If `a= -omega^(2) x ` represents the acceleration of a particle executing S.H.M.which of the following statement(s) is `//`are correct ?

Options

A. a' is maximum at the extreme position
B. Timeperiod is `T = 2pi sqrt( omega )`
C. At `x=0`, the potential energy is maximum
D. At x=0 , the K.E. is maximum

Answer

To solve the question regarding the statements about a particle executing Simple Harmonic Motion (S.H.M) with the acceleration given by the equation \( a = -\omega^2 x \), we will analyze each statement one by one.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( a = -\omega^2 x \) indicates that the acceleration \( a \) is directly proportional to the displacement \( x \) from the mean position and is directed towards the mean position (hence the negative sign). This is characteristic of S.H.M.

2. **Analyzing Statement 1**:
- **Statement**: "a is maximum at the extreme position."
- **Analysis**: At the extreme positions, the displacement \( x \) is at its maximum (equal to the amplitude \( A \)). Therefore, the acceleration \( a \) will also be maximum because \( a = -\omega^2 x \) will yield the largest value when \( x \) is at its maximum.
- **Conclusion**: This statement is **correct**.

3. **Analyzing Statement 2**:
- **Statement**: "Time period is \( T = 2\pi \sqrt{\omega} \)."
- **Analysis**: The correct formula for the time period \( T \) of a particle in S.H.M is given by \( T = \frac{2\pi}{\omega} \). The statement incorrectly states the formula.
- **Conclusion**: This statement is **incorrect**.

4. **Analyzing Statement 3**:
- **Statement**: "At \( x = 0 \), the potential energy is maximum."
- **Analysis**: The potential energy \( U \) in S.H.M is given by \( U = \frac{1}{2} k x^2 \) (where \( k = m\omega^2 \)). At the mean position \( x = 0 \), the potential energy is zero, not maximum. The potential energy is maximum at the extreme positions where \( x \) is maximum.
- **Conclusion**: This statement is **incorrect**.

5. **Analyzing Statement 4**:
- **Statement**: "At \( x = 0 \), kinetic energy is maximum."
- **Analysis**: The kinetic energy \( K \) in S.H.M is given by \( K = \frac{1}{2} m v^2 \). At the mean position \( x = 0 \), the velocity \( v \) is maximum, hence the kinetic energy is also maximum. This is consistent with the energy conservation in S.H.M.
- **Conclusion**: This statement is **correct**.

### Final Summary of Statements:
- Statement 1: Correct
- Statement 2: Incorrect
- Statement 3: Incorrect
- Statement 4: Correct

Thus, the correct statements are 1 and 4.