Hi, everyone. Welcome to our lecture on Dimensional Analysis. First, let's start off with the definition, what we mean by a dimension. A dimension is going to be some attribute that just measures a quantity. I understand this is a little bit weird definition, but it's just some inherent way that scientists want to measure something. It's very broad and that's intentional. We're going to talk about the things that we want to measure. So for example, what are some things that you want to measure? Well, perhaps you want to measure a distance. The thing you want to measure is the dimension, in this case distance, and we would use the idea, the notion of length. Another thing you might want to ask yourself is, well, how long does something take? Well, in this case, we have the notion or the dimension of time. So think about the basic questions that a scientist, or a researcher, or someone who's trying to model something would ask, and then those are your dimensions. So for example, if I wanted to know how hot something was, we have the dimension of temperature. If I wanted to know how many they were, we have the notion of population. I add this last one here to specify that you can have a dimension that's a scientific dimension, like a distance, how long, how much time, how hot it is, so you can have a non-scientific dimensions as well. In the scientific world, if I want to know how many they are or how much of the amount of something I have, we can also use the notion of amount. Chemists use this all the time. Why this is a weird definition for us is because we're not used to talking about these ideas in the abstract without immediately putting units to them, and this is the next definition that we're going to have. So units are a way to assign numerical values to the measurement of a quantity. This is what we're more familiar with. For example, let's go back to our dimension of length or distance. What are the units that we measure distance in? Well, of course we measure them in meters, or miles, or centimeters, or yards. Pick your favorite one, there are lots of them. The more science, perhaps you know, the more units you can add, for example, you can also have kilometers, or light-years, or inches. There's a ton of these things. Units of time, seconds, minutes, years, days, weeks. Units of temperature depending on what side of the pond you're on Fahrenheit, Celsius, your scientist, Kelvin. There's other ones of course as well. How many are population, this is interesting now because you have the scientific one and the non-scientific one, it shows populations you can count how many cats, and dogs, and people you have. This is your non-physical dimension. But usually in the scientific world, the amount of things you have, who remembers the basic chemistry? This is usually measured in moles. So the amount of something, and there are of course more than that too. So be careful, what I want to really do here is distinguish between dimensions and units. Unit is a way to assign numerical values. Dimension is more of the umbrella term, the abstraction, the unit less notion, less something we want to measure. There are two kinds of dimensions. There's fundamental dimensions and then there's a thing called derived dimensions. So let's define fundamental dimensions. Fundamental dimensions are dimensions that cannot be expressed in terms of other dimensions. These are foundations. These are the dimensions that build the other dimensions. So for example, let's say a couple of you wants your time. Time is measured in seconds or length. We use the notion of meters to measure these things. These are fundamental dimensions. There are more, but let's start with these two. What's the difference between them? Well, derive dimensions, these can be expressed in terms of other dimensions. So what's an example of a derived dimension? Well, how about velocity? If you think of a units for velocity, what is the velocity? It's a measure of distance over time. So in terms of its units, we would use the units for distance using meters and time per second. So velocity is a derived dimension from the fundamental dimensions of distance or length and time. Another example is acceleration. Acceleration is of course, velocity over time, and velocity, we said in terms of its dimensions, this is distance over time, over time. So if you put these together, you get something like distance over time squared. Oftentimes the units you'll see for this are meters per second squared. What's to follow, it's more common to write this as meters times seconds_negative two. We don't like working with fractions, we're using and write this with negative exponents. There's one caveat that I want to point out here for derived dimensions, and this is maybe just a little note. When you have a derived dimension, the powers, the exponents, they must be integers. So positive and negative is okay, zero is okay as well. When there's nothing written, of course it's an imaginary number 1, but you have to have integers. You can't have one-half, you can't have pi. You only get integers with these things for derived dimensions. That's an important note as we go forward through this video. Scientists recognize that there are seven fundamental dimensions, so they call them fundamental quantities, fundamental dimensions. The first one we talked about is length. You may remember from, I guess physics, the SI unit of length is the meter, denoted by the letter m, mass, another fundamental dimension, you have the kilogram, whose symbol is kg, time, how do we measure time? SI unit is the second denoted with an s. Other ones that you may not thought about, electric current, this is measured in amps known with a capital A. Temperature, now I have a problem here because I already used T for the temperature. So we go to Kelvin. Kelvin is a good one, not Celsius, not Fahrenheit, Kelvin, symbol's a K. Often the variable that we'll use for this will be Theta if we're just talking about it in general because I use T for time. So we introduced the Greek letter Theta for temperature. We said before we have the amount of a substance and the scientific way, this is a