newtonian mechanics - Rolling (without slipping) ball on a moving surface

I've been looking at examples of a ball rolling without slipping down an inclined surface. What happens if the incline angle changes as the ball is rolling?

More precisely I've been trying to find equations for programming a simulated (2D) ball rolling inside a swinging bowl/arc.

I thought I could still use the same equations for just having a ball rolling inside a still bowl (see below) and the changes in the incline angle (tangent at the contact point of the ball and the bowl) would take care of itself:

For a ball rolling inside a bowl: The only torque acting on the ball is the frictional force: $τ=Iα=fr$, using the rolling without slipping condition $a=rα$ and the moment of inertia for a solid sphere, $I = \frac{2}5 mR^2$, we get $f=\frac{2}5ma$. The net force acting on the system is gravity and the force of friction, $F=ma=mgsinθ−f$ and therefore, $a=\frac{5}7gsinθ$

I am speculating that due to the surface itself moving (swinging on a circular path), it's the relative motion that contributes to the friction? But, I don't know how to include that. Can someone please help me?