
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
Density of water =1000 kg/meter$3$ = 1 g/cm$3$ Density of air = 1.2 kg/meter$3$ = .0012 g/cm$3$
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/second^{2} Momentum Q = M V Mass * Speed kg meter/second Force Newtons F = M A Mass * Acceleration kg meter/second^{2} Energy Joule E = F X Force * Distance kg meters^{2}/second^{2} Power Watt P = E/T Energy / Time kg meters^{2}/second^{3} Area S = X^{2} Length$2$ meters$2$ Volume Υ = X^{3} Length$3$ meters$3$ Density ρ = M/Υ Mass / Volume kg / meters^{2} Pressure Pascal Φ = F/S Force / Area Newtons/meter$2$ = Joules/meter$3$ Angular momentum L = M V X Momentum*Length kg meters$2$/second Torque Γ = F X Force * Length kg meters$2$/second$2$ Frequency Hertz f = 1/T 1 / Time 1/second
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/inch^{2} (pounds interpreted as force) Pound/inch^{2} = 6895 Pascals Bar =101325 Pascals (Atmosphere pressure at sea level) = 14.50 pounds/inch^{2} (pounds interpreted as force) Earth gravity= 9.807 meters/second^{2} = 32.2 feet/second^{2} Standard sheet of paper = 11 x 8.5 inches = 27.94 x 21.59 cm
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⋅10^{7} 10.9 Moon distance 3.844⋅10^{8} 60.3 .00257 1.5 seconds Sun radius 6.957⋅10^{8} 109 .00474 2.3 seconds Earth orbit 1.496⋅10^{11} 23481 1.0 8 minutes Jupiter orbit 5.2 40 minutes Neptune orbit 30.1 3 days Light year 9.461⋅10^{15} 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
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 F35 Lightning 475 1.6 Stealth fighter F16 Falcon 590 2.0 Concorde 606 2.05 F22 Raptor 670 2.3 Stealth fighter F15 Eagle 740 2.5 SR71 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 10^{7} Fusion rocket 10^{7} Light 3⋅10^{8} 1020000Aircraft 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.
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⋅10^{22} .0123 Mars 6.42⋅10^{23} .107 Earth 5.92⋅10^{24} 1 Jupiter 1.90⋅10^{27} 318 .00096 Sun 1.99⋅10^{30} 330000 1.0 White dwarf max 2.9⋅10^{30} 1.44 Milky Way black hole7.4⋅10^{36} 4.2 million Milky Way 2.5⋅10^{42} 1.2 trillion Andromeda 2.5⋅10^{42} 1.2 trillion M87 galaxy 10 trillion Virgo galaxy cluster 1200 trillion
Ball Ball Court Court Ball diameter Mass length width density (mm) (g) (m) (m) (g/cm^{3}) 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 Pitcherbatter 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
grams/cm^{2} $/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
Mass Diameter Height Density Price/kg Copper Nickel Zinc Manganese g mm mm g/cm^{3} $/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 1000The above objects are all to scale. The dimensions of a dollar bill are 155.956 mm * 66.294 mm * .11 mm.
