Units and Dimensionality

Contents

  • Physical Quantities and their Associated Dimensions
  • Basic Physical Quantities
  • Mechanical Physical Quantities
  • Electrical Physical Quantities
  • The Algebra of Dimensionality
  • Conversion Between Systems of Units
  • Definitions of Fundemental Units
  • Definitions of Dervied Units
  • Units Conversion Constants
  • Physical Constants
  • Physics Equations
  • Physical Quantities and Their Associated Dimensions

       Errors can occur in writing equations to solve problems in classical
    physics. Many of these errors can be prevented by performing a dimensionality
    check on the equations. All physical quantities have a fundamental dimension
    that is independent of the units of measurement. The basic physical dimensions
    are: length, mass, time, electrical charge, temperature and luminous intensity.
    
       There are a number of systems of units for measuring physical dimensions. 
    The MKS system is based on meter, kilogram, second  measurement. The CGS system
    is based on centimeter, gram, second  measurement. The English system is based
    on feet, pound, second  measurement. A few physical dimensions and the
    associated measurement unit in these three systems are :
    
    
            Physical Quantity                           Unit System
              Dimension                         MKS         CGS          English
    
    
              length                            meter       centimeter   feet
    
              mass                              kilogram    gram         pound mass
    
              time                              second      second       second
    
              force                             newton      dyne         poundal
    
              energy                            joule       erg          B.t.u.
    
    
       The checking of a physical equation has two aspects. The first is to check
    the dimensionality. The dimensionality is independent of the unit system. The
    second is to check that a consistent system of units is used in the equation.
    
       An example of a dimensionality check is using the basic equation F=ma to
    determine that force has the dimension  mass x length / time squared, then
                  2
    check if  F=mv /r  is dimensionally correct. The check is performed by 
    expanding the dimensions, e.g.  mass x (length/time) x (length/time) / length.
    Combining terms and reducing yields  mass x length / time squared. This agrees
    with the dimensions expected for force from the basic equation F=ma. As
    expected, centripetal force has the same dimensionality as the force from
    Newton's second law of motion.
    
       The table below is organized to present the physical quantity name with
    associated information. The second column is one of the typical symbols used
    for the physical quantity. The third column is the dimension of the physical
    quantity expressed in terms of the fundamental dimensions. The fourth column
    is the name of the unit in the MKS measurement system. The fifth column
    is the typical MKS unit equation. An independent table presents conversion
    factors from the MKS measurement system to other measurement systems.
    
       Physics developed over a period of many years by many people from a variety
    of disciplines. Thus, there is ambiguity and duplication of symbols.
    

    Basic Physical Quantities

    PHYSICAL QUANTITY         SYMBOL  DIMENSION   MEASUREMENT UNIT  UNIT EQUATION
    _________________         ______  _________   ________________  ______________
    
    
    length                     s       L           meter              m
    
    mass                       m       M           kilogram           Kg
    
    time                       t       T           second             sec
    
    electric charge            q       Q           coulomb            c
    
    luminous intensity         I       C           candle             cd
    
                                                                      o
    temperature                T       K           degree kelvin       K
    
    angle                      theta   none        radians            none
    

    Mechanical Physical Quantities (derived)

    PHYSICAL QUANTITY         SYMBOL  DIMENSION   MEASUREMENT UNIT  UNIT EQUATION
    _________________         ______  _________   ________________  ______________
    
                                        2                              2
    area                       A       L           square meter       m
    
                                        3                              3
    volume                     V       L           stere              m
    
    velocity                   v       L/T         meter per second   m/sec
    
    angular velocity           omega   1/T         radians per second 1/sec
    
                                          2                                2
    acceleration               a       L/T         meter per square   m/sec
                                                   second
    
                                          2                                2
    angular acceleration       alpha   1/T         radians per        1/sec
                                                   square second
    
                                           2                                  2
    force                      F       ML/T        newton             Kg m/sec
    
                                         2  2                             2    2
    energy                     E       ML /T       joule              Kg m /sec
    work                       W                     "
    heat                       Q                     "
    
                                         2  2                             2    2
    torque                     T       ML /T       newton meter       Kg m /sec
    
    
                                         2  3 
    power                      P       ML /T       watt               joule/sec
    
                                          3                               3
    density                    D       M/L         kilogram per       Kg/m
                                                   cubic meter
    
