A particle is moving in a circle of radius R with constant speed. The time period of the particle is T. In a time `t=(T)/(6)`
Average velocity of the particle is…..

Options

A. `(3R)/(T)`
B. `(6R)/(T)`
C. `(2R)/(T)`
D. `(4R)/(T)`

Answer

To solve the problem, we need to find the average velocity of a particle moving in a circle of radius \( R \) with a time period \( T \) after a time \( t = \frac{T}{6} \).

### Step-by-Step Solution:

1. **Understanding Average Velocity**:
The average velocity \( \bar{v} \) is defined as the total displacement divided by the total time taken. Mathematically, it is given by:
\[ \bar{v} = \frac{\text{Total Displacement}}{\text{Total Time}} \]

2. **Determine the Total Time**:
We are given that the time \( t \) is:
\[ t = \frac{T}{6} \]

3. **Calculate Angular Displacement**:
The angular displacement \( \theta \) in radians can be calculated using the formula:
\[ \theta = \omega \cdot t \]
where \( \omega \) (angular velocity) is given by:
\[ \omega = \frac{2\pi}{T} \]
Substituting \( \omega \) into the equation for \( \theta \):
\[ \theta = \frac{2\pi}{T} \cdot \frac{T}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \text{ radians} \]

4. **Convert Angular Displacement to Degrees**:
The angular displacement \( \frac{\pi}{3} \) radians is equivalent to:
\[ \frac{\pi}{3} \times \frac{180}{\pi} = 60^\circ \]

5. **Determine the Displacement**:
The particle moves from its initial position to a final position after covering an angle of \( 60^\circ \). The displacement is the straight line distance between these two points on the circumference of the circle. This forms an equilateral triangle with each side equal to the radius \( R \).

Since the triangle is equilateral (each angle is \( 60^\circ \)), the length of the line segment (displacement) between the initial and final points is:
\[ \text{Displacement} = R \]

6. **Calculate Average Velocity**:
Now we can calculate the average velocity:
\[ \bar{v} = \frac{\text{Displacement}}{\text{Total Time}} = \frac{R}{\frac{T}{6}} = \frac{R \cdot 6}{T} = \frac{6R}{T} \]

### Final Answer:
The average velocity of the particle is:
\[ \bar{v} = \frac{6R}{T} \]