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Continuum mechanics |
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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.

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.

— Archimedes of Syracuse

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 into 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, that equals exactly to 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 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.

The pressure difference between the bottom and the top face is directly proportional to the height (difference in depth). Multiplying the pressure difference with the area of a face gives the net force on the cube - the buoyancy, or the weight of the fluid displaced. 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 water is less than the weight of object in air, because of the force acting on it which is called as 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 upon 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,

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

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:

(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. In increasing speed or driving a curve, the air moves in the opposite direction of the car's acceleration. The balloon however, is pushed due to buoyancy "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 buoyant force and it is proportional to the weight of 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.

Archimedes' principle does not consider the surface tension (capillarity) acting on the body.^{[2]}

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.

— Archimedes of Syracuse

^{[3]}

In other words, for a floating object on a liquid, the weight of the displaced liquid is 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 (as air is a fluid): a dirigible that weighs 100 tons displaces at least 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 floatation 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.

- "Eureka", reportedly exclaimed by Archimedes upon discovery that the volume of displaced fluid is equal to the volume of the submerged object (note that this idea is not Archimedes' principle)

**^**Acott, Chris (1999). "The diving "Law-ers": A brief resume of their lives.".*South Pacific Underwater Medicine Society journal***29**(1). ISSN 0813-1988. OCLC 16986801. Retrieved 2009-06-13.**^**"Floater clustering in a standing wave: Capillarity effects drive hydrophilic or hydrophobic particles to congregate at specific points on a wave" (PDF). 2005-06-23.**^**"The works of Archimedes". p. 257. Retrieved 11 March 2010. "Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced."