How much work will be done in increasing the diameter of a soap bubble from 2cm to 5cm? Surface tension solution is `3.0xx10^(-2)N//m.`

Explanation

To solve the problem of how much work will be done in increasing the diameter of a soap bubble from 2 cm to 5 cm, we can follow these steps:

### Step 1: Understand the problem
We need to calculate the work done in increasing the diameter of a soap bubble. A soap bubble has two surfaces (inside and outside), so we need to consider the change in surface area due to the change in diameter.

### Step 2: Convert diameters to radius
The initial diameter of the bubble is 2 cm, and the final diameter is 5 cm. We first convert these diameters to radii:
- Initial radius \( r_1 = \frac{2 \, \text{cm}}{2} = 1 \, \text{cm} = 0.01 \, \text{m} \)
- Final radius \( r_2 = \frac{5 \, \text{cm}}{2} = 2.5 \, \text{cm} = 0.025 \, \text{m} \)

### Step 3: Calculate the change in surface area
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4 \pi r^2 \]
Now, we calculate the initial and final surface areas:
- Initial surface area \( A_1 = 4 \pi (0.01)^2 = 4 \pi (0.0001) = 0.0004 \pi \, \text{m}^2 \)
- Final surface area \( A_2 = 4 \pi (0.025)^2 = 4 \pi (0.000625) = 0.0025 \pi \, \text{m}^2 \)

Now, we find the change in surface area \( \Delta A \):
\[ \Delta A = A_2 - A_1 = 0.0025 \pi - 0.0004 \pi = (0.0025 - 0.0004) \pi = 0.0021 \pi \, \text{m}^2 \]

### Step 4: Calculate the work done
The work done \( W \) in expanding the bubble is given by the formula:
\[ W = \text{Surface Tension} \times \Delta A \]
Given that the surface tension \( \gamma = 3.0 \times 10^{-2} \, \text{N/m} \), we can substitute the values:
\[ W = 3.0 \times 10^{-2} \times (0.0021 \pi) \, \text{J} \]

Calculating this:
\[ W = 3.0 \times 10^{-2} \times 0.0021 \times 3.14 \approx 3.0 \times 10^{-2} \times 0.0066 \approx 1.98 \times 10^{-4} \, \text{J} \]

### Final Result
Thus, the work done in increasing the diameter of the soap bubble from 2 cm to 5 cm is approximately:
\[ W \approx 1.98 \times 10^{-4} \, \text{J} \]