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80: Physical Dimension (Dimensional Analysis)
1.1k Plays1 year ago
The history of mathematics, in many ways, begins with counting. Things that needed, initially, to be counted were, and often still are, just that; things. We can say we have twelve tomatoes, or five friends, or that eleven days have passed. As society got more complex, tools that had been used since time immemorial, such as string and scales, became essential tools for counting not only concrete things, like sheep and bison, but more abstract things, such as distance and weight based on agreed-upon multiples of physical artifacts that were copied. This development could not have taken place without the idea of a unit: a standard of measuring something that defines what it means to have one of something. These units can be treated not only as counting numbers, but can be manipulated using fractions, and divided into arbitrarily small divisions. They can even be multiplied and divided together to form new units. So where does the idea of a unit come from? What's the difference between a unit, a dimension, and a physical variable? And how does the idea of physical dimension allow us to simplify complex problems? All of this and more on this episode of Breaking Math.
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[Featuring: Sofía Baca; Millicent Oriana, Jacob Urban]
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Transcript
00:00:00
Speaker
The
Foundation of Mathematics: Counting and Measuring
00:00:00
Speaker
history of mathematics, in many ways, begins with counting. Things that needed initially to be counted were, and often still are, just that, things. We can say that we have twelve tomatoes, or five friends, or that eleven days have passed. As society got more complex, tools that have been used since time immemorial, such as string and scales,
00:00:18
Speaker
became essential tools for counting not only concrete things like sheep and bison, but more abstract things such as distance and weight based on agreed upon multiples of physical artifacts that were copied. This development could not have taken place without the idea of a unit, a standard of measuring something that defines what it means to have one of something.
00:00:34
Speaker
These units can be treated not only as counting numbers, but can be manipulated using fractions and divided into arbitrarily small pieces. They can even be multiplied and divided together to form new units. So where does the idea of a unit come from? What's the difference between a unit, a dimension, and a variable? And how does the idea of physical dimension allow us to simplify complex problems? All this is more on this episode of Breaking Math. Episode 80, Physical Dimension.
Introducing the Hosts and Setting the Scene
00:01:06
Speaker
I'm Sofia, and you're listening to Breaking Math. With me I have on the host and co-host of Nerd Forensics, I have Millicent Oriana. Hey, how you doing? Oops, sorry. Hey, how is everybody doing? Yeah, I bumped Millie's microphone. And how about you do Jacob Urban? I'm good, yeah, I'm here. Let's get to the math. So let me ask both of you, name a unit or a dimension, a physical dimension.
00:01:34
Speaker
Hogs head hogs said yeah, which is I believe 31 gallons or 62 something like that. It's like it's a lot. I know that 63 gallons. That's what's really funny. The Simpsons when grandpa says I get three rods to the hogs head. Yeah, because it's like that car burns like 60 gallons to move like 10 feet.
00:01:56
Speaker
And actually, let's convert that. So here's the thing. And that's the thing that's cool. You know, rods to the Hogshead, it's in units of what? A rod is distance, right? Yes. Or length, whatever you want to call that. Often the literature that's referred
Humorous Units and Practical Conversions
00:02:10
Speaker
to as length, not distance. And Hogshead is a measure of what? Liquid. Well, specifically a measure of volume. Because you can have a Hogshead of air or a Hogshead of... Hogsheads.
00:02:23
Speaker
Yeah. Yes, you could. A hogshead of hogsheads. It would never speak of the hogshead of hogshead. But you can convert this into miles per gallon because a mile is also a length. I mean, the gallon is a volume measurement. Oh, it's OK. So 40 rods, 40 rods, logs, which is let's see.
00:02:45
Speaker
said 63 yeah so if we so if we have if we know that it's um 40 rods to the hogshead it's going to be less miles per hogshead right yeah well it depends actually because it says here a hogshead of wine is 63 gallons while hogshead of beers for you oh yeah but i'm just saying let's assume for the problem there's 63 yeah what's your intuition is uh if if 40 rocks to the hogshead would that be fewer miles to the hogshead or more miles to the hogshead to the hogshead
00:03:12
Speaker
Sure. Yeah. Oh, yeah. So
Historical Measurement Systems and Standardization
00:03:15
Speaker
then we're going to do is we're going to divide by 320. So now we're left with hogs heads and gallons. So if I have a certain number of, you know, distance per gallon per hogshead, am I going to be able to go a further distance or less distance with a gallon less?
