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Unformatted text preview: First Edition – Volume 5 Formulas and Conversions Published by IDC Technologies,
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Technical Director,
Steve Mackay
Dear Colleague,
Welcome to our latest engineering pocket guide focusing
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P.S. Don't forget our norisk guarantee on all our products – we give you a 100%
guarantee of satisfaction or your money back. Other books in this series
Volume 1 – INSTRUMENTATION
Automation using PLCs, SCADA and Telemetry, Process Control and
Data Acquisition
Volume 2 – COMMUNICATIONS
Data Communications, Industrial Networking, TCP/IP and Fiber Optics
Volume 3 – ELECTRICAL
Power Quality, Power Systems Protection and Substation Automation
Volume 4 – ELECTRONICS
Personal Computers, Digital Signal Processing and Analog/Digital Conversions 5.3.7
5.3.8
5.3.9
5.3.10
5.3.11
5.3.12
5.3.13
5.3.14
5.3.15
5.3.16
5.3.17
5.3.18
5.3.19
5.3.20
5.3.21
5.3.22
5.3.23 Table of Contents
Chapter 1
Definition and Abbreviations for Physical Quantities ...........1
Chapter 2
Units of Physical Quantities .................................................3
Chapter 3
System of Units ..................................................................23
5.4 General Mathematical Formulae........................................27
4.1
4.2
4.3
4.4
4.5
4.6 Algebra................................................................................. 27
Geometry ............................................................................. 29
Trigonometry ........................................................................ 39
Logarithm ............................................................................. 40
Exponents ............................................................................ 42
Complex Numbers ............................................................... 42 Chapter 5
Engineering Concepts and Formulae ................................44
5.1
5.2 Electricity.............................................................................. 44
Applied Mechanics ............................................................... 57
5.2.1
5.2.2
5.2.3
5.2.4
5.2.5 5.3 Newton's laws of motion ..........................................................57
Linear Velocity And Acceleration .............................................60
Force........................................................................................61
Centripetal (Centrifugal) Force.................................................62
Stress, Strain And Modulus Of Elasticity..................................64 Thermodynamics.................................................................. 64
5.3.1
5.3.2
5.3.3
5.3.4
5.3.5
5.3.6 Laws of Thermodynamics ........................................................64
Momentum...............................................................................65
Impulse ....................................................................................65
Elastic and Inelastic collision ...................................................65
Center of Mass ........................................................................65
Angular Motion.........................................................................65 Fluid Mechanics ................................................................... 77
5.4.1
5.4.2
5.4.3 Chapter 4 Conditions of Equilibrium .........................................................65
Gravity .....................................................................................66
Vibrations & Waves .................................................................66
Standing Waves.......................................................................66
Beats........................................................................................66
Temperature and Heat.............................................................67
Ideal Gases..............................................................................67
Elastic Deformation..................................................................68
Temperature Scales ................................................................68
Sensible Heat Equation ...........................................................68
Latent Heat ..............................................................................68
Gas Laws.................................................................................68
Specific Heats Of Gases..........................................................69
Efficiency of Heat Engines .......................................................70
Heat Transfer by Conduction ...................................................71
Thermal Expansion of Solids ...................................................72
Chemical Heating Value of a Fuel ...........................................72
Discharge from an Orifice ........................................................77
Bernoulli’s Theory ....................................................................78
Actual pipe dimensions ............................................................78 Chapter 6
References.........................................................................80
6.1
6.2 Periodic Table of Elements .................................................. 80
Resistor Color Coding .......................................................... 81 Formulas and Conversions Formulas and Conversions
Symbol Definition and Abbreviations for Physical Quantities Prefix Factor by which unit is
multiplied k Chapter 1 Kilo 103 h Hecto 102 da Deca 10 Quantity d Deci 101 meter Length c Centi 102 kilogram Mass m Milli 103 s second Time µ Micro 106 A ampere Electric current n Nano 109 K kelvin Thermodynamic temp p Pico 1012 cd candela Luminous intensity Symbol Unit m
kg Quantity Unit Symbol Equivalent Plane angle radian rad  Force newton N kg · m/s2 Work, energy heat joule J·N·m Power watt W J/s Frequency hertz Hz s1 Viscosity:
kinematic  m2/s 10 c St
(Centistoke) Viscosity:
Dynamic  Ns/m2 103 cP
(Centipoise) Pressure  Pa or N/m2 pascal, Pa Symbol Prefix Factor by which unit is
multiplied Quantity Electrical
unit Symbol Derived
unit Potential Volt V W/A Resistance Ohm Ώ V/A Charge Coulomb C A·s Capacitance Farad F A·s/V Electric field
strength  V/m  Electric flux
density  C/m2  Quantity Magnetic
unit Symbol Derived unit Magnetic flux Weber Wb V·s = N·m/A Inductance Henry H V·s/A = N·m/A2 T Tera 1012 A/m  Giga 109 Magnetic field
strength  G
M Mega 106 Magnetic flux density Tesla T Wb/m2 =
(N)/(Am) 1 2 Formulas and Conversions Formulas and Conversions Chapter 2 Multiply
by Name lb·s2/in4 kg/m3 3 3 Divide by 1.069E+07 9.357E08 Density 1.000E+07 1.602E19 6.242E+18 Ft·lbf J 1.3557 0.7376 kiloton TNT J 4.187E+12 2.388E13 KW·hr J 3.600E+06 2.778E07 Megaton TNT J 4.187E+15 2.388E16 Dyne N 1.000E05 1.000E+05 Lbf N 4.4484 0.2248 Ozf N 0.2780 3.5968 BTU/lbm · °F J/kg·°C 4188 2.388E04 Heat transfer coefficient Degrees Fahrenheit = (Degrees Celsius) (9/5) + 32 1.000E07 J Heat capacity Degrees Celsius = (Degrees Fahrenheit  32) (5/9) J eV Force 1 pound per square inch = 2.31 feet of water erg Force 1 pound = 0.454 kilograms 0.2389 Force 1 million gallons per day = 694 gallons per minute 4.1859 Energy 1 horsepower = 0.746 kilowatts J Energy 1 grain per gallon = 17.1 mg/L cal Energy 1 gallon = 8.34 pounds 9.478E04 Energy 1 gallon = 3.79 liters 1.940E03 1055 Energy 1 foot = 0.305 meters 515.40 J Energy 1 cubic foot = 7.5 gallons kg/m BTU Energy 1 acre = 43,560 square feet slug/ft Energy Conversion Factors (general): BTU/hr·ft2·°F W/m2·°C 5.6786 0.1761 Length To convert from
2 Multiply
by To
