Excellent answer! Thanks for going to all of the effort for doing the numerical solution and plotting the results.

@user628075 I would recommend accepting this answer.

@Dale This answer seems impossible, how can CoM move in almost straight line in force change orientation when rod rotate all the time?

@user628075 this answer is correct. I confirmed it numerically myself too. Think of a rocket. The change in the momentum of the rocket is the opposite of the momentum of the exhaust. Your rocket pushing on the rod is spraying exhaust nearly equally in every direction. So it is nearly no net force. It is always curving, but it gets asymptotically close to a straight line. The plots are correct here, and very well done

@Dale Does my rod rotate around itself(CoM) during this "straight" path? If net force is almost zero, then rod must stay at place and just rotate. This movement has nothing to do when we have air.

@Jono94 What result/trajectory you get when you include air force?

@user628075 yes, it rotates at very high speed

@Dale But why translate in one direction?

@user628075 I explained why with words in my comment above already and with math in my answer. I am not sure what else I can give you.

@Dale Is this classic physics answer or include relativistic effects because RPM rise toward infinity? Can I say that rod rotate with very high RPM but translate very very slowly?

@user628075 this is purely classical, no relativistic effects are included.

@Dale It is so weird if we just add air, rod completely differently moves..

@user628075 what makes you think that?

@Dale If you add air(drag), then rod just rotate around CoM, zero translation.

@user628075 Yes, this is completely classical, as my equations show. The question asks about the rod in "free space", so that's what I solved for. Solving for the motion with air drag sounds very difficult but interesting (though I don't think it would rotate with zero translation), I'll definitely put it into my to-do list!

@Jono94 In steady state thrust equal air drag, so net force=0, that mean zero translation...

@user628075 said “If you add air(drag), then rod just rotate around CoM, zero translation.” I am very skeptical of that. Of course, it is possible, but I would need to see the math supporting such a claim. 0 force means 0 acceleration, not 0 translation

@Dale I talk only for steady state, if that is not a case, monocopter couldnt fly at same place.

I’m still not totally convinced, but I think it’ll be a good idea to move this discussion into chat? (I don’t know how to do that though)

@Jono94 youtube.com/… Try add air in your simulation, if you get CoM translate then something is wrong..