I'm flying, turning in a stable orbit, i.e. at constant level with a constant angle of bank, at constant airspeed, with a constant radius of turn, as in the picture here (sorry it's my first post, cannot upload pictures).
I am flying along the black curved line. The lift is directed as the blue axis (orthogonal to the wings). The vertical component of the lift compensate exactly the gravity, and the horizontal component of the lift is responsible for the turn (forget the component in the direction of movement).
Now consider the inclinometer (aka the ball aka balance indicator aka slip indicator). The position of the ball is supposed to tell me whether my turn is "coordinated" or not. If the ball is centered the turn is coordinated, otherwise the normal practice is to use rudders ("step on the ball") in order to center the ball and obtain a "coordinated turn".
Since the orbit is stable, the ball is stationary with respect to me, so the ball must be subject to the same exact acceleration I feel. Consider the frame of reference attached to the airplane, as in the picture. In this non-inertial frame of reference the forces I experience are the gravity, the lift, and the (apparent) centrifugal force. Since neither me nor the ball are moving with respect to the reference frame, all these forces must be balanced. In my case the lift is exercised on me by the seat and in the case of the ball by the wall of the cage/vial in which it is contained. In both cases the force exercised on the bodies to compensate gravity and centrifugal force must be in the same exact direction and strength of the lift. This can only happen for the ball when the ball is at the bottom of the curved vial, because only then the wall of the vial is normal to the vertical direction (blue axis).
So my question is: how can the ball ever be "out of the cage"? And how can the use of rudder impact on the ball position if the airplane yaws by rotating around the blue axis (i.e. around the direction of the lift)?