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Question #42641

Difficulty: 8/10 | 🟢 Free

The Question

A long string under constant tension $T$ has a variable mass per unit length $\mu(x) = \mu_0 (1 + \alpha x)^4$ for $x \ge 0$ . A single positive transverse wave pulse (displacement $y \ge 0$ ) is generated at $x=0$ with an initial amplitude $A_0$ and propagates in the $+x$ direction. The string rests on a surface that exerts a constant transverse kinetic friction force of magnitude $f_k$ per unit length, which always opposes the transverse velocity of the string particles. Assuming $f_k$ is sufficiently small such that the spatial variation of the pulse amplitude is predominantly governed by the non-uniform density (WKB approximation), what is the magnitude of the total work done by friction on the string segment from $x=0$ to $x=L$ during the complete passage of the pulse?

Options

A.
$\frac{2 f_k A_0}{\alpha} \ln(1 + \alpha L)$
B.
$\frac{f_k A_0}{\alpha} \ln(1 + \alpha L)$
C.
$\frac{2 f_k A_0}{\alpha} \left( 1 - \frac{1}{1+\alpha L} \right)$
D.
$2 f_k A_0 L$

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

Particle Velocity vs. Wave Velocity|Work-Energy Theorem Validity|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of the Work-Energy theorem with wave propagation in non-uniform media.

A common trap is assuming the wave amplitude remains constant ( $A=A_0$ ), leading to the incorrect work calculation $2 f_k A_0 L$ .

Another frequent mistake is incorrectly applying energy conservation by assuming energy density $E \propto \mu A^2$ is constant, which yields an incorrect amplitude scaling of $A \propto \mu^{-1/2}$ .

Students must recognize that the local impedance $\sqrt{T\mu(x)}$ dictates the power flux, leading to the WKB scaling $A(x) \propto \mu(x)^{-1/4}$ .

Furthermore, they must correctly map the macroscopic wave amplitude to the microscopic transverse distance ( $2A$ ) traveled by the particles to compute the work done by the non-conservative friction force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Particle Velocity vs. Wave Velocity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free