Question
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 $r^{\alpha}$ and $T \propto S^{\alpha}A^{\beta}\rho^{\gamma}R^{\delta}$ then
Options
Solution
$T \propto S^{\alpha}A^{\beta}\rho^{\gamma}R^{\delta}$
$$\begin{matrix} & M^{0}{\text{ }L}^{0}{\text{ }T}^{1} = K\left( {MT}^{- 2} \right)^{\alpha}\left( L^{2} \right)^{\beta}\left( {ML}^{- 3} \right)^{\gamma}L^{\delta} \\ & M^{0}{\text{ }L}^{0}{\text{ }T}^{1} = K\left\lbrack M^{\alpha + \gamma}L^{2\beta - 3\gamma + \delta}T^{- 2\alpha} \right\rbrack \\ & \ - 2\alpha = 1\ \alpha = - \frac{1}{2} \\ & \alpha + \gamma = 0\ \gamma = \frac{1}{2} \\ & 2\beta - 3\gamma + \delta = 0 \\ & 2\beta - 3\left( \frac{1}{2} \right) + \delta = 0 \end{matrix}$$
By hit and trial (using option (1))
Put $\beta = - 1$
$$2( - 1) - \frac{3}{2} + \delta = 0\ \therefore\delta = \frac{7}{2}$$
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