and rushed home while crying “Eureka!” (“I’ve found it!”) He then made two objects – one gold and one silver – that were the same weight as the crown, and dropped each one into a vessel filled to the brim with water.

Archimedes observed that the silver mass caused more water to flow out of the vessel than the gold one. Next, he observed that his "gold" crown caused more water to flow out of the vessel than the pure gold object he had created, even though the two crowns were of the same weight. Thus, Archimedes demonstrated that his crown indeed contained silver.

Though this tale illustrates the principle of buoyancy, it may be a legend. Archimedes never wrote down the story himself. Furthermore, in practice, if a tiny amount of silver were indeed swapped for the gold, the amount of water displaced would be too small to reliably measure.

Prior to the discovery of buoyancy, it was believed that an object’s shape determined whether or not it would float.

## Buoyancy and Hydrostatic Pressure

The buoyant force arises from differences in** hydrostatic pressure **– the pressure exerted by a static fluid. A ball that is placed higher up in a fluid will experience less pressure than the same ball placed further down. This is because there is more fluid, and therefore more weight, acting on the ball when it is deeper in the fluid.

Thus, the pressure at the top of an object is weaker than the pressure at the bottom. Pressure can be converted to force using the formula Force = Pressure x Area. There is a net force pointing upward. This net force – which points upwards regardless of the object’s shape – is the buoyancy force.

The hydrostatic pressure is given by P = rgh, where r is the density of the fluid, g is acceleration due to gravity, and h is the *depth* inside the fluid. The hydrostatic pressure does not depend on the shape of the fluid.

## The Archimedes Principle

The **Archimedes principle** states that the buoyant force exerted on an object that is submerged partially or completely in a fluid is equal to the weight of the fluid that is displaced by the object.

This is expressed by the formula F = rgV, where r is the density of the fluid, g is acceleration due to gravity, and V is the volume of fluid that is displaced by the object. V only equals the volume of the object if it is completely submerged.

The buoyant force is an upward force that opposes the downward force of gravity. The magnitude of the buoyant force determines whether an object will sink, float, or rise when submerged in a fluid.

- An object will sink if the gravitational force acting on it is greater than the buoyant force.
- An object will float if the gravitational force acting on it is equal to the buoyant force.
- An object will rise if the gravitational force acting on it is less than the buoyant force.

Several other observations can be drawn from the formula, as well.

- Submerged objects that have equal volumes will displace the same amount of fluid and experience the same magnitude of buoyant force, even if the objects are made of different materials. However, these objects will differ in weight and will float, rise, or sink.
- Air, which has a density roughly 800 times lower than water’s, will experience a much lesser buoyant force than water.

### Example 1: A Partially Immersed Cube

A cube with a volume of 2.0 cm^{3} is submerged halfway into water. What is the buoyant force experienced by the cube?

- We know that F = rgV.
- r = density of water = 1000 kg/m
^{3} - g = gravitational acceleration = 9.8 m/s
^{2} - V = half of the cube’s volume = 1.0 cm
^{3}= 1.0*10^{-6}m^{3} - Thus, F = 1000 kg/m
^{3}* (9.8 m/s^{2}) * 10^{-6}m^{3}= .0098 (kg*m)/s^{2}= .0098 Newtons.

### Example 2: A Fully Immersed Cube

A cube with a volume of 2.0 cm^{3} is submerged fully into water. What is the buoyant force experienced by the cube?

- We know that F = rgV.
- r = density of water = 1000 kg/m3
- g = gravitational acceleration = 9.8 m/s
^{2} - V = the cube’s volume = 2.0 cm
^{3}= 2.0*10^{-6}m3 - Thus, F = 1000 kg/m
^{3}* (9.8 m/s^{2}) * 2.0*10-6 m^{3}= .0196 (kg*m)/s^{2}= .0196 Newtons.

## Sources

- Biello, David. “Fact or Fiction?: Archimedes Coined the Term ‘Eureka!’ in the Bath.”
*Scientific American*, 2006, https://www.scientificamerican.com/article/fact-or-fiction-archimede/. - “Density, Temperature, and Salinity.”
*University of Hawaii*, https://manoa.hawaii.edu/exploringourfluidearth/physical/density-effects/density-temperature-and-salinity. - Rorres, Chris. “The Golden Crown: Introduction.”
*New York State University*, https://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html.