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Question #83217

Difficulty: 5/10 | 🟢 Free

The Question

Two long, thin cylindrical rods, A and B, are joined end-to-end at $x=0$ . Both rods have the same uniform cross-sectional area $S$ , identical density $\rho$ , and identical Bulk Modulus $B$ . Rod A has a Poisson's ratio $\sigma_A = 0.25$ . Rod B is made of a nearly incompressible material such that its Poisson's ratio $\sigma_B \to 0.5$ . A longitudinal harmonic displacement wave travels in Rod A towards the junction. Four wave properties are evaluated in the order given below: (I) Acoustic impedance of Rod A (II) Phase difference between reflected and incident displacement waves at $x=0$ (III) Speed of the transmitted longitudinal wave in Rod B (IV) Ratio of reflected to incident average power

One of the theoretically calculated values for these properties is INCORRECT. Identify the INCORRECT value from the options below (the options correspond to the properties in the same order).

Options

A.
$\sqrt{\frac{3}{2}\rho B}$
B.
$0$
C.
$\sqrt{\frac{B}{\rho}}$
D.
$1$

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

Poisson's Ratio Limits|Boundary Condition Reflection|Waves

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Common Mistakes

This question tests the critical distinction between 1D stress (thin rods) and 1D strain (bulk media or fluids) in longitudinal wave propagation.

Students often blindly apply the formula $v = \sqrt{B/\rho}$ for longitudinal waves, failing to realize it only applies to fluids or laterally constrained bulk media.

In a thin rod, the lateral boundaries are free to expand, so the restoring force is governed by Young's Modulus $Y = 3B(1-2\sigma)$ .

As a material approaches the incompressible limit ( $\sigma \to 0.5$ ), it expands laterally with zero volume change, offering no resistance to longitudinal stretching if unconstrained.

Consequently, $Y \to 0$ and the rod wave speed $v \to 0$ .

The trap exploits the misconception that 'incompressible' implies a rigid medium with high wave speed, leading students to incorrectly select the bulk wave speed.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Poisson's Ratio Limits
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free