Question

State archimedes' principle and prove it mathematically

Answer 1

A mathematical proof of Archimedes' Principle, that the buoyant force pushing up on an object immersed in a fluid is equal to the weight of the fluid that is displaced by that object.
It can be proved mathematically using Newtons's Laws and Divergence Theorem. Consider a body of volume V having closed surface S submerged in liquid of density D.
Suppose the pressure at the surface of the fluid is zero then the pressure at a point 'P' at height ybelow the surface will be Dgy . Now, the net upward force exerted by the fluid on the body is the sum(integration) of vertical component of infinitesimal forces exerted on the body by the fluid over the surface. This surface integral becomes  volume integral of function 'Dg' (divergence of  'Dgy')  by using Divergence Theorem which gives DgV(weight of liquid displaced) as the net upward force exerted on the body by the fluid.
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Answer 2

Archimedes’ Principle:

Archimedes' principle states that the upward buoyant force that is utilized on a body immersed in a fluid, whether fully or partially plunged, is equal to the weight of the fluid that the body displaces.  

Mathematical proof:

Buoyant force Fb= F up –F down

As we know that F=P.A where P is a pressure exerted by the fluid and A is the area of the surface.

Therefore,

Fb= P bottom * A- Ptop A

Now,  

Density = pgH

Now,

Fb= pgH(bottom) A – pgH (top) A

Fb=pgA(H(bottom)- H(top))

As we know that  

H= H(bottom)- H(top)

So, Fb=pgAh

Also we know that A*H=Volume

So Fb= pgV

As p.V= m

So equation becomes  

Fb= m(displace fluid). g

This is the mathematical proof of Archimedes principle.

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