The phase difference between the instantaneous velocity and acceleration of a particle executing simple harmonic motion is

Options

A. ` 0.5 pi`
B. `pi `
C. `0.707 pi `
D. zero

Explanation

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

### Step 1: Understand the equations of SHM
In SHM, the displacement \( x \) of a particle can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

### Step 2: Derive the velocity
The instantaneous velocity \( v(t) \) is the time derivative of displacement:
\[ v(t) = \frac{dx}{dt} = A \omega \cos(\omega t + \phi) \]

### Step 3: Derive the acceleration
The instantaneous acceleration \( a(t) \) is the time derivative of velocity:
\[ a(t) = \frac{dv}{dt} = -A \omega^2 \sin(\omega t + \phi) \]

### Step 4: Analyze the phase of velocity and acceleration
From the equations derived:
- The velocity \( v(t) \) is proportional to \( \cos(\omega t + \phi) \).
- The acceleration \( a(t) \) is proportional to \( -\sin(\omega t + \phi) \).

### Step 5: Determine the phase difference
To find the phase difference between velocity and acceleration:
- The cosine function \( \cos(\theta) \) can be expressed as \( \sin(\theta + \frac{\pi}{2}) \). This indicates that velocity leads the displacement by \( \frac{\pi}{2} \).
- The acceleration, being proportional to \( -\sin(\theta) \), can be rewritten as \( \sin(\theta + \pi) \), which indicates that acceleration lags behind displacement by \( \pi \).

### Step 6: Calculate the phase difference between velocity and acceleration
Since velocity leads displacement by \( \frac{\pi}{2} \) and acceleration lags behind displacement by \( \pi \):
- The phase difference between velocity and acceleration is:
\[ \text{Phase difference} = \frac{\pi}{2} + \pi = \frac{3\pi}{2} \]
However, since we are interested in the phase difference between velocity and acceleration, we can also express it as:
\[ \text{Phase difference} = \frac{\pi}{2} \]

### Conclusion
Thus, the phase difference between the instantaneous velocity and acceleration of a particle executing simple harmonic motion is:
\[ \frac{\pi}{2} \text{ radians} \quad \text{or} \quad 0.5 \pi \]

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