Question

if you attach a mass to the spring from its initial equilibrium position, it vibrates forever in simple harmonic motion. Why doesn't it come to rest after stretching by a distance 'd'; proportional to the weight of the mass, when the spring's restoring force cancels out the weight of the mass? How will you measure the equilibrium position? How can you attach the mass to the spring so that it doesn't oscillate when you let go?

Answer 1

When the weight comes to the new equilbrium position, where weight = kd, it will still be moving because the spring pulled it there and inertia keeps it going.

To measure the new equilbrium position, stop the vibration.

To attach the mass with no vibration, do it slowly, lower your hand, and keep the mass in your hand until the spring lifts it off