J. J. Lodder wrote:
The formally correct version of this equation is
F = k m a
where k is a dimension-less constant depending only on the units chosen
for measuring the physical quantities force, mass and acceleration; by
suitable choice of measurements units, it can be chosen to be equal to 1.

I don't understand this remark.
Are you implying that every physical formula
should be thought of as containing an invisible extra constant,
dimensionless and of value 1?

This is how the formulas will be, if one does not settle for a specific
set of measurement units.

You can of course think so,
but what good would that do?

In order to be formally correct, but the formulas quickly become
unreadable. As illustration:

In theoretical physics, one may set c = 1, so all lengths can be
measured in seconds. So how about buying 20 ns (nano seconds) of rope?

In article <1ifbvtl.4kohut1kw41ddN@de-ster.xs4all.nl>,
nospam@de-ster.demon.nl (J. J. Lodder) writes:

Hans Aberg <haberg_20080406@math.su.se> wrote:

The formally correct version of this equation is
F = k m a
where k is a dimension-less constant depending only on the units chosen
for measuring the physical quantities force, mass and acceleration; by
suitable choice of measurements units, it can be chosen to be equal to 1.

I don't understand this remark.
Are you implying that every physical formula
should be thought of as containing an invisible extra constant,
dimensionless and of value 1?

You can of course think so,
but what good would that do?

I see what he is driving at. In this case, the extra constant is
dimensionless and has value 1. However, that is because m (the
intertial mass) is DEFINED as the proportionality constant between F and
a. However, let's assume that it has previously been defined in some
other way, say as the gravitational mass m'. Wonder of wonders, it is
STILL 1 and dimensionless, i.e. m = m'. This is a great mystery in
Newtonian mechanics.

In other cases, say gravitational attraction, we have

F = G m1 m2
-----------
r"

In this case, the constant G is neither dimensionless nor does it have
the value 1. In many cases, units are often used so that G is
dimensionless and of value 1. Such definitions are often made for many
such constants at once, and one sees notes like G = h = c = 1.

In other words, the quantity of interest (F in the examples above) is
proportional to various things we can measure (like r, or m). In
general, to make the proportionality an equality, we need a constant.

A trivial example is the maximal system of dimensions,
in which every physical quantity has its own dimension.

In theoretical physics, one may set c = 1, so all lengths can be
measured in seconds. So how about buying 20 ns (nano seconds) of rope?

That's of course just what we would do,
if we could start afresh with our unit systems.

Backward compatibility does have it's claims however.

And in actual fact you are buying rope measured in nanoseconds,
it is just that you (and everybody else)
prefer to use some inconvenient numerical constants as well.

The formally correct version of this equation is
F = k m a
where k is a dimension-less constant depending only on the units chosen
for measuring the physical quantities force, mass and acceleration; by
suitable choice of measurements units, it can be chosen to be equal to 1.
...

In this case, the extra constant is
dimensionless and has value 1. However, that is because m (the
intertial mass) is DEFINED as the proportionality constant between F and
a. However, let's assume that it has previously been defined in some
other way, say as the gravitational mass m'. Wonder of wonders, it is
STILL 1 and dimensionless, i.e. m = m'. This is a great mystery in
Newtonian mechanics.

In other cases, say gravitational attraction, we have

F = G m1 m2
-----------
r²

In this case, the constant G is neither dimensionless nor does it have
the value 1. In many cases, units are often used so that G is
dimensionless and of value 1. Such definitions are often made for many
such constants at once, and one sees notes like G = h = c = 1.

In other words, the quantity of interest (F in the examples above) is
proportional to various things we can measure (like r, or m). In
general, to make the proportionality an equality, we need a constant.

On Apr 8, 2:22=A0pm, John Forkosh <j...@please.see.sig.for.email.com
wrote:

. . . what you call data types
is what physics calls units.

I am not sure because I am trying to understand if force in F=ma is
type string or type number. Only type number has units and dimensions.
The distinction between string and number is called types too, correct?
[...]

Hi,

I submitted this question to sci.math.research but it was rejected for
being related to physics or computer science and not math perse.

I have been trying to classify data types in physics

http://www.densytics.com/wiki/index.php?title=Data_types_in_Physics

and I am confused about what is considered string (in the sense of
computer language terminology) and what is considered number or
quantity or magnitude in physics.

More specifically, looking at F in F=ma I see something like a pointer
to an address, not the address itself. Or if ma is number in a
spreadsheet cell, F is the name of that cell.

I know this is not how physicists see it. I am looking to find the
correct mathematical terminology so that I can state the problem
clearly. Do you know a field of physics or math that studies these
things?

Thank you for the help.

Phillip Helbig---remove CLOTHES to reply wrote:
The formally correct version of this equation is
F = k m a
where k is a dimension-less constant depending only on the units chosen
for measuring the physical quantities force, mass and acceleration; by
suitable choice of measurements units, it can be chosen to be equal to 1.
..

In this case, the extra constant is
dimensionless and has value 1. However, that is because m (the
intertial mass) is DEFINED as the proportionality constant between F and
a. However, let's assume that it has previously been defined in some
other way, say as the gravitational mass m'. Wonder of wonders, it is
STILL 1 and dimensionless, i.e. m = m'. This is a great mystery in
Newtonian mechanics.

There are several issues involved here. First, assume that mass is
measured in pounds, acceleration using the Paris acceleration in vacuum,
and F using the Paris archive spring - then there is good chance that k
will not be 1, though dimensionless. - Decades ago, it was mentioned to
me somebody trying to write out all these constants, but the formulas
because unreadable.