                                           2                                  2
    pressure                   P       M/LT        newton per         Kg/m sec
    elastic modulus                                square meter
    
    momentum                   p       ML/T        newton second      Kg m/sec
    impulse                                        
    
                                         2                                2
    inertia                    I       ML /T       joule second       Kg m /sec
    
    
    luminous flux              phi     C           lumen (4Pi candle  cd sr 
                                                   for point source)
                                          2                                  2
    illumination               E       C/L         lumen per          cd sr/m
                                                   square meter
    
                                         2  2                             2    2 o
    entropy                    S       ML /T K     joule per degree   Kg m /sec   K
    
                                        3                              3
    volume rate of flow        Q       L /T        cubic meter        m /sec
                                                   per second
    
                                        2                              2
    kinematic viscosity        nu      L /T        square meter       m /sec
                                                   per second
    
    dynamic viscosity          mu      M/LT        newton second      Kg/m sec
                                                   per square meter
    
                                          2 2                             2   2
    specific weight            gamma   M/L T       newton             Kg/m sec
                                                   per cubic meter
    
    

    Electrical Physical Quantities (derived)

    
    PHYSICAL QUANTITY         SYMBOL  DIMENSION   MEASUREMENT UNIT  UNIT EQUATION
    _________________         ______  _________   ________________  ______________
    
    electric current           I       Q/T         ampere             c/sec
    
                                         2  2                             2    2
    emf,voltage,potential      E       ML /T Q     volt               Kg m /sec c
    
                                         2   2                            2      2
    electric resistance        R       ML /TQ      ohm                Kg m /sec c
    
                                         2   3                             2     3
    conductivity               sigma   TQ /ML      mho per meter      sec c /Kg m
    
                                        2 2   2                          2 2     2
    capacitance                C       T Q /ML     farad              sec c /Kg m
    
                                         2  2                             2  2
    inductance                 L       ML /Q       henry              Kg m /c
    
                                           2                                 2
    current density            J       Q/TL        ampere per         c/sec m
                                                   square meter
    
                                          3                              3
    charge density             rho     Q/L         coulomb per        c/m
                                                   cubic meter
    
    magnetic flux,             B       M/TQ        weber per          Kq/sec c
       magnetic induction                          square meter
    
    magnetic intensity         H       Q/LT        ampere per meter   c/m sec
    
    magnetic vector potential  A       ML/TQ       weber/meter        Kg m/sec c
    
                                           2                                  2
    electric field intensity   E       ML/T Q      volt/meter or      Kg m/sec c
                                                   newton per coulomb
    
                                          2                              2
    electric displacement      D       Q/L         coulomb per        c/m
                                                   square meter
    
                                           2                                2
    permeability               mu      ML/Q        henry per meter    Kg m/c
    
                                        2 2   3                          2 2     3 
    permittivity,              epsi    T Q /ML     farad per meter    sec c /Kg m
      dielectric constant
    
                                                                         -1
    frequency                  f       Pi/T        hertz              sec
    
                                                                         -1
    angular frequency          omega   1/T         radians per second sec
    
    wave length                lambda  L           meters             m
    

    The Algebra of Dimensionality

    
       The dimension of any physical quantity can be written as
    
                      a   b   c   d   e   f
                     L   M   T   Q   C   K
    
    where a,b,c,d,e and f are integers such as -4, -3, -2 , -1, 0, 1, 2, 3, 4
    and L is length, M is mass, T is time, Q is charge, C is luminous intensity
    and K is temperature. An exponent of zero means the dimension does not apply
    to the physical quantity. The normal rules of algebra for exponents apply
    for combining dimensions.
    
       In order to add or subtract two physical quantities the quantities must
    have the same dimension. The resulting physical quantity has the same
    dimensions. Physical quantities with the same dimension in different
    systems of units can be added or subtracted by multiplying one of
    the quantities by a units conversion factor to obtain compatible units.
    
       The multiplication of two physical quantities results in a new physical
    quantity that has the sum of the exponents of the dimensions of the initial
    two quantities.
    
       The division of one physical quantity by another results in a new physical
    quantity that has the dimension of the exponents of the first quantity minus
    the exponents of the second quantity.
    
       Taking the square root of a physical quantity results in a new physical
    quantity having a dimension with exponents half of the initial dimension.
    