00:03:31
Speaker
Yeah, exactly. So we divide that further by the number of gallons in a hog's head. So it's 40 divided by 320 divided by 63, which gets us 0.00198 miles per gallon, which is equal to about 504 gallons per mile.
00:03:59
Speaker
Now let's say you wanted to convert this into feet per gallon, how would you do that? Remember, this is miles per gallon, so how do you do feet per gallon? You divide it again. Well, would it be fewer feet per gallon or more feet per gallon? Fewer. Oh, sorry. Sorry, more. Yeah. I've got to be careful with these. Yeah. So we multiply that by 5,280, which means it's 10 feet per gallon. If we collide.
00:04:26
Speaker
Oh, so, yeah, that's that gives us an idea of how we manipulate these numbers. You know, as we're using multiplication division, we're kind of seen like, you know, when we went from a large unit, the hogshead to a smaller unit, the gallon, it decreased the thing that we were dealing with. Right. Yeah. But only because it was the reciprocal. It was it was we were dividing by gallons. Right. Yes. So we're going to. So we're going to talk about, like, you know, why this is on this episode, basically.
00:04:58
Speaker
Jacob, you're on the episode about counting. So what can you tell us just briefly about counting? It's basically a way to organize like a system of units. Yeah. Knowing if you have more one thing. Yeah. Yeah. And what's interesting is that the history of measurement in early times, like, you know, you have to measure length for certain things. Right. If you're making a drill, for example, you're going to want the shell to have kind of the same curvature as the end of the the piece of wood that you have. Right. Yeah.
00:05:28
Speaker
Yeah, in one of those kind of bow drills. You're going to need to measure things all the time. You're going to need to like, you know, do weight intuitively. Like if somebody is like, you know, like this fish is bigger than that fish. So we use our bodies is like that was our first way of measuring length and it was just our bodies, our eyes, things like that. You know, time actually was the kind of the last thing to be measured accurately because it requires a clock to be measured accurately.
00:05:51
Speaker
It wasn't really measured accurately until Greek times where water clocks were used for court cases, which was standardized in Roman times. Have either of you heard of this? No. No. That's interesting. It was a conical vessel so that the pressure would be equal the whole time and a court had, like certain parts of the court had to last certain amounts of time on this water clock. But yeah, so the way that masks started being measured is a balanced scale is one of the simplest devices you can do. You just need something rigid and two things that you can hang from it that are the same. Like a teeter totter.
00:06:20
Speaker
Yeah, exactly. It's like a teeter-totter with two things hanging from it. So the Egyptians, they used what was called a deben as a standard weight in Egypt, and it was first used to kind of measure gold. One early mention of the deben talks about, it's on the juridical stella from the reign of Neburirab I. He had a debt of 60 gold deben, and so he gave up his governorship of el-Kab, early example of buying a government office.
00:06:46
Speaker
Then in the 19th Dynasty a slave girl, priced 4 Devon and 1 kite of silver, was paid for with 6 bronze vessels, 10 Devon of copper, 15 linen garments, a shroud, a blanket, and a pot of honey. So it's interesting basically to get all from all this, weight was necessary at first to measure things having to do with money.
00:07:06
Speaker
Yeah. Yeah. Then you have like, you know, the idea of a stone, which was used to measure like weights, and it was originally 14 pound stone, which was standardized. But that is to say that all these different ways of measuring things became more and more standardized. You had the qubit, which was a length between an elbow and the tip of the middle finger. Yeah.
00:07:29
Speaker
But the king's qubit, what's interesting is that they added another hand to it, another four inches, because the king has to have big arms.
