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 step by step, we need to analyze the motion of the particle in a circular path with the given conditions.

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

### Step 2: Identify Angular Velocity
Since the particle is moving clockwise, we can denote the initial angular velocity as:
\[ \omega_0 = \frac{v_0}{R} \]
This is derived from the relationship between linear velocity and angular velocity, where \( v = \omega R \).

### Step 3: Analyze Angular Acceleration
The angular acceleration \( \alpha \) is acting in the opposite direction (anticlockwise) to the angular velocity. This means that the particle is experiencing angular retardation.

### Step 4: Condition for Parallel Vectors
The problem states that we need to find the time \( t \) when the acceleration vector \( \mathbf{a} \) and the velocity vector \( \mathbf{v} \) become parallel. For this to happen, the centripetal acceleration \( a_c \) must be zero, as the tangential acceleration \( a_t \) will be in the direction of the velocity vector.

### Step 5: Set Centripetal Acceleration to Zero
The centripetal acceleration is given by:
\[ a_c = \omega^2 R \]
For \( a_c \) to be zero, we need:
\[ \omega = 0 \]

### Step 6: Use the Equation of Motion
Using the equation for uniformly retarded motion:
\[ \omega_f = \omega_i - \alpha t \]
where \( \omega_f = 0 \) (when the velocity and acceleration vectors become parallel), we have:
\[ 0 = \omega_0 - \alpha t \]
Rearranging gives:
\[ t = \frac{\omega_0}{\alpha} \]

### Step 7: Substitute for \( \omega_0 \)
Substituting \( \omega_0 = \frac{v_0}{R} \):
\[ t = \frac{v_0 / R}{\alpha} = \frac{v_0}{\alpha R} \]

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