When the net force F acting on the particle varies as its position along x-axis shown in the figure, which of the following cannot be oscillatory motion. (where x is the position of the particle)

Options

A.
B.
C.
D.

Answer

To determine which of the given force vs. position graphs cannot represent oscillatory motion, we need to analyze the relationship between force and displacement for each graph. In oscillatory motion, the force acting on the particle must always act in the opposite direction to the displacement from the equilibrium position. This means that when the displacement (x) is positive, the force (F) should be negative, and when the displacement is negative, the force should be positive.

### Step-by-Step Solution:

1. **Understand the Condition for Oscillatory Motion**:
- For a particle to be in oscillatory motion, the force must always act in the opposite direction to the displacement. Mathematically, this can be expressed as:
\[ F = -kx \]
- Here, \( k \) is a positive constant, and \( F \) and \( x \) must have opposite signs.

2. **Analyze Each Graph**:
- **Graph 1**: Check the relationship between force and displacement.
- If \( x < 0 \) (negative), then \( F > 0 \) (positive).
- If \( x > 0 \) (positive), then \( F < 0 \) (negative).
- This graph satisfies the condition for oscillatory motion.

- **Graph 2**: Check the relationship.
- If \( x < 0 \), then \( F > 0 \).
- If \( x > 0 \), then \( F < 0 \).
- This graph also satisfies the condition for oscillatory motion.

- **Graph 3**: Check the relationship.
- If \( x < 0 \), then \( F > 0 \).
- If \( x > 0 \), then \( F < 0 \).
- This graph satisfies the condition for oscillatory motion as well.

- **Graph 4**: Check the relationship.
- If \( x < 0 \), then \( F < 0 \) (negative).
- If \( x > 0 \), then \( F > 0 \) (positive).
- This graph does not satisfy the condition for oscillatory motion since the force and displacement are in the same direction for both positive and negative displacements.

3. **Conclusion**:
- The graph that cannot represent oscillatory motion is **Graph 4** because the force does not act in the opposite direction to the displacement.

### Final Answer:
The graph that cannot represent oscillatory motion is **Graph 4**.