The aspect which bugs me about this concept, is the inevitability of end-over-end tumbling. Knife throwers and axe throwers have to factor-in this phenomenon in order to ensure that the knife/axe hits the target "right" (or it will just bounce off). Such tumbling is not a problem for a hammer thrower, but if you watch a hammer-throw, you will see that the hammer continues to spin after it is released - and it is still gently spinning when it lands.
When SpinLaunch is winding-up to throw. the rotor-arm with the attached projectile build up an angular momentum ("Lrp" - say ). This surely can be split into the angular momentum of the rotor-arm ("Lr" - say) and the angular momentum of the projectile ("Lp" - say). Hence Lrp = Lr + Lp.
According to the laws of mechanics, angular momentum must be conserved. So, at the moment of release, the angular momentum of the projectile (Lp) does not suddenly become zero. This must surely mean that the projectile will continue to rotate - i.e. it will tumble end-over-end - in the same plane as the SpinLaunch rotor.
Let's go back to consider the revolution BEFORE the moment of release. At one whole revolution before release the projectile will be pointing "up". At 3/4 of a revolution before release, the projectile will be pointing "left". At 1/2 a revolution before release, the projectile will be pointing "down". At 1/4 of a revolution before release, the projectile will be pointing "right". For completeness, let's just remind ourselves, at the moment of release, the projectile will be pointing "up" again.
But it will CONTINUE TO ROTATE — even after it is released!!!
So, after it has travelled a distance of 1/4.Pi.2.radii, it will be pointing "left" again - i.e. it will be travelling sideways - not good aerodynamically!
And, after it has travelled a distance of 1/2.Pi.2.radii, it will be pointing "down" again - i.e. it will be travelling backwards!!!
Again, after it has travelled a distance of 3/4.Pi.2.radii, it will be pointing "right" again - i.e. travelling sideways again.
If the projectile had large tail fins like a playing dart (see link below — which it doesn't) it would merely tail-wobble violently without actually tumbling, before eventually stabilising. (I have actually tried this in my misspent youth)
https://cdn.britannica.com/76/125876-050-48F1FD76/Darts-dartboard.jpg
