Fluid Mechanics

Step 1: Find Extra Pressure Inside a BubbleThe extra (excess) pressure inside a soap bubble is $p = \; \frac{4 T}{r}$, where $T$ is surface tension and $r$ is the radius.So for bubble $A$ : $p_A = \; \frac{4 T}{r_1}$And for bubble $B$ : $p_B = \; \frac{4 T}{r_2}$Step 2: Use the Gas Law for BubblesThe usual gas equation is $p V = n R T$. Here, $p$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the gas constant, and $T$ is temperature.Since the temperature is constant, $p V \propto n$. The number of moles $n$ is related to the mass $m$ of the gas. So, $p V \propto m$.Step 3: Work Out the Mass for Each BubbleFor bubble $A$ :The volume of a sphere is $V_A = \; \frac{4}{3} \pi r_1^3$.So $p_A V_A \propto m_A$ becomes:$\; \frac{4 T}{r_1} \times \; \frac{4}{3} \pi r_1^3 \propto m_A$This simplifies to $m_A \propto r_1^2$.For bubble $B$ :Similarly, $V_B = \; \frac{4}{3} \pi r_2^3$.So $\; \frac{4 T}{r_2} \times \; \frac{4}{3} \pi r_2^3 \propto m_B$This gives us $m_B \propto r_2^2$.Step 4: Find the Ratio of the MassesThe ratio is $\; \frac{m_A}{m_B} = \; \frac{r_1^2}{r_2^2}$.