    
       Raising a physical quantity to a power results in a new physical quantity
    having a dimension with the exponents multiplied by the power.
    
              e.g. v has dimension L/T
    
                         2                2  2      2 -2
                   then v  has dimension L /T  or  L T
    
    
       The derivative of a physical quantity with respect to another physical
    quantity results in a new physical quantity with the exponents of the
    first dimension minus the exponents of the other dimension.
    
             e.g.  v has dimension L/T, t has dimension T,
    
                                               2
                   then dv/dt has dimension L/T    of acceleration
    
    
       The integral of a physical quantity over the range of another physical
    quantity results in a new physical quantity that has a dimension with the
    sum of the exponents of the two quantities.
    
            
             e.g.  v has dimension L/T, t has dimension T,
    
                   then  integral v dt  has dimension  L
    

    Conversion Between Systems of Units

    
       This section is organized to be consistent with the discussion of physical
    quantities and equations of physics. The definition of the six fundamental
    units of physical quantities is presented for the MKS system of units. The
    definition of some derived units is then presented in the MKS system. The
    definitions in other systems of units follow the MKS definitions. This is
    followed by a table of conversion factors between the MKS system and other
    systems of units.
    
    
       The MKS system based on the meter, kilogram second was augmented to allow
    force and energy from electrical quantities to be measured in one rationalized
    system of units. The system was proposed by Giorgi in 1904. It was adopted by
    the IEC in 1935 to take effect on January 1, 1940. The electrical to mechanical
    conversion was chosen to be based on the permeability of free space to be
    
            -7
    4Pi x 10   henry per meter. 
    

    Definition of Fundemental Units

    
    Meter, fundamental unit of length, defined as the distance between two
                                                              o
    specified lines on a specific bar of platinum-iridium at 0 C at standard
    atmospheric pressure supported at two neutral points 0.285 meter from the
    center of the bar. The bar is kept at the International Bureau of Weights
    and Measures near Paris France.
    
    
    Centimeter, cgs unit of length, defined as 1/100 meter.
    
    
    Feet, English unit of length, defined as 0.3048 meter in U.S.
    
    
    Inch, English unit of length, defined as 0.00254 meter in U.S.
    
                                           -10
    Angstrom, unit of length, defined as 10    meter.
    
    
    Kilogram, fundamental unit of mass, defined as the mass of a specific
    cylinder of platinum - iridium kept at the International Bureau of Weights and
    Measures.
    
    
    Gram, cgs unit of mass, defined as 1/1000 kilogram.
    
    
    Pound, English unit of mass, the avoirdupois pound is defined to be
    0.4535924277 kilogram in the U.S. The apothecary or troy pound is
    5760/7000 of the avoirdupois pound.
    
    
    Second, fundamental unit of time, defined as one 86,400th part of a mean
    solar day. Presently measured by an atomic clock based on the rate of nuclear
    decay.
    
    
    Coulomb, fundamental unit of charge, defined as the charge required to
    obtain one newton of force between two such charges at a distance of one
    meter.
    
    
    Candle, fundamental unit of luminous intensity, defined as the source
    intensity of 1/60 centimeter square opening of the standard light source
    of a glowing cavity with temperature equal to that of solidifying platinum.
    A point source of one candle radiates one lumen per steradian.
    
    
    Degrees kelvin, fundamental unit of temperature, defined as zero where
    the molecular activity of gases cease. The scale is based on zero degrees
    centigrade (Celsius) for the freezing point of water and 100 degrees
    centigrade at the boiling point of water. Zero degrees centigrade is 273.16
    degrees kelvin.
    
    
    Radians, fundamental unit of angle, defined as the angle formed by a
    length of circular arc being equal to the radius creating the arc. 
    

    Definition of Derived Units

    
    Newton, unit of force, defined as the force required to accelerate a mass
    of 1 kilogram at 1 meter per second per second when acting continuously. 
    
    
    Dyne, cgs unit of force, defined as the force required to accelerate a mass
                                                                      -5
    of 1 gram at at 1 centimeter per second per second. One dyne is 10   newton.
    
    
    Poundal, English unit of force, defined as the force required to accelerate
    a mass of 1 pound at 1 foot per second per second. One poundal is  
              -10
    7.23300 10    newton. A poundal based on earth's gravitation is 32.174 pounds
    avoirdupois.
    
    
    Joule, unit of energy, defined as work done by 1 newton acting through a
    distance of one meter.
    
    
    Erg, cgs unit of energy, defined as work done by 1 dyne acting through a
                                             -7
    distance of one centimeter. One erg is 10   joule.
    
