This is a question from a lab. I\'m not sure how to go about it, any help would

Question

This is a question from a lab. I'm not sure how to go about it, any help would be great.

In the introduction, we claimed that the gravitational potential energy could be ignored if the displacement used in the elastic potential energy was measured from the hanging equilibrium position. First write the total mechanical energy (kinetic, gravitational potential, and elastic potential energy) in terms of a coordinate system, position measured upward and labeled y, whose origin is located at the bottom of the relaxed spring of constant k (no force applied). Then determine the equilibrium position s when a mass m is suspended from the spring. This will be the new origin for a coordinate system with distance labeled h. Write a new expression for total energy in terms of h. Show that when the energy is written in terms of h rather than y, the gravitational potential energy cancels out.

Explanation / Answer

According to the given problem,

E = 1/2 m v2 + 1/2 k y2 + mg y

Equilibrium: mg = -ky

y = -mg/k

s = y + mg/k

E = 1/2 m v2 + 1/2 k (s-mg/k)2 + mg(s-mg/k)

= 1/2 mv2 + 1/2 ks2 + constant.

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