The question
Name: ______________________ Date: ____________ A tank drains through a narrowing pipe (10 points)
Diagram
A tank and a narrowing pipe
A wide, open tank holds water of density ρ. Water flows out through a horizontal pipe at the bottom. The pipe has cross-sectional area A₁ at section 1 and a smaller area A₂ at section 2, where it ends open to the air at atmospheric pressure P₀. The center of the pipe is a depth h below the water surface. The tank is so wide that its surface moves down negligibly slowly. Treat the water as an ideal fluid in a full pipe.
(a) Derive an expression for the speed v₂ of the water leaving section 2. Express your answer in terms of h and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. [2 points]
(b) Derive an expression for the gauge pressure P₁ − P₀ of the water in section 1. Express your answer in terms of ρ, g, h, A₁ and A₂. [3 points]
(c) Calculate the gauge pressure in section 1 when h = 0.80 m, A₂ = A₁/2 and the fluid is water (ρ = 1000 kg/m³). Use g = 9.8 m/s². [1 point]
(d) The tank is refilled to different depths. On the axes, sketch a graph of the exit speed v₂ as a function of the depth h. [2 points]
(e) The narrow section is removed, so the pipe has area A₁ all the way to its open end. A student claims that the gauge pressure in section 1 increases, because more water now flows through the pipe each second. Is the student correct? Justify your answer. [2 points]