Archimedes' principle

In simple words, Archimedes' principle states that, when a body is partially or completely immersed in a fluid, it experiences an apparent loss in weight that is equal to the weight of the fluid displaced by the immersed part of the body(s).

Consider a cuboid immersed in a fluid, its top and bottom faces orthogonal to the direction of gravity (assumed constant across the cube's stretch).

Multiplying the pressure difference by the area of a face gives a net force on the cuboid—the buoyancy—equaling in size the weight of the fluid displaced by the cuboid.

In simple terms, the principle states that the buoyant force (Fb) on an object is equal to the weight of the fluid displaced by the object, or the density (ρ) of the fluid multiplied by the submerged volume (V) times the gravity (g)[1][3] We can express this relation in the equation: where

Suppose a rock's weight is measured as 10 newtons when suspended by a string in a vacuum with gravity acting on it.

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.

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 drift in the same direction as the car's acceleration.

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: Then: Therefore, the shape of the open surface of a fluid equals the equipotential plane of the applied outer conservative force field.

In this case the field is gravity, so Φ = −ρfgz where g is the gravitational acceleration, ρf is the mass density of the fluid.

The buoyancy force exerted on a body can now be calculated easily, since the internal pressure of the fluid is known.

In other words, the "buoyancy force" on a submerged body is directed in the opposite direction to gravity and is equal in magnitude to The net force on the object must be zero if it is to be a situation of fluid statics such that Archimedes principle is applicable, and is thus the sum of the buoyancy force and the object's weight If the buoyancy of an (unrestrained and unpowered) object exceeds its weight, it tends to rise.

Once it fully sinks to the floor of the fluid or rises to the surface and settles, Archimedes principle can be applied alone.

For a sunken object, the entire volume displaces water, and there will be an additional force of reaction from the solid floor.

An object which tends to float requires a tension restraint force T in order to remain fully submerged.

An object which tends to sink will eventually have a normal force of constraint N exerted upon it by the solid floor.

To find the force of buoyancy acting on the object when in air, using this particular information, this formula applies: The final result would be measured in Newtons.

A simplified explanation for the integration of the pressure over the contact area may be stated as follows: Consider a cube immersed in a fluid with the upper surface horizontal.

Similarly, the downward force on the cube is the pressure on the top surface integrated over its area.

Angled surfaces do not nullify the analogy as the resultant force can be split into orthogonal components and each dealt with in the same way.

In this case, the net force has been found to be different from Archimedes' principle, as, since no fluid seeps in on that side, the symmetry of pressure is broken.

The deeper the iron bowl is immersed, the more water it displaces, and the greater the buoyant force acting on it.

Archimedes' principle, as stated above, equates the buoyant force to the weight of the fluid displaced.

[8][9][10] Archimedes reportedly exclaimed "Eureka" after he realized how to detect whether a crown is made of impure gold.

While he did not use Archimedes' principle in the widespread tale and used displaced water only for measuring the volume of the crown, there is an alternative approach using the principle: Balance the crown and pure gold on a scale in the air and then put the scale into water.