An incompressible fluid is flowing through a pipe with a constriction. The pipe is on an incline with an angle of 30.0 degrees. The narrow section is 10.0 m from the wide section and the narrow section is lower than the wide section. The velocity of the fluid in the wide section of the pipe is 4.00 m/s. The diameter of the wide section is 12.00 cm and the diameter of the narrow section is 9.00 cm. The pressure of the fluid in the wide section is 250 kPa. What is the pressure in the narrow section of the pipe? (density of the fluid is 1,000 kg/m³) A) 295 kPa B) 282 kPa C) 275 kPa D) 252 kPa E) 240 kPa

Question

Quizplus · Accepted Answer

To find the pressure in the narrow section, we can use Bernoulli's equation, which states that the total mechanical energy in a flowing fluid remains constant. The equation is $P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2$, where $P$ is pressure, $\rho$ is the fluid density, $v$ is the fluid velocity, $g$ is the acceleration due to gravity, and $h$ is the height above a reference point. Given the incline and the heights, we can calculate the change in height ($\Delta h$) using trigonometry, $h = 10 \sin(30^\circ) = 5$ m. The velocities can be related by the continuity equation $A_1v_1 = A_2v_2$, where $A$ is the cross-sectional area of the pipe. Using the given diameters, we can find $v_2$, and then apply Bernoulli's equation to solve for $P_2$, the pressure in the narrow section.

An Incompressible Fluid Is Flowing Through a Pipe with a Constriction