A particle starts moving in a circular path of radius R with an initial speed `v_(0)` clockwise. If the angular acceleration is `alpha rad//s^(2)` anticlockwise, the time when the acceleration and the velocity vectors of the particle become parallel, is

Options

A. `v_(0)//Ralpha`
B. `2v_(0)//Ralpha`
C. `v_(0)//2Ralpha`
D. this condition is impossible

Answer

To solve the problem, we need to determine the time when the acceleration and velocity vectors of a particle moving in a circular path become parallel. Here's a step-by-step solution:

### Step 1: Understanding the Motion
The particle is moving in a circular path with a radius \( R \) and an initial speed \( v_0 \) in the clockwise direction. The angular acceleration \( \alpha \) is acting in the anticlockwise direction.

### Step 2: Components of Acceleration
The total acceleration \( \vec{A} \) of the particle can be broken down into two components:
1. **Tangential Acceleration** (\( A_t \)): This is due to the change in speed along the circular path.
2. **Centripetal Acceleration** (\( A_c \)): This is due to the change in direction of the velocity vector as the particle moves along the circular path.

### Step 3: Conditions for Parallel Vectors
The velocity vector \( \vec{V} \) and acceleration vector \( \vec{A} \) will be parallel when the perpendicular component of acceleration (centripetal acceleration) becomes zero. This occurs when the tangential acceleration is the only component acting on the particle.

### Step 4: Expression for Angular Velocity
The angular velocity \( \omega \) at any time \( t \) can be expressed using the formula:
\[ \omega_f = \omega_i + \alpha t \]
where:
- \( \omega_f \) is the final angular velocity,
- \( \omega_i = \frac{v_0}{R} \) is the initial angular velocity,
- \( \alpha \) is the angular acceleration (negative in this case since it is anticlockwise).

### Step 5: Setting the Final Angular Velocity to Zero
To find the time when the acceleration and velocity vectors are parallel, we set the final angular velocity \( \omega_f \) to zero:
\[ 0 = \frac{v_0}{R} - \alpha t \]

### Step 6: Solving for Time
Rearranging the equation gives:
\[ \alpha t = \frac{v_0}{R} \]
Thus, we can solve for \( t \):
\[ t = \frac{v_0}{\alpha R} \]

### Conclusion
The time when the acceleration and velocity vectors of the particle become parallel is:
\[ t = \frac{v_0}{\alpha R} \]

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