When the load on a wire increased slowly from `2 kg wt.` to `4 kg wt.`, the elogation increases from `0.6 mm` to `1.00 mm`. How much work is done during the extension of the wire? `[ g = 9.8 m//s^(2)]`

Explanation

To find the work done during the extension of the wire when the load is increased from 2 kg to 4 kg, we can follow these steps:

### Step 1: Understand the problem
We have a wire that is subjected to a load that increases from 2 kg to 4 kg, causing an elongation change from 0.6 mm to 1.00 mm. We need to calculate the work done during this extension.

### Step 2: Convert units
First, we convert the elongation from millimeters to meters:
- Initial elongation \( x_1 = 0.6 \, \text{mm} = 0.6 \times 10^{-3} \, \text{m} \)
- Final elongation \( x_2 = 1.00 \, \text{mm} = 1.0 \times 10^{-3} \, \text{m} \)

### Step 3: Calculate the forces
Using the weight formula \( F = mg \):
- For the initial load (2 kg):
\[ F_1 = 2 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 19.6 \, \text{N} \]
- For the final load (4 kg):
\[ F_2 = 4 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 39.2 \, \text{N} \]

### Step 4: Calculate work done
The work done during the extension of the wire can be calculated using the formula for work done in stretching:
\[ W = \frac{1}{2} F_2 x_2 - \frac{1}{2} F_1 x_1 \]
Substituting the values:
\[ W = \frac{1}{2} (39.2 \, \text{N}) (1.0 \times 10^{-3} \, \text{m}) - \frac{1}{2} (19.6 \, \text{N}) (0.6 \times 10^{-3} \, \text{m}) \]

### Step 5: Simplify the expression
Calculating each term:
- Work done with the final load:
\[ W_2 = \frac{1}{2} \times 39.2 \times 1.0 \times 10^{-3} = 19.6 \times 10^{-3} \, \text{J} \]
- Work done with the initial load:
\[ W_1 = \frac{1}{2} \times 19.6 \times 0.6 \times 10^{-3} = 5.88 \times 10^{-3} \, \text{J} \]

### Step 6: Calculate the total work done
Now, substituting back into the work done equation:
\[ W = 19.6 \times 10^{-3} - 5.88 \times 10^{-3} = 13.72 \times 10^{-3} \, \text{J} \]

### Final Answer
Thus, the work done during the extension of the wire is:
\[ W = 13.72 \, \text{mJ} \, \text{(or } 13.72 \times 10^{-3} \, \text{J)} \]