Why Do We Use this Symbol for that?

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The
the
The Meaning Behind the
Symbols Common in
Symbols
in
Starring…
h p
c a s Ω λ
υ I π Q S e L
James Lincoln,
Tarbut V’ Torah
Irvine, CA, 2010
Introduction & Motivation
Often times in Physics we (teachers and students)
get so deep into solving a “problem” that the pursuit
becomes entirely mathematical.
We can sometimes forget that the formulas we are
working with actually stand for physical things.
This project is my attempt at an antidote for that, and
as a reminder that the symbols were chosen for a
reason; they actually symbolized something.
Introduction
First recall that the formulas do not actually exist
in nature. They are only a tool that assists in the
understanding.
I often have to remind my students that there are
no formulas in the real world. That the symbols
are representing real objects and interactions; and
that in the end it is the result of those interactions
that generates a predictable reality.
It is naïve to believe that the formulas themselves
are controlling reality, or that the physical world is
obeying the laws we have written down.
On the contrary, we are merely attempting to
approximate behaviors with these algebraic
formulae.
Some Quick and Easy Ones - 1
The symbol “equals” or “ = ” represents
two segments of the same length.
“…to avoide the tediouse repetition of these woordes: is equal
to: I will sette [a pair of] lines of one lengthe, thus: = bicause
noe 2 thinges can be moare equalle…”
-Robert Recorde, 1543
( The originator of “ = ” )
Are these two equals?
Source: E=mc2, Bodanis, pg 25
Some Quick and Easy Ones - 2
•In 1637, Rene Descartes in his La Geometrie used letters toward the
end of the alphabet, especially x, for unknowns, and letters near the
beginning of the alphabet for given quantities, ie a, b, c.
▪Descartes introduced the equation ax + by = c, which is still used to
describe the equation of a line.
Source: Return of Descartes
Johnson, pg 145
http://veling.nl/anne/templars/
constants.html
Some Quick and Easy Ones - 3
The symbol “ π ” was popularized by Euler who used it in the
year 1748 * in Introductio in Analysin Infinitorum. It was chosen
because it is the first letter of the greek word for perimeter.
περιμετρου
( “perimetron” )
Source:
http://www.maths.tcd.ie/pub/HistMath/People/Euler/RouseBall/RB_Euler.html
* Note it was first used by W. Jones, 1706
Some Quick and Easy Ones - 4
“Ask me a question in Latin and I will answer
you in Greek.” - J. Robert Oppenheimer
Some of the symbols slip from one language to another...
v=λ υ
A wave’s velocity is equal to the wave’s λength
multiplied by the υumber of times it passes per second.
Source:
Old Text: An Introductory Course in College Physics,
Black pg 577
Some Quick and Easy Ones - 5
The letter “ I ” has not always been the symbol for Current.
In fact, “ C ” was also an earlier convention.
As an example, Maxwell in 1876 writes Ohm’s Law: “ R = E/C ”
At other times, he would use to the cross-cultural symbol letter “ I ”
which he says represents the “Intensity” of the current. (Intensité is the
French word for intensity.) Here he is discussing Faraday’s Lines of Force.
 I have a hypothesis that this “intensité” is actually related to way to measure
current before there were galvanometers. When people actually had to feel the
“Intensity of the pain” themselves in order to measure it. Any Ideas, sources?
Source: The Scientific Papers of James Clerk Maxwell, V2, pg 534
V1, pg 182
Q: the symbol for Everything
The everpresent the letter “Q” in Physics Equations almost always has
its origin in the French word “ quantité.” For example:
The Quantity of heat:
Q=mcT
( Carnot )
The Quantity of charge: Q = CV
(Coulomb)
The Quantity of Motion q = m*v
( Descartes )
The Quantity of speed
( Planck )
q = s/T
The Quantity of displacement? q
(Hamilton?)
The Quantity of Mass & Force Net Q = Q a
(?)
