Archimedes' principle

Archimedes' principle is a law of physics stating that the upward buoyant force exerted on a body immersed in a fluid is equal to the weight of the fluid the body displaces. In other words, an immersed object is buoyed up by a force equal to the weight of the fluid it actually displaces. Archimedes' principle is an important and underlying concept in the field of fluid mechanics. This principle is named after its discoverer, Archimedes of Syracuse.

Explanation

Archimedes' two-part treatise on hydrostatics, called On Floating Bodies, states that:

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

with the clarifications that for a sunken object the volume of displaced fluid is the volume of the object. Thus, in short, buoyancy = weight of displaced fluid. Archimedes' principle is true of liquids and gases, both of which are fluids. If an immersed object displaces 1 kilogram of fluid, the buoyant force acting on it is equal to the weight of 1 kilogram (as a kilogram is unit of mass and not of force, the buoyant force is the weight of 1 kg, which is approximately 9.8 Newtons.) It is important to note that the term immersed refers to an object that is either completely or partially submerged. If a sealed 1-liter container is immersed halfway into the water, it will displace a half-liter of water and be buoyed up by a force equal to the weight of a half-liter of water, no matter what is in the container.

If such an object is completely submerged, it will be buoyed up by a force equivalent to the weight of a full liter of water (1 kilogram of force). If the container is completely submerged and does not compress, the buoyant force will equal the weight of 1 kilogram of water at any depth, since the volume of the container does not change, resulting in a constant displacement regardless of depth. The weight of the displaced water, and not the weight of the submerged object, is equal to the buoyant force.

Objects weigh more in air than they do in water. If a 30-kilogram object displaces 20 kilograms of fluid when it is immersed, its apparent weight will be equal to the weight of 10 kilograms (98 newtons). Similarly, when submerged, a 3-kilogram block may have an apparent weight of 1 kilogram. The "missing weight" is equal to the weight of the water displaced, the weight of 2 kilograms (19.6 newtons), which is the buoyant force. The apparent weight of a submerged object is its weight under standard conditions minus the buoyant force.

Consider a completely submerged cube. The fluid exerts pressure on all six faces, but as long as the cube is not tilted, the forces on the four vertical faces balance each other out. 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 give the net force on the cube - the buoyancy, or the weight of the fluid displaced. It makes no difference how deep the cube is placed because, although the pressures are greater with increasing depths, the difference between the pressure up against the bottom of the cube and the pressure down against the top of the cube is the same at any depth. Whatever the shape of the submerged body, the buoyant force is equal to the weight of the fluid displaced.

Formula

The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). 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.

Assuming Archimedes' principle to be reformulated as follows,

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

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

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

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. 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.

Refinements

Archimedes' principle does not consider the surface tension (capillarity) acting on the body.

Principle of floatation

Archimedes' principle relates 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

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 floatation": 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.

See also

• "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)