hello guys so this video is the theory part of the cpu of the cv lab first let's review the left setup so we have a jar with a tube on top of it and we have a tiny metal cylinder inside the tube which can move freely up and down and we assume there's no gas escaping from the side of the cylinder when the system is at rest the gravity balance style to the pressure difference inside and outside so the cylinder will stay somewhere in the tube but when there's a perturbation to the sitting the position let's say if the cylinder is being moved up a little bit then the pressure inside the jaw will drop a little bit so gravity will wing and it will and the center will start falling and when the cylinder is moved down a little bit then because the gas inside the jar is compressed then the pressure will increase and due to the increase of the pressure the ceiling that will start moving up so you can imagine when there's a perturbation to the position of the cylinder it will start oscillating around the equilibrium position okay so now we have the cylinder oscillating inside the tube and according to experience the period of the oscillation is relatively short by seeing relatively short timing during each period of the oscillation the heat conducting through the side of the jar can be treated as zero which means we can treat the compression and decompression process of the gas inside the jar as adiabatic process and for an adiabatic process we will have this equation which is pv gamma pv to the gamma equals some constant whereas gamma is the is the constant we are looking for so gamma equals cp over cv so we want to find gamma and pv to the gamma equals some constant so if we differentiate this equation we will get this line this is the differential equation of p and v and if we replace dv so remember v is the volume of the gas inside the jar so the volume will be changing due to the perturbation to the position of the cylinder and even can be returning to pi r squared times d h whereas r square r represents the radius of the cylinder th represents the perturbation to the position of the cylinder then by replacing dv with pi r squared times th we will get this equation over here okay so still this equation looks ugly but i it starts looking familiar to us because if we times pi r squared on both side then the left-hand side becomes df pressure times pi r squared pressure times area give us the force right so df represents the change of force the change of force applied to the cylinder from the gas inside the charge due to the perturbation to the position of deceiving there so df equals to negative gamma ray to the negative one times p times pi the square of pi r squared so i boxed all these parameters here and the left turn is the ph which is the perturbation to the position so if i replace all these parameters with a parameter k with a constant k because all these are sort of constant because we only have a small perturbation so v and p doesn't change a lot and r and gamma are definitely two constants so we can create this whole part so there's a sort of a constant and if we call it k then we will end up with df equals negative k times dh so i think all of you are familiar with this equation because it's just an equation the dynamical equation for the simple harmonic oscillation and for simple harmonic oscillation we have the period equals 2 pi times square root m over k so remember here m represents the mass of the cylinder k is this bunch of parameters and if we combine these two equations and plug k e this is what we will get we will have gamma equals 4 m v over p naught times r to the force times t squared and if we replace r with the diameter of the cylinder which is two times r then we will end up with gamma equals 64 times m v over p naught d to the fourth times t to the t square so to measure gamma which is cp over cv we only needed to measure the oscillating theorem of the tiny cylinder inside the tube okay so how do we find the period when the cylinder is oscillating inside the tube we would naively think that the change of the pressure due to the oscillation is given by this equation i let me not use p naught because it's not p node it's just the sum amplitude so the change of the pressure equals to some npq times this cosine function and the period is given by 2 pi over omega so actually that's what we do so if if we can measure the pressure inside the jar versus time and the we would expect it to be a sine or cosine function looks like this so we just defeat this function with this formula over here and we will get this fitted parameters like a omega 5 but the only thing matters is omega because omega can directly tells us what the t is but this is not the real case we have actually when you measure pressure versus time you will notice it's not a it's not with com stands amplitude what we will have is a decaying oscillation which looks like this so the mp2 keeps decaying so instead of using this dp equals a cosine something we should be using we should be using the 10 dollar solution function to fit the data we got so this is actually how you process the data so you will be offered with all the this data point and if you plot in some software like m style you will get this damped oscillation pattern and you need to do the fitting by yourself and using this equation this specific equation to fit this function and find out what omega is and using t equals 2 pi over omega to find out what's the period of the oscillation and use that period of oscillation plug it back into the equation i just proved it to you and use that equation to find out what gamma is okay so now so now let me show you the process of this lab as you can see we have a jar a very big jar here on top of it is a tube and we have a tiny cylinder inside the tube now it's just floating there and we have this l pump connected to the jar which is used to pumping air into the jar because we wanted the pressure inside the jar and to be stable if we don't have the l-pump and then due to the the leak of the air through the side of the cylinder the air is leaking slowly so if we don't have the l pump then the cylinder will start moving down slowly [Music] but you don't want to uh like turn uh turn on the air pump too much because if too much air is flowing in then the cylinder will be shoot out okay and we have a pumping squeeze bulb right over here so when we squeeze it then we suddenly change the pressure inside the tube and the cylinder will start oscillating let me show you so as i squeeze the pumping squeeze bulb then i perturb the the position of the cylinder and it i put actually i perturbed the pressure inside the jar so the cylinder starts oscillating just like this okay and we have this sparkling platform which is being used to do the data collection it's connected to the jar so it can measure the barometric pressure inside the jar and we use a software also called passcode to to get the data collected now i've created this plot which is pressure versus time as you can see if i start recording and do a very quick squeeze i will get this stamped oscillating pattern for the pressure versus time so now this is one data set like we've collected for l so it can be used to measure the gamma of l what you need to do then is just to fit this data with the oscillation function i give you but remember in that function we have dp equals something right and that dp represents the p that deviates from the average pressure so for example in this case like you can you can know what's the average pressure through this part where the oscillation has been damped out right or you can directly fit this uh this data with a function similar to that equation but uh plus an extra constant term which represents the average pressure over here so as you can see we also have some gas tanks over here we have like carbon dioxide argon and some other gases so instead of using the air pump we can also use this gas tank which means we fuse a jar with some kind of gas and this gas tank this high pressure gas tank serves as the uh the l pump so they they are used not only to fuel the jar with some kind of gas but also to keep the pressure stable and then we can measure gamma for other for gases other than l