A block of mass m is attached to a spring of force constant k along a rough incline of angle θ. Initially the spring is unstretched and the block is released downward from rest. The coefficients of kinetic and static friction are μₖ and μₛ respectively, and tanθ > μₖ. After reaching its first maximum extension, the block will start moving back upward only if:
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Question

A block of mass m is attached to a spring of force constant k along a rough incline of angle θ. Initially the spring is unstretched and the block is released downward from rest. The coefficients of kinetic and static friction are μₖ and μₛ respectively, and tanθ > μₖ. After reaching its first maximum extension, the block will start moving back upward only if:

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G1 Physics · Accepted Answer

During the initial downward motion, energy gives the first maximum extension x₁ = 2mg(sinθ-μₖcosθ)/k. To start moving upward from that point, spring force must exceed the downhill gravity component plus maximum downhill static friction: kx₁ > mg sinθ + μₛmg cosθ. Substitution gives tanθ > 2μₖ+μₛ.