Mass = M Diameter = D Height = H Volume = Vol = π H D^{2} / 4 Density = M / VolGold 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
For a 1 kg mass, Newton's law is:
Mass = M = 1 kg Gravitational acceleration = g = 9.8 m/s^{2} Gravitational force = F = M g = 9.8 Newtons = 2.203 poundsThe 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/s^{2} Gravitational force = F = M g = 4.448 Newtons = 1 pound
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
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 Innercore boundary Sun's surface 5780 Solar core 13.6 million Helium4 fusion 200 million Carbon12 fusion 230 million
Surface area = A Force = F Pressure = P = F / A (Pascals or Newtons/meter^{2} or Joules/meter^{3})
Mass of the Earth's atmosphere = M = 5.15e18 kg Surface area of the Earth = A = 5.10e14 m^{2} Gravitational constant = g = 9.8 m/s^{2} Pressure on Earth's surface = P = M g / A = 101000 Pascals = 15 pounds/inch^{2} = 1 BarOne bar is defined as the Earth's mean atmospheric pressure at sea level
Height Pressure Density (km) (Bar) (kg/m^{3}) 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 F22 Raptor 19.8 .056 .091 SR71 Blackbird 25.9 .022 .034 Space station 400 .000009 .000016
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
Energies in MJoules = 10^{6} 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, lithiumion, AAAA .0023 Battery, lithiumion, AAA .0047 Battery, lithiumion, AA .009 Battery, lithiumion, A .047 Battery, lithiumion, B .058 Battery, lithiumion, C .067 Battery, lithiumion, 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 Lithiumion 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⋅10^{7} = 20 kilotons of TNT Fusion bomb 8⋅10^{10} = 20 megatons of TNT World energy used in 1 year 6⋅10^{14}
Forms of energy:
Distance = X meters Force = F Newtons Mass = M kg Velocity = V meters/second Gravity constant = g = 9.8 meters/second^{2} Pressure = P Pascals Volume = U meters^{3} Mechanical energy= E_{w} = F X Joules Gravity energy = E_{g} = MgX Joules (X = height above ground) Kinetic energy = E_{k} = ½MV^{2} Joules Pressure energy = E_{p} = P U Joules
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⋅10^{6} Blue whale 2.5⋅10^{6} Boeing 747 1.4⋅10^{8} Hoover Dam 2.1⋅10^{9} U.S. power consumption 3.4⋅10^{12} World power consumption 1.5⋅10^{13} Earth geologic heat 4.4⋅10^{13} World photosynthesis 7.5⋅10^{13} Earth solar power 1.7⋅10^{17} Total solar power falling on the Earth
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, lithiumion .8 1200 .010 Yes 1 hour 1000 Supercapacitor .026 14000 .0005 Yes Instant Infinite Aluminum capacitor .010 50000 .0001 Yes Instant Infinite
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, aluminumair 4.68 130 Battery, LiS 1.44 670 Battery, Liion .8 1600 Battery, Lipolymer .6 4000 Battery, Alkaline .4 Battery, Lead acid .15 150 Lithium supercapacitor .054 15000 Supercapacitor .016 8000 Aluminum capacitor .010 10000 Spring .0003
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
Meters/second^{2} 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 longterm acceleration in the direction of blood rushing to your head Blackout 50 Max longterm acceleration while sitting Formula1 car 50 Highspeed breaking and cornering with a downforce wing Blackout with g suit 90 Max longterm acceleration while sitting with a gsuit Max longterm (front) 120 Max longterm acceleration while lying on one's front Max longterm (back) 170 Max longterm acceleration while lying on one's back Max shortterm 500 Max shortterm acceleration Bullet 310000 9x19 Parabellum handgun, average acceleration along the barrel
terameter = Tm = 10$12$ meters gigameter = Bm = 10$9$ meters megameter = Mm = 10$6$ meters kilometer = km = 10$3$ meters meter = m = 10$0$ 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$
Examples of scientific notation.
1 = 10^{0} = e0 10 = 10^{1} = e1 100 = 10^{2} = e2 123 = 1.23⋅10^{2} = 1.23e2 .123 = 1.23⋅10^{1} = 1.23e1 11000 * .012 = 1.1⋅10^{4} * 1.2⋅10^{2} = 1.32⋅10^{2} = 132The abbreviation "e" for "10^" comes from Fortran and is standard in all programming languages.
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 = .0032Rounding:
6.3424 → 6.342 6.3425 → 6.342 6.3426 → 6.343If 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
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
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 metersOne 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 nonSI units but care must be taken to make sure the units are consistent.
The logarithm is the inverse of the exponential function.
10^{2} = .01 log_{10} .01 = 2 10^{1} = .1 log_{10} .1 = 1 10^{0} = 1 log_{10} 1 = 0 10^{1} = 10 log_{10} 10 = 1 10^{2} =100 log_{10}100 = 2 10^{log10x} = log_{10} 10^{x} = x
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 40.2The "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 EverclearA table is usually accompanied by equations showing how the columns are related.
Volume of the drink = V Fraction of alcohol = F Volume of alcohol = V_{alc} = F V Volume of one beer = V_{beer} = 12 ounces Alcohol fraction of beer = F_{beer} = .05 Alcohol volume in one beer= V_{Bond} = .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 gallonsConclusions 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.
