3-2-30-7 Quantitative

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Abstract 3-2-30-9

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Chapter 1
MEASURING:
Measured Value and Measuring Unit
1.1 The Empirical Method
All scientific insight begins when a curious and attentive person wonders about some
phenomenon, and begins a detailed qualitative observation of this aspect of nature. This
observing process then can become more and more quantitative, the object of interest
increasingly idealized, until it becomes an experiment asking a well-defined question.
The answers to this experiment, the measured data, are organized into tables, and can
be graphically visualized in diagram form to facilitate the search for correlations and
dependencies. After calculating or estimating the precision of the measurement, the so-
called experimental error, one can interpolate and search for a description or at least
an approximation in terms of known mathematical curves or formulae From
such empirical connections, conformities to known laws may be discovered. These are
mostly formulated in mathematical language (e.g. as differential equations). Once one
has found such a connection, one wants to “understand” it. This means either one finds
a theory (e.g. some known physical laws) from which one can derive the experimentally
obtained data, or one tries using a “hypothesis” to guess the equation which underlies the
phenomenon. Obviously also for doing this task a lot of mathematics is necessary. Finally
mathematics is needed once again to make predictions which are intended to be checked
against experiments, and so on. In such an upward spiral science is progressing.
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1.2 Physical Quantities
In the development of physics it turned out again and again how difficult, but also impor-
tant it was to develop the most suitable concepts and find the relevant quantities (e.g.
force or energy) in terms of which nature can be described both simply and comprehen-
sively.
Insert: History: It took more than 100 years for the discussion among the “nat-
ural philosophers” (especially D´Alembert, Bruno, Newton, Leibniz, Boskovic and
Kant) to create our modern concepts of force and action from the old terms prin-
cipium, substantia, materia, causa efficiente, causa formale, causa finale, effectum,
actio, vis viva and vis insita.
Every physical quantity consists of a a measured value and a measuring unit, i.e.
a pure number and a dimension. All difficulties in conversations are avoided, if we treat
both parts like a product “value times dimension”.
Example: Velocity: In residential districts often a speed limit v = 30
km
h
is imposed, which
means 30 kilometers per hour. How many meters is that per second?
One kilometer contains 1000 meters: 1km = 1000m, thus v = 30 · 1000
m
h
.
Every hour consists of 60 minutes: 1h = 60min, consequently v = 30 · 1000
m
60min
.
One minute has 60 seconds: 1 min = 60 s , therefore v = 30 ·1000
m
60·60s
= 8.33
m
s
.
Even that may be too fast for a ball playing child.
Insert: Denotations: It is an accepted thing in international physics for long
time past to abbreviate as many of the physical quantities as possible by the first
letter of the corresponding English word, e.g. s(pace), t(ime), m(ass), v(elocity),
a(cceleration), F(orce), E(nergy), p(ressure), R(esistance), C(apacity), V(oltage),
T(emperature), etc..
Of course there are some exceptions from this rule: e.g. momentum p, angular
momentum l, electric current I or potential V
Whenever the Latin alphabet is not sufficient, we use the Greek one:
alpha α A
beta β B
gamma γ Γ
delta δ
epsilon E
zeta ζ Z
eta η H
theta θ Θ
iota ι I
kappa κ K
lambda λ Λ
my µ M
ny ν N
xi ξ Ξ
omikron o O
pi π Π
rho ρ P
sigma σ Σ
tau τ T
ypsilon υ Y
phi φ Φ
chi χ X
psi ψ Ψ
omega ω
In addition the Gothic alphabet is at our disposal.
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1.3 Units
The units are defined in terms of yardsticks. The search for suitable yardsticks and their
definition, by as international a convention as possible, is an important part of science.
Insert: Standard units: What can be used as a standard unit? - The an-
swers to this question have changed greatly through the centuries. Originally people
everywhere used easily available comparative quantities like cubit or foot as units of
length, and the human pulse beat as unit of time. (The Latin word tempora initially
meant temple!) But not every foot has equal length, and the pulse can beat more
quickly or slowly. Alone in Germany there have been more than 100 different cubit
and foot units in use.