2 Divide by AU m 1.496E+11 6.685E12 Length 1% = 10,000 mg/L Name To Density Units of Physical Quantities To convert from ft m 0.3048 3.2810 Length in m 2.540E02 39.3700 Length mile m 1609 6.214E04 Nautical mile m 1853 5.397E04 Acceleration ft/sec m/s 0.3048 3.2810 Length Area acre m2 4047 2.471E04 Length parsec m 3.085E+16 3.241E17 Area ft2 m2 9.294E02 10.7600 Mass amu kg 1.661E27 6.022E+26 Area hectare m2 1.000E+04 1.000E04 Mass lbm kg 0.4535 2.2050 Area in2 m2 6.452E04 1550 Mass lb·s2/in kg 1200.00 5.711E03 Density g/cm3 kg/m3 1000 1.000E03 Mass slug kg 14.59 6.853E02 Density lbm/ft3 kg/m3 16.02 6.243E02 Mass flow rate lbm/hr kg/s 1.260E04 7937 Density lbm/in3 kg/m3 2.767E+04 3.614E05 3 4 Formulas and Conversions Name To convert from
lbm/sec Multiply
by To Mass flow rate Formulas and Conversions kg/s Divide by Name To convert from To Multiply
by Divide by sidereal year S 3.156E+07 3.169E08 0.4535 2.2050 Time 2 2 Moment of inertia ft·lb·s kg·m 1.3557 0.7376 Torque ft·lbf N·m 1.3557 0.7376 Moment of inertia in·lb·s2 kg·m2 0.1130 8.8510 Torque in·lbf N·m 0.1130 8.8504 Moment of inertia oz·in·s2 kg·m2 7.062E03 141.60 Torque In·ozf N·m 7.062E03 141.61 Power BTU/hr W 0.2931 3.4120 Velocity ft/min m/s 5.079E03 196.90 Power hp W 745.71 1.341E03 Velocity ft/s m/s 0.3048 3.2810 Power tons of refrigeration W 3516 2.844E04 Velocity Km/hr m/s 0.2778 3.6000 Pressure bar Pa 1.000E+05 1.000E05 Velocity miles/hr m/s 0.4470 2.2370 2 2 Pressure dyne/cm Pa 0.1000 10.0000 Viscosity – absolute centipose N·s/m 1.000E03 1000 Pressure in. mercury Pa 3377 2.961E04 Viscosity – absolute g/cm·s N·s/m2 0.1000 10 2 47.87 2.089E02 2 Pressure in. water Pa 2 248.82 4.019E03 Viscosity – absolute 2 lbf/ft ·s N·s/m Pressure kgf/cm Pa 9.807E+04 1.020E05 Viscosity – absolute lbm/ft·s N·s/m 1.4881 0.6720 Pressure lbf/ft2 Pa 47.89 2.088E02 Viscosity – kinematic centistoke m2/s 1.000E06 1.000E+06 Pressure 2 lbf/in Pa 6897 1.450E04 Viscosity – kinematic Pressure mbar Pa 100.00 1.000E02 Volume Pressure microns mercury Pa 0.1333 7.501 Pressure mm mercury Pa 133.3 Pressure std atm Pa Specific heat BTU/lbm·°F Specific heat cal/g·°C ft /sec 2 m /s 9.294E02 10.7600 ft3 m3 2.831E02 35.3200 Volume in3 m3 1.639E05 6.102E+04 7.501E03 Volume Liters m3 1.000E03 1000 1.013E+05 9.869E06 Volume U.S. gallons m3 3.785E03 264.20 J/kg·°C 4186 2.389E04 Volume flow rate ft3/min m3/s 4.719E04 2119 J/kg·°C 4186 2.389E04 Volume flow rate U.S. gallons/min m3/s 6.309E05 1.585E+04 Temperature °F °C 0.5556 1.8000 Thermal conductivity BTU/hr·ft·°F W/m·°C 1.7307 0.5778 Thermal conductivity BTU·in/hr·ft2·°F W/m·°C 0.1442 6.9340 Thermal conductivity cal/cm·s·°C W/m·°C 418.60 2.389E03 2 Thermal conductivity cal/ft·hr·°F W/m·°C 6.867E03 day S 8.640E+04 1.157E05 DISTANCE (Length) Conversions
Multiply By To obtain LENGTH 145.62 Time A. 0.03280840 foot Centimeter 5 Centimeter 0.3937008 inch 6 Formulas and Conversions
Multiply By To obtain Foot
Foot
Foot 304.8 Inch 0.0254*
2.54
25.4 Inch
Kilometer *
* Centimeters Yards 0.01093613 Centimeters Feet 0.0328084 Centimeters Inches 0.3937008 meter(m) * 0.01 millimeter(mm) 30.48* Multiply By Meters meter(m) 0.3048* To Centimeters centimeter(cm) 1.8288* To Convert meter(m) Chains, (Surveyor's) Rods 4 centimeter(cm) Fathom Inch Formulas and Conversions Chains, (Surveyor's) Meters 20.1168 millimeter(mm) 0.6213712 Meter 39.37008 Meter 0.54680066 Chains, (Surveyor's) Feet 66 mile(USstatute) Fathoms Meters 1.8288 Inch Fathoms Feet 6 Fathom Feet Statute Miles 0.00018939 Meter 3.280840 Foot Feet Kilometers 0.0003048 Meter 0.1988388 Rod Feet Meters 0.3048 Meter 1.093613 Yard Feet Yards 0.3333333 Meter 0.0006213712 Microinch
micrometer(micron) mile(USstatute) Feet Inches 12 micrometer(micron)(µm) 0.0254* Feet Centimeters 30.48 39.37008 Microinch Furlongs Statute Miles 0.125 mile(USstatute) 1,609.344* meter(m) Furlongs Meters 201.168 mile(USstatute) * kilometer(km) Furlongs Yards 220 millimeter 0.003280840 1.609344 Foot Furlongs Feet 660 millimeter 0.0397008 Inch Furlongs Inches 7920 Rod 5.0292 * meter(m) Hands (Height Of Horse) Inches 4 Yard 0.9144* meter(m) Hands (Height Of Horse) Centimeters 10.16 Inches Meters 0.0254 Inches Yards 0.02777778 Inches Feet 0.08333333 Inches Centimeters 2.54 Inches Millimeters 25.4 To Convert To Multiply By Cables Fathoms 120 Cables Meters 219.456 Cables Yards 240 7 8 Formulas and Conversions Formulas and Conversions To Convert To Multiply By To Convert To Multiply By Kilometers Statute Miles 0.621371192 Miles, Statute Centimeters 160934.4 Kilometers Meters 1000 Millimeters Inches 0.039370079 Leagues, Nautical Nautical Miles 3 Mils Inches 0.001 Leagues, Nautical Kilometers 5.556 Mils Millimeters 0.0254 Leagues, Statute Statute Miles 3 Paces (US) Inches 30 Leagues, Statute Kilometers 4.828032 Paces (US) Centimeters 76.2 Links, (Surveyor's) Chains 0.01 Points (Typographical) Inches 0.013837 Links, (Surveyor's) Inches 7.92 Points (Typographical) Millimeters 0.3514598 Links, (Surveyor's) Centimeters 20.1168 Rods Meters 5.0292 Meters Statute Miles 0.000621371 Rods Yards 5.5 Meters Kilometers 0.001 Rods Feet 16.5 Meters Yards 1.093613298 Spans Inches 9 Meters Feet 3.280839895 Spans Centimeters 22.86 Meters Inches 39.370079 Yards Miles 0.00056818 Meters Centimeters 100 Yards Meters 0.9144 Meters Millimeters 1000 Yards Feet 3 Microns Meters 0.000001 Yards Inches 36 Microns Inches 0.0000394 Yards Centimeters 91.44 Miles, Nautical Statute Miles 1.1507794 Miles, Nautical Kilometers 1.852 Miles, Statute Kilometers 1.609344 Miles, Statute Furlongs 8 Miles, Statute Rods 320 Miles, Statute Meters 1609.344 Miles, Statute Yards 1760 Miles, Statute Feet 5280 Miles, Statute Inches 63360 9 Conversion
Length
1 ft = 12 in 1 yd = 3 ft 1 cm = 0.3937 in 1 in = 2.5400 cm 1 m = 3.281 ft 1 ft = 0.3048 m 1 m = 1.0936 yd 1 yd = 0.9144 m 1 km = 0.6214 mile 1 mile = 1.6093 km 1 furlong = 40 rods 1 fathom = 6 ft  10  Formulas and Conversions Formulas and Conversions Conversion Conversion 1 statute mile = 8 furlongs 1 rod = 5.5 yd Dry Volume 1 statute mile = 5280 ft 1 in = 100 mils 1 quart = 2 pints 1 quart = 67.2 in3 1 nautical mile = 6076 ft 1 light year = 9.461 x 1015 m 1 peck = 8 quarts 1 peck = 537.6 in3 1 bushel = 4 pecks 1 bushel = 2150.5 in3 5 1 league = 3 miles 1 mil = 2.540 x 10 m Area
1 ft2 = 144 in2 1 acre = 160 rod2 1 yd2 = 9 ft2 Area Conversions 1 acre = 43,560 ft2 2 1 rod = 30.25 yd 2 Multiply 1 mile2 = 640 acres 1 cm2 = 0.1550 in2
2 B. 1 in2 = 6.4516 cm2 2 1 m = 10.764 ft 1 ft2 = 0.0929 m2 1 km2 = 0.3861 mile2 1 mile2 = 2.590 km2 1 cm3 = 0.06102 in3 1 in3 = 16.387 cm3 1 m3 = 35.31 ft3 1 ft3 = 0.02832 m3 1 Litre = 61.024 in3 1 in3 = 0.0164 litre 1 Litre = 0.0353 ft3 1 ft3 = 28.32 litres 1 Litre = 0.2642 gal. (U.S.) 1 yd3 = 0.7646 m3 1 Litre = 0.0284 bu (U.S.) 1 gallon (US) = 3.785 litres 1 Litre = 1000.000 cm3 1 gallon (US) = 3.785 x 103 m3 1 Litre = 1.0567 qt. (liquid) or
0.9081 qt. (dry) 1 bushel (US) = 35.24 litres 1 oz (US fluid) = 2.957 x 105 m3 1 stere = 1 m3 2 centimeter2
2 hectare 0.1550003 inch2 0.001076391
* foot2
meter2 (m2) foot 0.09290304 foot2 929.03042 centimeter2 (cm2) 2 foot 92,903.04 millimeter2 (mm2) hectare 2.471054 acre inch2 645.16* millimeter2 (mm2) inch2 6.4516 centimeter2 (cm2) inch2 0.00064516 meter2 (m2) 2 meter 1,550.003 inch2 meter2 10.763910 foot2 1.195990 yard2 meter 1 gill = 4 fluid ounces 1 barrel = 31.5 gallons meter2 1 pint = 4 gills 1 hogshead = 2 bbl (63 gal) millimeter2 1 quart = 2 pints 1 tun = 252 gallons millimeter 1 gallon = 4 quarts 1 barrel (petrolum) = 42 gallons yard2  11  meter2 (m2) 0.4046856 acre 2 Liquid Volume To obtain 4,046.856 acre centimeter Volume By
AREA 0.0002471054 2 acre 0.00001076391 foot2 0.001550003 inch2 0.8361274  12  meter2 (m2) Formulas and Conversions C. Formulas and Conversions Volume Conversions
Metric Conversion Factors: Volume (including Capacity) To Multiply By Carat
