Dimensional analysis

davieddy · 2011-07-31, 07:59

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

xilman · 2011-07-31, 16:17

Quote:

Originally Posted by davieddy

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

It doesn't.

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

wblipp · 2011-07-31, 19:56

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.

davieddy · 2011-07-31, 23:33

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 10¹²⁰ ?"

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!

davieddy · 2011-08-01, 11:53

I prefer not to think about the number of beer molecules I consume

fivemack · 2011-08-01, 13:24

Quote:

Originally Posted by wblipp

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.

That's a property of unit systems in general; if you're working in grams and centimetres then your energy comes out in small multiples of ergs, if you're working in kilograms and metres then it comes out in small multiples of joules.

I think the constants are small because they're mostly the result of integrals of not-wildly-varying functions over small intervals.

(though the Stefan-Boltzmann constant is quite a long way from unity, and the argument that power is proportional to T^4 is quite a subtle one)

davieddy · 2011-08-01, 19:05

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.

davieddy · 2011-08-01, 20:18

This thread is veering rather close to the one which spawned

3 plates = 1 pipe

cmd · 2011-08-02, 09:34

137.036
..ropos~anth..

davieddy · 2011-08-02, 09:59

Quote:

Originally Posted by davieddy

This thread is veering rather close to the one which spawned

3 plates = 1 pipe

Quote:

Originally Posted by cmd

137.036
..ropos~anth..

Yes cmd.
We know this sort of puzzle is right up your street

2011-07-31, 07:59	#1
davieddy "Lucan" Dec 2006 England 2·3·13·83 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 2011-07-31 at 08:12

2011-07-31, 23:33	#4
davieddy "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 10¹²⁰ ?" 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 2011-07-31 at 23:55

2011-08-01, 19:05	#7
davieddy "Lucan" Dec 2006 England 14512₈ 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 2011-08-01 at 19:56

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2011-07-31, 19:56	#3
wblipp "William" May 2003 Near Grandkid 5³·19 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.

2011-08-01, 11:53	#5
davieddy "Lucan" Dec 2006 England 2·3·13·83 Posts	I prefer not to think about the number of beer molecules I consume

2011-08-01, 20:18	#8
davieddy "Lucan" Dec 2006 England 2×3×13×83 Posts	This thread is veering rather close to the one which spawned 3 plates = 1 pipe

2011-08-02, 09:34	#9
cmd "(^r'°:.:)^n;e'e" Nov 2008 ;t:.:;^ 1750₈ Posts	137.036 ..ropos~anth..