An incompressible fluid is flowing through a vertical pipe with a constriction. The wide section is 2.00 cm in diameter and is at the top of the pipe. The pressure of the fluid in the wide section at the top is 200 kPa. The velocity of the fluid in the wide section is 4.00 m/s. The narrow section is 1.00 cm in diameter and is located 2.00 m below the wide section. What is the pressure of the fluid in the narrow section? (density of the fluid is 1,000 kg/m³) A) 72.0 kPa B) 80.0 kPa C) 99.6 kPa D) 105 kPa E) 116 kPa

Question

Quizplus · Accepted Answer

We can use Bernoulli's equation to solve for the pressure in the narrow section of the pipe:

P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2

where P1 and v1 are the pressure and velocity in the wide section (given in the problem), P2 is the pressure we want to find, v2 is the velocity in the narrow section (which we can solve for using the continuity equation), and h1 and h2 are the heights of the two sections of the pipe relative to some reference point (which we can choose to be the narrow section).

First, we can use the continuity equation to relate the velocities in the two sections of the pipe:

A1v1 = A2v2

where A1 and A2 are the cross-sectional areas of the wide and narrow sections, respectively. We can solve for v2:

v2 = A1v1/A2 = (π/4)(0.02 m)^2(4 m/s)/[(π/4)(0.01 m)^2] = 16 m/s

Now we can plug in the given values and solve for P2:

200 kPa + 1/2(1000 kg/m^3)(4 m/s)^2 + (1000 kg/m^3)(9.81 m/s^2)(0 m) = P2 + 1/2(1000 kg/m^3)(16 m/s)^2 + (1000 kg/m^3)(9.81 m/s^2)(-2 m)

Simplifying and solving for P2:

P2 = 99.6 kPa

An Incompressible Fluid Is Flowing Through a Vertical Pipe with a Constriction