Work done by a spning force is

Options

A. Always negative
B. Always positive
C. Always zero
D. may be positive and negative

Answer

To solve the question regarding the work done by a spring force, we can follow these steps:

### Step 1: Understand the Spring Force
The spring force \( F \) is given by Hooke's Law:
\[ F = -kx \]
where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

### Step 2: Set Up the Work Done Formula
The work done \( W \) by the spring force as it moves from an initial position \( x_i \) to a final position \( x_f \) can be calculated using the integral of the force over the displacement:
\[ W = \int_{x_i}^{x_f} F \, dx \]

### Step 3: Substitute the Spring Force into the Work Formula
Substituting the expression for the spring force into the work formula:
\[ W = \int_{x_i}^{x_f} -kx \, dx \]

### Step 4: Perform the Integration
Now, we perform the integration:
\[ W = -k \int_{x_i}^{x_f} x \, dx \]
The integral of \( x \) is:
\[ \int x \, dx = \frac{x^2}{2} \]
Thus, we have:
\[ W = -k \left[ \frac{x^2}{2} \right]_{x_i}^{x_f} \]

### Step 5: Evaluate the Limits
Evaluating the limits gives:
\[ W = -k \left( \frac{x_f^2}{2} - \frac{x_i^2}{2} \right) \]
This simplifies to:
\[ W = -\frac{k}{2} (x_f^2 - x_i^2) \]
Rearranging this, we can write:
\[ W = \frac{k}{2} (x_i^2 - x_f^2) \]

### Step 6: Analyze the Result
From the expression \( W = \frac{k}{2} (x_i^2 - x_f^2) \):
- If \( x_i > x_f \) (the spring is compressed), then \( W > 0 \) (work done by the spring is positive).
- If \( x_i < x_f \) (the spring is extended), then \( W < 0 \) (work done by the spring is negative).

### Conclusion
Thus, the work done by a spring force can be either positive or negative depending on the initial and final positions of the spring.

### Final Answer
The work done by a spring force is **may be positive or may be negative**.

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