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Question #42372

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The Question

A thin rod is composed of two uniform cylindrical halves of equal mass $m$ and initial length $L_0$ each. The coefficients of linear expansion of the left and right halves are $\alpha_1$ and $\alpha_2$ respectively ( $\alpha_1 > \alpha_2$ ). The rod is placed inside a fixed smooth spherical cavity of radius $R$ ( $R > L_0$ ). Initially, at temperature $T_0$ , the rod rests horizontally in stable equilibrium. If the temperature of the system is increased by a small amount $\Delta T$ , the rod tilts to a new equilibrium position. Assuming $\Delta T$ is small enough that higher-order terms can be neglected, what is the angle $\theta$ the rod makes with the horizontal?

Options

A.
$\frac{L_0(\alpha_1 - \alpha_2)\Delta T}{4 \sqrt{R^2 - L_0^2}}$
B.
$\frac{L_0(\alpha_1 - \alpha_2)\Delta T}{2 \sqrt{R^2 - L_0^2}}$
C.
$\frac{L_0(\alpha_1 + \alpha_2)\Delta T}{4 \sqrt{R^2 - L_0^2}}$
D.
$\frac{L_0(\alpha_1 - \alpha_2)\Delta T}{\sqrt{R^2 - L_0^2}}$

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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

Rigid Body Constraints|Equilibrium Stability|Expansion of Cavities in Solids

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between thermal expansion, rigid body constraints, and equilibrium stability.

The primary trap lies in calculating the shift of the center of mass.

Students frequently assume that the center of mass remains at the physical junction of the two materials, leading to a shift of $\Delta L/2$ and an incorrect factor of 2 in the denominator.

However, because each uniform half expands about its own center, the true center of mass of the system shifts by only $\Delta L/4$ relative to the geometric center.

Furthermore, students must recognize the geometric constraint of the smooth spherical cavity, which forces the normal reactions to intersect at the center of the sphere, thereby dictating that the center of mass must align vertically with the sphere's center to maintain rotational equilibrium.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Rigid Body Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free