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.
This is the same classics mechanics kinematic problem. The difference is gravity force substituted by force which come from platform acceleration. The solution is very close to rolling body along incline.
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?
The spring doesn't push rolling object to theirs center of mass. Therefore the force of spring should no be fully spend to object translational motion. The spring force push platforms with rolling objects. Here's different case.
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.
Please look on solution of problem a rolling body along incline. There is just operate with forces which should equate to projection of gravity force. One oh them is translation motion force. Another came from torque with radius multiplication.
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.
What do you mean is not rotating? The spring is connecting to platforms. These platforms have acceleration and wheel is free moving on them. Both wheels have reverse motion relatively to platforms.
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.
You're correct with your model. However, the wheels have a free move on platforms. It means part of spring force with spend to translational motion of rolling object. Another part of force will spend to rotate this object which equate to torque and radius of object multiplication.
Hmm I kind of got a bit lost, sorry.
It's fine
It's hard to understand from first look
P.S.
Thank you for you meaningful answers. I know a lot of forums where opponents know physic but don't understand that. This forum is complete different. Thank you again.
Alex
