# Archimedes' principle

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Archimedes' principle indicates that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces. Archimedes' principle is a law of physics fundamental to fluid mechanics. Archimedes of Syracuse[1] formulated this principle, which bears his name.

## Explanation

In his treatise on hydrostatics, On Floating Bodies, Archimedes states:

Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

Practically seen, the Archimedes principle allows the volume of an object to be measured by measuring the volume of the liquid it displaces after submerging, and the buoyancy of an object immersed in a liquid to be calculated.

For any immersed object, the volume of the submerged portion equals the volume of fluid it displaces. E.g., by submerging in water half of a sealed 1-liter container, we displace a half-liter volume of fluid, regardless of the container's contents. If we fully submerge the same container, we then displace one liter of liquid, which exactly equals the volume of the 1-liter container.

An empty 1-litre plastic bottle released in the air will fall down due to the gravitational force of the Earth acting on it. If the same bottle is released under water, the same gravitational force acts on it, but it will be pushed upwards towards the surface of the water. The extra force that pushes the bottle upwards comes from the upthrust or Archimedes force.

## Formula

Consider a cube immersed in a fluid, with its sides parallel to the direction of gravity. The fluid will exert a normal force on each face, and therefore only the forces on the top and bottom faces will contribute to buoyancy. The pressure difference between the bottom and the top face is directly proportional to the height (difference in depth). Multiplying the pressure difference by the area of a face gives the net force on the cube - the buoyancy, or the weight of the fluid displaced. By extending this reasoning to irregular shapes, we can see that, whatever the shape of the submerged body, the buoyant force is equal to the weight of the fluid displaced.

The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). The weight of the object in the fluid is reduced, because of the force acting on it, which is called upthrust. In simple terms, the principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational constant, g. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy.

Suppose a rock's weight is measured as 10 Newtons when suspended by a string in a vacuum with gravity acting on it. Suppose that when the rock is lowered into water, it displaces water of weight 3 Newtons. The force it then exerts on the string from which it hangs would be 10 Newtons minus the 3 Newtons of buoyant force: 10 − 3 = 7 Newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor. It is generally easier to lift an object up through the water than it is to pull it out of the water.

For a fully submerged object, Archimedes' principle can be reformulated as follows,

$\text{apparent immersed weight} = \text{weight of object} - \text{weight of displaced fluid}\,$

then inserted into the quotient of weights, which has been expanded by the mutual volume

$\frac { \text{density of object}} { \text{density of fluid} } = \frac { \text{weight}} { \text{weight of displaced fluid} }$

yields the formula below. The density of the immersed object relative to the density of the fluid can easily be calculated without measuring any volumes:

$\frac { \text {density of object}} { \text{density of fluid} } = \frac { \text{weight}} { \text{weight} - \text{apparent immersed weight}}.\,$

(This formula is used for example in describing the measuring principle of a dasymeter and of hydrostatic weighing.)

Example: If you drop wood into water, buoyancy will keep it afloat.

Example: A helium balloon in a moving car. When increasing speed or driving in a curve, the air moves in the opposite direction to the car's acceleration. However, due to buoyancy, the balloon is pushed "out of the way" by the air, and will actually drift in the same direction as the car's acceleration.

When an object is immersed in a liquid, the liquid exerts an upward force, which is known as the buoyant force, that is proportional to the weight of the displaced liquid. The sum force acting on the object, then, is proportional to the difference between the weight of the object ('down' force) and the weight of displaced liquid ('up' force), hence equilibrium buoyancy is achieved when these two weights (and thus forces) are equal.

## Refinements

Archimedes' principle does not consider the surface tension (capillarity) acting on the body.[2] Moreover, Archimedes' principle has been found to break down in complex fluids. [3]

## Principle of flotation

Archimedes' principle shows buoyant force and displacement of fluid. However, the concept of Archimedes' principle can be applied when considering why objects float. Proposition 5 of Archimedes' treatise On Floating Bodies states that:

Any floating object displaces its own weight of fluid.

In other words, for an object floating on a liquid surface (like a boat) or floating submerged in a liquid (like a submarine in water or dirigible in air) the weight of the displaced liquid equals the weight of the object. Thus, only in the special case of floating does the buoyant force acting on an object equal the objects weight. Consider a 1-ton block of solid iron. As iron is nearly eight times denser than water, it displaces only 1/8 ton of water when submerged, which is not enough to keep it afloat. Suppose the same iron block is reshaped into a bowl. It still weighs 1 ton, but when it is put in water, it displaces a greater volume of water than when it was a block. The deeper the iron bowl is immersed, the more water it displaces, and the greater the buoyant force acting on it. When the buoyant force equals 1 ton, it will sink no farther.

When any boat displaces a weight of water equal to its own weight, it floats. This is often called the "principle of flotation": A floating object displaces a weight of fluid equal to its own weight. Every ship, submarine, and dirigible must be designed to displace a weight of fluid equal to its own weight. A 10,000-ton ship must be built wide enough to displace 10,000 tons of water before it sinks too deep in the water. The same is true for vessels in air: a dirigible that weighs 100 tons needs to displace 100 tons of air. If it displaces more, it rises; if it displaces less, it falls. If the dirigible displaces exactly its weight, it hovers at a constant altitude.

It is important to realize that, while they are related to it, the principle of flotation and the concept that a submerged object displaces a volume of fluid equal to its own volume are not Archimedes' principle. Archimedes' principle, as stated above, equates the buoyant force to the weight of the fluid displaced.