Buoyancy

Buoyancy acts against the force of gravity and so makes objects seem lighter with respect to gravity. To represent this effect, which is important for sedimentation, it is common to define a buoyant mass m_b that represents the effective mass of the object with respect to gravity

$m_{b}=m_{\mathrm {object} }\cdot \left(1-{\frac {\rho _{\mathrm {fluid} }}{\rho _{\mathrm {object} }}}\right)$

where m_object is the true (vacuum) mass of the object, whereas ρ_object and ρ_fluid are the average densities of the object and the surrounding fluid, respectively. Thus, if the two densities are equal, ρ_object = ρ_fluid, the object appears to be weightless. If the fluid density is greater than the average density of the object, the object floats; if less, the object sinks.

Forces and equilibrium

Buoyancy provides an upward force on the object(like floating). The magnitude of this force is equal to the weight of the displaced fluid. (Displacement is the term used for the weight of the displaced fluid and, thus, is an equivalent term to buoyancy.) The buoyancy of an object depends, therefore, only upon two factors: the object's volume, and the density of the surrounding fluid. The greater the object's volume and surrounding density of the fluid, the more buoyant force it will experience. If the buoyancy of an (unrestrained and unpowered) object exceeds its weight, it will tend to rise. An object whose weight exceeds its buoyancy will tend to sink.

The atmosphere's density depends upon altitude. As an airship rises in the atmosphere, therefore, its buoyancy reduces as the density of the surrounding air reduces. The density of water is essentially constant: as a submarine expels water from its buoyancy tanks (by pumping them full of air) it rises because its buoyancy stays the same (because the volume of water it displaces stays the same) while its weight is decreased.

As a floating object rises or falls the forces external to it change and, as all objects are compressible to some extent or another, so will the object's volume. Buoyancy depends on volume and so an object's buoyancy reduces if it is compressed and increases if it expands.

If an object's compressibility is less than that of the surrounding fluid, it is in stable equilibrium and will, indeed, remain at rest, but if its compressibility is greater, its equilibrium is unstable, and it will rise and expand on the slightest upward perturbation, or fall and compress on the slightest downward perturbation.

A submarine is more compressible than the surrounding water. As depth increases, the resulting pressure causes the submarine's volume to decrease more than the volume of the surrounding water decreases. Buoyancy depends upon the object's volume and the weight of the displaced fluid. Volume has decreased so the weight displaced has decreased which means a decrease in buoyancy and the submarine tends to sink further. A rising submarine expands more than the surrounding water, the submarine tends to rise further.

The height of a balloon tends to be stable. As a balloon rises it will tend to increase in volume with reducing atmospheric pressure. But the balloon's cargo will not expand. The average density of the balloon decreases less, therefore, than that of the surrounding air. The balloon's buoyancy reduces because the weight of the displaced air is reduced. A rising balloon tends to stop rising. Similarly a sinking balloon tends to stop sinking.

The buoyant force can be expressed using the following equation:

$F_{\mathrm {buoyant} }=-\rho Vg\,$

where

$\rho \,$ is the density of the fluid;

V is the volume of the object submerged;

g is the standard gravity ($\approx$ 9.81 N/kg on Earth). Negative sign must be used since the buoyancy is opposite in direction with the acceleration due to gravity.

Archimedes' principle

Archimedes' principle, or the law of upthrust, is,

The weight of fluid that a submerged object displaces is the same amount of buoyancy force applied to the submerged object.

In other words, when a body is partially or completely immersed in a liquid, then it experiences an upward buoyant force which is equal to the weight of the fluid displaced by the immersed part of the body.

It is named after Archimedes of Syracuse, who first discovered this law. Vitruvius (De architectura IX.9–12) recounts the famous story of Archimedes making this discovery while in the bath (for which see eureka) but the actual record of Archimedes' discoveries appears in his two-volume work, On Floating Bodies. The ancient Chinese child prodigy Cao Chong also applied the principle of buoyancy in order to measure the accurate weight of an elephant, as described in the Sanguo Zhi.

This is true only as long as one can neglect the surface tension (capillarity) acting on the body, see .

The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (specifically if the surrounding fluid is of uniform density). Thus, among objects with equal masses, the one with greater volume has greater buoyancy.

Suppose a rock's weight is measured as 10 newtons when suspended by a string in a vacuum. Suppose that when the rock is lowered by the string into water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs will be 10 newtons minus the 3 newtons of buoyant force: 10 − 3 = 7 newtons. This same principle even reduces the apparent weight of objects that have sunk completely to the sea floor, such as the sunken battleship USS Arizona at Pearl Harbor, Hawaii. It is generally easier to lift an object up through the water than it is to finally pull it out of the water.

The density of the immersed object relative to the density of the fluid is easily calculated without measuring any volumes:

${\mbox{Relative density}}={\frac {\mbox{Weight}}{{\mbox{Weight}}-{\mbox{Apparent immersed weight}}}}$

Density

If the weight of an object is less than the weight of the fluid the object would displace if it were fully submerged, then the object has an average density less than the fluid and has a buoyancy greater than its weight. If the fluid has a surface, such as water in a lake or the sea, the object will float at a level so it displaces the same weight of fluid as the weight of the object. If the object is immersed in the fluid, such as a submerged submarine or a balloon in the air, it will tend to rise. If the object has exactly the same density as the liquid, then its buoyancy equals its weight. It will tend neither to sink nor float. An object with a higher average density than the fluid has less buoyancy than weight and it will sink. A ship floats because although it is made of steel, which is more dense than water, it encloses a volume of air and the resulting shape has an average density less than that of the water.

Applications

• Anderton Boat Lift
• Falkirk Wheel
• Neutral Buoyancy Laboratory

See also

• Hydrostatics
• Buoyancy compensator
• Cartesian diver
• Diving weighting system
• Hull (ship)
• Hydrometer
• Lighter than air
• Naval architecture
• Negative buoyancy
• Pontoon
• Quicksand
• Submarine
• Thrust

External links

• Falling in Water (Animation 1)
• Falling in Water (Animation 2)
• Falling in Water