Which graph represents the difference between total energy and potential energy of a partical execution SHM vs it's distance from mean position

Options

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Answer

To solve the problem, we need to analyze the relationship between total energy, potential energy, and kinetic energy of a particle executing Simple Harmonic Motion (SHM).

### Step-by-step Solution:

1. **Understand the Energy in SHM**:
In Simple Harmonic Motion, a particle oscillates about a mean position. The total mechanical energy (E) of the system remains constant and is the sum of kinetic energy (K) and potential energy (U):
\[ E = K + U \]

2. **Define Potential Energy**:
The potential energy (U) of a particle in SHM can be expressed as:
\[ U = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the displacement from the mean position.

3. **Define Kinetic Energy**:
The kinetic energy (K) can be derived from the total energy:
\[ K = E - U \]
Therefore, substituting for U:
\[ K = E - \frac{1}{2} k x^2 \]

4. **Express the Difference**:
We need to find the difference between total energy and potential energy:
\[ E - U = K = E - \frac{1}{2} k x^2 \]

5. **Graphing Kinetic Energy**:
Since \( K = E - \frac{1}{2} k x^2 \), we can see that as \( x \) increases (moving away from the mean position), the potential energy increases, and consequently, the kinetic energy decreases. The graph of kinetic energy (K) versus displacement (x) will be a downward-opening parabola.

6. **Identify the Graph**:
The graph that represents the difference between total energy and potential energy (which is the kinetic energy) will be a downward parabola, reaching its maximum at the mean position (where \( x = 0 \)) and decreasing to zero when the particle reaches the maximum displacement (amplitude).

### Conclusion:
The graph that represents the difference between total energy and potential energy of a particle executing SHM versus its distance from the mean position is a downward-opening parabola.