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Topic: Surface energy, surface tension, angle of contact, excess pressure, drops, bubbles, capillary rise· Properties of Solids and Liquids

A balloon is made of a material of surface tension S and its inflation outlet (from where gas is filled in it) has small area A. It is filled with a gas of density ρ and takes a spherical shape of radius R. When the gas is allowed to flow freely out…

A balloon is made of a material of surface tension \(S\) and its inflation outlet (from where gas is filled in it) has small area \(A\). It is filled with a gas of density \(\rho\) and takes a spherical shape of radius \(R\). When the gas is allowed to flow freely out of it, its radius \(r\) changes from \(R\) to 0 (zero) in time \(T\). If the speed \(v(r)\) of gas coming out of the balloon depends on \(r\) as \(r^a\) and \(T \propto S^{\alpha}A^{\beta}\rho^{\gamma}R^{\delta}\) then

1. \(a=\frac12,\alpha=\frac12,\beta=-1,\gamma=+1,\delta=\frac32\)

2. \(a=-\frac12,\alpha=-\frac12,\beta=-1,\gamma=-\frac12,\delta=\frac52\)

3. \(a=-\frac12,\alpha=-\frac12,\beta=-1,\gamma=\frac12,\delta=\frac72\)

4. \(a=\frac12,\alpha=-\frac12,\beta=-1,\gamma=\frac12,\delta=\frac72\)

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Answer: 3. \(a=-\frac12,\alpha=-\frac12,\beta=-1,\gamma=\frac12,\delta=\frac72\)

Key idea: the excess pressure inside the balloon is proportional to \(S/r\) (it is \(4S/r\) in the standard treatment), and it drives the gas out through the outlet (Bernoulli).

Exit speed: \(\tfrac12\rho v^2=\dfrac{4S}{r}\Rightarrow v\propto\sqrt{\dfrac{S}{\rho r}}\propto r^{-1/2}\), so \(a=-\tfrac12\).

Volume outflow rate \(=Av\), so \(-4\pi r^2\dfrac{dr}{dt}=A\sqrt{\dfrac{8S}{\rho}}\,r^{-1/2}\). Rearranging, \(r^{5/2}\,dr\propto-A\sqrt{S/\rho}\,dt\), and integrating \(r\) from \(R\) to 0:

$$T\propto\frac{R^{7/2}}{A\sqrt{S/\rho}}=S^{-1/2}A^{-1}\rho^{1/2}R^{7/2}$$

So \(\alpha=-\tfrac12,\ \beta=-1,\ \gamma=\tfrac12,\ \delta=\tfrac72\), which is option 3. Options 1 and 4 have a positive \(a\) and option 2 has the wrong \(\gamma\) and \(\delta\).

Doubtly AI · 30 Oct 2026 AI-generated · not yet verified Report
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