Circle Geometry Formula

Two circles are joined together, creating a new circle. Those two circles move together in time, creating an angle $delta theta (t)$. I would like to get the length L (mentioned in the photo, with the red rectangular) in the function of each circle radius, time, and speed of rotation v and/or delta theta.
Also, the delta_theta, in the function of time, speed of rotation, and radius of the circle and/or length of Arc.
You will find the photo, explaining the concept.