A spring whose instretched 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. The ratio `k_(1)//k_(2)` of the corresponding force constant, `k_(1)` and `k_(2)` will be :

Options

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

Answer

To solve the problem, we need to find the ratio of the force constants \( k_1 \) and \( k_2 \) of two pieces of a spring that has been cut into two parts. The unstretched lengths of these parts are given as \( l_1 \) and \( l_2 \), where \( l_1 = n l_2 \).

### Step-by-Step Solution:

1. **Understanding the Spring Constant**:
The spring constant \( k \) is related to the unstretched length of the spring. For a spring of unstretched length \( l \) and force constant \( k \), the relationship is given by:
\[ k = \frac{c}{l} \]
where \( c \) is a constant.

2. **Finding the Spring Constants for Each Piece**:
For the first piece of the spring with unstretched length \( l_1 \):
\[ k_1 = \frac{c}{l_1} \]
For the second piece of the spring with unstretched length \( l_2 \):
\[ k_2 = \frac{c}{l_2} \]

3. **Setting Up the Ratio of the Spring Constants**:
We want to find the ratio \( \frac{k_1}{k_2} \):
\[ \frac{k_1}{k_2} = \frac{\frac{c}{l_1}}{\frac{c}{l_2}} = \frac{l_2}{l_1} \]
Here, the constant \( c \) cancels out.

4. **Substituting the Relationship Between \( l_1 \) and \( l_2 \)**:
We know from the problem statement that \( l_1 = n l_2 \). Substituting this into the ratio gives:
\[ \frac{k_1}{k_2} = \frac{l_2}{n l_2} \]

5. **Simplifying the Expression**:
The \( l_2 \) terms cancel out:
\[ \frac{k_1}{k_2} = \frac{1}{n} \]

6. **Conclusion**:
Therefore, the ratio of the spring constants \( k_1 \) and \( k_2 \) is:
\[ \frac{k_1}{k_2} = \frac{1}{n} \]

### Final Answer:
The ratio \( \frac{k_1}{k_2} \) is \( \frac{1}{n} \).

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