Friction between two bodies (theoretical situation)

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When a force greater than the static friction between body B and the floor, and also greater than the static friction between body A and B, but less than their sum, is applied to B, it will begin to move. Once B starts moving, the situation shifts to kinetic friction, which is typically lower than static friction. The net force acting on B will be the applied force minus the kinetic friction force. If this net force exceeds the static friction between A and B, then A will also begin to move backwards. Therefore, the relationship between static and kinetic friction is crucial in determining the motion of both bodies.
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What happens if we apply on B a force bigger than the static friction with the floor and also bigger than the static friction with the upper body A, but smaller than their sum? Will B move? what about A?
 

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You are applying the force to B? Then, since the force is greater than the static friction between B and the floor, B will start to move. Once it has started to move, the problem reverts to "kinetic friction". The net force on B will be the total force minus the kinetic friction force. If that is larger than the static friction between A and B, then A will slip backwards. Static friction is not, in general the same as kinetic friction so just knowing that the force is greater than the sum of the two static frictions is not enough. It is true that, for most materials, the kinetic friction is less than the static friction. In that case, if force> static friction + static friction> kinetic friction + static friction, yes, A will move.
 

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