Anne,
You want to know the physics behind what the turn coordinator ball is telling you. Coordinated flight happens when the ball is centered and it means that you are not in a skid (too much rudder) or a slip (too little or opposite rudder).
When you bank an airplane, a component of your lift acts towards the center of the turn. That is why you pull back (pitch up) a little more to support the weight of the airplane, if you want to maintain your same elevation (because some lift is used for banking and is unavailable to support the airplane). So, the plane rolls over (with aileron) and a component of lift acts straight up and a component acts straight towards the center of the turn radius. The rudder induces yaw and yaw creates side forces to pivot the plane about it's vertical axis. The yaw is used in a bank to counter (or exaggerat) side forces induced by a roll. So in the mix are forces related to weight, lift, bank, pitch, and yaw.
Coordinated (ball centered):
With the right amount of rudder, the forces are all balanced so that it feels as if the weight is acting through the vertical axis of the plane (which is not truely "vertical" at this time). Race car on track analogy with no feeling of side forces would be: side slope of the track just right for speed/radius = rudder input just right.
Uncoordinated - Slip (ball to the inside):
With the too little or opposite rudder, the forces are all balanced so that it feels as if the weight is acting through an axis angled to the inside of the turn. You and the little ball feel like you are being forced downward/inside of the turn. Race car on track analogy with inside/downward side forces: side slope of the track too steep for speed/radius = too little or opposite rudder input.
Uncoordinated - Skid (ball to the outside):
With too much rudder, the forces are all balanced so that it feels as if the weight is acting through an axis angled to the outside of the turn. You and the little ball feel like you are being forced upward/outside of the turn. Race car on track analogy with outside/upward side forces: side slope of the track too shallow for speed/radius = too much rudder input.
How did I do MTV?
Erik