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 higher 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) 210 kPa B) 205 kPa C) 195 kPa D) 184 kPa E) 175 kPa

Question

Quizplus · Accepted Answer

To find the pressure in the narrow section, we can use the Bernoulli's equation, which for an incompressible fluid is given by $P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2$, where $P$ is the pressure, $\rho$ is the density of the fluid, $v$ is the velocity of the fluid, $g$ is the acceleration due to gravity, and $h$ is the height above a reference point. Given that the fluid is incompressible, the volume flow rate must be constant, so $A_1v_1 = A_2v_2$, where $A$ is the cross-sectional area of the pipe. From this, we can solve for $v_2$, and then use Bernoulli's equation to find $P_2$.First, find the cross-sectional areas: $A_1 = \pi r_1^2 = \pi (0.06)^2$ m$^2$ and $A_2 = \pi r_2^2 = \pi (0.045)^2$ m$^2$. Then, find $v_2$ using the continuity equation: $v_2 = \frac{A_1}{A_2}v_1$.Next, calculate the height difference ($h_2 - h_1$) using trigonometry: $h_2 - h_1 = 10 \sin(30^\circ)$.Finally, substitute all known values into Bernoulli's equation to solve for $P_2$. After calculations, you'll find that the pressure in the narrow section corresponds to the value given in option D, 184 kPa.

An Incompressible Fluid Is Flowing Through a Pipe with a Constriction