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. The coefficient of kinetic friction is μₖ and tanθ > μₖ. The first maximum extension of the spring is:
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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. The coefficient of kinetic friction is μₖ and tanθ > μₖ. The first maximum extension of the spring is:

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G1 Physics · Accepted Answer

During downward displacement x, gravity does work mg sinθ·x and friction removes μₖmg cosθ·x. At the first maximum extension the speed is zero, so mg(sinθ-μₖcosθ)x = ½kx². Hence x_max = 2mg(sinθ-μₖcosθ)/k.