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 Page 7
Formulas and Conversions
- 1 -
Definition and Abbreviations for Physical Quantities
Symbol Unit
Quantity
m
meter
Length
kg
kilogram Mass
s
second
Time
A
ampere
Electric current
K
kelvin
Thermodynamic temp
cd
candela
Luminous intensity
Quantity
Unit
Symbol
Equivalent
Plane angle
-
Force
newton
N
kg · m/s
2
Work, energy
heat
joule
J·N·m
Power
watt
W
J/s
Frequency
hertz
Hz
s
-1
Viscosity:
kinematic
-
m
2
/s
10 c St
(Centistoke)
Viscosity:
Dynamic
-
Ns/m
2
10
3
cP
(Centipoise)
Pressure
-
Pa or N/m
2
pascal, Pa
Symbol Prefix
Factor by which unit is
multiplied
T
Tera
10
12
G
Giga
10
9
M
Mega
10
6
Chapter 1
Formulas and Conversions
- 2 -
Symbol Prefix
Factor by which unit is
multiplied
k
Kilo
10
3
h
Hecto
10
2
da
Deca
10
d
Deci
10
-1
c
Centi
10
-2
m
Milli
10
-3
μ
Micro
10
-6
n
Nano
10
-9
p
Pico
10
-12
Quantity
Electrical
unit
Symbol
Derived
unit
Potential
Volt
V
W/A
Resistance
Ohm
Ώ
V/A
Charge
Coulomb
C
A·s
Capacitance
F
A·s/V
Electric field
strength
-
V/m
-
Electric flux
density
-
C/m
2
-
Quantity
Magnetic
unit
Symbol
Derived unit
Magnetic flux
Weber
Wb
V·s = N·m/A
Inductance
Henry
H
V·s/A = N·m/A
2
Magnetic field
strength
-
A/m
-
Magnetic flux density Tesla
T
Wb/m
2
=
(N)/(Am)
Formulas and Conversions
- 3 -
Units of Physical Quantities
Conversion Factors (general):
1 acre = 43,560 square feet
1 cubic foot = 7.5 gallons
1 foot = 0.305 meters
1 gallon = 3.79 liters
1 gallon = 8.34 pounds
1 grain per gallon = 17.1 mg/L
1 horsepower = 0.746 kilowatts
1 million gallons per day = 694 gallons per minute
1 pound = 0.454 kilograms
1 pound per square inch = 2.31 feet of water
Degrees Celsius = (Degrees Fahrenheit - 32) (5/9)
Degrees Fahrenheit = (Degrees Celsius) (9/5) + 32
1% = 10,000 mg/L
Name
To convert from
To
Multiply
by
Divide by
Acceleration
ft/sec
2
m/s
2
0.3048
3.2810
Area
acre
m
2
4047
2.471E-04
Area
ft
2
m
2
9.294E-02
10.7600
Area
hectare
m
2
1.000E+04 1.000E-04
Area
in
2
m
2
6.452E-04
1550
Density
g/cm
3
kg/m
3
1000
1.000E-03
Density
lbm/ft
3
kg/m
3
16.02
6.243E-02
Density
lbm/in
3
kg/m
3
2.767E+04 3.614E-05
Chapter 2
Formulas and Conversions
- 4 -
Name
To convert from
To
Multiply
by
Divide by
Density
lb·s
2
/in
4
kg/m
3
1.069E+07 9.357E-08
Density
slug/ft
3
kg/m
3
515.40
1.940E-03
Energy
BTU
J
1055
9.478E-04
Energy
cal
J
4.1859
0.2389
Energy
erg
J
1.000E-07
1.000E+07
Energy
eV
J
1.602E-19
6.242E+18
Energy
Ft·lbf
J
1.3557
0.7376
Energy
kiloton TNT
J
4.187E+12 2.388E-13
Energy
KW·hr
J
3.600E+06 2.778E-07
Energy
Megaton TNT
J
4.187E+15 2.388E-16
Force
Dyne
N
1.000E-05
1.000E+05
Force
Lbf
N
4.4484
0.2248
Force
Ozf
N
0.2780
3.5968
Heat capacity
BTU/lbm · °F
J/kg·°C
4188
2.388E-04
Heat transfer coefficient
BTU/hr·ft
2
·°F
W/m
2
·°C
5.6786
0.1761
Length
AU
m
1.496E+11 6.685E-12
Length
ft
m
0.3048
3.2810
Length
in
m
2.540E-02
39.3700
Length
mile
m
1609
6.214E-04
Length
Nautical mile
m
1853
5.397E-04
Length
parsec
m
3.085E+16 3.241E-17
Mass
amu
kg
1.661E-27
6.022E+26
Mass
lbm
kg
0.4535
2.2050
Mass
lb·s
2
/in
kg
1200.00
5.711E-03
Mass
slug
kg
14.59
6.853E-02
Mass flow rate
lbm/hr
kg/s
1.260E-04
7937
Formulas and Conversions
- 5 -
Name
To convert from
To
Multiply
by
Divide by
Mass flow rate
lbm/sec
kg/s
0.4535
2.2050
Moment of inertia
ft·lb·s
2
kg·m
2
1.3557
0.7376
Moment of inertia
in·lb·s
2
kg·m
2
0.1130
8.8510
Moment of inertia
oz·in·s
2
kg·m
2
7.062E-03
141.60
Power
BTU/hr
W
0.2931
3.4120
Power
hp
W
745.71
1.341E-03
Power
tons of refrigeration
W
3516
2.844E-04
Pressure
bar
Pa
1.000E+05 1.000E-05
Pressure
dyne/cm
2
Pa
0.1000
10.0000
Pressure
in. mercury
Pa
3377
2.961E-04
Pressure
in. water
Pa
248.82
4.019E-03
Pressure
kgf/cm
2
Pa
9.807E+04 1.020E-05
Pressure
lbf/ft
2
Pa
47.89
2.088E-02
Pressure
lbf/in
2
Pa
6897
1.450E-04
Pressure