A few symbols I haven’t solved yet:
s for distance? …
Source:
s = spacio, spatio, latin ?
no source, Descartes, Einstein?
p for momentum= perpetuum, perpetuity ?
Source:
no source, Hamilton?
L for anguLar momentum = ???????
Source:
no source, Hamilton?
 Do you know the answers?
A More Thorough Discussion is
required of the following symbols
S – entropy
c – speed of light
a = ∑ F/m
Ω – Ohm’s Unit
k – Boltzmann
h – Planck
Here we go…….
Der Entropie und Rudolf Clausius: S
Rudolfo ClaSiuS – inventor of Entropie
Der Entropie von Rudolf Clausius: S
“I propose to name the quantity “ S ” the entropie of the
system, after the Greek word [τροπη trope], the
transformation. I have deliberately chosen the word
entropie to be as similar as possible to the word energie:
the two quantities to be named by these words are so
closely related in physical significance that a certain
similarity in their names appears to be appropriate.”
- Rudolf Clausius, 1865,
Clausius’ reason for choosing “ S ” is not provided
There are three possible reasons for this choice…
Source:
Mechanical Theory of Heat, Clausius [1850-1865]
Why did Clasius choose S for Entropie?
Guess #1
Clausius named it after Sadi Carnot.
This is a reasonable guess, especially because he often
refers to him as “ S . Carnot ” in his writing. Also,
much of Clausius is based closely around Carnot’s.
It seems even more reasonable when we find out that he
even invented his own Entropie unit, the Claud, “ Cl. ”
(no joke)
Source: Mechanical Theory of Heat, Clausius [1850-1865]
Why did Clasius choose S for Entropie?
Guess #2
Clausius chose it for the word summa, the integral.
This seems less likely, and only possible because of his previous
formulas of Entropie which often contain integrals.
Source: Mechanical Theory of Heat, Clausius [1850-1865]
Why did Clasius choose S for Entropie?
Guess #3 - My Guess
Clausius chose it as a symbol to match Entropie
with the other symbols that are already associated.
Remember Clausius said, “ I have deliberately chosen the word
entropie to be as similar as possible to the word energie.”
What letter would help make Entropie fit in with Energie?
p – Pressure
Q – Quantity of Heat
R – Ideal Gas Konstante
S – available
T – Temperature
U – Internal Energy
hmmmmmm….?
v – Volume
W – Work (Clausius both used L and W)
Source: Mechanical Theory of Heat, Clausius [1850-1865]
NEXT! The speed of light, c
The history of using “c” as the symbol for the speed
of light in Physics was chronicled by Kenneth
Mendelson in the article “The story of c” published in
the Nov. 2006 Am. J. Physics.
I will be referring heavily to that article and adding a
few bits of extra insight; especially with Einstein.
Mendelson found that c was first introduced by
Wilhelm Weber (German), who recognized that it was a
constant but did not connect it to light or EM waves.
Also, Weber’s “ c ” was actually off by a factor of √2 .
“Nevertheless it is from Weber that the letter c
eventually came to denote the speed of light.”
[Mendelson]
the speed of light, c
“Weber did not say why he chose c.” It is unlikely
that he chose it to represent the word “celerity” or
“celeritas” as is often believed. ( This idea was
introduced by Asimov, with no historical context).
In fact, Weber’s papers show that he originally
assigned this constant the letter “a” then later
switched to c. Not knowing it was the speed of light.
It is possible he was using c to represent Constante.
Often this German word is spelled with a K,
Konstante, however in Weber’s original papers it is
spelled with a “c”. (This flip-flop is not unusual).
Reference:
Isaac Asimov – C for Celeritas, Mag of Fantasy Sci Fi 1959
What about E=mc2
Initially, Einstein would follow Maxwell’s notation, using V
for the speed of light (probably because he referred directly
to Maxwell’s work so frequently).
