Water flows downwards through a vertical tapering pipe. The cross-sectional area of the top end is 4 cm ^2 and that of the bottom end is 2 cm ^2 . The vertical distance between the two ends is 15 cm . If the pressure of water at the top end is equal to the pressure at the bottom end, what is the volume flow rate of water through the pipe? (Take g = 10 m s ⁻² )

Step-by-step solution

Let the top end be section 1 and the bottom end be section 2. Given: A₁ = 4 cm ^2 A₂ = 2 cm ^2 h = 15 cm g = 10 m s ⁻² = 1000 cm s ⁻² P₁ = P₂ Applying Bernoulli's equation between the top and bottom ends: P₁ + g h + 1 2 v₁^2 = P₂ + 0 + 1 2 v₂^2 Since P₁ = P₂ , the pressure terms cancel out: g h + 1 2 v₁^2 = 1 2 v₂^2 2 g h = v₂^2 - v₁^2 From the equation of continuity, the volume flow rate Q = A₁ v₁ = A₂ v₂ . Thus, v₁ = Q 4 and v₂ = Q 2 . Substitute these into the simplified Bernoulli's equation: 2 1000 15 = ( Q 2 )