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Question #43475

Difficulty: 8/10 | 🟢 Free

The Question

A solid truncated cone of length $L$ (in $\text{m}$ ) is made of an isotropic material with Young's modulus $Y$ (in $\text{N/m}^2$ ) and coefficient of volume expansion $\gamma$ (in $\text{K}^{-1}$ ). The radii of its circular ends are $R$ and $2R$ (in $\text{m}$ ). The cone is rigidly fixed between two unyielding walls at an initial temperature $T_0$ . If the temperature of the cone is uniformly increased to $T_0 + \Delta T$ (where $\Delta T$ is in $\text{K}$ ), what is the maximum compressive thermal stress developed in the cone? Assume the material remains in the elastic limit and $\gamma \Delta T \ll 1$ .

Options

A.
$\frac{1}{3} Y \gamma \Delta T$
B.
$\frac{2}{3} Y \gamma \Delta T$
C.
$\frac{3}{4} Y \gamma \Delta T$
D.
$\frac{1}{6} Y \gamma \Delta T$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Principle of Homogeneity Exceptions|Thermal Stress in Constrained Rods|Thermal Expansion of Area and Volume

Why This Question Is Tricky

Common Mistakes

This question tests the misconception that thermal stress is a local property given by the standard formula $\sigma = Y\alpha\Delta T$ , which is only valid for prismatic bars (constant cross-section).

Students often fall into the trap of assuming stress is independent of geometry and directly apply the standard formula, leading to Option A.

The correct physical framework requires recognizing that in a constrained non-uniform rod, it is the internal restoring force $F$ that remains constant across all sections (due to Newton's laws in equilibrium), not the stress.

Students must apply the macroscopic boundary condition (total elongation $\int \epsilon(x) dx = 0$ ) to find this constant force $F$ , and subsequently evaluate the local stress $\sigma(x) = F/A(x)$ at the minimum cross-section to find the maximum stress.

Furthermore, it tests the relation between volume and linear expansion coefficients ( $\alpha = \gamma/3$ ) for isotropic materials.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Principle of Homogeneity Exceptions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free