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Question #43539

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The Question

A uniform metal rod of length $L$ and square cross-section of side $a$ is fixed to a rigid wall at $x=0$ . The other end passes through a frictional clamp of length $d$ ( $d \ll L$ ). The clamp consists of two rigid, immovable pads that perfectly constrain the rod in the $y$ -direction, while the $z$ -direction is free to expand. Each pad exerts an initial normal force $N_0$ on the rod. The coefficient of static friction between the rod and the pads is $\mu_s$ . The rod has Young's modulus $Y$ , coefficient of linear thermal expansion $\alpha$ , and Poisson's ratio $\nu$ . The entire rod is heated uniformly. Assume the longitudinal stress is uniform throughout the rod before slipping. At what minimum temperature increase $\Delta T$ will the rod transition from static to kinetic friction and slip through the clamp?

Options

A.
$\frac{2 \mu_s N_0}{Y \alpha a [a - 2 \mu_s (1 + \nu) d]}$
B.
$\frac{2 \mu_s N_0}{Y \alpha a^2}$
C.
$\frac{2 \mu_s N_0}{Y \alpha a [a - 2 \mu_s \nu d]}$
D.
$\frac{\mu_s N_0}{Y \alpha a [a - \mu_s (1 + \nu) d]}$

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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

Thermal Expansion of Area and Volume|Thermal Stress in Constrained Rods|Kinetic Friction Transition

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the generalized Hooke's law in 2D with thermal expansion, specifically targeting the interplay between longitudinal thermal stress and the Poisson effect.

A common misconception is to either ignore the lateral thermal expansion entirely (assuming normal force remains constant, leading to Option B) or to incorrectly calculate the lateral stress by forgetting that thermal expansion occurs in all directions (omitting the $\alpha\Delta T$ term in the lateral strain equation, leading to Option C).

Furthermore, students must recognize that the longitudinal thermal stress acts as the driving force, while the lateral thermal stress increases the normal force and thus the static friction threshold.

The transition occurs when the linearly increasing driving force overtakes the linearly increasing friction force.

Quick Facts About This Question

Question Information

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