A string of length L and force constant k is stretched to obtain extension l. It is further stretched to obtain extension `l_(1)`. The work done in second stretching is

Options

A. `(1)/(2)kl_(1)(2l+l_(1))`
B. `(1)/(2)kl_(1)^(2)`
C. `(1)/(2)k(l^(2)+l_(1)^(2))`
D. `(1)/(2)k(l_(1)^(2)-l^(2))`

Explanation

To solve the problem of finding the work done in the second stretching of a string, we can follow these steps:

### Step 1: Understand the Work Done in Stretching a String
The work done (W) in stretching a string to an extension \( x \) is given by the formula:
\[ W = \frac{1}{2} k x^2 \]
where \( k \) is the force constant of the string.

### Step 2: Calculate the Work Done for the First Extension
For the first extension \( l \), the work done \( W_1 \) is:
\[ W_1 = \frac{1}{2} k l^2 \]

### Step 3: Define the Total Extension After Further Stretching
After the first extension \( l \), the string is further stretched to a new extension \( l_1 \). The total extension after the second stretching becomes:
\[ l' = l + l_1 \]

### Step 4: Calculate the Work Done for the Total Extension
The work done \( W_2 \) to stretch the string to the total extension \( l' \) is:
\[ W_2 = \frac{1}{2} k (l + l_1)^2 \]

### Step 5: Find the Work Done in the Second Stretching
The work done in the second stretching (from extension \( l \) to \( l' \)) is the difference between the work done to stretch to \( l' \) and the work done to stretch to \( l \):
\[ W_{\text{second}} = W_2 - W_1 \]
Substituting the expressions for \( W_2 \) and \( W_1 \):
\[ W_{\text{second}} = \frac{1}{2} k (l + l_1)^2 - \frac{1}{2} k l^2 \]

### Step 6: Simplify the Expression
Now, we can simplify the expression:
\[ W_{\text{second}} = \frac{1}{2} k \left[(l + l_1)^2 - l^2\right] \]
Expanding \( (l + l_1)^2 \):
\[ (l + l_1)^2 = l^2 + 2ll_1 + l_1^2 \]
Thus,
\[ W_{\text{second}} = \frac{1}{2} k \left[l^2 + 2ll_1 + l_1^2 - l^2\right] \]
The \( l^2 \) terms cancel out:
\[ W_{\text{second}} = \frac{1}{2} k (2ll_1 + l_1^2) \]

### Final Result
Therefore, the work done in the second stretching is:
\[ W_{\text{second}} = \frac{1}{2} k l_1 (2l + l_1) \]