Einstein originally wrote “if a body emits energy L then its
mass decreases by L / V2 .” Where he called L the
available energy to be lost of radiation. [Source: Einstein, 1905 #4]
How do you feel about L = m V2 ? …..
NOTE: The first person to use c exclusively for the speed of light
was Max Abraham* (also German) in 1903, he also published a
popular Electromagnetism textbook in 1904, which – translated to
English – became and stayed popular up to the 1950’s.
* Abraham studied under Max Planck
What about E=mc2
Q. So why in 1907 did Einstein switch from L to ε and V to c ?
It wasn’t just because it was
the first letter of his name.
A. Einstein made a new friend, someone who used c for the
speed of light, and used ε for Energy.
………………Can you Guess who that friend was?
Source: Einstein, Walter Isaacson
What about E=mc2
Q. So why in 1907 did Einstein switch from L to ε and V to c ?
It wasn’t just because it was
the first letter of his name.
A. Einstein made a new friend, someone who used c for the
speed of light, and used ε for Energy.
………………Can you Guess who that friend was?
Source: Einstein, Walter Isaacson
Max Planck
Die Universalkonstanten von Max Planck und von Ludwig Boltzmann
Mr. h
Planck, 1901
when he wrote
ε=hν
Mr. k
Boltzmann, 1844-1906
note the equation S = k log W
In 1899 Planck solves the equation for the Black Body
Spectrum & finds two distinct constants in the solution.
At first he calls them α and β
Later he redoes the solution and
republishes, changing the letters.
He solves for the values of the constants,
and interprets their units, calling both of
them either “natural” or “universal” constants.
Source: Planck 1900, Planck 1901
The Boltzmann Konstante
He recognizes one of the constants from Boltzmann’s work.
(Boltzmann and Planck corresponded frequently).
It is the gas constant R divided by Avogadro’s Number NA.
k = R / NA
Planck gives it a new symbol, k, probably for konstante.
…we continue…
Source:
Planck 1900, Planck 1901
The Planck Constant
There’s also that other constant - a new one -he doesn’t recognize.
Initially, Planck didn’t know what to call his new “Naturconstanten.”
[Source:tp://isi.cbs.nl/NLet/NLet081.htm ]
Thus, he named it h for hilfsgröße (auxiliary quantity) and made no
further interpretation at this time.
[Source:tp://isi.cbs.nl/NLet/NLet081.htm ]
“In this situation, Planck had no reason to question the continuity
of the resonator energy.” [Source: pg 75 Darrigol from C to Q Numbers ]
Therefore, Planck probably did not fully understand that h was the
Quantum of Action until Einstein explained it in 1906.
[Source: Planck's Theory of Radiation and the Theory of Specific Heat" ]
The Planck Constant
=================
In this house
taught
MAX PLANCK
The discoverer
of the elementary
Quantum of Action h
from to
1889
1928
=================
Fun Fact:
In German “h-bar” is
Called “h-cross.”
Retranslating Newton’s 2nd Law
Newton wrote his second
law in sentences, not
equations. He also wrote
the Principia in Latin.
Latin does not translate
very well into English.
Therefore, some
translations will differ
slightly. Some will be
more helpful than others.
Retranslating Newton’s 2nd Law
Here is one translation of N 2nd
that I found to be enlightening:
The alteration of motion* is ever
proportional to the motive force
impressed; and is made in the
direction of the right line in
which that force is impressed.
* where “Motion” = mass x velocity
Source: Principia, Translated by Andrew Motte
Retranslating Newton’s 2nd Law
The alteration of motion is ever
proportional to the motive force
impressed; and is made in the
direction of the right line in
which that force is impressed.
Essentially, a ~ F . The constant of
proportionality is obviously 1/m.
Thus: a = F/m can be read not as
“acceleration” but as change or
“alteration” of motion.
Why is this helpful?
