Simple Harmonic Motion Clarification

1. The problem statement, all variables and given/known data

I was wanting to get some clarification on some of the simple harmonic motion equations. So, say for example there is a box of mass "m" undergoing simple harmonic motion attached to a spring of spring constant K on a horizontal surface. To find where the box is, as a function of time, I would use the following equation:


My first question is, will the "x" of this equation be the displacement from the maximum extension point, or the equilibrium point?

Second question: In regards to the phase constant "ø", if at t=0, x=A, then ø would have to be 0, correct? and this is because x(0)=Acos(ø)–>A=Acos(ø), cos(ø)=1 ∴ ø=0

Third question: I have seen in many place a formula used for the amplitude:

A = sqrt((x0)^2 + (v0/ω)^2)

I’m not sure how this is derived. I know it has to do with the velocity function: -Aωsin(ωt + φ)
It just says to square both sides and simplify, but I still don’t see how that would give the amplitude equation…



*EDIT* I believe I figured that last part out, simply by rearranging some terms.

So, I could use the equation
x(t)=Acos(ωt+ø) to get the displacement from equilibrium for ANYTHING undergoing SHM, and the associated velocity and acceleration functions to get the velocity or acceleration of ANYTHING undergoing simple harmonic motion, i.e mass on a spring etc.

Thanks 🙂

2. Relevant equations

See above

3. The attempt at a solution

See above

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