Question

A soap bubble of surface tension \(0.04\,\text{N/m}\) is blown to a diameter of \(7\,\text{cm}\). If \((15000 - x)\,\mu\text{J}\) of work is done in blowing it further to make its diameter \(14\,\text{cm}\) \((\pi = 22/7)\), then the value of \(x\) is ________.

Hint:

Soap bubbles have two surfaces, hence energy involves a factor of 2.

Answer 1

Correct Answer: 11304

Step 1: Surface energy of a soap bubble.
A soap bubble has two surfaces, so surface energy is \[ E = 2 \times T \times 4\pi r^2 = 8\pi T r^2 \]
Step 2: Radii before and after expansion.
Initial diameter \(= 7\,\text{cm} \Rightarrow r_1 = 3.5\,\text{cm} = 0.035\,\text{m}\)
Final diameter \(= 14\,\text{cm} \Rightarrow r_2 = 7\,\text{cm} = 0.07\,\text{m}\)
Step 3: Work done in expansion.
\[ W = 8\pi T (r_2^2 - r_1^2) \]
Step 4: Substitute values.
\[ W = 8 \times \frac{22}{7} \times 0.04 \left(0.07^2 - 0.035^2\right) \] \[ W = 0.006304\,\text{J} = 6304\,\mu\text{J} \]
Step 5: Compare with given expression.
\[ 15000 - x = 6304 \Rightarrow x = 11304 \]