Multiply To Convert Milligrams 200 Drams, Avoirdupois Avoirdupois Ounces 0.06255 Drams, Avoirdupois Grams 1.7718452 inch3 Drams, Avoirdupois Grains 27.344 meter3 (m3) Drams, Troy Troy Ounces 0.125 liter Drams, Troy Scruples 3 meter3 (m3) Drams, Troy Grams 3.8879346 litre Drams, Troy Grains 60 meter3 (m3) By To obtain VOLUME (including CAPACITY)
centimeter3 0.06102376 foot3 0.028311685 foot3 28.31685 gallon (UK liquid) 0.004546092 gallon (UK liquid) 4.546092 gallon (US liquid) 0.003785412 Grains Kilograms 6.47989E05 gallon (US liquid) 3.785412 liter Grains Avoirdupois Pounds 0.00014286 inch3 16,387.06 millimeter3 (mm3) Grains Troy Pounds 0.00017361 inch3 16.38706 centimeter3 (cm3) Grains Troy Ounces 0.00208333 inch3 0.00001638706 meter3 (m3) Grains Avoirdupois Ounces 0.00228571 Liter 0.001* meter3 (m3) Grains Troy Drams 0.0166 Liter 0.2199692 gallon (UK liquid) Grains Avoirdupois Drams 0.03657143 Liter 0.2641720 gallon (US liquid) Grains Pennyweights 0.042 Liter 0.03531466 foot3 Grains Scruples 0.05 meter3 219.9692 gallon (UK liquid) Grains Grams 0.06479891 3 meter 264.1720 gallon (US liquid) Grains Milligrams 64.79891 meter3 35.31466 foot3 Grams Kilograms 0.001 meter3 1.307951 yard3 Grams Avoirdupois Pounds 0.002204623 meter3 1000.* liter Grams Troy Pounds 0.00267923 meter3 61,023.76 inch3 Grams Troy Ounces 0.032150747 0.00006102376 inch3 Grams Avoirdupois Ounces 0.035273961 meter3 (m3) Grams Avoirdupois Drams 0.56438339 Grams Grains 15.432361 millimeter3
Yard3 D. 0.7645549 Mass and Weight Conversions  13   14  Formulas and Conversions Formulas and Conversions To Convert To Multiply By To Convert To Multiply By Grams Milligrams 1000 Ounces, Avoirdupois Avoirdupois Drams 16 Hundredweights, Long Long Tons 0.05 Ounces, Avoirdupois Grams 28.34952313 Hundredweights, Long Metric Tons 0.050802345 Ounces, Avoirdupois Grains 437.5 Hundredweights, Long Short Tons 0.056 Ounces, Troy Avoirdupois Pounds 0.06857143 Hundredweights, Long Kilograms 50.802345 Ounces, Troy Troy Pounds 0.0833333 Hundredweights, Long Avoirdupois Pounds 112 Ounces, Troy Avoirdupois Ounces 1.097143 Hundredweights, Short Long Tons 0.04464286 Ounces, Troy Troy Drams 8 Hundredweights, Short Metric Tons 0.045359237 Ounces, Troy Avoirdupois Drams 17.55429 Hundredweights, Short Short Tons 0.05 Ounces, Troy Pennyweights 20 Hundredweights, Short Kilograms 45.359237 Ounces, Troy Grams 31.1034768 Hundredweights, Short Avoirdupois Pounds 100 Ounces, Troy Grains 480 Kilograms Long Tons 0.0009842 Pennyweights Troy Ounces 0.05 Kilograms Metric Tons 0.001 Pennyweights Grams 1.55517384 Kilograms Short Tons 0.00110231 Pennyweights Grains 24 Kilograms Short Hundredweights 0.02204623 Pounds, Avoirdupois Long Tons 0.000446429 Kilograms Avoirdupois Pounds 2.204622622 Pounds, Avoirdupois Metric Tons 0.000453592 Kilograms Troy Pounds 2.679229 Pounds, Avoirdupois Short Tons 0.0005 Kilograms Troy Ounces 32.15075 Pounds, Avoirdupois Quintals 0.00453592 Kilograms Avoirdupois Ounces 35.273962 Pounds, Avoirdupois Kilograms 0.45359237 Kilograms Avoirdupois Drams 564.3834 Pounds, Avoirdupois Troy Pounds 1.215278 Kilograms Grams 1000 Pounds, Avoirdupois Troy Ounces 14.58333 Kilograms Grains 15432.36 Pounds, Avoirdupois Avoirdupois Ounces 16 Milligrams Grains 0.015432358 Pounds, Avoirdupois Avoirdupois Drams 256 Ounces, Avoirdupois Kilograms 0.028349523 Pounds, Avoirdupois Grams 453.59237 Ounces, Avoirdupois Avoirdupois Pounds 0.0625 Pounds, Avoirdupois Grains 7000 Ounces, Avoirdupois Troy Pounds 0.07595486 Pounds, Troy Kilograms 0.373241722 Ounces, Avoirdupois Troy Ounces 0.9114583 Pounds, Troy Avoirdupois Pounds 0.8228571  15   16  Formulas and Conversions Formulas and Conversions To Convert To Multiply By To Convert To Multiply By Pounds, Troy Troy Ounces 12 Tons, Short Long Tons 0.8928571 Pounds, Troy Avoirdupois Ounces 13.16571 Tons, Short Metric Tons 0.90718474 Pounds, Troy Avoirdupois Drams 210.6514 Tons, Short Long Hundredweights 17.85714 Pounds, Troy Pennyweights 240 Tons, Short Short Hundredweights 20 Pounds, Troy Grams 373.2417216 Tons, Short Kilograms 907.18474 Pounds, Troy Grains 5760 Tons, Short Avoirdupois Pounds 2000 Quintals Metric Tons 0.1 Quintals Kilograms 100 Quintals Avoirdupois Pounds 220.46226 Scruples Troy Drams 0.333 Scruples Grams 1.2959782 Scruples Grains 20 Tons, Long (Deadweight) Metric Tons 1.016046909 Tons, Long (Deadweight) Short Tons 1.12 Tons, Long (Deadweight) Long Hundredweights 20 Tons, Long (Deadweight) Short Hundredweights 22.4 Tons, Long (Deadweight) Kilograms 1016.04691 Tons, Long (Deadweight) Avoirdupois Pounds 2240 Tons, Long (Deadweight) Avoirdupois Ounces 35840 Tons, Metric Long Tons 0.9842065 Tons, Metric Short Tons 1.1023113 Tons, Metric Quintals 10 Tons, Metric Long Hundredweights 19.68413072 Tons, Metric Short Hundredweights 22.04623 Tons, Metric Kilograms 1000 Tons, Metric Avoirdupois Pounds 2204.623 Tons, Metric Troy Ounces 32150.75  17  E. Density Conversions
To Convert To Multiply By Grains/imp. Gallon Parts/million 14.286 Grains/US gallon Parts/million 17.118 Grains/US gallon Pounds/million gal 142.86 Grams/cu. Cm Pounds/milfoot 3.405E07 Grams/cu. Cm Pounds/cu. in 0.03613 Grams/cu. Cm Pounds/cu. ft 62.43 Grams/liter Pounds/cu. ft 0.062427 Grams/liter Pounds/1000 gal 8.345 Grams/liter Grains/gal 58.417 Grams/liter Parts/million 1000 Kilograms/cu meter Pounds/milfoot 3.405E10 Kilograms/cu meter Pounds/cu in 0.00003613 Kilograms/cu meter Grams/cu cm 0.001 Kilograms/cu meter Pound/cu ft 0.06243 Milligrams/liter Parts/million 1 Pounds/cu ft Pounds/milfoot 5.456E09 Pounds/cu ft Pounds/cu in 0.0005787  18  Formulas and Conversions Formulas and Conversions To Convert To Multiply By Pounds/cu ft Grams/cu cm 0.01602 Pounds/cu ft Kgs/cu meter 16.02 Pounds/cu in Pounds/milfoot 0.000009425 Pounds/cu in Gms/cu cm 27.68 Pounds/cu in Pounds/cu ft 1728 Pounds/cu in Kgs/cu meter 27680 Substance Relative
Density Sand (dry) 1.42 Carbon (graphite) 2.3 Silicon 2.6 Carbon (charcoal) 1.8 Relative Density (Specific Gravity) Of Various Substances
Substance Relative
Density 10.57 Chromium F. Silver 6.5 Slate 2.12.8 Clay 1.9 Sodium 0.97
1.361.4
7.87 Water (fresh) 1.00 Coal Mica 2.9 Steel (mild) Water (sea average) 1.03 Cobalt 8.6 Nickel 8.6 Sulphur 2.07 Aluminum 2.56 Copper 8.77 Oil (linseed) 0.94 Tin 7.3 Antimony 6.70 Cork 0.24 Oil (olive) 0.92 Tungsten 19.1 Bismuth 9.80 Glass (crown) 2.5 Oil (petroleum) 0.760.86 Wood (ash) 0.75 Brass 8.40 Glass (flint) 3.5 Oil (turpentine) 0.87 Wood (beech) 0.70.8 Brick 2.1 Gold 19.3 Paraffin 0.86 Wood (ebony) 1.11.2 Calcium 1.58 Iron (cast) 7.21 Platinum 21.5 Wood (elm) 0.66 3.4 Iron (wrought) 7.78 Carbon (diamond)  19   20  Formulas and Conversions
Substance Relative
Density Wood (lignumvitae) 1.3 Formulas and Conversions Name Lower
Case Upper
Case Eta η Η Lead 11.4 Theta θ Θ Magnesium 1.74 Iota ι Ι Manganese 8.0 Kappa κ Κ Mercury 13.6 Lambda λ Λ Lead 11.4 Mu µ Μ Magnesium 1.74 Nu ν Ν Manganese 8.0 Xi ξ Ξ Omicron ο Ο Pi π Π Wood (oak) 0.71.0 Wood (pine) 0.56 Wood (teak) 0.8 Rho ρ Ρ
Zinc 7.0 Sigma σ and ς Σ Wood (oak) 0.71.0 Tau τ Τ Wood (pine) 0.56 Upsilon υ Υ Wood (teak) 0.8 Phi φ Φ Zinc 7.0 Chi χ Χ Mercury 13.6 Psi ψ Ψ Omega ω Ω G. Greek Alphabet
Name Lower
Case Upper
Case Alpha α Α Beta β Β Gamma γ Γ Delta δ ∆ Epsilon ε Ε Zeta ζ Ζ  21   22  Formulas and Conversions Formulas and Conversions Chapter 3 Multiply
by Into
Milli Into
Centi Into
Deci Into
MGL* Into
Deca Into
Hecto Into
Kilo System of Units To
convert
Hecto 105 104 103 102 101 1 101 To
convert
Deca 104 103 102 101 1 101 102 To
convert
MGL* 103 102 101 1 101 102 103 To
convert
Deci 102 101 1 101 102 103 104 To
convert
Centi 101 1 101 102 103 104 105 To
convert
Milli 1 101 102 103 104 105 106 The two most commonly used systems of units are as follows:
• SI
• Imperial
SI: The International System of Units (abbreviated "SI") is a scientific method of expressing
the magnitudes of physical quantities. This system was formerly called the meterkilogramsecond (MKS) system.