mbar
Pa
100.00
1.000E-02
Pressure
microns mercury
Pa
0.1333
7.501
Pressure
mm mercury
Pa
133.3
7.501E-03
Pressure
std atm
Pa
1.013E+05 9.869E-06
Specific heat
BTU/lbm·°F
J/kg·°C
4186
2.389E-04
Specific heat
cal/g·°C
J/kg·°C
4186
2.389E-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·ft
2
·°F
W/m·°C
0.1442
6.9340
Thermal conductivity
cal/cm·s·°C
W/m·°C
418.60
2.389E-03
Thermal conductivity
cal/ft·hr·°F
W/m·°C
6.867E-03
145.62
Time
day
S
8.640E+04 1.157E-05
Formulas and Conversions
- 6 -
Name
To convert from
To
Multiply
by
Divide by
Time
sidereal year
S
3.156E+07 3.169E-08
Torque
ft·lbf
N·m
1.3557
0.7376
Torque
in·lbf
N·m
0.1130
8.8504
Torque
In·ozf
N·m
7.062E-03
141.61
Velocity
ft/min
m/s
5.079E-03
196.90
Velocity
ft/s
m/s
0.3048
3.2810
Velocity
Km/hr
m/s
0.2778
3.6000
Velocity
miles/hr
m/s
0.4470
2.2370
Viscosity – absolute
centipose
N·s/m
2
1.000E-03
1000
Viscosity – absolute
g/cm·s
N·s/m
2
0.1000
10
Viscosity – absolute
lbf/ft
2
·s
N·s/m
2
47.87
2.089E-02
Viscosity – absolute
lbm/ft·s
N·s/m
2
1.4881
0.6720
Viscosity – kinematic
centistoke
m
2
/s
1.000E-06
1.000E+06
Viscosity – kinematic
ft
2
/sec
m
2
/s
9.294E-02
10.7600
Volume
ft
3
m
3
2.831E-02
35.3200
Volume
in
3
m
3
1.639E-05
6.102E+04
Volume
Liters
m
3
1.000E-03
1000
Volume
U.S. gallons
m
3
3.785E-03
264.20
Volume flow rate
ft
3
/min
m
3
/s
4.719E-04
2119
Volume flow rate
U.S. gallons/min
m
3
/s
6.309E-05
1.585E+04
A. DISTANCE (Length)
Conversions
Multiply
By
To obtain
LENGTH
Centimeter
0.03280840
foot
Centimeter
0.3937008
inch
Formulas and Conversions
- 7 -
Multiply
By
To obtain
Fathom
1.8288
*
meter(m)
Foot
0.3048
*
meter(m)
Foot
30.48
*
centimeter(cm)
Foot
304.8
*
millimeter(mm)
Inch
0.0254
*
meter(m)
Inch
2.54
*
centimeter(cm)
Inch
25.4
*
millimeter(mm)
Kilometer
0.6213712
mile(USstatute)
Meter
39.37008
Inch
Meter
0.54680066
Fathom
Meter
3.280840
Foot
Meter
0.1988388
Rod
Meter
1.093613
Yard
Meter
0.0006213712
mile(USstatute)
Microinch
0.0254
*
micrometer(micron)(μm)
micrometer(micron)
39.37008
Microinch
mile(USstatute)
1,609.344
*
meter(m)
mile(USstatute)
1.609344
*
kilometer(km)
millimeter
0.003280840
Foot
millimeter
0.0397008
Inch
Rod
5.0292
*
meter(m)
Yard
0.9144
*
meter(m)
To Convert
To
Multiply By
Cables
Fathoms
120
Cables
Meters
219.456
Cables
Yards
240
Formulas and Conversions
- 8 -
To Convert
To
Multiply By
Centimeters
Meters
0.01
Centimeters
Yards
0.01093613
Centimeters
Feet
0.0328084
Centimeters
Inches
0.3937008
Chains, (Surveyor's)
Rods
4
Chains, (Surveyor's)
Meters
20.1168
Chains, (Surveyor's)
Feet
66
Fathoms
Meters
1.8288
Fathoms
Feet
6
Feet
Statute Miles
0.00018939
Feet
Kilometers
0.0003048
Feet
Meters
0.3048
Feet
Yards
0.3333333
Feet
Inches
12
Feet
Centimeters
30.48
Furlongs
Statute Miles
0.125
Furlongs
Meters
201.168
Furlongs
Yards
220
Furlongs
Feet
660
Furlongs
Inches
7920
Hands (Height Of Horse)
Inches
4
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
Formulas and Conversions
- 9 -
To Convert
To
Multiply By
Kilometers
Statute Miles
0.621371192
Kilometers
Meters
1000
Leagues, Nautical
Nautical Miles
3
Leagues, Nautical
Kilometers
5.556
Leagues, Statute
Statute Miles
3
Leagues, Statute
Kilometers
4.828032
Chains
0.01
Inches
7.92
Centimeters
20.1168
Meters
Statute Miles
0.000621371
Meters
Kilometers
0.001
Meters
Yards
1.093613298
Meters
Feet
3.280839895
Meters
Inches
39.370079
Meters
Centimeters
100
Meters
Millimeters
1000
Microns
Meters
0.000001
Microns
Inches
0.0000394
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
Formulas and Conversions
- 10 -
To Convert
To
Multiply By
Miles, Statute
Centimeters
160934.4
Millimeters
Inches
0.039370079
Mils
Inches
0.001
Mils
Millimeters
0.0254
Paces (US)
Inches
30
Paces (US)
Centimeters
76.2
Points (Typographical)
Inches
0.013837
Points (Typographical)
