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Question #42461

Difficulty: 8/10 | 🟢 Free

The Question

A steady, incompressible fluid of density $\rho$ flows out of a circular pipe of radius $R$ . The velocity profile of the fluid across the pipe's cross-section is non-uniform and given by $v(r) = v_0 \left(1 - \frac{r^2}{R^2}\right)$ , where $r$ is the radial distance from the central axis and $v_0$ is the maximum velocity. The fluid stream strikes a stationary hemispherical deflector placed coaxially, which completely reverses the direction of the fluid flow ( $180^\circ$ deflection). Assuming the fluid leaves the deflector with the same speed it had upon striking for each streamline, what is the magnitude of the horizontal force exerted by the fluid on the deflector?

Options

A.
$\frac{2}{3} \pi \rho R^2 v_0^2$
B.
$\frac{1}{2} \pi \rho R^2 v_0^2$
C.
$\pi \rho R^2 v_0^2$
D.
$\frac{1}{3} \pi \rho R^2 v_0^2$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Impulse-Momentum Theorem|Venturi Meter Logic|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Impulse-Momentum Theorem for continuous fluid flow with a non-uniform velocity profile.

A common 'Lethal Collision' occurs when students incorrectly assume they can use the average velocity $v_{avg} = v_0/2$ to calculate the force as $F = 2 \dot{m} v_{avg} = 2(\rho A v_{avg})v_{avg} = \frac{1}{2}\pi\rho R^2 v_0^2$ .

However, because momentum flux depends on $v^2$ , the integral of $v^2$ over the cross-section is not equal to the square of the average velocity.

Students must recognize that the rate of change of momentum requires a spatial integral $\int 2v \, d\dot{m}$ over the contact area, distinguishing between bulk kinetic energy assumptions and rigorous momentum flux integration.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Impulse-Momentum Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free