An incompressible fluid is flowing through a horizontal pipe with a constriction. The velocity of the fluid in the wide section of the pipe is 5.00 m/s. The diameter of the wide section is 10.0 cm and the diameter of the narrow section is 8.00 cm. The pressure of the fluid in the wide section is 200 kPa. What is the pressure in the narrow section of the pipe? (density of the fluid is 680 kg/m³) A) 188 kPa B) 175 kPa C) 160 kPa D) 152 kPa E) 148 kPa

Question

Quizplus · Accepted Answer

To find the pressure in the narrow section, we can use the Bernoulli's equation, which for a horizontal flow states that $P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$, where $P_1$ and $P_2$ are the pressures in the wide and narrow sections respectively, $v_1$ and $v_2$ are the velocities in the wide and narrow sections respectively, and $\rho$ is the density of the fluid. The continuity equation, $A_1v_1 = A_2v_2$, where $A_1$ and $A_2$ are the cross-sectional areas of the wide and narrow sections respectively, allows us to find $v_2$. Given $d_1 = 10.0$ cm and $d_2 = 8.00$ cm, we find $A_1 = \pi(0.05)^2$ m$^2$ and $A_2 = \pi(0.04)^2$ m$^2$. With $v_1 = 5.00$ m/s, we solve for $v_2$ and then use Bernoulli's equation to find $P_2$. The calculation shows that the pressure in the narrow section is lower than in the wide section, consistent with the increase in velocity and the conservation of energy, leading to the correct answer of 188 kPa.

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