An experiment is conducted to measure the pressure drop P across a fixed horizontal constriction in a pipe for various steady volume flow rates Q of an ideal fluid. Which of the following graphs best represents the variation of P (on the y-axis) with Q (on the x-axis)?

Step-by-step solution

Let the cross-sectional area of the wider part of the pipe be A₁ and that of the constriction be A₂ . From the equation of continuity, the volume flow rate is Q = A₁ v₁ = A₂ v₂ . The velocities at the two sections are v₁ = Q A₁ and v₂ = Q A₂ . Applying Bernoulli's principle for a horizontal pipe: P₁ + 1 2 v₁^2 = P₂ + 1 2 v₂^2 The pressure drop P is given by: P = P₁ - P₂ = 1 2 (v₂^2 - v₁^2) Substituting the expressions for velocities: P = 1 2 [ ( Q A₂ )^2 - ( Q A₁ )^2 ] P = [ 1 2 ( 1 A₂^2 - 1 A₁^2 ) ] Q^2 Since the