0687: "Dimensional Analysis"

[wintermute]

Brits: "There certainly is a big difference between English, Welsh, Scotch, ... . Why don't these non-Brits understand?!"
non-Brits: "Whatever difference between English, Welsh, Scotch, ... . Why do these Brits think we care?!"

Beef, eggs and whisky are Scotch; people are Scots.

[finlay]

^ This, or "Scottish" to be consistent with "English", "Welsh".

[gmalivuk]

I think the people who posted their calculation understood something wrong. They say the calculation are not equal to Pi, well I personally think they are not using the best Prius available. Maybe there are some current prototypes that makes the equation equal pi?

No, that only makes it worse. If you read the thread, you'll see that most people get numbers that are already bigger than pi. Which means that, if mileage improves, being on the top of the fraction, the total will only move farther away from pi.

Unless something gives, of course, like the width of the Channel or the pressure at Earth's core.

[masterwizard]

Clearly though, if your numbers don't match up, it's due to an error in your measurements, not the formula. This is actually useful, because if we can accurately and precisely measure the narrowest part of the english channel, and the fuel economy of a prius, then we can use that to calculate, with equal precision, the pressure of the earth's core. Randall should publish this so scientists can use this in their calculations.

[skeeto]

The units don't actually cancel because it's confusing a common name for different things. Specifically, "miles per gallon". The bottom unit is not units of volume! If you write it out, it becomes "miles per gallon of gasoline", where "gallon of gasoline" is a unit of energy, not volume. If it was merely a unit of volume we could measure car efficiency in units of 1/distance^2, which is how it's being used in the comic.

If we just look at the types of units, they don't cancel. This is how I initially approached it, checking for error.

(energy / pressure) * ((distance/energy) / distance)

Which leaves 1/pressure, which is definitely not unitless. (That's the whole reason I came to the forum to see if anyone had noticed this yet. )

Of course, making mistakes is the point of the comic.

[Diadem]

I bet this is related to the "God Equation" nonsense.

Surely that piece is parody, right? Please tell me it's parody?

[jdkoftinoff]

Wonder if it is at all related to: http://www.scientificblogging.com/hammo ... se_gravity

Derivation of Newtonian Gravity from information theory assuming a holographic universe.

Actually, I'd like to think that it is related to my posting in reply to someone else on slashdot about that article on Verlinde's paper:
http://slashdot.org/comments.pl?sid=150 ... d=30718694

RGOOMH!

--jeffk++

[MotorToad]

This bugs me because when I tutor people in fizzix I tell them that if the units cancel then the equation is always going to be correct. I guess it'll still work in Phys 1 & 2 homework problems.

[MrJinks]

England: 51.5 million people.
Great Britain: 61.5 million people.
Percentage of a damn the distinction is worth giving: 16.3%

Why not just call the UK "London"? It seems to work for acts visiting the larger UK venues.

[Diadem]

The units don't actually cancel because it's confusing a common name for different things. Specifically, "miles per gallon". The bottom unit is not units of volume! If you write it out, it becomes "miles per gallon of gasoline", where "gallon of gasoline" is a unit of energy, not volume.

This has already been remarked earlier in this thread *and* debunked.

If it's 'miles per gallon of gasoline', then we do we measure fuel efficiency of cars using LPG in miles per gallon as well? The figure says how many miles your car goes on one gallon of fuel. It doesn't care about the type of fuel. And indeed cars which use different fuels typically have different MPG figures. So it's not energy content.

[gmalivuk]

The bottom unit is not units of volume!

Except, yes it is, since that's how we actually measure it. How much distance do you get for a particular volume of fuel?

(As already pointed out, the European L/100km method, which gives a result in area, is actually kind of physically meaningful, that being the cross-sectional area you'd get if you string the fuel out along your path.)

[skeeto]

This has already been remarked earlier in this thread *and* debunked.

Ah, I skimmed the second page and missed it. However, the "debunking" was completely wrong. Those fuel differences (except diesel) are differences in octane, not usable energy density (and is partly why using these higher grade fuels is generally useless).

then we do we measure fuel efficiency of cars using LPG in miles per gallon as well?

Very similar usable energy density if the numbers are the same. That's all.

And indeed cars which use different fuels typically have different MPG figures.

Exactly because it is energy content. You made my (and AL_'s) point.

The point is that "gallons of gasoline" (or even "gallons of fuel") is not a unit of volume and is not the same unit as "gallons", so they can't cancel each other. Just look at the names. Here's a simple demo of this: It could just as easily be changed to "kilograms of gasoline", representing the same useful measure for car efficiency, but completely break the equation in the comic. Not a unit of volume. If you can't account for "miles per kilogram of fuel" then you know you're looking at it wrong.

