A block of mass m is attached to a horizontal spring of force constant k and rests on a rough horizontal floor. The coefficients of kinetic and static friction are μₖ and μₛ respectively. Initially the spring is compressed by x₀ and the block is released from rest. Assume the block crosses the natural-length position and reaches its first turning point on the other side. The block will remain permanently at rest at this first turning point if:
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Question

A block of mass m is attached to a horizontal spring of force constant k and rests on a rough horizontal floor. The coefficients of kinetic and static friction are μₖ and μₛ respectively. Initially the spring is compressed by x₀ and the block is released from rest. Assume the block crosses the natural-length position and reaches its first turning point on the other side. The block will remain permanently at rest at this first turning point if:

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G1 Physics · Accepted Answer

The first extension on the opposite side follows from energy with kinetic friction: x₁ = x₀ - 2μₖmg/k. At that turning point the block remains stuck if the spring force can be balanced by static friction: kx₁ ≤ μₛmg. Therefore k[x₀ - 2μₖmg/k] ≤ μₛmg, giving x₀ ≤ (μₛ+2μₖ)mg/k.