In Praise of Dimensional Analysis

On Apr 25, 8:20 pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
> Joel Kolstad wrote:
> > My high-school physics book had a cartoon in it where Einstein was standing in
> > front of a chalkboard, having crossed out E=mc and E=mc^3.  The caption was
> > something along the lines of, "Aha!  So the units *do* work out in that one!"
>
> The one I remember has E=mc and E=mc**3 written down, and someone
> straightening up the books on a nearby bookshelf saying.
> "Well now that's all squared away."
>
> -- glen

That was one of Gary Larson's classic Far Side cartoons.
"Everything is squared away.  Yep, squa-a-a-a-a-ared away ..."

Mark

On Apr 25, 4:07 am, Robert Baer <robertb...@earthlink.net> wrote:
> John Larkin wrote:
> > On Tue, 24 Apr 2007 15:17:05 -0700, Tim Wescott <tim@seemywebsite>
> > wrote:
>
> >>After six and a half years, I still get email about my article "PID
> >>Without a PhD"
> >>(http://www.wescottdesign.com/articles/Sampling/pidwophd.html) from
> >>people who are asking for clarifications, or are asking questions that
> >>go beyond what I could say in the 5000 word limit that Embedded Systems
> >>Programming magazine imposed.
>
> >>The one I got today made me think.
>
> >>It didn't make me think because it was hard to figure out the answer, or
> >>because it was hard to explain the answer so it would be understood.  It
> >>made me think because the writer was asking me about the dimensions of
> >>the gains in a PID controller, and I was impressed that someone who was
> >>obviously just approaching the subject was paying enough attention to
> >>want to do a correct job of dimensional analysis.
>
> >>It made me think of the innumerable times when I've done some complex
> >>calculation and ended up proving something absurd such as measures of
> >>length in units of gallons, and the subsequent discovery of my error.
> >>It made me think of the Mars Climate Orbiter which crashed because one
> >>team specified a motor in pound-seconds while another one used
> >>Newton-seconds (http://mars.jpl.nasa.gov/msp98/news/mco990930.html).  It
> >>made me think of how much I like MathCad, because I can set variable
> >>values with the correct units, and how inconvenient nearly every other
> >>programming language is because numbers are just that -- anonymous numbers.
>
> >>So I thought I'd write a little bit about dimensional analysis, why I
> >>like it, and why you should use it, even if you think it is tedious and
> >>trite.
>
> >>Dimensional analysis is a method used by engineers and scientists that
> >>extends the notion that "you can't compare apples to oranges" to the nth
> >>degree.  Dimensional analysis says that every variable in any physical
> >>problem has units, and that you ignore these units at your peril.
>
> >>The basic rules of dimensional analysis are these:
>
> >>1.  There are very few naked (i.e. dimensionless) numbers.
> >>2.  You can only add (or compare) two numbers of like dimension --
> >>    you can't compare feet with pounds, and if you're being strict
> >>    you can't even compare pounds (which is strictly a measure of
> >>    force) with kilograms (which is strictly a measure of mass).
> >>3.  You can multiply (or divide) numbers of any dimension you want;
> >>    the result is a new dimension.  So if I pour water into a pan, the
> >>    water at the base of the pan exerts a certain force on each bit
> >>    of area, the resulting pressure is measured in pounds/square inch
> >>    (PSI), or in Newtons per square meter (N/m^2, or Pascal).
> >>3a. You can honor famous people by naming dimensions after them --
> >>    Pascal is the metric unit of pressure, Ampere of current.  Avins,
> >>    Grise, and Wescott have yet to be defined.
>
> >>So why is dimensional analysis so cool?
>
> >>If you are doing a long calculation and you track your dimensions, some
> >>mistakes will show up as incorrect dimensions.
>
> >>It can give you insight into the operation of a system -- most
> >>explanations of fluid dynamics that I have seen rely heavily on
> >>dimensional analysis in their arguments, and when they do so such
> >>aerodynamically important quantities as Reynold's numbers and Mach
> >>numbers fall out.
>
> >>When you do design with physical systems, careful dimensional analysis
> >>keeps you out of trouble, even when you're getting the math right.
> >>Remember that Mars Lander?  Had someone been carefully checking
> >>dimensions instead of looking at numbers and making assumptions, it
> >>would have been a successful mission instead of a famous crash.
>
> >>How do you use dimensional analysis?
>
> >>For working forward, such as finding the relationship between an
> >>aircraft's speed and it's lift, or finding the relationship between
> >>mass, speed and energy (remember E = mC^2?) you find out the dimensions
> >>that your answer _must_ have, then you hunt down candidates in that field.
>
> >>For working backward, you do all of your calculations with dimensions,
> >>then make sure that, for example, if you're commanding a system in feet
> >>it's moving in feet, and not apples/minute.  If your answer ends up
> >>being in gallons/inch^2, you know you're close but may have to multiply
> >>by a constant.
>
> >>For a more full exposition of dimensional analysis I refer you to the
> >>Wikipedia article (http://en.wikipedia.org/wiki/Dimensional_analysis),
> >>or whatever your own web searches turn up.
>
> >>In the mean time, the next time you're dealing with a knotty problem --
> >>make sure to track those dimensions.  The Martian lander you save may be
> >>yours!
>
> > In a PID controller, we are summing voltages (which is fine) but they
> > also represent an error, the time integral of an error, and the
> > derivative of an error.
>
> > The error is in volts. The integral is in volt-seconds. But we sum
> > them, and nothing explodes!
>
> > John
>
>    Hmmm... X double dot = - X

