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Q.
If the work done in blowing a soap bubble of volume $V$ is $W$ , then the work done in blowing a soap bubble of volume $2V$ under the same conditions will be
NTA AbhyasNTA Abhyas 2020
Solution:
The work done will be against surface tension and we know that surface tension is related to work as $W=T\times \Delta A$
Since, the surface area of sphere is $4\pi R^{2}$ and there are two free surfaces, we have
$W=T\times 8\pi R^{2}$ ... $\left(\right.i\left.\right)$
and volume of sphere $=\frac{4}{3}\pi R^{3}$
i.e. $V=\frac{4}{3}\pi R^{3}$
$\Rightarrow R=\left(\frac{3 V}{4 \pi }\right)^{1 / 3}$ ... $\left(\right.ii\left.\right)$
From Eqs. $\left(\right.i\left.\right)$ and $\left(\right.ii\left.\right)$ , we get
$W=T\times 8\pi \times \left(\frac{3 V}{4 \pi }\right)^{2 / 3}$
$⇒ \, \, W \propto V^{2 / 3}$
$\therefore \, W_{1} \propto V_{1}^{2 / 3}$
and, $W_{2} \propto V_{2}^{2 / 3}$
$\frac{W_{2}}{W_{1}}=\left(\frac{2 V_{1}}{V_{1}}\right)^{2 / 3}$
$\Rightarrow W_{2}=2^{2 / 3}W_{1}=4^{1 / 3}W$