A block of mass m rests on a plank of mass M. The plank is attached to a horizontal spring of force constant k and lies on a smooth floor. The coefficient of static friction between block and plank is μ. The system is displaced horizontally by amplitude A and released. The maximum A for which no slipping occurs is:
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Question

A block of mass m rests on a plank of mass M. The plank is attached to a horizontal spring of force constant k and lies on a smooth floor. The coefficient of static friction between block and plank is μ. The system is displaced horizontally by amplitude A and released. The maximum A for which no slipping occurs is:

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G1 Physics · Accepted Answer

If there is no slipping, the two bodies move together as mass M+m. Thus ω² = k/(M+m). The upper block requires friction f = m|a|, whose maximum value occurs at the extreme position: f_req,max = mω²A = mkA/(M+m). For no slip, mkA/(M+m) ≤ μmg. Therefore A_max = μ(M+m)g/k.