20110731, 07:59  #1 
"Lucan"
Dec 2006
England
6474_{10} Posts 
Dimensional analysis
Joe Hatton (one of my Oxford tutors) threw me this one:
When you figure out a formula with the method of dimensions, how come the "right" numerical constant always has the order of unity? (2, pi or summat). I assumed at the time that he knew the answer, but it could equally well have been a genuine query. Any ideas? David Last fiddled with by davieddy on 20110731 at 08:12 
20110731, 16:17  #2  
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
5^{2}×7×67 Posts 
Quote:
The most egregious example I know of is the cosmological constant. The observed value is about 120 orders of magnitude smaller than that calculated. Perhaps I'm misunderstanding you and Hatton (who I also knew, BTW). Paul 

20110731, 19:56  #3 
"William"
May 2003
Near Grandkid
100101000110_{2} Posts 
Paul points out that "always" is an exaggeration. I suspect that to the extent it is true, it is because units are chosen to be convenient for people, and getting answers with small numbers is an important part of convenience. In astronomy, the A.U and the parsec come to mind as measures that exist primarily because they result in small numbers for some discussions.

20110731, 23:33  #4 
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Yep.
Much of it comes down the fact that when you are strolling across Hampstead heath at twilight and encounter a puddle, it doesn't take that much longer to walk round it than through it. Assuming you can see it of course. (I had just mistaken an aircraft about to land for Venus at the time). BTW I think Joe was probably thinking about things like "why is fine structure constant ~1/137 and not 10^{120} ?" David PS Joe and Gwynneth were still living in their home in North Oxford last time I heard. He's in his 90s and prone to blackouts, she is blind but their minds are still sharp! Last fiddled with by davieddy on 20110731 at 23:55 
20110801, 11:53  #5 
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
I prefer not to think about the number of beer molecules I consume

20110801, 13:24  #6  
(loop (#_fork))
Feb 2006
Cambridge, England
2·7·461 Posts 
Quote:
I think the constants are small because they're mostly the result of integrals of notwildlyvarying functions over small intervals. (though the StefanBoltzmann constant is quite a long way from unity, and the argument that power is proportional to T^4 is quite a subtle one) 

20110801, 19:05  #7 
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
Here's a real coincidence, I think:
Unless the MksA (SI) system was adopted with this in mind, is it not remarkable that the electric potential at an atomic radius from a proton is a handful of volts? Can you think of similar examples? David OTOH, the emf of a Voltaic cell might explain a lot of it. Yep. I think keeping the Amp and the Volt convenient/appropriate units might have a lot to do with the adoption of the m and kg, so as to get the Joule right. Last fiddled with by davieddy on 20110801 at 19:56 
20110801, 20:18  #8 
"Lucan"
Dec 2006
England
194A_{16} Posts 
This thread is veering rather close to the one which spawned
3 plates = 1 pipe 
20110802, 09:34  #9 
"(^r'ยฐ:.:)^n;e'e"
Nov 2008
;t:.:;^
2^{3}×5^{3} Posts 
137.036
..ropos~anth.. 
20110802, 09:59  #10 
"Lucan"
Dec 2006
England
2·3·13·83 Posts 

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