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

Options

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

Explanation

To solve the problem of finding 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 motion in SHM
In simple harmonic motion, 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: Find the expression for velocity
The instantaneous velocity \( v(t) \) is the time derivative of displacement:
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}[A \sin(\omega t + \phi)] \]
Using the derivative of sine, we get:
\[ v(t) = A \omega \cos(\omega t + \phi) \]

### Step 3: Find the expression for acceleration
The instantaneous acceleration \( a(t) \) is the time derivative of velocity:
\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}[A \omega \cos(\omega t + \phi)] \]
Using the derivative of cosine, we find:
\[ a(t) = -A \omega^2 \sin(\omega t + \phi) \]

### Step 4: Relate velocity and acceleration to phase
Now we can express the velocity and acceleration in terms of sine and cosine:
- The velocity can be rewritten as:
\[ v(t) = A \omega \cos(\omega t + \phi) = A \omega \sin\left(\frac{\pi}{2} + \omega t + \phi\right) \]
- The acceleration can be rewritten as:
\[ a(t) = -A \omega^2 \sin(\omega t + \phi) \]

### Step 5: Determine the phase difference
From the expressions for velocity and acceleration, we can see that:
- The velocity \( v(t) \) is represented by a sine function with a phase shift of \( \frac{\pi}{2} \).
- The acceleration \( a(t) \) is represented by a sine function without any additional phase shift.

The phase difference \( \Delta \phi \) between the velocity and acceleration is:
\[ \Delta \phi = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]

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