Question

If the work done in blowing a bubble of volume $V$ is $W$ , then the work done in blowing the bubble of volume $2V$ from the same soap solution will be

• A. $\text{W}$
• B. $\sqrt{2} \text{W}$
• C. $\sqrt[3]{2} \text{W}$
• D. $\sqrt[3]{4} \text{W}$

Answer 1

Answer: (D)

As volume of the bubble
v $= \frac{4}{3} \pi \left(\text{R}\right)^{3} ⇒ \text{R} = \left(\frac{3}{4 \pi }\right)^{1 / 3} \left(\text{V}\right)^{1 / 3} ⇒ \left(\text{R}\right)^{2} = \left(\frac{3}{4 \pi }\right)^{2 / 4} \left(\text{V}\right)^{2 / 3}$
$⇒ \text{R}^{2} \alpha \text{V}^{2 / 3}$
Work done in blowing a soap bubble
$\text{W} = 8 \pi \text{R}^{2} \text{T} ⇒ \text{W} ∝ \text{R}^{2} ∝ \text{V}^{2 / 3}$
$∴ \frac{\left(\text{W}\right)_{2}}{\left(\text{W}\right)_{1}} = \left(\frac{\left(\text{V}\right)_{2}}{\left(\text{V}\right)_{1}}\right)^{2 / 3} = \left(\frac{2 \text{V}}{\text{V}}\right)^{2 / 3} = \left(2\right)^{2 / 3} = \left(4\right)^{1 / 3} ⇒ \left(\text{W}\right)_{2} = \sqrt[3]{4} \text{W}$