 We're now going to take a look at the topic of buoyancy, buoyancy within fluids, as well as we'll briefly take a look at stability with the buoyancy force. In the buoyancy force within a submerged object, what we'll do is we'll calculate that and we'll be using the hydrostatic relationship in order to determine what the buoyancy force is. So let's assume that we have a free surface shown like this and we have some object underneath that free surface. What we're now going to do, this is a three-dimensional object and we're going to break that object into a little differential parallel pipe bed. So that would be a differential element within this object and we know that we have a hydrostatic pressure at the top, P1 and one at the bottom, P2. What we will do is we will say the lower surface is at H2 and the upper surface is at H1. And so this is some arbitrary volume, we denote as V and we have an area here, DAH and we will also have an area on the bottom, DA and then the volume formed by this parallel pipe bed we will denote as being DV. And finally, the difference in height between the upper and the lower surface is H2 minus H1. So that's the scenario, we have this parallel pipe bed within a submerged body and what we're going to do now is we're going to go through and we're going to try to determine what is the net force on this little parallel pipe bed in a submerged body. So we'll begin that analysis. So we will refer to that differential force on the differential element as being DFB and the force acting on that body is just going to be due to the pressures, the lower pressure and the upper pressure multiplied by the areas. And this is where we can bring in our hydrostatic relationship and we can rewrite it in terms of depth instead of pressures. The fluid is consistent in this system so the density is the same and if you recall our hydrostatic relationship we have P2 minus P1 therefore we have H2 minus H1 and that is being multiplied by this differential area AH. If we look at our object we have H2 minus H1 times H that's essentially equal to the volume of that differential element. So this here is nothing more than just DV so we can rewrite this expression in the following way actually that would be minus DV because we have H2 minus H1 and H2 is lower so that this is further down than H1. So we end up with rho G times DV and if we integrate that we want to integrate that across that entire body so what we end up with for the buoyancy force is rho G times the volume of the displaced body and I'll put a little comment here H2 minus H1 is negative because H2 is more negative than H1 so if you're wondering how I did that sign change that was how it was done and that was using the hydrostatic equation. Now if the body is floating then the buoyancy force needs to be equal to it's equal to the volume displaced and that also needs to be equal to the weight of the object. So this would give the equilibrium for a floating body and we know that the buoyancy force is this but it also needs to be equal to the weight so that would be for the case of a floating body that's not moving up or down but it's just sitting in the water and with this we can write out a few rules so with that we can write out a couple of different rules one is a body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces which we can see from this expression here that's essentially what that is telling us the other one is that if there is a object floating in water the amount of water that it is displacing or floating in any liquid is equal to its own weight and and that's where you get the stability so if you have some object floating under here and it's not moving what happens is you have the weight but you also have the buoyancy force acting there and if those two are equal the object won't move the buoyancy force is equal to the volume displaced of the liquid and the weight is the weight of the object and that gives you the equilibrium and so those are a couple of the rules that we have for buoyancy and the last thing that we want to take a look at is relates to stability and in this diagram here I've drawn a very stable system that doesn't always occur and so a few words about stability it's important in terms of where the buoyancy force acts and where the center of gravity of the object is and so what I'm going to do is sketch a couple of scenarios so let's say we have an object like that and we will have the exact same object also tilted so you can imagine this perhaps as a ship haul or something like that and we have a liquid that it is sitting in now imagine the scenario where the gravity or the weight so the center of gravity of that object is there and let's say the buoyancy force is acting here that would be a case where you would have a moment acting in this direction and that would turn the object upright or upwards so that would be a stable situation versus if we had something like this again we have the weight so at the center of gravity we have mass times gravity but if it turns out that our buoyancy force was over here then the moment that we would get would look something like this and consequently that would continue to rotate the object that would be an unstable situation and consequently so the location with where the center of gravity is with respect to the center of buoyancy is very important for stability of an object that is floating and and that leads to either a stable configuration or an unstable one so obviously the desirable one for ships and things like that is to have stability so that it does not overturn when it has for example wind blowing from the side or something like that so that's a word on stability and buoyancy you can use them to calculate the forces and submerged objects