Conservation of angular momentum of system of two objects

Question

This may be a silly question, but I just want to clarify something.

Say we have two spinning spheres, each spinning about an axis going through their own center of mass:

All of the conservation of angular momentum problems I have done have been simple problems where the two objects spin about the same axis. How would you find the angular momentum of this system?

Do you find the angular momentum of the system by using an axis going through the center of mass of the system as your reference axis and use the parallel axis theorem to find the new moments of inertia of the spheres? Can you calculate the angular momentum by considering either of the axis going through a sphere as the reference axis and then use the parallel axis theorem on the other sphere to calculate its new moment of inertia? Can you pick any random axis and use the parallel axis theorem? Or do you simply add the angular momentum of the obects about the axis they are rotating?

Answer 1

"Say we have two spinning spheres, each spinning about an axis going through their own centre of mass.
I have highlighted a very important statement in your question.

If you have a body rotating about its centre of mass then if the centre of mass does not move the angular momentum of the body will be the same about any point.

So the angular momentum of your two spheres which are rotating about their respective centres of mass is just the vector sum of their individual angular momentums about their centre of mass.

Choosing an axis through the centre of mass of the two sphere does not change the total angular momentum provided that the centres of mass of the two spheres do not move.