    
    Kilogram calorie, large calorie, unit of energy, is the heat required to
    raise the temperature of 1 kilogram of water from 1 degree centigrade at a
    stated temperature. i.e. Kg Cal(22 C). The mean kilogram calorie is defined as
    1/100 of the heat required to raise the temperature of 1 kilogram of water
          o        o
    from 0 C to 100 C. The small calorie is the gram calorie equal to 1/1000 of
    a large calorie. One mean kilogram calorie is 0.000238889 joule .
    
    
    British thermal unit, B.t.u , unit of energy, the heat required to raise
    the temperature of 1 pound of water 1 degree Fahrenheit at a stated 
                               o
    temperature. i.e. B.t.u.(39 F). The mean British thermal unit is defined as
    1/180 of the heat required to raise the temperature of 1 pound of water from
      o        o
    32 F to 212 F. One mean B.t.u. is 0.00009480 joule.
    
    
    Mole, kilogram molecule, is the number of kilograms of a substance that
    corresponds to its molecular weight divided by 1000. In the cgs system of
    units a mole, gram molecule, is the number of grams of a substance that
    corresponds to its molecular weight. The mass of a single molecule in
    kilograms is the kilogram molecule divided by Avogadro's number. For atoms
    the molecular weight is the atomic weight.
    
    
    Steradian, sr, is the ratio of the area of the intercepted surface of
    a sphere to the radius of the sphere squared. 4Pi steradians means the
    total area of the sphere is intercepted.
    
    
    Watt, unit of power, defined as work done at a constant rate of one
    joule per second.
    
    
    Horsepower ( mechanical ), English unit of power, defined as work done
    at a rate of 550 foot-pounds per second. One mechanical horsepower is
    745.705 watt.
    
    
    Horsepower ( electrical ), English unit of power, by definition exactly
    760 watt.
    
    
    Ampere, unit of electric current, defined as the current that will flow
    through a circuit with a resistance of one ohm when one volt is applied. The
    international standard is defined as the current which will deposit silver
    at a rate of 0.00111800 gram per second. One international ampere is about
    0.999835 absolute ampere. International electrical units are based on physical
    standards whose specifications are slightly in error. Instruments made after
    January 1, 1948 are calibrated in absolute units.
    
    
    Notes:
            The singular form of units is used with the exception of foot and feet.
            Proper names appearing in units and constants are not capitalized.
    
    
    References:
                 Conversion Factors and Tables by Zimmerman and Lavine
                 Electric and Magnetic Fields by Stephen Attwood
                 Elements of Physics by Shortley and Williams
    
    

    UNITS CONVERSION CONSTANTS

    
    to get MKS units from other units       to get other units from MKS units
    
    value      value                        value        value
    in MKS  =  in other  x  constant        in other  =  in MKS  x  constant
    units      units                        units        units
    
                                     length
    
    meter = angstrom x 1.0E-10              angstrom = meter x 1.0E10
    
    meter = mil x 0.254E-4                  mil = meter x 39370.07874
    
    meter = centimeter x 0.01               centimeter = meter x 100
    
    meter = inch x 0.0254                   inch = meter x 39.37007874
    
    meter = feet x 0.3048                   feet = meter x 3.280839895
    
    meter = yard x 0.9144018288             yard = meter x 1.0936111
    
    meter = fathom x 1.8288036              fathom = meter x           
    
    meter = rod x 5.0292099                 rod = meter x 0.19883839
    
    meter = chain(surveyor) x 20.12         chain(surveyor) = meter x 66 ft
    
    meter = chain(engineer) x 30.48006      chain(engineer) = meter x 100 ft
    
    meter = furlong x 0.2011684E+3          furlong = meter x 0.49709597E-2
    
    meter = mile(statute) x 1.6093472E+3    mile(statute) = meter x 0.6213699E-3  *
    
    meter = mile(nautical) x 1.8532487E+3   mile(nautical) = meter x 0.539593E-3
    
    meter = league(land) x 4.82804E+3       league(land) = meter x 
    
    meter = league(marine) x 5.5596E+3      league(marine) = meter x
    
    meter = light year x 9.459936E+15       light year = meter x
    
    
    
