 But, we now have a fourth center also and that is the aerodynamic center or the AC as we call it. So, what is this AC? AC is basically a theoretical point it is not a practical point it is a theoretical point such that the effect of angle of attack or the effect of the aerodynamic moment on the angle of attack is removed. So, the aerodynamic moment remains same at all angle of attack about the aerodynamic center. Center of pressure moves generally it moves back, okay, if you have a flight which is supersonic it may even move forward because it is a function of pressure distribution. But now we are not bothered whatever be the pressure distribution there is some lift there is some drag there is some moment, so the moment does not change about the aerodynamic center. So, at all angle of attack you have the same moment, so that is the beauty of this theoretical center called as the aerodynamic center. Interestingly if you take a flat plate or a very thin aerofoil the approximate location of this particular place theoretical place is one-fourth of the chord. So, if you want to do a very quick and simple design and if you want the aircraft to fly you should try to adjust your center of gravity near the quarter chord of the wing because that is where the aerodynamic center is. Now, I will show you how to decide the exact location, okay. So, we will take a toy plane I will be doing I will be doing two experiments in the class one is not an experiment but a demonstration using only sketches a toy plane. So, these are basically figures taken from a textbook and this textbook by the way very interesting textbook by Professor Eberhardt and Anderson Professor Eberhardt was a visiting professor in our department for one semester in 2011. He worked in the Boeing company for many years and then he joined Washington University and then he now travels all over the world taking up assignments as a visiting professor. So, he was in Spain, he was in Australia, so now he has gone to some other location. He has written a very nice book along with D.F. Anderson this is not the same Anderson that you know this is some other Anderson, so Anderson and Eberhardt one of the best books to understand the basics of flight in my opinion even far better than Anderson available online just by a simple Google search copies are available. In fact, I have autographed copy also by Professor Eberhardt but we have in our departmental library as well as in the main library. I recommend all of you to go through this book and I have borrowed this particular example from his textbook using a toy plane, so here is a toy plane. Now this toy plane assume that through some magic I am throwing it in the air and it is in equilibrium at some condition okay and now some disturbance acts on it because of atmospheric disturbances whatever and there is a nose up pitching moment but before we go ahead let us first look at the forces acting on this aircraft. So as we see the point marked by cross is the neutral point, how do you find it? You cannot find it, it is not easy to find it but it is there. In this example as is typically the case the lift of the wings acts slightly ahead of the neutral point, the center of gravity is far ahead okay, so center of gravity is far ahead of the neutral point. Now what will happen tell me, so what tell me why do we have the tail loads and the way it is shown? So take moments about center of gravity, lift alone is going to give you a nose down pitching moment but I said the aircraft is in equilibrium therefore the tail has to carry a down load. Now why is the load on the tail much smaller? Because it is far away from CG, it is the moment that is important not the force because we are trying to take moments about CG. So this aircraft will fly in a trim condition only when the moments about CG are 0. So in this condition the lift force into that distance is equal to and opposite to the tail force and the tail distance of the tail aerodynamic center from the center of gravity. So something like Lw into Xw equal to LT into Xt and there is no net moment, so therefore about center of gravity you have some net lift which is the brown vertical line which is the lift of the wing minus tail and the total weight of the aircraft is W, they are both equal, balanced, opposite and hence this aircraft is in trimmed horizontal flight. Now for some reason there is a perturbation in the flight because of which the angle of attack increases. So what will happen as the alpha increases and if the alpha is below alpha stall then the lift will increase. So as the angle of attack increases the lift is going to increase so the nose down pitching moment is going to be created because of the delta L. But what will happen to the angle of attack of the tail? If the aircraft was at some angle in trimmed condition and the tail had negative load that means tail was deflected little bit down and now the alpha is increased what is going to be the angle of attack of the tail, it is going to decrease little bit. So if it decreases its lift will not remain the same therefore the tail down moment created by the tail will be lower than the nose down moment created by the wing. So we disturb the aircraft, we created a angle of attack which means the nose went up but the nose will come down because the nose down moment is more than the tail down moment. So what can you say about this aircraft? This is positively stable or stable but we can only comment about