00:07:38
Speaker
Yes, yeah, because he's important. Look at how long my arms are. Yes, we were talking about you, Noxy, that's why. Yeah, Noxy has a king's cubit. Yeah, the cat is meowing. And the Chinese actually used a type of clock called the incense clock as well, where powdered incense was put on kind of a spiral thing around. Interesting.
00:08:00
Speaker
Yeah, and what was cool about it, too, is that if it had different periods of stuff, it used different types of incense for different periods of time. Which, you know, I think is pretty cool, actually. I kind of wish I had that.
00:08:12
Speaker
Yeah, they would like burn at different. So like different types of like. Yeah, it's. Depending on. Not even. Let me clarify. Actually, what I mean. And what I mean is like, let's say you had, you know, one cent for the first two hours and then another cent for the last three hours. So then when you smell. Oh, yeah. The time has reached that. That's pretty cool. Yeah. That's a cool little
Dimensions, Units, and Balancing Equations
00:08:36
Speaker
little thing.
00:08:37
Speaker
Yeah. And, um, so yeah, this kind of, you know, start this continued for a while. Um, essentially you have all these units being standardized over time, eventually getting to the point where you're standardizing other things. So not just immediate things that you could measure, such as, uh, you know, with your eyes, such as like, you know, a volume, you know, which is a cube of length, you have length times length times length is volume or light, which is a derived unit, obviously from length.
00:09:03
Speaker
or weight, but now you're measuring things such as temperature. Temperature was originally thought in the 1500s to be something that you couldn't really measure because it was so subjective. People pointed to the idea of dry colds versus wet colds and stuff like that, but it turns out that Newton and Romer figured that out in the early 1700s.
00:09:25
Speaker
The way that you measure temperature is by seeing how much your thermo-metric material changes volume. Some sort of thermo meter. Some kind of thermo meter for measuring thermos. Thermo means Greek for heat. So this is a newton scale, zero being the heat of air in winter, which water begins to freeze.
00:09:45
Speaker
And he called these degrees of heat. So then at 40 degrees, you have a melting point of an alloy of one part lead, four parts tin, and five parts bismuth, and so on. And then Romer's system was a little bit refined past that because it was based on the freezing point of water being 7.5 degrees and the boiling point being 60 degrees. Interesting.
00:10:09
Speaker
which it might seem a little bit weird, but 7.5 is 60 divided by eight. Yeah. So it does make some kind of sense, especially because this was before decimals were used all the time in science. Okay.
00:10:23
Speaker
And then you have metrification in the 1600s through the 1800s. And we've gotten to the point now where none of the, originally a kilogram was this platinum weight called the kilogram. And the meter was two scratches on a, I think it was like a gold pole.
00:10:40
Speaker
Yeah, it sounds funny, but you have to start somewhere, right? Yeah. Yeah. It's just really funny. It's just like, uh, uh, let's put some scratches on a pole and that'll be the length we need. Yeah. And it was originally defined as one 10,000, um, thousand of the, the, um, equal.
00:11:00
Speaker
Interesting. But now it is defined exactly as how quickly light travels in 1,299,792,458th of a second. And second is defined as taking the fixed numerical value of the cesium frequency, the unperturbed ground state hyperfine transitions frequency of the cesium 133 atom to be exactly 7,192,631,770 hertz.
00:11:27
Speaker
That is to say that now all of our metric units are based on things you could find anywhere in the universe Which I think is pretty cool. I think we should just go back to using like a king Yeah, this is the size of Phillips nose No, we should just use King Philip is all lengths of measurement. Let's dig up King Philip
00:11:50
Speaker
All right, so now we got derived units. So let me just kind of go through a few of these right here with you. Do you know the difference between a dimension and a unit is?
00:12:01
Speaker
Dimension is going to be like the size of something. You think so? But turns out that dimensions actually just talks about what unit is being measured, what unit you need. But the unit talks about the specific unit. So for example, a foot and a meter are both used to measure length. And length is given the symbol L for dimensions. So they would still use the same dimension. I mean, you can even use rods, you can use miles, you can use meters.
