Find x(t) and y(t) then plot x vs y.
You did not mention friction so we assume there is none. So the equations of motion could be written
The forces include the inward spring force and the centrifugal force in this rotating frame of reference.
The spring force and centrifugal force are radial. Therefore, rather than trying to develop the equations above, I would express both forces in terms of r, and solve the differential equation for r(t).
where %20=%20f_spring(r)%20+%20f_centrifugal(r).png)
You did not say if the angular velocity is constant. The angular velocity versus time will determine what the centrifugal force looks like and will determine theta versus time. Once you know r(t) and theta(t), convert to x(t) and y(t) in the standard way. If the angular velocity is constant then it is pretty easy. If the initial angle is not specified, you might as well assume angle theta=0 at t=0.
That should be enough hints to get you going.
