If a particle is moving along a circle of radius 3 m with a constant speed `9 m//s`, then it covers a quarter of the circle in time of

Options

A. `pi //3 s`
B. `pi //6 s`
C. `pi //2 s`
D. `pi//8 s`

Explanation

To solve the problem of how long it takes for a particle moving along a circle of radius 3 m at a constant speed of 9 m/s to cover a quarter of the circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Radius of the circle, \( r = 3 \, \text{m} \)
- Constant speed of the particle, \( v = 9 \, \text{m/s} \)

2. **Determine the total circumference of the circle:**
- The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
- Substituting the radius:
\[ C = 2\pi \times 3 = 6\pi \, \text{m} \]

3. **Calculate the distance covered for a quarter of the circle:**
- A quarter of the circle means covering \( \frac{1}{4} \) of the total circumference:
\[ \text{Distance for quarter circle} = \frac{1}{4} C = \frac{1}{4} \times 6\pi = \frac{3\pi}{2} \, \text{m} \]

4. **Use the formula for time:**
- Time \( t \) can be calculated using the formula:
\[ t = \frac{\text{Distance}}{\text{Speed}} \]
- Substituting the distance covered and the speed:
\[ t = \frac{\frac{3\pi}{2}}{9} \]

5. **Simplify the expression:**
- Simplifying the time:
\[ t = \frac{3\pi}{2 \times 9} = \frac{3\pi}{18} = \frac{\pi}{6} \, \text{s} \]

6. **Final answer:**
- The time taken to cover a quarter of the circle is:
\[ t = \frac{\pi}{6} \, \text{s} \]