A spring has length'l' and spring constant 'k'. It is cut into two pieces of lengths `l_(1) and l_(2)` such that `l_(1)=nl_(2)`. The force constant of the spring of length `l_(1)` is

Options

A. k(l+n)
B. k
C. `(k)/((n+1))`
D. `(k(n+l))/(n)`

Explanation

To solve the problem, we need to find the spring constant of the piece of spring with length \( l_1 \) after the original spring of length \( l \) and spring constant \( k \) is cut into two pieces where \( l_1 = n l_2 \).

### Step-by-Step Solution:

1. **Define the Lengths**:
- Let the total length of the spring be \( l \).
- The spring is cut into two parts: \( l_1 \) and \( l_2 \).
- According to the problem, \( l_1 = n l_2 \).

2. **Express Total Length**:
- The total length of the spring can be expressed as:
\[ l = l_1 + l_2 \]
- Substituting \( l_1 \) into the equation gives:
\[ l = n l_2 + l_2 = (n + 1) l_2 \]

3. **Solve for \( l_2 \)**:
- Rearranging the equation to find \( l_2 \):
\[ l_2 = \frac{l}{n + 1} \]

4. **Find \( l_1 \)**:
- Now, substituting \( l_2 \) back into the equation for \( l_1 \):
\[ l_1 = n l_2 = n \left(\frac{l}{n + 1}\right) = \frac{n l}{n + 1} \]

5. **Relate Spring Constant to Length**:
- The spring constant \( k \) of the original spring is related to its length by:
\[ k \propto \frac{1}{l} \]
- For the new spring of length \( l_1 \), the spring constant \( k_1 \) can be expressed as:
\[ k_1 = k \frac{l}{l_1} \]

6. **Substituting \( l_1 \)**:
- Substitute \( l_1 \) into the equation for \( k_1 \):
\[ k_1 = k \frac{l}{\frac{n l}{n + 1}} = k \cdot \frac{(n + 1)}{n} \]

7. **Final Expression for \( k_1 \)**:
- Thus, the spring constant \( k_1 \) of the spring of length \( l_1 \) is given by:
\[ k_1 = k \cdot \frac{(n + 1)}{n} \]

### Conclusion:
The force constant of the spring of length \( l_1 \) is:
\[ k_1 = k \cdot \frac{(n + 1)}{n} \]