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Question #84803

Difficulty: 8/10 | 🟢 Free

The Question

A uniform heavy wire of mass $M$ , length $L$ , cross-sectional area $A$ , and Young's modulus $Y$ hangs vertically from a rigid support under its own weight. Let $U_{\text{true}}$ be the correct total elastic potential energy stored in the wire. A student INCORRECTLY calculates the stored energy by assuming the elastic energy density is uniform throughout the wire and equal to the local energy density at the wire's mid-point. If this incorrectly calculated total energy is $U_{\text{wrong}}$ , what is the value of the difference $(U_{\text{true}} - U_{\text{wrong}})$ ?

Options

A.
$\frac{M^2 g^2 L}{24 Y A}$
B.
$\frac{M^2 g^2 L}{12 Y A}$
C.
$\frac{M^2 g^2 L}{8 Y A}$
D.
$\frac{M^2 g^2 L}{6 Y A}$

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

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Concept Bridge

Cross-chapter synthesis (e.g., using Logarithms in Electrochemistry).

Difficulty Boost : +5

Common Mistakes

This question targets the misconception of treating distributed parameters as lumped or average values in non-linear relationships.

Students often incorrectly apply the global elastic potential energy formula using an average tension or the tension at the center of mass.

Because elastic energy density depends on the square of the local stress ( $\sigma^2$ ), the average of the energy density is not equal to the energy density of the average stress ( $\langle x^2 \rangle \neq \langle x \rangle^2$ ).

The problem forces students to rigorously integrate the local energy density $dU = \frac{T(x)^2}{2YA} dx$ and explicitly compares it to the trap of using the mid-point stress, highlighting the spatial variance of internal restoring forces.

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