[flguy1980]

Yeah, but when we measure fuel efficiency--whether in mpg or l/100 km, we also have to remember that it could be gasoline, diesel, or for some fleet vehicles, LP or NG. These four fuel types have different energy conents in J/L, thus, we'd have to know that too.

If Randall used the fuel efficiency of a VW Jetta TDI, that wouldn't be gasoline now, would it?

The units don't actually cancel because it's confusing a common name for different things. Specifically, "miles per gallon". The bottom unit is not units of volume! If you write it out, it becomes "miles per gallon of gasoline", where "gallon of gasoline" is a unit of energy, not volume. If it was merely a unit of volume we could measure car efficiency in units of 1/distance^2, which is how it's being used in the comic.

If we just look at the types of units, they don't cancel. This is how I initially approached it, checking for error.

(energy / pressure) * ((distance/energy) / distance)

Which leaves 1/pressure, which is definitely not unitless. (That's the whole reason I came to the forum to see if anyone had noticed this yet. )

Of course, making mistakes is the point of the comic.

[SocialSceneRepairman]

This bugs me because when I tutor people in fizzix I tell them that if the units cancel then the equation is always going to be correct. I guess it'll still work in Phys 1 & 2 homework problems.

I don't know, even in Physics 1 & 2 that led me to be off by factors of 2 and/or pi pretty often.

[minke]

My favorite part is that you can tell Randall's a mathematician and not an engineer because he's writing on a chalkboard, not a whiteboard.

[Tardyon]

Alt-text: Or the pressure at the Earth's core will rise slightly.


Or both.

[Tardyon]

My favorite part is that you can tell Randall's a mathematician and not an engineer because he's writing on a chalkboard, not a whiteboard.

A physicist. No Oompa-loompa is he.

[neoliminal]

So Wales, Scotland and Ireland stay where they are? England's not going to move much, then...

Are you suggesting that if the Channel got bigger that the Europe would be moved instead of Wales and Scotland? We know that the Channel is getting bigger!!

[gmalivuk]

Well the obvious way for the channel to get wider isn't for any land mass to move, so much as for a bit of England to fall into the water (or France, of course).

[Monika]

I bet this is related to the "God Equation" nonsense.

Surely that piece is parody, right? Please tell me it's parody?

Probably not, but if you want one, read "My paranormal bicycle" by Gero von Randow.

[mochaholic]

England isn't an island, it's as simple as that.

Well, actually it's a bit more complicated than that. Here's a handy Venn diagram I found: http://fuckyeahbritain.tumblr.com/post/325241942/so-i-guess-a-lot-of-people-are-in-some-doubt-as-to

[Hitaro0]

My favorite part is that you can tell Randall's a mathematician and not an engineer because he's writing on a chalkboard, not a whiteboard.

Actually might just be because he's teaching children.
I presume that most schools have chalkboards everywhere.
Screw chalkboards.

I'm just sad I can't work on chalkboards because it's way too dusty ):

[gmalivuk]

England isn't an island, it's as simple as that.

Well, actually it's a bit more complicated than that. Here's a handy Venn diagram I found: http://fuckyeahbritain.tumblr.com/post/325241942/so-i-guess-a-lot-of-people-are-in-some-doubt-as-to

So, the bigger island of the British Isles is "Great", right? That means Irish people wouldn't mind if I called their island "Lesser Britain"?

[nirvana_grace]

Hmmmm. Using a transport device to increase pressure at the core of the planet? If the pressure increases sufficiently, we'd get fusion. Which means . . . "Any efficiency-advanced Prius is indistinguishable from Monoliths."

-with apologies to Mr. Clarke, for abusing both his Law and his novel.

[Makri]

And indeed cars which use different fuels typically have different MPG figures.

Exactly because it is energy content. You made my (and AL_'s) point.

Why should cars that use different fuels require different amounts of energy?

[Robstickle]

"correct to within experimental error", so i wonder if that means it's accurate to 0 or possibly 1 sf

[sagenought]

I'm surprised no one has posted yet that they have a problem with the spacing between the words "and" and "the".

The spacing of the words "and" and "the" was removed to add more space between "Fundamental" and "Law". Just ask Randall.
Thus spacing was conserved throughout the comic.

[ianf]

So, the bigger island of the British Isles is "Great", right? That means Irish people wouldn't mind if I called their island "Lesser Britain"?

Actually, "Lesser Britain" is Brittany in France. That's why the term "Great Britain" was used, to distinguish it from Brittany.

[samf]

The units don't actually cancel because it's confusing a common name for different things. Specifically, "miles per gallon". The bottom unit is not units of volume! If you write it out, it becomes "miles per gallon of gasoline", where "gallon of gasoline" is a unit of energy, not volume.

