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 for centripetal force:

\[ F_c = m \cdot a_c \]

where:
- \( F_c \) is the centripetal force,
- \( m \) is the mass of the particle,
- \( a_c \) is the centripetal acceleration.

The centripetal acceleration can be expressed in terms of angular speed \( \omega \) and radius \( r \):

\[ a_c = r \cdot \omega^2 \]

### 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. **Calculate the centripetal acceleration \( a_c \):**
\[ a_c = r \cdot \omega^2 \]
Substituting the values:
\[ a_c = 1 \cdot (2\pi)^2 \]
\[ a_c = 1 \cdot 4\pi^2 \]
\[ a_c = 4\pi^2 \, \text{m/s}^2 \]

3. **Calculate the centripetal force \( F_c \):**
\[ F_c = m \cdot a_c \]
Substituting the values:
\[ F_c = 2 \cdot 4\pi^2 \]
\[ F_c = 8\pi^2 \, \text{N} \]

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