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Question #42201

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical metal rod of Young's modulus $E$ , Poisson's ratio $\nu$ , and coefficient of linear thermal expansion $\alpha$ is constrained longitudinally between two rigid, immovable walls. Laterally, it is enclosed by a non-rigid sleeve that exerts a radial pressure $p = \beta E \epsilon_r$ , where $\epsilon_r$ is the lateral strain and $\beta$ is a positive dimensionless constant. If the rod is heated by a temperature $\Delta T$ , what is the fractional change in its density, $\frac{\Delta \rho}{\rho_0}$ ? (Assume small strains and $\rho_0$ is the initial density).

Options

A.
$\frac{-2 \alpha \Delta T (1+\nu)}{1 + \beta(1+\nu)(1-2\nu)}$
B.
$-3 \alpha \Delta T$
C.
$\frac{-2 \alpha \Delta T (1-\nu)}{1 + \beta(1-2\nu)}$
D.
$\frac{-3 \alpha \Delta T (1-2\nu)}{1 + \beta(1+\nu)}$

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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

Poisson's Ratio Limits|Thermal Stress Calculation|Variation of Density with Temperature

Why This Question Is Tricky

Common Mistakes

This question tests the coupling between constrained thermal expansion, 3D Hooke's Law, and density variation.

A common trap is assuming the fractional density change is simply $-3\alpha \Delta T$ (free expansion) or ignoring the Poisson contribution of the induced longitudinal stress on the lateral strain.

Students often fail to realize that the rigid longitudinal constraint forces a compressive stress $\sigma_z$ , which in turn causes an additional lateral expansion via Poisson's ratio.

Furthermore, it highlights the 'Poisson Limit': as $\nu \to 0.5$ (incompressible limit), the mechanical stress terms in the volume change vanish, and the density change reverts to the free expansion value $-3\alpha \Delta T$ , regardless of the lateral stiffness $\beta$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Poisson's Ratio Limits
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free