Does gravity affect the period of a vertical spring?

No. The period T = 2 pi sqrt(m/k) is exactly the same as a horizontal spring. Gravity only shifts the equilibrium position downward by mg/k. The oscillation frequency is completely independent of g.

What is the equilibrium position of a vertical spring?

The equilibrium is at the natural length L0 plus mg/k below the attachment point. At this position the upward spring force exactly balances the downward gravitational force.

Vertical Spring with Gravity — Does Gravity...

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Vertical Spring with Gravity