                                        mass
    
    
    kilogram = gram x 0.001                 gram = kilogram x 1000
    
    kilogram = grain(troy) x 0.6480E-4      grain(troy) = kilogram x
    
    kilogram = pennyweight(troy) x 1.5552E-3 pennyweight(troy) = kilogram x 
                  24 grains
    
    kilogram = carat(troy) x 0.2E-3  3086 grains
    
    kilogram = scruple x 1.296E-3           scruple = kilogram x
    
    kilogram = dram(avdp) x 1.772E-3        dram(avdp) = kilogram x
    
    kilogram = ounce(avdp) x 0.02834952     ounce(avdp) = kilogram x 35.27
    
    kilogram = ounce(troy) x 0.031103481    ounce(troy) = kilogram x 32.15
    
    kilogram = pound(troy) x 0.37324177     pound(troy) = kilogram x 2.6792285
    
    kilogram = pound(avdp) x 0.45359244     pound(avdp) = kilogram x 2.204622341  *
    
    kilogram = ton(short) x 907.18486       ton(short) = kilogram x 2000lbs       *
    
    kilogram = ton(long) x 1016.047         ton(long) = kilogram x 0.9842064E-3
    
    kilogram = ton(metric) x 1000           ton(metric) = kilogram x 0.001
    
    
    
                                         time
    
    
    second = minute x 60                    minute = second x
    
    second = hour x 3600                    hour = second x
    
    second = day x 0.86400E+5               day = second x
    
    second = fortnight x 1.2096E+6          fortnight = second x
    
    second = month x 2.628E+6               month = second x 
    
    second = year x                         year = second x
    
    
    
                                 electric charge
    
    
    
    coulomb = electron charge x             electron charge = coulomb x 1.60193E-19
    
    coulomb = faraday x                     faraday = coulomb x 96.480
    
    coulomb = ampere hours x                ampere hours = coulomb x 3600
    
    
    
                                    temperature
    
    
    o    o                                  o    oL
     K =  C + 273.16                         C =  K - 273.16
    
    o                                       o     oL
     K =                                     F = ( K - 273.16) x 1.8 + 32.0
    
    
                                        angle
    
    
    
    radian = second(angular) x 4.84814E-6   second(angular) = radian x
    
    radian = minute(angular) x 0.000290888  minute(angular) = radian x
    
    radian = degree(angular) x 0.0174533    degree(angular) = radian x
    
    radian = revolution x 6.2831853         revolution = radian x
    
    radian = bam x
    
    
    
                                area
    
    
    square meter = square centimeter        square centimeter = square meter 
                   x 1.0E-4                                     x 10,000
    
    square meter = square inch              square inch = square meter
                   x                                      x
    
    square meter = square feet              square feet = square meter
                   x 0.09290341                           x 
    
    square meter = square yard              square yard = square meter
                   x                                      x
    
    square meter = square mile(statute)     square mile(statute) = square meter
                   x                                               x
    
    square meter = acre x 4046.873         acre = square meter x
    
    square meter = circular mil x           circular mil = square meter x 1.97352E+6
    
    square meter = hectare x 1.0E+4         hectare = square meter x
    
    square meter = township x 93.24E+6      township = square meter x
    
    square meter = barn x 1.0E-28
    
    
                                  volume
    
    
    cubic   cubic                           cubic        cubic
    meter = centimeter x 1.0E-6             centimeter = meter x 1.0E+6
    
    cubic   cubic                           cubic   cubic
    meter = inch  x 0.16387162E-4           inch  = meter x
    
    cubic meter = cubic feet x 0.028317017  cubic feet = cubic meter x
    
    cubic meter = cubic yard x              cubic yard = cubic meter x
    
    cubic   cubic                           cubic           cubic
    meter = mile(statute) x                 mile(statute) = meter x
    
    cubic meter = liter x 0.001             liter = cubic meter x 1000
    
    cubic meter = fluid ounce x 0.295737E-4 fluid ounce = cubic meter x 0.33814E+7
    
    cubic meter = cup x
    
    cubic                                                  cubic
    meter = pint(liquid) x 0.4731798E-3     pint(liquid) = meter x 21113.4
    
    cubic meter = quart(liquid) x           quart(liquid) = cubic meter x
    
    cubic meter = gallon x 0.003785         gallon = cubic meter x
    
    cubic meter = barrel x 1/0.1589873      barrel = cubic meter x 0.1589873
    
    cubic meter = pint(dry) x 0.03524/64
    
    cubic meter = quart(dry) x 0.03524/32
    
    cubic meter = peck x 0.03524/4
    
    cubic meter = bushel x 0.03524          bushel = cubic meter x
    
    cubic meter = keg x (less than 10 gal)
    
    cubic meter = cord x 3.625
    
    barrel = gallon x 31.5 (food)  x 42 (petroleum)
    
