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This is the no-nonsense module
with the hard facts about units, constants and transformations from one system
of units into an another one (after this paragraph, that is). |
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No explanations, historical
roots, really outdated or unusual units are given - for the
fun part use the link. |
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First, the
basics: |
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In physics we always have two things: a physical quantity - e.g. the speed of something, or
the strain of something under load - and
some units to measure the quantity in question. |
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The physical quantity is
what it is - it does not depend on how you
express it in numbers. Somebody on some other planet will for sure do it
differently from you and me. |
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The number you will give
to the physical quantity is strictly a function of the units you chose. You might use m/s, oder
lightyears/s, or wersts/year - that will just change the number for the speed of the moving object a lot, but
not the speed itself. Trivial, but often forgotten. |
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To make life easier for everybody
(at least for scientists), the choice of units was taken away from you and me,
and everybody is now required to strictly
adhere to the international standard
system, abbreviated in any language as
SI units. |
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Well, by now you, and I, and most others scientists, do comply
with the SI system (which was not always the case) - but the
public at large does not give shit; especially in the USA. Tell the gas
station attendant any number you like in Pascal or bar for the
tire pressure, and he (or she) will just look at you as if you escaped from the
lunatic asylum. Its psi or
bust! And on occasion, even engineers or scientists do not use SI units - with
disastrous consequences if you have
tough luck. |
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The question now is: how many
basic units do we need, so we can express everything else in these units? And which ones do we
take? |
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This is one of the deeper questions of humankind. Physicists
claim that we just need one more truly basic constant of nature - and we do not need
units at all anymore. Velocities, for
instance, can always be given using the absolutely constant speed of light (in
vacuum) as the unit; your typical car speed than would be something like
0,000001. |
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But redundancy tends to make life easier (just look at your
typical Sheik and his harem), and the SI system gives us 7 basic units which are independent of each
other. |
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Quantity |
Name |
Unit |
Length |
Meter |
m |
Mass |
Kilogram |
kg |
Time |
Second |
s |
Electrical current |
Ampere |
A |
Thermodynamic temperature |
Kelvin |
K |
Amount of substance |
Mol |
mol |
Luminous intensity |
Candela |
cd |
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From this basic units all other
SI units can be derived. Below are tables with the more important
secondary units. |
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First, we look at some secondary units just
invoking basic units and a length. While we
often do use special symbols for these quantities (e.g. r for density), these symbols are not really necessary
and thus were not pronounced immutable and sacred as, e.g., the
"m" for meter or the "s" for second. |
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Quantity |
Name |
Unit |
Area |
Square meter |
m2 |
Volume |
Cubic meter |
m3 |
Velocity |
Meter per second |
m/s; ms1 |
Acceleration |
Meter per square second |
m/s2; ms2 |
Wave number |
reciprocal meter |
m1 |
Density |
Kilogram per cubic meter |
kg/m3 |
Specific volume |
Cubic meter per Kilogram |
m3/kg |
Electrical current density |
Ampere per square meter |
A/m2 |
Magnetic field strength |
Ampere per meter |
A/m |
Substance concentration |
Mol per cubic meter |
mol/m3 |
Luminance |
Candela per sqare meter |
cd/m2 |
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Now some more involved units -
including important quantities like energy,
voltage, and magnetic things. |
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They are more involved, because we usually do
not express them in SI basic units -
which is perfectly possible - but in secondary units. We will also find one case where
there is no unit - it just cancels
out. |
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These units often have their own
symbols for reasons that become clear if you look at the SI units, and
these symbols should not be used for something else |
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Quantity |
Name |
Symbol |
Unit |
In secondary units |
In basic units |
Plane angle |
Radiant |
rad |
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m / m = 1 |
Frequency |
Hertz |
Hz |
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s1 |
Force |
Newton |
N |
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m · kg · s2 |
Pressure, stress |
Pascal |
Pa |
N/m2 |
m1 · kg ·
s2 |
Energy, work, quantity of heat |
Joule |
J |
N·m |
m2 · kg · s2 |
Power, energy flux |
Watt |
W |
J/s |
m2 · kg · s3 |
Quantity of electricity
Electric charge |
Coulomb |
C |
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A·s |