Imperial: A unit of measure for capacity officially adopted in the British Imperial System;
British units are both dry and wet Metric System
Exponent
value Numerical
equivalent Representation Example Tera 1012 1000000000000 T Thz (Tera
hertz) MGL = meter, gram, liter Giga 109 1000000000 G Ghz (Giga
hertz) Example: Mega 106 1000000 M Mhz (Mega
hertz) Unit
quantity 1 1 Micro 106 0.001 µ µF (Micro
farads) Nano 109 0.000001 n nF (Nano
farads) p pF (Pico
farads) Pico 12 10 hz (hertz)
F (Farads) To convert Kilogram Into Milligram → (1 Kilo X 106 ) Milligrams Physical constants Conversion Chart Symbolic
Representation Numerical Equivalent Avogadro's number N 6.023 x 1026 /(kg mol) Bohr magneton B 9.27 x 1024 Am 252 Boltzmann's constant k 1.380 x 1023 J/k StefanBoltzmann constant 0.000000000001 Name d 5.67 x 108 W/(m2K4) Multiply
by Into
Milli Into
Centi Into
Deci Into
MGL* Into
Deca Into
Hecto Into
Kilo Characteristic impedance of free
space Zo (µo/Eo)1/2=120ΠΩ To
convert
Kilo 106 105 104 103 102 101 1 Electron volt eV 1.602 x 1019 J Electron charge e 1.602 x 1019 C  23   24  Formulas and Conversions Formulas and Conversions Name Symbolic
Representation Numerical Equivalent Name Symbolic
Representation Numerical Equivalent Electronic rest mass me 9.109 x 1031 kg Acceleration due to gravity on
Earth g 9.80 m s2 Electronic charge to mass ratio e/me 1.759 x 1011 C/kg Acceleration due to gravity on the
Moon gM 1.62 m s2 Faraday constant F 9.65 x 107 C/(kg mol) Radius of the Earth RE 6.37 x 106 m Permeability of free space µ0 4Π x 107 H/m Mass of the Earth ME 5.98 x 1024 kg Permittivity of free space Eo 8.85 x 1012 F/m Radius of the Sun RS 6.96 x 108 m Planck's constant h 6.626 x 1034 J s Mass of the Sun MS 1.99 x 1030 kg Radius of the Moon RM 1.74 x 106 m Proton mass mp 1.672 x 1027 kg Mass of the Moon MM 7.35 x 1022 kg Proton to electron mass ratio mp/me 1835.6 EarthMoon distance  3.84 x 108 m Standard gravitational
acceleration g 9.80665 m/s2, 9.80665 N/kg EarthSun distance  1.50 x 1011 m Speed of light in air c 3.00 x 108 m s1 Universal constant of gravitation G 6.67 x 1011 N m2/kg2 Electron charge e 1.60 x 1019 C Universal gas constant Ro 8.314 kJ/(kg mol K) Mass of electron me 9.11 x 1031 kg 2.9979 x 10 m/s Planck's constant h 6.63 x 1034 J s C 5/9(0F  32) Universal gravitational constant G 6.67 x 1011 N m2 kg2 K 5/9(0F + 459.67), 5/90R, 0C +
273.15 Electron volt 1 eV 1.60 x 1019 J Mass of proton mp 1.67 x 1027 kg Acceleration due to gravity on
Earth g 9.80 m s2 Acceleration due to gravity on the
Moon gM 1.62 m s2 Ton 1 ton 1.00 x 103 kg Velocity of light in vacuum
Temperature
Temperature C
0 8 Speed of light in air c 3.00 x 108 m s1 Electron charge e 1.60 x 1019 C Mass of electron me 9.11 x 1031 kg Planck's constant h 6.63 x 1034 J s Universal gravitational constant G 6.67 x 1011 N m2 kg2 Electron volt 1 eV 1.60 x 1019 J Mass of proton mp 1.67 x 1027 kg  25   26  Formulas and Conversions Formulas and Conversions Identity a+0 = 0+a = a Inverse a + (a) = 0, a(1/a) = 1 Cancellation If a+x=a+y, then x=y Zerofactor a0 = 0a = 0 Negation (a) = a, (a)b= a(b) = (ab), (a)(b) = ab Chapter 4
General Mathematical Formulae
4.1 Algebra
A. Expansion Formulae
Square of summation • (x + y) 2 = x2 + 2xy + y2 Square of difference Algebraic Combinations • (x – y) 2 = x2 – 2xy + y2 Factors with a common denominator can be expanded:
a+b a b
=+
c
cc Difference of squares • x2 – y2 = (x + y) (x – y)
Cube of summation • (x + y) 3 = x3 + 3x2y + 3xy2 + y3 Fractions can be added by finding a common denominator:
a b ad + bc
+=
cd
cd Summation of two cubes • x3 + y3 = (x + y) (x2  xy + y2) Products of fractions can be carried out directly:
a b ab
×=
c d cd Cube of difference • (x – y) 3 = x3 – 3x2y + 3xy2 – y3
Difference of two cubes • x3 – y3 = (x – y) (x2 + xy + y2) Quotients of fractions can be evaluated by inverting and multiplying:
a
b = a × d = ad
c
b c bc
d B. Quadratic Equation • If ax2 + bx + c = 0,
Then x = −b ± b 2 − 4ac
2a Radical Combinations The basic algebraic properties of real numbers a, b and c are: a + b = b + a, ab = ba Associative a = a1/ n n a
=
b a + b and ab are real numbers Commutative ab = n a n b Description Closure n n Property (a+b) + c = a + (b+c), (ab)c = a(bc)
(a+b)c = ac+bc b am = a n nm  27  a n m
n Distributive n a = mn a  28  2 (L + B) Circumference
/ Perimeter s1 + s2 + s3
where s1, s2, s3
are the 3 sides
of the triangle Rectangle Item Triangle s1 + s2 + s3 4s Square Right
triangle Circumference
/ Perimeter Item 4.2 Geometry 2 1
× B× H
2 1
× B× H
2 Area  29  NA NA Surface Area  30  NA NA Surface Area NA NA Volume NA NA Volume Formulas and Conversions (Length)(Breadth)
= L·B s Area Formulas and Conversions Figure Figure Circle C = 2πr
C = πd where Ө and Φ are
the 2 base angles Circumference
/ Perimeter Item Trapezoid 3s
where s is the
length of each
side s1 + s2 + s3 Circumference
/ Perimeter Equilateral
triangle Generic
triangle Item A = πr2 ⎛a +b⎞
A=⎜
⎟h
⎝2⎠ Area 1
bh
2 a+b+c
2 A= s= where  31  NA NA Surface Area  32  NA NA Surface Area NA NA Volume NA NA Volume Formulas and Conversions s ( s − a)( s − b)( s − c) Area Formulas and Conversions Figure Figure Circumference
/ Perimeter Sum of all sides 6s Trapezoid Hexagon where D and d
are the two axis Item Ellipse 2r + (arc
length) Circle
Sector (1/4)·D·d·∏ Circumference
/ Perimeter Item 4 π
Dd 2 θ °r
2 1
(b1 + b2 )h
2 A = 2.6s2
Where s is the
length of 1 side A= Area  33  NA NA Surface Area  34  NA NA Surface Area NA NA Volume NA NA Volume Formulas and Conversions D is the larger
radius and d is the
smaller radius A= A= A= arc × r
2
θ°
A=
× πr 2
360 Area Formulas and Conversions Figure Figure NA Area NA Circumference
/ Perimeter NA NA Cube Item Rectangular
solid Right
cylinder NA NA NA Volume 2  35  6s s3 NA Surface Area  36  S = 2πrh +
2πr2 2 l h + 2wh + 2 Surface Area V = πr2h l ×w ×h Volume Formulas and Conversions A = 4.83 s2
Where s is the
length of 1 side 8s Octagon Area Circumference
/ Perimeter Item Formulas and Conversions Figure Figure NA Area NA Circumference
/ Perimeter NA NA Pyramid Item Rectangular
prism Cone NA NA NA NA Sphere Area Circumference
/ Perimeter Item perpendicular
height 1
base area·
3  38  pi·r(r+sh) 2lh+2lw+2wh Surface Area 43
πr
3 Volume 12
πr h
3 V = lwh Volume Formulas and Conversions  37  ½.perimeter·
slant height +
B S = 4πr2 Surface Area Formulas and Conversions Figure Figure Formulas and Conversions Formulas and Conversions
Tangent, Secant and CoSecant 4.3 Trigonometry
A. Pythagoras' Law sin θ
cosθ
1
secθ =
cosθ
1
cscθ =
sin θ
tan θ = c2 = a2 + b2 B. Basic Ratios • Sin θ = a/c
• Cos θ = b/c
• Tan θ = a/b
• Cosec θ = c/a
• Sec θ = c/b
• Cot θ = b/a c a
θ
b C. Trigonometric Function Values
Euler’s Representation e jθ = cos(θ ) + j sin(θ ) Degrees versus Radians • A circle in degree contains 360 degrees
• A circle in radians contains 2π radians e− jθ = cos(θ ) − j sin(θ ) e jnθ = cos(nθ ) + j sin(nθ )
hypotenuse θ cosθ = e jθ + e − jθ
2 sin θ = opposite e jθ − e − jθ
2j adjacent 4.4 Logarithm Sine, Cosine and Tangent sin θ = opposite
hypotenus cosθ = adjacent
hypotenus Sine, Cosine and the Pythagorean Triangle [sin θ ] + [cosθ ]
2 2 = sin 2 θ + cos 2 θ = 1 tan θ = opposite
adjacent Definition The logarithm of a number to a particular base is the power (or index) to which that
base must be raised to obtain the number.
The number 8 written in index form as 8 = 23
The equation can be rewritten in logarithm form as log 2 8 = 3
Logarithm laws The logarithm laws are obtained from the index laws and are:
• loga x + loga y = loga xy  39   40  Formulas and Conversions
• loga x – loga y = loga (x/y) Formulas and Conversions 4.5 Exponents
Summary of the Laws of Exponents • loga xy = y loga x Let c, d, r, and s be any real numbers. • loga (1/x) = loga x c r ⋅ c s = c r+s cr
= c r−s , c ≠ 0
cs • loga a = 1
• a (log a x ) cr
⎛c⎞
⎜ ⎟ = r , d ≠0
d⎠
d
⎝ ( c r ) s = c r ⋅s • loga 1 = 0 (c ⋅ d ) r = c r ⋅ d r c −r = r =x Note: It is not possible to have the logarithm of a negative number. All logarithms must
have the same base.
Euler Relationship The trigonometric functions are related to a complex exponential by the Euler
relationship:
e jx = cos x + j sin x
− jx e = cos x − j sin x
From these relationships the trig functions can be expressed in terms of the complex
exponential: Basic Combinations
Since the raising of a number n to a power p may be defined as multiplying
n times itself p times, it follows that n p1 + p 2 = n p1 n p 2
The rule for raising a power to a power can also be deduced
(na)b = nab
(ab)n = anbn
am/an = amn e jx + e − jx
cos x =
2
e jx − e − jx
sin x =
2 where a not equal to zero 4.6 Complex Numbers
A complex number is a number with a real and an imaginary part, usually
expressed in Cartesian form Hyperbolic Functions The hyperbolic functions can be defined in terms of exponentials.
Hyperbolic sine = sinh x = 1
cr e x − e− x
2 Hyperbolic cosine = cosh x = e x + e− x
2 Hyperbolic tangent = tanh x = sinh x e x − e − x
=
cosh x e x + e x a + jb where j = √1 and j · j = 1
Complex numbers can also be expressed in polar form Aejθ where A = √a2 +b2 and θ = tan1 (b/a)
The polar form can also be expressed in terms of trigonometric functions using the Euler
relationship
ejθ = cos θ + j sin θ
Euler Relationship
The trigonometric functions are related to a complex exponential by the
Euler relationship
ejx = cos x + j sin x  41   42  Formulas and Conversions Formulas and Conversions Chapter 5
ejθ = cos x  j sin x Engineering Concepts and Formulae From these relationships the trigonometric functions can be expressed in terms of the
complex exponential: 5.1 Electricity
e jx + e − jx
cos x =
2
e jx − e − jx
sin x =
2 Ohm's Law
I= This relationship is useful for expressing complex numbers in polar form, as
well as many other applications.
Polar Form, Complex Numbers
The standard form of a complex number is
a + jb where j = √1
But this can be shown to be equivalent to the form Aejθ where A = √a2 +b2 and θ = tan1 (b/a)
which is called the polar form of a complex number. The equivalence can be shown by
using the Euler relationship for complex exponentials. Ae jθ b⎤
b⎤
⎡
⎡
= a + b (cos ⎢ tan −1 ⎥ + j sin ⎢ tan −1 ⎥ )
a⎦
a⎦
⎣
2 2 Ae jθ = a 2 + b 2 ( a
a2 + b2 +j b
a2 + b2 ) = a + jb V
R Or
V = IR
Where
I = current (amperes)
E = electromotive force (volts)
R = resistance (ohms) Temperature correction
Rt = Ro (1 + αt)
Where
Ro = resistance at 0ºC (.)
Rt = resistance at tºC (.)
