This answer appears to not explain how the slip-skid ball can be centered in a normal steady-state "coordinated" turn at a constant airspeed and turn rate. The only unbalanced force-- i.e. the net force-- and net acceleration -- is the centripetal one-- toward the center of the turn. And this does indeed contain a lateral component in the aircraft's reference frame. Yet the slip-skid ball is centered. Another answer has observed that the net AERODYNAMIC force vector-- as opposed to the NET force vector-- does not contain a lateral component in this situation.

@quiet flyer BANKED TURN sort of sums it up. The ball is centered because the Net vector of lateral acceleration from yaw and gravity. Notice that the "ball" works best in shallow turns and level flight. Very steep, "bank and yank" turns pull the plane through the earth referenced yaw plane with the elevator. Rudder and elevator swap functions. Remember, a constant speed turn is not steady state, it is constantly accelerating in the direction of the earth referenced yaw plane (for a level turn). The combination of gravity and yaw produce the direction and magnitude of G force.

I know a constant turn involves an acceleration. Yaw is not a force. Not true that the ball doesn't work well in a steep turn.

If you keep trying to say that the ball responds to ALL forces including gravity, pretty soon you'll find you need to drag "centrifugal force" into your explanation, like the FAA does, in which case you'll find it has no explanatory power at all. For any body moving on any trajectory, the sum of the real forces plus the apparent "centrifugal" forces is ALWAYS zero, so this tells us nothing.

"The combination of gravity and yaw produce the direction and magnitude of G force"-- no, the felt G-force is simply the mirror image of the real aerodynamic force produced by the plane. That's all it is. Maybe grounds for another ASE question-- "Is the felt G-force simply the reflection of the real aerodynamic force produced by the aircraft, or is it due to the sum of the real aerodyamic force plus gravity?"

You are correct to discuss centripetal force, which (by accelerating one way) produces G force in the other direction, which is where the ball tries to go. In a very steep turn (90 degrees) there is no plane reference lateral acceleration. The elevator pulls the plane through the earth referenced yaw plane. Hence, the ball can be towards the ground without issue.

But you seem confused about force aerodynamic acceleration forces and steady state. Your work is very strong, with much glider experience. In powered flight, it is a bit different, as thrust can produce acceleration over a period of time without altitude loss (a level turn). Constant acceleration?? Ha, ha. This stuff is good.

"Hence, the ball can be towards the ground without issue."-- if the ball is deflected toward the ground, the plane is sideslipping, no matter how steep the turn. The airflow is striking the side of the fuselage, making an aerodynamic force that is the real cause of the deflection of the ball.

No, I'm not confused. Maybe "steady state" wasn't the ideal phrase, maybe I should have just said "stabilized". Certainly I'm well aware that there is a net acceleration component in a turn. Looking for a simple word or phrase to describe a turn with constant airspeed, bank angle, rotation rates, etc. Thanks for the suggestion, I'll edit my recent answers to change "steady state" to "stabilized". Unless you can suggest some better word or phrase. Does that remove all your objections?

Why does the term "centrifugal force" carry such an apparent stigma? It explains the workings of the balance ball PERFECTLY. Don't overthink things - you guys are talking in circles about a simple concept that a 5 year old would understand if you gave him/her a bucket of water and had them perform a brief demo of the phenomenon. Seriously.

@Michael Hall "Centrifugal force" is the NET G force vector of gravity and lateral acceleration. Thats it. The inclinometer ball falls to the "lowest" point in the curved tube. I'm sure my esteemed colleague will realize this once he does a 1 G loop and finds himself "weightless" at the top. Gravity cannot be ignored, and inclinometer only measures acceleration and gravity in the lateral plane. If the aircraft banks into a side slip, wind holding straight ground track, the ball does not tell you "I'm in a slip", it only tells you "0 lateral acceleration forces, earth is that way".

Please, save your fingertips... I get it!

