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Question #43542

Difficulty: 8/10 | 🟢 Free

The Question

A straight rod of length $L$ is rigidly fixed between two unyielding parallel walls at $x = 0$ and $x = L$ . The cross-sectional area of the rod varies linearly as $A(x) = A_0 \left(1 + \frac{x}{L}\right)$ . The rod is subjected to a non-uniform temperature increase given by $\Delta T(x) = T_0 \left(1 + \frac{x}{L}\right)$ , where $T_0$ is a positive constant. If the Young's modulus of the material is $Y$ and its coefficient of linear thermal expansion is $\alpha$ , what is the magnitude of the compressive force $F$ exerted by the walls on the rod? Assume the material remains in the elastic limit and $\alpha$ is independent of temperature.

Options

A.
$\frac{3 Y A_0 \alpha T_0}{2 \ln 2}$
B.
$\frac{9}{4} Y A_0 \alpha T_0$
C.
$\frac{3 Y A_0 \alpha T_0}{2}$
D.
$\frac{Y A_0 \alpha T_0}{\ln 2}$

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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

Thermal Stress in Constrained Rods|Stress vs. Strain Definition|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply Hooke's Law and thermal expansion principles locally rather than globally.

A common misconception is to apply the standard formula $F = Y A \alpha \Delta T$ using average values for area and temperature, or to assume that the local thermal stress $\sigma(x) = Y \alpha \Delta T(x)$ directly dictates the force $F(x) = \sigma(x) A(x)$ .

This incorrectly implies a varying internal force, violating Newton's laws for a static rod with no distributed axial loads.

The correct framework requires recognizing that the internal force $F$ is uniform, and the net displacement constraint ( $\Delta L = 0$ ) must be satisfied by integrating the differential thermal strains and mechanical strains over the entire length of the rod.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Thermal Stress in Constrained Rods
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free