α = temperature coefficient which has an average value for copper of 0.004
28 (Ω/Ω ºC) R2 = R1 (1 + αt2 )
(1 + αt1 ) Where R1 = resistance at t1
R2 = resistance at t2
Values of
alpha
Copper 0.00428 Platinum 0.00358 Nickel 0.00672 Tungsten  43  Ω/Ω ºC 0.00450  44  Formulas and Conversions
Aluminum Current, I = Formulas and Conversions 0.0040 Where EG = generated e.m.f.
EB = generated back e.m.f.
Ia = armature current
Ra = armature resistance nqvtA
= nqvA
t Alternating Current Conductor Resistivity
R= ρL
a Where
ρ = specific resistance (or resistivity) (ohm meters, Ωm)
L = length (meters)
a = area of crosssection (square meters)
Quantity Equation Resistance R of a uniform
conductor L
R=ρ
A Resistors in series, Rs Rs = R1 + R2 + R3 Resistors in parallel, R p 1
1
1
1
=
+
+
R p R1 R2 R3 Power dissipated in resistor: Potential drop across R P = VI = I 2 R = Slip of Induction Motor
[(Slip speed of the field – Speed of the rotor) / Speed of the Field] × 100 V2
R V=IR Dynamo Formulae 2ϕNpZ
Average e.m.f. generated in each conductor =
60c
Where
Z = total number of armature conductors
c = number of parallel paths through winding between positive and negative brushes
Where c = 2 (wave winding), c = 2p (lap winding)
Φ = useful flux per pole (webers), entering or leaving the armature
p = number of pairs of poles
N = speed (revolutions per minute)
Generator Terminal volts = EG – IaRa
Motor Terminal volts = EB + IaRa  45  RMS value of sine curve = 0.707 of maximum value
Mean Value of Sine wave = 0.637 of maximum value
Form factor = RMS value / Mean Value = 1.11
pN
cycles per second
Frequency of Alternator =
60
Where p is number of pairs of poles
N is the rotational speed in r/min Inductors and Inductive Reactance
Physical Quantity
Inductors and Inductance Equation
VL = L di
dt Inductors in Series: LT = L1 + L2 + L3 + . . . . Inductor in Parallel: 1
1
1
1
=
+
+
+ .....
L T L1 L 2 L 3 Current build up
(switch initially closed after having
been opened) At v L ( t) = E e t τ − t v R ( t) = E(1  e τ )
i(t) = E R
L
τ=
R
Current decay
(switch moved to a new position)  (1 − e   t τ ) t i(t) = I o e τ ′
vR(t) = R i(t)
vL(t) = − RT i(t)  46  Formulas and Conversions
Physical Quantity Formulas and Conversions Equation Quantity L
τ' =
RT Current Divider Rule Equation Alternating Current f = 1/T
ϖ=2πf Two impedance values in
parallel Complex Numbers: C=a+jb
C = M cos θ + j M sin θ I x = IT
ZT = Capacitance M = a 2 + b2
⎛b⎞
θ = tan 1 ⎜ ⎟
⎝a⎠ Capacitors Polar form: C=M∠θ Capacitor in Series Inductive Reactance XL = ω L 1
1
1
1
=
+
+
+ .....
C T C1 C 2 C 3 Capacitive Reactance XC = 1 / (ω C) Capacitors in Parallel C T = C1 + C 2 + C 3 + ..... Resistance R Charging a Capacitor Impedance Resistance: ZR = R ∠0°
Inductance: ZL = XL ∠90° = ω L ∠90°
Capacitance: ZC = XC ∠90° = 1 / (ωC)
∠90° C= Q
V [F] (Farads) t i(t) = E  RC
e
R v R ( t) = E e  t
RC v C ( t) = E(1  e  t
RC ) τ = RC
Quantity Equation Ohm’s Law for AC Discharging a
Capacitor V=IZ
v(t) = Vm sin (ω t ± φ)
i(t) = Im sin (ω t ± φ) Phasor Notation V = Vrms ∠ φ
V = Vm ∠ φ Components in Series ZT = Z1 + Z2 + Z3 + .
. Vx = VT Components in Parallel Vo  τ ′
e
R v R ( t) = − Vo e Time Domain Voltage Divider Rule t i(t) = − Zx
ZT   t τ′ t v C ( t) = Vo e τ ′
τ' = RTC Quantity Capacitance Equation C= Q
V 1
1
1
1
=
+
+
+ ...
Z T Z1 Z 2 Z 3  47   48  ZT
Zx Z1 Z 2
Z1 + Z 2 Formulas and Conversions
Quantity Equation Capacitance of a
Parallelplate Capacitor C= Isolated Sphere Current in AC Circuit
RMS Current
In Cartesian
form εA
d E= Formulas and Conversions V
d C = 4πεr I= V
1 ⎞⎤
⎡
⎛
⋅ ⎢ R − j ⎜ ωL −
⎟
2
ωC ⎠ ⎥
⎡2 ⎛
⎝
⎦
1⎞⎤ ⎣
⎟⎥
⎢ R + ⎜ ωL −
ωC ⎠ ⎥
⎝
⎢
⎣
⎦ Amperes
In polar form V I= 2 1⎞
⎛
[ R + ⎜ ωL −
⎟]
ωC ⎠
⎝ ∠ − φ s Amperes 2 Capacitors in parallel Capacitors in series Energy stored in a
charged capacitor C = C1 + C2 + C3 Modulus Q 2 1
1
W=
= CV 2 = QV
2C 2
2 W= Q If the capacitor is
connected to a battery W= 1
CV 2
2 Charging a capacitor
Discharging a capacitor R ⎢
⎣ 1
1
1
1
=
+
+
C C1 C 2 C 3 If the capacitor is
isolated For R C circuits ⎡
⎢ ωL − where φ s = tan −1 ⎢ I= V
1⎞
⎛
R + ⎜ ωL −
⎟
ωC ⎠
⎝ 2 1⎤ ωC ⎥
⎥
⎥
⎦ Amperes 2 2 2C Q = Qo (1  et/RC);
V = Vo
(1  et/RC) Complex Impedance
In Cartesian
form
In polar form Q = Qo e t/RC
V = Vo et/RC • If the capacitor is isolated, the presence of the dielectric decreases the potential
difference between the plates
• If the capacitor is connected to a battery, the presence of the dielectric increases the
charge stored in the capacitor.
• The introduction of the dielectric increases the capacitance of the capacitor  49  Modulus 1⎞
⎛
Z = R + j ⎜ ωL −
⎟ Ohms
ωC ⎠
⎝
2 1⎞
⎛
Z = R 2 + ⎜ ωL −
⎟ ∠φ s Ohms
ωC ⎠
⎝
1⎤
⎡
ωL −
−1 ⎢
ωC ⎥
Where φ s = tan ⎢
⎥
R
⎥
⎢
⎦
⎣
2 1⎞
⎛
Z = [ R 2 + ⎜ ωL −
⎟ ] Ohms
ωC ⎠
⎝  50  Formulas and Conversions Formulas and Conversions Power dissipation Three Phase Alternators Average power, P = VI cos φ Watts Power dissipation in a
resistor P = I R Watts
2 Rectification
Controlled half wave
rectifier Average DC voltage =
Volts Controlled full wave
rectifier Average DC voltage =
Volts Vm
(1 + cos α )
2π
Vm π (1 + cos α ) Star connected
Line voltage = 3 · phase voltage
Line current = phase current
Delta connected
Line voltage = phase voltage
Line current = 3 · phase current
Three phase power
P = 3 EL IL cos Φ
EL = line voltage
IL = line current
cos Φ = power factor
Electrostatics
Quantity Power Factor Instantaneous current,
DC
Power
AC
Power Pdc = VI = I 2 R = Equation I= 2 V
R Permittivity of free space Pac = Re(V .I ) = VI cos φ Quantity The mean power = P = Irms Vrms = Irms2 R Inductance The instantaneous power = (Io sin wt) (Vo sin (wt +
π) The mean power P =0 Capacitance P =0 Formula for a.c.
power The mean power = P = Irms Vrms cos φ = 1
CV 2 Joules
2 The instantaneous power = (Io sin (wt + π/2)) (Vo sin
wt ) The mean power 10 −9
= 8.85 × 10 −12 Farads
36π Equation Resistance ε0 = (meters)1
Energy stored in a
capacitor Power in ac circuits dv
dq
Amperes
=C
dt
dt Quantity Equation Coulomb’s law F =k Electric fields E= Due to a conducting sphere carrying charge
Q Inside the sphere  52  F
q E= Due to a point charge  51  Q1Q2
r2 Q
4πε o r 2 E=0 Formulas and Conversions
Quantity Formulas and Conversions Equation Outside the sphere Quantity E= Just outside a uniformly charged conducting
sphere or plate Q
4πε o r 2 Relation between E and V E= σ
εo Equation For uniform electric field • An electric field E is a vector
• The electric field strength is directly proportional to the number of electric field lines
per unit crosssectional area,
• The electric field at the surface of a conductor is perpendicular to the surface.
• The electric field is zero inside a conductor. Quantity Equation Suppose a point charge Q is at A. The work done in
bringing a charge q from infinity to some point a distance
r from A is
Electric potential Qq
4πε o r W= V= Due to a point charge W
q V= Due to a conducting sphere, of radius a, carrying charge
Q:
Inside the sphere
Outside the sphere Q
4πε o r V= Q
4πε o a V= E=− E= Physical Quantity Equation Magnetic flux density (also called the Bfield) is defined as the force acting per unit
current length. B= Force on a currentcarrying conductor in a
magnetic field Force on a moving charged particle in a
magnetic field
Circulating Charges Work done in bringing charge q from A of potential VA to
point B of potential VB W = q (VB – VA) F = I l BF = I l · B
And Magnitude of F = F = I l B
sin θ
F=q v · B mv 2
r Calculation of magnetic flux density
Physical Quantity Equation Magnetic fields around a long straight wire
carrying current I B= µo I
2πa where a = perp. distance from a
very long straight wire.