Millimeters
0.3514598
Rods
Meters
5.0292
Rods
Yards
5.5
Rods
Feet
16.5
Spans
Inches
9
Spans
Centimeters
22.86
Yards
Miles
0.00056818
Yards
Meters
0.9144
Yards
Feet
3
Yards
Inches
36
Yards
Centimeters
91.44
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
Formulas and Conversions
- 11 -
Conversion
1 statute mile = 8 furlongs
1 rod = 5.5 yd
1 statute mile = 5280 ft
1 in = 100 mils
1 nautical mile = 6076 ft
1 light year = 9.461 x 10
15
m
1 league = 3 miles
1 mil = 2.540 x 10
-5
m
Area
1 ft
2
= 144 in
2
1 acre = 160 rod
2
1 yd
2
= 9 ft
2
1 acre = 43,560 ft
2
1 rod
2
= 30.25 yd
2
1 mile
2
= 640 acres
1 cm
2
= 0.1550 in
2
1 in
2
= 6.4516 cm
2
1 m
2
= 10.764 ft
2
1 ft
2
= 0.0929 m
2
1 km
2
= 0.3861 mile
2
1 mile
2
= 2.590 km
2
Volume
1 cm
3
= 0.06102 in
3
1 in
3
= 16.387 cm
3
1 m
3
= 35.31 ft
3
1 ft
3
= 0.02832 m
3
1 Litre = 61.024 in
3
1 in
3
= 0.0164 litre
1 Litre = 0.0353 ft
3
1 ft
3
= 28.32 litres
1 Litre = 0.2642 gal. (U.S.)
1 yd
3
= 0.7646 m
3
1 Litre = 0.0284 bu (U.S.)
1 gallon (US) = 3.785 litres
1 Litre = 1000.000 cm
3
1 gallon (US) = 3.785 x 10
-3
m
3
1 Litre = 1.0567 qt. (liquid) or
0.9081 qt. (dry)
1 bushel (US) = 35.24 litres
1 oz (US fluid) = 2.957 x 10
-5
m
3
1 stere = 1 m
3
Liquid Volume
1 gill = 4 fluid ounces
1 barrel = 31.5 gallons
1 pint = 4 gills
1 hogshead = 2 bbl (63 gal)
1 quart = 2 pints
1 tun = 252 gallons
1 gallon = 4 quarts
1 barrel (petrolum) = 42 gallons
Formulas and Conversions
- 12 -
Conversion
Dry Volume
1 quart = 2 pints
1 quart = 67.2 in
3
1 peck = 8 quarts
1 peck = 537.6 in
3
1 bushel = 4 pecks
1 bushel = 2150.5 in
3
B. Area
Conversions
Multiply
By
To obtain
AREA
acre
4,046.856
meter
2
(m
2
)
acre
0.4046856
hectare
centimeter
2
0.1550003
inch
2
centimeter
2
0.001076391
foot
2
foot
2
0.09290304
*
meter
2
(m
2
)
foot
2
929.0304
2
centimeter
2
(cm
2
)
foot
2
92,903.04
millimeter
2
(mm
2
)
hectare
2.471054
acre
inch
2
645.16
*
millimeter
2
(mm
2
)
inch
2
6.4516
centimeter
2
(cm
2
)
inch
2
0.00064516
meter
2
(m
2
)
meter
2
1,550.003
inch
2
meter
2
10.763910
foot
2
meter
2
1.195990
yard
2
meter
2
0.0002471054
acre
millimeter
2
0.00001076391
foot
2
millimeter
2
0.001550003
inch
2
yard
2
0.8361274
meter
2
(m
2
)
Formulas and Conversions
- 13 -
C. Volume
Conversions
Metric Conversion Factors: Volume (including Capacity)
Multiply
By
To obtain
VOLUME (including CAPACITY)
centimeter
3
0.06102376
inch
3
foot
3
0.028311685
meter
3
(m
3
)
foot
3
28.31685
liter
gallon (UK liquid)
0.004546092
meter
3
(m
3
)
gallon (UK liquid)
4.546092
litre
gallon (US liquid)
0.003785412
meter
3
(m
3
)
gallon (US liquid)
3.785412
liter
inch
3
16,387.06
millimeter
3
(mm
3
)
inch
3
16.38706
centimeter
3
(cm
3
)
inch
3
0.00001638706
meter
3
(m
3
)
Liter
0.001
*
meter
3
(m
3
)
Liter
0.2199692
gallon (UK liquid)
Liter
0.2641720
gallon (US liquid)
Liter
0.03531466
foot
3
meter
3
219.9692
gallon (UK liquid)
meter
3
264.1720
gallon (US liquid)
meter
3
35.31466
foot
3
meter
3
1.307951
yard
3
meter
3
1000.
*
liter
meter
3
61,023.76
inch
3
millimeter
3
0.00006102376
inch
3
Yard
3
0.7645549
meter
3
(m
3
)
D. Mass and Weight
Conversions
Formulas and Conversions
- 14 -
To Convert
To
Multiply By
Carat
Milligrams
200
Drams, Avoirdupois
Avoirdupois Ounces
0.06255
Drams, Avoirdupois
Grams
1.7718452
Drams, Avoirdupois
Grains
27.344
Drams, Troy
Troy Ounces
0.125
Drams, Troy
Scruples
3
Drams, Troy
Grams
3.8879346
Drams, Troy
Grains
60
Grains
Kilograms
6.47989E-05
Grains
Avoirdupois Pounds
0.00014286
Grains
Troy Pounds
0.00017361
Grains
Troy Ounces
0.00208333
Grains
Avoirdupois Ounces
0.00228571
Grains
Troy Drams
0.0166
Grains
Avoirdupois Drams
0.03657143
Grains
Pennyweights
0.042
Grains
Scruples
0.05
Grains
Grams
0.06479891
Grains
Milligrams
64.79891
Grams
Kilograms
0.001
Grams
Avoirdupois Pounds
0.002204623
Grams
Troy Pounds
0.00267923
Grams
Troy Ounces
0.032150747
Grams
Avoirdupois Ounces
0.035273961
Grams
Avoirdupois Drams
0.56438339
Grams
Grains
15.432361
Formulas and Conversions
- 15 -
To Convert
To
Multiply By
Grams
Milligrams
1000
Hundredweights, Long
Long Tons
0.05
Hundredweights, Long
Metric Tons
0.050802345
Hundredweights, Long
Short Tons
0.056
Hundredweights, Long
Kilograms
50.802345
Hundredweights, Long
Avoirdupois Pounds
112
Hundredweights, Short
Long Tons
0.04464286
Hundredweights, Short
