It's quite a different problem. However the entire force of gravity is applied to the roller, then there are reaction forces pushing it other ways too.
I treat your roller on the platform as having a horizontal force applied to its center (equal to spring force), and additionally a moment applied about its center. That's fine. You can do that. Who says you can't?
I don't need to do it, because it's simple wheel, which has standard equation. Kind of flywheel.
You need it if you're summing angular momentums to apply the law of conservation of momentum. When you do that you have to include the angular momentum of every part, all measured about the _same_ axis. The simple flywheel formula only applies to angular momentum about the wheel's own axis. Even tho it's not rotating it still has angular momentum about the other wheel's center, and it has a different rotational inertia about that axis too.
I think this is a crucial part which needs to be incorporated. You can't just ignore the angular momentum of wheel 1 because it's not rotating.
I disagree, because just part Fs is using for translational motion. It would be
Imagine you break the connection between wheel 2 and the platform. Then install a lever fixed to the center of the wheel, and its other end is pin-jointed to the platform. From this it's obvious that the entire force is transmitted to the center of the wheel - there's nowhere else it can go. It also shows there's an additional moment applied about the center of the wheel.
Hmm I kind of got a bit lost, sorry.