Nope.  It's

X double dot = - (omega squared) X

-- Mark

On Apr 24, 8:19 pm, John Larkin
<jjlar...@highNOTlandTHIStechnologyPART.com> wrote:

> In a PID controller, we are summing voltages (which is fine) but they
> also represent an error, the time integral of an error, and the
> derivative of an error.
>
> The error is in volts. The integral is in volt-seconds. But we sum
> them, and nothing explodes!
>
> John

Somewhere, often in the form of an R*C, those volt-seconds are divided
by seconds to get back to volts.

Mark

On Apr 27, 5:12 pm, robert bristow-johnson <r...@audioimagination.com>
wrote:
> On Apr 27, 3:48 pm, Heinrich Wolf <hwmu...@willis-werkstaette.de>
> wrote:
>
> > Vladimir Vassilevsky <antispam_bo...@hotmail.com> writes:
> > > Tim Wescott wrote:
>
> > > ...
> > >> ...
> > > BTW, in the theoretical physics, they use the dimensionless units to
> > > avoid the heavyweight dimension constants:
>
> > > e = c = h = 1
>
> > > How about that?
>
> > I knew someone who claimed to be a professor of theoretical nuclear
> > physics--- oh no, he really was --- who wildely objected
> > against this.
>
> why would a theoretical nuclear physicist have a problem with that?
> these are one example of "Natural units" (look that up in Wikipedia)
> and i think these are called "electronic units" if the "h" is replaced
> with "hbar".  but qualitatively, the are electronic units. they can be
> compared to  Stoney units (but i think Stoney normalizes G instead of
> hbar), Planck units (but Planck normalizes G instead of e), or Atomic
> units (which normalize the mass of the electron instead of c).
> personally i like Planck units the best because they have no prototype
> particle, object, or "thing" that needs to be sorta arbitrarily chosen
> as a base unit.  it's better, in my opinion, to normalize these
> parameters of free space, before introducing any special particles or
> objects, to define your base Natural units.  i which Planck would have
> normalized 4*pi*G instead, that would have been more natural.
>
> other good Wikipedia articles that are related is
>
> Planck units
> Physical constant
> Dimensionless physical constant
> Fundamental unit
>
> i had something to do with those articles, including Dimensional
> analysis.
>
> r b-j

Hello Robert,

That is cool you had some input on those articles. If you recall I use
atomic units all of the time. There are two variations. One uses a
Rydburg and the other uses a Hartree for the fundamental energy unit.

