The potential energy of a particle execuring S.H.M. is 2.5 J, when its displacement is half of amplitude. The total energy of the particle be

Options

A. 18 J
B. 10 J
C. 12J
D. 2.5J

Explanation

To find the total energy of a particle executing simple harmonic motion (S.H.M.) given its potential energy at a specific displacement, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The potential energy (PE) of the particle is 2.5 J.
- The displacement (x) is half of the amplitude (A), i.e., \( x = \frac{A}{2} \).

2. **Use the Formula for Potential Energy in S.H.M.**:
- The potential energy in S.H.M. is given by the formula:
\[ PE = \frac{1}{2} k x^2 \]
- Here, \( k \) is the spring constant.

3. **Substitute the Displacement**:
- Since \( x = \frac{A}{2} \), we can substitute this into the potential energy formula:
\[ PE = \frac{1}{2} k \left(\frac{A}{2}\right)^2 \]
- This simplifies to:
\[ PE = \frac{1}{2} k \left(\frac{A^2}{4}\right) = \frac{1}{8} k A^2 \]

4. **Set Up the Equation**:
- We know from the problem that \( PE = 2.5 \, \text{J} \), so we set up the equation:
\[ \frac{1}{8} k A^2 = 2.5 \]

5. **Solve for Total Energy**:
- The total energy (E) in S.H.M. is given by:
\[ E = \frac{1}{2} k A^2 \]
- From our previous equation, we can express \( k A^2 \):
\[ k A^2 = 2.5 \times 8 = 20 \, \text{J} \]
- Now substituting this back into the total energy formula:
\[ E = \frac{1}{2} (20) = 10 \, \text{J} \]

6. **Final Answer**:
- The total energy of the particle is \( E = 10 \, \text{J} \).

### Summary:
The total energy of the particle executing S.H.M. is **10 J**.