A block of mass m rests on a plank of mass M connected to a horizontal spring of force constant k. The floor is smooth. Initially the spring is at natural length and both bodies are given the same horizontal speed V. The coefficient of static friction between them is μ. The maximum V for which no slipping occurs throughout the subsequent motion is:
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Question

A block of mass m rests on a plank of mass M connected to a horizontal spring of force constant k. The floor is smooth. Initially the spring is at natural length and both bodies are given the same horizontal speed V. The coefficient of static friction between them is μ. The maximum V for which no slipping occurs throughout the subsequent motion is:

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G1 Physics · Accepted Answer

With no slipping, the pair behaves as a single mass M+m attached to the spring. Hence ω = √[k/(M+m)]. Starting from equilibrium with speed V, the oscillation amplitude is A = V/ω, so maximum acceleration is ω²A = ωV. The upper block needs friction mωV. Requiring mωV ≤ μmg gives V ≤ μg/ω = μg√[(M+m)/k].