No, it's distance over volume.

If you wanted distance over energy, you would want the Miles per gallon gasoline equivalent.

The article explains the distinction.

To think of it another way: if you were actually calculated the distance traveled per energy consumed, something like a solar car or a plug-in electric would have no better mpg ratings than anything else, because it would still be consuming (roughly) the same amount of energy.

[gmalivuk]

Right. The EPA's gas mileage numbers are just miles per gallon, as stated. Distance per volume.

That the gallons happen to be of gasoline in a Prius, rather than, say, ice cream, is simply a function of the fact that the Prius runs on gasoline and not ice cream. Which is good, because otherwise it would be way less economical to drive.

[Eternal Density]

This one burned my cheese to a far greater degree than any xkcd for quite a while. It was actually clever and not about girlfriends or people acting like computers.

[SEE]

So, the bigger island of the British Isles is "Great", right? That means Irish people wouldn't mind if I called their island "Lesser Britain"?

The West Britons don't even like the terminology "British Isles".

[qrs_iii]

Ok... I think that the comic is right that it is correct within experimental error . I used:
Planck Energy = 1.956e9 J (Wikipedia)
Pressure at Earth's Core = 360e9 Pa (Wikipedia, states 340-360 at time of post)
Prius Combined EPA Gas Mileage = 46 mi/gal (feuleconomy.gov, this was a bit difficult to find, but it is the official value at time of post)
Minimum Width of English Channel = 34 km (WIkipedia)
Three of these have 2 significant figures, so the answer will have 2 significant figures.

(1.956e9 J)/(360e9 Pa)*(46 mi/gal)/(34 km) = (1.956 N*m)/(360 N/m^2)*(46 mi/gal)/(21.1 mi) = (1.956/360 m^3)*(46/21.1 1/gal)*(1 gal)/(.00379 m^3) = 3.1
And 3.1 = pi to 2 significant figures.

Of course, all of the comments about the use of mpg in this way are right: it doesn't work. I assume that's why it's entitled abusing dimensional analysis.

[MotorToad]

I'm laughing at "gallons of gasoline" being interpreted as some amount of energy density. Why not just mention how many standard cubic feet of oxygen the car uses, since that's no less important to the car's running. When you determine a car's efficiency you measure the distance it traveled and compare it to the consumption of fuel by volume. Miles/kms vs. gallons/liters.

Also, I think using observed mpg for the Prius would get a result a lot closer to pi than the EPA's average numbers. Perhaps that could even be a bragging standard for Prius owners (since the only people that will talk to Prius owners are other Prius owners). "I'm 5% over pi for the trip here! Too bad the return trip is up-hill."

[HighwoodFool]

hahaha...One of the best xkcd's in a while.

In response to the "England" vs. "UK" dispute:
Technically, only England would have to recede to widen the English Channel. Besides, the statement that "England will drift out to sea" does not necessarily entail that the rest of the UK won't drift with it.

[Zomfgcrazy]

http://www.wolframalpha.com/input/?i=((1.956+GJ)+/+(360+GPa))+*+((46.24+mpg)/(34+km))

(yes, I set all of that equal to pi with x for the milage so as to solve for the milage to make it pi. sue me.)

[9squirrels]

If the UK drifted out to sea making the English Channel wider, I imagine that there'd be a great deal of rejoicing on both sides of the Channel. Up until the Brits realised that their easy access to cheap booze had been made harder, and the supermarket owners in Calais realised that they'd just lost almost their entire customer base.
Beyond that I don't think anyone would care.

The easiest way for the channel to widen (which is happening I believe) is for the South East of England to continue to sink, putting more of Kent under water, making the Channel wider. From memory it had been noted some years ago that Scotland was rising slowly and that the South East was actually sinking, the theory being that it was due the release of weight from the glaciers that melted after the last ice age.

[GurnBlanston]

The EPA has changed their rating system. The 2008 Prius under the old system had a combined rating of 55, the 2009 model under the new system has a combined rating of 46. Using 55 works pretty good:
Planck Energy = 4.63e108 bar
Pressure at the Earth's core = 3.8e6 bar
Prius combined mileage = 55
Width of the Channel = 21
\frac{4.63x10^{108}}{3.8x10^6} \times \frac{55}{21}=3.19 which is within experimental error

[rateguard]

hahaha, i don't get it but i feel smart when i read this

[squareroot]

A much more accurate result for the right side would be:

(\alpha^{2} 2 \pi e(\pi+\pi)^{\pi})^{-1}

, where "e" is the base of the natural logarithm and α is the fine structure constant.