Magnetic fields inside a long solenoid,
carrying current I: B = µo n I, where n = number of
turns per unit length. Hall effect
At equilibrium Q The current in a material is given by  53  F
Il qvB = 4πε o r U = qV V
d Magnetostatics Q If the potential at a point is V, then the potential energy
of a charge q at that point is dV
dx VH
= QvB and
d I = nQAv  54  VH = B v d Formulas and Conversions Formulas and Conversions Physical Quantity Equation Quantity The forces between two currentcarrying
conductors µIIl
F21 = o 1 2
2πa Energy stored in an inductor: Equation Transformers:
Physical Quantity Equation The torque on a rectangular coil in a magnetic
field T = F b sin θ
= N I l B b sinθ
= N I A B sinθ If the coil is in a radial field and the plane of the
coil is always parallel to the field, then T = N I A B sin θ
= N I A B sin 90o
=NIAB Magnetic flux φ φ = B A cos θ
and
Fluxlinkage = Current Sensitivity SI = θ
I = Lenz's law
The direction of the induced e.m.f. is such that it tends to
oppose the fluxchange causing it, and does oppose it if
induced current flows. I= When a great load (or smaller
resistance) is connected to
the secondary coil, the flux in
the core decreases. The
e.m.f., εp, in the primary coil
falls. Vp εp = I R; I = d
φ
dt Equation Power E.m.f. induced in a straight conductor
E.m.f. induced between the center and the rim of a spinning
disc ε = B πr2f E.m.f. induced in a rotating coil Ε = N A B w sin
wt P= W
= VI
t I= q
t ε =BLv ε
dI / dt Nφ =LI  55  Electric current
Work W = qV Ohm’s Law V = IR Resistances in Series Equation L=− R Kirchoff's second law (Loop Theorem)
The net e.m.f. round a circuit is equal to the sum of the p.d.s round the loop. Physical Quantity Selfinduction VP − ε p Kirchoff's first law (Junction Theorem)
At a junction, the total current entering the junction is equal to the total
current leaving the junction. EMF Equations Quantity E
(1 − e − Rt / L )
R Kirchoff’s laws NAB
c ε = −N VS N S
=
VP N P The L R (d.c.) circuit: Nφ 12
LI
2 U= R T = R1 + R 2 K Resistances in Parallel 1
1
1
=
+
K
R T R1 R 2 Magnetic flux Φ = BA  56  Formulas and Conversions Formulas and Conversions Impulse = force · time = change of momentum
Ft=mv–mu (Φ 2 − Φ 1 )
t
emf = l v B Electromagnetic
induction Emf = − N Magnetic force F=I l B Transformer turns ratio Vs = Vp Newton's third law of motion When two objects interact, they exert equal and opposite forces on one another.
"Thirdlaw pair" of forces act on two different bodies.
Universal Law
F = Gmsmp/d2 Ns
Np ms is the mass of the sun.
mp is the mass of the planet.
The Universal law and the second law must be consistent Electromagnetic spectrum Newton’s Laws of Motion and Their Applications Wavelength
102 λ (m) 10 101 102 103 104 105 106 107 108 109 1010 1011 1 radio frequencies Physical Quantity v av = Average velocity Xrays Area of
Spectrum Equations Acceleration
visible microwaves ultraviolet
radiation s v+u
=
t
2 a= vu
t Momentum
Force f(Hz) 6 10 7 10 10 8 9 10 10 10 10 11 12 13 10 10 10
Frequency 14 15 10 16 10 10 17 18 10 19 10 10 20 Note: 1. Shaded areas represent regions of overlap.
2. Gamma rays and Xrays occupy a common region. 5.2 Applied Mechanics F = ma Weight gamma rays
infrared radiation p = mv weight = mg Work done W = Fs Kinetic energy 1
E k = 2 mv 2 Gravitational potential energy E p = mgh Equations of motion a= v−u
;
t 1
s = ut + 2 at 2 ; 5.2.1 Newton's laws of motion
Centripetal acceleration a= Newton' first law of motion The inertia of a body is the reluctance of the body to change its state of rest or motion.
Mass is a measure of inertia. v2
r F = ma = Newton’s Law of Universal
Gravitation Newton's second law of motion
mvmu
;
F=
∆t Centripetal force F=G F=ma  57   58  mv 2
r m1m2
r2 v 2 = u 2 + 2as Formulas and Conversions
Physical Quantity Formulas and Conversions Equations Gravitational field strength g=G Physical Quantity
Moment of a force M
r2 Equations Stress 1 ft
m
= 3.28 2
s2
s Acceleration due to gravity, g is 9.81 m/s2 5.2.2 Linear Velocity and Acceleration
Quantity Equations If u initial velocity and v final velocity,
then displacement s, M = rF
∑M = 0 Principle of
moments Conversion: ⎛v+u⎞
s=⎜
⎟
⎝2⎠ Stress = Young’s Modulus If t is the elapsed time Strain = Strain F
A
∆l
l If a is the acceleration Y= F/A
∆ l/ l s = ut + 12
at
2 v 2 = u 2 + 2as Angular Velocity and Acceleration
Quantity Vector: a property described by a magnitude and a direction θ= • ω angular velocity (radians/s); ω1 = initial, ω2 = final Velocity: vector property equal to displacement / time
The magnitude of velocity may be referred to as speed
In SI the basic unit is m/s, in Imperial ft/s
Other common units are km/h, mi/h
Conversions:
1m/s = 3.28 ft/s
1km/h = 0.621 mi/h ω1 + ω 2
2 ×t 1
2 θ = ω 1t + αt 2 α angular acceleration
(radians/s2) ω 2 2 = ω 1 2 + 2αθ Linear displacement s=rθ Linear velocity v=rω Linear, or tangential
acceleration Speed of sound in dry air is 331 m/s at 0°C and increases by about 0.61 m/s for each °C
rise.
Speed of light in vaccum equals 3 x 108m/s Equations θ angular displacement
(radians) Scalar: a property described by a magnitude only aT = r α Tangential, Centripetal and Total Acceleration 2 In SI the basic unit is m/s
In Imperial ft/s2  59  Quantity Equations Tangential acceleration aT is due to angular acceleration
α Acceleration: vector property equal to change in velocity time. aT = rα  60  Formulas and Conversions Formulas and Conversions Centripetal (Centrifugal) acceleration ac is due to change
in direction only ac = v2/r = r ω2 Total acceleration, a, of a rotating point experiencing
angular acceleration is the vector sum of aT and ac a = aT + ac Kinetic Energy 1
mk 2ω 2
2
Where k is radius of gyration, ω is angular velocity in rad/s
ER = Kinetic Energy of Rotation Er = 5.2.3 Force
Vector quantity, a push or pull which changes the shape and/or motion of an object
In SI the unit of force is the newton, N, defined as a kg m
In Imperial the unit of force is the pound lb
Conversion: 9.81 N = 2.2 lb
Weight The gravitational force of attraction between a mass, m, and the mass of the Earth
In SI weight can be calculated from Weight = F = mg, where g = 9.81 m/s2
In Imperial, the mass of an object (rarely used), in slugs, can be calculated from the
known weight in pounds
weight
m=
g
ft
g = 32.2 2
s
Torque Equation T = I α where T is the acceleration torque in Nm, I is the moment of inertia in kg m2 and
α is the angular acceleration in radians/s2
Momentum Vector quantity, symbol p,
p = mv [Imperial p = (w/g)v, where w is weight]
in SI unit is kgm / s
Work Scalar quantity, equal to the (vector) product of a force and the displacement of an
object. In simple systems, where W is work, F force and s distance
W=Fs
In SI the unit of work is the joule, J, or kilojoule, kJ
1 J = 1 Nm
In Imperial the unit of work is the ftlb
Energy Energy is the ability to do work, the units are the same as for work; J, kJ, and ftlb 1
Iω 2
2 Where I = mk2 is the moment of inertia 5.2.4 Centripetal (Centrifugal) Force
mv 2
r
Where r is the radius
Where ω is angular velocity in rad/s
Fc = Potential Energy
Quantity Equation Energy due to position in a force
field, such as gravity Ep = m g h In Imperial this is usually expressed Ep = w h
Where w is weight, and h is height
above some specified datum Thermal Energy In SI the common units of thermal energy are J, and kJ, (and kJ/kg for specific
quantities)
In Imperial, the units of thermal energy are British Thermal Units (Btu)
Conversions 1 Btu = 1055 J
1 Btu = 778 ftlb
Electrical Energy In SI the units of electrical energy are J, kJ and kilowatt hours kWh. In Imperial, the unit
of electrical energy is the kWh
Conversions 1 kWh = 3600 kJ
1 kWh = 3412 Btu = 2.66 x 106 ftlb
Power  61   62  Formulas and Conversions A scalar quantity, equal to the rate of doing work
In SI the unit is the Watt W (or kW)
J
1W = 1
s
In Imperial, the units are:
Mechanical Power – (ft – lb) / s, horsepower h.p.
Thermal Power – Btu / s
Electrical Power  W, kW, or h.p. Formulas and Conversions • 1 atmosphere (atm) = 101.3 kPa = 14.7 psi
Simple Harmonic Motion Velocity of P = ω R 2 − x 2 m
s 5.2.5 Stress, Strain And Modulus Of Elasticity
Young’s modulus and the breaking stress for selected materials Conversions 746W = 1h. p.
1h. p. = 550 Material ft − lb
s Breaking stress
x 108 Pa Aluminium 0.70 2.4 Copper 1.16 4.9 Brass Btu
1kW = 0.948
s 0.90 4.7 Iron (wrought) In Imperial, the basic unit is the pound per square inch, psi
Atmospheric Pressure At sea level atmospheric pressure equals 101.3 kPa or 14.7 psi
Pressure Conversions 1 psi = 6.895 kPa
Pressure may be expressed in standard units, or in units of static fluid head, in both SI
and Imperial systems
Common equivalencies are:
• 1 kPa = 0.294 in. mercury = 7.5 mm mercury
• 1 kPa = 4.02 in. water = 102 mm water
• 1 psi = 2.03 in. mercury = 51.7 mm mercury
• 1 psi = 27.7 in. water = 703 mm water
• 1 m H2O = 9.81 kPa
Other pressure unit conversions:
• 1 bar = 14.5 psi = 100 kPa
• 1 kg/cm2 = 98.1 kPa = 14.2 psi = 0.981 bar  63  11.0 Glass 0.55 10 4.10 20 Bone N
m2 3.0 2.10 Tungsten A vector quantity, force per unit area
In SI the basic units of pressure are pascals Pa and kPa 1.93 Mild steel Pressure 1Pa = 1 Young modulus
x 1011 Pa 0.17 1.8 5.3 Thermodynamics
5.3.1 Laws of Thermodynamics
• W = P∆V
• ∆U = Q – W
• W= nRT lnVf/Vi
• Q = Cn∆T
• Cv= 3/2R
• Cp= 5/2R
• Cp/Cv = γ= 5/3
• e = 1 – Qc/Qh = W/Qh
• ec = 1 – Tc/Th
• COP = Qc/W (refrigerators)
• COP = Qh /W (heat pumps)
• Wmax= (1Tc/Th)Qh
• ∆S = Q/T  64  Formulas and Conversions Formulas and Conversions • ∑ Fy = 0
• ∑τ = 0 5.3.2 Momentum
• p = mv
• ∑F = ∆p/∆t (any axis) 5.3.8 Gravity
5.3.3 Impulse
I = Fav∆ t = mvf – mvi 5.3.4 Elastic and Inelastic collision • miv1i + m2v2i = m1v1f + m2v2f
• (½) miv1i2 + (½) m2v2i2 = ½ m1v1f2 + ½ m2v2f2
• miv1i + m2v2i = (m1 + m2)vf 5.3.5 Center of Mass
• xcm = ∑mx/M
• Vcm = ∑mv/M
• Acm = ∑ma/M
• MAcm = Fnet • F = Gm1m2/r2
• T = 2π / √r3 /GMs
• G = 6.67 x 1011Nm2/kg2
• g = GME / R2E
• PE =  Gm1m2 / r
• ve = √2GME / RE
• vs = √GME / r
• ME = 5.97 x 1024 kg
• RE = 6.37 x 106 m 5.3.9 Vibrations & Waves
• F = kx
• PEs = ½kx2
• x = Acosθ = Acos(ωt)
• v = Aωsin(ωt)
• a = Aω2cos(ωt)
• ω = √k / m
•f = 1 / T
• T = 2π√m / k
• E = ½kA2
• T = 2π√L / g
• vmax = Aω
• amax = Aω2
•v = λ f
v = √FT/µ
• µ = m/L
• I = P/A
• β = 10log(I/Io)
• Io = 1 x 1012 W/m2
• f’ = f[(1 ± v0/v)/(1 m vs/v)]
• Surface area of the sphere = 4πr2
• Speed of sound waves = 343 m/s 5.3.6 Angular Motion
• s = rθ
• vt = rω
• at = rα
• ac = vt2/r = rω2
• ω = 2π/T
• 1 rev = 2π rad = 360o For constant α
• ω = ω o + αt
• ω2 = ωo2 +2αθ
• θ = ω o t + ½α t 2
• θ = (ωo + ω)·t/2
• I = ∑mr2
• KER = ½Iω2
• τ = rF
• ∑τ = Iα
• WR = τθ
• L = Iω
• ∑τ = Iα
• WR = τθ
• L = Iω
• Li = Lf 5.3.10 Standing Waves
• fn = nf1
• fn = nv/2L (air column, string fixed both ends) n = 1,2,3,4…….