Metric Tons
0.045359237
Hundredweights, Short
Short Tons
0.05
Hundredweights, Short
Kilograms
45.359237
Hundredweights, Short
Avoirdupois Pounds
100
Kilograms
Long Tons
0.0009842
Kilograms
Metric Tons
0.001
Kilograms
Short Tons
0.00110231
Kilograms
Short Hundredweights
0.02204623
Kilograms
Avoirdupois Pounds
2.204622622
Kilograms
Troy Pounds
2.679229
Kilograms
Troy Ounces
32.15075
Kilograms
Avoirdupois Ounces
35.273962
Kilograms
Avoirdupois Drams
564.3834
Kilograms
Grams
1000
Kilograms
Grains
15432.36
Milligrams
Grains
0.015432358
Ounces, Avoirdupois
Kilograms
0.028349523
Ounces, Avoirdupois
Avoirdupois Pounds
0.0625
Ounces, Avoirdupois
Troy Pounds
0.07595486
Ounces, Avoirdupois
Troy Ounces
0.9114583
Formulas and Conversions
- 16 -
To Convert
To
Multiply By
Ounces, Avoirdupois
Avoirdupois Drams
16
Ounces, Avoirdupois
Grams
28.34952313
Ounces, Avoirdupois
Grains
437.5
Ounces, Troy
Avoirdupois Pounds
0.06857143
Ounces, Troy
Troy Pounds
0.0833333
Ounces, Troy
Avoirdupois Ounces
1.097143
Ounces, Troy
Troy Drams
8
Ounces, Troy
Avoirdupois Drams
17.55429
Ounces, Troy
Pennyweights
20
Ounces, Troy
Grams
31.1034768
Ounces, Troy
Grains
480
Pennyweights
Troy Ounces
0.05
Pennyweights
Grams
1.55517384
Pennyweights
Grains
24
Pounds, Avoirdupois
Long Tons
0.000446429
Pounds, Avoirdupois
Metric Tons
0.000453592
Pounds, Avoirdupois
Short Tons
0.0005
Pounds, Avoirdupois
Quintals
0.00453592
Pounds, Avoirdupois
Kilograms
0.45359237
Pounds, Avoirdupois
Troy Pounds
1.215278
Pounds, Avoirdupois
Troy Ounces
14.58333
Pounds, Avoirdupois
Avoirdupois Ounces
16
Pounds, Avoirdupois
Avoirdupois Drams
256
Pounds, Avoirdupois
Grams
453.59237
Pounds, Avoirdupois
Grains
7000
Pounds, Troy
Kilograms
0.373241722
Pounds, Troy
Avoirdupois Pounds
0.8228571
Formulas and Conversions
- 17 -
To Convert
To
Multiply By
Pounds, Troy
Troy Ounces
12
Pounds, Troy
Avoirdupois Ounces
13.16571
Pounds, Troy
Avoirdupois Drams
210.6514
Pounds, Troy
Pennyweights
240
Pounds, Troy
Grams
373.2417216
Pounds, Troy
Grains
5760
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
Metric Tons
1.016046909
Short Tons
1.12
Long Hundredweights
20
Short Hundredweights
22.4
Kilograms
1016.04691
Avoirdupois Pounds
2240
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
Formulas and Conversions
- 18 -
To Convert
To
Multiply By
Tons, Short
Long Tons
0.8928571
Tons, Short
Metric Tons
0.90718474
Tons, Short
Long Hundredweights
17.85714
Tons, Short
Short Hundredweights
20
Tons, Short
Kilograms
907.18474
Tons, Short
Avoirdupois Pounds
2000
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/mil-foot
3.405E-07
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/mil-foot
3.405E-10
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/mil-foot
5.456E-09
Pounds/cu ft
Pounds/cu in
0.0005787
Formulas and Conversions
- 19 -
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/mil-foot
0.000009425
Pounds/cu in
Gms/cu cm
27.68
Pounds/cu in
Pounds/cu ft
1728
Pounds/cu in
Kgs/cu meter
27680
F. Relative Density (Specific Gravity) Of Various Substances
Substance
Relative
Density
Water (fresh)
1.00
Mica
2.9
Water (sea average)
1.03
Nickel
8.6
Aluminum
2.56
Oil (linseed)
0.94
Antimony
6.70
Oil (olive)
0.92
Bismuth
9.80
Oil (petroleum)
0.76-0.86
Brass
8.40
Oil (turpentine)
0.87
Brick
2.1
Paraffin
0.86
Calcium
1.58
Platinum
21.5
Carbon (diamond)
3.4
Formulas and Conversions
- 20 -
Substance
Relative
Density
Sand (dry)
1.42
Carbon (graphite)
2.3
Silicon
2.6
Carbon (charcoal)
1.8
Silver
10.57
Chromium
6.5
Slate
2.1-2.8
Clay
1.9
Sodium
0.97
Coal
1.36-1.4
Steel (mild)
7.87
Cobalt
8.6
Sulphur
2.07
Copper
8.77
Tin
7.3
Cork
0.24
Tungsten
19.1
Glass (crown)
2.5
Wood (ash)
0.75
Glass (flint)
3.5
Wood (beech)
0.7-0.8
Gold
19.3
Wood (ebony)
1.1-1.2
Iron (cast)
7.21
Wood (elm)
0.66
Iron (wrought)
7.78
Formulas and Conversions
- 21 -
Substance
Relative
Density
Wood (lignum-vitae)
1.3
11.4
Magnesium
1.74
Manganese
8.0
Mercury
13.6
11.4
Magnesium
1.74
Manganese
8.0
Wood (oak)
0.7-1.0
Wood (pine)
0.56
Wood (teak)
0.8
Zinc
7.0
Wood (oak)
0.7-1.0
Wood (pine)
0.56
Wood (teak)
0.8
Zinc
7.0
Mercury
13.6
G. Greek Alphabet
Name
Lower
Case
Upper
Case
Alpha
α
Α
Beta
β
Β
Gamma
γ
Γ
Delta
δ
Δ
Epsilon
ε
Ε
Zeta
ζ
Ζ
Formulas and Conversions
- 22 -
Name
Lower
Case
Upper
Case
Eta
η
Η
Theta
θ
