A particle of mass m oscillates with simple harmonic motion between points `x_(1)` and `x_(2)`, the equilibrium position being O. Its potential energy is plotted. It will be as given below in the graph

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the correct graph for the potential energy of a particle oscillating in simple harmonic motion (SHM) between points \( x_1 \) and \( x_2 \) with the equilibrium position at \( O \), we can follow these steps:

### Step 1: Understand the Potential Energy in SHM
The potential energy \( U \) of a particle in simple harmonic motion is given by the formula:
\[ U = \frac{1}{2} m \omega^2 x^2 \]
where:
- \( m \) is the mass of the particle,
- \( \omega \) is the angular frequency,
- \( x \) is the displacement from the equilibrium position.

### Step 2: Analyze the Formula
From the formula, we can observe that:
- The potential energy \( U \) is proportional to the square of the displacement \( x \).
- This means that as \( x \) increases (either positively or negatively), the potential energy increases.

### Step 3: Identify the Shape of the Graph
Since \( U \) is proportional to \( x^2 \), the graph of potential energy versus displacement will be a parabola that opens upwards. The vertex of this parabola will be at the equilibrium position \( O \) (where \( x = 0 \)), and the potential energy at this point will be zero.

### Step 4: Determine the Characteristics of the Graph
- At the equilibrium position \( O \) (i.e., \( x = 0 \)), the potential energy \( U \) is zero.
- As the particle moves towards \( x_1 \) and \( x_2 \), the potential energy will increase, reflecting the quadratic relationship.

### Step 5: Evaluate the Given Options
Based on the characteristics of the potential energy graph:
- The correct graph should show a parabola opening upwards with the lowest point (minimum potential energy) at the equilibrium position \( O \).
- The potential energy should be zero at \( x = 0 \) and increase as \( x \) moves away from zero in either direction.

### Conclusion
After analyzing the options provided in the question, the graph that matches the described characteristics (a parabola opening upwards with the vertex at the equilibrium position) is the correct answer.

### Final Answer
The correct option is **Option 3**, which represents the potential energy of the particle in simple harmonic motion.

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