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Question #42233

Difficulty: 8/10 | 🟢 Free

The Question

A long thin-walled elastic cylindrical tube of unstretched radius $R$ and thickness $h$ ( $h \ll R$ ) is made of a material with Young's modulus $Y$ . The tube is completely filled with a compressible fluid of density $\rho$ and bulk modulus $B$ . A piston at one end suddenly pushes inward, generating a pressure pulse that propagates along the tube. Assuming the tube expands only radially (longitudinal strain is negligible), what is the speed $v$ of the pressure wavefront propagating through the fluid?

Options

A.
$\left[ \rho \left( \frac{1}{B} + \frac{2R}{hY} \right) \right]^{-1/2}$
B.
$\sqrt{\frac{B}{\rho}}$
C.
$\left[ \rho \left( \frac{1}{B} + \frac{R}{hY} \right) \right]^{-1/2}$
D.
$\left[ \rho \left( \frac{1}{B} + \frac{R}{2hY} \right) \right]^{-1/2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Pascal’s Law Limitations|Stress vs. Strain Definition|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

Students often incorrectly assume that Pascal's Law implies instantaneous and perfect pressure transmission in a fluid, leading them to use the standard sound speed formula $v = \sqrt{B/\rho}$ which assumes rigid boundaries.

This question tests the transition from simple hydrostatics to Fluid-Structure Interaction (FSI).

By introducing a deformable elastic boundary, the pressure becomes a function of the boundary strain.

The trap lies in ignoring the work done against the deforming elastic boundary, which increases the effective compressibility of the system.

Recognizing this requires coupling the fluid's bulk modulus with the tube's hoop stress and Young's modulus to find the effective wave speed (Moens-Korteweg equation).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Pascal’s Law Limitations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free