Θ
Iota
ι
Ι
Kappa
κ
Κ
Lambda
λ
Λ
Mu
μ
Μ
Nu
ν
Ν
Xi
ξ
Ξ
Omicron
ο
Ο
Pi
π
Π
Rho
ρ
Ρ
Sigma
σ and ς
Σ
Tau
τ
Τ
Upsilon
υ
Υ
Phi
φ
Φ
Chi
χ
Χ
Psi
ψ
Ψ
Omega
ω
Ω
Formulas and Conversions
- 23 -
System of Units
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 meter-kilogram-
second (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
10
12
1000000000000
T
Thz (Tera
hertz)
Giga
10
9
1000000000
G
Ghz (Giga
hertz)
Mega
10
6
1000000
M
Mhz (Mega
hertz)
Unit
quantity
1
1
hz (hertz)
Micro
10
-6
0.001
μ
μF (Micro
Nano
10
-9
0.000001
n
nF (Nano
Pico
10
-12
0.000000000001
p
pF (Pico
Conversion Chart
Multiply
by
Into
Milli
Into
Centi
Into
Deci
Into
MGL*
Into
Deca
Into
Hecto
Into
Kilo
To
convert
Kilo
10
6
10
5
10
4
10
3
10
2
10
1
1
Chapter 3
Formulas and Conversions
- 24 -
Multiply
by
Into
Milli
Into
Centi
Into
Deci
Into
MGL*
Into
Deca
Into
Hecto
Into
Kilo
To
convert
Hecto
10
5
10
4
10
3
10
2
10
1
1
10
-1
To
convert
Deca
10
4
10
3
10
2
10
1
1
10
-1
10
-2
To
convert
MGL*
10
3
10
2
10
1
1
10
-1
10
-2
10
-3
To
convert
Deci
10
2
10
1
1
10
-1
10
-2
10
-3
10
-4
To
convert
Centi
10
1
1
10
-1
10
-2
10
-3
10
-4
10
-5
To
convert
Milli
1
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
MGL = meter, gram, liter
Example:
To convert Kilogram Into Milligram → (1 Kilo X 10
6
) Milligrams
Physical constants
Name
Symbolic
Representation
Numerical Equivalent
N
6.023 x 10
26
/(kg mol)
Bohr magneton
B
9.27 x 10
-24
Am 25
2
Boltzmann's constant
k
1.380 x 10
-23
J/k
Stefan-Boltzmann constant
d
5.67 x 10
-8
W/(m
2
K
4
)
Characteristic impedance of free
space
Zo
o
/E
o
)
1/2
=120ΠΩ
Electron volt
eV
1.602 x 10
-19
J
Electron charge
e
1.602 x 10
-19
C
Formulas and Conversions
- 25 -
Name
Symbolic
Representation
Numerical Equivalent
Electronic rest mass
m
e
9.109 x 10
-31
kg
Electronic charge to mass ratio
e/m
e
1.759 x 10
11
C/kg
F
9.65 x 10
7
C/(kg mol)
Permeability of free space
μ
0
4Π x 10
-7
H/m
Permittivity of free space
E
o
8.85 x 10
-12
F/m
Planck's constant
h
6.626 x 10
-34
J s
Proton mass
m
p
1.672 x 10
-27
kg
Proton to electron mass ratio
m
p
/m
e
1835.6
Standard gravitational
acceleration
g
9.80665 m/s
2
, 9.80665 N/kg
Universal constant of gravitation
G
6.67 x 10-11 N m
2
/kg
2
Universal gas constant
R
o
8.314 kJ/(kg mol K)
Velocity of light in vacuum
C
2.9979 x 10
8
m/s
Temperature
0
C
5/9(
0
F - 32)
Temperature
K
5/9(
0
F + 459.67), 5/9
0
R,
0
C +
273.15
Speed of light in air
c
3.00 x 10
8
m s
-1
Electron charge
e
-1.60 x 10
-19
C
Mass of electron
m
e
9.11 x 10
-31
kg
Planck's constant
h
6.63 x 10
-34
J s
Universal gravitational constant
G
6.67 x 10
-11
N m
2
kg
-2
Electron volt
1 eV
1.60 x 10
-19
J
Mass of proton
m
p
1.67 x 10
-27
kg
Formulas and Conversions
- 26 -
Name
Symbolic
Representation
Numerical Equivalent
Acceleration due to gravity on
Earth
g
9.80 m s
-2
Acceleration due to gravity on the
Moon
g
M
1.62 m s
-2
Radius of the Earth
R
E
6.37 x 10
6
m
Mass of the Earth
M
E
5.98 x 10
24
kg
Radius of the Sun
R
S
6.96 x 10
8
m
Mass of the Sun
M
S
1.99 x 10
30
kg
Radius of the Moon
R
M
1.74 x 10
6
m
Mass of the Moon
M
M
7.35 x 10
22
kg
Earth-Moon distance
-
3.84 x 10
8
m
Earth-Sun distance
-
1.50 x 10
11
m
Speed of light in air
c
3.00 x 10
8
m s
-1
Electron charge
e
-1.60 x 10
-19
C
Mass of electron
m
e
9.11 x 10
-31
kg
Planck's constant
h
6.63 x 10
-34
J s
Universal gravitational constant
G
6.67 x 10
-11
N m
2
kg
-2
Electron volt
1 eV
1.60 x 10
-19
J
Mass of proton
m
p
1.67 x 10
-27
kg
Acceleration due to gravity on
Earth
g
9.80 m s
-2
Acceleration due to gravity on the
Moon
g
M
1.62 m s
-2
Ton
1 ton
1.00 x 10
3
kg
Formulas and Conversions
- 27 -
General Mathematical Formulae
4.1 Algebra
A. Expansion Formulae
Square of summation
•(x + y)
2
= x
2
+ 2xy + y
2
Square of difference
•(x – y)
2
= x
2
– 2xy + y
2
Difference of squares
•x
2
– y
2
= (x + y) (x – y)
Cube of summation
•(x + y)
3
= x
3
+ 3x
2
y + 3xy
2
+ y
3
Summation of two cubes
•x
3
+ y
3
= (x + y) (x
2
- xy + y
2
)
Cube of difference
•(x – y)
3
= x
3
– 3x
2
y + 3xy
2
– y
3
Difference of two cubes
•x
3
– y
3
= (x – y) (x
2
+ xy + y
2
)
•If ax
2
+ bx + c = 0,
Then
2
4
2
b
b
ac
x
a
− ±
=
The basic algebraic properties of real numbers a, b and c are:
Property
Description
Closure
a + b and ab are real numbers
Commutative
a + b = b + a, ab = ba
