Question

The excess pressure inside a spherical soap bubble X, is six times that of another spherical soap bubble Y. The ratio of mass of bubble X to that of bubble Y is

• A. \(\frac{1}{6}\)
• B. \(\frac{1}{36}\)
• C. \(\frac{1}{216}\)
• D. \(\frac{1}{1296}\)

Hint:

Excess pressure in a bubble is inversely proportional to its radius, and the mass of enclosed air is proportional to its volume.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The excess pressure in a soap bubble is \(\Delta P=\frac{4S}{r}\), so \(r\propto\frac{1}{\Delta P}\).

Step 2: Calculation
\(\frac{\Delta P_{X}}{\Delta P_{Y}}=6\Rightarrow\frac{r_{X}}{r_{Y}}=\frac{1}{6}\). The mass of the enclosed air at the same density is proportional to volume, i.e. to \(r^{3}\).

Step 3: Compare the options
\(\frac{m_{X}}{m_{Y}}=\left(\frac{1}{6}\right)^{3}=\frac{1}{216}\).

Final Answer:
The ratio of the masses is \(\frac{1}{216}\). \[ \boxed{\text{(C) }\dfrac{1}{216}} \]