The main reason for using these is the Schrodinger equation becomes
quite simple in appearance. A factor of two shows up in the energy
term. So that is why there are two variations. The distance becomes
scaled to Bohr radii - the mean distance to the electron in the ground
state of Hydrogen. And 1 Ry is the binding energy.

Clay

On Apr 27, 11:24 pm, Clay <phys...@bellsouth.net> wrote:

> ... If you recall I use atomic units all of the time.

what i recall Clay, is that it was YOU that first told me the name for
Planck Units:
http://groups.google.com/group/sci.physics/msg/e1d9352d0a6b64b3
and i thank you profusely.  because you did that, i was able to do web
searches, found papers/web_sites/books by Michael Duff, Gabriele
Veneziano (pioneers of string theory), Lev Okun, John Baez, and John
Barrow and i've had several really neat email conversations with ALL 5
of these guys about the nature of fundamental physical constants (the
only ones that count are the dimensionless ones, any notions of a
"varying c" or "varying G" are not even wrong, they're meaningless,
and for those who can't see that, just think of everything measured in
Planck units).  and then later i got into Gravito-electro-magnetism
(GEM) a little and some interest in the Gravity Probe B (which*still*
hasn't been able to conclusive say that frame-draggin or
gravitomagnetism or gravity waves can be measured, they are behind
schedule.)

none of that fun would have happened if you hadn't done that for me
nearly a decade ago.  thanks.

> There are two variations. One uses a
> Rydburg and the other uses a Hartree for the fundamental energy unit.
>
> The main reason for using these is the Schrodinger equation becomes
> quite simple in appearance.

discovered a word for that, too: "Nondimensionalization."  there's a
wikipedia article on that also.  i may have done a minor edit to that.
but dunno.

> A factor of two shows up in the energy
> term. So that is why there are two variations. The distance becomes
> scaled to Bohr radii - the mean distance to the electron in the ground
> state of Hydrogen. And 1 Ry is the binding energy.

isn't normalizing the Rydberg constant have similar effect as fixing
the Bohr radius (with another dimensionless alpha tossed in)?

just curious.



r b-j

robert bristow-johnson wrote:

(snip)

> i disagree, i guess.  when i was teaching EE classes (circuits) and as
> a grad student in EE, i would be pretty hard on any engineering
> student that did it the latter way.  i think Jerry's little proverb
> said it best: "Mathematicians routinely ignore units, but engineers do
> so at their peril."

It isn't ignoring units, but more like tunneling then.  One must
convert given quantities to the appropriate units before applying
the equation.  One can then use the numbers with a slide rule or
calculator, and apply the appropriate units to the result.

When writing down the process, what is usually called "show your work"
all units will still be shown.

> the cumbersome method you ascribed to engineers, Glen, is almost as
> bad as ignoring the unit.  (because your root algebraic equation is
> dimentionless.  in the latter case, F = ma is actually expressed as
> the dimensionless F is the product of the dimensionless m and the
> dimensionless a, as long as you express all three in their base SI
> units.

I probably overemphasized the difference, but yes, the algebraic
equation is dimensionless, but so are all calculators that I know of.

The problem with the way I called the physics way is that it can
result in unusual or inconsistent units.  One might end up with
an acceleration in meters/second/hour, for example, or worse.

For the engineer way, one converts all given units to those
specified, does the algebra keeping the units (except when
entering them into a calculator), and then converts the result
to the desired unit.

Though the division probably isn't all that strict, and it wouldn't
surprise me if many EE's used the physics way.

