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Science fundamentals
Dr. Jay Maron


Units
Distancre
Mass
Time
Charge

The fundamental units are the meter, second, kilogram, and Coulomb. They were defined in 1793 as the "Standard International" (SI) units, or "MKS" units.


Quantity   Unit       Definition

Length     Meter      The Earth's circumference is 40 million meters
Time       Second     There are 86400 seconds in one Earth day
Mass       Kilogram   The mass of a cube of water 10 cm on a side is 1 kilogram
Charge     Coulomb    The force between two charges of one Coulomb each and
                      separated by 1 meter is 9 billion Newtons

1 kg of water
1 kg of air
Density of water  =1000   kg/meter3  = 1      g/cm3
Density of air    =   1.2 kg/meter3  =  .0012 g/cm3

Derived units

The fundamental units are length, mass, time, and charge, and all other units are derived from these.


Quanity         Unit       Composition

Length          meter      X                                  meter
Mass            kg         M                                  kg
Time            second     T                                  second
Charge          Coulomb    C                                  Coulomb
Speed                      V = X/T     Length / Time          meter/second
Acceleration               A = V/T     Speed  / Time          meter/second2
Momentum                   Q = M V     Mass   * Speed         kg meter/second
Force           Newtons    F = M A     Mass   * Acceleration  kg meter/second2
Energy          Joule      E = F X     Force  * Distance      kg meters2/second2
Power           Watt       P = E/T     Energy / Time          kg meters2/second3
Area                       S = X2      Length2                meters2
Volume                     Υ = X3      Length3                meters3
Density                    ρ = M/Υ     Mass   / Volume        kg / meters2
Pressure        Pascal     Φ = F/S     Force  / Area          Newtons/meter2  =  Joules/meter3
Angular momentum           L = M V X   Momentum*Length        kg meters2/second
Torque                     Γ = F X     Force  * Length        kg meters2/second2
Frequency       Hertz      f = 1/T     1      / Time          1/second

Unit conversions
Meter        =    3.281 feet
             =   39.37  inches
Mile         = 5280     feet       (exact)
             = 1609     meters
Foot         =   12     inches     (exact)
Inch         =   25.4   mm         (exact)

Minute       =   60     seconds    (exact)
Hour         =   60     minutes    (exact)
Day          =   24     hours      (exact)
Year         =  365.25  days

Ton          = 1000     kg         (exact)
Kilogram     = 1000     grams      (exact)
             =    2.205 pounds     (pounds interpreted as mass)
Newton       =     .225 pounds     (pounds interpreted as force)
Pound        =   16     ounces     (exact)  (interpreted as mass)
             =     .454 kg
                  4.448 Newtons    (Newtons interpreted as force)
Ounce        =   28.35  grams      (ounces interpreted as mass)
Meter/second =    2.24  miles/hour
Km/hour      =     .621 miles/hour
Miles/hour   =    1.609 km/hour
Pascal       = .0001450 pounds/inch2   (pounds interpreted as force)
Pound/inch2  = 6895     Pascals
Bar          =101325    Pascals    (Atmosphere pressure at sea level)
             =   14.50  pounds/inch2     (pounds interpreted as force)
Earth gravity=    9.807 meters/second2
             =   32.2   feet/second2
Standard sheet of paper  =  11 x 8.5 inches  =  27.94 x 21.59 cm

Magnitudes


Distance

Larger Picture
                    Meters       Earth      Earth        Light travel
                                 radii      orbits           time
                                            (AU)
Nucleus                2⋅10-15
Atom                   2⋅10-10
Green light            5.5⋅10-7
Neuron                    .00002
Dime thickness            .00135
Dime diameter             .0178
Quarter diameter          .024
Tennis ball diameter      .067
Soccer ball diameter      .22
Average person           1.78
Central Park width     800
Mount Everest         8848
Moon radius        1737000           .273
Mars radius        3390000           .532
Earth radius       6371000          1.0
Jupiter radius    6.991⋅107         10.9
Moon distance     3.844⋅108         60.3         .00257      1.5 seconds
Sun radius        6.957⋅108        109           .00474      2.3 seconds
Earth orbit       1.496⋅1011     23481          1.0          8   minutes
Jupiter orbit                                  5.2         40   minutes
Neptune orbit                                 30.1          3   days
Light year        9.461⋅1015                63241            1   year
Alpha Centauri                                              4.4 years      Nearest star
Galaxy thickness                                         1000   years
Galaxy center                                           27200   years
Galaxy diameter                                        100000   years
Andromeda distance                                          2.5 million years
Virgo cluster distance                                     54   million years
Size of universe                                           14   billion years
A photon traveling from the Earth to the moon

