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Question #43497

Difficulty: 8/10 | 🟢 Free

The Question

A uniform rod of mass $M$ , length $L$ , cross-sectional area $A$ , and Young's modulus $E$ is rigidly fixed between two horizontal supports at $x=0$ (bottom) and $x=L$ (top). The rod is in a uniform downward gravitational field $g$ . The coefficient of linear thermal expansion of the rod varies with height $x$ as $\alpha(x) = \alpha_0 \left(\frac{x}{L}\right)^2$ , where $\alpha_0$ is a positive constant. The temperature of the rod is uniformly increased by $\Delta T$ . Assuming the supports are perfectly rigid and the rod does not buckle, at what distance $x$ from the bottom support is the internal tension in the rod equal to zero? (Given that $2EA \alpha_0 \Delta T < 3Mg$ )

Options

A.
$L \left( \frac{1}{2} + \frac{EA \alpha_0 \Delta T}{3 Mg} \right)$
B.
$L \left( \frac{1}{2} + \frac{EA \alpha_0 \Delta T}{2 Mg} \right)$
C.
$L \left( \frac{1}{2} - \frac{EA \alpha_0 \Delta T}{3 Mg} \right)$
D.
$L \left( \frac{1}{2} + \frac{EA \alpha_0 \Delta T}{Mg} \right)$

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

Tension in Massless vs Massive Strings|Thermal Stress in Constrained Rods|Breaking Stress Concept

Why This Question Is Tricky

Common Mistakes

This question tests the superposition of gravitational and thermal stresses in a statically indeterminate system with a spatially varying property.

A common trap is treating the thermal stress as a uniform scalar or evaluating it locally without integrating the strain over the entire constrained length.

Students often incorrectly assume local strain is zero ( $\epsilon(x) = 0$ ) rather than the global displacement being zero ( $\int \epsilon(x) dx = 0$ ).

Furthermore, using an incorrect average for the non-linear thermal expansion coefficient $\alpha(x)$ leads to incorrect boundary reaction forces, testing the student's ability to set up and solve the correct boundary-value integral.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Tension in Massless vs Massive Strings
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free