00:12:29
Speaker
meters you can use you can use smoots um smoot really that's an actual word yeah uh smoot uh smoot was um it was measured by um there's this guy uh Oliver R Smoot who laid down repeatedly on the harvard bridge between boston and cambridge so his fraternity brothers could use his height to measure the length of the bridge and they defined one smoot uh the bridge's length to be measured to be exactly six 364.4 smoots
00:12:56
Speaker
See, that's the type of measurement systems we need. I agree. Yeah. So yeah, like you said, life can be foot smoots. Let's see. Uh, we also have, so here we have velocity. Um, but you see right here, LT to the negative one. What that means is L divided by T. Is that clear? Yes. Yes. And what that means is length over time. It makes sense, right? My miles per hour, meters per second, feet per second. Yeah. Yeah.
00:13:20
Speaker
And then you have acceleration, feet per second per second, or miles per hour per second, meters per second per second, whatever, and so on. But the idea is that you have these certain base dimensions that you can divide all these other dimensions into, right? Yeah. Yeah, exactly. Even temperature, which is a base dimension, that's actually an interesting thing to bring up. Temperature
00:13:43
Speaker
itself describes the average energy of something, which you'd think would be mass times length squared over time squared, right? You'd think that since temperature can be reduced to that, you'd be able to change the units, right? And get that even to a more base unit. Yeah.
00:14:00
Speaker
But it turns out that it's it's about the macroscopic view of the system. So if temperature is not if we're not dealing with the actual energy of the particles, we can treat temperature as an abstract quantity. And the same thing goes for, for example, mass and energy. There's a mass energy equivalence, but we're not going to use it in every problem. There's also the equivalence between time and space. You know, in most problems, we don't use that equivalence, right? No, not at all.
00:14:29
Speaker
Yeah, because I mean, we don't talk about like, you know, I'm gonna, I'm gonna, for example, Jacob, you're about six feet away from me, right? Which means that you're about six nanoseconds away from me in light terms. So I was, it'll list the sources at the end of this episode, but one of the sources I was reading talks about how in fluid mechanics, the primary or fundamental dimensions are mass, length, time, and temperature, which means that everything can be broken into these four things. In other areas of physics, you might even have luminous intensity,
00:14:57
Speaker
amount of substance or electric current be base units. Okay. But the important thing is that you can't add a foot to a pound, right? No. You can multiply them, you can have a foot pound, you can't add them. And that's the first thing we're gonna talk about is the idea of commeasure ability. When we're talking about physical quantities, we have to be able to compare them in some way, right? Physically.
00:15:19
Speaker
Yeah. And so, you know, it's easy to say we could add things together, but adding lengths together is different than adding weights together, right? Yeah. Like just physically like so different things. Yeah, exactly. Yeah. So I'm going to do this. I'm going to give you these two pencils and I'm going to I want you to express the sum of their lengths by just moving around in space. How would you add their lengths together? I just stack them one on the other.
00:15:47
Speaker
Yeah, right. How would you weigh the pencils against each other? I'd hold my hands like this and I'd just pretty much use the weight against them. Yeah. And if you wanted to add them together, you could put them both in one spot, right? Yeah, I'd put them both in one hand. Oh yeah. So if we did a balance scale, we could weigh like, you know, two pencils against a third that weighed as much as both of them combined, right? Yeah, there's like a heavier pencil. Yeah, I could, you know, two of these and one of those.
00:16:11
Speaker
Yeah. So that's so we've so that's how you'd show equality is by using a scale. Right. If two weights are equal and they will be level on the scale. Yeah. If two lengths are equal, then they all match up geometrically. Right. Yes. Yeah. They'll both compare to one another.
00:16:27
Speaker
Yeah. And so that, so that means that we have a few rules. So the comparison operation equality must obey the identity law. So if two lengths are equal to each other and the second one is equal to a third one, then the first and the third are the same length, right? Yeah. Or mass or whatever unit, right?
The Buckingham Pi Theorem and Its Applications
00:16:43
Speaker
Yeah. Whatever, whatever we're adding, whatever you're measuring. Yeah.