Speed

Cheetah
Needletail
Concorde
F-16 Falcon

F-15 Eagle
F-35 Lightning
F-22 Raptor
SR-71 Blackbird

            meters/second   Mach

Walk                 1.5
Running sprint      10
Cycling sprint      20
Cheetah             30             Fastest land animal
70 miles/hour       31
Baseball pitch      45             100 miles/hour
Human neuron       100
Sound at altitude  295             Speed of sound at altitude 10 km to 20 km
747 airplane       266      .9
Sound at sea level 340     1.0     At sea level and 15 degrees Celsius
F-35 Lightning     475     1.6     Stealth fighter
F-16 Falcon        590     2.0
Concorde           606     2.05
F-22 Raptor        670     2.3     Stealth fighter
F-15 Eagle         740     2.5
SR-71 Blackbird    980     3.3
Orbit speed       7800    26.4     Minimum speed to orbit the Earth
Escape speed     11200    38.0     Minimum speed to escape the Earth's gravity
Ion rocket      100000             Fastest spacecraft we can build
Fission rocket     107
Fusion rocket      107
Light            3⋅108  1020000
Aircraft typical fly at altitude 10 km to 20 km, where the speed of sound is 295 m/s. Mach 1 for aircraft is defined using this speed.


Mass

                       kg         Earth      Solar
                                  masses     masses
Electron          9.109⋅10-31
Proton            1.673⋅10-27
Neutron           1.675⋅10-27
1 ounce                    .0283
Tennis ball                .058
Soccer ball                .44
1 pound                    .454
Typical human            70
Sumo wrestler           200
Ton                    1000
Honda Civic            1200
Elephant               5000
Bradley tank          27000
Argentinosaurus       70000                                   Largest dinosaur
Blue whale           200000
Moon               7.35⋅1022         .0123
Mars               6.42⋅1023         .107
Earth              5.92⋅1024        1
Jupiter            1.90⋅1027      318           .00096
Sun                1.99⋅1030   330000          1.0
White dwarf max     2.9⋅1030                   1.44
Milky Way black hole7.4⋅1036                   4.2 million
Milky Way           2.5⋅1042                   1.2 trillion
Andromeda           2.5⋅1042                   1.2 trillion
M87 galaxy                                   10   trillion
Virgo galaxy cluster                       1200   trillion

Balls

Ball sizes are in scale with each other and court sizes are in scale with each other.
Ball sizes are magnified by 10 with respect to court sizes.
The distance from the back of the court to the ball is the characteristic distance the ball travels before losing half its speed to air drag.

             Ball    Ball   Court   Court    Ball
           diameter  Mass   length  width   density
             (mm)    (g)     (m)     (m)    (g/cm3)

Ping pong      40      2.7    2.74    1.525   .081
Squash         40     24      9.75    6.4     .716
Golf           43     46                     1.10
Badminton      54      5.1   13.4     5.18    .062
Racquetball    57     40     12.22    6.10    .413
Billiards      59    163      2.84    1.42   1.52
Tennis         67     58     23.77    8.23    .368
Baseball       74.5  146                      .675   Pitcher-batter distance = 19.4 m
Hockey puck    76    163     61      26      1.44    25 mm thick
Whiffle        76     45                      .196
Football      178    420     91.44   48.76    .142
Rugby         191    435    100      70       .119
Bowling       217   7260     18.29    1.05   1.36
Soccer        220    432    105      68       .078
Basketball    239    624     28      15       .087
Cannonball    220  14000                     7.9     For an iron cannonball

Density

               grams/cm2     $/kg   Year of discovery

Magnesium         1.74         2.8     1808
Aluminum          2.70         1.7     1827
Titanium          4.51        10       1910
Zinc              7.14         2.0     1300
Manganese         7.21         2.3     1774
Iron              7.9           .3    -1200
Nickel            8.91        15       1751
Copper            8.96         6      -5000
Silver           10.49       640    Ancient
Lead             11.3          2      -6500
Tungsten         19.25        50       1783
Gold             19.30     43000    Ancient
Platinum         21.45     37000       1735
Osmium           22.59     12000       1803    Densest element

Air at Everest     .0004                       10 km altitude
Air at Denver      .001                        1 Mile altitude
Air at sea level   .00127
Ice                .92
Water             1.0
Rock              2.8

Earth             5.52
Moon              3.35
Mars              3.95
Europa            3.10
Ganymede          1.94
Callisto          1.83
Titan             1.88

Balsa              .12
Corkwood           .21
Cedar              .32
Pine               .37
Spruce, red        .41
Oak, red           .66
Hickory            .81
Bamboo             .85
Oak, live          .98
Ironwood          1.1
Lignum Vitae      1.26

Coins

            Mass  Diameter  Height  Density  Price/kg  Copper    Nickel     Zinc    Manganese
             g       mm       mm     g/cm3     $/kg   fraction  fraction  fraction  fraction

