Archimedes principle and buoyant force

so let's say we have a cup of water let me draw the cup it's one side of the cup that's the bottom of the cup it's the other side of the cup and it's water well let me say say it's something liquid doesn't have to be water some arbitrary liquid it could be water so that's the surface of it so we've already learned that the pressure at any point within this liquid is its it's dependent on how deep we go into the liquid and one point I want to make before we we move on and I touched on this a little bit before is that the pressure at some point it isn't just acting downwards or it isn't just acting in one direction it's acting in all directions on that point so although how far we go down determines how much pressure there is the pressure is actually acting in all directions including up and the reason why that makes sense is because I'm assuming that this is this is a static system or that the fluids in this liquid are are stationary or you could imagine an object down here and it's stationary and the fact that it's stationary tells us that the pressure in every direction must be equal especially let's think about a molecule of water a molecule of water we'll say it's roughly a sphere if I have a molecule of water let me do it in this corner because I'm going to need this space later if the pressure were different in one direction or let's say the prêt if the pressure down were greater than the pressure up then the object would start accelerating downwards right because its surface area pointing upwards is the same as the surface area pointing downwards so the force upwards would be more and would start accelerating downwards so even though the pressure is a function of how far down we go at that point the pressure is acting in every direction so let's remember that and now let's let's keep that in mind too to learn a little bit about Archimedes principle so let's say I submerge a cube into this liquid and let's say this cube this cube and let me draw the cube up here so you let's say this cube has dimensions D so every side is d D a B D is dimensioned D so what I want to do is I want to figure out if there's any force or what is the net force acting on this cube due to the water well let's think about what the pressure on this cube is at different points well at the depths along the side of the cube we know that the pressures are equal right because we know at this step say right here the pressure is going to be the same as at that depth and they're going to offset each other so these are going to be the same but one thing we do know just based on the fact that pressure is a function of depth is that at this point the pressure is going to be higher I don't know how much higher then at this point right because this point is deeper into the water so let's call this p1 that's the pressure on let's call that pressure on top PT let's call this point down here let's call that PD P no pressure on the bottom PB right and what's going to be the net force on this object on this cube well it's going to be the net force the net force let's call it f sub n is going to be equal to the force acting upwards on this object and what's the force acting upwards on the object is going to be this pressure at the bottom of the object pressure the bottom of the object times the surface area at the bottom of the object well what's the surface area of the bottom of the object well that's just d squared right the any any surface of a cube is d squared so the bottom is going to be d squared and then minus and I'm doing this because I actually know we know that the pressure down here is higher than the pressure here so this is going to be the larger quantity and that the net force is actually going to be upwards so that's why I can you know do the minus confidently up here minus the pressure at the top so what's the force at the top the force at the top is going to be the pressure on the top times the surface area of the top of the cube right times d squared and we can even separate out the d squared already at that point so the net force is equal to pressure of the bottom minus the pressure of the top or the difference in pressure times the surface area of either the top or the bottom or really any of the sides of the cube now let's see if we can figure what these are let's say the cube is submerged H units or H meters into the water H so what's the pressure at the top also the pressure at the top is going to be equal to the density of the liquid which I keep saying water but it could be any liquid the density of the liquid times how far down we are so we're H units down maybe H meters times gravity and what's the pressure at the bottom well the pressure at the bottom similarly will be the density of the liquid times the depth so what's the depth it would be this H and then we're another d-down right so H plus D that's our total depth times gravity all right so let's just substitute both of those back into our net force let me switch colors to keep from getting monotonous so I get the net force is equal to the pressure at the bottom which is this and let's just let's just multiply it out so we get P times H times G P H G Plus D times P times G D P G I just distributed this out multiply this out that's the pressure at the bottom and then minus the pressure at the top - pH G and then we learned it's all of that times d squared all right so immediately we see something cancels out ph g ph g subtract cancels out so we're just left with what's the net force the net force is equal to DP g times d squared or that equals d cubed times the density of the of the liquid times gravity well let me ask you a question what is d cubed d cubed is the volume of this cube right and what else is it well it's also the volume of the water displaced right if I stick this cube into the water and the cube isn't shrinking or anything that and let me I mean you could even imagine it being empty doesn't have to be empty but that that amount of water has to be moved out of the way in order for that cube to go in right so this is the volume of the water displaced volume it's also the volume of the cube volume of water displaced this is what this is the density I keep saying water well let's just say it's water but it could be any liquid this is the density of the liquid this is gravity so what is this volume times density is the mass of the liquid displaced right so the net force is also equal to the mass of liquid displace so let's just say mass times gravity or we could say that the net force acting on this object is what's the mass of the liquid displaced times gravity well that's just the weight of weight of liquid displaced so that's a pretty interesting thing so if I submerge anything I the the net force acting upwards on it or the amount that I'm lighter by is equal to the weight of the water being displaced and that's actually called Archimedes principle and that net upward force due to the fact that there's more pressure on the bottom than there is on the top that's called the buoyant force and that's what makes things float so I'll leave you there with with right now just just to ponder that and we'll use this concept in the next couple of videos to actually solve some problems I'll see you soon