Water is flowing through the tube depicted in the figure. The cross-sectional areas of the wide and narrow portions of the tube are 5 cm ^2 and 2 cm ^2 , respectively. If the rate of flow of water through the tube is 500 cm ^3 s ⁻¹ , determine the difference in the mercury levels in the U-tube.

Step-by-step solution

Given: Cross-sectional area of wide portion, A₁ = 5 cm ^2 Cross-sectional area of narrow portion, A₂ = 2 cm ^2 Rate of flow of water, Q = 500 cm ^3 s ⁻¹ Density of water, _w = 1 g cm ⁻³ Density of mercury, _m = 13.6 g cm ⁻³ Acceleration due to gravity, g = 980 cm s ⁻² Velocity of water in the wide portion: v₁ = Q A₁ = 500 5 = 100 cm s ⁻¹ Velocity of water in the narrow portion: v₂ = Q A₂ = 500 2 = 250 cm s ⁻¹ According to Bernoulli's principle for a horizontal tube: P₁ + 1 2 _w v₁^2 = P₂ + 1 2 _w v₂^2 P₁ - P₂ = 1 2