A spring of length `'l'` has spring constant `'k'` is cut into two parts of length `l_(1)` and `l_(2)`. If their respective spring constahnt are `K_(1)` and `k_(2)`, then `(K_(1))/(K_(2))` is:

Options

A. `(l_(2))/(l_(1))`
B. `(2l_(2))/(l_(1))`
C. `(l_(1))/(l_(2))`
D. None

Answer

To solve the problem, we need to find the ratio of the spring constants \( K_1 \) and \( K_2 \) of two parts of a spring that has been cut into lengths \( l_1 \) and \( l_2 \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a spring of total length \( l \) and spring constant \( k \).
- The spring is cut into two parts: one of length \( l_1 \) and the other of length \( l_2 \).
- We need to find the ratio \( \frac{K_1}{K_2} \) where \( K_1 \) is the spring constant of the first part and \( K_2 \) is the spring constant of the second part.

2. **Using the Spring Constant Formula**:
- The spring constant of a spring is inversely proportional to its length. This means:
\[ K \propto \frac{1}{l} \]
- Therefore, if we have two parts of the spring, we can express their spring constants as:
\[ K_1 = \frac{k \cdot l}{l_1} \quad \text{and} \quad K_2 = \frac{k \cdot l}{l_2} \]

3. **Setting Up the Ratio**:
- Now, we can find the ratio of the spring constants \( K_1 \) and \( K_2 \):
\[ \frac{K_1}{K_2} = \frac{\frac{k \cdot l}{l_1}}{\frac{k \cdot l}{l_2}} \]

4. **Simplifying the Ratio**:
- The \( k \) and \( l \) terms cancel out:
\[ \frac{K_1}{K_2} = \frac{l_2}{l_1} \]

5. **Final Answer**:
- Thus, the ratio of the spring constants is:
\[ \frac{K_1}{K_2} = \frac{l_2}{l_1} \]