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Question #42862

Difficulty: 8/10 | 🟢 Free

The Question

A uniform heavy wire of length $L$ , cross-sectional area $A$ , volume mass density $\rho$ , Young's modulus $Y$ , and coefficient of linear thermal expansion $\alpha$ is clamped vertically between two perfectly rigid horizontal supports. Initially, at a temperature $\theta_0$ , the wire is straight and taut such that the tension at its lowest point is exactly zero. The temperature of the wire is then uniformly decreased by $\Delta \theta$ . Assuming the cross-sectional area remains approximately constant and $g$ is the acceleration due to gravity, which of the following expressions gives the fundamental frequency of transverse vibrations of the wire at the new temperature?

Options

A.
$\frac{1}{4L} \left( \sqrt{\frac{Y \alpha \Delta \theta}{\rho} + g L} + \sqrt{\frac{Y \alpha \Delta \theta}{\rho}} \right)$
B.
$\frac{1}{2L} \sqrt{\frac{Y \alpha \Delta \theta}{\rho}}$
C.
$\frac{1}{2L} \sqrt{\frac{Y \alpha \Delta \theta}{\rho} + \frac{g L}{2}}$
D.
$\frac{1}{4L} \left( \sqrt{\frac{Y \alpha \Delta \theta}{\rho} + g L} - \sqrt{\frac{Y \alpha \Delta \theta}{\rho}} \right)$

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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

Fundamental Frequency in Strings|Stress vs. Strain Definition|Thermal Properties of Matter

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of thermal properties of matter and wave mechanics in a non-uniform medium.

A common trap is assuming the tension is uniform and simply using $f = \frac{1}{2L}\sqrt{T/\mu}$ (Option B), ignoring the wire's own weight.

Another trap is incorrectly using the average tension to find the wave speed (Option C), which violates the exact integral time-of-flight required for non-uniform wave speeds.

Students must recognize that clamped boundaries convert thermal contraction entirely into a uniform additive thermal stress, which superimposes on the gravity-induced variable tension, necessitating an integral approach to find the true transit time and resonant frequency.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Fundamental Frequency in Strings
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free