The phase difference between displacement and acceleration of a particle in a simple harmonic motion is:

Options

A. `(3pi)/(2)` rad
B. `(pi)/(2)` rad
C. Zero
D. `pi` rad

Explanation

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

### Step-by-Step Solution:

1. **Write the equation for displacement in SHM:**
The displacement \( x \) of a particle in SHM can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

**Hint:** Recall the standard form of SHM equations.

2. **Determine the expression for velocity:**
The velocity \( v \) is the derivative of displacement with respect to time:
\[ v(t) = \frac{dx}{dt} = A \omega \cos(\omega t + \phi) \]

**Hint:** Remember that velocity is the rate of change of displacement.

3. **Determine the expression for acceleration:**
The acceleration \( a \) is the derivative of velocity with respect to time:
\[ a(t) = \frac{dv}{dt} = -A \omega^2 \sin(\omega t + \phi) \]

**Hint:** Acceleration is the rate of change of velocity.

4. **Identify the phases of displacement and acceleration:**
- The phase of displacement \( x(t) \) is \( \omega t + \phi \).
- The phase of acceleration \( a(t) \) can be rewritten as:
\[ a(t) = -A \omega^2 \sin(\omega t + \phi) = A \omega^2 \sin\left(\pi + (\omega t + \phi)\right) \]
This indicates that the phase of acceleration is \( \omega t + \phi + \pi \).

**Hint:** Note how the negative sign in the sine function affects the phase.

5. **Calculate the phase difference:**
The phase difference \( \Delta \phi \) between acceleration and displacement is given by:
\[ \Delta \phi = \text{Phase of acceleration} - \text{Phase of displacement} \]
Substituting the phases:
\[ \Delta \phi = (\omega t + \phi + \pi) - (\omega t + \phi) = \pi \]

**Hint:** Simplify the expression carefully to find the difference.

6. **Conclusion:**
The phase difference between displacement and acceleration in simple harmonic motion is:
\[ \Delta \phi = \pi \text{ radians} \]

Therefore, the correct answer is **option 4: \( \pi \) radians**.