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Dimensional Analysis and Uncertainty - Example 4

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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Video Transcript

welcome to our final example video on how to propagate uncertainty through a calculation. Now uncertainty is generally determined when you're taking a measurement. For example, um, if you are measuring with a ruler and the best that you can do is to the nearest centimeter, then you might say that you have a measurement of your distance X plus or minus 0.0 1 m. If you're going to say I can't determine between one centimeter and the next, more likely you're going to say plus or minus 0.5 or +005 m, saying that you can approximately tell the difference between one centimeter and 1.5 centimeters. So you have determined your uncertainty here to be 0.5 m. On the other hand, if you're taking a measurement on a scale or some sort of digital device, you know that the last number is probably rounded. So you would say, for example, if your scale gives you a measurement of 9.35 g, then you know that's gonna be plus or minus 0.5 g because you're not sure if it's 3.9 point 359 or if he was 9.35 two or whatever it might be, um, so we wanna make sure that we're assigning the correct amount of uncertainty here Now. The difficulty, then comes when we want to talk about what do we do with that uncertainty If we have to do a calculation? Well, it turns out there are some basic rules that you can follow. For example, if you are adding up a number of different variables, so say you have a Prada in amount. Q. Which is the sum of two measurements? X and Y and X and Y both have some uncertainty on them, so we have X plus or minus. We're gonna put a Delta X and why, plus or minus Delta Y where the Delta X and Delta, where the uncertainty on those two measurements if I wanted to know the uncertainty on cue, it turns out all you have to do is add up the uncertainties on your two measurements. Now this is pretty easy. In fact, if it was a X minus, why we could do the same thing, it would still be uncertainty on X plus three. Uncertainty on why, whether it's addition or subtraction, it doesn't matter now if we were taking product or a question say that we had. Q is equal to X over y UM or X Times. Why even it'd be a little more complicated. So for X over y or for Q equals X times y. If we wanted to know the uncertainty on cue, we would actually calculate it using this equation where we're taking the ratio of all the different elements here. So we have Delta Q. The uncertainty on cue over Q is equal to the uncertainty on X over X, mostly uncertainty on why over why so notice? In order to get the uncertainty on cue, you'd actually have to multiply the Q value over itself where Q is equal to X over y or X times. Why, I, um, Now there's other, uh, other rules here, For example, if you have Q is equal to some constant times X, then that means that the uncertainty on cue is going to be to equal to the constant times the uncertainty on X or if you have Q is equal to X to some power n than the uncertainty on cue is going to be calculated from an equation that's gonna look like this. Okay, so all this is a little complicated. Let me show you how it applies in a actual scenario here. So say we have an equation that looks something like this. We have a force f is equal to mass times acceleration. Plus, we'll call this f not here, some other force. And we want to calculate What's the uncertainty on f here? So we would say, Well, Delta F Well, we have a summation here, so it's gonna be equal to the uncertainty on the amount m A plus the uncertainty on F Not presumably, we haven't uncertainty for each of these measurements over here. So now we need to know what's the uncertainty of this product? Well, we go back and look at our product rule, and we can see that the uncertainty on F then is going to be equal to m a times The uncertainty on em over em plus the uncertainty on a over a All of that added to the uncertainty on f not so. This whole block here came from our product rule for propagating uncertainty. Thes takes some practice to get right. Make sure that your classes actually using these rules before you really try practicing with them.