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Question #95891

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical rod of length $L$ and cross-sectional area $A$ is oriented along the body diagonal of a cubical room of side $a$ (so $L = \sqrt{3}a$ ). The rod is fixed between two rigid walls at its ends $x=0$ and $x=L$ . The rod has a uniform volume charge density $\rho_e$ and is placed in a uniform magnetic field $\vec{B} = B_0 \hat{k}$ . The rod is initially at a uniform temperature $T_0$ . The temperature is then increased such that the change in temperature at a distance $x$ from the $x=0$ end is $\Delta T(x) = \Delta T_0 \frac{x}{L}$ . The coefficient of linear thermal expansion varies as $\alpha(x) = \alpha_0 \frac{x}{L}$ . The Young's modulus of the rod varies with temperature as $Y(T) = \frac{Y_0}{1 + \beta(T - T_0)}$ . Which of the following statements regarding the rod in its final equilibrium state is INCORRECT?

Options

A.
The net displacement of the cross-section at $x = L/2$ is zero.
B.
The compressive force developed is $\frac{A Y_0 \alpha_0 \Delta T_0}{3(1 + \frac{1}{2}\beta \Delta T_0)}$ .
C.
The total free thermal expansion of the rod is $\frac{1}{3} \alpha_0 \Delta T_0 L$ .
D.
The magnitude of local mechanical strain at $x = L$ is $\frac{\alpha_0 \Delta T_0 (1 + \beta \Delta T_0)}{3(1 + \frac{1}{2}\beta \Delta T_0)}$ .

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Interatomic Force Curve|Dipole Potential Angular Dependence|Electric Field Symmetry

Why This Question Is Tricky

Examiner Levers

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the advanced application of Hooke's Law and thermal expansion in a system with continuously varying properties, deliberately trapping students who rely on lumped-parameter formulas like $\Delta L = L \alpha \Delta T$ .

The primary trap lies in the assumption of symmetry or zero internal displacement.

Students frequently assume that if a rod is fixed at both ends, its midpoint remains stationary ( $u(L/2) = 0$ ).

However, because the temperature gradient, thermal expansion coefficient, and Young's modulus all vary with position, the local thermal strain and mechanical strain do not cancel out locally, only globally.

The right side of the rod expands significantly more and is mechanically softer, causing the internal cross-sections to shift towards the cooler, stiffer end to maintain equilibrium.

The inclusion of the magnetic field and charge density serves as a conceptual red herring to test confidence in static equilibrium conditions.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Interatomic Force Curve
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free