**Archimedes' principle **states that when a body is immersed partially or completely in a liquid,it experiences an upthrust or buoyant force ,which is equal to the weight of the liquid displaced by it.

When a body is immersed in a liquid,the liquid exerts an upward force on the body.This force is called the **upthrust or buoyant force.**

or we can say,

the upward force exerted on a body by the fluid in which it is submerged is called **the upthrust or buoyant force.**

Now we shall prove mathematically that when a body is immersed in a liquid,the upthrust on body due to liquid is equal to the weight of liquid displaced by the submerged part of the body.

For the convenience of calculation,let us consider a cylindrical body of cross sectional area A immersed in a liquid of density d.Let the upper surface of the body be at a depth h1 below the free surface of liquid and the lower surface of the body be at a depth h2 below the free surface of liquid.

At depth h1,pressure on the upper surface

p1=h1dg

therefore,downward thrust on the upper surface

F1=pressure*area=h1dgA .....(1)

At depth h2 pressure on the lower surface

F2=h2dgA .................(2)

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The horizontal thrust at various points on the vertical sides of the cylinder get balanced because the liquid pressure ia same at all points at the same depth.

From above equations (1) and (2), it is clear that F2>F1 because h2>h1 and therefore, the cylinder will experience a net upward force.

resultant upward thrust on the body,

F=F2-F1

=h2dgA-h1dgA

=A(h2-h1)dg

But A(h2-h1)=V,THE VOLUME OF CYLINDER SUBMERGED IN LIQUId

therefore,upthrust F="Vdg.........(3)

since a solid when immersed in a liquid ,displaces liquid equal to the volume of its submerged part,therefore

Vdg=Volume of solid immersed *density of liquid*acceleration due to gravity.

=volume of liquid displaced*density of liquid*acceleration due to gravity.

=mass of liquid displaced*acceleration due to gravity.

=weight of the liquid displaced by the submerged part of the body.

hence,Upthrust=weight of the liquid displaced by the submerged part of the body. .....(4)

note that if the body is completely immersed in a liquid,the volume of liquid displaced will be equal to its own volume.Although above relation (4) is derived for a cylindrical body,but it is equally true for a body of any shape and size.

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