Penny        2.5     19.05   1.52    5.77       4.0     .025               .975
Nickel       5.000   21.21   1.95    7.26      10.0     .75      .25
Dime         2.268   17.91   1.35    4.62      44.1     .9167    .0833
Quarter      5.670   24.26   1.75    6.29      44.1     .9167    .0833
Half dollar 11.340   30.61   2.15    7.90      44.1     .9167    .0833
Dollar       8.100   26.5    2.00    7.53     123.5     .885     .02       .06      .035
Dollar bill  1.0              .11     .88    1000
The above objects are all to scale. The dimensions of a dollar bill are 155.956 mm * 66.294 mm * .11 mm.
For a coin,
Mass     =  M
Diameter =  D
Height   =  H
Volume   =  Vol  =  π H D2 / 4
Density  =  M / Vol
Gold was the densest element known until the discovery of tungsten in 1783 and was hence valuable as an uncounterfeitable currency. Silver can be counterfeited with lead because lead is more dense and cheaper than silver.

The price of the metal in a penny is

Metal price  =  Penny mass * (Copper fraction * Copper price/kg + Zinc fraction * Zinc price/kg)
             =  .0025 kg   * (     .025              6 $/kg           .975           2 $/kg    )
             =  .0052 $
For a penny made of pure copper the price of the metal is 1.5 cents. A penny made of gold, silver, or zinc has a value of:
       Price/Mass   Price
          $/kg        $

Zinc        2        .005
Copper      6        .015
Silver    640       1.6
Gold    43000     108

Kilograms and pounds

For a 1 kg mass, Newton's law is:

Mass                       =  M          = 1   kg
Gravitational acceleration =  g          = 9.8 m/s2
Gravitational force        =  F  =  M g  = 9.8 Newtons  =  2.203 pounds
The pound is a unit of force. 1 pound = 4.448 Newtons. In Earth gravity a mass of .454 kg produces a force of 1 pound.
Mass                       =  M          =  .454 kg
Gravitational acceleration =  g          = 9.8   m/s2
Gravitational force        =  F  =  M g  = 4.448 Newtons  =  1 pound

Frequency

                        Frequency   Note
                         (Hertz)

Whale songs                 10
Human ear lower limit       20
Bass lowest note            41       E
Bass guitar lowest note     41       E
Cello lowest note           65       C
Bass singer lowest note     82       E
Viola & tenor lowest note  131       C
Violin & alto lowest note  196       G
Soprano lowest note        262       C
Violin D string            293       D
Violin A string            440       A
Violin E string            660       E
Human ear upper limit    20000

Temperature

                        Kelvin   Celsius   Fahrenheit

Absolute zero             0      -273.2     -459.7
Water melting point     273.2       0         32
Room temperature        294        21         70
Human body temperature  310        37         98.6
Water boiling point     373.2     100        212


                       Kelvin

Absolute zero             0
Helium boiling point      4.2
Hydrogen boiling point   20.3
Triton                   38
Pluto                    44
Titania                  70
Nitrogen boiling point   77.4
Oxygen boiling point     90.2
Titan                    94
Europa                  102
Hottest superconductor  135          HgBaCaCuO
Ceres                   168
Mars                    210
Water melting point     273.15
Earth average           288
Room temperature        293
Water boiling point     373.15
Venus                   740
Wood fire              1170
Copper melting point   1358
Iron melting point     1811
Bunsen burner          1830
Tungsten melting point 3683          Highest melting point among metals
Earth's core           5650          Inner-core boundary
Sun's surface          5780
Solar core             13.6 million
Helium-4 fusion         200 million
Carbon-12 fusion        230 million

Pressure

Surface area =  A
Force        =  F
Pressure     =  P  =  F / A    (Pascals or Newtons/meter2 or Joules/meter3)

Atmospheric pressure

Mass of the Earth's atmosphere  =  M              =  5.15e18 kg
Surface area of the Earth       =  A              =  5.10e14 m2
Gravitational constant          =  g              =  9.8 m/s2
Pressure on Earth's surface     =  P  =  M g / A  =  101000 Pascals
                                                  =  15 pounds/inch2
                                                  =  1 Bar
One bar is defined as the Earth's mean atmospheric pressure at sea level
              Height   Pressure   Density
               (km)     (Bar)     (kg/m3)

Sea level         0      1.00     1.225
Denver            1.6     .82     1.05        One mile
Everest           8.8     .31      .48
Airbus A380      13.1     .16      .26
F-22 Raptor      19.8     .056      .091
SR-71 Blackbird  25.9     .022      .034
Space station   400       .000009   .000016

