Bernoulli's equation derivation part 1

let's say we have a pipe again this is the opening and we have fluid going through it and the fluid is going with a velocity of V 1 the pressure entering the pipe is P 1 and then the area of this opening of the pipe is a 1 and then it's a pipe and it could even go up it goes up and the other end is actually even smaller and let's say that the the fluid the liquid is exiting the pipe with velocity V 2 the pressure that it exerts as it goes out let's say if you know if there was like a membrane on the outside how much pressure would exert on it as it pushes it out or whatever or on the adjacent water is P 2 and the area of the smaller opening it doesn't have to be smaller as a 2 and let's say that this opening is at a height of I don't know let's pick ax let's say this opening is at a height on average of h1 and let's say that this opening so the water exiting this opening is on average at a height of h2 and we won't worry too much about the differential between the top of the pipe in the bottom of the pipe we will assume that it these HS are much bigger relative to the size of the pipe let's say this is h2 so with that set up and remember there's fluid going through this thing it's fluid going through this thing so with that set up let's let's go back to what would keep showing up which is the law of conservation of energy which is in any closed system the amount of energy that you put into something is equal to the amount of energy that you get out so energy in energy in is equal to energy out well what's the energy that you put into a system or that the system starts off with at this end well it's the work that you input Plus the potential plus the potential energy at that point of the system plus the kinetic energy of that at that point of the system and then we know from the conservation of energy that that has to equal that has to equal the output work plus the output potential energy plus the output kinetic energy a lot of times in the past we just said that the potential energy input plus the kinetic energy input is equal to the potential energy output plus a kinetic energy output but we could also so the initial energy in the system can also be done by work so we just added work to this equal this equation that says that the energy in is equal to the energy out so with that information let's see if we can do anything interesting with this pipe that I've drawn so what's the work that's being put into this into the system well work is Force Times distance so let's just focus on this it's the force in times the distance in and let's say that so over a period of time T what is what what has been done so we learned in the last video that over a period of time T the fluid here might have moved this far and what is this distance this distance is the input velocity times whatever amount of time we're dealing with so T so that's the distance and then what's the force well the force is just pressure times area and we can figure that out by just dividing force we could divide this F by area and then multiply by area so we get input force divided by area input times area input right I just divided and multiplied by the same number that's pressure that's area is equal to the input the input distance over that amount of time and that's velocity times time times velocity input times time so the work input is equal to just so the work input is equal to pressure the input pressure times the input area times the input velocity times time right and what is what is this area this area times velocity times time times this distance well that's the volume of fluid that flowed in over that amount of time right so that equals the volume of fluid fluid over that period of time so we could call that volume in or volume I I keep switching between the 1 and the I but that's the input volume right and we know what we know that density is just is just mass per volume or that or that volume times density is equal to mass or we know that volume is equal to mass divided by density so the work that I'm putting into the system I know I'm doing a lot of crazy things but it'll make sense so far the work that I'm putting into the system is equal to the input pressure times the amount of volume of fluid that moved over that period of time and that volume of fluid is equal to the mass of the fluid that went in at that period of time so we could call that the input mass divided by divided by the density right hopefully hopefully that that makes a little bit of sense and as we know the input the input volume is going to be equal to the output volume so the input mass because the density doesn't change is equal to the output mass we don't have to write an input and output for the mass the mass is going to be constant in any given amount of time the mass that enters the system will be equivalent to the mass exit system so there we go we have an expression an interesting expression for the work being put into the system what is the potential energy of the system on the left hand side well the potential energy of the system potential energy input is going to be equal to what it's going to be equal to that same mass of fluid right that I talked about same mass fluid times gravity times the this input height the initial height times h1 and what's the initial kinetic energy of the fluid well that equals the mass of the fluid this mass right here that of that same cylindrical volume that I keep pointing to times the velocity of the fluid squared this is just we remember this from kinetic energy divided by two so what's the total energy at this point of the system over this period of time how much energy has gone into the system well it's going to be the work done which is the input pressure I'm running out of space the actually let me erase all of this and I'll probably have to run out of time too that's okay better than being confused let me erase all of this I want to erase everything dum de dum de dum de dum hope I'm not boring you okay we're ready to go now so back to what we were doing so what's so the total energy going into the system is the work being done into the system and I rewrote it in this format which is the input pressure the input pressure will call that p1 times the mass divided by the density of the of the liquid whatever it is so this is work in plus and what's the potential energy well I wrote it right here that's just MGH where m is the mass of this volume of fluid H is its average height you can almost think of the center of mass how high is the center of mass above above the surface of the planet where since we have a G here we assume we're on earth so this is and this is h1 because the height actually changes so this is potential energy input plus the kinetic energy M v1 squared over 2 so that is the kinetic energy input all right and we know we know that this has to equal the energy coming out of the system right so let me clear up some more space here now that we wrote all that down neatly you can get rid of it on this side and we'll have space for our equations I didn't have to delete this I probably won't use this space up here so this is going to be equal to the same thing on the output side right so this is going to be equal to the work out so that'll be the output pressure times the mass divided by the density plus the out the output potential energy which will just be mg h2 plus the outbound kinetic energy which would be M v2 squared divided by 2 I just realize I'm out of time so I will continue this in the next video see you soon