Electric potential, voltage |
Volt |
V |
W/A |
m2·kg·s-3·A1 |
Capacitance |
Farad |
F |
C/V |
m2·kg1·s4·A2 |
Electric resistance |
Ohm |
W |
V/A |
m2·kg·s3·A2 |
Conductance |
Siemens |
S |
A/V |
m2·kg-1·s3·A2 |
Magnetic flux |
Weber |
Wb |
V·s |
m2·kg·s2·A-1 |
Magnetic flux density |
Tesla |
T |
Wb/m2 |
kg·s-2·A-1 |
Inductance |
Henry |
H |
Wb/A |
m2·kg·s2·A2 |
Celsius temperature |
Degree Celsius
("Centigrade") |
oC |
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K |
Radioactivity |
Becquerel |
Bc |
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1/s |
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Mercifully, the members of the
"Comite International des Poids and Mesures" are human (up to a
point, at least). In consequence they did not outlaw all older units in one fell stroke, but
sorted them into three groups: |
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"Old" units which may be
used together with SI units without
restrictions. |
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Old units which may be used for some time in parallel to SI units. |
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Old units which are definitely out and must not be used at all any more. |
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Some of the units in the second
category are regional and you probably have never heard of them. We will not
include them here. The number of outlawed units is legion, we just include the
still tempting ones. |
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Here is the first
category: Some of the non-SI units you still
may use without restrictions: |
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Quantity |
Name |
Unit |
Minute |
min |
1 min = 60 s |
Hour |
h |
1h = 60 min = 3600 s |
Day |
d |
1 d = 24 hr = 86400 s |
Angle Degree
Angle minute
Angle second |
o
'
'' |
1o = (p/180) rad
1 ' = (1/60) o
1 '' = (1/60) ' = (1/3600) o |
Liter |
l, L |
1 l = 1 dm3 = 103
m3 |
Ton |
t |
1 t = 103 kg |
Electronvolt |
eV |
1 eV = 1,602 540 2 · 1019 J |
Atomic mass unit |
u |
1 u = 1,660 540 2 · 1027 kg |
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What a relief! |
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Now to the old units you may use for some more time to come in
parallel to the SI units: |
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Quantity |
Name |
Unit |
Ångstrom |
Å |
1 Å = 0,1 nm |
Ar |
a |
1 a = 100 m2 |
Hectar |
ha |
1 ha = 100 a |
Bar |
bar |
1 bar = 0,1 MPa |
Barn |
b |
1 b = 100 fm = 1028 m2 |
Curie |
Ci |
1 Ci = 3,7 · 1010 Bq |
Roentgen |
R |
1 R 0 2,58 · 104 Ci/kg |
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Now to the units you must not use anymore!. We might put them
into two groups: |
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1. The forerunners of the SI units,
the cgs
units; i.e. the units based on the
centimeter, the
gram and the
second. |
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2. The simple old fashioned no-no's. |
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While it may appear that the
cgs system is practically the same as the SI system, this is not so! |
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Of course, the cm, g, and
s are essentially the same basic units as
in the SI system, the abbreviation "cgs", however, does
not tell you anything about the other necessary basic units in this system -
and that is where the problems come
in! |
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In fact, there were several cgs systems - the electrostatic, the electromagnetic, and the Gauss cgs system! |
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We will not unravel all the intricacies for
cgs systems and the conversion to SI units here - this is done in
its own module - but just give some of
the more common units and their conversion. |
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Quantity |
Name |
Unit |
Erg |
erg |
1 erg = 107 J |
Dyn |
dyn |
1 dyn = 105 N |
Poise |
P |
1 P = 1 dyn·s/cm2 = 0,1
Pa·s |
Gauss |
Gs, G |
1 G corresponds to 104 T |
Oersted |
Oe |
1 Oe corresponds to (1000/4p)
A/m |
Maxwell |
Mx |
1 Mx corresponds to 108 Wb |
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The "corresponds to" instead of simply
"=" is an indication that while the three quantities in
question do have SI units that correspond to magnetic flux density,
magnetic field strength, and magnetic flux, they are not exactly the same thing. |
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Finally, some still fondly remembered
old units you simply do not use
anymore: |
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Quantity |
Name |
Unit |
Torr |
torr |
1 Torr = (101 325/760) Pa
= 133.32 Pa |
Physical atmosphere |
atm |
1 atm = 101 325 Pa |
Kilopond |
kp |
1 kp = 9,806 65 N |
Calory |
cal |
1 cal = 4,186 8 J |
Micron
(Micrometer is what you use!) |
µ |
1 = 1 µm |
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Fundamental constants are some numbers with units that cannot
(yet) be calculated from some physical theory, but must be measured. |
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This may have three possible
reasons:
- There is presently no theory, and there never will be a theory, that allows to calculate
fundamental constants. They have the value they have because an act
of one or more gods
and/or godesses, or they are purely random (i.e we just happen to live in
an universe, where the value is what we measure. In some other universe, or
some other corner of our universe, it will be arbitrarily different).
