Ok so I've got a couple of questions regarding CG, CP and pitch
1) When the aircraft is taking off, is the tail down force required to pitch up the aircraft high? Because the aircraft always has a natural tendency to pitch down.
2) When the aircraft increases its AoA, the CP moves forward (towards the leading edge). Does this mean it would go ahead of the CG or move closer to the CG.
Reading the heated discussion between Robert and JZYL makes me sad. Both are right and still they cannot agree. Maybe a longer and more detailed description might help.
1) When the aircraft is taking off, is the tail down force required to pitch up the aircraft high? Because the aircraft always has a natural tendency to pitch down.
The aircraft should always have a natural tendency to return to its trimmed state. This might be by pitching up or down, depending on the actual state. There are conditions where that is not true, therefore I say 'should'.
In a take-off, the aircraft is rolling on its wheels and needs to rotate around the main wheels in order to increase angle of attack for liftoff. This means either to decrease tail lift (say, on a taildragger where the pilot has lifted the tail off the runway earlier in order to reduce drag) or to increase tail downforce. In order to pitch up the aircraft, the center of pressure needs to move forward of the main wheel location and the center of gravity, which is done by pulling on the stick. So what is needed is not an absolute tail downforce but a decrease in tail lift. This might well be a downforce, but could also be a lower positive tail lift. Most likely, it is a change from around zero to negative lift on the tail.
As soon as the correct pitch attitude is reached, the pitch rotation has to be stopped by pushing briefly on the stick. This shifts the center of pressure briefly back, behind the center of gravity. After that, the aircraft is in the correct attitude for its initial climb and the center of pressure is at the same lengthwise location as the center of gravity. If it isn't, the aircraft accelerates into some pitching motion which is not desired in straight flight.
Disclaimer: This all assumes no pitch moment contribution from either engines or drag. Any additional moment from other than lift forces adds to the momentum equilibrium and requires the pilot to shift the center of pressure a bit fore or aft of the center of gravity for straight flight.
2) When the aircraft increases its AoA, the CP moves forward (towards the leading edge). Does this mean it would go ahead of the CG or move closer to the CG.
When the angle of attack of an airfoil with positive camber is increased, its center of pressure moves forward.
In potential flow theory, lift can be calculated as the linear superposition of a contribution from camber and one from angle of attack. While the camber-related part of lift is constant, the angle-of-attack related part varies linearly with this parameter. The center of pressure of the camber part is somewhere at mid-chord (details depend on the camber line; with a Joukowski airfoil the center of pressure is precisely at mid chord). The center of pressure of the angle-of-attack dependent part is at the quarter chord (the center of the area below the chordwise Birnbaum distribution of lift). The important part is the self-similarity of Birnbaum distributions for different angles of attack: The center of pressure of the angle-of-attack dependent part is constant and at 25% of chord for 2D flow and wings of large aspect ratio. Since lift at zero angle of attack is solely from camber, the center of pressure is at around mid chord and moves forward towards the quarter chord point as angle of attack increases. But this would make an airplane with only a main wing using a positively cambered airfoil unstable in pitch.
Hence the addition of a tail surface with less loading than the main wing. Since it operates on a point of the lift curve slope that is lower than that of the wing, a change in angle of attack means a larger relative lift change on the tail which shifts the center of pressure of the whole aircraft back as angle of attack is increased. However, as this happens, the pilot will pull back on the stick, reducing some of the additional tail lift in order to keep the center of pressure close to the center of gravity.
In short: The center of pressure is always very close to the center of gravity unless the pilot commands some pitch acceleration.
1) When the aircraft is taking off, is the tail down force required to pitch up the aircraft high? Because the aircraft always has a natural tendency to pitch down.
For rotation, yes. Aft stick increases elevator/stabilator trailing edge up, which increases the downward force on the tail, thereby producing a positive pitching moment and increasing pitching angle/AOA.
2) When the aircraft increases its AoA, the CP moves forward (towards the leading edge). Does this mean it would go ahead of the CG or move closer to the CG.
No, as AOA increases from the trimmed flight condition, the CP of the whole airplane moves rearward assuming the pitch control is fixed. This is a requirement for pitch stability: as AOA increases, we would like the airplane to automatically pitch down (via aerodynamics without pilot compensation). This means that CP has to move aft relative to the CG to produce this negative torque.
I'm not sure how these two questions are related. Maybe if you can clarify a bit further, we can produce better answers.
As far as the takeoff goes, the tail is doing its hardest work then, because its downforce must overcome the direct mechanical leveraging of the center of mass about the gear's runway contact point until the wings are supporting the weight, and it has to be able to do this at the most extreme case of being able to drag the tail on the ground at minimum takeoff speed. "Minimum Unstick Speed" (Vmu) testing, where the plane is flown off with the tail dragging on the ground, establishes this.
How hard the tail has to work depends on the gear's fore/aft location. The worst case is on airplanes with tail mounted engines that have a more rearward C of G when not loaded with passengers than one with wing mounted engines. The gear has to be far enough aft to keep it from tipping on its tail when unloaded (otherwise, a tail strut would have to be deployed whenever the airplane was unloaded - very inconvenient) and as a result the tail's job on rotation is harder than on the same airplane with wing mounted engines and gear that is not so far aft. This is one of the significant downsides to rear mounted engines because more tail power, more area or more speed, than would otherwise be needed is required.
