Two identical massless springs A and B consist spring constant `k_(A)` and `k_(B)` respectively. Then :

Options

A. it they are compressed by same force, work done on A is more expanded when `k_(A) gt k_(B)`
B. if they are compressed by same amount, work done on A is more expanded when `k_(A)lt k_(B)`
C. if they are compressed by same amount, work done on A is more expanded when `k_(A) gt k_(B)`
D. both (1) and (2) are correct

Answer

To solve the problem regarding the two identical massless springs A and B with spring constants \( k_A \) and \( k_B \), we need to analyze the work done on these springs under different conditions.

### Step-by-Step Solution:

1. **Understanding Spring Constants**:
- The spring constant \( k \) is a measure of a spring's stiffness. A higher spring constant means the spring is stiffer and requires more force to compress or extend it by a certain distance.

2. **Work Done on Springs**:
- The work done on a spring when it is compressed or extended can be expressed using the formula:
\[ W = \frac{1}{2} k x^2 \]
where \( W \) is the work done, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position.

3. **Case 1: Compressed by the Same Force**:
- If both springs are compressed by the same force \( F \), the displacement \( x \) for each spring can be calculated using Hooke's Law:
\[ F = k x \implies x = \frac{F}{k} \]
- The work done on each spring can then be expressed as:
\[ W_A = \frac{1}{2} k_A \left(\frac{F}{k_A}\right)^2 = \frac{F^2}{2 k_A} \]
\[ W_B = \frac{1}{2} k_B \left(\frac{F}{k_B}\right)^2 = \frac{F^2}{2 k_B} \]
- From these equations, we can see that if \( k_A > k_B \), then \( W_A < W_B \). Thus, the work done on spring A is less than that on spring B when compressed by the same force.

4. **Case 2: Compressed by the Same Amount**:
- If both springs are compressed by the same distance \( x \), the work done on each spring becomes:
\[ W_A = \frac{1}{2} k_A x^2 \]
\[ W_B = \frac{1}{2} k_B x^2 \]
- In this case, if \( k_A > k_B \), then \( W_A > W_B \). Therefore, the work done on spring A is greater than that on spring B when compressed by the same amount.

5. **Conclusion**:
- Based on the analysis, we can summarize:
- If compressed by the same force, \( W_A < W_B \) if \( k_A > k_B \).
- If compressed by the same amount, \( W_A > W_B \) if \( k_A > k_B \).
- Therefore, the correct option is that if they are compressed by the same amount, the work done on A is more than that on B when \( k_A > k_B \).

### Final Answer:
The correct option is **C**: If compressed by the same amount, work done on A is more than work done on B when \( k_A > k_B \).