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Dimensional Analysis and Uncertainty - Example 3

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 first example video on significant figures and uncertainty. So in this video, we are going to look specifically at the rules for using significant figures because it is such a common way of handling precision and uncertainty and physics classes and in science classes in general. Um, you've probably seen some of these before. We're just gonna go over them really briefly here. So first of all, everyone knows that the significant figures in a number like 135 is three. We have three SIG figs here, and the reason for that is because we have three digits, none of which are zero. Now it's the zeros that tend to confuse people. So let's go over those quickly. First of all, there's a type of zero that we would call a leading zero. Something like 00.0 25 We didn't even put another 0.25 It turns out, because thes air leading zeros, none of them contribute to the number of significant figures. There are only two sig figs in this example, and that's because the two digits here, all that the zeros air doing, is telling you where exactly? Those appear in the number line. Okay, What decimal places they occupy. On the other hand, if I were to have what we might call a captive zero, which might look something like this. 0.2005 Notice we still have one leading zero. But now these two zero's air in between two non zero digits. That means in this case, we have four significant figures. Okay, So zeros that are captive or included between non zero digits are counted into the total number of significant figures. And lastly, we have something called trailing zero. So, for example, if I have 25 0.0 versus 25 25 that's just gonna have to Sig Figs. But 25.0 that has three sig figs. The reason being we have a decimal here. By the same token, if we have 100 versus 100 decimal point, this is a one sigfig number, whereas 100 decimal point is a three sig fig number. Okay, including the decimal point means that you want to count the that particular zero. Now, um, that doesn't mean that I could write 25.0 And that would be reasonable, necessarily. There are better ways to do that. We might use scientific notation or something like that instead. Um, but for this basic example, here you can get the idea. Now, the difficulty with six figure, significant figures usually comes when you're trying to do some sort of calculation. So, for example, if you have 1.4 times 4.56 4.56 if we want to multiply these two things together when we multiply two numbers that have different numbers of sig figs, what we're gonna do is we're going to keep the number of significant figures of the least precise numbers. So in this case, that's 1.4. It only has two significant figures. So in our final product, we're only going to keep two significant figures. Similarly, if we're talking about addition or subtraction, I should say this applies to division as well. You only keep the number of significant figures of the least precise number in the calculation. On the other hand, if you're doing addition or subtraction, say you have 12.11 plus 18 0.0 plus one point 013 and we want to know what am I going to keep here? What we're actually gonna look at is the decimal place of the least precise number. So, for example, here we have the least precise number is 18 because it has three significant figures, 12.11 has 41.13 also has four. So this is the least precise digit. Here is this zero. And so we're going to say 31 are actually to hear one than 89 10 11, three. So this is our answer without the correct number of sig Figs. Because this is, uh because this is our least precise. It's are limiting term. It's got the smallest decimal place. Then what we're gonna do is we're gonna say 31.1. We're gonna only stopped at the tens place because this number stops at the tens place. If it had gone to the hundreds, say it had been 18.1 so we'd end up with three here. Then the hundreds place would be where we had stopped, and we have 31.13 So these are basic rules for using significant figures. Um, in Siris of calculations, make sure you pay attention to them.