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v = omega r

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The equation $v = \omega r$ is a fundamental relationship in circular motion.

The formula to calculate the **linear speed** expressed in meters per second [ $m.s^{-1}$ ] from the **angular velocity** expressed in radians per second [ $rad.s^{-1}$ ] is:

$v = r \omega$

where:

- $v$ is the linear (tangential) velocity expressed in meters per second ( $m.s^{-1}$ )
- $\omega$ (Greek letter omega) is the angular velocity expressed in radians per second ( $rad.s^{-1}$ )
- $r$ is the radius expressed in meters ( $m$ )

This formula expresses the connection between the rotational motion (angular velocity) and the linear motion (tangential velocity) of an object. It tells us that the linear velocity is directly proportional to both the angular velocity and the radius of the circle.

You'll find on this page an online converter from angular to linear velocity.

**For example**, if you know the angular velocity of a rotating object and the radius of its path, you can calculate how fast a point on the object is moving through space using this formula.

Let's consider a vehicle with wheels having a radius of 10 centimetres. If the rotational velocity of the wheels is 20 rad/s, the linear speed of the vehicle is given by:

$v = r \omega = 1.0 \times 20 = 2~m.s^{-1}$