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Question #42879

Difficulty: 8/10 | 🟢 Free

The Question

A standing acoustic wave of wavelength $\lambda$ and displacement amplitude $A_0$ is established in a closed cylindrical tube filled with an ideal rigid diatomic gas. The space-and-time averaged mechanical energy density of this standing wave is found to be exactly equal to a small fraction $\alpha$ of the thermal energy density associated with \textit{only the translational} degrees of freedom of the gas molecules. Which of the following expressions correctly gives the displacement amplitude $A_0$ ?

Options

A.
$\frac{\lambda}{\pi} \sqrt{\frac{15 \alpha}{14}}$
B.
$\frac{5\lambda}{\pi} \sqrt{\frac{\alpha}{14}}$
C.
$\frac{\lambda}{2\pi} \sqrt{\frac{15 \alpha}{7}}$
D.
$\frac{5\lambda}{2\pi} \sqrt{\frac{\alpha}{7}}$

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

Energy Distribution in Standing Waves|Equipartition Theorem|Molecular Degrees of Freedom

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge macroscopic wave mechanics with microscopic kinetic theory.

The primary trap lies in the misapplication of the Equipartition Theorem: students often reflexively use the total degrees of freedom ( $f=5$ for a rigid diatomic gas) when calculating thermal energy, ignoring the explicit constraint isolating translational kinetic energy ( $f=3$ ).

A secondary trap involves the mechanical energy density of a standing wave; students frequently confuse it with a traveling wave ( $\frac{1}{2}\rho\omega^2A^2$ ) or fail to properly time-and-space average the energy, missing the crucial $\frac{1}{4}$ factor.

Finally, it requires correctly linking the macroscopic wave speed to the thermodynamic properties ( $\gamma P_0/\rho_0$ ) to eliminate the unknown pressure and density.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Energy Distribution in Standing Waves
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free