A block of mass m rests on a plank of mass M. The plank lies on a smooth horizontal floor. Coefficient of static friction between block and plank is μ. A horizontal force F is applied to the plank. The maximum value of F for which the block does not slip is:
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Question

A block of mass m rests on a plank of mass M. The plank lies on a smooth horizontal floor. Coefficient of static friction between block and plank is μ. A horizontal force F is applied to the plank. The maximum value of F for which the block does not slip is:

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G1 Physics · Accepted Answer

For no slipping, both bodies have common acceleration a = F/(M+m). The upper block is accelerated only by friction, so required friction f = ma. Since f ≤ μmg, ma ≤ μmg ⇒ a ≤ μg. Therefore F/(M+m) ≤ μg, giving Fₘₐₓ = μ(M+m)g.