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Question #42857

Difficulty: 8/10 | 🟢 Free

The Question

A uniform wire of unstretched length $L_0$ , cross-sectional area $A_0$ , mass $m$ , and coefficient of linear expansion $\alpha$ is suspended vertically from a rigid support. A block of mass $M$ ( $M \gg m$ ) is attached to its lower end. A vibrator at the top end drives the wire at a constant frequency $f$ , forming a standing wave. The $n$ -th node from the top is at a distance $x_n$ (Assume the top end acts as a node). If the temperature of the wire is increased by $\Delta T$ , the fractional change in the position of the $n$ -th node, $\frac{\Delta x_n}{x_n}$ , is given by: (Assume $\alpha \Delta T \ll 1$ , initial stress $\sigma = \frac{Mg}{A_0} \ll Y$ , where $Y$ is the Young's modulus of the wire)

Options

A.
$\frac{1}{2} \alpha \Delta T \left( 1 - \frac{2\sigma}{Y} \right)$
B.
$\frac{1}{2} \alpha \Delta T \left( 1 - \frac{\sigma}{Y} \right)$
C.
$\alpha \Delta T \left( 1 - \frac{2\sigma}{Y} \right)$
D.
$\frac{1}{2} \alpha \Delta T$

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

Standing Wave Nodes/Antinodes|Thermal Expansion of Area and Volume|Calculus (Chain Rule)

Why This Question Is Tricky

Common Mistakes

This question tests the coupled effects of thermal expansion and elasticity on wave propagation.

The primary trap is assuming that thermal expansion only affects the unstretched length, leading to the naive answer $\frac{1}{2}\alpha\Delta T$ .

Students often fail to account for the fact that as the wire expands thermally, its cross-sectional area increases ( $A = A_0(1+2\alpha\Delta T)$ ), which decreases the elastic stretch for a constant tension.

Furthermore, students might incorrectly assume $\mu$ is constant or use unstressed density, missing that $\mu = m/L$ dynamically couples the wave speed to the total stretched length.

Differentiating the full thermo-elastic length equation is required to capture the $(1 - 2\sigma/Y)$ correction factor.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Standing Wave Nodes/Antinodes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free