If we start taking into account the free body diagram, we then start damping motion of a body by means of this equation:
m d^2 x/dt^2 + c dx/dt + k x = 0
free undamped motion is when (c =0)
therefore
50 d^2 x/dt^2 + 200 x = 0
w = omega = sqrt(k/m) = sqrt(200/50) = 2
Then
Acos(wt) + Bsin(w*t) = x
find A and B using intial conditions and you will get the rest that you need. Hope this helps
