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 speed of the particle is ……

Options

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

Answer

To solve the problem step by step, we will find the average speed of a particle moving in a circular path with a given radius and time period.

### Step-by-Step Solution:

1. **Understand the given parameters**:
- The radius of the circular path is \( R \).
- The time period of the particle is \( T \).
- The time interval we are considering is \( t = \frac{T}{6} \).

2. **Identify the formula for average speed**:
- The average speed \( v_{avg} \) is defined as:
\[ v_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} \]

3. **Calculate the distance traveled in time \( t \)**:
- The distance traveled by the particle in time \( t \) can be calculated using the formula:
\[ \text{Distance} = \text{Velocity} \times \text{Time} \]
- Since the particle is moving with constant speed, let the speed be \( v \). Therefore, the distance traveled in time \( t \) is:
\[ \text{Distance} = v \times t = v \times \frac{T}{6} \]

4. **Determine the time taken**:
- The time taken to cover this distance is simply \( t = \frac{T}{6} \).

5. **Substitute into the average speed formula**:
- Now, substituting the distance and time into the average speed formula:
\[ v_{avg} = \frac{v \times \frac{T}{6}}{\frac{T}{6}} \]
- The \( \frac{T}{6} \) terms cancel out:
\[ v_{avg} = v \]

6. **Relate speed \( v \) to the radius and time period**:
- The speed \( v \) can be expressed in terms of the radius \( R \) and the time period \( T \):
\[ v = \frac{2\pi R}{T} \]
- This is because the distance traveled in one complete revolution (circumference of the circle) is \( 2\pi R \), and it takes time \( T \) to complete that distance.

7. **Final expression for average speed**:
- Therefore, substituting this back, we find:
\[ v_{avg} = \frac{2\pi R}{T} \]

### Conclusion:
The average speed of the particle in the time \( t = \frac{T}{6} \) is:
\[ \boxed{\frac{2\pi R}{T}} \]