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Static Force Model

The simplistic ‘four-forces’ model (lift equals weight and thrust equals drag) doesn't account for off-center moment-inducing forces and can't model the effect of shifting center of gravity. Use the following model instead:

Static model of forces with typical CG

In straight-and-level unaccelerated flight there are two physical quantities that must cancel:

Forward CG

With this model in mind, we can predict the effect of a more forward center of gravity (again, while maintaining straight-and-level unaccelerated flight): Static model of forces with forward CG

Aft CG

Conversely, with a more aft CG: Static model of forces with aft CG

Mathematical Insight

Starting with the conditions for straight-and-level unaccelerated flight we can derive the lift and tail forces as a function of the weight and respective arms:

Mathematical derivation of lift and tail forces as a function of weight and moment arms

The difference in lift and tail arms is a constant for the aircraft; we can further clarify these if we substitute as a parameter the ratio of the lift arm to this constant as:

Substitution of parameter to describe relationship of moment arms

Restatement of lift and tail forces in terms of the moment arms parameter

Then, α = 0 represents some ‘extreme aft’ condition where CG has come to coincide with CL; in this case, lift exactly counteracts weight and the tail-down force required is zero. Conversely, α = 1 represents the ‘extreme forward’ condition where CG is forward at an equal distance from the wings as the tail is rear to it; now, the tail-down force must equal weight and the wings must therefore lift double the aircraft's weight.

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