The particle performing S.H.M. about mean position, displacement and acceleration have initial phase difference of

Options

A. `pi//2 rad`
B. `3pi//2 rad`
C. `pi rad`
D. `2pi rad`

Explanation

To solve the problem of finding the initial phase difference between the displacement and acceleration of a particle performing Simple Harmonic Motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Equations of SHM**:
- The displacement \( x \) of a particle in SHM can be expressed as:
\[ x(t) = A \sin(\omega t) \]
where \( A \) is the amplitude and \( \omega \) is the angular frequency.

2. **Determine the Acceleration**:
- The acceleration \( a \) of the particle is given by the second derivative of displacement with respect to time. The equation for acceleration can be derived as follows:
\[ a(t) = \frac{d^2x}{dt^2} = -A\omega^2 \sin(\omega t) \]
- This can also be written as:
\[ a(t) = -\omega^2 x(t) \]

3. **Rearranging the Acceleration Equation**:
- To analyze the phase difference, we can express the acceleration in terms of sine:
\[ a(t) = -A\omega^2 \sin(\omega t) = A\omega^2 \sin(\omega t - \pi) \]
- Here, we have introduced a phase shift of \( \pi \) to express the negative sign.

4. **Comparing Phase of Displacement and Acceleration**:
- From the equations:
- Displacement: \( x(t) = A \sin(\omega t) \)
- Acceleration: \( a(t) = A\omega^2 \sin(\omega t - \pi) \)
- We can see that the phase of the displacement is \( \omega t \) and the phase of the acceleration is \( \omega t - \pi \).

5. **Calculating the Phase Difference**:
- The phase difference \( \Delta \phi \) between the displacement and acceleration is given by:
\[ \Delta \phi = \text{Phase of acceleration} - \text{Phase of displacement} = (\omega t - \pi) - (\omega t) = -\pi \]
- Since phase differences are often expressed as positive angles, we can say that the phase difference is \( \pi \).

### Final Answer:
The initial phase difference between the displacement and acceleration of a particle performing SHM is \( \pi \) radians.