 Let us look at some of the turning flight equations. So here we have an aircraft this is just a sketch so do not assume that aircraft fly like this which is at an angle phi. So it is banking at an angle phi. So the lift force is going to be perpendicular to its reference line a component of that is going to overcome the weight and the remaining component is going to give you the centrifugal force. So l cos phi will be w that is our first equation and l sin phi will be mv square by r which is the centripetal acceleration force acting on it, centrifugal force acting on it. So we take a ratio of these 2 by 1 so you will get tan phi will be v square by gr. So if you want the angle phi to be small then let your speed be small or let your r be large. This is what is there in transports. In transport aircraft we go for a very long turn because banking it by a large angle can cause discomfort to passengers. So there the value of phi is small but military aircraft either want to turn very fast they want to have a high v in turn or they would like to have very small turn radius. So they have to bank it larger. So for a given speed the angle of bank is directly connected to the radius of turn or I would say inversely proportional to the radius of turn and the ratio of l by w is going to be the load factor. Now this need not be equal to 1 in turning flight. Actually this will be equal to secant of phi 1 by cos of phi. So larger your turning speed or smaller turning radius larger is the angle of bank and angle of bank directly gives you the value. So if phi is 60 degrees n is equal to 2 straight away. So the banking angle is a very important parameter it is cos inverse 1 by n. So you will not be able to unless you have thrust vectoring. You will not be able to turn like this. You always have to bank to turn. If you have direct thrust vectoring then you can do you are flying like this and now you have a thrust vectoring so you can do this and then you can do this. So that is not the topic of our discussion. We are looking at the conventional aircraft which does not have thrust vectoring. And that too thrust vectoring normally is only in the vertical horizontal plane. Side thrust vectoring is normally not provided in aircraft. But there are some other aerospace vehicles for example in airships it is very common to provide a direct side force in which case no banking is needed. You can go this way and then you can go this way. So load factor depends on the aircraft design parameters. Higher load factor means more load on the structure. The wing has to carry n times the weight. So imagine if W is 50 tons and if n is equal to 5 that means you have to design the structure to carry 250 tons. So that leads to weight penalty. And normally the load factor is spoken in terms of G's. How many times the weight? So 2G, 3G, 5G. This is not telecommunication G. Yes. Actually I do not know what is thrust vectoring. Thrust vectoring is a word in English thrust vectoring. Thrust can be vectored. Means we are producing additional thrust in the opposite direction. Thrust is basically a force produced by the engine which is normally on the back of the engine. Yes. Now suppose I tilt the engine then that force I am vectoring it. That is called thrust vectoring. But sir how can we provide it in a perpendicular direction to the flight? By putting a special engine or any other force or any other device that produces force in that direction. That is why I said normally you do not see it in aircraft. That is why I dismissed it. I mentioned it. But if you have theoretically speaking if you have an engine mounted let us say on the rear of the fuselage and now you can swivel the engine any direction suppose. Then you can provide thrust at any angle you want. But direct side force as I said is not very common in aircraft. So that is why it is not relevant to our discussion right now. So when I say that an aircraft cannot turn horizontally without banking I want to be sure that I do not make it a statement true for every aircraft. So that is why I said unless you can vector the thrust. If you can vector the thrust then you can make it move this way, this way, this way. So a helicopter for example it can do a turn like this. Because there you can tilt the blades so helicopter is stationary tilt it it goes this way. That is the point I was making. But good it is good that you pointed out. Such kind of interruptions are helpful for me because I may assume as somebody mentioned on the module page also in the feedback that there are some terms which are not commonly known to the non aerospace engineers. So it is very good to interrupt me. Then I can qualify my statements or make them more apparent and clear to other people okay. So thanks a lot for interrupting right. So let us look at the turn radius now. So this force mv square by r m can be replaced by w upon g. So it becomes w by g v square by r and we have already seen can be replaced. So in other words the radius of turn is directly proportional to square of the velocity and inversely proportional to the root of n square minus 1 g being a constant. So what does it tell us? It tells us that if you want to have a tighter turn small r either you fly at a higher speed. Same aircraft flying at a higher speed will give you a tighter turn or you can increase the load factor. So when n is large then r is going to be small. So just v and n are the two factors that define the radius of the turn and that is the turn radius, the radius at which you are turning. So that is this particular definition is for the tightest turn or a shortest radius. The other aspect that we need is called as a turn rate. That means when you are trying to close into an enemy you would like to have larger value of change in this omega, d omega by dt. You want it to be larger. So turning a smaller radius is meaningless if you are not turning fast enough. So degrees travelled per second is the turn rate omega which is d theta by dt. So that will be v by r. So omega will be g root n square minus 1 by v because that is a divided v by r. So here this is proportional to n, n square minus 1 under root and inversely proportional to v. So if you want to turn at a faster rate then you have to be at a lower velocity. This is an interesting thing. Load factor is one aspect but load factor you cannot