 Deshmukh Sachin, I am working as assistant professor in Valachan Stop Technology Civil Engineering department. Today we are going to learn about analytical method used for calculation of metacentric height. There are two methods actually that is one is experimental method and another is analytical method. Experimental method we used in the laboratory. If you have done the experiment you know it very well, we have taken a ship model which is floating and we are going to add the weights in that and with the inclination, with the inclination why I am telling you because this particular total methods are depending on the tilting you can say angle. So you have added the weight in that and then you have calculated the metacenter and metacentric height. What is going to happen in analytical method that we are going to see over? So at the end of this topic you will be able to calculate the metacentric height by using analytical method. Just concentrate on this, concentrate on this particular figure this is a you can say it is a floating body, it is a floating body, the uppermost part, uppermost point it is M which is a metacenter, G is a center of gravity, G is a center of gravity and B is a center of buoyancy, center of buoyancy. See these are acting in a straight line, they are acting in a straight line. This particular figure already you have studied but again I am repeating this to know the exact locations exact you can say point of applications of these points, then some tilting is given then some tilting is given or we can say the body is sure the you can say angle of count changed you can say so that this particular line is here only it is not going to change but this particular point it is shifted over here, it is shifted over here and when you are going to draw a straight line through this point and where it cuts to this original line this point is called your metacenter, this point is called as a metacenter. So metacentric height it is defined as the distance between the center of gravity of the floating body and its metacenter, a larger metacentric height implies greater initial stability against overturning that is the most important thing and we are going to study with taking some examples, there are three stability conditions that you are knowing or equilibrium condition that you are knowing that is a you can say stable, unstable and neutral. So we will see the two conditions we will find out through the problems. So one of the problem just concentrate on this wooden block in the form of rectangular prism floats with its sharpest or you can say shortest axis vertical the block is 40 centimeter long 20 centimeter wide and 15 centimeter deep the dimensions are given and the depth of immersion is 12 centimeter this is very important depth of immersion is very important. So calculate the position of metacenter and we have to comment on the stability. So concentrate will just go through these all the dimensions and then first step we have to calculate the weight of the body, weight of the body is weight of displaced volume of water okay that is 93.98 you will get then you can calculate the OB that is height of center of buoyancy above the base of the block you have to see this okay OB what is OM what is OG all these things okay. So OB that you have to calculate so that will be 6 centimeter because it is 12 centimeter dipped in water 12 divided by 2 it is 6 OG that you have to take the total height 15 centimeter is the height so 15 divided by 2 it is 7.5 centimeter and then you can calculate if M is the metacenter it is BM is equal to I upon VI is the moment of inertia it is bdq by 12 so that you can calculate it is it comes to be 1.278 BM is equal to 2.278 centimeter and MG MG is most important that is metacentric height BM minus BG so it is 2.778 minus 7.5 minus 6.0 it is 1.278 so it is a positive so G you can say it is stable equilibrium it is stable equilibrium then you can say you can go for the second second one second problem see the figure a solid cylinder 2 meter in diameter 2 meter high is floating in water with its axis vertical if the specific gravity of the material of cylinder is 0.65 so that it is floating calculate its metacentric height and also its equilibrium condition okay again concentrate on the figure same case that diameter of cylinder you have to calculate height which is given specific gravity is 0.65 depth of cylinder in water that is specific gravity multiplied by height you will get 1.3 meter distance of center of buoyancy B from O which is 1.3 divided by 2 that is 0.65 meter so center of distance of center of gravity from O it is 2 divided by 2 it is 1 meter BG is equal to OG minus OB from the figure we will get 0.35 bm is moment of inertia divided by volume it is a circular pi d raised to 4 by 64 you will get moment of inertia is 0.75 volume you will get 4.084 meter cube so bm you will get 0.192 meters metacentric height is equal to bm minus BG it is 0.192 minus 0.35 now it comes to be minus okay it comes to be minus negative sign means that is the metacentric height is below okay and the previous example we have seen the it is a positive so the metacentric height is above so okay it is a unstable equilibrium when it is minus it is unstable equilibrium okay there are three conditions stable unstable and neutral keep in your mind there are some review questions very interesting a floating body is in stable condition when when the metacenter is below its center of gravity metacenter is above center of gravity metacenter height is 0 okay and center of gravity below the center of buoyancy see when such questions will be there that rapid questions will be there you must know what is the center of buoyancy what is a metacenter what is a metacentric height what are the different stable conditions all these things you must know very well how the buoyant force is acting how the gravitational force is acting how the behavior of the you can say tilting is given to the particular body how the body is going to tilt okay so everything that you have to visualize yourself visualize yourself the when the axis is when the tilting is given the buoyant the center of buoyancy shifted okay but the center of gravity will be there in that same the previous axis but the shifted the shifted buoyant the center of buoyancy then you have to find out the vertical line from that and which it cuts which it cuts to that particular previous axis that is a metacenter and then you have to find out gm that is a metacentric height okay so all these things you have to keep in your mind then related with that the archimedes principle also archimedes principle what is the archimedes principle that also you have to you must know it very well because if that we already studied in the school days when a where a pot is taken which is full of water a body is dipped in that the water comes out water comes out from that particular utensil or you can say whatever the pot is taken so the weight of the weight of the water collected the volume of the water collected is equal to weight of that body is a simple archimedes principle is given to you and based on that all the theory is depending that is buoyancy floatation everything okay so wherever the questions that have been asked you must know the previous theory behind this and the second question is very interesting and ice cube it's a very common normal thing that is an ice cube is floating in a glass of water as the cube melts the water level what happens to the water level as it remains same it falls rises definitely you are going to visualize it definitely okay so you you just do yourself at home also and the answers are a floating body is in a stable condition when the metacentric high meta center is above the center of gravity okay and an ice cube is floating in a glass of water as the cube melts the water level falls okay the water level falls okay these are some references you can use Bodhisattva you can use Bansal you can use Roy you can use Gupta and you can use Shrutri many many books are available and very different different examples are there so that you can solve many examples related with stable condition unstable condition you will come to know how the analytical method is useful okay if you find any difficulties again you can contact me thank you