Therefore, since 1795 people referred to the ten millionth part of the earth meridian
quadrant as the “meter” and represented this length by the well-known rod made out
of an alloy of platinum and iridium. The measurement of time was referred to the
earth’s rotation: for a long time the second was defined as the 86400th part of an
average solar day.
In the meantime more exact atomic standards have been introduced: One meter is
now the distance light travels within the 1/299 792 485 part of a second. One second
is now defined in terms of the period of a certain oscillation of cesium 133 atoms in
“atomic clocks”. Perhaps some day these standards will also be improved.
Today, these questions are solved after many error ways by the conventions of the SI-units
(Syst`eme International d’Unit´es) The following fundamental quantities are specified:
length measured in meters: m
time in seconds: s
mass in kilograms: kg
electric current in ampere: A
temperature in kelvin: K
luminous intensity in candelas: cd
even angle in radiant: rad
solid angle in steradiant: sr
amount of material in mol: mol
All remaining physical quantities are to be regarded as derived, thus by laws, definitions
or measuring regulations traced back to the fundamental quantities: e.g.
7
frequency measured in hertz: Hz := 1/s
force in newton: N := kg m/s
2
energy in joule: J := Nm
power in watt: W := J/s
pressure in pascal: Pa := N/m
2
electric charge in coulomb: C := As
electric potential in volt: V := J/C
electric resistance in ohm: := V/A
capacitance in farad: F := C/V
magnetic flux in weber: Wb := Vs
Exercise 1.1 SI-units
a) What is the SI-unit of momentum?
b) From which law can we deduce the unit of force?
c) Who formulated this law first?
d) What is the dimension of work?
e) What is the unit of the electric field strength?
Insert: Old units: Some examples of units which are still widely in use in spite
of the SI-convention:
grad:
= (π/180)rad = 0.01745 rad
kilometer per hour: km/h = 0.277 m/s
horse-power: PS = 735.499 W
calorie: cal ' 4.185 J
kilowatt-hour: kW h = 3.6 · 10
6
J
elektron volt: eV ' 1.6 · 10
19
J
Many non-metric units are still used especially in England and the USA:
inch = Zoll: in = ” = 2.54 cm
foot: ft = 12 in ' 0.30 m
yard: yd = 3 ft ' 0.9144 m
(amer.) mile: mil = 1760 yd ' 1609 m
ounce: oz ' 28.35 g
(engl.) pound: lb = 16 oz ' 0.454 kg
(amer.) gallon: gal ' 3.785 l
(amer.) barrel: bbl = 42 gal ' 158.984 l
8
Exercise 1.2 Conversion of units
a) You are familiar with the conversion of angles from degrees to radiants using your
pocket calculator: Calculate 30
, 45
, 60
, and 180
in radiant and 1 rad and 2 rad in
degrees.
b) How many seconds make up one sidereal year with 12 months, 5 days, 6 hours, 9
minutes and 9.5 seconds?
c) How much does it cost with an “electricity tariff” of 0.112 /kWh, if you burn one night
long a 60-Watt bulb for six hours and your PC runs needing approximately 200 watts?
d) Maria and Lucas measure their training distance with a stick, which is 5 feet and 2
inches long. The stick fits in 254 times. What is the run called in Europe?
How many rounds do Maria and Lucas have to run, until they put a mile back?
e) Bill Gates said: “If General Motors had kept up with technology like the computer
industry has, we would all be driving twenty-five dollar cars that go 1000 miles per gallon.”
Did he mean the “3-litre car”?
