Suppose I have the system on the attached file, where two slender rods collide. I want to calculate the angular momentum of the system before collision relative to the center of mass of the two rods. I'm however not sure how to do this. Definately the formula L=Iw doesn't apply, since I don't know from which point I should measure the angular velocity. Equivalently I don't know where what R to use if I use the formula: L = mvR. It seems intuitive if it was the center of mass, but why it is, I have no idea. Who can explain what R to use and why?
You can always treat the total angular momentum of a system as composed of the angular momentum due to the motion of its center of mass plus the angular momentum about its center of mass. If you wish to use the center of mass as your origin for computing angular momentum, just transform your first diagram to a frame in which the center of mass is at rest. Since neither rod is rotating (at first), all you have to worry about is the motion of each rod's center of mass.