00:16:46
Speaker
Yeah, and also it has to not matter what order we add things in. So A plus B plus C has to be the same as B plus A plus C. They seem like, you know, self-explanatory things, but there are certain things that don't act this way. For example, matrices, when you multiply them together, there's certain operations that just don't work that way. And then finally, if you add things together a certain number of times, so if you add it together three times, you have to be able to divide that back out. So those are the rules for a physical dimension.
00:17:14
Speaker
So now we have like, so E equals MC squared is a famous equation, right? Yes. F equals MA. What's interesting about any physical formula, no matter what, the units have to be the same on both sides, no matter what. It sounds like something that you could take for granted.
00:17:32
Speaker
But if you actually analyze this fact about physical quantities, you get some pretty deep stuff. So Joseph, just for example, Joseph Fourier, he introduced the concept of physical dimension in his 1822 book, Terri Analytic de la Calure, or the Analytical Theory of Heat. This is the first time that physical dimension was mentioned in any text, and this is the table from his book. We have x, which is one of the variables being dimension one of length, zero of duration, and zero of temperature. Right? Yeah.
00:18:02
Speaker
We have specific conductivity, surface conductability, capacity period, et cetera. So this is a way, so what's interesting is that by breaking down, by listing all of our different factors that we have in a problem and breaking them down, there's actually a way that we can significantly simplify most problems. So that brings us to...
00:18:25
Speaker
So we're going to talk about the Buckingham Pi theorem. So it says that any any physically meaningful equation with n variables can be rewritten as n minus k dimensionless parameters where k is the number of independent dimensions. So what does that mean? So what that means is that any physical thing, so like let's say we get the period of a pendulum, right?
00:18:45
Speaker
or the height of a splash in liquid. Any formula that takes in a bunch of dimensions, like height, heat, whatever, and spits something else out, whether it be a frequency, a weight, a force, you can always express it in dimensionless parameters. And dimensionless parameters are the parameters combined together. So let's explore what that means. And a pendulum is swinging, right?
00:19:13
Speaker
Yeah. Yeah. Let's assume that there's no air, nothing like that. We don't care about the friction of the string, anything like that. We just have a pendulum swinging. Do you know anything about the period of a pendulum either of you? What's it dependent on? It's the gear itself. The gear has to turn it, right?
00:19:27
Speaker
Oh, well, not not not a clock pendulum, per se, just like a swinging ball on. Yeah, it's isn't something about the leverage or. Well, we'll figure it out. Let's pretend it will always like it will have to eventually the swinging will have to go down. Yeah. Oh, yeah. But that's only to continue on friction. We're talking about an ideal pendulum and swings forever. So what we're going to talk about is let's say we want to know how long the pendulum takes to swing back and forth. Right. The period. Yes. What is that measured in?
00:19:55
Speaker
Second time. The exact both of you. Right. The unit is seconds and the dimension is time. So let's let's think about certain things that might be able to affect a pendulum swing. The length of the string, obviously. Right. Yeah. So let's write these down. So the length of the string. So we're trying to get the period in seconds. We have the length of the string. What else? What are the weight? The weight. Yeah. The weight of the string. Well, and of the pendulum of the pendulum itself at the end.
00:20:24
Speaker
Oh, that's what I mean. It's not the string yet, the pendulum. We're assuming the string is massless. So the mass is going to be measured in kilograms. It's going to be mass. Length is going to be measured in meters. And it's going to be L for length period. It's measured in seconds. It's measured in time. Let's see what else. Oh, we have, obviously, how much gravity there is on Earth, right? Yes. So we have G, which is measured in meters per second squared, which is length times time to the negative 2.
00:20:53
Speaker
Anything else that we could think of that might affect this problem? I mean since we're taking out things like air and of that nature Like we're focused on just the pendulum itself. I think that might be it. I think we're good
00:21:06
Speaker
All right, so let's take a look at this. The Buckingham Pi Theorem, so we have how many variables here? Four. Four variables, right? Period, length, mass, and g, where g is pretty much the same everywhere on Earth, we're assuming. That gives us four variables. Now, how many base dimensions do we have here? One, two. We have time. We have length.
