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Question #42527

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The Question

A uniform rod of length $L$ and cross-sectional area $A$ is rigidly clamped between two fixed walls at $x=0$ and $x=L$ . The Young's modulus of the rod is $Y$ . The thermal conductivity of the rod varies with position as $K(x) = K_0 \left(1 + \frac{x}{L}\right)$ , and the coefficient of linear thermal expansion varies as $\alpha(x) = \frac{\alpha_0}{1 + \frac{x}{L}}$ . Initially, the entire rod is at a uniform temperature $T_i$ . The end at $x=0$ is maintained at $T_i$ , and the end at $x=L$ is maintained at $T_i + T_0$ . Assuming steady-state heat flow is established and the walls do not yield, what is the thermal stress $\sigma$ developed in the rod?

Options

A.
$\frac{1}{2} Y \alpha_0 T_0 \ln 2$
B.
$Y \alpha_0 T_0 \ln 2$
C.
$\frac{1}{2} Y \alpha_0 T_0$
D.
$\frac{Y \alpha_0 T_0}{\ln 2}$

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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

Steady State Temperature Gradient|Stress vs. Strain Definition|Young's Modulus Sensitivity

Why This Question Is Tricky

Common Mistakes

This question tests the principle of mechanical equilibrium in a non-uniform temperature field.

A common lethal mistake is assuming that thermal stress is a local property defined by $\sigma(x) = Y \alpha(x) \Delta T(x)$ , which would imply a varying stress along the rod.

However, for a rod in static equilibrium with no external longitudinal forces along its length, Newton's laws dictate that the internal stress must be strictly uniform ( $d\sigma/dx = 0$ ).

Students must realize that the correct constraint is global: the integral of the local free thermal expansions must be exactly canceled by the total elastic strain induced by a uniform compressive stress.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Steady State Temperature Gradient
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free