State and prove Archimedes principle.

Explanation

Archimedes principle states that when a body is partially or wholly immersed in a fluid, it experiences an upward thrust equal to the weight of the fluid displaced by it and its upthrust acts through the centre of gravity of the liquid displaces.
Proof: Consider a body of height h lying inside a liquid of density `rho`, at a depth x below the free surface of the liquid. Area of cross-section of the body is a. The forces on the sides of the body cancel out.

Pressure at the upper face of the body,
`P_(1)=xrhog`
Pressure at the lower face of the body,
`P_(2)=(x+h)rhog`
Thrust acting on the upper face of the body is
`F_(1)=P_(1)a=xrhoga,`
acting vertically downwards.
Thrust acting on the lower face of the body is
`F_(2)=P_(2)a=(x+h)rhoga`
acting vertically upwards.
The resultant force `(F_(2)-F_(1))` is acting on the body is the upward direction and is called Upthrust (U).
`therefore" "U=F_(2)-F_(1)=(x+h)rhog-xrhoga=ahrhog`
But, ah = V, the volume of the body = volume of liquid displaced.
`U=Vrhog=Mg`
[`thereforeM=Vrho=` mass of liquid displaced]
i.e., upthrust or buoyant force. = Weight of liquid displaced.
This proves the Archimedes principle.