A particle executing SHM. The phase difference between acceleration and displacement is

Options

A. 0
B. `pi/2`
C. `pi`
D. `2pi`

Explanation

To find the phase difference between acceleration and displacement for a particle executing simple harmonic motion (SHM), we can follow these steps:

### Step-by-Step Solution:

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

2. **Differentiate to Find Velocity**: To find the velocity \( v \), we differentiate the displacement with respect to time:
\[ v = \frac{dy}{dt} = \frac{d}{dt}(A \sin(\omega t)) = A \omega \cos(\omega t) \]

3. **Differentiate to Find Acceleration**: Next, we differentiate the velocity to find the acceleration \( a \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}(A \omega \cos(\omega t)) = -A \omega^2 \sin(\omega t) \]

4. **Express Acceleration in Terms of Displacement**: We can rewrite the acceleration in terms of displacement:
\[ a = -\omega^2 y \]
This shows that acceleration is proportional to displacement but in the opposite direction.

5. **Identify the Phase of Displacement and Acceleration**:
- The phase of displacement \( y \) is given by \( \phi_1 = \omega t \).
- The phase of acceleration \( a \) can be expressed as \( \phi_2 = \omega t + \pi \) (since \( a \) is negative when \( y \) is positive).

6. **Calculate the Phase Difference**: The phase difference \( \Delta \phi \) between acceleration and displacement is:
\[ \Delta \phi = \phi_2 - \phi_1 = (\omega t + \pi) - \omega t = \pi \]

7. **Convert to Degrees**: The phase difference in degrees is:
\[ \Delta \phi = 180^\circ \]

### Final Answer:
The phase difference between acceleration and displacement in SHM is \( 180^\circ \).