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 and takes a spherical shape of radius R. When the gas is allowed to flow freely out of it,

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Solution

Q.
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 of it, its radius $r$ changes from R to 0 (zero) in time T. If the speed $ν (r)$ of gas coming out of the balloon depends on $r$ as $r^{a}$ and $T \propto S^{α} A^{β} ρ^{γ} R^{δ}$ then [2025]

1 $a = - \frac{1}{2}, α = - \frac{1}{2}, β = - 1, γ = \frac{1}{2}, δ = \frac{7}{2}$

2 $a = \frac{1}{2}, α = \frac{1}{2}, β = \frac{- 1}{2}, γ = \frac{1}{2}, δ = \frac{7}{2}$

3 $a = \frac{1}{2}, α = \frac{1}{2}, β = - 1, γ = + 1, δ = \frac{3}{2}$

4 $a = - \frac{1}{2}, α = - \frac{1}{2}, β = - 1, γ = - \frac{1}{2}, δ = \frac{5}{2}$

Ans.
(1)

$T \propto S^{α} A^{β} ρ^{γ} R^{δ}$

$[T] =$ ${[M T^{- 2}]}^{α}$ ${[L^{2}]}^{β}$ ${[M L^{- 3}]}^{^{γ}}$ ${[L]}^{δ}$

$α + γ = 0,$ $2 β - 3 γ + δ = 0,$ $- 2 α = 1$

$∴$ $α = - \frac{1}{2}, γ = \frac{1}{2}$

Now, $2 β - \frac{3}{2} + δ = 0$

$\Rightarrow 2 β + δ = \frac{3}{2}$

By checking with options

$β = - 1, δ = \frac{7}{2} \Rightarrow Option (a) holds$

So, $v \propto \sqrt{Δ P} \propto \sqrt{\frac{S}{r}} \propto r^{- \frac{1}{2}}$

So, $a = - \frac{1}{2}$

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