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
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/second^{2} 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 WattsThere 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
*) Unitsstyle equation
*) Numerical example
*) Units
Most unitsstyle calculations can be done using this recipe.
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/kgThe maximum power/mass that a lithium battery is capable of producing is 750 Watts/kg.
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/kgIn this example we added a column for nonSI units (days and Calories), which have to be converted to SI units.
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/$
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 = ½MV^{2} = 32 MJoules Electricity energy/$ = c = 72 MJoules/$ Electricity cost = C = E/c = .44 $
Speed of light 2.9979e8 m/s Gravitational constant 6.6738e11 m^{3}/kg/s^{2} Planck constant 6.6261e34 J s Earth surface gravity 9.8067 m/s Electric force constant 8.9876e9 N m^{2} / C^{2} Magnetic constant 4 Pi e7 N/A^{2} Proton mass 1.6726e27 kg = 938.272 GeV Neutron mass 1.6749e27 kg = 939.565 GeV Electron mass 9.1094e31 kg Electron charge 1.6022e19 C Atomic mass unit 1.6605e27 kg Bohr radius 5.2918e11 m = hbar^{2} / (ElectronMass*ElectronCharge^{2}*Ke) Boltzmann constant 1.3806e23 J/K Avogadro number 6.0221e23 particles/mole Gas constant 8.3145 J/K/mole StefanBoltzmann constant 5.6704e8 Watts/m^{2}/K^{4} Wein constant 2.8978e3 m K Mole of Carbon12 .012 kg Exact Planck length 1.6162e35 m Planck mass 2.1765e8 kg Planck time 5.3911e44 s Planck charge 1.8755e18 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 StefanBoltzmann 5.67e8 Watts/meter^{2}/Kelvin^{4} = (2π^{5}/15) Boltzmann^{4} / SpeedOfLight^{2} / PlanckConstant^{3} Wein 2.898e3 Kelvin meters Electron spin 5.2729e35 Joule seconds = PlanckConstant / (4 Pi) Pi 3.14159 Euler number 2.71828
For gases, the density at boiling point is used. Size data
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/m^{3} Atomic mass unit= M_{0} = 1.661⋅10^{27} kg Atomic mass = M_{A} = 63.55 Atomic mass units Mass = M = M_{A}⋅M_{0} = 9.785⋅10^{26} kg Number density = N = D / M = 9.096⋅10^{28} atoms/m^{3} Cube volume = Υ_{cube}= 1 / N = 1.099⋅10^{29} m^{3} Volume/atom if the atoms are cubes Cube length = L = Υ^{1/3}_{cube} = 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 = Υ_{sph}= Υ_{cube} f = 8.14⋅10^{30} m^{3} = ^{4}⁄_{3}πR^{3} Volume/atom if the atoms are spheres Sphere radius = R = 1.25⋅10^{10} m
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 meters^{2}/second^{2}) Dimensionless constant = K (Unitless)Assume the formula as the form
E = K M^{m} V^{v}For some value of m and v. The values that gives units of energy are:
E = K M^{1} V^{2}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 V^{2}.
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 meters^{2} Density of air = D = 1.22 kg/meter^{3} Drag force = F Newtons = kg meters/second^{2} Dimensionless constant= K UnitlessAssume the formula has the form
F = K D^{d} A^{a} V^{v}for some value of {d,a,v}. The values that give units of force are
F = K D^{1} A^{1} V^{2}
Aerodynamic drag force = 1/2 Density CrossSection Velocity^{2} Aerodynamic drag power = 1/2 Density CrossSection Velocity^{3} Gravitational force = G Mass_{1} Mass_{2} / Distance^{2} Gravitational energy = G Mass_{1} Mass_{2} / Distance Gravitational selfenergy = 3/5 G Mass^{2} / Radius For a sphere of uniform density Kinetic energy = 1/2 Mass Velocity^{2} Sound Speed = [Γ Pressure / Density]^{1/2} Γ=7/5 for air Wave speed for a string = [Tension Length / Mass]^{1/2}
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.
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 = V^{2}/R = I^{2}R Ohm's Law: V = IR
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