Associative
(a+b) + c = a + (b+c), (ab)c = a(bc)
Distributive
(a+b)c = ac+bc
Chapter 4
Formulas and Conversions
- 28 -
Identity
a+0 = 0+a = a
Inverse
a + (-a) = 0, a(1/a) = 1
Cancellation
If a+x=a+y, then x=y
Zero-factor
a0 = 0a = 0
Negation
-(-a) = a, (-a)b= a(-b) = -(ab), (-a)(-b) = ab
Algebraic Combinations
Factors with a common denominator can be expanded:
c
b
c
a
c
b
a
+
=
+
Fractions can be added by finding a common denominator:
cd
bc
d
b
c
a
+
=
+
Products of fractions can be carried out directly:
cd
ab
d
b
c
a
=
×
Quotients of fractions can be evaluated by inverting and multiplying:
bc
c
d
b
a
d
c
b
a
=
×
=
n
n
n
b
a
ab =
n
n
a
a
/1
=
n
n
n
b
a
b
a
=
n
m
n
m
a
a =
mn
n m
a
a =
Formulas and Conversions
- 29 -
4.2 Geometry
Item
Circumference
/ Perimeter
Area
Surface Area
Volume
Figure
Square
4s
s
2
NA
NA
Rectangle
2 (L + B)
= L·B
NA
NA
Formulas and Conversions
- 30 -
Item
Circumference
/ Perimeter
Area
Surface Area
Volume
Figure
Triangle
s
1
+ s
2
+ s
3
where s
1,
s
2,
s
3
are the 3 sides
of the triangle
H
B×
×
2
1
NA
NA
Right
triangle
s
1
+ s
2
+ s
3
H
B×
×
2
1
NA
NA
Formulas and Conversions
- 31 -
Item
Circumference
/ Perimeter
Area
Surface Area
Volume
Figure
Generic
triangle
s
1
+ s
2
+ s
3
)
)(
)(
(
c
s
b
s
a
s
s
where
2
c
b
a
s
+
+
=
NA
NA
Equilateral
triangle
3s
where s is the
length of each
side
bh
A
2
1
=
NA
NA
Formulas and Conversions
- 32 -
Item
Circumference
/ Perimeter
Area
Surface Area
Volume
Figure
Trapezoid
where Ө and Φ are
the 2 base angles
h
b
a
A
⎛ +
=
2
NA
NA
Circle
C = 2πr
C = πd
A = πr
2
NA
NA
Formulas and Conversions
- 33 -
Item
Circumference
/ Perimeter
Area
Surface Area
Volume
Figure
Circle
Sector
2r + (arc
length)
2
r
arc
A
×
=
2
360
r
A
π
θ
×
°
=
2
2
r
A
°
=
θ
NA
NA
Ellipse
(1/4)·D·d·∏
where D and d
are the two axis
Dd
A
4
π
=
D is the larger
radius and d is the
NA
NA
Formulas and Conversions
- 34 -
Item
Circumference
/ Perimeter
Area
Surface Area
Volume
Figure
Trapezoid
Sum of all sides
h
b
b
A
)
(
2
1
2
1
+
=
NA
NA
Hexagon
6s
A = 2.6s
2
Where s is the
length of 1 side
NA
NA
Formulas and Conversions
- 35 -
Item
Circumference
/ Perimeter
Area
Surface Area
Volume
Figure
Octagon
8s
A = 4.83 s
2
Where s is the
length of 1 side
NA
NA
Cube
NA
NA
6s
2
s
3
Formulas and Conversions
- 36 -
Item
Circumference
/ Perimeter
Area
Surface Area
Volume
Figure
Rectangular
solid
NA
NA
2
l
h + 2wh + 2
l × w × h
Right
cylinder
NA
NA
S = 2πrh +
2πr
2
V = πr
2
h
Formulas and Conversions
- 37 -
Item
Circumference
/ Perimeter
Area
Surface Area
Volume
Figure
Sphere
NA
NA
S = 4πr
2
3
4
πr
3
Pyramid
NA
NA
½.perimeter·
slant height +
B
3
1
base area·
perpendicular
height
Formulas and Conversions
- 38 -
Item
Circumference
/ Perimeter
Area
Surface Area
Volume
Figure
Rectangular
prism
NA
NA
2lh+2lw+2wh
V = lwh
Cone
NA
NA
pi·r(r+sh)
3
1
πr
2
h
Formulas and Conversions
- 39 -
4.3 Trigonometry
A. Pythagoras' Law
c
2
= a
2
+ b
2
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
•A circle in degree contains 360 degrees
•A circle in radians contains 2π radians
Sine, Cosine and Tangent
sin
opposite
hypotenus
θ
=
cos
hypotenus
θ
=
tan
opposite
θ
=
Sine, Cosine and the Pythagorean Triangle
[
] [
]
2
2
2
2
sin
cos
1
sin
cos
θ
θ
θ
θ
+
=
+
=
c
a
b
θ
θ
opposite
hypotenuse
Formulas and Conversions
- 40 -
Tangent, Secant and Co-Secant
sin
tan
cos
θ
θ
θ
=
1
sec
cos
θ
θ
=
1
csc
sin
θ
θ
=
C. Trigonometric Function Values
Euler's Representation
cos( )
sin( )
j
e
j
θ
θ
θ
=
+
cos( )
sin( )
j
e
j
θ
θ
θ
=
cos( )
sin( )
jn
e
n
j
n
θ
θ
θ
=
+
cos
2
j
j
e
e
θ
θ
θ
+
=
sin
2
j
j
e
e
j
θ
θ
θ
=
4.4 Logarithm
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 = 2
3
The equation can be rewritten in logarithm form as
2
log 8 = 3
Logarithm laws
The logarithm laws are obtained from the index laws and are:
•log
a
x + log
a
y = log
a
xy
Formulas and Conversions
- 41 -
•log
a
x – log
a
y = log
a
(x/y)
•log
a
xy = y log
a
x
•log
a
(1/x) = -log
a
x
•log
a
1 = 0
•log
a
a = 1
x
a
x
a
=
)
(log
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:
x
j
x
e
jx
sin
cos +
=
x
j
x
e
jx
sin
cos −
=
From these relationships the trig functions can be expressed in terms of the complex
exponential:
2
cos
jx
jx
e
e
x
+
=
2
sin
jx
jx
e
e
x
=
Hyperbolic Functions
The hyperbolic functions can be defined in terms of exponentials.
