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Question #85060

Difficulty: 8/10 | 🟢 Free

The Question

A heavy, thick wire of total mass $M$ , length $L$ , and uniform cross-sectional area $A$ is suspended vertically from a rigid support. The material of the wire has a Young's modulus $Y$ . The linear mass density $\lambda$ of the wire is not uniform; it varies with the downward vertical distance $x$ from the top support according to the relation $\lambda(x) = kx^2$ , where $k$ is a positive constant. If $g$ is the acceleration due to gravity, what is the total elongation of the wire due to its own weight?

Options

A.
$\frac{3MgL}{4AY}$
B.
$\frac{MgL}{4AY}$
C.
$\frac{MgL}{2AY}$
D.
$\frac{MgL}{3AY}$

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|Continuous Charge Distribution|Integration

Why This Question Is Tricky

Examiner Levers

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Boundary Condition

Tests behavior at limits like 0, 1, infinity, or empty sets.

Difficulty Boost : +4

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to apply Hooke's Law to a continuous system with a non-uniform mass distribution.

A common trap is blindly applying the standard formula for a uniform wire ( $\Delta L = MgL/2AY$ ) without recognizing that the variable density shifts the effective center of gravity.

Another frequent mistake is setting up the tension integral incorrectly by integrating from the top ( $0$ to $x$ ) instead of calculating the weight of the segment strictly below the cross-section ( $x$ to $L$ ), which yields the inverted result $MgL/4AY$ .

Students must construct the local stress-strain relationship from first principles and execute the boundary conditions properly.

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	8
Target Solve Time	180 sec
Access	Free