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State and prove Archimedes principle.

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#### Solution

** Archimedes principle**

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 center of gravity of the liquid displaced.

**Proof:** Consider a body of height h lying inside a liquid of density p, at a depth x below the free surface of the liquid. The area of a cross-section of the body is a. The forces on the sides of the body cancel out.

** Buoyant force on a body**

Pressure at the upper face of the body,

P_{1} = xρg

Pressure at the lower face of the body,

P_{2} = (x + h)pρg

Thrust acting on the upper face of the body is F_{1} = P_{1}a = xρga acting vertically downwards.

Thrust acting on the lower face of the body is F_{2} = P_{2}a = (x+h)ρga acting vertically upwards.

The resultant force (F_{2} – F_{1}) is acting on the body in the upward direction and is called Upthrust (U).

∴ U = F_{2} – F_{1} = (x + h)ρg – xρga = ahρg

But, ah = V, the volume of the body = volume of liquid displaced.

U = Vρg = Mg

[∴ M = Vρ = mass of liquid displaced]

i.e., upthrust or buoyant force = Weight of liquid displaced.

This proves the Archimedes principle.

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