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Question #42883

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical organ pipe of length $L$ and radius $r$ is closed at one end. The pipe is made of a solid material with a coefficient of linear expansion $\alpha$ . The temperature of the pipe is increased by $\Delta T$ . To isolate the effect of the pipe's expansion, assume the air inside the pipe is maintained at a constant temperature so that the speed of sound $v$ remains constant. The end correction for the pipe is given by $e = 0.6r$ . A student calculates the magnitude of the fractional change in the fundamental frequency, $\left| \frac{\Delta f}{f} \right|$ , but incorrectly assumes that the end correction does not change with temperature. What is the absolute difference between the true magnitude of the fractional change and the student's calculated magnitude?

Options

A.
$\frac{L}{L + 0.6r} \alpha \Delta T$
B.
$\alpha \Delta T$
C.
$\frac{0.6r}{L + 0.6r} \alpha \Delta T$
D.
$\frac{0.6r}{L} \alpha \Delta T$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Harmonic Content in Pipes|Expansion of Cavities in Solids|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply thermal expansion to all relevant geometric dimensions of a system.

A common trap in acoustics problems involving thermal expansion is to only consider the linear expansion of the pipe's length ( $L$ ) while ignoring the radial expansion of the cavity ( $r$ ).

Since the end correction ( $e = 0.6r$ ) depends on the radius, it also expands.

By asking for the difference between the true fractional change and the incorrectly calculated one, the problem explicitly isolates this misconception, forcing students to recognize that isotropic thermal expansion scales all linear dimensions (including the radius of a cavity) by the same factor $(1 + \alpha \Delta T)$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Harmonic Content in Pipes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free