                                  velocity
    
    
    meter per second = centimeters per second x
    
    meter per second = kilometer per hour x
    
    meter per second = inches per second x
    
    meter per second = feet per second x
    
    meter per second = miles per second x
    
    meter per second = inches per minute x
    
    meter per second = feet per minute x
    
    meter per second = miles per hour x
    
    meter per second = knots x
    
    
    
                                acceleration
    
    
    meter per second squared = centimeter per second squared x
    
    meter per second squared = feet per second squared x
    
    meter per second squared = miles per hour squared x
    
    
    
                                  force
    
    
    
    newton = dyne x 1.0E-5
    
    newton = poundal x 7.23300E-10
    
    newton = pound x 7.23300E-10/32.17  g
    
    
    
                                 energy
    
    
    
    joule = erg x 1.0E-7
    
    joule = gram calorie x 0.238889E-6
    
    joule = kilogram calorie x 0.238889E-3
    
    joule = gram calorie x 0.238889E-6
    
    joule = B.t.u x 0.9480E-4
    
    joule = foot pounds x 1.356
    
    joule = kilowatt hour x 3.6E+6
    
    joule = horsepower hours x 2.684E+6
    
    
    
                                  power
    
    
    watt = kilogram calorie per second x
    
    watt = kilogram calorie per minute x
    
    watt = horsepower(mechanical) x 745.705
    
    watt = horsepower(electrical) x 760
    
    watt = horsepower(metric)  1.014 ?
    
    watt = horsepower(boiler) x 9.804E+3  33,520 Btu per hour
    
    watt = B.t.u per minute x 17.57
    
    watt = B.t.u per hour x 17.57*60
    
    watt = foot pound per minute x 0.2260E-3   33000 HP
    
    watt = foot pound per second x 1.356   550 HP
    
    
    
                                 density
    
    kilogram per cubic meter = pound per cubic foot
    
    
                                 pressure
    
    
    
    pascal = newton per square meter x 1
    
    pascal = pounds per square foot x
    
    pascal = ton per square foot x
    
    pascal = atmosphere(standard) x 1.013250E5
    
    pascal = feet of water x
    
    pascal = inches of mercury
    
    pascal = millimeters of mercury x 1/133.3
    
    pascal = bar x 1.0E5
    
    pascal = millibar x
    
    pascal = torr x
    
    
                                 torque
    
    
    newton meter = foot pound x
     
    
                                flow rate
    
    
    
    cubic meter per second = gallon per minute x 0.6309E-8
    
    cubic meter per second = cubic feet per minute x 0.4719E-3
    
    
                              specific heat, entropy
    
    
                       o                     oL
    joule per kilogram  K = B.t.u. per pound  F x 4.187E+3
    
    
    
                              dynamic viscosity
    
    poise = dyne second per square centimeter
    
    
                               kinematic viscosity
    
    stoke = square centimeter per second
    
    
                               electric current
    
    
    ampere = abampere x 10
    
    ampere = statampere x 0.333333E-9
                                         
                             magnetic flux B
    
                             magnetic induction
    
                            magnetomotive force
    
                           magnetic field strength H
    
                            dielectric constant
    
                            permittivity constant
    
                              rotation rate
    
    
    radians per second = revolutions per second x
    
    radians per second = revolutions per minute x
    
    
    

    PHYSICAL CONSTANTS

    
       There are a number of physical constants that are used in equations
    
    to solve problems in physics. Errors may occur because the dimensionality
    
    and/or units of the physical constant are not known. The table below
    
    presents some physical constants with their typical symbol, dimension,
    
    nominal value and unit of measure in the MKS system.
    