Hyperbolic sine = sinh x =
2
x
x
e
e
Hyperbolic cosine = cosh x =
2
x
x
e
e
+
Hyperbolic tangent = tanh x =
x
x
x
x
e
e
e
e
x
x
+
=
cosh
sinh
Formulas and Conversions
- 42 -
4.5 Exponents
Summary of the Laws of Exponents
Let c, d, r, and s be any real numbers.
s
r
s
r
c
c
c
+
=
r
r
r
d
c
d
c
=
⋅ )
(
0
=
c,
c
c
c
s
r
s
r
0
=
d,
d
c
d
c
r
r
r
s
r
s
r
c
c
=
)
(
r
r
c
c
1
=
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
2
1
2
1
p
p
p
p
n
n
n
=
+
The rule for raising a power to a power can also be deduced
(n
a
)
b
= n
ab
(ab)
n
= a
n
b
n
a
m
/a
n
= a
m-n
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
a + jb where j = √-1 and j · j = -1
Complex numbers can also be expressed in polar form
Ae
where A = √a
2
+b
2
and θ = tan
-1
(b/a)
The polar form can also be expressed in terms of trigonometric functions using the Euler
relationship
e
= cos θ + j sin θ
Euler Relationship
The trigonometric functions are related to a complex exponential by the
Euler relationship
e
jx
= cos x + j sin x
Formulas and Conversions
- 43 -
e
-jθ
= cos x - j sin x
From these relationships the trigonometric functions can be expressed in terms of the
complex exponential:
2
cos
jx
jx
e
e
x
+
=
2
sin
jx
jx
e
e
x
=
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
Ae
where A = √a
2
+b
2
and θ = tan
-1
(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.
)
tan
sin
tan
(cos
1
1
2
2
+
+
=
a
b
j
a
b
b
a
Ae
j
θ
jb
a
b
a
b
j
b
a
a
b
a
Ae
j
+
=
+
+
+
+
=
)
(
2
2
2
2
2
2
θ
Formulas and Conversions
- 44 -
Engineering Concepts and Formulae
5.1 Electricity
Ohm's Law
R
V
I =
Or
V = IR
Where
I = current (amperes)
E = electromotive force (volts)
R = resistance (ohms)
Temperature correction
R
t
= Ro (1 + αt)
Where
Ro = resistance at 0ºC (.)
R
t
= resistance at tºC (.)
α = temperature coefficient which has an average value for copper of 0.004
28 (Ω/Ω ºC)
)
1(
)
1(
1
2
1
2
t
t
R
R
α
α
+
+
=
Where R
1
= resistance at t
1
R
2
= resistance at t
2
Values of
alpha
Ω/Ω ºC
Copper
0.00428
Platinum
0.00358
Nickel
0.00672
Tungsten
0.00450
Chapter 5
Formulas and Conversions
- 45 -
Aluminum
0.0040
Current,
nqvA
t
nqvtA
I
=
=
Conductor Resistivity
a
L
R
ρ
=
Where
ρ = specific resistance (or resistivity) (ohm meters, Ωm)
L = length (meters)
a = area of cross-section (square meters)
Quantity
Equation
Resistance R of a uniform
conductor
A
L
R
ρ
=
Resistors in series,
s
R
s
R
= R
1
+ R
2
+ R
3
Resistors in parallel,
p
R
3
2
1
1
1
1
1
R
R
R
R
p
+
+
=
Power dissipated in resistor:
R
V
R
I
VI
P
2
2
=
=
=
Potential drop across R
V = I R
Dynamo Formulae
Average e.m.f. generated in each conductor =
c
NpZ
60
2
ϕ
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
Formulas and Conversions
- 46 -
Where EG = generated e.m.f.
EB = generated back e.m.f.
Ia = armature current
Ra = armature resistance
Alternating Current
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
Frequency of Alternator =
60
pN
cycles per second
Where p is number of pairs of poles
N is the rotational speed in r/min
Slip of Induction Motor
[(Slip speed of the field – Speed of the rotor) / Speed of the Field] × 100
Inductors and Inductive Reactance
Physical Quantity
Equation
Inductors and Inductance
V
L
= L
td
id
Inductors in Series:
L
T
= L
1
+ L
2
+ L
3
+ . . . .
Inductor in Parallel:
.....
L
1
L
1
L
1
L
1
3
2
1
T
+
+
+
=
Current build up
(switch initially closed after having
been opened)
At
τ
t
-
L
e
E
t)(
v
=
)
e-
E(1
t)(
v
t
R
τ
=
τ
t
-
e
1(
R
E
i(t)
=
)
τ =
R
L
Current decay
(switch moved to a new position)
τ
=
t
-
o
e
I
i(t)
v
R
(t) = R i(t)
v
L
(t) = − R
T
i(t)
Formulas and Conversions
- 47 -
Physical Quantity
Equation
τ' =
T
R
L
Alternating Current
f = 1/T
ϖ = 2 π f
Complex Numbers:
C = a + j b
C = M cos θ + j M sin θ
2
2
b
a
M
+
=
θ
=
a
b
tan
1-
Polar form:
C = M ∠ θ
Inductive Reactance
|X
L
| = ω L
Capacitive Reactance
|X
C
| = 1 / (ω C)
Resistance
R
Impedance
Resistance: Z
R
= R ∠0°
Inductance: Z
L
= X
L
∠90° = ω L ∠90°
Capacitance: Z
C
= X
C
∠-90° = 1 / (ωC)
∠-90°
Quantity
Equation
Ohm's Law for AC
V = I Z
Time Domain
v(t) = V
m
sin (ω t ± φ)
i(t) = I
m
sin (ω t ± φ)
Phasor Notation
V = V
rms
∠ φ
V = V
m
∠ φ
Components in Series
Z
T
= Z
1
+ Z
2
+ Z
3
+ .
.
Voltage Divider Rule
T
x
T
x
Z
Z
V
V =
Components in Parallel
...
Z
1
Z
1
Z
1
Z
1
3
2
1
T
+
+
+
=
Formulas and Conversions
- 48 -
Quantity
Equation
Current Divider Rule
x
T
T
x
Z
Z
I
I =
Two impedance values in
parallel
2
1
2
1
T
Z
Z
Z
Z
Z
+
=
Capacitance
Capacitors
C =
V
Q
Capacitor in Series
.....
C
1
C
1
C
1
C
1
3
2
1
T
+
+
+
=
Capacitors in Parallel
.....