• fn = nv/4L (open at one end) n = 1,3,5,7……… 5.3.11 Beats 5.3.7 Conditions of Equilibrium • fbeats =  f1 – f2  • ∑ Fx = 0 • Fluids
 65   66  Formulas and Conversions • ρ = m/V
• P = F/A
• P2 = P1 + ρgh
• Patm = 1.01 x 105Pa = 14.7 lb/in2
• FB = ρfVg = Wf (weight of the displaced fluid)
• ρo/ρf = Vf /Vo (floating object)
3
• ρwater = 1000 kg/m • Wa=WFB Equation of Continuity: Av = constant
Bernoulli’s equation: P + ½ ρv2 + ρgy = 0 5.3.12 Temperature and Heat
• TF= 9/5TC+32
• TC= 5/9(TF32)
• ∆TF = 9/5∆TC
• T= TC+273.15
• ρ= m/v
• ∆L = αLo∆T
• ∆A = γAo∆T
• ∆V = βVo∆T β=3α
• Q = mc∆T
• Q = mL
• 1 kcal = 4186 J
• Heat Loss = Heat Gain
• Q = (kA∆T)t/L,
• H = Q/t =(kA∆T)/L
• Q = eσT4At
• P = Q/t
• P = σAeT4
• P net= σAe(T4TS4)
• σ = 5.67 × 108 W/m 2K4 Formulas and Conversions 5.3.14 Elastic Deformation • P = F/A
• Y = FLo/A∆L
• S = Fh/A∆x
• B = –Vo∆F / A∆V
• Volume of the sphere = 4πr3/3
• 1 atm = 1.01 × 105 Pa 5.3.15 Temperature Scales
• °C = 5/9 (°F – 32)
• °F = 5/9 (°C + 32)
• °R = °F + 460 (R Rankine)
• K = °C + 273 (K Kelvin) 5.3.16 Sensible Heat Equation
• Q=mc∆T
• M=mass
• C=specific heat
• ∆T=temperature chance 5.3.17 Latent Heat
• Latent heat of fusion of ice = 335 kJ/kg
• Latent heat of steam from and at 100°C = 2257 kJ/kg
• 1 tonne of refrigeration = 335 000 kJ/day = 233 kJ/min 5.3.18 Gas Laws
Boyle’s Law When gas temperature is constant
PV = constant or
P1V1 = P2V2
Where P is absolute pressure and V is volume
Charles’ Law 5.3.13 Ideal Gases When gas pressure is constant,
V
= const.
T
or • PV = nRT
• R = 8.31 J/mol K
• PV = NkT
• NA = 6.02 × 1023 molecules/mol
• k = 1.38 × 1023 J/K
• M=NAm
• (KE)av=(1/2mv2 )av= 3/2kT
• U= 3/2NkT = 3/2nRT V1 V2
=
T1 T2
where V is volume and T is absolute temperature  67   68  Formulas and Conversions Formulas and Conversions GayLussac's Law When gas volume is constant,
P
= const.
T GAS Specific Heat
at Constant
Volume
kJ/kgK or
kJ/kg oC Ratio of
Specific
γ= cp / cv Helium 1.105 0.85 1.30 2.177 1.675 1.30 1.043 0.745 1.40 Oxygen 0.913 0.652 1.40 Sulphur Dioxide 0.632 0.451 1.40 5.3.20 Efficiency of Heat Engines Carnot Cycle T1 − T2
T1
where T1 and T2 are absolute temperatures of heat source and sink Also
PV = nRoT where P = absolute pressure (kPa)
V = volume (m3)
T = absolute temperature K
N = the number of kmoles of gas
Ro = the universal gas constant 8.314 kJ/kmol/K η= Air Standard Efficiencies 5.3.19 Specific Heats Of Gases
Specific Heat
at Constant
Pressure
kJ/kgK or
kJ/kg oC Specific Heat
at Constant
Volume
kJ/kgK or
kJ/kg oC Ratio of
Specific
γ= cp / cv Air 1.005 0.718 1.40 Ammonia 2.060 1.561 1.32 Carbon Dioxide 0.825 0.630 1.31 Carbon
Monoxide 1.051 0.751 1.40  69  1.41 Nitrogen P1V1 P2V 2
=
= const.
T1
T2
P V = m R T where P = absolute pressure (kPa)
V = volume (m3)
T = absolute temp (K)
m = mass (kg)
R = characteristic constant (kJ/kgK) 1.66 10.096 Methane General Gas Law 3.153 14.235 Hydrogen
Sulphide where P is absolute pressure and T is absolute temperature 5.234 Hydrogen or
P1 P2
=
T1 T2 GAS Specific Heat
at Constant
Pressure
kJ/kgK or
kJ/kg oC Spark Ignition Gas and Oil Engines (Constant Volume Cycle)
1
η = 1 − (γ −1)
rv
rv= compression ratio
γ = specific heat (constant pressure) / Specific heat (constant volume)
Diesel Cycle η =1− Rγ − 1)
γ −1 rv γ ( R − 1)
Where r = ratio of compression
R = ratio of cutoff volume to clearance volume
High Speed Diesel (DualCombustion) Cycle η =1 kβ γ − 1 rv γ −1 [(k − 1) + γk ( β − 1)]
 70  Formulas and Conversions Where rv= cylinder volume / clearance volume
k = absolute pressure at the end of constant V heating (combustion) / absolute pressure at
the beginning of constant V combustion
β = volume at the end of constant P heating (combustion) / clearance
volume
Gas Turbines (Constant Pressure or Brayton Cycle) 1 η =1−
r ⎛ γ −1 ⎞
⎜
⎜γ⎟
⎟
⎠
p⎝ where rp = pressure ratio = compressor discharge pressure / compressor intake pressure 5.3.21 Heat Transfer by Conduction
Material Coefficient of Thermal
Conductivity
W/m °C Formulas and Conversions 5.3.22 Thermal Expansion of Solids
Increase in length = L α (T2 – T1)
Where L = original length
α = coefficient of linear expansion
(T2 – T1) = rise in temperature
Increase in volume = V β (T2 – T1)
Where V = original volume
β = coefficient of volumetric expansion
(T2 – T1) = rise in temperature
Coefficient of volumetric expansion = Coefficient of linear expansion × 3
β = 3α 5.3.23 Chemical Heating Value of a Fuel
Chemical Heating Value MJ per kg of fuel = 33.7C + 144( H 2 −
C is the mass of carbon per kg of fuel
H2 is the mass of hydrogen per kg of fuel
O2 is the mass of oxygen per kg of fuel
S is the mass of sulphur per kg of fuel Air 0.025 Brass 104 Concrete 0.85 Cork 0.043 Glass 1.0 Iron, cast 70 Steel 60 Wallboard,
paper 0.076 Aluminum 206 Brick 0.6 Copper 380 Boiler Efficiency Felt 0.038 m s (h1 − h2 )
mf × (calorificvalue) Theoretical Air Required to Burn Fuel ⎡8 ⎤ 100 Air (kg per kg of fuel) = ⎢ C + 8( H 2 − O2 ) + S ⎥
⎣3
⎦ 23
Air Supplied from Analysis of Flue Gases Air in kg per kg of fuel = N2
×C
33(CO2 + CO) Boiler Formulae m s (h1 − h2 )
2257 kj / kg
(h1 − h2 )
Factor of evaporation =
2257 kj / kg Equivalent evaporation = Glass, fibre 0.04 Plastic, cellular 0.04 Where Wood 0.15 ms = mass flow rate of steam
h1 = enthalpy of steam produced in boiler
h2 = enthalpy of feedwater to boiler
mf = mass flow rate of fuel  71   72  O2
) + 9.3S
8 0 1 Constant
pressure
P=Pressure Isothermal
T=Constant Polytropic
PVn =
Constant γ  T1 P
=1
T2 P2 TP P ⎡V2 ⎤
1
=⎢ ⎥
P2 ⎣ V1 ⎦ n T1 ⎡ P ⎤
= ⎢ 1⎥
T2 ⎣ P2 ⎦ T1 ⎡ P ⎤
= ⎢ 1⎥
T2 ⎣ P2 ⎦ P V2
1
=
P2 V1
P ⎡V2 ⎤
1
=⎢ ⎥
P2 ⎣ V1 ⎦    PV n −l
n γ −l
γ  T1 V1
=
T2 V2  TV P(V2V1) 0 Work done  73  mc n (T2 − T1 ) mR
(T1 − T2 )
n −1 mc v (T1 − T2 ) ⎛P⎞
mRT log e ⎜ 1 ⎟
⎜P ⎟
⎝ 2⎠
0 ⎛P⎞
mRT log e ⎜ 1 ⎟
⎜P ⎟
⎝ 2⎠ mc p (T2 − T1 ) mc v (T2 − T1 ) Heat added  74  ⎛γ − n⎞
cm = Specific heat for polytropic process = cv ⎜
⎟kJ / kgK
⎝ 1− n ⎠
H = Enthalpy, kJ
γ = Isentropic Exponent, cp/cv
n = polytropic exponent
P = Pressure, kPa
R = Gas content, kJ/kgK
S = Entropy, kJ/K
T = Absolute Temperature, K = 273+˚C
U = Internal Energy, kJ
V = Volume, m3
m = Mass of gas, kg Formulas and Conversions *Can be used for reversible adiabatic processes
cv = Specific heat at constant volume, kJ/kgK
cp = Specific heat at constant pressure, kJ/kgK n −1 γ −1 T1 ⎡V2 ⎤
=⎢ ⎥
T2 ⎣ V1 ⎦ T1 ⎡V2 ⎤
=⎢ ⎥
