A uniform spring whose unstretched length is l has a force constant k. the spring is cut into two pieces of unstretched lengths `l_(1)` and `l_(2)`, where `l_(1)=nl_(2)` and n is an integer. What are the corresponding force constant `k_(1)` and `k_(2)` in terms of n and k?

Options

A. `k eta and k(eta +1)`
B. `(k(eta+1))/(eta) and k(eta-1)`
C. `(k(eta-1))/(eta) and k(eta+1)`
D. `(k(eta+1))/(eta) and k(eta+1)`

Explanation

To solve the problem, we need to find the force constants \( k_1 \) and \( k_2 \) of the two pieces of the spring after it has been cut. The original spring has a force constant \( k \) and an unstretched length \( L \). The lengths of the two pieces after cutting are given as \( l_1 \) and \( l_2 \), where \( l_1 = n l_2 \).

### Step-by-step Solution:

1. **Define the lengths of the pieces**:
Since we know that \( l_1 + l_2 = L \) and \( l_1 = n l_2 \), we can substitute \( l_1 \) into the equation:
\[ n l_2 + l_2 = L \]
This simplifies to:
\[ (n + 1) l_2 = L \]

2. **Solve for \( l_2 \)**:
Rearranging the equation gives us:
\[ l_2 = \frac{L}{n + 1} \]

3. **Find \( l_1 \)**:
Using the relation \( l_1 = n l_2 \):
\[ l_1 = n \left(\frac{L}{n + 1}\right) = \frac{nL}{n + 1} \]

4. **Relate the force constants to the lengths**:
The force constant \( k \) of a spring is inversely proportional to its length. Therefore, we can express \( k_1 \) and \( k_2 \) in terms of \( k \):
\[ k_1 = \frac{L}{l_1} k \quad \text{and} \quad k_2 = \frac{L}{l_2} k \]

5. **Calculate \( k_1 \)**:
Substituting \( l_1 \) into the equation for \( k_1 \):
\[ k_1 = \frac{L}{\frac{nL}{n + 1}} k = \frac{(n + 1)L}{nL} k = \frac{n + 1}{n} k \]

6. **Calculate \( k_2 \)**:
Substituting \( l_2 \) into the equation for \( k_2 \):
\[ k_2 = \frac{L}{\frac{L}{n + 1}} k = (n + 1) k \]

### Final Results:
- The force constant of the first piece is:
\[ k_1 = \frac{n + 1}{n} k \]
- The force constant of the second piece is:
\[ k_2 = (n + 1) k \]