One end of a light spring of spring constant k is fixed to a wall and the other end is tied to a block placed on a smooth horizontal surface. In a displacement, the work done by the spring is `+(1/2)kx^(2)`. The possible cases are.

Options

A. the spring was initially stretched by a distance x and finally was in its natural length
B. the spring was initially in its natural length and finally it was compressed by a distance x
C. the spring was initially compressed by a distance x and finally was in its natural length
D. the spring was initally in its natural length and finally stretched by a distance x

Answer

To solve the problem, we need to analyze the work done by the spring in different scenarios based on its displacement. The work done by the spring is given by the formula:

\[ W = \frac{1}{2} k x^2 \]

where \( k \) is the spring constant and \( x \) is the displacement from the spring's natural length.

### Step-by-Step Solution:

1. **Understanding the Spring System**:
- One end of the spring is fixed to a wall, and the other end is attached to a block on a smooth horizontal surface.
- The spring can either be compressed or extended from its natural length.

2. **Work Done by the Spring**:
- The work done by the spring is positive when the spring is either compressed or extended, as long as the displacement and the force exerted by the spring are in the same direction.
- The formula for work done by the spring indicates that the work is dependent on the square of the displacement \( x \).

3. **Analyzing Different Cases**:
- **Case A**: The spring is initially stretched (extended) and then released. As it returns to its natural length, the displacement and the spring force are in the same direction, resulting in positive work done.
- **Case B**: The spring is initially at its natural length. In this case, there is no displacement, so the work done is zero. This case does not yield positive work.
- **Case C**: The spring is initially compressed and then released. Similar to Case A, as it returns to its natural length, the displacement and the spring force are in the same direction, resulting in positive work done.
- **Case D**: Again, the spring is initially at its natural length, which results in zero work done.

4. **Conclusion**:
- The possible cases where the work done by the spring is positive are:
- Case A: Spring initially stretched.
- Case C: Spring initially compressed.
- Therefore, the correct answers are A and C.

### Final Answer:
The correct cases where the work done by the spring is positive are:
- A: Spring was initially stretched.
- C: Spring was initially compressed.