How to solve: the system as a pic. determine the magnitude of the force $\vec F$ that must be applied to M so that m remains in a fixed position relative to M ( that means m doesn't move)

Question

loser66 · Answer

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loser66 · Answer

the total force in M is $ F_t = F +F_{Mg} +F_{MN} + F_{mN}$ I am not sure about $F_{mN}$ does it affect M?? @ybarrap

ybarrap · Answer

If F = 0, what happens to block m? It will slide if friction forces are smaller than its tangential force due to its own weight. Should we assume no friction? If so, then we just need to keep up with block m, which will want to slide along the ramp. 

With this assumption (that is, no friction), F just needs to provide sufficient acceleration to accelerate block M at the same rate that block m is accelerating down the ramp but not too much that block m begins to slide back up the ramp. So we need to figure out block m's acceleration down the ramp and make block M accelerate at that same rate.

Does this make sense?

loser66 · Answer

yes.

ybarrap · Answer

Also, we should assume no friction on  the horizontal on which block M sits?

loser66 · Answer

yes, no friction

ybarrap · Answer

I was thinking that if force (F) is constant on bock M then the increase in velocity required to sustain block m would not be possible past the speed of light. In essence, the acceleration required is not sustainable and therefore the Force required is not sustainable once the speed of light is reached. Once speeds get this high, there could be no further acceleration and block m should just slide off.  But then block m would be going faster than the speed of light (because it is sliding ahead of block M, which is also going at the speed of light), which is not possible. Therefore, it is possible to hold block m in place for a finite amount of time, but not forever. 

I'd be interested to hear what others think about this problem.