Question

A bubble having surface tension $T$ and radius $R$ is formed on ring of radius $r(r<<R)$. Air of density $\rho$ is blown inside the tube with velocity $v$ such that air molecule collides perpendicularly with the wall of the bubble and stops. The radius at which the bubble separates from the ring is $\frac{ZT}{\rho v^2}$. Find $Z$.

Answer 1

Answer: 4

Excess pressure inside a bubble $=\frac{4T}{R}$
Let area of wall of bubble where air strikes be $A$.

$\therefore$ Force due to excess pressure $=\frac{4TA}{R}$
Now, force due to striking air $=\rho A{v}^2$ For bubble to separate from the ring,
$\rho A{v}^2=\frac{4TA}{R}$
$\Rightarrow R=\frac{4T}{\rho v^2}$
$\Rightarrow Z=4$