Equation of motion and normal modes of a coupled oscillator

[VapeL]

Homework Statement

Find the equation of motion of a mass on a spring

Relevant Equations

x(t) = Acos(ωt + φ)

This is a question from an exercise I don't have the answers to.

I have been trying to figure this out for a long time and don't know what to do after writing

mx''¨(t)=−kx(t)+mg
I figure that the frequency ω=√(k/m) since the mg term is constant and the kx term is the only term that changes.

I have the equation of motion written as x(t) = Acos(ωt + φ) but do not know how to go about finding the amplitude either.

Any help would be great thanks!

 

[S.G. Janssens]

It looks like this is a simple (not coupled) oscillator.

It also looks like you have this backwards, in a sense. The equation of motion is not the expression for $x$ as a function of time, but rather it is Newton's second law, involving gravity and the restoring force that the spring exerts on the mass. In other words: It is a differential equation.