Unit of $v$ in $v=rω$

Question

unit of velocity is m/sec but in $v=rω$ unit of $r$ is metre and unit of $ω$ is radian/sec so unit of $v$ should be radian*metre/sec but it is m/sec how I know this is dimensionally correct but I dont understand the unit

Answer 1

The unit radian is dimensionless.
1 radian = 1 m/m = 1.
So the unit of $\omega$ is 1/s. And the unit of $v=r\omega$ is m/s, as it should be.

Answer 2

Thomas Fritsch's answer is completely correct. I thought it might also be nice to give a quick derivation of $v=r\omega$ to show the unit isn't mysterious.

For a dot moving on the rim of the circle, we can calculate the velocity by taking the circumference of the circle $C$ and dividing by the time it takes to do one revolution, $T$ \begin{equation} v = \frac{C}{T} \end{equation} The units here are hopefully clear; circumference has a unit of length, and $T$ has a unit of time.

Then multiply and divide by $2\pi$ \begin{equation} v = \frac{C}{2\pi} \times \frac{2\pi}{T} \end{equation} and use the following facts \begin{equation} \frac{C}{2\pi} = r \end{equation} where $r$ is the radius of the circle, and \begin{equation} \frac{2\pi}{T} = \omega \end{equation} The left hand side has units of 1/time. So does the right hand side. We often add the word "radians" when talking about $\omega$ to make it clear we are talking about an angular quantity -- it is helpful for humans doing physics to know when something is an angle. But, radians are fundamentally dimensionless, so it is correct (from the point of view of dimensional analysis) to ignore them.

Combining these facts yields \begin{equation} v=r\omega \end{equation}