Buoyancy

When a body is either wholly or partially immersed in a fluid, a lift is generated due to the net vertical component of hydrostatic pressure forces experienced by the body.

This lift is called the buoyant force and the phenomenon is called buoyancy
 Consider a solid body of arbitrary shape completely submerged in a homogeneous liquid as shown in Fig. 5.4. Hydrostatic pressure forces act on the entire surface of the body.
Fig 5.4 Buoyant Force on a Submerged Body
To calculate the vertical component of the resultant hydrostatic force, the body is considered to be divided into a number of elementary vertical prisms. The vertical forces acting on the two ends of such a prism of crosssection dA_{z} (Fig. 5.4) are respectively

(5.19a) 

(5.19b) 
Therefore, the buoyant force (the net vertically upward force) acting on the elemental prism of volume is 

(5.19c) 
Hence the buoyant force F_{B} on the entire submerged body is obtained as

(5.20) 
Where is the total volume of the submerged body. The line of action of the force F_{B} can be found by taking moment of the force with respect to zaxis. Thus

(5.21) 
Substituting for dF_{B} and F_{B} from Eqs (5.19c) and (5.20) respectively into Eq. (5.21), the x coordinate of the center of the buoyancy is obtained as

(5.22) 
which is the centroid of the displaced volume. It is found from Eq. (5.20) that the buoyant force F_{B} equals to the weight of liquid displaced by the submerged body of volume . This phenomenon was discovered by Archimedes and is known as the Archimedes principle.
ARCHIMEDES PRINCIPLE
The buoyant force on a submerged body

The Archimedes principle states that the buoyant force on a submerged body is equal to the weight of liquid displaced by the body, and acts vertically upward through the centroid of the displaced volume.

Thus the net weight of the submerged body, (the net vertical downward force experienced by it) is reduced from its actual weight by an amount that equals the buoyant force.
The buoyant force on a partially immersed body

According to Archimedes principle, the buoyant force of a partially immersed body is equal to the weight of the displaced liquid.

Therefore the buoyant force depends upon the density of the fluid and the submerged volume of the body.
 For a floating body in static equilibrium and in the absence of any other external force, the buoyant force must balance the weight of the body.
