I'm having troubles to understand the difference between kinetic energy and potential centrifugal energy in some situations. I make an example where I am confused.
A object moves on a rod attached to a spring (rest lenght $l$), the rod is inclined ad an angle $\alpha$ and rotates at constant angular velocity $\omega$. Initially the spring is compressed of a lenght $\delta$. Find the maximum elongation of the spring.
Here is what I have tried. I used the rotating rod frame, where potential centrifugal energy has to be considered.
$mg[(l-\delta) Cos(\alpha)]+ \frac{1}{2} k (\delta)^2 -\frac{1}{2} m\omega^2 [(l-\delta) Sin(\alpha)]^2 = mg[(l+x) Cos(\alpha)] + \frac{1}{2} k x^2 - \frac{1}{2} m\omega^2 [(l+x)Sin(\alpha)]^2$
But this does not lead to the correct result. Besides my possible mistakes the unclear thing here is that, if I use a steady inertial frame instead, I do not have potential centrifugal energy (which is negative), but kinetic energy or rotational energy, which is exactly the same but positive and the result would be different of course. How can that be? Are these two ways to solve the problem equivalent?
And in general what is the difference between potential centrifugal energy and rotational/kinetic energy?