 right now our focus is property number 1 which is thermal expansion right now this is a property of every substance and it says that the substance right now the substance will expand okay substance will expand if the temperature of the substance is increased the temperature of the substance is increased what will happen to the substance if the temperature is decreased you will contract okay so whatever was the situation before you will come back to that right so this is what we are going to learn now when I say expand expansion what do you mean by that expansion is expansion of length expansion of width expansion of area or expansion of volume what do you mean expansion of volume expansion of everything all the dimensions each and every dimension will expand fine with length width area or volume everything will expand okay so we will talk about it one by one first we will take the most simplest scenario that is linear expansion right now linear expansion and in a way you know area is nothing but multiplication of two linear dimensions so if the expansion is there volume expansion is anyway there and if expansion is there volume anyway is there okay so they are interlinked linear expansion so have you learned a little bit about calculus little bit like for example something is varying a curve sort of thing then you assume a very very small interval and then you say that in that small interval you can assume it to be linear even though it is a curve okay similarly here the expansion is so less expansion is so less that you can assume that it changes linearly with the temperature what I mean to say is that if you consider length you consider length l if this length changes by delta l for delta t increase in temperature if you change the temperature by 2 delta t how much is my increase in length 2 delta l 2 delta l okay so you assume it to be sequential increase or linear increase with the temperature fine now let's get into little bit more detail suppose I take a rod let's say a rod has length l okay you have provided heat to the rod and its temperature has changed from t naught from t naught its temperature has gone to t naught plus delta t t naught plus delta t so increase in temperature is delta t and length is increased to l plus delta l okay now can you tell me this delta l increase in the length should be proportional to what delta t in immunity that will come in on my delta t it should be proportional to delta t what else delta l should be proportional to what else it should be like it should depend on delta t delta l should depend on material of course because it is a property of a material it has to depend on what is a material otherwise it is not a property of substrates okay what else delta l depends on what else that does it depend on l also delta l if I take half the length l by 2 will delta l be still the same or it will be different if I just take l by 2 length l by 2 okay I divide this rod into l by 2 and l by 2 okay if I increase the temperature what should be the expansion here if delta l was initial it should be delta l by 2 here and delta l by 2 there are you getting it or let me give you a simple way to visualize you take a rubber band and you stretch it if shorter the rubber band same amount of force small extension is there longer the rubber band same amount of force longer extension okay each and every part will expand this l by 2 will expand now so each and every part here will expand but the total length is l by 2 right now so total expansion to delta l by 2 so in a way the expansion depends on the original length also fine so delta l is proportional to whatever was the initial length let's call it as l0 fine now these are the two mathematical relations okay so I can combine these two and I can say that delta l is proportional to l0 into delta t I can say that okay now I will put a proportionality constant okay I will put a proportionality constant and I will say that that propositive constant is specific to the material so this depends on material this thing is taken care by the constant itself getting a point so delta l is equal to l0 alpha delta t where alpha is coefficient of linear expansion alpha is coefficient of linear expansion it is typically of the order of 10 x 1 minus 5 it is very very less and if you see the variation of alpha let's say with temperature this is alpha it goes something like this okay so after 500 Kelvin after 500 Kelvin alpha becomes a constant this proportionality constant alpha even though we are treating it a constant it doesn't remain constant at a lesser temperature it becomes constant after 500 Kelvin or so for a particular material itself okay but then whenever we talk about thermal expansion our temperature will be typically more than this okay so until and unless specified we will assume alpha is a constant for a given material can I doubt now using this let's say you have a rectangular plate you have a rectangular plate of length l and width b fine you are increasing the temperature by delta t okay coefficient of linear expansion is given as alpha okay initial area is what l into b l into b is the initial area find out the final area when temperature is increased by delta t try to derive it what will happen to l how much delta l will be l alpha delta t okay so l will increase by delta l what will happen to b increase by delta b so b will increase little bit like this so the new rectangle will be like this okay so this much is the extra area that comes in because of the expansion fine I want to find out what is you know new area in terms of lb alpha and delta t assume alpha to be very less should I do it most of you have exams going on they have exams going on right before we start the next week this step is coming yeah you will ignore it ignore alpha square because alpha is up to the order of 10 is 1 minus 5 yeah so you ignore alpha square if it is like alpha square plus alpha you ignore alpha square it's like adding 0.0001 with 100 still 100 okay so the new length will be equal to the original length plus delta l can it out here delta l is what l alpha delta t okay so you will get l times 1 plus alpha delta t right and similarly you can find the new width to be equal to original width 1 plus alpha delta t okay so the new area is what length into width again so the new area new length into new width this is equal to l into b 1 plus alpha delta t whole square l into b is what original area right so the new area is equal to the original area and then expand this 1 plus alpha square delta t square plus 2 alpha delta t okay this just ignore alpha square is very very less so you will get new area is equal to the original area in the 1 plus 2 alpha delta t fine so it looks very similar to l is equal to l naught 1 plus alpha delta t okay or you can also write here a n minus a naught which is delta area change in area is equal to a naught into 2 alpha delta t fine so this formula looks very similar to delta l is equal to l naught alpha delta t okay so if alpha is coefficient of linear expansion 2 alpha should be coefficient of aerial expansion fine so we we can also name this constant as beta and we see that this is equal to a naught beta delta t okay where beta is equal to 2 alpha and it is coefficient of aerial expansion end outs but then now similarly what do you think the formula will be for volumetric expansion can you draw parallel and write down the formula for volumetric expansion delta v is equal to 3 alpha delta so you not only multiply length and width you also multiply height so there comes 1 plus alpha delta t whole cube so you ignore cube and square terms and 3 alpha delta t fine so here delta v is equal to original volume into 3 alpha delta t this 3 alpha you can write it as gamma so v naught gamma delta t is gamma is coefficient of volumetric expansion and it out still now so this is the quantitative analysis of expansion we have quantified all the formulas over here no doubt