Dimensions and Units

Okay, hi everybody. So today I'm going to talk about the dimensions and units. It's kind of important because, you know, I already started writing it to save some time for you, but the goal here is this. As you know, we have an SI unit. It's SI standard, Standard International. We also have a British gravitational, okay, and it's called USCS as well, United States Customary System. So the question is this. We have to communicate kind of clearly with each other, okay. So what will be the, the, the, which one am I gonna pick, and why, etc. So I'm gonna discuss that in this particular segment, okay. You kind of know this. Let's start writing about the units in the SI. So the Newton, and I designate with N, right. Length is in meters, right, and I have m. Time, seconds. Not minutes, not hours, seconds, right? Mass is kilogram, and temperature, we'll talk about that, but actually Kelvin is the absolute temperature for this, but this will do just fine, Celsius. Okay, let's look at the equivalent in the British gravitational units. So the first is the force is the pounds, okay. And this is the pound force, right. You'll see why I put an f here. You know, this pound is coming from Italian libra, but, you know, this f, we'll talk in a minute why there's an f there. The length, foot. Okay, now here's the, here's the problem. With the British gravitational units, I don't mean problem as deal breaker, but you need to be careful. Here is it. What is the unit of pressure in SI? Pascals, right. What's the unit of pressure in British gravitational? It is psi. But that psi is not pounds divided by foot square. It's pounds divided by inch square. So there's a conversion there, so you need to be careful when you deal with British gravitational. Okay, let me continue my coverage. Thankfully, the time is the same, okay. I'm glad we don't use different time scales, right. The mass. Okay, this is a tricky one now. So, pound mass. Okay, pound mass. But in thermodynamics books, majority of the books, it will be dealing with pound mass. And as you may have noticed already, I'm a fluid mechanics guy, okay. And in the fluid mechanics, we use slugs. All right, just pick a book from the, you know, fluid mechanics domain. So you're gonna see slug, and I'll talk about the distinction in a minute, okay. Just want to, right now, go over it. And looking at the temperature, it will be the Rankine. We'll talk about that, but we mostly use the Fahrenheit. That is going to do just fine for us. Okay, now for the units of stuff that we will deal with in thermodynamics, I will obtain some primary dimensions. Okay. In theory, there are seven of them, but I need, in this course, four of them, okay. Mass, length, time, and temperature. Make sure that you differentiate time and temperature, okay. With these four primary dimensions, I can obtain the rest of the dimensions. I gave you the example of pascal, which is newton per meter square, right. So I simply obtain. You can see over here, force is ma. Well, why don't I do it, right, instead of talking. So let's talk about, you know, I say that this is pascal, right. So let's derive the pascal. So pascal is equal to, basically, force divided by area, isn't it? And the force is m, which is mass, times the acceleration, divided by the area, right. So the mass is M, that is a primary dimension, given to me, I can use it. How about acceleration? Remember, acceleration is length divided by time. So the distance that I travel in a certain amount of time is called the velocity, right, or the speed. And then the acceleration is another derivative of it. So I'm going to get myself L, T minus 2. You can also write L divided by T square. In SI, this will be meter per second square, right? And the area that I have over here is going to be length square, okay. So you can see here, what I obtain, it's kind of simple at this point, but I'm going to get my pascal as M, L minus 1, T minus 2. Okay, and instead of saying, you know, like, let's say, use SI, kilogram divided by length, divided by second square, we kind of go ahead and say pascal, as an example. Okay, but I want to also talk about two important things. And we're going to use this pretty much every day that ends with a Y, okay. I'm going to let you think for a second what I said. Is there any days without the Y, ending with the Y? Yeah. Okay, the first is the energy. And I told you in the previous segment that, well, the thermodynamics course name should have been called energy to begin with, right. So the energy dimension, the unit, is joule in SI and Btu. Okay, actually, let me, I kind of forget that bonus on me. So I didn't discuss why I need to use, you know, why am I dealing with both of them. Okay, so at this point in time, if this was a face-to-face class, what I would do is I would pose this question to students. Okay, you've seen the SI, you've seen the BG. Which one should I use? And typically, in a class of 120 or so, what students typically tell me is 110 of them, or around 115 of them, say that you're going to use the SI. And the rest say that I want to use British gravitational, five of them or something. And then I make a joke, say that, okay, exclusively we are going to use BG in this class. Right, so I pretend that I just made that joke. Anyways, okay, I got it. You like SI, right? I like SI. For my research, I will use microfluidics. We don't have micro inches. We don't use it. We use micrometers, right. But anyways, the point is this, though. You're an engineer, when you go and talk to, let's say that you go to a big box store, right, and you want to buy something, a pipe, right, you're not going to say that I need a pipe with 2.5 54 centimeter diameter. Here in the US is what I'm talking about, right. So you're