When an object is totally or partially immersed in a liquid, an upthrust acts on it equal to the weight of the liquid it displaces.
An air filled balloon immediately shoots up to the surface when released under water. The same would happen if a piece of wood is released under water. We might have noticed that a mug filled with water feels light under water but feels heavy as soon as we take it out of water.
More than two thousand years ago, the Greek scientist, Archimedes noticed that there is an upward force which acts on an object kept inside a liquid.As a result an apparent loss of weight is observed in the object is called the upthrust of the liquid.
Consider a solid cylinder of cross-sectional area A and height h immersed in a liquid. Let h1 and h2 be the depths of the top and bottom faces of the cylinder respectively from the surface of the liquid.
Then h2 – h2 = h
If P1 and P2 are the liquid pressure at depths h1 and h2 respectively and ρ is the density, then according to equation (2):
P1 = ρ g h 1
P2 = ρ g h2
Then the force F1 is exerted at the cylinder top by the liquid due to pressure P1 and the force F2 is exerted at the bottom of the cylinder by the liquid due to P2.
F1 = P1 A = ρ g h1 A
and F2 = P2 A = ρ g h2 A
F1 and F2 are acting on the opposite faces of the cylinder. Therefore, the net force F will be F2 – F1 in the direction of F2. This net force F on the cylinder is called the upthrust of the liquid.
F2 – F1 =ρ g h2 A – ρ g h1 A
=ρ gA(h2 – h1)
Upthrust of liquid =ρ gAh
or =ρ gV
Here Ah is the volume V of the cylinder and is equal to the volume of the liquid displaced by the cylinder. Therefore, ρ g V is the weight of the liquid displaced. In equation that an upthrust acts on the body immersed in a liquid and is equal to the weight of liquid displaced, which is Archimedes principle.
Density Of an Object:
Archimedes principle is also helpful to determine the density of an object. The ration in the weights of a body with an equal volume of liquid is the same as in their densities.
Let Density of the object = D
Density of the liquid = ρ
Weight of the object = w1
Weight of equal volume of liquid = w = w1 – w2
Here w2 is the weight of the solid in liquid. According to Archimedes principle, w2 is less than its actual weight w1 by an amount w.
Thus finding the weight of the solid in air w,and its weight in water w2,we can calculate the density of solid by above equation.
Principle Of Floatation:
A floating object displaces a fluid having weight equal to the weight of the object.
An object sinks if its weight is greater than the upthrust acting on it. An objects floats if its weight is equal or less than the upthrust. When an object floats in a fluid, the upthrust acting on it is equal to the weight of the object. In case of floating object, the object may be partially immersed. The upthrust is always equal to the weight of the fluid displaced by the object. This is the principle of floatation.