Question

If the work done in blowing a soap bubble of volume $V$ is $W$ , then the work done in blowing is soap bubble of volume $2V$ is:
(A) $4W$
(B) $8W$
(C) ${2^{1/3}}W$
(D) ${4^{1/3}}W$

Answer 1

Hint
The work done is given as the product of surface tension and change in surface area. Among those two, the surface tension is fixed because the liquid is the same. But as the volume doubles, the surface area also changes. So, by getting the relationship between surface area and volume, we can find the new work done.
Formula used: Work done, $W = T \times \Delta A$ where $T$ is the surface tension and $\Delta A$ is the change in surface area.
Surface area of the soap bubble, $A = 4\pi {r^2}$ where $r$ is the radius of the soap bubble.
Volume of the soap bubble, $V = \dfrac{4}{3}\pi {r^3}$ .

Complete step by step answer
Since, it is given that the volume of the bubble doubles, we should be able to find the relation between the surface area and the volume of the soap bubble. Therefore, we can write:
$\Rightarrow A \propto {r^2}$ and
$\Rightarrow V \propto {r^3}$ because the area and volume depends only on the radius of the soap bubble.
Therefore, we can also write that:
$\Rightarrow r \propto {V^{1/3}}$ and since $W \propto \Delta A \propto {r^2}$ , we can elucidate that:
$\Rightarrow W \propto {V^{2/3}}$ .
Let us assume that the work done in blowing a soap bubble of volume $V$ be ${W_1}$ and that of volume $2V$ be ${W_2}$ , then the ratio:
$\Rightarrow \dfrac{{{W_1}}}{{{W_2}}} = \dfrac{{\Delta {A_1}}}{{\Delta {A_2}}}$ where $\Delta {A_1}$ and $\Delta {A_{_2}}$ are the change in area during the blowing of soap bubbles of volumes $V$ and $2V$ respectively. Hence, we can write the ratio again as:
$\Rightarrow \dfrac{{{W_1}}}{{{W_2}}} = {\left( {\dfrac{V}{{2V}}} \right)^{2/3}} = \dfrac{1}{{{2^{2/3}}}}$
Therefore, the work done to blow the soap bubble of $2V$ volume is:
$\Rightarrow {W_2} = {2^{2/3}}{W_1}$
Hence, the correct answer is option (C).

Note
In this case, only the radius of the soap bubble is dependent on the work done. In a general case, the work done to blow the soap bubble depends also on the surface tension. But here, since the liquid is the same, we have to consider only the relationship between radius and the volume.