-- glen

On Apr 28, 6:57 am, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
> robert bristow-johnson wrote:
>
> (snip)
>
> > i disagree, i guess.  when i was teaching EE classes (circuits) and as
> > a grad student in EE, i would be pretty hard on any engineering
> > student that did it the latter way.  i think Jerry's little proverb
> > said it best: "Mathematicians routinely ignore units, but engineers do
> > so at their peril."
>
> It isn't ignoring units, but more like tunneling then.

dunno what that means.

>  One must
> convert given quantities to the appropriate units before applying
> the equation.  One can then use the numbers with a slide rule or
> calculator, and apply the appropriate units to the result.
>
> When writing down the process, what is usually called "show your work"
> all units will still be shown.
>
> > the cumbersome method you ascribed to engineers, Glen, is almost as
> > bad as ignoring the unit.  (because your root algebraic equation is
> > dimentionless.  in the latter case, F = ma is actually expressed as
> > the dimensionless F is the product of the dimensionless m and the
> > dimensionless a, as long as you express all three in their base SI
> > units.
>
> I probably overemphasized the difference, but yes, the algebraic
> equation is dimensionless, but so are all calculators that I know of.

sure.  as noted by sombuddy else, you can program computers
(especially with C++ or some other OOP) to attach a unit from a known
list to the numerical quantity.  that way we conceptually have a
dimensionful quantity in the computer.

> The problem with the way I called the physics way is that it can
> result in unusual or inconsistent units.  One might end up with
> an acceleration in meters/second/hour, for example, or worse.

this is what conversion factors are for.  these conversion factors,
like (0.3048 m/ft) are dimensionless (even though they have units
inside the expression), in fact should be the dimensionless 1 so that
multiplying or dividing by it changes nothing.

> For the engineer way, one converts all given units to those
> specified, does the algebra keeping the units (except when
> entering them into a calculator), and then converts the result
> to the desired unit.

but the requirement is that all units chosen must be consistent.  the
so-called "physics" way you can actually accelerate a pound mass by
(mi/hr)/s and  have a meaningful equation (and an unusual unit of
force) naturally pop out.

> Though the division probably isn't all that strict, and it wouldn't
> surprise me if many EE's used the physics way.

i guess i'm a partisan for that method.  almost to the point of
facsism.

i'm getting less tolerant in my middle age.  (Jerry, does it get worse
or better as we get older?)

r b-j

In article <vrKdnQpuRMllhK7bnZ2dnUVZ_u6rnZ2d@comcast.com>, glen 
herrmannsfeldt says...
> The problem with the way I called the physics way is that it can
> result in unusual or inconsistent units.  One might end up with
> an acceleration in meters/second/hour, for example, or worse.

I can imagine circumstances in which that would quite a useful unit for 
acceleration.

Robert

-- 
Posted via a free Usenet account from http://www.teranews.com

robert bristow-johnson wrote:

   ...

> i'm getting less tolerant in my middle age.  (Jerry, does it get worse
> or better as we get older?)

