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Question #42218

Difficulty: 8/10 | 🟢 Free

The Question

A uniform heavy elastic wire of mass $M$ , natural length $L$ , cross-sectional area $A$ , and Young's modulus $Y$ is suspended vertically from a rigid ceiling. A block of mass $m$ is attached to its lower end. The block is pulled down slightly and released, causing longitudinal oscillations. Assuming the dynamic displacement profile along the wire is linear, what is the angular frequency $\omega$ of the oscillations?

Options

A.
$\sqrt{\frac{YA}{L\left(m + \frac{M}{3}\right)}}$
B.
$\sqrt{\frac{YA}{Lm}}$
C.
$\sqrt{\frac{YA}{L(m + M)}}$
D.
$\sqrt{\frac{YA}{L\left(m + \frac{M}{2}\right)}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Wire Elongation under Self-Weight|Torsional Oscillations|Effective Spring Constant

Why This Question Is Tricky

Common Mistakes

This question tests the concept of mass-inertia coupling in distributed systems using the energy method (Rayleigh's principle).

The lethal trap is treating the heavy wire either as a massless spring (ignoring $M$ ) or assuming its entire mass $M$ (or $M/2$ based on the center of mass) contributes directly to the inertia.

Students must recognize that the linear velocity gradient along the wire necessitates integrating the kinetic energy, which yields an effective inertial mass of $M/3$ for the wire.

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