    PHYSICAL CONSTANT           SYMBOL  DIMENSION   MKS VALUE        UNIT
    _________________           ______  _________   _________        ____
    
                                            3                             3
    air density, normal          rho     M/L         1.293            Kg/m
       conditions
    
    
    air molecule, mass           m       M           4.81E-26         Kg
                                  a
    
    
    air molecule,                w       M           0.028952         Kg/mole
     kilogram molecular weight
    
    
                                             2                                2
    atmospheric pressure         A       M/LT        1.01325          newton/m
    
    
    Avogadro's number            N       none        6.023E+23        molecules in
      molecules in a mole                                               a mole
    
    
                                           2  2                             o
    Boltzmann's constant         k       ML /T K     1.380E-23        joule/ K
    
    
                                           2  2                          
    electron volt                e       ML /T       1.60210E-10      joule
    
    
                                           3  2 2                         2       2
    electrostatic constant       k       ML /T Q     8.987E+9         nt m/coulomb
      reciprocal permittivity                                          m/farad
    
    
    elementary charge            e       Q           1.6021892E-19    coulomb
    
    
    electron mass                m       M           9.1066E-31       Kg
                                  e
    
    faraday                      f       L/T         9.648456E+4      coulomb/mole
    
    
                                           2  2                             o
    gas constant of a mole       R       ML /T K     8.3144           joule/ K
    
                                            2                              2
    gravity (earth)              g       L/T         9.80665          m/sec
    
    
    hydrogen atom mass           m       M           1.6734E-27       Kg
                                  h
    
    
    hydrogen atom                w       M           1.0079E-3        Kg/mole
      kilogram atomic weight
    
    
                                           2   2                          
    impedance of free space      Z       ML /TQ      120Pi            ohm
                                  0
    
    
    mechanical equivalent        J       none        4186.05          joule/
       of heat                                                          Kg calorie
    
    
                                          2 2   3                        
    permittivity (vacuum)        epsi    T Q /ML     8.854E-12        farad/meter
                                     0
    
    
                                             2                           
    permeability (vacuum)        mu      ML/Q        4Pi E-7          henry/meter
                                   0
    
    
    Pi, ratio of circumference   Pi      none        3.14159265       radians
          to diameter
    
                                           2                              
    Planck's constant            h       ML /T       6.624E-34        joule second
    
    
    speed of light (vacuum)      c       L/T         2.99792458E+8    meter/second
    
    
    speed of sound (air)         s       L/T         331.45           meter/second
    
    
                                          2   2                           2   2
    universal gravitational      G       L /MT       6.6720E-12       nt m /Kg
        constant
    
    
    
    Note: some constants are related to combinations of other constants :
          electrostatic constant = 1/ 4Pi permittivity (vacuum)
          speed of light = 1/ sqrt( permittivity x permeability )
          impedance of free space Z  = sqrt( permeability / permittivity )
                                   0
    
    

    PHYSICS EQUATIONS

    
    
                         SOME EQUATIONS OF PHYSICS 
    
    
    F = m a              force equals mass times acceleration,
                         Newton's second law of motion
    
    
           2  
    F = m v /r           force equals mass times velocity squared over radius,
                         centripetal force of a mass traveling in a circle
    
    
                 2 
    F = G m  m /s        gravitational force between mass  and mass  at distance s
           1  2                                          1         2
                         with universal gravitational constant G
    
    
                   2
    g = G m      /r        acceleration due to gravity on earth
           earth   earth
    
    
                 2 
    F = k Q  Q /s        electrical force between charge  and charge  at distance s
           1  2                                         1           2
                         with electrostatic constant k . If there is a dielectric
                         then multiply by the non dimensional dielectric constant.
    
    
    F = 1/2Pi  mu I  I  /s
                   1  2
                         electrical force between two parallel wires carrying
                         currents I  and I  with a spacing s with permeability
                                   1      2
                         mu. This is the force for one meter of wire length.
    
    
             2 
    F = B H s
                         electrical force in a magnetic field equals the magnetic
                         flux times the magnetic intensity applied to an area
    
             2 
    F = E D s
                         electrical force in an electric field equals the electric
                         field intensity times the electric displacement applied
                         to an area
    
    
    s = v t              distance equals velocity times time (linear)
    
    
    v = a t              velocity equals acceleration times time (linear)
    
    
                          2
    s = s  + v t + 1/2 a t 
         0    0  
                         linear distance equals initial distance plus
                         initial velocity times time plus one half acceleration
                         times time squared
    
                2 
    v  = sqrt( v  + 2as)
     f          0
                        the final velocity equals the square root of the
                        initial velocity squared plus two times the acceleration
                        times the distance traveled
    
    
    v  = sqrt( s g )    the critical velocity for any object to orbit at a
     c
                        distance s from the source of gravitational field g
    
    
    theta = omega t     angle equals angular velocity times time (rotational)
    
    
    omega = alpha t     angular velocity equals angular acceleration times time
                        (rotational)
    