00:21:31
Speaker
We have mass. We have mass. And we have this. This is just a combination of length and time, right? Yeah. Yeah. All right. So I would say three very or three units. All right. Yeah. So the bucking and pie theorem says that we can combine this into a few into into only one unit list thing. Right. Yeah. Yes. All right. So we have a period as being a function of mass length and G. Right. Because we're trying to figure out the formula for period. Right.
00:21:59
Speaker
Yes. Do you notice that period, length, and G, none of those have a kilogram in them? The weight is the only thing with a kilogram in it. Time appears both in period and G. Length appears in both in G and the length of the string itself.
00:22:16
Speaker
But the mass, if we're doing any kind of unitless thing, this has to be zero, right? We can't have mass be part of the thing because we can multiply G by the period squared and get a different quantity. We could multiply G by the, divide G by the length and get something without units. But the kilogram, there's nothing else in the problem with the kilogram. So what might that tell you?
00:22:41
Speaker
Yeah, we don't need it. Exactly. Yeah, exactly. Turns out that the mass of the thing in the pendulum does not matter. And what's fascinating is that we got to that with dimensional analysis, not by scientific measurement, right? Yeah. And that's the power of dimensional analysis, really, is that you can simplify these problems to much, much simpler problems.
00:23:05
Speaker
All right, so we're trying to get period out of length n g right now. Yes. So we need a second out of a length n g, which is length per second squared, right? So if we take length and divide it by g, we're left with seconds squared, right? Yes. And obviously, we'd have to take the square root of that because we need to get back to regular seconds, right?
00:23:28
Speaker
Yeah. And that's actually one of the things that's interesting about dimensional analysis is you always need, unless you're doing some weird fractal stuff, which I don't know how to do, but that honestly sounds like something I should look into. You're going to want a whole number or at least a rational number of fractions because basically you just have to always get the dimensions to match up. As soon as you get the dimensions to match up, you know what affects what.
00:23:53
Speaker
So now we're gonna do a problem together for the Buckingham Pi theorem and how about we do the problem of let's say you throw an object into a liquid How high does it splash? Do you want to do that problem?
00:24:07
Speaker
Yeah. Yeah. Absolutely. That sounds great.
Dimensional Analysis: Simplifying Science
00:24:10
Speaker
All right. So let's say, uh, so we, let's say we have a few things that matter. Let's say the radius of the object, right? So you have the radius of the object that much matter, right? Because if something is 10 times the radius, it can cause a different splash, right? Yeah. The mass of the object, we're saying the velocity of the object, the density of the medium, the viscosity of the medium and the height of the splash. Yeah.
00:24:30
Speaker
So it turns out that the way that you make dimensionless quantities out of that is if we take the viscosity multiplied by the radius squared divided by the velocity and divided by the mass of the object, you get a dimensionless quantity. If you take the density of the medium multiplied by the radius cubed and divided by the mass, you get another dimensionless quantity. And then finally, the height of the splash divided by the radius of the object is another dimensionless quantity.
00:25:00
Speaker
So what we get here is this formula, h over r is some function of these two independent quantities, w r squared over v m and p r cubed over m. So it might seem like we're just reordering variables around, right? Yeah. But what's interesting is that this actually tells some fundamental truths about the problem. So for example, since it's a function of these two variables, right,
00:25:24
Speaker
No matter what, if this stays constant, and while this stays constant, this is not going to change, right? Here, if we make the medium twice as viscous, we have to have twice the mass to have the same splash. And that tracks, right?
00:25:42
Speaker
Yeah. Now we can actually redo the problem with a completely different set of dimensions. Um, radius of object density of the object this time, right? So this time, because now the radius and mass would get assuming a spherical thing. So radius of mass would give us the density, right? Yeah. But now we're just treating the density as a separate quantity itself. And we're got rid of mass as a quantity. And if it turns out that this actually simplifies a problem slightly, you see?
