2566: Decorative Constants

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Decorative Constants

Title text: Arguably, the '1/2' in the drag equation is purely decorative, since drag coefficients are already unitless and could just as easily be half as big. Some derivations give more justification for the extra 1/2 than others, but one textbook just calls it 'a traditional tribute to Euler and Bernoulli.'

Explanation

This explanation may be incomplete or incorrect: Created by a DECORATIVE BOT - What is the formula representing when removing the two decorative constants? - Do NOT delete this tag too soon. If you can address this issue, please edit the page! Thanks.

This is another one of Randall's Tips, this time a Math Tip.

He gives an example of a complex looking equation (4-15 in the book it is taken from):

T = Dm₀(r_out - r_in)^μ

T: Net rate

m₀: Unit mass

(r_out-r_in): Flow balance

D, μ: Decorative

But since D and μ are decorative (and μ=1 is the only situation where it does not change everything), then the equation can be reduced to:

T = m₀(r_out - r_in)

Also note that the D is written with two horizontal lines and μ with a bar over the top, to further spice up the formula.

This all leads up to the Math tip of the day. If one of your equations ever looks too simple, try adding some purely decorative constants.

Other examples of well known equations that are profound but look simple are:

E = Mc² (from Special Relativity)

PV = NRT (the Ideal Gas Law)

F = ma (Newton's Second Law)

V = IR (Ohm's Law)

These could all use some spicing up with extra constants.

Maybe this was what Albert Einstein was thinking when he included the Cosmological Constant in General Relativity, but it ended up being meaningful.

In the title text Randall mentions the Drag equation. In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid.

The equation is:

Fd = 1/2ρu2CdA. 

Randall jokes about the 1/2 in front is meaningless and thus purely decorative, since the drag coefficients, C_d, are already unitless and could just as easily be half as big thus leaving out the 1/2 in front of the equation. The 1/2 is thus just an example of a Decorative Constant according to Randall.

He continues to claim that in some text books the derivations of the formula give more justification for the extra 1/2 than others. But finishes with an example of a book that just calls it "a traditional tribute to Euler and Bernoulli".

This is in reference of Leonhard Euler and the Bernoulli family, possibly mainly Daniel Bernoulli who Euler worked together with to formulate their Euler–Bernoulli beam theory. Euler was a friend of the entire family. Euler is held to be one of the greatest mathematicians in history. The Bernoulli family was a patrician family, notable for having produced eight mathematically gifted academics. Both Euler and Daniel also worked on fluids.

But! First of all the drag equation is attributed to Lord Rayleigh, so that in it self has nothing to do with the two names mentioned.

Second the 1/2ρu² also called P_D is the dynamic pressure due to kinetic energy of fluid experiencing relative flow velocity u with ρ being the mass density of the fluid. Thus this is analogues to kinetic energy E = 1/2mv², thus the 1/2 comes from this.

Third the force F is proportional to P_DA where A is the area over which the pressure P_D is applied. The proportionality factor is thus the one called C_d, the drag coefficient. And thus the 1/2 belongs to a part of the equation, that could be taken out to give a specific value for the dynamic pressure. And it is thus relevant to keep it apart from the otherwise unitless drag coefficient.

So it seems that this is just another joke by Randall. But possibly he did read the statement in a text book, but a citation for that is needed.

Transcript

[A small panel only with text. Written as an excerpt from a mathematical text book. Begins with a number for an equation, then follows the equation written in larger letters and symbols. And below are explanations of each term in the equation. The μ has a bar over the top and the D has a double vertical line.]

Eq. 4-15

T = Dm₀(r_out - r_in)^μ

T: Net rate

m₀: Unit mass

(r_out-r_in): Flow balance

D, μ: Decorative

[Caption below the panel:]

Math tip: If one of your equations ever looks too simple, try adding some purely decorative constants.

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Discussion

I don't have any idea what to put in the actual description, but whoever does should probably note that r(in) - r(out) equals zero, not one. And multiplying by a constant 0 absolutely changes the value! GreatWyrmGold (talk) 21:59, 10 January 2022 (UTC)

r_out and r_in are different values. The subscripts represent different instances of the same variable at different point. In the same way, you might calculate something happening over a time interval t_end - t_start . 172.69.71.77 23:02, 10 January 2022 (UTC)

Yes for sure they are two different values. On the other hand if μ is not 1 then the it is not just decorative! D on the other hand is just a proportionality constant, which may have a value other than 1. I have tried to put something in the explanation here. Quite a bit. Do not really now anything about Drag, so just took it from the wiki page. Also I hope someone can explain the formula in the image, as I'm sure it is just something about the flow, that would relate it to a drag equation. --Kynde (talk) 23:41, 10 January 2022 (UTC)

Note that the title text is pretty much word-for-word a repeat from Randall's book *How To*. In Chapter 11: *How to Play Football*, he misuses the drag equation, and mentions this fact in more depth, in a footnote. Bit of trivia! --162.158.134.79 23:13, 10 January 2022 (UTC)

