 Deshmukh Sachin, working as an assistant professor in Walsh and Storpe technology in civil engineering department. Today we are going to learn about what is a metacenter and what is a metacentric height and what is its importance. At the end of this topic, you are able to calculate the metacentric height of a ship model which is very important and all the construction of the ship models are totally depending on this and you can classify the equilibrium conditions so that you will come to know how to stabilize or how to make the ships in stable conditions and how to construct it. Now first of all, before starting this metacenter, we must know what is the center of gravity? What is the center of buoyancy? Center of buoyancy, here the upward force is acting. When the body is immersed in water, there is upward force is acting and that force it is located at some place. Here just go through this figure, you will come to know. And the point M which we call now, it is a metacenter. It is located above that center of gravity if it is in a stable condition. Metacenter, it is a theoretical point at which an imaginary vertical line passing through the center of buoyancy intersects the imaginary vertical line through a new center of buoyancy created when the body is displaced. What body is displaced? As you see in the second figure, body is displaced or tipped in the water. Just concentrate on this figure. This is a water and this is a body. This is the center of gravity. G is the center of gravity. This is the original vertical line passing through center of gravity. Now this is a buoyancy, center of buoyancy. You might have remember the Archimedes principle also. This is the center of buoyancy, the center of gravity and this is the M metacenter. Now this particular body is just displaced or tipped. We have written in its definition, it is just displaced. Somehow it is displaced like this. Now this G is over here. Now this B is the center of buoyancy. It is shifted to this place. It is shifted from this point to this point. And from this point, again you have to draw a vertical line. And this vertical line where it intersects to this previous line, this one. Center of gravity M and B which is passing through. This is the previous line where it cuts, where it cuts. That point is a metacenter. That point is called as a metacenter. Now what is a metacentric height? The metacentric height, it is a measurement of the initial static stability of the floating body. And it is the distance between the center of gravity of the ship and its metacenter. That means G and M, that means G and M. A larger metacentric height implies always greater initial stability against over turning. Here, another model I have drawn for you. This is the B, the center of buoyancy. It is the force of the buoyant or I can say buoyant force is there. G and weights, the weight of the body which is acting downwards side. B is displaced over here. This is the M. And this G and M, this G and M is a metacentric height. This G and M is metacentric height. See here, the action over here in the side, body is dipping in the side. So that to make it in a stable condition, we have to go like this. The force of buoyant force is acting over here. And it is going to meet over here M, M and G. The distance between M and G, metacenter and center of gravity, it is called as metacentric height. Which is again a very important word in the ship model or construction of ship model. Now, what are its applications actually? The metacentric height and metacenter, actually these words are used in ship construction. And also to measure the stability of the vessel which is kept in water. Now see, this figure is an old ship model. This is an old ship model. We will come to next figure afterwards. This is an old ship model. And all these strings are there, these strings to make the ship stable. Here, probably here, when the ship is in a stable condition, probably the M will be above G and the buoyant force will be here. It is acting through this axis. And we know that as the metacentric height will be more, more stability is there. The same concept from ancient days to the new modern technology is coming in, providing the metacenter and metacentric height. Just go through this figure. Observe, a multi-storey building is there on a ship. 3-4 floors are there which are constructed on one side. Never matter. If you are keeping the metacenter above center of gravity, it is in a stable condition. It is in a stable condition. Just go through it. Here, you can observe, no such more weight. But here, on the backside, there is a building, small building. And here, more weight is there. But it does not matter if you are going to keep the metacenter above center of gravity and maximum metacentric height. It will give you a stable condition. Metacentric height is determined by two methods, analytical method and experimental method. Here also, one figure I have kept. Here also, you can see, there is a building. These big ships, they are traveling in the sea for passenger traveling and also for cargo, also for wars. The metacenter is maintained so that there is no question of sinking. So metacentric height can be determined by two methods, analytical and experimental method. We will just flash on both the methods. Now pause the video and give the answer for this. What is the proper explanation for the metacenter? A is point at which line of action of force meets the normal axis of body when it is given an angular displacement. B, intersection of line passing through a new center of buoyancy and center of gravity. Point about which body starts oscillating when it is given a small angular displacement. The answer is all, all, all the choices. Here, point at which the line of action of force meets the normal axis of body when it is given an angular displacement. It is a phenomenon where you can see the line of action of forces meets in the normal axis. Then the intersection of line passing through a new, the center of buoyancy is shifted. So from the new center of buoyancy, again you are going for a vertical line which is going to meet that previous one. And point about which body starts oscillating that is m. The body starts oscillating and again coming to its original place when a small angular displacement is given. So all these choices are correct. Now we will shift to this analytical method. This is the analytical method. This is a shift model and this is as m is given over here. And here in this figure we are concentrating on this particular prismized shape. That is o dash to, q dash to q and n dash o and n. This particular small prism. This b is shifted to b1. This is angle. Pb is a buoyant force, center of gravity. Again when it is tilted, the center of gravity is over. This weight of the body is going to act. This is w. This is center of gravity. Now see it is to find the metacentric height. Consider that element is strip, q, q dash and portion of n dash o at a distance x from o. This is the x distance from o. So calculate the volume of this elementary strip. Then the upward force, that buoyant force we can calculate. And the moment of this buoyant force we are going to calculate about o. For the total portion of that n o n, we are getting this particular integration. Because we are concentrating on a small strip, there are different many strips are there. So we are going for the integration of this. And we can find this where i is moment of inertia of the sectional area about the water line. Put these values. Put these values. And finally, bm. You are getting as i upon v. That is moment of inertia upon volume. So once you relate bm, gm is our metacentric height. So bm plus or minus bg. b that is buoyant force where it is acting and g is the center of gravity. So positive sign when g is below b and negative sign when g is above b. Keep this in mind. M must be always above when it is below. It is unstable. Experimental method is also there. From experimental method we can calculate this gm by this formula. Reference books are there. Thank you.