00:26:07
Speaker
Yeah. Yes. And it turns out that it's a function of the ratio between the viscosity, between the density of the medium and the density of the object. As long as that stays constant, the problem of the splash will stay the same height. Yeah. I should say the same exact height. No, that makes sense.
00:26:24
Speaker
Oh, yeah. And and notice this could be any formula with these two things. So, I mean, it could be as complex as we want because, OK, the sign of something tells you goes from an angle to basically how far something is from the center of the circle sideways. Right. I mean, not the sign, the cosine. Yeah. Yeah.
00:26:43
Speaker
Um, so, but you can't take the cosine of a meter because the, the cosine formula is, um, if you break it down, okay. So like X plus minus X cubed divided by three factorial plus X Q X to the fifth divided by five factorial minus. So you have all these different powers and we can't add like a kilogram to a kilogram cubed to a kilogram to the fifth. Right. No. No.
00:27:06
Speaker
And that's why when we're actually dealing with physical quantities, we can only really add, divide, square, take square roots, do that algebraic things. And what's interesting here that I wanted to bring to your attention
00:27:20
Speaker
Oh yeah, you're revising the formula and making it more simplistic. This is how the variables choke out exactly with the new set of variables. It shakes out differently. But yeah, so you simplify the formula by sometimes trying different sets of variables. Yeah. Yeah. And when you get something that doesn't make sense, that's usually physics way of telling you, you forgot a variable. You forgot something that matters. And that's another huge, huge benefit of using dimensional analysis.
00:27:51
Speaker
Yeah, because you sit there going, this doesn't make sense. That's not right. If it doesn't add up, then, you know, you know, something's wrong.
00:27:59
Speaker
Yeah. And, um, exactly. Yeah. And, uh, to, and the Buckingham by theorem, the way that you get these new variables is, uh, it's actually the rank of the dimension matrix. So remember earlier, I was showing you that, um, that, uh, list of numbers from that paper. Yeah. Yeah. Uh, it turns out that if you treat this like a matrix and you see how many independent independent linearly independent, um, factors it has, uh, that turns out to be the number of, uh, dimensionless quantities that you can have.
00:28:25
Speaker
Oh yeah, I was also going to say that the base quantities, so let's say we have a problem where we have something, one of the variables is force, which can be broken down into kilograms times meters per second squared, and the other variables of importance are
00:28:44
Speaker
Let's say one of them is mass and one of them is acceleration, right? F equals ma. Okay, let's just use that. The base units of that formula aren't, there's not three base units. You might think that there are because of kilograms, meters, and seconds, right? Yeah. But it turns out that because on one side you have kilogram times meters per second squared, and the other side you have kilograms and meters per second squared.
00:29:10
Speaker
So the fact that you can't really break these down further, like I can't get a second out of this, right? Like I can't divide these units to get a second. And that's basically the definition of a base unit in any problem. It's not about the actual meters and seconds. Those can help us find it, but it's about how many actual physically independent things there are, whether that be force or even a more derived quantity, you know, like moment of area or something weird.
00:29:37
Speaker
Yeah. And yeah, this is a really, really useful thing. Milly, were you on the episode about back of the envelope estimations? No, I think that wasn't you. But yeah, this can be used with back of the envelope estimations. So like if I reduce the problem into a bunch of simpler problems, I can do stuff like play with balloons in the sink and be like, okay, that's how a submarine might work or something like that.
00:30:01
Speaker
It allows us to really, so if we have six different variables and we need to kind of like test the different numerical quantities that matter, we're hit with the curse of dimensionality, which means that, let's say if you just wanted to have 10 samples in each direction, you get 10 times 10 times 10, like six times, which is a million different samples.
00:30:23
Speaker
But if you reduce it to two dimensionless quantities, the number of samples you need to really get a good idea of how the problem works goes down to 100. OK. Yeah, that makes sense. Yes. Ultimately, it just makes things simpler, easier to work like it makes these big, gigantic like problems simpler and easier to work on.