1.4 Order of Magnitude
Natural phenomena are so various and cover so many orders of magnitude, that in
relation to a standard unit, e.g. meter, tiny or enormous numbers often result. Just think
of the diameter of an atom or the size of our Milky Way expressed in meters. In both
cases “useless” zeros arise. One has therefore introduced powers of ten and as well as
abbreviations and easily remembered names: e.g. the kilogram 1000 g = 103 g = kg. The
decimal prefixes, too, are today internationally standardized. We indicate the most
important ones:
tenth 10
1
= d dezi- ten 10
1
= D deka-
hundredth 10
2
= c centi- hundred 10
2
= h hecto-
thousandth 10
3
= m milli- thousand 10
3
= k kilo-
millionth 10
6
= µ mikro- million 10
6
= M mega-
billionth 10
9
= n nano- billion 10
9
= G giga-
trillionth 10
12
= p pico- trillion 10
12
= T tera-
quadrillionth 10
15
= f femto- quadrillion 10
15
= P peta-
Examples: In order to give you an idea of orders of magnitude, we give some examples
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from the field of length measurement:
The diameter of the range, within which scattered electrons feel a proton, amounts
to about 1.4 fm, atomic nuclei are between 3 and 20 fm thick.
The wavelengths of gamma-rays lie within the range of pm. Atomic diameters reach
from 100 pm to 1 nm.
Important molecules are about 10 nm thick. 100 nm is the order of magnitude of
viruses, and also the wavelengths of visible light lie between 300 and 800 nm.
Bacteria have typical diameters of µm, our blood corpuscles of 10µm, and protozoan
measure some 100µm.
Thus we already come to your everyday life scale of pinheads: 1 mm, hazel-nuts: 1
cm and grapefruits: 1 dm.
Electromagnetic short waves are 10 to 100 m long, medium waves 100 m to 1 km
and oscillate with 1 MHz. The distance e.g. of the bridges over the Neckar river in
Heidelberg amounts to 1 km. Flight altitudes of the large airliners are about 10 km.
The diameter of the earth is to 12.7 Mm and that of the Jupiter is about 144 Mm.
The sun’s diameter is with 1.4 Gm, the average distance of the earth from the sun
is approximately 150 Gm, and Saturn circles at a distance of approximately 1.4 Tm
around the sun.
Finally, light travels 9.46 Pm in one year.
Insert: Billion: While these prefixes of the SI system are internationally fixed,
this is by no means so with our familiar number words . The Anglo-American
and also French expression “billion” in the above table means the German “Mil-
liarde” = 10
9
and is different from the German Billion = 10
12
. “The origin of our
sun system 4,6 billion years ago...” must be translated as “die Entstehung unseres
Sonnensystems vor 4,6 Milliarden Jahren...”. Similar things apply to the Anglo-
American “trillion” = 10
12
, while the German “Trillion” = 10
18
.
Insert: Other unit names: Special names are also still used for some metric
units: You know perhaps 10
2
m
2
as are, 10
4
m
2
as hectare, 10
3
m
3
as litre, 10
2
kg
as quintal and 10
3
kg as ton.
Do you also know 10
5
P a as bar, 10
28
m
2
= bn as barn, 10
5
N = dyn, 10
7
J = erg,
10
15
m = fm under the name of Fermi, 10
10
m = 1
˚
A after
˚
Angstr
¨
om or 10
8
W b
under the name of Maxwell?
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Exercise 1.3 Decimal prefixes
a) Express the length of a stellar year (365 d + 6 h + 9 min + 9.5 s) in megaseconds.
b) The ideal duration of a scientific seminar talk amounts to one microcentury.
c) How long does a photon need, in order to fly with the speed of light
c = 2.997 924 58 ·10
8
m/s 21 m far through the lecture-room?
d) With the Planck energy of E
p
= 1.22 · 10
16
TeV gravitation effects for the elementary
particles are expected. Express the appropriate Planck mass M
P
in grams.
In the following we are only concerned with the numerical values of the examined
physical quantities, which we read off usually in the form of lengths or angles from our
measuring apparatuses, these being calibrated for the desired measuring range in appro-
priate units of the measured quantities.


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