OK, if you understand...

Here's how it would go from the viewpoint of a rotating reference frame:

The centrifugal "force" and the gravity force put opposing torques on the ball about an axis passing through the point of contact where the ball will roll out.  The gravity holds the ball in the hole until the centrifugal "force" is large enough to overcome the torque due to gravity.

The moment arm of gravity is the radius of the hole; the moment arm of the centrifugal "force" is the height of the center of the ball above the plane of the hole.  The ball lifts out of the hole when the torques are equal; that is, when:

mg x (radius of the hole) = mw2r x (height of center).

That makes w2 equal to: (g/r) x (radius of hole)/(height of center).

The fraction, radius of hole divided by the height of center, is the tangent of the angle q.

That gives the answer for the "fleeing" value of w:
 

As the angular velocity, w, increases, the direction of the force exerted on the ball by the edge of the hole moves from vertical toward the horizontal.  The ball rolls out of the hole when that line passes through the center of the ball.  That value of w is wflee.

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