Consider a vessel filled with a liquid upto height H . The bottom of the vessel lies in the X - Y plane passing through the origin. The density of the liquid varies with Z -axis as ρ z = ρ 0 2 - z H 2 . If P 1 and P 2 are the pressures at the bottom surface and top surface of the liquid, the magnitude of P 1 - P 2 is :

Step-by-step solution

Given that, P Z = ρ 0 2 - Z H 2 Let a small column of liquid thickness be d z . Pressure due to column, d P 1 = ρ 0 2 - Z H 2 g d z + P 2 P 1 = ∫ 0 π ρ 0 2 - Z H 2 g d z + P 2 = ρ 0 g ∫ 0 π 2 - Z 2 H 2 d z + P 2 ' = ρ 0 g 2 H - 1 3 H 2 H 3 + P 2 P 1 = ρ 0 g 2 H - H 3 + P 2 P 1 - P 2 = ρ 0 g 5 H 3 = 5 3 ρ 0 g H