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Question #80139

Difficulty: 8/10 | 🟢 Free

The Question

A spherical soap bubble of surface tension $\gamma$ is blown at the end of a short cylindrical nozzle of cross-sectional area $A$ . A pump delivers air such that the radius $R$ of the bubble grows at a constant rate $v_0$ , starting from $R=0$ at $t=0$ . The density of the air is $\rho$ . Assuming the air is incompressible and the nozzle is short enough to neglect the unsteady acceleration term ( $\rho \int \frac{\partial u}{\partial t} dx \approx 0$ ), which of the following expressions represents the instantaneous mechanical power $P(t)$ required from the pump at time $t$ ? (Assume the pressure outside the bubble is atmospheric pressure $P_0$ , and the kinetic energy of the air dissipates entirely upon entering the bubble).

Options

A.
$16\pi \gamma v_0^2 t + \frac{32\pi^3 \rho v_0^9 t^6}{A^2}$
B.
$16\pi \gamma v_0^2 t$
C.
$8\pi \gamma v_0^2 t + \frac{32\pi^3 \rho v_0^9 t^6}{A^2}$
D.
$16\pi \gamma v_0^2 t + \frac{64\pi^3 \rho v_0^9 t^6}{A^2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Excess Pressure|Power-Velocity Relationship|Differential Equations

Why This Question Is Tricky

Examiner Levers

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the coupling between fluid dynamics (Bernoulli's principle) and surface tension (Laplace's Law) in a time-varying flow regime.

A common trap is assuming quasi-static equilibrium, where students equate the pump pressure directly to the excess pressure of the bubble ( $\Delta P = 4\gamma/R$ ), completely ignoring the dynamic pressure ( $\frac{1}{2}\rho u^2$ ) required to accelerate the fluid through the nozzle.

Furthermore, because the expansion rate $v_0$ is constant, the volume flow rate $Q$ and nozzle velocity $u$ are highly non-linear with respect to time ( $u \propto t^2$ ).

By requiring the calculation of instantaneous mechanical power ( $P = \Delta P \cdot Q$ ), the problem forces candidates to synthesize thermodynamic work against surface tension with the kinetic energy flux of the fluid, penalizing those who rely solely on the rate of change of surface energy or misapply the Young-Laplace equation for a single surface.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Excess Pressure (Laplace Law)
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free