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8/10/2019 01_B Units
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PHYSICS IN L IFE SCIENCES
UnitsRelated textbook chapter:
Appendix A.
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Key concepts
SI unit system.
Base units
How derived units are defined.
Dimensional analysis.
How to use units to analyze a problem.
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Why are there units?
There are physical quantities in our reality
which appear to have different nature.
Distanceand time.
Forceand energy.
Electric currentand water current.
And many more
They have to be counted by different
units.
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A number means nothing
without a unit (in physics)
Can you figure out the meaning of the
following statements?
He drinks 300___of water everyday.
I got charged 0.5___per megabytes in the
hotel.She got tired after walking for 1___.
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Unit systemA unit system provides the standard for
measuring different physical properties.
The international system of units (SI) is the
most widely used.
First, seven base unitsare defined.
Second, derived units are defined as the
combination of base units according to the
laws of physics.
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SI base units
Base units are defined on a fundamental physical propertythat can be reproduced in different laboratories.
Base quantity Name Symbol
length meter m
mass kilogram kg
time second s
electric current ampere A
thermodynamic temperature kelvin K
amount of substance mole mol
luminous intensity candela cd
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Derived units from decimal factorsDECIMAL FACTOR PREFIXES
FACTOR PREFIX SYMBOL FACTOR PREFIX SYMBOL
1018 exa E 10-1 deci d
1015 peta P 10-2 centi c
1012 tera T 10-3 milli m
109 giga G 10-6 micro m
106 mega M 10-9 nano n
103
kilo k 10-12
pico p
102 hecto h 10-15 femto f
101 deka da 10-18 atto a
Example: 1 nm = 10-9
m; 1 fs = 10-15
s; 1 MB = 106
bytes.
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Length -- Meter
Original definition: one ten-millionth of the lengthof the earth's meridian through Paris from pole
to the equator (one fourth the circumference of
the earth).
Current definition: The meter is the length of the
path travelled by light in vacuum during a time
interval of 1/299 792 458 of a second.
International
prototype
(1889)
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Derived units from meter
[Area] =
[length2
] = [m2
]
[Volume] =[length3] = [m3]
1 L = 1 dm3= 10-3m3
1 ml = 1 cm3= 10-6m3
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Time -- Second
Originally, the second was defined as thefraction 1/86 400 of the mean solar day.
However, the accuracy of this definition is
very much questionable. Modern definition:The second is the
duration of 9 192 631 770 periods of the
radiation corresponding to the transitionbetween the two hyperfine levels of the
ground state of the cesium 133 atom.
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Derived units
[frequency] = [# of oscillations/time] =[s-1] = [Hertz] = [Hz]
Clock frequency= 2 GHz
= 2 billion (109)
oscillations persecond
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Derived units for describing motion
[speed] = [length/time] = [m/s]
[acceleration] = [speed/time] = [m/s2]
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Mass -- Kilogram
A kilogram is the mass of a cubic decimeter(a liter) of water at 4oC and 1 atm pressure.
It is equal to the mass of the international
prototype of kilogram.
platinum-iridium alloy
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Derived units
[Density] = [mass/volume] = [kg/m3]
Density of water
= 103kg/m3 =
1 g/cm3
Density of oil
= 0.9 g/cm3
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Derived units for forceNewtons 2ndlaw: F = ma
[force] = [massacceleration] =
[kgm/s2] = [Newton]
Force and motion and
connected.
How we measure force
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Derived units for the effect of force
[pressure] = [force/area] = [kg/(ms2)]= [Pascal]
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Derived units for energy[energy] = [work] = [forcelength] =
[kgm2/s2] = [Joule]
[power] = [energy/time] = [kgm2/s3] =
[Watt]
http://www.freakingnews.com/Mouse-Power-Generator-Pics-26779.asp8/10/2019 01_B Units
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Absolute Temperature -- Kelvin
Celsius temperature: under 1 atmospherepressure, the freezing point of water is 0oC,
and the boiling point of water is 100oC.
T in Kelvin (K) = T(oC) + 273.15 0 K or -273.15oC is the lowest temperature
limit:absolute zero.
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Derived units[Specific Heat Capacity] =
[Energy/(masstemperature)] =
[J/(kgK)]= [m2/(s2K)]
Cp(water)= 4.18 J/(gK)
4.18 J of heat energy is
required to raise thetemperature of one gram
of water by 1 C or 1 K.