going to say one inch. So we got to know this. I'm sorry to say that because some instructors don't do it, but I will is we will have to go through this process as well. It's not going to be as common as this, but I will still be exposing you to that. Okay, that's why you can see Btu is posted over here. British Thermal Unit is what Btu is. Now this energy, I don't want to go too far ahead of myself, but related to work and there will be another called heat transfer, we'll talk about that soon. But let's just simply energy is work. Okay, and we kind of know this. Actually, the work is force times distance, right? Force times distance. And the force, you know, let's use SI for the time being. This is Newtons, right? And the distances need a meter. And we can call it the Newton meter for torque purposes, for instance. Okay. But we can also call this joule. Very important. You can use this. Okay, so that's something to know. Now I want to also talk about a little bit about the British Gravitational Units in terms of the force. The reason why I want to do it now, as you can see I have a force term over here. Like I said, it’s Newtons, right? And pound-force is what I have over there. But let me do this. You know, it's kind of interesting because the way that we define pound-force is this way, okay? Acceleration due to gravity is 32.2 feet per second square in the British Gravitational Units. Okay. And here is how we define the. You can see ma, so basically I'm doing mass times acceleration. Mass times acceleration. So this acceleration term is 32.2. And this is what we call pound-foot. Okay. And previously I mentioned that I will also use something called slug, and this is called the slug. Okay. So again, if you're taking fluid mechanics from me next semester, you will be exposed to this. Now you're be getting exposed to this. Sorry, it's not my decision, it's how the books are, you know, using them. Okay. Oh, I should write it, pound-force times foot, right? We call it foot-pounds of torque as an example, for a car, etc. Okay? Another very very important thing that we will definitely confuse, no question about that one, is energy versus power. Okay. And I can almost hear you now. Power is Watt which is joule per seconds. In the British Gravitational, sometimes it's called as Btu per hour. You can see this in the air conditioning units as an example. Okay, but also another one that you see is power. How powerful the car is. Horsepower, right? You know, you deal with it. So we discussed this foot-pounds, right? So I want to talk about this horsepower for a minute here. The reason is, it's kind of, you know, interesting. So basically, this horsepower figure that we use all day long for the cars is coming from this this conversion, actually is 550 foot-pounds per second. Okay, so apparently, which I, you know, I'm just teaching here, is a horse can move 550 pounds, increase its height or potential energy by one foot in one second. Okay. So that's what it turns out to be. So we kind of like an off unit. I'm not sure how they obtained that one, but that’s... Let me repeat: 550 pounds of mass, what I do is I can move it up one foot in a second. That's how much, you know, one horse can do. Again, I don't know whether that's entirely accurate or not, but it is what it is. Okay, the next thing I want to talk about is the weight. This also comes in handy, right? It's force, and force is in basically Newtons or pound-force, right? If I'm using British Gravitational Units, the same thing is what I covered already. So I don't want to talk a lot about it, but it I also gave you the 32.2 feet per second square as the gravitational acceleration. But, you know, I just want to let you know, sometimes students confuse weight and mass. Okay. So this is the relation between those two. So let me write it. Weight is equal to mg. Okay. I don't have a lot to say about this. Okay. But the thing that is important is dimensional homogeneity. So if I have an equation, right, the hardest thing about this dimensional homogeneity is writing the word "homogeneity" down properly without making a typo. Okay. That is the hardest thing. The rest is easy. “Homogeneity.” I think that's written that way. So, but simply, it's a fancy term, but at the end of the day, it says that every term in any equation must have the same units. And we kind of know this from, you know, a long time back. Right? And I'll start with some formula that we have in the fluid mechanics, but it will do. This is the shear stress is equal to viscosity times the rate of change of velocity in the perpendicular direction. Okay. So my question to you would be: what is the unit of this guy? It's called viscosity. When you look at the engine oils, for instance, the first thing that you see is 10W-30, 0W-16, whatever. So this is the viscosity value. So let's try. What is the unit of it? Okay, so the shear stress, as we know, is Newton per meter square, right? And the viscosity is... well, I don't know. So I can use the advantage of dimensional homogeneity and find this. Okay. The reason is the unit of this must be equal to the unit of that. Right? Okay. So then, du, as you know, the mathematical symbols doesn't have units. Okay. So that is going to be meter per second. Why? That's a velocity. u is velocity, and you know, the distance is in meters, right? So you see here what happens. The meters cancel out. So I get myself viscosity unit is Newton-second per meter square. That's actually the unit of viscosity. Okay, so that's something to know. So this is important for all your courses, not just this course you're taking from me. But make sure that the units does match. Okay, in an equation. All right, that's all I have about this particular module. Thank you for listening.