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Dimensional Analysis and Uncertainty - Overview

In physics, dimensional analysis is the study of the relationships between different fundamental physical quantities. It is a method of checking the consistency of the basic physical quantities used to describe a system, and to discover possible errors in a set of derived units.

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welcome to Section Four of our math review in this section will be looking at dimensional analysis and uncertainty. We're going to combine these topics because they appear in most science courses and so you've probably seen them before. And also I want Thio. I'm only going to go over uncertainty briefly because every class has a slightly different way of handling it. So I'm going to go over some of the standard ways here. But you should definitely check with your professor and see what way they want you to use. And it may be something completely different that I don't even cover. That's a possibility. So let's start out by thinking about dimensional analysis. So in dimensional analysis, what we're talking about are what are the units that were used that were attaching toe are different measurements and how do we keep track of them? So, for example, we might measure time and seconds distance and meters and mass in kilograms. In fact, these air what are called the S I units there the standard units that we report in in physics notice that in chemistry a lot of times they'll use the Graham instead of the kilogram have to be careful not to make that mistake. Here. In physics, we use the kilogram because we're dealing with larger objects, so that's gonna be 1000 times larger than your standard Graham. Now, the reason we want to look at dimensional analysis is because it helps us with converting between these. You all know that there is more than just seconds that we use to measure time. Sometimes we might use ours instead. So say we had 3000 seconds and we wanted to know how many hours that is. Well, the way we'd convert, as we'd say. I know there are 60 seconds in one minute, and I know there are 60 minutes in one hour. So what I've done here is I've set up some ratios to multiply 3000 seconds and notice that these air technically one because one minute is 60 seconds. So I'm Onley multiplying this by one. All it's doing is changing the units. So I've canceled out seconds. I've canceled out minutes and I'm left with just hours. So I know that 3000 seconds then is going to be equal to 3000 divided by 3600 hours. So a little less than an hour. So this is one way that we use dimensional analysis. Another way is when we have something like meters per second, which is the way that we measure speed or velocity, and we want to convert it to something like, say, miles per hour. What we have to know here is how Maney meters Aaron a mile. How many seconds are in an hour? So I happen to know that there are 1608 m in one mile, and I know there are 3600 seconds in one hour. If you don't have all these committed to memory, don't worry too much. Most classes will provide them to you, but find out if yours does. And if it doesn't check on the inside the front pages and the back pages of your book, there's certain to be a list somewhere. So in order to convert, I'll say I have meters per second and I'm gonna multiply this by 1608 m in one mile. So notice I've set it up. So the meter cancels and I'm gonna multiply it by 3600 seconds in one hour, so notice that the seconds will cancel is well, so I'm left with 1 m per second is equal to 30. 600 by two by 16 08 miles per hour. Uh, two more things to say about dimensional analysis. When you are reading a problem and it tells you to give an answer that is a distance or a mass, make sure that you come up with a solution to your calculation that has the correct unit. The easiest way to see if you've screwed something up is toe watch the units through. A lot of students get lazy and they only write down the numbers as they're doing their calculations. Don't fall into that trap. If you track the units as well, you'll always have a really quick sanity check to see if you've gotten the right answer. The last thing to know about dimensional analysis, in particular with S I units, are the different prefixes that are available. So, for example, we know that kilo means tend to the three, and many of you have probably heard of senti as in centimeter, which means 10 to the negative too. So what we're gonna do here is a centimeter is 10 to the negative. 2 m a kilogram is tend to the 3 g. A kilometer is tend to the 3 m, so it's a pretty simple system to use now. Ah, lot of students don't bother to learn what all the different prefixes are, but I found that it's really helpful. And it's something that some professors will use to just test and see how much you're actually paying attention. Undoubtedly in your book, there is a list of what all these prefixes are. I recommend that you go learn what the prefixes are, what values are associated with them and perhaps even the symbols that are associated them. So, for example, Nano uses a small, uh Nano, not Nana Nano uses a lower case N, whereas Kilo uses a K and mega uses a capital M. Make sure you know what all of these are, and if you do, then you'll never get caught in the easiest of physics traps. Let's move on to uncertainty. The way that many students are introduced to some uncertainty is through something called significant figures. So some of you may have grown when I said that, um and it is something that tends to give people fits and starts later on. In one of the example videos, I'll go over the different rules of significant figures and how to do calculations with them. Um, and it is a legitimate way to track what what your answer is and how maney, how maney significant figures. That is to say how precise your different calculations are. Um, speaking of precision, though, another way that we look at precision is in how many digits we put on the measurements that we take. So, for example, if we have a ruler that only has tick marks at the inches, then we can probably only distinguish between an inch and a half inch and not so much between an inch and a quarter or a third of an inch. With that being said generally, what we would write is that something is either one inch or it's 1.5 inches or it's two inches. So you can see here that the most number of significant figures we could have would be to in this case if we say that is very close to the middle, though we understand with this precision that we're rounding. Okay? In fact, often, instead of reporting as one inch, we might report it as 1.0 inches, understanding that the last digit is being rounded. Um, we can again when you see that and you go back and look at the rules for using significant figures, you'll be able to tell how to use the precision of your measurement. Thio actually get your final calculation correct. Another way that professors will use to handle uncertainty in their classes is when you have, ah, a list of numbers So safe, for example, you did a bunch of different measurements and you got five seconds for one and 5.3 seconds for another. And maybe you have an outlier 6.1 seconds and 4.9 seconds. You have all these different numbers and you say, Okay, what am I going to do? Well, first of all, you've taken average and you report that value well, it's a little more complicated than that. Some professors have different ways of doing it. The way I prefer is to calculate something called the standard deviation. So the standard deviation is a formula that you can look up online or you can type it into your calculator. It definitely has it programmed in. Or you can go to any spreadsheet like Excel or Google sheets, and it will help you to calculate the standard deviation off a list of numbers. All you need to do in a spreadsheet is type equals S T E V, and then list the cell. So if you have a one through a five that you want to take the standard deviation off when you typed that in, it'll give you the standard deviation automatically. What standard deviation is doing is telling you what the average will not necessarily average, but it's giving you an idea of how much the different measurements depart from the average. And that could be a really useful number. Two have and can assign to you what the uncertainty off your measurement is from a list of numbers. The other way you can think about uncertainty is using something that is called propagation of uncertainty. Now, generally speaking, this is a technique that's only going to be used in more advanced classes. But I've also seen it pop up in the introductory physics Siri's. So I'm gonna address it briefly in one of the example Varios later on. But what it's gonna do is it's going to take a more sophisticated look than simple, significant figure calculations about how to take the precision of a measurement. The uncertainty on a measurement So say you have say, Well, I know it's an inch, but that's Ah plus or minus 0.25 inches because I can't determine any better than that. So we're gonna be able to take something like this, this 0.25 inches and then do calculations with it and determine what the uncertainty on your final result is. So there's a lot of rules involved with that. It's a pretty technical video. Um, if you are not necessarily using this technique in your class, I don't recommend that you watch it.