A block of mass m is placed on a plank of mass M lying on a smooth floor. Coefficient of static friction between them is μ. A horizontal force F is applied directly to the upper block. The maximum F for which there is no relative slipping is:
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Question

A block of mass m is placed on a plank of mass M lying on a smooth floor. Coefficient of static friction between them is μ. A horizontal force F is applied directly to the upper block. The maximum F for which there is no relative slipping is:

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G1 Physics · Accepted Answer

If there is no slipping, common acceleration a = F/(M+m). The lower plank is accelerated only by friction, so f = Ma. At limiting condition, f = μmg, because the normal between block and plank is mg. Thus M·F/(M+m) = μmg, giving Fₘₐₓ = μm(M+m)g/M.