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)/(2)k`
B. `(3k)/(5)`
C. `(2k)/(5)`
D. `k`

Explanation

To find the stiffness of spring A after cutting the original spring into two parts A and B in the ratio of their lengths \( l_A : l_B = 2 : 3 \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Length of the Spring**:
Let the total length of the original spring be \( L \).

2. **Determine the Lengths of Springs A and B**:
Given the ratio \( l_A : l_B = 2 : 3 \), we can express the lengths of A and B in terms of \( L \):
\[ l_A = \frac{2}{5}L \quad \text{and} \quad l_B = \frac{3}{5}L \]

3. **Use the Relationship Between Stiffness and Length**:
The product of stiffness \( k \) and length \( L \) is a constant. Therefore, we can write:
\[ k \cdot L = k_A \cdot l_A = k_B \cdot l_B \]
where \( k_A \) and \( k_B \) are the stiffnesses of springs A and B, respectively.

4. **Set Up the Equation for Spring A**:
From the relationship, we can express the stiffness of spring A:
\[ k \cdot L = k_A \cdot \left(\frac{2}{5}L\right) \]

5. **Solve for \( k_A \)**:
Rearranging the equation gives:
\[ k_A = \frac{k \cdot L}{\frac{2}{5}L} = \frac{k \cdot L \cdot 5}{2L} = \frac{5}{2}k \]

6. **Conclusion**:
The stiffness of spring A is:
\[ k_A = \frac{5}{2}k \]