Nice, I will have to check up on that. Thanks. --Kynde (talk) 23:41, 10 January 2022 (UTC)

Can confirm this, the book mentions that the "traditional tribute to Euler and Bernoulli" comes from Frank White's Fluid Mechanics textbook. Clam (talk) 01:08, 11 January 2022 (UTC)

There it is, page 266 in the 1986 2nd edition: "They both have a factor ½ as a traditional tribute to Bernoulli and Euler, and both are based on the projected area..." https://books.google.com/books?id=wGweAQAAIAAJ&q=traditional -- 172.70.162.5 02:13, 11 January 2022 (UTC)

Great thanks have included both references in the explanation. --Kynde (talk) 08:32, 11 January 2022 (UTC)

Wait, wouldn't the values be twice as big (rather than half as big) if you left off the 1/2? 141.101.69.154 12:43, 11 January 2022 (UTC)

No. If 1/2Cd = Constant, then the new constant would be half as big as Cd since Cd=2x constant. You would just put in the 1/2 in the new version of Cd, so the new Cd is half as big as the old, and the final result the same.--Kynde (talk) 10:44, 12 January 2022 (UTC)

Of course, the c^2 im e=mc^2 is just as decorative, when using natural units where c=1.... 172.68.50.171 00:29, 11 January 2022 (UTC)

And the resulting equation is then just e=m - or m=e which is beautiful and profound. "Mass is Energy". Without the complications, you stop thinking of it as a PROCESS for converting one into the other and get the more profound point that Mass and Energy are the exact same thing. SteveBaker (talk) 03:33, 11 January 2022 (UTC)

I respectfully disagree. The c² isn't decorative; mc² is a measure of energy and m is not. e=mc², like f=ma, still works even if you change the size of any of the basic units (of length, time, mass) from which the units of energy and force are derived. As I see it, an equation that ties you to any definition of unit size is less profound, not more. Tom239 (talk) 17:21, 12 January 2022 (UTC)

To the sort of person who (thoughtfully) uses c=1, this feels a bit like saying that the "f" is profound in dist=sqrt[x^2+y^2+(z/f)^2], where of course I've measured xy-distances in miles and z-distances in feet, so f=5280ft/mi. Yes, it's entirely possible to choose different units for different coordinates, and if you're very accustomed to that then the conversion factors can be deeply important for your understanding of the system (and provide extra flexibility in your choice of units: you can easily use "f=1760yd/mi" if you'd prefer). But there's still a very well-defined sense in which sqrt[x^2+y^2+z^2] is the more fundamental equation, and the "f" is an unnecessary complication (however convenient it may be). Whether I'd call it "decorative"... I'm not sure. But I don't see this "f" as profound. Steuard (talk) 17:59, 29 May 2022 (UTC)

I think the 1/2 in the drag equation is intuitive. I understand that it is technically superfluous, but F=Pd*A and Pd=1/2*rho*u^2 so the 1/2 carries over intuitively. 172.70.98.15 (talk) (please sign your comments with ~~~~)

Agrees I had this written down in an early version of the explanation but that was edited out. Maybe I will put it in again.--Kynde (talk) 10:44, 12 January 2022 (UTC)

Drag coefficients could just as easily be half as big. This is true but how is their being unitless relevant? It's more about how defining constants is partially arbitrary. Lev (talk) 08:07, 12 January 2022 (UTC)

If Cd had a unit, say it was an energy which represented some relevant value for a given material, then it would not be correct to say that it was half as much, just because 1/2 came into the equation. But if it has no units, then it is just a constant saying something about the material, and then the 1/2 could in principle be absorbed without changing anything. But as stated above 1/2 actually has physical meaning in the way it enters the equation. --Kynde (talk) 10:44, 12 January 2022 (UTC)

It doesn't make any difference. For instance, Coulomb's law works fine whether we write it F = -q₁q₂/(4πε₀r²) or F = -kq₁q₂/r². Similarly, if we had a factor of 2 in the gas law for some reason, that would just change the values of the gas constants.

I've seen the double-struck capital "D" used commonly as a symbol for the Domain of a function (While the double-struck "R" was used for the range in that context) 162.158.63.243 21:16, 17 January 2022 (UTC)

Any use of double-struck or bold capital R to mean something besides the set of real numbers can be considered nonstandard, like using + to represent a non-commutative function or using a fraktur lowercase c to represent anything other than the cardinality of the continuum. It happens of course, but it's not any kind of standard. EebstertheGreat (talk) 03:18, 20 August 2023 (UTC)

Does anybody know enough math to figure out what that equation is supposed to do? I really want to delete that tag.New editor (talk) 19:13, 25 January 2022 (UTC)

The r terms are used in describing things like water treatment plants or dialysis machines, where you're trying to use fluid flow to regulate some solute. If fluid balance is large, it means the "tank" is going to empty or dry out. I guess T is the rate at which this happens. Not really a math thing, more of an engineering thing, seems to me.

Count down clock

See Countdown in header text. Discussion has been moved here Talk:Countdown_in_header_text. --Kynde (talk) 11:10, 12 January 2022 (UTC)