- There is presently no theory, but some day there will be one. Some
fundamental constants will then be calculated and then are no longer
fundamental.
- There already is a theory, or at least a general theoretical framework; we
just are not yet smart enough to see the obvious or to do the numerics. Masses
of elementary particles, e.g., might be "fundamental constants" that
fall into this category.
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Hot-shot physicists have some ideas, which
constant might fall into which category. Speculations along this line are a lot
of fun - but of no consequence so far. So I
will not dwell on this. (Of course, you may
check for yourself which one of the three possibilities you are going to
embrace and thus get some idea of what kind of person you are). |
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Fundamental physical theories usually
introduce one new fundamental constant. Mechanics (including gravitation) needs
the gravity constant G, quantum theory has Plancks constant h,
statistical thermodynamics introduces Boltzmanns constant k, the special
theory of relativity (or Maxwells theory of electromagnetism which is really
part of the relativity theory) needs the speed of light c. |
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New theories sometimes "explain" old
constants of nature because they can calculate them, or replace them by
something more fundamental. Boltzmann's constant k, for example, is more
fundamental than the "fundamental" gas constant R, because it
relates its number to a fundamental unit of matter (1 particle) and not
to an arbitrary one like 1 Mol. |
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How many truly fundamental constants
are there? Why do they have the values they have? (Just slight deviations in
the values of some constants would make carbon based life impossible; this is
where the so-called "anthropic
principle" comes in). Will we eventually be able, with a "Theory
of Everything" (TOE) to calculate all
natural constants? |
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Nobody knows. We run against the deepest physical
questions at this point. |
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So let's just look at what we have. Since it is
customary to list as natural constants some quantities that are actually
computable from others, we include some of these "constants" here,
too (together with the conversion formula). |
Symbol and formula |
Numerical value |
Magnitude and unit |
Remarks |
Speed of light in
vacuum |
c0, c |
2,997 924 58 |
108 m·s1 |
Truly fundamental |
Gravitational
constant |
G |
6,673 |
1011
m3·kg1·s2 |
Truly fundamental |
Planck's
constant |
h |
6,626 068 76 |
1034 J·s |
Truly fundamental |
4,1356 |
1015 eV·s |
Elementary
charge |
e |
1,602 176 462 |
1019 C |
Truly fundamental ?
Maybe not |
Fine structure
constant |
a =
µ0·c·e2/2h |
7,297 352 533 |
103 |
Unitless, maybe more
fundamental than others. |
Mass of a electron at
rest |
me |
9,109 381 88 |
1031 kg
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Not truly fundamental; can be
calculated in principle |
0,510 998 902 |
MeV |
Mass of a proton at
rest |
mp |
1,672 621 58 |
1027 kg |
Not truly fundamental, can be
calculated in principle |
1,007 276 466 |
u |
938,271 998(38) |
MeV |
Avogadro
constant |
NA |
6,022 141 99(47) |
1023 mol1 |
Not truly fundamental any more |
Faraday
constant |
F = e·NA |
96 485,3415(39) |
C·mol1 |
Not truly fundamental any more |
Universal gas constant |
R |
8,314 472(15) |
J·mol1·K1 |
Not truly fundamental any more |
Boltzmann
constant |
k = R/NA |
1,380 6503 |
1023 J·K1 |
Truly fundamental |
8,617269 |
105 eV·K1 |
Magnetic
permeability of vacuum |
µ0 = 1/e0c2 |
12,566 370 614 |
107
V·s·A1m1 |
Not truly fundamental |
Electric
susceptibility of vacuum |
e0 =
1/µ0c2 |
8,854 187 817 |
1012
A·s·V1m1 |
Not truly fundamental |
Magnetic flux
quant |
P = h/2e |
2,067 833 636 |
1015 Wb |
Smallest possible magnetic flux
Not truly fundamental |
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© H. Föll (MaWi 1 Skript)