When a particle is moving in a circular path of radius r with angular speed ω, its linear speed v is given by the formula:

```

v = rω

```

where:

* v is the linear speed in meters per second (m/s)

* r is the radius of the circular path in meters (m)

* ω is the angular speed in radians per second (rad/s)

Derivation of the Formula

The linear speed of a particle moving in a circular path can be derived using the concept of tangential velocity. Tangential velocity is the velocity of a particle that is moving along a tangent to a circular path at a given point. In the case of a particle moving in a circular path, the tangential velocity is equal to the linear speed of the particle.

The tangential velocity of a particle moving in a circular path can be calculated using the formula:

```

v = rω

```

where:

* v is the tangential velocity in meters per second (m/s)

* r is the radius of the circular path in meters (m)

* ω is the angular speed in radians per second (rad/s)

The angular speed of a particle moving in a circular path is defined as the rate at which the particle changes its angular position. The angular speed is measured in radians per second (rad/s).

Example

A particle is moving in a circular path of radius 2 meters with an angular speed of 3 radians per second. What is the linear speed of the particle?

```

v = rω

v = (2 m)(3 rad/s)

v = 6 m/s

```

Therefore, the linear speed of the particle is 6 meters per second.