See, we are going to continue our talk on the dynamic equations of motion. Actually, Last lecture we ended up getting this newton equation in this way, I will add those extra terms later on. See this well of course, the equation if I the origin at center of Gravity but, if my origin what located at some other distance and typically if I thought. r G to be X G 0 0. Because, normally you would always choose body coordinate at a distance with X G.. Normally, the center of gravity is on the center line and origin also you will put on the canter line just X will be so. If just to not 0 X then the extra terms that comes out are. that is why I left it free you see etcetera. So, this is what was my newton’s equation of motion that is what it says that is all essentially, this this is what it says but, now as I was mentioning today, which going to be a rather important from the maneuvering point of view see now I need to figure out a way to represent the fluid forces that is these forces, X Y N are the fluid forces coming on the hull remember and all the forces external forces to which it is excited. Now this force X Y and N. So, this I can divide in terms of let me call this to be generically as X bar just to show vectorially. This will consist of I can break it in components you know one will be what we can call hull forces, one will be because of rudder, so one will be because of propulsion and there can be other environmental forces. I will just explain what all this see there is this trajectory the ship is going this way velocity vector this it has u here it has v here etcetera. Now, there is a rudder here there is a certain amount of force coming on the hull, why because? Remember that this hull is going in this direction whereas, it is oriented here. So, the flow comes like that flow is coming like that to it. So, it is not really collinear like it is not center line. So, that will give you some force on the hull and moment. So, I call this to be the hull forces now of course, here in this see I am trying to figure out all force components that is arising here. Remember, this is a situation, where the body is undergoing a motion, it has certain u certain v maybe certain psi dot. So, all this thing if it I look at that. that are my inertia forces this side that should balance with my total force acting on the hull. What are the total forces? I have first of all a forces arising on the hull because of water is flowing this way see as an example suppose the ship was going on a straight line what would happen the net force that come is I am calling resistance force. Remember that is what I call resistance is not it? fluid forces acting on the hull in along x direction because the ship is going along x direction was resistance force it is a fluid force. So, here it is not going along x direction but, with some kind of an angle. So, there will be some forces created I am calling it hull forces now, obviously you have got this orientation because, you have actually cause rudder to some angle, which has given some force on that. In fact, that is what has started a hull ship to turn. So, that is rudder forces now, propulsion force because I have propeller here, the propeller is going to give me a thrust so that gives me propulsion forces. And if I have for example, waves, currents, wind etcetera. and we can call them environmental forces, now what we are going to consider only the hull forces why because, consider this case I want to investigate first before the control surface that straight line stability remember, where my hull was going on a straight line and I just cause it to turn by a small angle disturb the hull environment because, calm water just like resistance I presume the study is in calm water point one. So, this I remove it propulsion and resistance cancels out because, when it goes on a steady state at a given speed exactly re propulsion minus resistance will become 0 that is why it is going on a forward speed. So, I will not have a propulsion force here for my this consideration I only want to know this side rudder I do not have, because I am not applying rudder. Let us say actually, we are trying to find out firstly hull forces we will model them eventually. But, rudder also when I want to study just a bare hull with rudder fixed not apply, rudder is kept fixed. I do not it becomes the part of . Normally, the center of gravity is on the center line and origin also you will put on the canter line just X will be so. If just to not 0 X then the extra terms that comes out are. that is why I left it free you see etcetera. So, this is what was my newton’s equation of motion that is what it says that is all essentially, this this is what it says but, now as I was mentioning today, which going to be a rather important from the maneuvering point of view see now I need to figure out a way to represent the fluid forces that is these forces, X Y N are the fluid forces coming on the hull remember and all the forces external forces to which it is excited. Now this force X Y and N. So, this I can divide in terms of let me call this to be generically as X bar just to show vectorially. This will consist of I can break it in components you know one will be what we can call hull forces, one will be because of rudder, so one will be because of propulsion and there can be other environmental forces. I will just explain what all this see there is this trajectory the ship is going this way velocity vector this it has u here it has v here etcetera. Now, there is a rudder here there is a certain amount of force coming on the hull, why because? Remember that this hull is going in this direction whereas, it is oriented here. So, the flow comes like that flow is coming like that to it. So, it is not really collinear like it is not center line. So, that will give you some force on the hull and moment. So, I call this to be the hull forces now of course, here in this see I am trying to figure out all force components that is arising here. Remember, this is a situation, where the body is undergoing a motion, it has certain u certain v maybe certain psi dot. So, all this thing if it I look at that. that are my inertia forces this side that should balance with my total force acting on the hull. What are the total forces? I have first of all a forces arising on the hull because of water is flowing this way see as an example suppose the ship was going on a straight line what would happen the net force that come is I am calling resistance force. Remember that is what I call resistance is not it? fluid forces acting on the hull in along x direction because the ship is going along x direction was resistance force it is a fluid force. So, here it is not going along x direction but, with some kind of an angle. So, there will be some forces created I am calling it hull forces now, obviously you have got this orientation because, you have actually cause rudder to some angle, which has given some force on that. In fact, that is what has started a hull ship to turn. So, that is rudder forces now, propulsion force because I have propeller here, the propeller is going to give me a thrust so that gives me propulsion forces. And if I have for example, waves, currents, wind etcetera. and we can call them environmental forces, now what we are going to consider only the hull forces why because, consider this case I want to investigate first before the control surface that straight line stability remember, where my hull was going on a straight line and I just cause it to turn by a small angle disturb the hull environment because, calm water just like resistance I presume the study is in calm water point one. So, this I remove it propulsion and resistance cancels out because, when it goes on a steady state at a given speed exactly re propulsion minus resistance will become 0 that is why it is going on a forward speed. So, I will not have a propulsion force here for my this consideration I only want to know this side rudder I do not have, because I am not applying rudder. Let us say actually, we are trying to find out firstly hull forces we will model them eventually. But, rudder also when I want to study just a bare hull with rudder fixed not apply, rudder is kept fixed. I do not it becomes the part of