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.

Answer

To solve the problem of determining the potential energy graph of a particle of mass \( m \) 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:
\[ U = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position \( O \).

### Step 2: Identify the displacement
In this case, the particle oscillates between points \( x_1 \) and \( x_2 \). The equilibrium position \( O \) is the midpoint between these two points. Thus, the maximum displacement from the equilibrium position is:
\[ A = \frac{x_2 - x_1}{2} \]

### Step 3: Analyze the potential energy at maximum displacement
At maximum displacement (either at \( x_1 \) or \( x_2 \)), the potential energy will be:
\[ U_{\text{max}} = \frac{1}{2} k A^2 \]
At the equilibrium position \( O \), the potential energy is:
\[ U(O) = 0 \]

### Step 4: Determine the shape of the graph
Since the potential energy is proportional to the square of the displacement, the graph of potential energy \( U \) versus displacement \( x \) will be a parabola opening upwards. The vertex of this parabola (where \( U = 0 \)) is at the equilibrium position \( O \).

### Step 5: Identify the correct graph
From the options provided, we need to find a graph that:
- Is a parabola opening upwards.
- Has its vertex at the origin (equilibrium position \( O \)).
- Passes through the points corresponding to maximum displacements \( x_1 \) and \( x_2 \) where the potential energy is at its maximum.

After analyzing the options, we find that option C fits these criteria.

### Final Answer
Thus, the correct option representing the potential energy of the particle oscillating in simple harmonic motion is **Option C**.

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