I'm curious what happens if you put some extra weight in the back of the airplane (let's say a small GE turbine or something). I'm new to Stability and Control side of things and curious how the new moments can be counteracted and if we need to add extra weight to the airplane in order to counter-balance it. So far I'm of the understanding that if you add, let's say 1000lbs to the tail of the airplane, you'll probably need to add thrice as much weight close to the center of the airplane to balance it. I'd appreciate some info and help! Illustrations might help too :)
If you put extra weight in the tail of the airplane, the pilot then has to add extra downforce on the controls to pull the tail up and push the nose down so the plane can fly level. In doing so, there is now less control authority remaining to make the plane descend on purpose and beyond a certain point, the plane becomes impossible to properly control and hence unsafe to fly.
There are several spectacular youtube videos of cargo planes in which the cargo comes loose during takeoff and all slides into the tail of the plane, tipping the nose of the plane violently up and causing the wing to stall. The plane then basically falls out of the air with its nose pointing up and crashes into the ground tail-first.
It's fairly simple. The airplane is certified to fly with a center of mass, or Center of Gravity, that is between point X and Y, or a Forward and Aft limit. You have to load it so the C of G falls in that range.
If you add weight at the back, it shifts the C of G back toward the weight you added some amount, depending on the weight relative to the total mass, and the distance away the weight was added. If the weight you added didn't move the C of G out of the allowable range, and you don't exceed the maximum total weight of the plane, you are good to go.
If the weight you added did move the C of G out of the allowable range, you have to add compensating weight on the opposite side of the C of G until it's moved back into range. How much and where you add that weight doesn't matter as long as you move the C of G forward into the allowable range, and you don't exceed the maximum all up weight. 1000 lbs added to the tail 50 ft from the C of G would require 1000 lbs added forward 50 ft from the C of G to keep the C of G where it was originally. Or some smaller amount if it can be added more forward than that. You're just talking weights, levers and moments in the end, why it's called Weight and Balance.
As far as what the limits of the C of G are, the aft limit is a function of static stability margin. The center of mass, the rotation center of the body in a fluid, has to be forward of the aerodynamic center of the entire body as a whole, the Neutral Point, the distance between the two points being the static margin. Think of a weathervane held sideways, with the pivot of the weathervane equivalent to the C of G. The weathervane "trails" into the wind with the aerodynamic center (the Neutral Point in the airplane's case) behind the pivot axis (the C of G in the plane's case). The airplane weathervanes about its C of G, as the Neutral Point wants to trail behind the C of G in the airstream. It's statically stable.
Moving the C of G too far aft moves it closer to, at, or beyond the Neutral Point. If you move the C of G to the Neutral Point, it simply loses its ability to point into wind. Move it aft of the Neutral Point, and it wants to flip around and point the other way, because its static margin, the C of G to Neutral Point relationship, is now in that direction.
Trimming forces generated by the elevator surfaces provide the ability to make the airplane more than a statically stable lawn dart when it moves through the air. Trimming forces allow the "weathervane" to align itself at angles to the flow off of zero, so that lift can be generated by the lifting surfaces (held at a positive Angle of Attack) to hole the whole thing up instead of just proceeding in a ballistic arc. It's the same as bending the trailing edge of a weathervane to force the weathervane to point itself into wind with its axis offset from the airstream, that offset being its Angle of Attack.
The forward limit is a function of the ability of the tail to generate trimming downforce in the most extreme-demand case, which is usually raising the nose for takeoff and landing around the fulcrum of the landing gear's contact patch with most tricycle gear airplanes, or a just desirable minimum flying speed (maximum AOA) in the case of a taildragger.
So, too far aft and the thing becomes a handful to fly as it no longer wants to "point" into the airstream, or in the extreme case wants to point in the opposite direction if you let it. Too far forward and your minimum flying speed, or the takoff and landing speed, will be higher than the airplane's specification.
Because of this, exceeding forward limit (within reason) is less dangerous than exceeding the aft limit.