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? what is the ratio `k_(1)//k_(2)`

Options

A. `1/n^(2)`
B. `n^(2)`
C. `1/n`
D. n

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, and then determine the ratio \( \frac{k_1}{k_2} \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a uniform spring of unstretched length \( l \) and spring constant \( k \).
- The spring is cut into two pieces with unstretched lengths \( l_1 \) and \( l_2 \).
- We know that \( l_1 = n l_2 \), where \( n \) is an integer.

2. **Total Length Equation**:
- The total length of the spring can be expressed as:
\[ l = l_1 + l_2 \]
- Substituting \( l_1 = n l_2 \) into the equation gives:
\[ l = n l_2 + l_2 = (n + 1) l_2 \]
- From this, we can express \( l_2 \) in terms of \( l \):
\[ l_2 = \frac{l}{n + 1} \]
- Consequently, we can find \( l_1 \):
\[ l_1 = n l_2 = n \left(\frac{l}{n + 1}\right) = \frac{n l}{n + 1} \]

3. **Finding the Spring Constants**:
- The spring constant \( k \) is inversely proportional to the length of the spring. Therefore:
\[ k_1 \propto \frac{1}{l_1} \quad \text{and} \quad k_2 \propto \frac{1}{l_2} \]
- We can express \( k_1 \) and \( k_2 \) in terms of \( k \):
\[ \frac{k_1}{k} = \frac{l}{l_1} \quad \text{and} \quad \frac{k_2}{k} = \frac{l}{l_2} \]

4. **Calculating \( k_1 \)**:
- Substitute \( l_1 \):
\[ \frac{k_1}{k} = \frac{l}{\frac{n l}{n + 1}} = \frac{(n + 1)}{n} \]
- Therefore:
\[ k_1 = k \cdot \frac{(n + 1)}{n} \]

5. **Calculating \( k_2 \)**:
- Substitute \( l_2 \):
\[ \frac{k_2}{k} = \frac{l}{\frac{l}{n + 1}} = n + 1 \]
- Therefore:
\[ k_2 = k(n + 1) \]

6. **Finding the Ratio \( \frac{k_1}{k_2} \)**:
- Now we can find the ratio:
\[ \frac{k_1}{k_2} = \frac{k \cdot \frac{(n + 1)}{n}}{k(n + 1)} = \frac{1}{n} \]

### Final Answers:
- The force constants are:
\[ k_1 = k \cdot \frac{(n + 1)}{n} \]
\[ k_2 = k(n + 1) \]
- The ratio of the spring constants is:
\[ \frac{k_1}{k_2} = \frac{1}{n} \]