change dramatically because that affects the structure. So there may be a limit. What is the value of n max? It could be 6, it could be 8, it could be 9. That is it. Beyond that we do not normally design the aircraft. So the load factor which they can take without any permanent deformation is approximately 9 for the most agile fighter aircraft. You cannot go beyond that. But v is in your hands. However you cannot go below a v because then you will stall. So omega depends on v and n. r depends also on v and n. Both of them are interrelated. So let us see the aspect of bank angle. Now the same aircraft, if you are banking at a lower angle you will have a larger radius. If you are banking at a smaller angle you will have a larger angle you will have a smaller radius. That is why you saw the video that I showed you where the aircraft was into a very tight turn. It was almost 90 degree bank, almost. So increasing the bank angle in the turn it results in higher turn rate, it results in smaller radius of turn but it also causes higher loading on the wings and the stall feed also increases. So it is dangerous because you will be stalling at a higher speed. You are going to have a higher stall feed because lesser speed is now available for your overcoming the weight and the remaining is being consumed in taking care of the acceleration. So fastest turn will be max turn rate. This is a measure of its maneuverability of the aircraft and it depends upon v and n. So do you agree with this v should be minimum and n should be maximum if you want to have a fastest turn rate. Lower speed and higher n give you faster turn rate, higher speed and higher n give you tighter turn. Now there are two types of turns. One is called as a sustained turn, the other is called as a instantaneous turn. The sustained turn and instantaneous turn the main difference is does your operating condition remain same or you are allowed to change them. So let us first take instantaneous turn. So what it means is you are flying at some speed, at some altitude and now you want to turn and during this turn you do not mind if your altitude reduces or speed reduces or a both reduce if you do not mind. This is called as instantaneous turn and this is governed only by aerodynamic parameters because you are allowing the aircraft to reduce in speed and reduce in the altitude. So at any given instant, so can you tell me a scenario for a military pilot where it will be helpful to have a high instantaneous turn rate. Under what situation or for what application for a military aircraft it will be useful to have a high instantaneous turn rate. Yes, to dodge missiles very good, excellent answer okay, not in dark fights that is a wrong answer. In dark fights you need to have high sustained rate. Yes, to dodge the enemy missiles because at that time you do not mind losing speed or velocity you want to save your skin basically and you want to be away from the missile. So at an instant you want to quickly turn but if you are in the combat situation and if you are either pursuing an enemy or you are being pursued by somebody at that time it is important that you do not lose velocity or altitude because in a dark fight situation the one who is behind you and above you is at an advantage always. Always in dark fights the pilots try to come behind the other guy and above the other guy because that is the best condition to shoot. Now if in this situation you go for instantaneous turn your velocity will reduce you will become a sitting duck target. If your height drops you will give a bigger advantage. So in a dark fight we are not interested to have instantaneous turn but sustained turn okay. Let us also look at tighter turn in this you want to have the minimum turn radius. Now what could be the application for this in a military aircraft? When would you like to have a minimum turn radius? Rate is not important. Radius is important in which situation? Yes, come back okay. So after completing the task when you want to come back that could be one application. The other application will be to avoid any obstacle. Let us say there is a mountain in front of you and you are now going at high speed what do you want to do? I do not want to turn fast. I want to turn at the shortest radius so that I avoid the obstacle. So this is also measure of maneuverability okay. This also depends upon design parameter but here we should be maximum not minimum correct. We should be maximum and n should also be maximum. Is it correct or is it wrong? We should be minimum in a tightest turn with minimum turn radius. Why is it so? Yeah so it is so the tightest turn so in both cases you want minimum speed sure. Let us go back to the formula. So this is the turn rate. So here you want V to be low and to be high. This is turn radius. So turn radius you want R to be less. So again you want V to be less and n to be high. So the condition is same for both. Now obviously when you have high instantaneous turn rate for that thrust to weight ratio is useful wing loading altitude and these two what is the minimum speed and what is the maximum load factor? This is something that you have to tell me. For a given aircraft you would like to have V to be minimum. How minimum can you go? You want n to be maximum. How maximum can you go? That is something that you will tell me and here is actually the answer. These are the limits which are imposed on the velocity. So turn radius is going to reduce and you want it to be less but you will hit two limits. On the right hand side you are going to hit the structure limit as the velocity increases. The dynamic pressure half rho V square on the structure will increase and the load that the structure can carry has to be balanced. On the lower side you have aerodynamic limit because power required, power available you know it also starts increasing. So the bank angle is more when you are going to turn at a lower velocity. So this is a very nice curve which shows that on the left hand side it is the aerodynamic limits. On the right side the structure limits and that sets you a boundary for the V versus radius. Similarly you have also another curve which tells you thrust available and drag will limit V infinity. CL max from aerodynamics you can get a maximum value and W by S can be driven by materials or by the size consideration.