Second, is k above always dimensionless?...

J. J. Lodder wrote:

A trivial example is the maximal system of dimensions,
in which every physical quantity has its own dimension.

That would be the most general dimension system, in the sens that the
others arise by imposing some equivalence relations.

In theoretical physics, one may set c = 1, so all lengths can be
measured in seconds. So how about buying 20 ns (nano seconds) of rope?

That's of course just what we would do,
if we could start afresh with our unit systems.

I think one use say the charge of the electron as basic unit.

Backward compatibility does have it's claims however.

That is one reason. Another is the ability to measure the physical
constants. For example, mass, time and length can be measured to a
higher accuracy than the gravitational constant, making it unsuitable as
a conversion constant.

And in actual fact you are buying rope measured in nanoseconds,
it is just that you (and everybody else)
prefer to use some inconvenient numerical constants as well.

There is another factor: even though time and length can be treated as
the same in a physical theory, they have in practical experience quite
different characteristics. So giving them different dimensions is in
line with that experience.

F = k m a
...
Second, is k above always dimensionless?...

Once again: being or not being dimensionless
is NOT a property of a physical quantity,
or of a term in an equation.

It is a choice you can freely make.

In the example above you can for example
choose F to have a dimension of it's own. [F]
Then the constant in F = kma must have the dimension
[k] = [F][M]^{-1}[L]^{-1}[T]^2

Dimensional analysis has nothing to do with reality.
It is a meta-analysis, of systems of equations
that are used to describe reality.

Dimensions can be assigned in any way we please,
subject only to consistency.

A trivial example is the maximal system of dimensions,
in which every physical quantity has its own dimension.
That would be the most general dimension system, in the sens that the
others arise by imposing some equivalence relations.

All dimension systems are equally general.
(all are finite dimensional algebras, mathematically speaking)

It just that some have more dimensions than others.
Some are more trivial than others though.
The maximal system of dimensions
(every physical quantity its own dimension)
is as trivial as the one-dimensional system,
(every physical quantity dimensionless)
for both can't be used for anything useful.

And to pick a nit: length can't be measured at all, nowadays.
All we can measure is time, with either a ruler or a stopwatch.

And to pick a nit: length can't be measured at all, nowadays.
All we can measure is time, with either a ruler or a stopwatch.
The reason is the same: time and speed (of light)
can be measured to greater accuracy than length,
so the obvious solution is to define c,
and not to measure length independently anymore.

H. Whitney: "The mathematics of physical quantities, Part I:
Mathematical models for measurement", Am. Math. Monthly, 75, 2 (1968),
pp. 115-38,
H. Whitney: "The mathematics of physical quantities, Part II, Quantity
structures and dimensional analysis", Am. Math. Monthly, 75, 3 (1968),
pp. 227-56

that the OP might find useful, although Whitney did not invoke "data
types" in the modern sense.

Also relevant,

M. Parkinson: "An Axiomatic Approach to Dimensions in Physics", Am. J.
Phys., 32, 3 (1964), pp. 200-5.

J.F. Price: "Dimensional analysis of models and data sets", Am. J.
Phys., 71, 5 (2003), pp. 437-47.

AB

============================================================================
Alain Bossavit
LGEP, CNRS,91192 Gif sur Yvette CEDEX, Francehttp://natrium.em.tut.fi/~bossavit/

In article <rvPOj.58991$097.58918@newsfe21.lga>, Paul Danaher
snip
There are a few different approaches to redefining the basic measurement
references. In the case of the kilogram, it won't be replaced by a
similar reference object, but will be defined by, say, a certain number
of atoms of a certain isotope. Perhaps it will be tied to something
else, say multiply the number of electrons corresponding to a coulomb by
a certain number to get the number of electrons in a kilogram.

There are two tendencies, which sometimes compete: One is to find a
(conceptually) simple measurement. The other is to make sure that if
something else which is defined by a measurement becomes more exact in
the future, that the other quantities don't shift too much.

A while back, there was an article in the Physikalische Blätter (now
called Physik Journal---the member magazine of the German Physical
Society) which reviewed the status of the various proposals.

At http://www.ptb.de/en/wegweiser/einheiten/si/kilogramm.html one can
read:
snip

Here are 3 references with more information about this:

There was a pretty good (actually I though _excellent_ for a general
newspaper!) article about this in the Los Angeles Times last week:
http://www.latimes.com/news/nationworld/nation/la-sci-kilogram17apr17,0,7998161,print.story

physorg.com also had a pretty good short piece in 2005:
http://www.physorg.com/printnews.php?newsid=3178

There's also a nice Fermilab colloquium presentations by Richard Steiner
(NIST) on "How Measuring the Planck Constant gets to an Electronic Kilogram
Standard" on 1 Aug 2007: search for Steiner in
http://www-ppd.fnal.gov/EPPOffice-w/colloq/colloq_06_07.html

There was a pretty good (actually I though _excellent_ for a general
newspaper!) article about this in the Los Angeles Times last week:

http://www.latimes.com/news/nationworld/nation/la-sci-kilogram17apr17,0,7998161,print.story

physorg.com also had a pretty good short piece in 2005:
http://www.physorg.com/printnews.php?newsid=3178

Uh-oh - to quote:
"Electrical power can be related to the Planck constant, defined as the
ratio between the frequency of an electromagnetic particle such as a
photon of light and its energy. This experimental method of defining the
kilogram relies on selecting a fixed value for the Planck constant,
which is currently determined experimentally based on the fixed value of
the kilogram artifact."

The circularity is disturbing.