State and prove 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,
P1 = xρg
Pressure at the lower face of the body,
P2 = (x + h)pρg
Thrust acting on the upper face of the body is F1 = P1a = xρga acting vertically downwards.
Thrust acting on the lower face of the body is F2 = P2a = (x+h)ρga acting vertically upwards.
The resultant force (F2 – F1) is acting on the body in the upward direction and is called Upthrust (U).
∴ U = F2 – F1 = (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.