Question

Air (density $\rho$) is being blown on a soap film (surface tension $T$) by a pipe of radius $R$ with its opening right next to the film. The film is deformed and a bubble detaches from the film when the shape of the deformed surface is a hemisphere. Given that the dynamic pressure on the film due to the air blown at speed $\upsilon$ is $\frac{1}{2}\rho\upsilon^{2},$ the speed at which thle bubbe formed is

• A. $\sqrt{\frac{T}{\rho R}}$
• B. $\sqrt{\frac{2T}{\rho R}}\:$
• C. $\sqrt{\frac{4T}{\rho R}}$
• D. $\sqrt{\frac{8T}{\rho R}}$

Answer 1

Answer: (D)

Bubble detached from the look at ends of circumference, when force of dynamic pressure exceeds force of surface tension.

$\Rightarrow p_{\text{dynamic}} \times Area \ge$ Force of surface tension
$\Rightarrow \frac{1}{2}\rho\upsilon^{2}\times\pi R^2 \ge 2\left(2\pi RT\right)$
Note that factor $2$ on right hand side appears as there are two sides of film surface.
$\Rightarrow \upsilon\ge\sqrt{\frac{8T}{\rho R}}$
$\Rightarrow \upsilon_{min}=\sqrt{\frac{8T}{\rho R}}$