A spring, which is initially in its unstretched condition, is first stretched by a length x and then again by a further length x. The work done in the first case is `W_(1)` and in the second case is `W_(2)`.

Options

A. `W_1=4W`
B. `W_1=3W`
C. `W_1=W`
D. `W_1=2W`

Answer

To solve the problem, we need to calculate the work done on a spring when it is stretched in two different scenarios.

### Step-by-Step Solution:

1. **Understanding the Work Done on a Spring**:
The work done on a spring when it is stretched is given by the formula:
\[ W = \frac{1}{2} k x^2 \]
where \( W \) is the work done, \( k \) is the spring constant, and \( x \) is the extension of the spring.

2. **Calculating Work Done in the First Case**:
In the first case, the spring is stretched by a length \( x \):
\[ W_1 = \frac{1}{2} k x^2 \]
This is our equation (1).

3. **Calculating Work Done in the Second Case**:
In the second case, the spring is first stretched by \( x \) and then further stretched by another \( x \), making the total stretch \( 2x \). The work done in this case can be calculated as follows:
\[ W_2 = \frac{1}{2} k (2x)^2 \]
Simplifying this gives:
\[ W_2 = \frac{1}{2} k (4x^2) = 2k x^2 \]

4. **Finding the Work Done in the Second Stretch**:
However, we need to find the work done during the second stretch (from \( x \) to \( 2x \)). This can be calculated as:
\[ W_{\text{second stretch}} = W_2 - W_1 \]
Substituting the values we found:
\[ W_{\text{second stretch}} = 2k x^2 - \frac{1}{2} k x^2 = \frac{4k x^2}{2} - \frac{1}{2} k x^2 = \frac{3}{2} k x^2 \]

5. **Relating the Two Works**:
Now we can relate \( W_2 \) to \( W_1 \):
\[ W_2 = 2k x^2 = 4 \cdot \frac{1}{2} k x^2 = 4W_1 \]
Therefore, we find that:
\[ W_2 = 3W_1 \]

### Conclusion:
The relationship between the work done in the first and second cases is:
\[ W_2 = 3W_1 \]