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Question #42319

Difficulty: 8/10 | 🟢 Free

The Question

A liquid flows steadily downwards through a vertical pipe of constant cross-sectional area $A$ . Due to a steady thermal gradient, the density of the liquid varies with depth $z$ as $\rho(z) = \rho_0 (1 + \alpha z)$ , where $\rho_0$ and $\alpha$ are positive constants. The mass flow rate of the liquid is $\dot{m}$ . If the pressure at $z=0$ is $P_0$ , what is the pressure $P_L$ at depth $z=L$ ?

Options

A.
$P_0 + \rho_0 g L \left(1 + \frac{\alpha L}{2}\right) + \frac{\dot{m}^2 \alpha L}{A^2 \rho_0 (1+\alpha L)}$
B.
$P_0 + \rho_0 g L \left(1 + \frac{\alpha L}{2}\right) + \frac{\dot{m}^2 \alpha L}{2 A^2 \rho_0 (1+\alpha L)}$
C.
$P_0 + \rho_0 g L \left(1 + \frac{\alpha L}{2}\right)$
D.
$P_0 + \rho_0 g L \left(1 + \frac{\alpha L}{2}\right) - \frac{\dot{m}^2 \alpha L}{A^2 \rho_0 (1+\alpha L)}$

Check Your Answer

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

Potential Energy Reference|Bernoulli’s Energy Balance|Steady State Temperature Gradient

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental limits of Bernoulli's equation.

A common trap is applying the integrated Bernoulli equation ( $P + \frac{1}{2}\rho v^2 + \rho g z = \text{constant}$ ) across streamlines when density is spatially variant.

Because density changes with depth, the work-energy theorem must be applied differentially ( $dP + \rho v dv - \rho g dz = 0$ ).

Integrating this with a constant mass flux $J = \rho v$ yields a dynamic pressure term proportional to $J \Delta v$ , which lacks the $1/2$ factor found in the standard Bernoulli equation.

Students who blindly apply the standard formula will incorrectly choose the option with the $1/2$ coefficient.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Potential Energy Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free