mol. Scientific unit of substance is a mol. I'm bringing back some memories of maybe chemistry class there. The last one is luminous intensity, how bright something is and they use candelas, denoted by cd for that. These are the seven fundamental quantities. Now, from these, scientists go off and measure everything and you say, "Well, how many derived dimensions are there?" As many as you want. Here's a table from a standard textbook that has some derives dimensions in addition to some fundamental dimensions. In particular, things that you might think that are fundamental are not, unless they appear on that table on the prior slide with the seven fundamental dimensions. In particular, energy is not fundamental. You can think of it as a force over a distance. You can measure it different ways, electronvolt is one of them. We saw before that mass and length, they are fundamental, but speed or velocity is another derived one. The velocity is your distance over time. Some other ones here, electron charge, you can measure this in terms of coulombs, but that is another force. We have many different derived dimensions. You start putting anything together that's not on that seven fundamental and they're all going to come from those. It's really interesting how just seven fundamental dimensions gives rise to all these units in physics or chemistry or any other science. This is a little bit like how the prime numbers build all the other numbers, or how with the music give a scale, and from that scale you can create symphonies. These are your building blocks to go off and calculate your derived dimensions. One last little story about why we care about units. If you've taken another math classes before, then perhaps the teacher always says the questions don't have units or we don't deal with units, you deal with that in another class. It's important for a couple of reasons. One is that units really matter. If the number doesn't make any sense, if you're just spitting out the number that the computer is calculating, then you don't really understand the problem. Usually, the focus in your other courses are just to the theory. The idea is the methods, but these quantities have dimensions, and the dimensions have units associated to them, and they play an important role in understanding the difference between what the human brings to the problem versus what the computer brings to the problem. The calculation of the number solving for x is something that a computer does very well. Interpreting the numbers, understanding what that number means, is the number reasonable, throwing out extraneous solutions because they don't make sense, that is something a computer does poorly, and you, the human, have to come in and look at that number and see if it makes sense. When the human fails to do that, you can get some really disastrous results. For example, in every lecture or story about units is always going to include this one so I don't want to be any different. But there's the wonderful story of the Mars Climate Orbiter in 1999 at the cost of a $125 million. They learned that NASA was using the metric system which is standard in the scientific community, but the contractors, Lockheed Martin, which is an American company, they used English units of measurement. The units were off and that led to the Rover getting too close to the atmosphere. So $125 million in your years of production and testing and the units were off from the contractor to the company and they didn't check it, and so when they were doing their calculations, when they are close to Mars, the Rover just bounced off, they got too close to the atmosphere, bounced off or either broke on impact or crashing of the planet. But it was just gone. Just gone because they forgot to care about units. This one is like this story that every textbook tells, and if you want to read about it, go read about it more. No one was onboard, no one got hurt. It's just a very expensive, costly mistake. Another example that perhaps is not told as much is the conversion, is the story of the Gimli glider. If you want to look that up and get all the background behind it, but it's an Air Canada. I'll write it down here, Gimli glider. It was a passenger jet that was refueling in Canada and was using different units for how much fuel that had. It was once again, switching between imperial and metric. The flight went onboard. They were flying, I think, across Canada, and they thought they had twice as much fuels they had. It turns out they ran out of fuel. In mid-flight, the left engine went and they're like, "Okay, one engine goes, I guess that happens. They practiced for it and they trained for it." Then all of a sudden, the second engine left. So mid-flight just absolutely no fuel. This is a long time ago when this happened. This was back in 1983. You had a full passenger jet with no engines flying, and they basically became a glider. They did an emergency landing at a Royal Air Force Base called Gimli, which had been converted, I think, to something like a race track, if you read the story, where there were people and children all around. It landed like a glider. Thankfully, a couple things going right for this flight. The pilot was an experienced glider pilot and was able to land the plane. Nobody was killed in the incident. The plane actually landed and I think two people had minor injuries or something. But it was an amazing story, it inspired a movie. But imagine being up in flight and having both engines go out, and this plane, this massive passenger plane have to make an emergency landing as a glider plane. If you want look at the Gimli glider, and again, just another problem of when people stopped remembering that units matter, and it led to an emergency plane landing. Well, thankfully, no one was hurt. These are the idea of dimensions. We're going to see another video that we can use these dimensions to actually calculate some units. But I want to introduce this concept that is often skipped in other courses. If you want, go read about the Mars Rover, go read about the Gimli glider, and there's other examples as well. Look out for those and we'll see you in the next video.