Energy and power

Energy          =  E          Joules
Time            =  T          seconds
Power           =  P  =  E/T  Watts
Mass            =  M          kilograms
Energy/Mass     =  e  =  E/M  Joules/kilogram
Power/Mass      =  p  =  P/M  Watts/kilogram

Energy

Energies in MJoules = 106 Joules

1 Watt hour                     .0036      1 Watt * 3600 seconds
Sprinting person                .004       (80 kg moving at 10 m/s)
1 food calorie                  .0042

Battery, lithium, CR1216        .00033     Smallest button cell
Battery, lithium, CR2032        .0030      Most common button cell
Battery, lithium-ion, AAAA      .0023
Battery, lithium-ion, AAA       .0047
Battery, lithium-ion, AA        .009
Battery, lithium-ion, A         .047
Battery, lithium-ion, B         .058
Battery, lithium-ion, C         .067
Battery, lithium-ion, D         .107

Battery, iPhone 7   (5 inch)    .040
Battery, Samsung S6 (5 inch)    .052
Battery, iPad mini  (8 inch)    .059
Battery, iPad Pro  (10 inch)    .100
Battery, iPad Pro  (13 inch)    .148

1 kg of Lithium-ion battery     .80
1 kg of TNT                    4.2
1 kg of sugar                 20     =  5000 Food Calories
1 kg of protein               20     =  5000 Food Calories
1 kg of alcohol               25     =  7000 Food Calories
1 kg of fat                   38     =  9000 Food Calories
1 kg of gasoline              48     = 13000 Food Calories

Tesla Model 3 battery        270
Fission bomb, uranium        8⋅107    = 20 kilotons of TNT
Fusion bomb                  8⋅1010   = 20 megatons of TNT
World energy used in 1 year  6⋅1014

Forms of energy:

Distance         =  X          meters
Force            =  F          Newtons
Mass             =  M          kg
Velocity         =  V          meters/second
Gravity constant =  g  =  9.8  meters/second2
Pressure         =  P          Pascals
Volume           =  U          meters3
Mechanical energy=  Ew =  F X  Joules
Gravity energy   =  Eg =  MgX  Joules    (X = height above ground)
Kinetic energy   =  Ek =  ½MV2 Joules
Pressure energy  =  Ep =  P U  Joules

Power
                         Watts

Human cell                 10-12
iPhone 7, standby            .05
iPhone 7, audio              .3
iPhone 7, video              .9
iPhone 7, talk               .9
iPad Pro 10 inch, idle      3
Human brain                20
Incandescent Light bulb    60
Human at rest             100
Unstrenuous cycling       200
1 horsepower              746
Strenuous cycling         600
Maximum human power      1600
World power per person   2500
Tesla S Ludicrous      397000       532 horsepower
Wind turbine               1⋅106
Blue whale               2.5⋅106
Boeing 747               1.4⋅108
Hoover Dam               2.1⋅109
U.S. power consumption   3.4⋅1012
World power consumption  1.5⋅1013
Earth geologic heat      4.4⋅1013
World photosynthesis     7.5⋅1013
Earth solar power        1.7⋅1017    Total solar power falling on the Earth

Fuel
Black: Carbon    White: Hydrogen    Red: Oxygen

Methane (Natural gas)
Ethane
Propane
Butane (Lighter fluid)
Octane (gasoline)
Dodecane (Kerosene)

Hexadecane (Diesel)
Palmitic acid (fat)
Ethanol (alcohol)

Glucose (sugar)
Fructose (sugar)
Galactose (sugar)
Lactose = Glucose + Galactose
Starch (sugar chain)
Leucine (amino acid)

ATP (Adenosine triphosphate)
Phosphocreatine
Nitrocellulose (smokeless powder)
TNT
HMX (plastic explosive)

Lignin (wood)
Coal

Medival-style black powder
Modern smokeless powder
Capacitor
Lithium-ion battery
Nuclear battery (radioactive plutonium-238)
Nuclear fission
Nuclear fusion
Antimatter


Vehicle power

Tesla Roadster

The energy sources that can be used by vehicles are:

              Energy/Mass   Power/mass   Energy/$   Rechargeable   Charge   Maximum charging
               MJoule/kg     Watt/kg     MJoule/$                  time          cycles

Gasoline            45                   60
Battery, aluminum    4.6       130                      No
Battery, lithium-ion  .8      1200         .010         Yes        1 hour      1000
Supercapacitor        .026   14000         .0005        Yes        Instant     Infinite
Aluminum capacitor    .010   50000         .0001        Yes        Instant     Infinite

