Question

A balloon is made of a material of surface tension S and has a small outlet. It is filled with air of density \( \rho \). Initially the balloon is a sphere of radius R. When the gas is allowed to flow out slowly at a constant rate, its radius shrinks as \( r(t) \). Assume that the pressure inside the balloon is \( P(r) \) and is more than the outside pressure (\( P_0 \)) by an amount proportional to the surface tension and inversely proportional to the radius. The balloon bursts when its radius reaches \( r_0 \). Then the speed of gas coming out of the balloon at \( r = R \) is :

• A. \( \sqrt{\frac{S}{\rho R}} \)
• B. \( \sqrt{\frac{2S}{\rho R}} \)
• C. \( \sqrt{\frac{4S}{\rho R}} \)
• D. \( \sqrt{\frac{S}{2\rho R}} \)

Hint:

The excess pressure inside a spherical balloon due to surface tension is \( \Delta P = \frac{2S}{r} \). Apply Bernoulli's equation to relate the pressure difference to the speed of the escaping gas. Assume the initial speed of the gas inside the balloon is negligible.