Main article: Archimedes' principle
Archimedes' principle is named after Archimedes of Syracuse, who first discovered this law in 212 B.C. For more objects, floating and sunken, and in gases as well as liquids (i.e. a fluid), Archimedes' principle may be stated thus in terms of forces:
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, and for a floating object on a liquid, the weight of the displaced liquid is the weight of the object.
More tersely: Buoyancy = weight of displaced fluid.
Archimedes' principle does not consider the surface tension (capillarity) acting on the body, but this additional force modifies only the amount of fluid displaced, so the principle that Buoyancy = weight of displaced fluid remains valid.
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 buoyancy force on an object is going to be 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 acceleration, g. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy.This is also known as upthrust.
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 buoyancy 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,
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.
 Forces and equilibrium
This is the equation to calculate the pressure inside a fluid in equilibrium. The corresponding equilibrium equation is:
where f is the force density exerted by some outer field on the fluid, and σ is the stress tensor. In this case the stress tensor is proportional to the identity tensor:
Here is the Kronecker delta. Using this the above equation becomes:
Assuming the outer force field is conservative, that is it can be written as the negative gradient of some scalar valued function:
Therefore, the shape of the open surface of a fluid equals the equipotential plane of the applied outer conservative force field. Let the z-axis point downward. In this case the field is gravity, so Φ = −ρfgz where g is the gravitational acceleration, ρf is the mass density of the fluid. Taking the pressure as zero at the surface, where z is zero, the constant will be zero, so the pressure inside the fluid, when it is subject to gravity, is
So pressure increases with depth below the surface of a liquid, as z denotes the distance from the...
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