Energy and power sources

                   Energy/Mass   Power/mass
                    MJoule/kg     Watt/kg

Antimatter          90000000000
Fusion bomb           250000000                 Max for d+t fusion
Fission bomb           83000000                 Max for a uranium bomb
Nuclear battery, Pu238  2265000          10     88 year half life
Nuclear battery, Sr90    589000          10     29 year half life
Hydrogen ( 0 carbons)       141.8
Methane  ( 1 carbon )        55.5               Natural gas
Ethane   ( 2 carbons)        51.9
Butane   ( 4 carbons)        49.5
Octane   ( 8 carbons)        47.8
Kerosene (12 carbons)        46
Diesel   (16 carbons)        46
Oil      (36 carbons)        46
Fat      (20 carbons)        37                 9 Calories/gram
Pure carbon                  32.8
Coal                         32                 Similar to pure carbon
Ethanol                      29                 7 Calories/gram
Wood                         22
Sugar                        17                 4 Calories/gram
Protein                      17                 4 Calories/gram
Plastic explosive             8.0               HMX
Smokeless powder              5.2               Modern gunpowder
TNT                           4.7
Black powder                  2.6               Medieval gunpowder
Phosphocreatine                .137             Recharges ATP
ATP                            .057             Adenosine triphosphate
Battery, aluminum-air         4.68      130
Battery, Li-S                 1.44      670
Battery, Li-ion                .8      1600
Battery, Li-polymer            .6      4000
Battery, Alkaline              .4
Battery, Lead acid             .15      150
Lithium supercapacitor         .054   15000
Supercapacitor                 .016    8000
Aluminum capacitor             .010   10000
Spring                         .0003

Battery sizes

Energies and powers are for lithium batteries, which have a voltage of 3.7 Volts. The "ID #" is often used instead of cell size.

Cell   Energy  Power  Current  Mass  Diameter  Length  Charge   Price    ID #
size   kJoule  Watt   Ampere   gram     mm       mm    AmpHour    $

D       107     220     60     138      32       67     8.0      13      32650
C        67     220     60      92      26       50     5.0       8      26650, 25500
B        58     160     45      72      22       60     4.5       5      21700, 20700
A        47     110     30      49      18       50     3.5       3      18650
AA        9      22      6      15      14       53      .70      1      14500
AAA       4.7    11      3       7.6    10       44      .35       .5    10440
AAAA      2.3     6      1.5     3.8     8       42      .17       .25   75400

Acceleration
                    Meters/second2

Ceres gravity            .27
Europa gravity          1.31
Titan gravity           1.35
Moon gravity            1.62
Mars gravity            3.8
Venus gravity           8.87
Earth gravity           9.8
Bugatti Veyron         15.2  0 to 100 km/h in 2.4 seconds
Red out                30    Max long-term acceleration in the direction of blood rushing to your head
Blackout               50    Max long-term acceleration while sitting
Formula-1 car          50    High-speed breaking and cornering with a downforce wing
Blackout with g suit   90    Max long-term acceleration while sitting with a g-suit
Max long-term (front) 120    Max long-term acceleration while lying on one's front
Max long-term (back)  170    Max long-term acceleration while lying on one's back
Max short-term        500    Max short-term acceleration
Bullet             310000    9x19 Parabellum handgun, average acceleration along the barrel

Science fundamentals

Prefixes

terameter   =  Tm  =  1012  meters
gigameter   =  Bm  =  109   meters
megameter   =  Mm  =  106   meters
kilometer   =  km  =  103   meters
meter       =   m  =  100   meters
centimeter  =  cm  =  10-2  meters
millimeter  =  mm  =  10-3  meters
micrometer  =  μm  =  10-6  meters
nanometer   =  nm  =  10-9  meters
picometer   =  pm  =  10-12 meters
femtometer  =  fm  =  10-15 meters

1 million kg       =  1 Mkg
1 million dollars  =  1 M$

Scientific notation

Examples of scientific notation.

    1  =     100  =  e0
   10  =     101  =  e1
  100  =     102  =  e2
  123  = 1.23⋅102  = 1.23e2
 .123  = 1.23⋅10-1 = 1.23e-1

11000 * .012  =  1.1⋅104 * 1.2⋅10-2  =  1.32⋅102  =  132
The abbreviation "e" for "10^" comes from Fortran and is standard in all programming languages.
Precision

A measurement consists of a quantity and an estimated error. For example, you might measure the length of a room to be

Length  =  6.35 ± .02 meters
"6.35" is the measurement and ".02" is the estimated error.

