Board and Spool

Suppose that a cylinder is rotating about a point A with an angular speed $\omega$ and use the fact that an angular speed is the same for all points of the rigid body. 

There is another way of solving this problem. Look at the motion of spool as two seperate motions: the spool moves horizontally as its all mass is at point B and rotates about the point B.

The angular speed is the rate of rotation $\omega = \frac{\delta \Theta}{\delta t}$.