Since the "X" is accelerated in the frame you are using, and the "X" is painted on the belt, the BELT is accelerated in the frame. There is no way out of that for you.
Now, since the belt is a continuous loop, and it undergoes acceleration in some places, how can it not undergo acceleration in all places? Is it because the acceleration takes place only at the ends, where it goes around the cylinders?
The motor is only at one end of the belt, yet when it accelerates the belt, the entire belt must accelerate, or do you not agree with this?
Since the ends of the belt going around the cylinders undergo acceleration, that acceleration must be transmitted along the entire belt, in the same way the motor transmits motion along the entire belt.
There is no way you can talk around this with your babble! You cannot isolate the top of the belt from the forces acting on the belt elsewhere. You cannot "attach" an IFOR to a belt that is rotating in that frame, as the "X" would definitely be doing. You cannot base a FOR on a projection. Give it up; the treadmill is dead.
Sorry PPG, you are still confused.
For the purpose of discussion let's assume the following are the axis of our reference frame:
x axis=origin is associated with the location with the first pass of one of your X's (once the TM is rolling, it is only valid for one rotation. From the point of view of the IFOR, when the X appears again, it will be effectively a different X moved the length of the belt in the IFOR. +x direction is up TM. (If you don't get this, ask or give it some thought!)
ETA: I see that Jadebox just expressed the same idea in the post above about the painted "X".
y axis=origin-the center of the belt's upper surface. +y direction is away from the observer on the right side of the TM with upTM being in the direction of the observer's right hand.
z axis=origin-the upper surface of the belt TM. +z direction is towards the ceiling.
The IFOR is in space as a mathematical entity moving (smoothly translating only in the direction of down TM at belt speed wrt earth) It's not attached to the belt.
Now, this is definitive. While smoothly operating, the belt's upper and lower surface is unaccelerated and when the X appears there, it is unaccelerated in both the IFOR just defined and an IFOR indexed to the earth. The part of the belt that moves around the roller is accelerated in the x and z direction as it moves around the roller. (x because it changes direction and z because it goes from top to bottom on one end and bottom to top on the other.) The top and bottom of the belt in a smoothly running TM are never accelerated in the x axis. The belt is only operating under a constant tension in these areas.
Any object that remains stationary (wrt earth-like the cart) on the top of the TM is unaccelerated (don't confuse this with not moving) in the IFOR that has been defined above and one indexed to the earth. Why do we know this: because no false forces exist in either IFOR defined and they are smoothly translating wrt to each other. Galilean relativity holds and physics is completely valid in both.
The TM is alive and well (figuratively, of course.)!
PS, just because you can't understand a clear explanation, that the other posters do, does not make it babble (a derogatory term which you say you are never guilty of!) simply because you don't understand. All IFORs are based on projections of their axis into three space. I'm sorry you are having a problem understanding this concept but it is a fact.
WindGrins
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