Care should be taken to use an appropriate number of digits. For example,

Length  =  6.3     ± .02 meters                   Not enough digits in the measured quantity
Length  =  6.34    ± .02 meters                   Minimum number of digits to state the measured quantity
Length  =  6.342   ± .02 meters                   It is wise to to include an extra digit
Length  =  6.3421  ± .02 meters                   Too many digits.  The last digit is unnecessary.
The fractional error is defined as
Fractional error  =  Error  /  Measured quanitity
                  =   .02   /       6.34
                  =       .0032
Rounding:
6.3424  →  6.342
6.3425  →  6.342
6.3426  →  6.343
If the last digit is even then round down, and if odd then round up. This prevents bias in rounding. For example:
6.05  →  6.0
6.15  →  6.2
6.25  →  6.2
6.35  →  6.4
6.45  →  6.4
6.55  →  6.6
6.65  →  6.6
6.75  →  6.8
6.85  →  6.8
6.95  →  7.0

Unit conversion
1 mile  =  1609 meters
1 hour  =  3600 seconds

                                1609 meters        1 hour
1 mile/hour  =  1 mile/hour  *  -----------  *  ------------  =  .447 meters/second
                                  1 mile        3600 seconds

SI units

If you have data that is not in SI units, then the safest procedure is to convert everything to SI units do the calculation. You can't go wrong with this. For example, if a car moving at 70 mph travels for 2 hours, how far does it go?

Speed of a car     =  V          =   70 mph    =    31.3 meters/second
Time traveled      =  T          =    2 hours  =    7200 seconds
Distance traveled  =  X  =  V T  =  140 miles  =  225360 meters
One first converts 70 mph and 2 hours to SI units, then apply X=VT to arrive at X=225360 meters, and then convert this to mph.

Alternatively, you can do the calculation in non-SI units but care must be taken to make sure the units are consistent.


Logarithm

The logarithm is the inverse of the exponential function.

10-2  =   .01               log10 .01 = -2
10-1  =   .1                log10 .1  = -1
100   =  1                  log10  1  =  0
101   = 10                  log10 10  =  1
102   =100                  log10100  =  2

10log10x  =  log10 10x  =  x

2D tables of numbers

Many things are best understood by constructing a 2D table of numbers. For example, suppose you're wondering how to compare the alcohol content of a 6 pack of beer, a bottle of wine, and a bottle of Scotch.

A standard bottle of beer is 12 ounces, has an alcohol fraction of .05, and the volume of alcohol is .6 ounces. For other drinks,

              Alcohol   Drink   Alcohol   Alcohol
              fraction  volume  volume    volume
                        (oz)     (oz)     (Beers)

Beer (12 oz)      .05     12       .6      1
Wine glass        .13      4.6     .6      1
Scotch shot       .40      1.5     .6      1
Beer pitcher      .05     64      3.2      5.3
Beer keg          .05   1984     99.2    165.3
Wine bottle       .13     25.4    3.3      5.5
Scotch bottle     .40     25.4   10.1     16.9
Distilled ethanol .95     25.4   24.1     40.2
The "alcohol volume" column provides a comparison between different drinks.

In a 2D table, columns are units and rows are items. Each column has a label and a unit. For example in the alcohol table, one of the columns is labeled "alcohol volume" and it has units of ounces. Monospace fonts are helpful for aligning columns.

Numbers in columns are easier to compare than numbers in rows, and this is why units are aligned as columns rather than rows. Most often it is units that are being compared.

We define a strategic unit, the "beer", which is the volume of alcohol in a standard beer. 1 beer = .6 ounces. In the table's last column we show the alcohol volume in units of beers.

We can expand the table to show the cost of each form of alcohol.

              Alcohol   Volume  Alcohol  Alcohol   $    $/Beer
              fraction   (oz)    (oz)    (Beers)

Beer (12 oz)      .05     12       .6       1       .67   .67   Budweiser
Wine glass        .13      4.6     .6       1      8     8.0    Napa Valley
Scotch shot       .40      1.5     .6       1      8     8.0    Laphroaig
Beer pitcher      .05     64      3.2       5.3   16     3.0    Budweiser
Beer keg          .05   1984     99.2     165.3  100      .60   Budweiser
Wine bottle       .13     25.4    3.3       5.5    3      .55   Charles Shaw
Vodka bottle      .40     25.4   10.1      16.9   15      .89   Smirnoff
Scotch bottle     .40     25.4   10.1      16.9   50     3.0    Laphroaig
Distilled ethanol .95     25.4   24.1      40.2   15      .37   Everclear
A table is usually accompanied by equations showing how the columns are related.
Volume of the drink       =  V
Fraction of alcohol       =  F
Volume of alcohol         =  Valc  = F V
Volume of one beer        =  Vbeer = 12    ounces
Alcohol fraction of beer  =  Fbeer =   .05
Alcohol volume in one beer=  VBond =   .6  ounces
Ounce                     =  29.6 mL
One "Beer" of alcohol     =    .6 ounces     (Volume of alcohol in a 12 ounce beer)
Pint                      =  16   ounces
Wine or Scotch bottle     =  25.4 ounces  =  750 ml
Pitcher                   =  64   ounces
Gallon                    = 128   ounces
Keg                       =1984   ounces  =  15.5 gallons
Conclusions can be read off from the table, such as the fact that the cheapest alcohol in terms of price per alcohol content is Charles Shaw wine.