00:30:47
Speaker
Oh, yeah, exactly. Yeah. And the problem in one in the book, one of the books I reference that I'm referencing, I'll give the name at the end of the episode. It talks about the problem of throwing a ball against a wall and how much if the ball is covered in ink, how big is the spot on the ball compared to the ball itself? Yeah. Like the spot on the wall. I mean, that that from the splash and all that.
00:31:08
Speaker
Yeah.
Conclusion: The Art of Measurement and Analysis
00:31:09
Speaker
And as you can see here, he takes all these different variables, E row, whatever, and reduces it to two variables. And you can see how they behave in terms of one another. It'd be great to live in like the 1700s and just dip a ball in ink and throw it out a wall and say, I'm a mathematician. Now, this is more recent stuff. This is actually it's the utility hundred. OK, it goes back to the 1600s.
00:31:36
Speaker
But the utility of this stuff really got big and people like Feynman during the atomic bomb, they used to do this thing where you had 10 seconds to give somebody a math problem. They had 10 seconds to answer it with one digit of accuracy, which is 10 different numbers.
00:31:53
Speaker
And exercises like that and kind of just our full understanding of units and just kind of like our fuller understanding of mathematics in the 20th century really led to this coming into like, you know, this is big stuff now comparatively.
00:32:10
Speaker
Yeah. I mean, one interesting thing about Oppenheimer is that when the atomic bomb went off, he dropped a handful of paper. The paper shards blew away and he was able to calculate the approximate megatonnage of the bomb within a few seconds before all the calculations were given. Wow. Yeah. Just by looking at the paper blowing. Yeah. He saw how far the paper blew. Yeah.
00:32:33
Speaker
Oh, and really quick, I'm just going to talk about the steps of the Buckingham Pi theorem. So let's say you have n relevant variables in m independent dimensions, right? Yeah. So you have these parameters that you want to talk about the relationships between.
00:32:49
Speaker
What you do is you choose M scaling variables. So like, you know, just variables that represent like, you know, just like base variables. So you choose like a few of them that are linearly independent. And there's going to be a guarantee they're going to be able to find M because it's the amount of linearly independent rows and columns, the rank of the matrix. So it's guaranteed to exist.
00:33:11
Speaker
And then for each of the N minus M remaining variables, what you do is you multiply them by each of those variables. So you go through the variables and you make them unitless. And that's the procedure. That's the entire procedure to determining the pi groups, is what they're called. Okay.
00:33:31
Speaker
Physical dimension allows us to talk about the world in terms of numbers. This was the basis of physics, since physics is just math about the world around us. We've explored how dimensions can be reduced to dimensionless quantities, and the power that this gives us to talk about seemingly hopelessly complex physical problems in straightforward terms.
00:33:47
Speaker
I'm Sophia, and this has been Breaking Math. The big source we used for this episode today was the Physical Bases of Dimensional Analysis by Ayn A. Sonnen and the lectures by David Apsley from the University of Manchester on his website. I would give both of those a check out if I were y'all. So Millie, Jacob, what did you think?
00:34:09
Speaker
Well, if I ever need to know like the radius of like, um, well, an ink ball splashing on a wall, I think I know now have the tools I need to figure it out.
00:34:21
Speaker
Yeah, like if I want to know how big the splash is going to be from that artificial lake I'm going to make filled with maple syrup and when I chuck bowling balls in it all day, I can figure out how big the splash is going to be. You can call yourself a physicist and I can call myself a physicist for doing it. Oh, one important thing, actually, that I forgot to bring up on the main episode, but you kind of brought up here is
00:34:47
Speaker
This only helps us in as much as we know how to measure quantities, right? So like if we don't know the units for viscosity, talking about viscosity won't help us, right? Exactly. Oh, yeah. So it's just one of the... Oh, and another thing we didn't quite talk about is that sometimes if you include too many variables in your analysis, you get too many dimensionless quantities and that can make the problem complex again. So it's all about playing around with... It's all about trial and error in math and science. We're back to the scales of trying to find a balance.
00:35:17
Speaker
Yeah, that is the scale related term, isn't it? Yeah. And to find a balance. Any last thoughts? Oh, yeah, I already asked that. Good night, everyone.