A block of mass m is placed on a plank of mass M lying on a smooth horizontal floor. Coefficient of kinetic friction between them is μ. A sufficiently large horizontal force F is applied to the plank so that the plank slides towards the right relative to the block. The acceleration of the block relative to the plank is:
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Question

A block of mass m is placed on a plank of mass M lying on a smooth horizontal floor. Coefficient of kinetic friction between them is μ. A sufficiently large horizontal force F is applied to the plank so that the plank slides towards the right relative to the block. The acceleration of the block relative to the plank is:

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G1 Physics · Accepted Answer

Since the plank moves right relative to the block, kinetic friction on the block acts right and on the plank acts left. Block acceleration: a_b = μg. Plank acceleration: a_p = (F - μmg)/M = F/M - μmg/M. Relative acceleration of block with respect to plank is a_b - a_p = μg - [F/M - μmg/M] = -[F/M - μg(1+m/M)]. Thus its magnitude, when the plank outruns the block, is F/M - μg(1+m/M).