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 by the spring is `1/2kx^2`. The possible cases are

Options

A. the spring was initially compresed bya distance x and was finaly in its natural length
B. it was initially in its natural distance x and finally was in its natural length
C. it was initially in its natural lenth and finaly in a compressed position.
D. it was initially in its natural length and finally in a stretched positon.

Answer

To solve the problem regarding the work done by a spring, we need to analyze the scenarios described in the question. Let's break it down step by step.

### Step 1: Understanding the Work Done by the Spring
The work done by a spring when it is either compressed or extended is given by the formula:

\[ W = -\frac{1}{2} k (x_f^2 - x_i^2) \]

where:
- \(W\) is the work done by the spring,
- \(k\) is the spring constant,
- \(x_f\) is the final displacement from the natural length,
- \(x_i\) is the initial displacement from the natural length.

The negative sign indicates that the work done by the spring is in the opposite direction to the displacement.

### Step 2: Analyzing the Cases
We will analyze the three possible cases mentioned in the question:

**Case 1:** The spring was initially compressed by a distance \(x\) and was finally at its natural length.

- Here, \(x_i = x\) and \(x_f = 0\).
- Substituting into the work formula:

\[ W = -\frac{1}{2} k (0^2 - x^2) = -\frac{1}{2} k (-x^2) = \frac{1}{2} k x^2 \]

This case is possible as the work done is positive.

**Case 2:** The spring was initially at its natural length and was finally compressed by a distance \(x\).

- Here, \(x_i = 0\) and \(x_f = x\).
- Substituting into the work formula:

\[ W = -\frac{1}{2} k (x^2 - 0^2) = -\frac{1}{2} k x^2 \]

This case is also possible as the work done is negative, indicating that energy is stored in the spring.

**Case 3:** The spring was initially at its natural length and finally at a distance \(x\) (either compressed or extended).

- Here, \(x_i = 0\) and \(x_f = x\) (for extension) or \(x_f = -x\) (for compression).
- For extension:

\[ W = -\frac{1}{2} k (x^2 - 0^2) = -\frac{1}{2} k x^2 \]

- For compression:

\[ W = -\frac{1}{2} k ((-x)^2 - 0^2) = -\frac{1}{2} k x^2 \]

In both cases, the work done is negative, which indicates that energy is stored in the spring. Thus, this case is also possible.

### Conclusion
Based on the analysis of the three cases:
- Case 1 is possible (work done is positive).
- Case 2 is possible (work done is negative).
- Case 3 is possible (work done is negative).