A particle moves along a circle of radius R =1m so that its radius vector `vecr` relative to a point on its circumference rotates with the constant angular velocity `omega=2rad//s`. The linear speed of the particle is

Options

A. `4m//s`
B. `2m//s`
C. `1m//s`
D. `0.5m//s`

Explanation

To find the linear speed of a particle moving along a circle with a given radius and angular velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Radius of the circle, \( R = 1 \, \text{m} \)
- Angular velocity of the radius vector, \( \omega = 2 \, \text{rad/s} \)

2. **Understand the Motion**:
- The particle moves along the circumference of the circle.
- The radius vector rotates with a constant angular velocity \( \omega \).

3. **Determine the Effective Angular Velocity**:
- The angular velocity of the particle moving along the circle can be derived from the geometry of the situation.
- The angle \( \theta \) subtended by the radius vector at the center of the circle is related to the angle \( \theta' \) at the circumference.
- The relationship is given by \( \theta' = 2\theta \) due to the properties of the triangle formed by the radius and the tangential line.

4. **Relate Angular Velocities**:
- If \( \omega' \) is the angular velocity of the particle, we have:
\[ \omega' = 2\omega \]
- Substituting the value of \( \omega \):
\[ \omega' = 2 \times 2 = 4 \, \text{rad/s} \]

5. **Calculate the Linear Speed**:
- The linear speed \( v \) of the particle is given by the formula:
\[ v = R \cdot \omega' \]
- Substituting the values:
\[ v = 1 \, \text{m} \times 4 \, \text{rad/s} = 4 \, \text{m/s} \]

6. **Final Answer**:
- The linear speed of the particle is \( 4 \, \text{m/s} \).