# Angular velocity in dimensional analysis

Hullo was wondering if anyone could help me. In dimensional analysis using then buckingham pi theorum, i'm not sure how to express an angular velocity in terms of basic dimensions (i.e M (mass), L (length), T(time), $$\Theta$$ (temp).

I know an angular velocity is revs/s or rad/s so its going to be 'whatever radians/ revs are in basic dimensions * T^(-1)

But I'm not sure how to express revs/rads in basic dimensions? Are they just dimensionless?

Yes. Radian is a dimensionless measure. If there is a circle, and a small part of it's circumference [an arc] subtends an angle $\theta$ radians at the center, then $\theta$ is the ratio of the length of the arc to the radius of that circle. As you can see, a radian is a ratio of two lengths: the arc length and the radius and hence, it is dimensionless as the length dimension of both these quantities cancel each other out.
The 'rad' in 'rad/s' is there to denote that we specifically mean a unit of angular velocity, to seperate it from '/s', which is a unit of frequency. However, angular velocity is also the angular frequency of rotation in a uniform circular motion i.e. it differs from the no. of rotations by a factor of $2\pi$.
As far as revolution is considered, n revolutions just mean $n \times 2\pi$ radians. Even then, revolutions are counted in pure basic numbers. And as you know, there is nothing like a 'kilo numbers' or a 'mega numbers'.. hence revolutions is also a dimensionless unit.