@MichaelHall -- if the 5-year-old understands, then they are doing better than the FAA. The CURRENT "Airplane Flying Handbook" has been improved, but I've seen plenty of past written test question had very flawed depictions of the "forces" acting on the slip-skid ball. The concept of apparent "centrifugal force" can explain how forces generated by the aircraft get transferred to a deflection of the slip-skid ball, but if you don't understand that it's the lateral component of the aerodynamic force generated by the aircraft that is ultimately running the show, then you don't understand...

@MichaelHall -- The concept of apparent centrifugal force can explain how forces generated by the aircraft get transferred to a deflection of the slip-skid ball, but if you don't understand that it's the lateral component of the aerodynamic force generated by the aircraft that is ultimately running the show, then you don't understand anything about what is really going on. I have yet to see the lateral aerodynamic force (sideforce) -- the real physical force of the airflow hitting the side of the aircraft-- be mentioned in any FAA ground school material or written test question.

@MichaelHall -- instead we read useless information such as "the turn rate is wrong for the bank angle", which sheds no light on anything that is really going on. That's why my hang glider buddies think they'll slip sideways through the air if they don't adequately load up the wing with a nose-up pitch input as they bank. Not actually true.

So we "recenter the ball" by a)rolling to wings level and centering rudder. b) moving rudder past center in the direction of the roll c) either d) neither ?

@MichaelHall-- only when upward or downward acceleration (curvature) is constrained to be zero does the concept of the slip-skid ball indicating whether the turn rate is "too high" or "too low" for the bank angle make any sense. Better to drop that constraint, and recognize that what the slip-skid ball really tells you is whether or not the aircraft is generating any lateral aerodynamic force.

I don't understand why the system is not offering me the "move this conversation to chat" option, I'm sure it will happen soon...

@RobertDiGiovanni -- actually, gravity MUST be ignored. Let's make a new instrument that is just a ball in a vertical tube. When the ball is at the bottom of the tube, we say the G-loading is "positive". Means the net aerodynamic force is upward in the aircraft's reference frame. When the ball is at the top of the tube, we say the G-loading is "negative". Means the net aerodynamic force is downward in the aircraft's reference frame. Gravity has nothing to do with it. Where is the ball when we float over the top of the loop at a reduced, but still positive, G-loading?

@RobertDiGiovanni -- It's at the bottom. But where is the ball when allow the aircraft to experience a slight NEGATIVE G-loading at or near the top of the loop, as described in this related ASE question? aviation.stackexchange.com/a/55876/34686 . Obviously the net direction of acceleration is earthward-- the flight path is deflecting earthward -- yet the ball will be deflected in the earthward direction (toward the top of the canopy), along with everything else in the glider. Reflecting the fact that the glider is actually making skyward lift.

@RobertDiGiovanni -- so as far as the deflection of the ball is concerned, all that matters is the direction of the net aerodynamic force generated by the aircraft. As long as we don't impose the artificial constraint that the trajectory of the vehicle must have no upward or downward curvature, then the force of gravity has no effect on new instrument described above, or on a traditional G-meter, or on a slip-skid ball. All that matters is the direction of the net aerodynamic force generated by the aircraft.

Lost in all the chatter: This answer does not explain how the slip-skid ball can be centered in a normal stabilized "coordinated" turn at a constant airspeed and turn rate. The net force-- and net acceleration -- are centripetal -- toward the center of the turn. And this does indeed include a lateral component in the aircraft's reference frame. Yet the slip-skid ball is centered.

Another answer has noted that the net AERODYNAMIC force vector-- as opposed to the NET force vector including the effect of gravity-- does not contain a lateral component in this situation. Please reply in chat.

No, let's not make s new instrument, let's use the one you have. "All that matters is the aerodynamic force". The aerodynamic force creates acceleration. Acceleration creates G force in the opposite direction. The ball is not weightless. Gravity also affects it. In a loop the magnitude and direction of the resultant vector is a combination of gravity and acceleration G forces. Sorry, not your best day. Try common sense first.