If a spring of stiffness `'k'` is cut into two parts `'A'` and `'B'` of length `l_(A):l_(B)=2:3`, then the stiffness of spring `'A'` is given by

Options

A. `5/2k`
B. `3k/5`
C. `2k/5`
D. k

Explanation

To find the stiffness of spring A after cutting a spring of stiffness \( k \) into two parts A and B with a length ratio of \( l_A : l_B = 2 : 3 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Length Ratio**:
Given the ratio \( l_A : l_B = 2 : 3 \), we can express the lengths of parts A and B in terms of a common length. Let the total length of the spring be \( L \). Then:
\[ l_A = \frac{2}{5}L \quad \text{and} \quad l_B = \frac{3}{5}L \]

2. **Relationship Between Stiffness and Length**:
The stiffness \( k \) of a spring is inversely proportional to its length. This means:
\[ k \propto \frac{1}{l} \]
Therefore, if we denote the stiffness of spring A as \( k_A \) and the stiffness of spring B as \( k_B \), we can write:
\[ k_A \cdot l_A = k \cdot L \]

3. **Express Stiffness of Spring A**:
From the relationship above, we can rearrange to find \( k_A \):
\[ k_A = \frac{k \cdot L}{l_A} \]

4. **Substituting the Length of Spring A**:
Now substitute \( l_A = \frac{2}{5}L \) into the equation:
\[ k_A = \frac{k \cdot L}{\frac{2}{5}L} \]

5. **Simplifying the Expression**:
The \( L \) cancels out:
\[ k_A = \frac{k}{\frac{2}{5}} = k \cdot \frac{5}{2} \]

6. **Final Result**:
Thus, the stiffness of spring A is:
\[ k_A = \frac{5}{2}k \]

### Conclusion:
The stiffness of spring A after cutting the original spring is \( \frac{5}{2}k \).