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Question #42876

Difficulty: 8/10 | 🟢 Free

The Question

A spherical soap bubble of equilibrium radius $R_0$ and surface tension $T$ pulsates radially in an atmosphere of constant ambient pressure $P_0$ , emitting a sound wave. The intensity of the emitted sound wave depends on the effective stiffness $k$ of the bubble. The gas inside undergoes small adiabatic oscillations such that its radius at any instant is $R = R_0 + x$ , where $x \ll R_0$ . The specific heat ratio of the gas is $\gamma$ . Defining the net radial restoring force on the bubble surface as $F = -kx$ , which of the following expressions correctly represents the effective spring constant $k$ ?

Options

A.
$4\pi R_0 \left[ 3\gamma P_0 + \frac{4T}{R_0}(3\gamma - 1) \right]$
B.
$4\pi R_0 \left[ 3\gamma P_0 + \frac{12\gamma T}{R_0} \right]$
C.
$4\pi R_0 \left[ 3\gamma P_0 + \frac{4T}{R_0}(3\gamma + 1) \right]$
D.
$4\pi R_0 \left[ 3\gamma P_0 + \frac{2T}{R_0}(3\gamma - 1) \right]$

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

Wave Intensity and Medium|Excess Pressure (Laplace Law)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to model the dynamic restoring forces on an oscillating boundary.

The primary trap is neglecting the dynamic variation of the Laplace pressure (surface tension induced excess pressure) during the bubble's expansion and contraction.

Students often incorrectly assume the Laplace pressure remains constant at its equilibrium value $4T/R_0$ , leading to the missing $-4T/R_0^2 x$ term in the pressure differential.

This oversight ignores the non-linear spring-like behavior of the surface tension itself, which actively reduces the effective stiffness of the bubble.

Additionally, the options test for sign errors in the Taylor expansion and confusion between a soap bubble (two surfaces) and a liquid drop (one surface).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Wave Intensity and Medium
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free