the static stability. We do not know what will happen in the end but at least the tendency is to come down. In other words when the center of gravity is ahead of the neutral point then we have a situation when the aircraft is stable. Now one more question I want to ask you, suppose I move the center of gravity further by adding a weight in the nose, now what will happen to the aircraft will it become more stable or less stable? Who will answer this question? More stable, less stable, none of the above, unstable, I do not care, I do not want to hear I do not care, tell me will it become more stable or less stable? More stable? Will it become more stable? Because the nose down moment will be further but remember even the tail also will have a longer tail arm but the numerical value of the lift is far more than the tail. So if you give more moment arm, larger force with that delta x will give you more moment than a smaller force with that delta x, agreed, Manikandar not happy, okay. So if you move CG forward it will be more stable and more stable and more stable, correct, okay. Let us see now when CG is moved behind and it is at neutral point, it is at neutral point now. So when the CG is at neutral point, the wing is still such that its lift will be acting slightly ahead of the CG, now it is ahead of the CG because the location of the lift on the wing is a function of the wing geometry and the wing location, we have not changed the wing location, I did not move the wing behind, I just move the CG behind. So the wing lift will now give you a nose up moment and the tail has to have upload to cancel that because I am saying that the aircraft is flying in a steady level flight even in this condition. So when the CG is at NP and the aircraft is still in a stable flight, now the position of Radomix center does not depend upon angle of attack, okay, so it remains the same place. So as the angle of attack increases, lift on the wing will go up but the tail lift will also increase and because it is neutral is because the two moments are going to balance your CG's at neutral point, therefore there will be no restoring torque, so therefore this aircraft is neutrally stable because the moments are balanced. As alpha increases, it increases both for tail and for the wing. Now let us look at the situation when the CG is behind neutral point. So now when the nose pitches up, the increase in the lift on the wing will be more than that on the tail, why will the delta L wing be more than delta L tail? Both will increase because alpha has increased but why is it so that change in the lift of the wing is more than the change in the lift of the tail. Let me give you a hint, suppose we assume the same aerofoil for both because airfoils can be different, so just to remove that confusion, both have the same aerofoil. Therefore the DCL by D alpha is same, change in the CL with change in the angle of attack is same. So it is not because of any change in the lift coefficient, what is the reason? Because the, I would like to hear not surface area but reference area, yes that is one reason wing is larger in size, reference area is larger. Any other, any other reason? Take a theoretical case when both have the same surface area, still I will say wing has been going to have more, the reason is actually the aspect ratio of the tail is normally kept lower than that of the wing and because of that also you will get higher lift, fine. So when you have larger lift from the wing, then the nose will pitch up further, so now you will not have restoring moment, you will actually have imbalance moment. The nose of the aircraft has gone up, it will tend to go up and tend to go up and tend to go up, so therefore the aircraft is unstable, so it is very straight forward. The relative location of the center of gravity and the neutral point is what decides the stability in longitudinal direction. If CG is ahead of neutral point, aircraft is stable, at neutral point, neutrally stable, behind neutral point unstable, the farther behind, more unstable, the farther ahead more stable, okay. But we forgot there is another position that the third center is there, yes you have a question. Neutral point is a theoretical point, okay. It is difficult to explain it as a physical quantity, like center of gravity we say, oh where the net weight acts, so it has a physical significance. Center of pressure, that is the physical significance, that is the center of the pressure which acts, but aerodynamic center and neutral points do not have any physical significance. These are some theoretical points, okay. So you can just assume that there is a point called as neutral point. Its location depends on the geometry of the aircraft, such that if the CG is ahead of that point, the aircraft is dynamically sorry longitudinal is stable, if the CG is behind it is unstable, if the CG is exactly there it is neutrally stable, this is how you define neutral point, okay. There is no other significance, similarly aerodynamic center is a theoretical point at which the moments are going to be the same at any angle of attack, lift and drag forces will change, yes it is a location on the aircraft, but you cannot say you can calculate it in this manner, CG you can calculate, you can say take any point, take moments ahead, take moments behind, if this is more move, move, move, move till moments are equal, but aerodynamic center and neutral point are not such quantities, they are