When the load on a wire is slowly increased from `3kgwt` to `5 kg wt`, the elongation increases from `0.61` to `1.02 mm`. The work done during the extension of wire is

Options

A. `16 xx10^(-3)J`
B. `8xx10^(-2)J`
C. `20xx10^(-2)J`
D. `11 xx 10^(-3)J`

Explanation

To solve the problem of finding the work done during the extension of a wire when the load is increased from 3 kgwt to 5 kgwt, we can follow these steps:

### Step 1: Understand the Given Data
- Initial load (mass) = 3 kg
- Final load (mass) = 5 kg
- Initial elongation (extension) = 0.61 mm = \(0.61 \times 10^{-3}\) m
- Final elongation (extension) = 1.02 mm = \(1.02 \times 10^{-3}\) m

### Step 2: Calculate the Forces
- The force due to the initial load (F1) can be calculated as:
\[ F_1 = m_1 \cdot g = 3 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 29.43 \, \text{N} \]

- The force due to the final load (F2) can be calculated as:
\[ F_2 = m_2 \cdot g = 5 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 49.05 \, \text{N} \]

### Step 3: Calculate the Work Done
The work done (W) during the extension of the wire can be calculated using the formula for work done in stretching a material:
\[ W = \frac{1}{2} \times (F_2 \cdot \Delta x_2 - F_1 \cdot \Delta x_1) \]
Where:
- \(\Delta x_2 = 1.02 \times 10^{-3} \, \text{m}\)
- \(\Delta x_1 = 0.61 \times 10^{-3} \, \text{m}\)

Substituting the values:
\[ W = \frac{1}{2} \times (49.05 \cdot 1.02 \times 10^{-3} - 29.43 \cdot 0.61 \times 10^{-3}) \]

Calculating the individual terms:
- \(49.05 \cdot 1.02 \times 10^{-3} = 0.0500001 \, \text{J}\)
- \(29.43 \cdot 0.61 \times 10^{-3} = 0.0179343 \, \text{J}\)

Now calculate the difference:
\[ W = \frac{1}{2} \times (0.0500001 - 0.0179343) = \frac{1}{2} \times 0.0320658 \approx 0.0160329 \, \text{J} \]

### Step 4: Convert to Standard Form
To express this in standard form:
\[ W \approx 16.0329 \times 10^{-3} \, \text{J} \approx 16 \times 10^{-3} \, \text{J} \]

### Final Answer
The work done during the extension of the wire is approximately:
\[ W \approx 16 \times 10^{-3} \, \text{J} \]