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Question #42133

Difficulty: 8/10 | 🟢 Free

The Question

An unconstrained isotropic solid block contains a spherical cavity of radius $R$ centered at the origin $(0,0,0)$ . The block is subjected to a steady-state temperature distribution $T(x,y,z) = T_0 + \beta x$ , where $T_0$ and $\beta$ are positive constants. The rigid body motion of the block is constrained such that the theoretical displacement and rotation at the origin are zero. If $\alpha$ is the coefficient of linear thermal expansion of the material, what is the displacement vector of the point on the cavity boundary originally located at $(0, R, 0)$ ? (Assume $\alpha \beta R \ll 1$ ).

Options

A.
$-\frac{1}{2} \alpha \beta R^2 \hat{i}$
B.
$\frac{1}{2} \alpha \beta R^2 \hat{i}$
C.
$\alpha \beta R^2 \hat{j}$
D.
$\vec{0}$

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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

Expansion of Cavities in Solids|Superposition Principle|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of thermal gradient-induced anisotropy and the superposition of displacement fields.

The lethal trap is the assumption that local thermal expansion depends only on the local temperature change.

Since the temperature at $(0, R, 0)$ is $T_0$ (meaning $\Delta T = 0$ ), most students will incorrectly conclude the point does not move (Option D), or they will assume the cavity expands radially (Option C).

However, thermal expansion is a cumulative, tensor-based deformation.

The expansion of the hotter region ( $x>0$ ) and contraction of the colder region ( $x<0$ ) forces the material at $x=0$ to bow towards the colder side to maintain geometric compatibility (zero shear stress).

This causes the points on the y-axis to displace in the negative x-direction, demonstrating that the cavity undergoes non-conformal distortion rather than simple scalar scaling.

Quick Facts About This Question

Question Information

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