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Question #42217

Difficulty: 8/10 | 🟢 Free

The Question

A uniform heavy wire of mass $M$ , length $L$ , cross-sectional area $A$ , and Young's modulus $Y$ is suspended vertically from a rigid support. Acceleration due to gravity is $g$ . What is the total elastic potential energy stored in the wire due to its own weight?

Options

A.
$\frac{M^2 g^2 L}{6 A Y}$
B.
$\frac{M^2 g^2 L}{8 A Y}$
C.
$\frac{M^2 g^2 L}{4 A Y}$
D.
$\frac{M^2 g^2 L}{2 A Y}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Wire Elongation under Self-Weight|Energy Density Locality|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize that elastic potential energy in a massive wire is non-uniformly distributed due to a linear gradient in internal tension.

A common trap is treating the wire as a massless spring with an 'average' tension ( $Mg/2$ ) and applying the global formula $U = \frac{1}{2} \frac{T_{avg}^2}{AY} L$ , which incorrectly yields $\frac{M^2 g^2 L}{8 A Y}$ .

Students must realize that energy density depends on the square of the local stress, necessitating integration of the local energy density $dU = \frac{\sigma^2}{2Y} dV$ over the wire's volume.

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