derived quantities, okay, is it answer to your question? No, no that is not the reason, the reason is the third one which I am going to come now, the aircraft is stable depending on the relative location of the CG and neutral point, CG ahead stable, CG behind unstable, but it assumes that the aerodynamic center is between the CG and neutral point, if the center is ahead of CG or behind neutral point the things go haywire, so that is why we need to now look at the location of the aerodynamic center also, okay, so the first one is CG ahead of neutral point and ahead of aerodynamic center that is the point which I have circled, okay, now location of center of gravity of the aircraft by the way, this is not about wing center of gravity, this is the aircraft center of gravity, interesting point, so typically the center of gravity of the aircraft lies somewhere on the wing, okay, it will be rarely a situation when the center of gravity of the aircraft is behind the wing and ahead of the wing, it is on the wing somewhere, so what we do is the center of gravity location, when I say CG location I am talking only about longitudinal, all our discussion is only about longitudinal, no lateral and no roll right now, so the longitudinal CG location of the aircraft is somewhere on the wing, so normally we express it in terms of the percentage of a term called as the mean aerodynamic chord, what is a chord you know, chord is the distance between the nose radius and the tail radius, correct or the tail junction, but the wing is not always of the same chord, the chord can reduce, it may even increase there are aircraft in which the chord at the center of the wing is less than at the tips, have you seen certain aircraft, okay, there are, so why do not you locate and put on modal picture of an aircraft in which the root chord is less than the tip chord and not just search, explain why, is there an advantage of doing that because nobody will build an aircraft with the configuration unless there is some advantage, so this is the interesting assignment, find out aircraft where the tip chord is more than the root chord and explain the reason and why do not we see it very often, why do we typically see that the tip chord is equal to root chord and less than the root chord, okay, so therefore in general the chord changes along the span, so therefore we define one particular chord called as a mean aerodynamic chord, it is not a geometric mean but it is a aerodynamic mean, there is a formula, now that goes into aerodynamics we will not get into that, assume that there is some way to calculate the mean aerodynamic chord, so the center of gravity of the aircraft is going to locate along the mean aerodynamic chord at some location, so we normally call it as percentage of MAC, 5%, 10%, 15%, 20%, please understand the mean aerodynamic chord may be larger than the chord at the root or smaller, we do not know, so this is just a illustration, so to avoid confusion assume that the aircraft has got the same chord throughout a rectangular wing, no confusion, now the MAC is equal to the C at the root, so the CG will be located at some position, we measure it in terms of the percentage of the MAC, so there are some examples here 5%, 25%, 35%, 40%, 45%, okay and we want to see what happens, so the first one is if the aerodynamic center which is typically at quarter chord, not always but typically at quarter chord and if the CG is at 5% of the MAC, then the aircraft is very, very stable because the CG is far ahead of the aerodynamic center, neutral point is behind, we do not know where it is right now but aerodynamic center also it is far ahead, so this condition is a condition where you have a saucer and a steel ball exactly as the center highly stable, undesirable, not desirable, then we have CG at aerodynamic center, so progressively we will move the CG behind, first 5%, now CG is at 25%, it is above aerodynamic center or at aerodynamic center, neutral point is behind, in this example neutral point is at 35% of MAC, just happens to be, you do not know how to calculate it, it happens to be, so what will happen now, will the aircraft be stable, unstable, neutrally stable, stable, sure it is stable, why is it stable, because ahead of neutral point that is it, it is stable and this my friend is a very desirable location for the CG of your aircraft, approximately quarter chord, okay, let us move it behind, aircraft moves behind aerodynamic center but still ahead of aerodynamic, still ahead of neutral point, aircraft is stable because NP is still behind but the level of stability is now reducing because you are moving towards neutral point, the moment you go behind aerodynamic center your CG is going to be, your stability is going to be less, for you and your glider it is a undesirable condition, your glider will still fly but not fly well, it is less stable but still stable but less stable, now let us say take the CG behind AC behind AC at neutral point, it will be neutrally stable, now this is the most rear position of CG which is acceptable for flight, I would say we do not want to go so back, in a practical aircraft we never fly at neutral point or even near it, we are sufficiently ahead of it but behind this is disaster, okay, ahead of this is better, best will be at aerodynamic center but here it is neutrally stable, once you go behind then you are unstable, alright, now this was a theoretical exercise, so this is in summary the position.