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Question #42219

Difficulty: 8/10 | 🟢 Free

The Question

A uniform heavy wire of length $L$ , cross-sectional area $A$ , density $\rho_w$ , and Young's modulus $Y$ hangs vertically from a rigid support. The wire is completely submerged in a non-homogeneous liquid whose density varies with depth $y$ (measured downwards from the support) as $\rho(y) = \rho_0(1 + \alpha y)$ , where $\rho_0$ and $\alpha$ are positive constants. Assuming the wire remains taut, what is the total steady-state elongation of the wire?

Options

A.
$\frac{g L^2}{2 Y} \left( \rho_w - \rho_0 - \frac{2}{3} \rho_0 \alpha L \right)$
B.
$\frac{g L^2}{2 Y} \left( \rho_w - \rho_0 - \frac{1}{2} \rho_0 \alpha L \right)$
C.
$\frac{g L^2}{2 Y} \left( \rho_w - \rho_0 - \frac{1}{3} \rho_0 \alpha L \right)$
D.
$\frac{g L^2}{2 Y} \left( \rho_w - \rho_0 - \rho_0 \alpha L \right)$

Check Your Answer

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

Wire Elongation under Self-Weight|Archimedes' Buoyancy Dynamics|Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to model local stress in a non-uniform force field.

A common trap is treating buoyancy as a global force acting on the center of mass or using the average liquid density to find an 'effective gravity' (leading to Option B).

However, because tension dictates local strain, the buoyant force must be integrated locally from the bottom up to find the exact tension distribution $T(y)$ before performing a second nested integration to find the total elongation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Wire Elongation under Self-Weight
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free