T2 ⎣ V1 ⎦ PVT Relationships Thermodynamic Equations for perfect gases n γ ∞ Constant
Volume
V=Constant Isentropic
S=Constant Value
of n Name of
process Formulas and Conversions mc v (T2 − T1 ) mc v (T2 − T1 ) 0 mc v (T2 − T1 ) mc v (T2 − T1 ) Change in
Internal
Energy mc p (T2 − T1 ) mc p (T2 − T1 ) 0 mc p (T2 − T1 ) mc p (T2 − T1 ) Change in
Enthalpy ⎞
⎟
⎟
⎠
⎛T
mc n log e ⎜ 2
⎜T
⎝1 ⎛T
mc n log e ⎜ 2
⎜T
⎝1 0 ⎞
⎟
⎟
⎠ ⎛P⎞
mR log e ⎜ 1 ⎟
⎜P ⎟
⎝ 2⎠ ⎞
⎟
⎟
⎠ ⎛T
mc v log e ⎜ 2
⎜T
⎝1 Change in
Entropy 0.909
0.209
0.125
0.383
0.795
0.402 Aluminum
Antimony
Bismuth
Brass
Carbon
Cobalt 0.130 Glass
Gold 12.0
29.0 0.465 Iron (cast)
Iron (wrought) 0.389 Zinc 1.800
4.183 1.633 Olive oil Water 0.139 Mercury Turpentine 3.643 Carbon Dioxide 2.135 1.138 Benzine 2.093 0.473 Ammonia Gasoline 2.470 Alcohal Petroleum Specific Heat
(at 20 o C )
KJ/kgK or kJ/kg o C
Liquid Specific Heat and Volume Expansion for Liquids  76  16.5 26.7 12.0 8.6 3.7 9.4 12.0 1.80 1.82 12.4 11.0 Coefficient of Volume Expansion
(Multiply by 104) Formulas and Conversions  75  0.230 Tin 19.5 0.235
0.494 Silver 0.741 Silicon Steel (mild) 7.8 0.134 Platinum 13.0 0.131
0.452 Lead
Nickel 10.4 0.544 Ice (between 20 C & 0 C ) 50.4 14.2 9.0 16.5 12.3 7.9 18.4 12.4 17.5 23.8 Coefficient of Linear Expansion
between 0 o C and 100 o C
(multiply by 106) 2.135 o 0.388
0.896 Copper o Mean Specific Heat between 0 o C
and 100 o C kJ/kgK or kJ/kg o C Specific Heat and Linear
Expansion of Solids Formulas and Conversions Formulas and Conversions Formulas and Conversions 5.4 Fluid Mechanics
5.4.1 Discharge from an Orifice
Let A = crosssectional area of the orifice = π
4 And Ac = crosssectional area of the jet at the vena
conrtacta
Then Ac = CcA π
4 Where B = breadth (m)
H = head (m above sill)
Triangular Right Angled Notch: Q = 2.635 H5/2
Where H = head (m above sill) d2 5.4.2 Bernoulli’s Theory
dc 2 Or C c = Ac ⎛ d c ⎞
=⎜ ⎟
A ⎝d⎠ H =h+ 2 P v2
+
w 2g H = total head (meters)
w = force of gravity on 1 m3 of fluid (N)
h = height above datum level (meters)
v = velocity of water (meters per second)
P = pressure (N/m2 or Pa)
Loss of Head in Pipes Due to Friction
L v2
Loss of head in meters = f
d 2g
L = length in meters
v = velocity of flow in meters per second
d = diameter in meters
f = constant value of 0.01 in large pipes to 0.02 in small pipes Where Cc is the coefficient of contraction 5.4.3 Actual pipe dimensions
Nominal
pipe size
(in) 10.3 6.8 1.73 3.660 × 105 13.7 9.2 2.24 6717 × 105 17.1 12.5 2.31 1.236 × 104 1/2  77  Flow area
(m2) 3/8 • Rectangular notch: Q = 0.62 (B · H) 2/3 √2gh Wall
thickness
(mm) 1/4 • Or Q = C c AC v 2 gh
• Typically, values for Cd vary between 0.6 and 0.65
• Circular orifice: Q = 0.62 A √2gh
3
2
• Where Q = flow (m /s) A = area (m ) h = head (m) Inside
diameter
(mm) 1/8 At the vena contracta, the volumetric flow rate Q of the fluid is given by
• Q = area of the jet at the vena contracta · actual velocity = AcV Outside
diameter
(mm) 21.3 15.8 2.77 1.960 × 104 3/4 26.7 20.9 2.87 3.437 × 104 1 33.4 26.6 3.38 5.574 × 104 1¼ 42.2 35.1 3.56 9.653 × 104 1½ 48.3 40.9 3.68 1.314 ×103 2 60.3 52.5 3.91 2.168 × 103  78  Formulas and Conversions Formulas and Conversions Nominal
pipe size
(in) Outside
diameter
(mm) Inside
diameter
(mm) Wall
thickness
(mm) Flow area
(m2) 2½ 73.0 62.7 5.16 3.090 × 103 3 88.9 77.9 5.49 4.768 × 103 3½ 101.6 90.1 5.74 6.381 × 103 4 114.3 102.3 6.02 8.213 × 103 5 141.3 128.2 6.55 1.291 × 102 6 168.3 154.1 7.11 1.864 × 102 8 219.1 202.7 8.18 3.226 × 102 10 273.1 254.5 9.27 5.090 × 102 12 323.9 303.2 10.31 7.219 × 102 14 355.6 333.4 11.10 8.729 × 102 16 406.4 381.0 12.70 0.1140 18 457.2 428.7 14.27 0.1443 20 508.0 477.9 15.06 0.1794 24 609.6 574.7 17.45 0.2594 Chapter 6
References
6.1 Periodic Table of Elements
A
1
1
H
1.00
8 8A
18
2A
2 3A
13 4
3
Li
Be
6.94 9.01
1
2
11
12
Na
Mg
22.9 24.3
9
1 4A
14 5A
15 6A
16 7A
17 2
He
4.00
3 5
6
7
8
9
10
B
C
N
O
F
Ne
10.8 12.0 14.0 16.0 19.0 20.1
1
1
1
0
0
8
3B
3 4B
4 5B
5 6B
6 7B
7 8B
8 8B
9 8B
10 1B
11 2B
12 13
14
15
16
17
18
Al
Si
P
S
Cl
Ar
26.9 28.0 30.9 32.0 35.4 39.9
8
9
7
7
5
5 19
31
32
33
34
35
36
20
21
22
23
24
25
26
27
28
29
30
K
Ga
Ge
As
Se
Br
Kr
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
39.1 40.0 44.9 47.9 50.9 52.0 54.9 55.8 58.9 58.7 63.5 65.3 69.7 72.5 74.9 78.9 79.9 83.8
0
2
9
2
6
0
0
8
6
0
4
0
4
5
3
0
5
8
37
49
50
51
52
53
54
38
39
40
41
42
43
44
45
46
47
48
Rb
In
Sn
Sb
Te
I
Xe
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
85.4 87.6 88.9 91.2 92.9 95.9 97.9 101. 102. 106. 107. 112. 114. 118. 121. 127. 126. 131.
7
8
7
8
6
9
3
2
1
2
1
4
1
9
4
9
4
56
55
81
82
83
84
85
86
57
72
73
74
75
76
77
78
79
80
Cs
Tl
Pb
Bi
Po
At
Rn
Ba
La
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
132. 137. 138. 178. 180. 183. 186. 190. 192. 195. 197. 200. 204. 207. 209. (209) (210) (222)
9
4
2
0
3
9
5
9
8
2
2
2
1
0
6
87
88
89
104 105 106 107 108 109
Fr
Ra
Db
Sg
Bh
Hs
Mt
Ac
Rf
(223) 226. 227. (261) (262) (266) (264) (265) (268)
0
0 58
62
67
68
69
70
71
59
63
60
64
61
65
66
Ho
Er
Tm
Yb
Lu
Ce
Sm
Pr
Eu
Nd
Gd
Pm
Tb
Dy
140. 140. 144. (145) 150. 152. 157. 158. 162. 164. 167. 168. 173. 175.
9
3
9
0
0
1
4
9
0
2
3
9
5
90
95
96
97
98
99
100 101 102 103
91
92
93
94
Es
Fm
Md
No
Lr
Th
Am
Cm
Bk
Cf
Pa
U
Np
Pu
232. 231. 238. 237. (244) (243) (247) (247) (251) (252) (257) (258) (259) (262)
0
0
0
0  79   80  Formulas and Conversions 6.2 Resistor Color Coding
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ACI PACKAGING • AJ GREAVES • ANCHOR PRODUCTS • AUCKLAND REGIONAL COUNCIL • BALLANCE AGRI NUTRIENTS •
CONTACT ENERGY • ENZAFOODS NZ LTD • ERICCSON • FISHER & PAYKEL • GEC ALSTHOM • JAMES HARDIE • METHANEX NZ
LTD • NATURAL GAS NZ • NZ MILK PRODUCTS • NZ WATER AND WASTE ASSOC • NORSKE SKOG • NZ ALUMINIUM SMELTERS •
NZ REFINING CO • PAN PAC FOREST PRODUCTS • POWERCO • ROCKWELL NZ • ROTORUA DISTRICT COUNCIL • ROYAL NEW
ZEALAND NAVY • THE UNIVERSITY OF AUCKLAND • SAUDI ARABIA
SAUDI ELECTRIC COMPANY SINGAPORE COMPANY MISSION
“To provide our clients with
measurable and significant
productivity gains through
excellence in cutting edge,
practical engineering and
technology training” ...
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This note was uploaded on 11/24/2010 for the course STRUCTURAL 1 taught by Professor 2 during the Spring '10 term at Indian Institute of Technology, Roorkee.
 Spring '10
 2

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