In the following sections we show more examples of 2D tables.


Caffeine

                Caffein  Density  Volume
                  mg      mg/oz   mg/oz

Coffee, brewed     163     20.4      8
Mtn. Dew Game Fuel 121      6.0     20
Red Bull            80      9.5      8.5
Espresso            77     51        1.5
Mountain Dew        54      4.5     12
Mello Yello         51      4.2     12
Tea (black)         42      5.2      8
Sunkist             41      3.4     12
Pepsi Cola          38      3.2     12
Arizona iced tea    38      1.9     20
Coca Cola           34      2.8     12
Coffee, decaf        6       .7      8
Sprite               0      0       12

Lion Slash problem-solving style

Suppose you measure the power exerted in climbing a set of stairs.


Time to climb stairs    =  T          =    10  seconds
Mass of climber         =  M          =   100  kg
Gravity constant        =  g          =    10  meters/second2
Height of stairs        =  H          =    20  meters
Vertical speed          =  V  =  H/T  =     2  meters/second
Gravity energy          =  E  =  MgH  = 20000  Joules
Power                   =  P  =  E/T  =  2000  Watts
There is a row for each variable and there are 5 columns showing the properties of each variable. The columns are:

*) Description of the variable
*) Symbol for the variable
*) Units-style equation
*) Numerical example
*) Units

Most units-style calculations can be done using this recipe.


Battery energy and power

If a smartphone is being used to play League of Legends, typical values for the lithium battery are

Energy      =  E  =       =  20000  Joules       (Typical smartphone battery energy)
Life        =  T  =       =  14400  seconds      (While playing League of Legends)
Power       =  P  =  E/T  =   1.39  Watts
Mass        =  M          =    .027 kg
Energy/Mass =  e  =  E/M  =    .75  MJoules/kg   (Typical value for lithium batteries)
Power/Mass  =  p  =  P/M  =     52  Watts/kg
The maximum power/mass that a lithium battery is capable of producing is 750 Watts/kg.
Human energy and power

It takes .7 kg of rice to feed one human for one day.

Energy in 1 Calorie               = 4200   Joules
Time                   =  T       =86400   seconds    =     1 day
Food energy in one day =  E       =   10.5 MJoules    =  2500 Calories
Power                  =  P = E/T =  121.5 Watts
Rice mass              =  M       =     .7 kg
Rice energy/mass       =  e = E/M =   15   MJoules/kg
In this example we added a column for non-SI units (days and Calories), which have to be converted to SI units.
Electricity

The price of electricity is 5 cents per kiloWatt hour.

Power    = P       = 1000 Watts       =  1 kiloWatt
Time     = T       = 3600 seconds     =  1 hour
Energy   = E = PT  =  3.6 MJoules
Price    = C = E/c =  .05 $
Energy/$ = c       =   72 MJoules/$

Orbital energy

The kinetic energy of an object in orbit is 32 MJoules/kg, and it costs .44 dollars for this much energy in electricity. The real launch cost is 2000 dollars/kilogram.

Orbital velocity     =  V         =8000 meters/second
Mass                 =  M         =   1 kg
Kinetic energy       =  E  = ½MV2 =  32 MJoules
Electricity energy/$ =  c         =  72 MJoules/$
Electricity cost     =  C  = E/c  = .44 $

Physical constants
Speed of light               2.9979e8   m/s
Gravitational constant       6.6738e-11 m3/kg/s2
Planck constant              6.6261e-34 J s
Earth surface gravity        9.8067     m/s
Electric force constant      8.9876e9   N m2 / C2
Magnetic constant            4 Pi e-7   N/A2
Proton mass                  1.6726e-27 kg  =  938.272 GeV
Neutron mass                 1.6749e-27 kg  =  939.565 GeV
Electron mass                9.1094e-31 kg
Electron charge              1.6022e-19 C
Atomic mass unit             1.6605e-27 kg
Bohr radius                  5.2918e-11 m           = hbar2 / (ElectronMass*ElectronCharge2*Ke)
Boltzmann constant           1.3806e-23 J/K
Avogadro number              6.0221e23  particles/mole
Gas constant                 8.3145     J/K/mole
Stefan-Boltzmann constant    5.6704e-8  Watts/m2/K4
Wein constant                2.8978e-3  m K
Mole of Carbon-12            .012       kg          Exact
Planck length                1.6162e-35 m
Planck mass                  2.1765e-8  kg
Planck time                  5.3911e-44 s
Planck charge                1.8755e-18 C
Planck temperature           1.4168e32  K
Water heat capacity          4200       J/kg/K
Steam heat capacity          2080       J/kg/K      At 100 Celsius
Ice heat capacity            2110       J/kg/K      At -10 Celsius
Air heat capacity            1004       J/kg/K
Stefan-Boltzmann             5.67e-8    Watts/meter2/Kelvin4
                                        = (2π5/15) Boltzmann4 / SpeedOfLight2 / PlanckConstant3
Wein                         2.898e-3   Kelvin meters
Electron spin                5.2729e-35 Joule seconds  =  PlanckConstant / (4 Pi)

Pi                           3.14159
Euler number                 2.71828

Density

Size of atoms
Dot size corresponds to atom size.

