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Question #43857

Difficulty: 8/10 | 🟢 Free

The Question

A metallic rod of length $L$ , uniform density $\rho$ , Young's modulus $Y$ , and coefficient of linear thermal expansion $\alpha$ is rigidly clamped between two immovable walls at $x=0$ and $x=L$ . The cross-sectional area of the rod varies linearly from $A_0$ at $x=0$ to $2A_0$ at $x=L$ . Initially, the rod is in an unstrained state at temperature $\theta_0$ . The temperature of the rod is then uniformly decreased by $\Delta\theta$ . A small transverse wave pulse is generated at the end $x=0$ . Assuming the dimensional changes due to thermal contraction are negligible for calculating the wave speed and density, what is the time $t$ taken by the pulse to reach the other end $x=L$ ?

Options

A.
$\frac{2L}{3} (2\sqrt{2} - 1) \sqrt{\frac{\rho \ln(2)}{Y \alpha \Delta\theta}}$
B.
$\frac{2L}{3} (2\sqrt{2} - 1) \sqrt{\frac{\rho}{Y \alpha \Delta\theta}}$
C.
$L \sqrt{\frac{\rho}{Y \alpha \Delta\theta}}$
D.
$L \sqrt{\frac{\rho \ln(2)}{Y \alpha \Delta\theta}}$

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

String Tension and Wave Speed|Thermal Stress in Constrained Rods|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple thermal stress with wave mechanics in a non-uniform medium.

The primary trap lies in the assumption that thermal tension $T$ can be calculated using a local or average cross-sectional area (e.g., $T = Y A_0 \alpha \Delta\theta$ ).

Students often misapply Hooke's Law by failing to recognize that in a statically determinate constrained system with varying cross-section, the tension is uniform, but the stress varies.

Consequently, the total zero-displacement boundary condition must be enforced via an integral of the local strain $\epsilon(x) = T/(A(x)Y) - \alpha \Delta\theta$ .

Furthermore, students must correctly identify that the linear mass density $\mu(x)$ also varies with position, requiring a second integration to find the wave transit time.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	String Tension and Wave Speed
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free