Source: Principia, Translated by Andrew Motte
Retranslating Newton’s 2nd Law
Why is this helpful?
If you tell students that force is proportional to “ a ” the alteration of motion - they will quickly understand
that forces can do more than just speed things up.
Forces can change direction – an alteration.
Forces can reduce speeds – an alteration.
Forces can add perpendicular components of velocity.
It eliminates the confusion of “Does accelerating mean
getting faster?” Is this ever an issue in your classroom?
Ohm’s Unit
Let’s do some quick unit analysis on Ohm’s Law.
What is an Ohm? From Ohm’s law you might think that it is a Volt
per Ampere. V / I = R
Very well, then what is an ampere? What is a Volt?
An Amp is 1 Coulomb/sec
A Volt is a Joule/Coulomb.
Very well, then by plugging,
an Ohm is a Joule*sec/Coulomb2.
This makes an Ohm…………….
The Resistible
G.S. Ohm
Ohm’s Unit
Makes an Ohm = 1 kilogram meter2/sec/ (.624*1019 e)
( where e = 1 electronic charge )
That does not look very standard.
Where did we get this “Ohm?”
Georg Simon Ohm (1789-1854) lived before
Thompson and Millikan and therefore could not have
possibly known the electronic charge.
The Ohm is a derived unit. This is how he derived
it in 1827*
*source: metrication timeline .
The Original Ohm Unit
“Taking wire of different material, but of the same thickness,
he found that the following lengths possessed equal
conductivities: copper 1000, gold 574, silver 356*, zinc 333,
brass 280, iron 174, platinum 171, tin 168, lead 97.”
The measurements were made by using a torsion balance
with a constant current running through it, via by Ampere’s
attraction of two parallel currents. (Cajori)
*An experimental error. Silver is actually a better conductor that copper,
the real value is actually 1057
Source: A History of Physics Cajori
The Original Ohm Unit
It is important to note that Ohm’s Unit of
resistance, “1000 units” of no. 10 [ American
Standard Guage] copper wire*, when the unit
is a foot results in 1 Ohm of resistance.
1 Ohm
I do not know if Ohm’s unit was the “foot”
(although it is possible because Germany
didn’t go metric until the European
“Convention du Mètre” in 1875.
Hence there actually should be an apostrophe in the unit Ohms.
From now on you may wish to refer to it as “Ohm’s Units for
Resistance” or simply “Ohm’s.”
*Note we now define the Ohm using Mercury rather than copper,
because copper changes resistance quickly when heated.
~ FIN ~
Thank you.
James Lincoln: [email protected]
Some Quick and Easy Ones
The letter “ G ” was of course first measured by
Cavendish, but he did call it that. Instead he usd the
symbol k, just like everyone else.
G is a good example of when someone establishing a very good
measurement assigns their own symbol to a constant. G was
measured precisely in ________ in a german paper called
_____________. (Popularized by Planck? Check constants of the
universe book).
Source:
http://www.maths.tcd.ie/pub/HistMath/People/Euler/RouseBall/RB_Euler.html
* Note it was first used by W. Jones, 1706
Introduction and Major Points
Formulas were not always present throughout the
history of physics. Consider Hooke’s Law as he
states it:
Latin: “ Ut tensio, sic vis ”
( As the extension, so the force )
Introduction and Major Points
Change in motion are proportionate to the motive
force impressed and takes place on a the straight
line along which that force is impressed.
Where previously Newton defined motion as the
product of mass and velocity, ie momentum.
Newton would also use segments for symbols,
being highly geometrical in his writing
For example, his formula for centripetal
acceleration was written as bd2/sb.
Introduction and Major Points
Newton also often spoke in terms. Consider his
statement of the second Law:
Mutationem motus proportionalem esse vi motrici
impressae et fieri secundum lineam rectam qua vis
illa imprimitur.
Change in motion are proportionate to the motive
force impressed and takes place on a the straight
line along which that force is impressed.
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