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?

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

The reason the mass attached to the spring does not come to rest at a stretched position even when the restoring force cancels out the weight of the mass is due to the concept of inertia. Inertia is the tendency of an object to resist changes in its state of motion. Once the mass is set into motion, it will continue to move back and forth due to its inertia, even if the restoring force and weight cancel each other out.

To measure the equilibrium position of the spring, you can use a ruler or a measuring tape. Start by holding the spring vertically with the bottom end fixed in place. Attach the mass to the spring but ensure it is not touching the ground or any other surface. Now, measure the distance from the fixed end of the spring to the bottom of the mass. This distance will represent the equilibrium position.

To attach the mass to the spring so that it does not oscillate when you let go, you can use a small clamp or a hook. First, hang the spring from a sturdy support or stand, making sure it is vertical. Then, attach the clamp or hook to the bottom end of the spring. Carefully attach the mass to the clamp or hook, ensuring that it is securely fastened. This setup will prevent the mass from moving when you let go, allowing you to observe the equilibrium position without any initial oscillations.

The reason the mass attached to the spring does not come to rest after stretching by a distance 'd' (proportional to the weight of the mass) even when the spring's restoring force cancels out the weight of the mass is due to inertia.

In simple harmonic motion, the restoring force from the spring is proportional to the displacement from the equilibrium position. As the mass is displaced from its initial equilibrium position, the spring generates a restoring force that opposes the displacement. However, when the mass reaches the furthest point in its oscillation, the spring's restoring force is at its maximum while the velocity of the mass is at its minimum (or zero). The inertia of the mass causes it to momentarily overshoot the equilibrium position, resulting in it oscillating back and forth around the equilibrium position instead of coming to rest immediately.

To measure the equilibrium position of the spring, you can use a reference point such as a ruler or a marked scale. Place the spring vertically or horizontally, ensure it is at rest, and note the position where it remains stable without any additional forces acting on it.

To attach the mass to the spring so that it doesn't oscillate when you let go, you can use a clamping mechanism or a support structure. This can be done by securing the mass to the bottom of the spring using a clamp or by fixing the top end of the spring to a rigid support. This way, the mass will be held in place and won't experience any initial displacement that triggers oscillation when released.