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Dimensional Analysis

The dimensional analysis is a technique for understanding the relationships between different physical quantities by identifying their fundamental dimensions (such as length, mass, time, and electric charge) and units of measure (such as miles versus kilometers, or pounds versus kilograms) and tracking these dimensions as calculations or comparisons are performed.

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dimensional analysis is used to understand how units can cancel to obtain new relationships or information between different quantities. It's very useful for its innovations, especially for obtaining contents like the gas constant or the planes constant and so on. It's also helpful for converting one quantity to another while retaining the meaning of the quantity. So, for example, you can convert one unit of energy to another unit of energy. So in this video, all be going over a couple of examples that you'll see in your country. Of course, eso. You're more familiar with dimensional analysis and how to use it. So for my first example, will be converting a value from kilograms 2 g. So let's say you have 2 kg of some material and you want it in grass. It is out this problem. We want to right our initial value on the side and then multiply these by different ratios literature to get our desired value. So we know that there is 1 kg in 1000 g of material, so if we set up this ratio, you can see that the units of kilograms canceled. So we're left with an answer that is Ingram's so this will be our final answer. So for next problem will be converting a value from millimeters two centimeters. So let's say you want to convert 23.4 nanometers in two centimeters using the same method as before. We'll write our initial value on the side and then multiply this by different ratios to obtain the units that we want. So we know that there is one nanometer in 10 to the negative 9 m. So then if we multiply this r nanometers canceled and currently our answer will be the meeting. But our final answer should be in centimeters, so we need to multiply this by another ratio. So we know that in one centimeter there is 10 to the negative thio meters. So then, if we cancel out the units again, our final answer should be lots in senators. So if we work out the math, our final answer should be 2.34 times ton to the negative six centimeters. And using dimensional analysis, we have been able to convert our value from nanometers to center. So in our third example, will be converting from grams to Adam and you will see this problem a lot, Um, later on in the course. So let's say we have 5 g of carbon and we want to figure out how many atoms are in programs department. So before we begin this problem, we need to know a few pieces of information. So we need to know Theo Atomic mass of carbon, which is 12 g Permal. Now we need to figure out a relationship between moles and number of atoms. So you will use avocados, constant lunch. I'll be going over later on in the chorus. But for now, all we need to know that this constant has this value and the units are atoms. Permal. So using all these pieces of information we can get from grams to Adams. So again, first off, we want to write our initial value on side and then we will be multiplying this by different nations to get the units that we want. So we'll be multiplying this so that our grams cancel out first to get moles so we can write the atomic mass in this manner so that the grams cancel out in our current answer is impulse. And now we want our answer and atoms and that moles. So we wanna make sure that malls are on the bottom and Adams are on the top, in which case is 6.22 times 10 to the 23rd. Oh, so if we cancel out our units, we should be left with our desired unit, which is Adams. And when you do the math, the answer should come out to be 2.5 09 times time to the 23 Adams. So this problem we've been able to successfully convert Grams Thio and now for my last example will be driving the units for the gas concept. Eso There are actually multiple units you can use for the gas constant, but I will specify which one amusing in this particular example. Um so the gas constant is very important for the ideal gas law which is represented by this formula which will carry a bit more meaning later on in this course. But for now, I just wanted to use it as an example because it is very useful and you'll see this later on in the course. So, um are represents the gas constant. He represents pressure. The represents volume and represents the number of bulls and he represents temperature. So if you want to figure out, how do you get are using dimensional analysis? We need to first isolate this. So in a sense, we're kind of working backwards from the result, Um, by using the starting pieces. So Thio isil er, we can divide, um, decide by anti so that we have a new formula p Times V is over on times t. So here we can write our on the side since that is what our internal week. And now we have this ratio different physical quantities. Eso pressure can be in units of Paschal's. Volume is like cube, so we can say meters cube and is moles and temperature is usually in Calvin. So here we're kind of using dimensional analysis in a way that we can derive a constant so that if you're missing a piece of information like pressure or volume or temperature, and you want to use the other quantities to find that this constant allows us to do that because of the units that it has