A block of mass m rests on a plank of mass M on a smooth horizontal floor. The upper block, not the plank, is attached to a horizontal spring of force constant k. The coefficient of static friction between the two blocks is μ. The maximum oscillation amplitude for which the blocks move together without slipping is:
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Question

A block of mass m rests on a plank of mass M on a smooth horizontal floor. The upper block, not the plank, is attached to a horizontal spring of force constant k. The coefficient of static friction between the two blocks is μ. The maximum oscillation amplitude for which the blocks move together without slipping is:

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G1 Physics · Accepted Answer

For common motion, ω² = k/(M+m). The lower plank is accelerated only by friction, so f r e q f_req f r ​ e q = Mω²|x|. At the extreme position, f r e q , m a x f_req,max f r ​ e q , ma x = MkA/(M+m). Limiting static friction is μmg. Thus MkA/(M+m) ≤ μmg, giving A m a x A_max A m ​ a x = μmg(M+m)/(Mk).