/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 A gas having density \(\rho\) fl... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 29

A gas having density \(\rho\) flows with a velocity \(v\) along a pipe of cross- sectional area \(s\) and bent at an angle of \(90^{\circ}\) at a point \(A\). The force exerted by the gas on the pipe at \(A\) is (A) \(\frac{\sqrt{2} s v}{\rho}\) (B) \(s v^{2} \rho\) (C) \(\frac{\sqrt{3} s v^{2} \rho}{2}\) (D) \(s v^{2} \rho\)

Short Answer

Expert verified
The force exerted by the gas on the pipe at point A is: \( F_A = \rho s v \sqrt{2} \)

Step by step solution

01

In this problem, we are given the density of the gas (\(\rho\)), the velocity of the gas along the pipe (v), and the cross-sectional area of the pipe (s). The force exerted by the gas on the pipe at point A is what we are trying to find. #Step 2: Calculate the momentum change of the gas#

Since the gas is flowing within the pipe and the pipe is bent at a 90掳 angle, the change in momentum of the gas at point A will result in a force acting on the pipe. We can calculate the momentum change by finding the difference in momentum before and after the bend. The initial momentum before the bend is given by the mass flow rate (峁�) multiplied by the initial velocity (\(v_x\)): \(峁� v_x = \rho s v_x\) After the bend, the momentum is given by the mass flow rate multiplied by the final velocity (\(v_y\)): \(峁� v_y = \rho s v_y\) Since the angle of the bend is 90掳, the initial velocity components are perpendicular to each other and the magnitudes of \(v_x\) and \(v_y\) are equal to the original velocity \(v\): \(v_x = v\) and \(v_y = v\) #Step 3: Determine the net force acting on the pipe at point A#
02

To calculate the net force exerted by the gas on the pipe at the point of bending, we'll need to combine the forces resulting from the changes in the x- and y-directions caused by the change in momentum. Using Newton's second law of motion, we can write the equation for net force as: \( F_x = \Delta ( \rho s v_x) = \rho s v \) \( F_y = \Delta ( \rho s v_y) = \rho s v \) Now we can find the net force acting on the pipe at point A (\(F_A\)) by combining the x- and y-components using the Pythagorean theorem: \( F_A = \sqrt{ F_x^2 + F_y^2 } = \sqrt{(\rho s v)^2 + (\rho s v)^2} \) #Step 4: Solve for the force exerted by the gas on the pipe at point A#

To find the force exerted by the gas on the pipe at point A, we'll simplify the equation from Step 3: \( F_A = \sqrt{2 (\rho s v)^2 } = \rho s v \sqrt{2} \) This matches the answer choice (A), so the force exerted by the gas on the pipe at point A is: \( F_A = \frac{\sqrt{2} s v}{\rho} \)

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