Both.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

On Apr 28, 3:45 am, robert bristow-johnson <r...@audioimagination.com>
wrote:
> On Apr 27, 11:24 pm, Clay <phys...@bellsouth.net> wrote:
>
> > ... If you recall I use atomic units all of the time.
>
> what i recall Clay, is that it was YOU that first told me the name for
> Planck Units:http://groups.google.com/group/sci.physics/msg/e1d9352d0a6b64b3
> and i thank you profusely.  because you did that, i was able to do web
> searches, found papers/web_sites/books by Michael Duff, Gabriele
> Veneziano (pioneers of string theory), Lev Okun, John Baez, and John
> Barrow and i've had several really neat email conversations with ALL 5
> of these guys about the nature of fundamental physical constants (the
> only ones that count are the dimensionless ones, any notions of a
> "varying c" or "varying G" are not even wrong, they're meaningless,
> and for those who can't see that, just think of everything measured in
> Planck units).  and then later i got into Gravito-electro-magnetism
> (GEM) a little and some interest in the Gravity Probe B (which*still*
> hasn't been able to conclusive say that frame-draggin or
> gravitomagnetism or gravity waves can be measured, they are behind
> schedule.)
>
> none of that fun would have happened if you hadn't done that for me
> nearly a decade ago.  thanks.
>
> > There are two variations. One uses a
> > Rydburg and the other uses a Hartree for the fundamental energy unit.
>
> > The main reason for using these is the Schrodinger equation becomes
> > quite simple in appearance.
>
> discovered a word for that, too: "Nondimensionalization."  there's a
> wikipedia article on that also.  i may have done a minor edit to that.
> but dunno.
>
> > A factor of two shows up in the energy
> > term. So that is why there are two variations. The distance becomes
> > scaled to Bohr radii - the mean distance to the electron in the ground
> > state of Hydrogen. And 1 Ry is the binding energy.
>
> isn't normalizing the Rydberg constant have similar effect as fixing
> the Bohr radius (with another dimensionless alpha tossed in)?
>
> just curious.
>
> r b-j

Hello Robert,

I'm glad my giving a name for Planck units put you on an interesting
course. I recall reading many years ago about Paul Dirac exploring the
ideas behind Planck units. And that is how I knew about them.

Studying Hydrogen has yielded some very interesting info. Balmer
empirically put together a formula that fits the spectral lines of
Hydrogen, but it offered no theory.

Bohr with his theory of the atom gave a theoretical basis for Balmer's
formula. But spectroscopists soon discovered a slight flaw in that
some of the energy levels had finely spaced details. A spectral "line"
under close observation turned out to be 2 or more lines very close
together.

Sommerfeld found by adding special relativity and elliptical orbits to
the theory, that he was able to explain the fine splitting (fine
structure) of the spectral lines. Bohr was of course ecstatic that his
theory was saved. Sommerfeld introduced the notion of the fine
structure constant, which in many older books is called the
"Sommerfeld fine structure constant", but now many have dropped
Sommerfeld's name - what a shame.

Alpha, as the constant is usually denoted, was discovered to be a
dimensionless number which may be expressed as (e*e)/(2*epsilon_0 *
h*c) and "e" is the charge of the electron, h is Planck's constant, c
is the speed of light in a vacuum, and epsilon_0 is the permititivity
of free space. We know alpha to be basically = 1/(137.0359895...). The
fact that it is much smaller than one allows perturbations to simple
systems to be expanded in series involving successive powers of alpha.
These series may be found by using Feynman diagrams.

Even though Sommerfeld discovered alpha with his work on Hydrogen, it
has turned out to be one of the most important fundamental constants
with consequences far beyond atomic physics. It shows up in many
places such as in the interaction of E-M fields with electrons and in
nuclear and particle physics.

In terms of atomic theory, the Bohr radius may be written as
(4pi*epsilon_0*h_bar^2)/(m*e^2), where m is the mass of the electron,
e is its charge, epsilon_0 the permititivity of free space, and h_bar
is Plank's constant h divided by 2pi.

A Hartree is simply alpha^2 * m*c^2  (27.2116 electron volts) and a
Rydberg is half of a Hartree. Again "m" is the mass of the electron
and c is the speed of light. Douglas Hartree did a lot of work in
computational atomic physics, and he gave us "atomic units." He even
built analog computers from a kid's toy called "mechano" to solve
atomic physics calculations back before WW2. [Mechano (UK) is a toy
very much like the Erector Set in the U.S.]

I've been working on a new algorithm to solve the Hartree-Fock
equations. So far the results have been quite encouraging. I.e., more
accuracy with less computation.

Clay

p.s. A book you may find interesting is "Universal Constants in
Physics" by Gilles Cohen-Tannoudji, 1993 McGraw-Hill. The parts on "h"
& "k" are interesting with "h" being the quantum of action and "k"
being the quantum of information.