The displacement-time equation of a particle executing SHM is `A-Asin(omegat+phi)`. At tme t=0 position of the particle is x=`A/2` and it is moving along negative x-direction. Then the angle `phi` can be

Options

A. `pi/6`
B. `(pi)/3`
C. `(2pi)/3`
D. `(5pi)/6`

Answer

To solve the problem step by step, we will analyze the given information and derive the angle \( \phi \).

### Step 1: Write the displacement-time equation
The displacement-time equation is given as:
\[ x(t) = A - A \sin(\omega t + \phi) \]

### Step 2: Substitute \( t = 0 \) into the equation
At time \( t = 0 \), the position of the particle is given as \( x = \frac{A}{2} \). Substituting \( t = 0 \) into the equation:
\[ x(0) = A - A \sin(\phi) = \frac{A}{2} \]

### Step 3: Rearranging the equation
Now we can rearrange the equation:
\[ A - A \sin(\phi) = \frac{A}{2} \]
Subtract \( A \) from both sides:
\[ -A \sin(\phi) = \frac{A}{2} - A \]
\[ -A \sin(\phi) = \frac{A}{2} - \frac{2A}{2} = -\frac{A}{2} \]

### Step 4: Divide by \(-A\)
Dividing both sides by \(-A\) (assuming \( A \neq 0 \)):
\[ \sin(\phi) = \frac{1}{2} \]

### Step 5: Find possible angles for \( \phi \)
The angles for which \( \sin(\phi) = \frac{1}{2} \) are:
\[ \phi = \frac{\pi}{6} \quad \text{(or 30 degrees)} \]
and
\[ \phi = \frac{5\pi}{6} \quad \text{(or 150 degrees)} \]

### Step 6: Determine the correct angle based on direction of motion
The problem states that the particle is moving in the negative x-direction. To find out which angle is correct, we need to analyze the velocity.

### Step 7: Differentiate the displacement equation to find velocity
The velocity \( v(t) \) is given by the derivative of displacement:
\[ v(t) = \frac{dx}{dt} = -A\omega \cos(\omega t + \phi) \]

### Step 8: Substitute \( t = 0 \) into the velocity equation
At \( t = 0 \):
\[ v(0) = -A\omega \cos(\phi) \]

### Step 9: Analyze the sign of the velocity
Since the particle is moving in the negative x-direction, \( v(0) \) must be negative:
\[ -A\omega \cos(\phi) < 0 \]
This implies that \( \cos(\phi) > 0 \).

### Step 10: Determine which angle satisfies the condition
- For \( \phi = \frac{\pi}{6} \): \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} > 0 \) (valid)
- For \( \phi = \frac{5\pi}{6} \): \( \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} < 0 \) (not valid)

### Conclusion
The correct angle \( \phi \) that satisfies both the position and the direction of motion is:
\[ \phi = \frac{\pi}{6} \]