We can test the effects by carrying out a few calculations.
Let’s suppose that we have the following initial conditions:
2000 lbs of lift being generated at the wing C of P.
C of G is 1 ft forward of the wing C of P.
Tail plane down force 200 lbs acting at the tail plane C of P.
C of G is 10 ft forward of the tail plane C of P.
In the initial condition we have the following:
Nose-down moment = wing lift multiplied by distance from wing C of P to C of G
Nose-down moment = 2000 lbs x 1 ft = 2000 ft lbs.
Nose-up moment = tail plane down force multiplied by distance from tail plane C of P to C of G
Nose-up moment = 200 lbs x 10 ft = 2000 ft lbs.
So we have an equilibrium condition with nose-up moments equal to nose-down moments.
If we now move the C of G 1 ft forward and employ the same elevator deflection to generate the same tail plane down force we have:
Wing lift = 2000 lbs acting at the wing C of P.
Distance from wing C of P to C of G = 2 ft.
Tail plane down force 200 lbs acting at the tail plane C of P.
Distance from tail plane C of P to C of G = 11 ft.
Nose-down moment = 2000 lbs x 2 ft = 4000 ft lbs. This means that the nose down moment has doubled.
Nose-up moment = 200 lbs x 11 ft = 2200 ft lbs.
This means that the nose up moment has increased by only 10%
The 10% increase in the nose-up moment that has been generated by the tail plane means that the effectiveness of the elevators has increased.
But to restore balance we must increase the nose-up moment to 4000 ft lbs to match the 4000 ft lbs nose-down moment that is being generated by the wing lift. This means that we require a 100% increase in the moment generated by the elevator deflection.
Although the elevators have become more effective in generating a nose-up moment, this improvement is insufficient to offset the increased nose-down moment generated by the wing lift. To see why this has occurred we need to look at the way in which the C of G shift has affected the moment arms of the wing and the tail plane. The wing moment arm increased by 100% (from 1 ft to 2 ft) but the tail plane moment arm increased by only 10% (from 10 ft to 11 ft). So the wing has become 100% more effective in generating nose-down moments, but the tail plane has only become 10% more effective in generating nose-up moments.