                                          2
    theta = theta  + omega t + 1/2 alpha t 
                 0        0  
                        angular rotation equals initial angle plus
                        initial angular velocity times time plus one half
                        angular acceleration times time squared
    
               2 
    w  = sqrt(w  + 2 alpha * angle) 
     f         0
                        the final angular velocity equals the square root of
                        the initial angular velocity squared time twice the
                        angular acceleration times the angle traveled
    
    
    E = I R             voltage equals current through a resistor times the
                        resistance
    
    I = C (E  - E )/(t  - t ) 
            2    1    2    1
                        the current through a capacitor equals the capacitance
                        times the change in voltage over the change in time
    
    E = L (I  - I )/(t  - t ) 
            2    1    2    1
                        the voltage across an inductor equals the inductance 
                        times the change in current over the change in time
    
    C = epsi  A/s
                        the capacitance in farad of a parallel plate capacitor
                        equals the permittivity times the area divided by the
                        spacing.
    
    L = n mu r (ln 8r/d  - 7/4)
                        the inductance in henry of n turns of wire with diameter
                        d closely wrapped in a coil of radius r with permeability
                        mu is approximately given by this equation.
    
    H = 1/2 I / r
                        the magnetic intensity at the center of a current loop
                        equals 1/2 the current divided by the radius of the loop
    
    B = mu H            the magnetic flux equals the permeability times the
                        magnetic intensity
    
    D = epsi E          the electric displacement equals the permittivity
                        times the electric field intensity
    
    P = E I             power equals an electrical potential causing a current
    
    P = F s             power equals a force applied over a distance
    
           2 L
    E = m c             energy from converting a mass to energy
                           ( c = speed of light)
    
               2 L
    E = 1/2 m v         kinetic energy of a mass traveling at a velocity
    
    E = m g s           potential energy of a mass in a gravitational field
                        at a height s
    
    E = 1/2 B H V       energy of a magnetic field in the volume V with magnetic
                        flux B and magnetic intensity H. This is usually an
                        integral of an incremental volume times B times H
                        in the incremental volume.
    
    E = 1/2 D E V       energy of an electric field in the volume V with electric
                        displacement D and electric field intensity E. This is
                        usually an integral of an incremental volume times D
                        times E in the incremental volume.
    
               2
    E = 1/2 C V         energy stored in a capacitor with capacitance C having
                        a voltage V
    
               2
    E = 1/2 L I         energy stored in an inductor with inductance L having
                        a current I
    
    
    T = F s             torque equals the force applied at radius s
    
    T = I alpha         torque equals the rotational inertia times the angular
                        acceleration
    
    
                                     2
    E = P V = R T = N k T = 1/3 N m v             ideal gas law 
                                     rms
                        These relations are for one mole (kilogram molecule) of
                        an ideal gas at an absolute pressure P, volume V,
                        gas constant R, Avogadro's number N, Boltzmann's 
                        constant k, temperature T in degrees kelvin, gas
                        molecule mass m, root mean square speed of the molecules
                        v    in meters per second. Each section of the equation
                         rms
                        represents energy in joule.
    
                 2                          2
    P + 1/2 rho v + rho g z  = P + 1/2 rho v + rho g z
     1           1         1    2           2         2
    
                        This equation relates pressure P, velocity v and relative
                        height z for a non compressible fluid in a pipe, observed
                        at location 1 and location 2. rho is the density of the
                        fluid and g is the gravitational constant.
    
                2
    L = C  rho v  A / 2
         LL
                        the lift force equals the dimensionless coefficient of
                        lift times the air density times the velocity squared times
                        the surface area divided by 2.
    
                2
    D = C  rho v  A / 2
         D
                        the drag force equals the dimensionless coefficient of
                        drag times the air density times the velocity squared times
                        the surface area divided by 2.
    
    nu = mu / rho
                        the kinematic viscosity equals the dynamic viscosity over
                        the density in a fluid
    
    P = Q (p - p )
            1   2       the power, P, required to drive a volume rate of flow, Q,
                        from pressure p  to pressure p  .
                                       1              1
    o    o
     C =  K - 273.16
                        degrees centigrade equals degrees kelvin minus 273.16
    
    o     oL
     F = ( K -273.16) x 9/5 + 32
                        degrees Fahrenheit as a function of degrees kelvin
    
    

    Go to top

    Last updated 9/8/01