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My questions:

1. How can we prove that we can use the equation w=(k/m)^(1/2) for this problem. Normally, you can just go:

ma=-ky

a=-k/m * y

a=-w^2*y

y's cancel out

w=(k/m)^(1/2)

but in this case you should have to account for the mg force, but in most solutions, I do not seem mg accounted for?

In one problem, I was asked to solve for the maximum amplitude the shm could have in order to not surpass a certain acceleration. Once again, all answers were along the lines:

ma=-kA

mg=-kA

A=-mg/k

Once again, how can you neglect the mg force?

My only idea is that since we determine the equilibrium point for most of these problems at the beginning - the point where the spring force matches the gravitational force - that they treat this equilibrium point like the spring's equilibrium point and can somehow, magically, neglect the spring force?