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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.asp

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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).