A body is subjected to a conservative force given by F=b-2ax where a and b are constant, then

Options

A. the body is in neutral equilibrium depending on the value of b
B. the body is in stable equilibrium if a is negative
C. the body is in unstable equilibrium if a positive
D. the body is in stable equilibrium if a is positive

Explanation

To solve the problem, we need to analyze the given conservative force \( F = b - 2ax \) where \( a \) and \( b \) are constants. We will determine the conditions for stable equilibrium based on the characteristics of conservative forces.

### Step-by-Step Solution:

1. **Understanding Conservative Forces**:
- A conservative force is one where the work done by the force on an object moving from one point to another is independent of the path taken. The work done is related to the change in potential energy.

2. **Identify the Force**:
- The force acting on the body is given by:
\[ F = b - 2ax \]
- Here, \( b \) is a constant and \( -2ax \) indicates that the force is dependent on the position \( x \).

3. **Finding the Equilibrium Position**:
- To find the equilibrium position, we set the force \( F \) to zero:
\[ 0 = b - 2ax \]
- Rearranging gives:
\[ 2ax = b \implies x = \frac{b}{2a} \]
- This is the position of equilibrium.

4. **Analyzing Stability**:
- To determine if this equilibrium is stable, we need to analyze the behavior of the force around this point. We can do this by taking the derivative of the force with respect to \( x \):
\[ \frac{dF}{dx} = -2a \]
- Since \( -2a \) is negative (assuming \( a > 0 \)), this indicates that the force decreases as \( x \) increases.

5. **Condition for Stable Equilibrium**:
- For stable equilibrium, the force must act to restore the object back to equilibrium when it is displaced. This means that if \( x \) increases, \( F \) must decrease, pulling the object back towards \( x = \frac{b}{2a} \).
- Since the slope \( \frac{dF}{dx} \) is negative, the equilibrium position is stable.

6. **Conclusion**:
- Therefore, for the body to be in stable equilibrium under the influence of the conservative force \( F = b - 2ax \), the constant \( a \) must be positive.

### Final Answer:
- The condition for stable equilibrium is that \( a > 0 \).