A particle of mass 2 kg is moving along a circular path of radius 1 m. If its angular speed is `2pi" rad s"^(-1)`, the centripetal force on it is

Options

A. `4piN`
B. `8piN`
C. `4pi^(4)N`
D. `8pi^(2)N`

Explanation

To find the centripetal force acting on a particle moving in a circular path, we can use the formula:

\[ F_c = m \omega^2 r \]

where:
- \( F_c \) is the centripetal force,
- \( m \) is the mass of the particle,
- \( \omega \) is the angular speed,
- \( r \) is the radius of the circular path.

### Step-by-Step Solution:

1. **Identify the given values:**
- Mass of the particle, \( m = 2 \, \text{kg} \)
- Radius of the circular path, \( r = 1 \, \text{m} \)
- Angular speed, \( \omega = 2\pi \, \text{rad/s} \)

2. **Substitute the values into the centripetal force formula:**
\[ F_c = m \omega^2 r \]
\[ F_c = 2 \times (2\pi)^2 \times 1 \]

3. **Calculate \( \omega^2 \):**
\[ (2\pi)^2 = 4\pi^2 \]

4. **Now substitute \( \omega^2 \) back into the equation:**
\[ F_c = 2 \times 4\pi^2 \times 1 \]
\[ F_c = 8\pi^2 \]

5. **Final result:**
The centripetal force \( F_c \) is:
\[ F_c = 8\pi^2 \, \text{N} \]

### Conclusion:
The centripetal force acting on the particle is \( 8\pi^2 \, \text{N} \).

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