For gases, the density at boiling point is used.   Size data


Density

Copper atoms stack like cannonballs. We can calculate the atom size by assuming the atoms are shaped like either cubes or spheres. For copper atoms,

Density         = D              =       8900 kg/m3
Atomic mass unit= M0             = 1.661⋅10-27 kg
Atomic mass     = MA             =      63.55 Atomic mass units
Mass            = M   =  MA⋅M0    = 9.785⋅10-26 kg
Number density  = N   =  D / M   = 9.096⋅1028  atoms/m3
Cube volume     = Υcube=  1 / N   = 1.099⋅10-29 m3            Volume/atom if the atoms are cubes
Cube length     = L   =  Υ1/3cube = 2.22⋅10-10  m             Side length of the cube
Sphere fraction = f   =  π/(3√2) =     .7405                Fraction of volume occupied by spheres in a stack o spheres
Sphere volume   = Υsph=  Υcube f  = 8.14⋅10-30  m3 = 43πR3    Volume/atom if the atoms are spheres
Sphere radius   = R              = 1.25⋅10-10  m

History of physics

Art of Order of Magnitude Physics

Order of magnitude physics is a style for generating numerical estimates with a minimum of calculation, and using units arguments to obtain formulae.
Using units to derive formulae

Equations can often be derived using units. For example, what is the formula for kinetic energy? The variables that will be present in the formula are:

Mass                   =  M  (kg)
Velocity               =  V  (meters/second)
Kinetic energy         =  E  (Joules  =  kg meters2/second2)
Dimensionless constant =  K  (Unitless)
Assume the formula as the form
E = K Mm Vv
For some value of m and v. The values that gives units of energy are:
E = K M1 V2
Units arguments often give the right formula up to a dimensionless constant and a more involved derivation is usually required to produce the constant. The formula with the dimensionless constant included can always be found on Wikipedia. For the kinetic energy, K=½ and E = ½ M V2.

Another example of using units to derive formulae is the aerodynamic drag force. The variables that will be present in the formula are:

Velocity              =  V        meters/second
Cross sectional area  =  A        meters2
Density of air        =  D = 1.22 kg/meter3
Drag force            =  F        Newtons = kg meters/second2
Dimensionless constant=  K        Unitless
Assume the formula has the form
F  =  K Dd Aa Vv
for some value of {d,a,v}. The values that give units of force are
F  =  K D1 A1 V2

Examples of equations that can be obtained with units arguments
Aerodynamic drag force    =  1/2  Density  CrossSection  Velocity2

Aerodynamic drag power    =  1/2  Density  CrossSection  Velocity3

Gravitational force       = -G  Mass1  Mass2  /  Distance2

Gravitational energy      = -G  Mass1  Mass2  /  Distance

Gravitational self-energy =  3/5  G  Mass2  /  Radius                 For a sphere of uniform density

Kinetic energy            =  1/2  Mass  Velocity2

Sound Speed               =  [Γ Pressure / Density]1/2         Γ=7/5 for air

Wave speed for a string   =  [Tension Length / Mass]1/2

Electric current

A moving charge is an "electric current". In an electric circuit, a battery moves electrons through a wire.

Charge            =  Q
Time              =  T
Electric current  =  I  =  Q / T   (Coulombs/second)
The current from a positive charge moving to the right is equivalent to that from a negative charge moving to the left.
Circuits

Battery

A battery moves charge upwards in voltage
A resistor dissipates energy as charges fall downwards in voltage

Charge          =  Q           Coulombs
Voltage         =  V           Volts
Energy          =  E  =  VQ    Joules
Time            =  T           seconds
Current         =  I  =  Q/T   Amperes
Resistance      =  R  =  V/I   Ohms
Power           =  P  =  QV/T  Watts
                      =  IV
                      =  V2/R
                      =  I2R

Ohm's Law:  V = IR

Resistance

Superconductor
Resistor

In a superconductor, electrons move without interference.
In a resistor, electrons collide with atoms and lose energy.

                 Resistance (Ohms)

Copper wire            .02          1 meter long and 1 mm in diameter
1 km power line        .03
AA battery             .1           Internal resistance
Light bulb          200
Human             10000

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