I am trying to understand angular momentum about the center of mass of a centripetally accelerating object. My scenario has a meter stick being swung horizontally from the end of a rigid rod, somewhat similar to how an Olympic discus thrower spins around to throw the discus. The rod uses a vise clamp to attach to the center of mass of the meter stick, and I am assuming that the meter stick is infinitely thin. There is also a switch that can be pressed to open the clamp and release the meter stick.
My three questions relate to the diagram that I have attached:
Does the concept of angular momentum around the center of mass of the meter stick even make sense, as the center of mass is undergoing centripetal acceleration? I see that the rod is changing its orientation but I don't know if that means that the rod has angular momentum about its center of mass. In this scenario, the center of mass is accelerating, and thus a reference frame moving with the center of mass would be a non-inertial reference frame.
If there is angular momentum about the center of mass of the meter stick, how was the angular momentum generated? My intuition is that no torque around the center of mass is generated throughout the swing since the radius arm would be zero, so I don't understand how the angular momentum could have gotten started.
If the rod is allowed to detach from the meter stick, will the meter stick spin about its center of mass in its now straight-line trajectory?