## 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

```

### 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.

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 = ampere hours x                ampere hours = coulomb x 3600

temperature

o    o                                  o    oL
K =  C + 273.16                         C =  K - 273.16

o                                       o     oL
K =                                     F = ( K - 273.16) x 1.8 + 32.0

angle

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                     oL
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

elementary charge            e       Q           1.6021892E-19    coulomb

electron mass                m       M           9.1066E-31       Kg
e

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
LL
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     oL
F = ( K -273.16) x 9/5 + 32
degrees Fahrenheit as a function of degrees kelvin

```

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Last updated 9/8/01