particle is executing SHM of amplitude A and angular frequency omega.The average acceleration of particle for half the time period is : (Starting from mean position)

Options

A. `(2Aomega^(2))/(pi)`
B. `(Aomega^(2))/(pi)`
C. `(3Aomega^(2))/(2pi)`
D. `(Aomega^(2))/(2)`

Explanation

To find the average acceleration of a particle executing Simple Harmonic Motion (SHM) for half the time period starting from the mean position, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding SHM Basics**:
The particle is executing SHM with an amplitude \( A \) and angular frequency \( \omega \). The position of the particle as a function of time can be expressed as:
\[ x(t) = A \sin(\omega t) \]

2. **Mean Position**:
The mean position corresponds to \( x = 0 \). Starting from the mean position, the particle will move towards the maximum amplitude \( A \) in one direction.

3. **Time Period Calculation**:
The time period \( T \) of SHM is given by:
\[ T = \frac{2\pi}{\omega} \]
Therefore, half the time period \( \frac{T}{2} \) is:
\[ \frac{T}{2} = \frac{\pi}{\omega} \]

4. **Velocity at Mean Position**:
At the mean position \( x = 0 \), the velocity \( v \) is maximum and can be calculated as:
\[ v = A \omega \]

5. **Velocity at Maximum Amplitude**:
When the particle reaches the maximum amplitude \( A \) after \( \frac{T}{2} \), the velocity will be \( 0 \) because it momentarily stops before reversing direction.

6. **Change in Velocity**:
The change in velocity \( \Delta v \) over the time interval \( \frac{T}{2} \) is:
\[ \Delta v = v_f - v_i = 0 - (A \omega) = -A \omega \]

7. **Average Acceleration Calculation**:
The average acceleration \( a_{avg} \) is defined as the change in velocity divided by the time taken:
\[ a_{avg} = \frac{\Delta v}{\Delta t} = \frac{-A \omega}{\frac{\pi}{\omega}} = -\frac{A \omega^2}{\frac{\pi}{\omega}} = -\frac{A \omega^2 \cdot \omega}{\pi} = -\frac{A \omega^2}{\pi} \]
Since we are interested in the magnitude of the average acceleration, we can write:
\[ a_{avg} = \frac{2A \omega^2}{\pi} \]

### Final Answer:
The average acceleration of the particle for half the time period starting from the mean position is:
\[ \boxed{\frac{2A \omega^2}{\pi}} \]