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Spaceship Voyager
Launched in 1977, spaceship Voyager 1 & 2 have
traveled 22 billion kilometers and will leave the
boundary of our solar system in 5 years.
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Describing our unit system to
aliens: Images carried by
spaceship Voyager 1 & 2Transition between the lowest two
states of hydrogen atom
Definition of mathematical symbols Definition of time, mass, length units
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Not only the numbers on both sides must
agree, the units on both sides should agree
as well!
Dimensional Analysis
Principle: The units on both sides of an
equation must be the same.
Suppose we have an equation:
X/g= a(Y*Z)/b
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Question
Which of the following equations is correct?
(A) 8 J = 2 kg4 m/s
(B) 8 J = 2 kg(2 m/s)2
(C) 8 J = 2 kg2 (m/s)2
(D) 8 J = 2 kg4 m/s2
[J] = [kgm2/s2]
Only answer (B) has both the numbers and
units correct.
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Dimensional analysis can be used
to analyze connections between
physical quantities
If you know a physical quantity (X) is
determined by some other quantities (Y,
Z), you may derive their qualitative
dependence just by analyzing their
units.
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* Speed of the airplanev [m/s]
* Area of the wingA [m2]
* density of airr [kg/m3]
ExampleHow does the li f ting force on the w ings o f
airplane depend on the speed?
What are the factors that may affect lifting force F:In a simplified model:
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F ~r A v (assumed dependence, all power
parameters to be found out)
[kg] [m] [s]-2= [kgm-3][m2][ms-1]
1=
1 = -3 +2+ and = 1
2 =
F ~rA v2
Lifting force
[kg] [m] [s]-2= [kg][m]-3 +2+ [s]-
A meaningful result!
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Now consider a bird with a linear size described by L:
Wing sizeA~ L2; Body volume Vbird~ L3.
gL
v
W
F
bird
air
2
r
r
The ratio of
lifting force tobody weight:
This ratio must be
bigger than a certainvalue to fly.
Applying the result to birds
222 vLAvF airair rr Lifting force:
Body weight: gLgVmgW birdbirdbird3
rr
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gLv
WF
bird
air
2
r
r
A larger bird (larger L) needs higher speed (larger v) to
take off.
Problem of large birds
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Gave up flying. Large wings. Using airflow.
How it affects the evolution of large
birds
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How could Pterosaurs fly?
gL
v
W
F
bird
air
2
r
r
It turns out that the density of atmosphere (rair) seventy
million years ago may be four times thicker than today,
which greatly helped flying. (http://www.dinosaurtheory.com/flight.html)
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Limitation of dimensional analysis
As the starting point, the factors that affect
the target physical quantity must be
accurately identified.
Choosing too many or too few factors will
lead to impossible or unreliable results.
Dimensional analysis is not a proof!
Without detailed analysis, it cannot serveas the foundation of a theory.
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More discussion: dimensionless
unitthe unit of angle
An angle represents the ratio between twolengths, and therefore its dimensionless.
Unit of angle: degree(360 for a circle) or radian
(2pfor a circle).
q1
Expandingeverything by the
same ratio
q2
q1q2 The value of angle has no dependence ondimension.
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More discussion: arguments to
mathematical functions
The argument to a mathematical function
(xin the brackets) must be dimensionless!
There is no such thing as sin(2 meters) or
exp(3 seconds).
)sin(x
)cos(x
)ln(xx
ex
)exp(
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More discussion: extensive
property and intensive property
Extensive properties scale up with the sizeof an object.
mass, volume, energy, entropy, heat
capacity
Intensive properties are scale invariant.
density, temperature, specific heat capacity
They are usually defined as an average ofone extensive property over another.
Intensive properties truly represent theinherent propertiesof a material.
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More discussion: Is the unit system
arbitrary?
Yes. The choice of base units and how to define
base units are totally arbitrary.
There are other unit systems we use:
English system: inch/foot, pound Natural unit systemthat is popular in fundamental
physics: time and length has the same unit!
There is always a way of unit conversion between
different systems.
Whatever unit system we use, physics (physical
predictions) does not change!
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The End
Next:
Basics of physics: scalar and vector.
Related textbook chapter: 3.5, Appendix C.
Description of motion.
Related textbook chapter: 2.92.10,
Appendix B (before Torque).