in section 1.6 we're going to explain dimensional analysis sometimes called the factor label approach to mathematical calculations [Music] and then we're going to use dimensional analysis to carry out some unit conversions which is a good example of how to do dimensional analysis that we can start off with for a given property and computations involving two or more properties let's take a look at this so a quantity of Interest may not be easy or even possible to measure directly but instead must be calculated from other indirect directly measured properties an appropriate mathematical relationship so this is very common we can't necessarily measure something right directly for instance we can't necessarily measure density directly we can measure mass and volume and then calculate what the density is the mathematical approach we will be using is known as dimensional analysis dimensional analysis is based on the premise that the units of quantities must be subjected to the same mathematical operations as their Associated numbers so let's take a look at dimensional analysis from the perspective of conversion factors a conversion factor or unit conversion factor is a ratio of two equivalent quantities expressed with different measurement units okay so when we're converting between two units what we do is we start with just the fact okay so it's a fact that there's 2.54 centimeters is equal to one centimeter and then we can express those as ratios okay so I can literally read this as 2.54 centimeters per one inch or I can read this as one inch per 2.54 centimeters okay and this is 2.54 divided by 1 1 divided by 2.54 so we get our ratio and there's a whole bunch of them that we can get right so there's all kinds of different true facts out there and you can actually just like Google these or you know write down a little scratch paper to be able to know different facts of what equals something else and anytime that you so you just need any kind of fact to be able to convert between them for instance if you know how many pounds are you want to know how many kilograms you could just start with the fact like one kilogram is 2.2046 pounds right once you know the fact that's going to allow you to convert between them you just go ahead and use that ratio to do that so say you have 34 inches and you want to know how many centimeters that is you go 34 inches times 2.54 centimeters and then you put the inch down here in the bottom okay and just like you saw when you were learning fractions and stuff you actually are able to cancel these units so this inch that is in essentially the numerator of this fraction is going to cancel with the inches at the bottom and the unit you're going to be left with is going to be centimeters and then you're going to get 86 centimeters okay and you're going to be able to see a lot of examples of this in our homework and the solution guide that I provided to you so temperature refers to the hotness or coldness of a substance okay and we already talked about there's different ones so we have our Celsius scale we have our Fahrenheit scale um and so we can see that if there's zero and 100 is where water freezes where water boils that's 100 degrees and that if in Fahrenheit it freezes at 32 and boils at 212. well they're 180 degrees here but they're only 100 degrees here so we can see that the unit of Celsius is a little bigger than the unit of Fahrenheit okay and we can use our dimensional analysis to allow us to convert between them because that's important for us because we live in America where everything's in Fahrenheit um so we can do that the SI unit of temperature is Kelvin we mentioned that unlike Celsius and Fahrenheit scales the Kelvin scale is an absolute temperature scale zero Kelvin corresponds to the lowest temperature that can theoretically be achieved okay so on the Kelvin scale water freezes at 273.15 Kelvin and water boils at 373.15 Kelvin um 100 degrees Celsius covers the same temperature interval as 100 degrees Kelvin okay so that means that the change in temperature in degrees Celsius is going to equal the change in temperature in degrees Kelvin but that's not true for Fahrenheit now to enter convert between them between Celsius and Fahrenheit it's a little tricky and that's because zero degrees Celsius is not zero degrees Fahrenheit okay zero inches is zero centimeters zero grams is zero pounds but zero cell degrees Celsius is not zero degrees Fahrenheit because we're allowing for negative numbers with these scales okay so their conversion uh equations get to be a little bit more complicated so if you want the temperature in degrees Fahrenheit and you have the temperature in degrees celsius first you have to multiply by nine uh fifths that's because there's nine degrees Fahrenheit for every five degrees uh Celsius that's because there's 180 degrees Fahrenheit that we saw before for every 100 degrees Celsius here that they're just reducing that ratio down um and then we're going to have to add 32 degrees Fahrenheit and so this is the difference between the zero of Celsius right zero degrees Celsius is 32 degrees Fahrenheit so we have to add that in when we're converting between them if we want the temperature in degrees Celsius and we know it in Fahrenheit we have to subtract out that 32 degrees Fahrenheit and then we have to multiply by 5 9 this time so the reverse of this guy and you can see how all these units are going to play out right this is going to be in degrees celsius after you multiply it by this you're going to cancel the Celsius you're going to be in degrees uh Fahrenheit and then you're going to add degrees Fahrenheit to it your final answer will be in degrees Fahrenheit over here you're going to start off with degrees Fahrenheit you're going to subtract degrees Fahrenheit you're going to multiply by this ratio you're going to cancel those degrees Fahrenheit oops and then you're going to have degrees Celsius for your final answer Celsius to Kelvin is a little bit easier because they are the same size right so we don't have to deal with this ratio anymore because they're the same size so you could just imagine if a degree Fahrenheit was the same size as degrees Celsius this ratio would go to 1 which is the case in degrees Kelvin so now all we have to do is deal with the difference in their zero okay so zero degrees Celsius is 273.15 degrees Kelvin so you have to add those degrees Kelvin to the degrees Celsius in order to get degrees Kelvin you most often are going to see it like this but the units don't actually work out because really we need to multiply by that ratio and that ratio is one degree Kelvin for every one degree Celsius but you see them right at really lazy I did mention this in one of the homework problems you can see where I worked it out the proper way and that's what's happening with these units here when you see that written and I've seen that written in well it seems like every science textbook I've ever seen they never covered that and so right when we're teaching you about dimensional analysis we stop talking about it with this conversion and I don't really like that so that that's what's happening there um so you have your temperature in degrees celsius if you had degrees Kelvin you're just going to subtract that 273.15 and here you can kind of graphically see what's happening with these uh different scales and you can also note that negative 40 degrees Fahrenheit is the same as negative 40 degrees celsius there's actually a clever way of using that fact to interconvert between them and have a different formula as well for this um and maybe I will post a slide with that too but this is this is also interesting right here that they just happen to correspond at negative 40 degrees Fahrenheit negative 40 degrees Celsius otherwise you can see the difference between them here say 0 degrees Celsius is 32 Fahrenheit is 273.15 Kelvin