A particle is performing S.H.M. Its total energy ie E When the displacement of the particle is half of its amplitude, its K.E. will be

Options

A. E/2
B. E/4
C. 3E/4
D. E/8

Explanation

To find the kinetic energy (K.E.) of a particle performing Simple Harmonic Motion (S.H.M.) when its displacement is half of its amplitude, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Total Energy in S.H.M.**:
The total energy (E) of a particle in S.H.M. is given by the formula:
\[ E = \frac{1}{2} m \omega^2 A^2 \]
where:
- \( m \) is the mass of the particle,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude.

2. **Identify the Displacement**:
We need to find the kinetic energy when the displacement \( x \) is half of the amplitude. Therefore:
\[ x = \frac{A}{2} \]

3. **Use the Kinetic Energy Formula**:
The kinetic energy (K.E.) in S.H.M. can be expressed as:
\[ K.E. = \frac{1}{2} m \omega^2 A^2 - \frac{1}{2} m \omega^2 x^2 \]
Substitute \( x = \frac{A}{2} \) into the formula:
\[ K.E. = \frac{1}{2} m \omega^2 A^2 - \frac{1}{2} m \omega^2 \left(\frac{A}{2}\right)^2 \]

4. **Calculate \( x^2 \)**:
Calculate \( \left(\frac{A}{2}\right)^2 \):
\[ \left(\frac{A}{2}\right)^2 = \frac{A^2}{4} \]

5. **Substitute and Simplify**:
Substitute \( x^2 \) back into the K.E. equation:
\[ K.E. = \frac{1}{2} m \omega^2 A^2 - \frac{1}{2} m \omega^2 \frac{A^2}{4} \]
This simplifies to:
\[ K.E. = \frac{1}{2} m \omega^2 A^2 - \frac{1}{8} m \omega^2 A^2 \]

6. **Combine the Terms**:
Convert \( \frac{1}{2} \) into eighths:
\[ \frac{1}{2} = \frac{4}{8} \]
Therefore:
\[ K.E. = \frac{4}{8} m \omega^2 A^2 - \frac{1}{8} m \omega^2 A^2 = \frac{3}{8} m \omega^2 A^2 \]

7. **Relate to Total Energy**:
Recall that the total energy \( E = \frac{1}{2} m \omega^2 A^2 \). Thus:
\[ K.E. = \frac{3}{4} E \]

8. **Final Result**:
Therefore, the kinetic energy when the displacement is half of the amplitude is:
\[ K.E. = \frac{3E}{4} \]

### Conclusion:
The correct answer is \( \frac{3E}{4} \).