C
C
C
C
3
2
1
T
+
+
+
=
Charging a Capacitor
RC
t
-
e
R
E
i(t) =
RC
t
-
R
e
E
t)(
v
=
)
e-
E(1
t)(
v
RC
t
-
C
=
τ = RC
Discharging a
Capacitor
τ
=
t
-
o
e
R
V
i(t)
τ
=
t
-
o
R
e
V
t)
v (
τ
=
t
-
o
C
e
V
t)
v (
τ' = R
T
C
Quantity
Equation
Capacitance
V
Q
C =
Formulas and Conversions
- 49 -
Quantity
Equation
Capacitance of a
Parallel-plate Capacitor
d
A
C
ε
=
d
V
E =
Isolated Sphere
C = 4πεr
Capacitors in parallel
C = C
1
+ C
2
+ C
3
Capacitors in series
3
2
1
1
1
1
1
C
C
C
C
+
+
=
Energy stored in a
charged capacitor
QV
CV
C
Q
W
2
1
2
1
2
2
2
=
=
=
If the capacitor is
isolated
C
Q
W
2
2
=
If the capacitor is
connected to a battery
2
2
1
CV
W =
For R C circuits
Charging a capacitor
Q = Q
o
(1 - e
-t/RC
);
V = V
o
(1 - e
-t/RC
)
Discharging a capacitor
Q = Q
o
e
- t/RC
V = V
o
e
-t/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
Formulas and Conversions
- 50 -
Current in AC Circuit
RMS Current
In Cartesian
form
+
=
C
L
j
R
C
L
R
V
I
ω
ω
ω
ω
1
1
2
2
Amperes
In polar form
s
C
L
R
V
I
φ
ω
ω
+
=
]
1
[
2
2
Amperes
where
=
R
C
L
s
ω
ω
φ
1
tan
1
Modulus
2
2
1
+
=
C
L
R
V
I
ω
ω
Amperes
Complex Impedance
In Cartesian
form
+
=
C
L
j
R
Z
ω
ω
1
Ohms
In polar form
s
C
L
R
Z
φ
ω
ω
+
=
2
2
1
Ohms
Where
=
R
C
L
s
ω
ω
φ
1
tan
1
Modulus
=
Z
2
2
1
[
+
C
L
R
ω
ω
] Ohms
Formulas and Conversions
- 51 -
Power dissipation
Average power,
φ
cos
VI
P =
Watts
Power dissipation in a
resistor
R
I
P
2
=
Watts
Rectification
Controlled half wave
rectifier
Average DC voltage
(
)
α
π
cos
1
2
+
=
m
V
Volts
Controlled full wave
rectifier
Average DC voltage
(
)
α
π
cos
1+
=
m
V
Volts
Power Factor
DC
Power
R
V
R
I
VI
P
dc
2
2
=
=
=
AC
Power
φ
cos
).
Re(
VI
I
V
Pac
=
=
Power in ac circuits
Quantity
Equation
Resistance
The mean power =
P
= I
rms
V
rms
= I
rms
2
R
Inductance
The instantaneous power = (Io sin wt) (Vo sin (wt +
π)
The mean power
P
= 0
Capacitance
The instantaneous power = (Io sin (wt + π/2)) (V
o
sin
wt )
The mean power
P
= 0
Formula for a.c.
power
The mean power =
P
= I
rms
V
rms
cos
φ
Formulas and Conversions
- 52 -
Three Phase Alternators
Star connected
Line voltage =
Voltage
Phase
3 •
Line current = phase current
Delta connected
Line voltage = phase voltage
Line current =
Current
Phase
3•
Three phase power
P =
φ
Cos
3
L
L
I
E
Where:
P is the active power in Watts
E
L
is the Line Voltage in Volts
I
L
is the line current in Amperes
Cos
φ
is the power factor
Electrostatics
Quantity
Equation
Instantaneous current,
dt
dv
C
dt
dq
I
=
=
Amperes
Permittivity of free space
12
9
0
10
85
.8
36
10
×
=
=
π
ε
(meters)
-1
Energy stored in a
capacitor
2
2
1
CV
=
Joules
Quantity
Equation
Coulomb's law
2
2
1
r
Q
Q
k
F =
Electric fields
q
F
E =
Due to a point charge
2
4
r
Q
E
o
πε
=
Formulas and Conversions
- 53 -
Quantity
Equation
Due to a conducting sphere carrying charge
Q Inside the sphere
E = 0
Outside the sphere
2
4
r
Q
E
o
πε
=
Just outside a uniformly charged conducting
sphere or plate
o
E
ε
σ
=
•An electric field E is a vector
•The electric field strength is directly proportional to the number of electric field lines
per unit cross-sectional 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
r
Qq
W
o
πε
4
=
Electric potential
q
W
V =
Due to a point charge
r
Q
V
o
πε
4
=
Due to a conducting sphere, of radius a, carrying charge
Q:
Inside the sphere
a
Q
V
o
πε
4
=
Outside the sphere
r
Q
V
o
πε
4
=
If the potential at a point is V, then the potential energy
of a charge q at that point is
U = qV
Formulas and Conversions
- 54 -
Quantity
Equation
Work done in bringing charge q from A of potential V
A
to
point B of potential V
B
W = q (V
B
– V
A
)
Relation between E and V
dx
dV
E
=
For uniform electric field
d
V
E =
Magnetostatics
Physical Quantity
Equation
Magnetic flux density (also called the B-
field) is defined as the force acting per unit
current length.
λI
F
B =
Force on a current-carrying conductor in a
magnetic field
F = I
λ
B
F
= I
λ
·
B
And Magnitude of
F
= F = I
λ
B
sin θ
Force on a moving charged particle in a
magnetic field
F = q
v
·
B
Circulating Charges
r
mv
qvB
2
=
Calculation of magnetic flux density
Physical Quantity
Equation
Magnetic fields around a long straight wire
carrying current I
a
I
B
o
π
μ
2
=
where a = perp. distance from a
very long straight wire.
Magnetic fields inside a long solenoid,
carrying current
I: B = μ
o
n I, where n = number of
turns per unit length.
Hall effect
At equilibrium
QvB
d
V
Q
H
=
and V
H
= B v d
Formulas and Conversions
- 55 -
Physical Quantity
Equation
The current in a material is given by
I = nQAv
The forces between two current-carrying
conductors
a
I
I
F
o
π
μ
2
2
1
21
λ
=
Physical Quantity
Equation
The torque on a rectangular coil in a magnetic
field
T = F b sin θ
= N I
λ
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 90
o
= N I A B
Magnetic flux φ
φ = B A cos θ
and
φ
N
Current Sensitivity
c
NAB
I
S
I
=
=
θ
Lenz's law
The direction of the induced e.m.f. is such that it tends to
oppose the flux-change causing it, and does oppose it if
induced current flows.
φ
ε
dt
d
N
=
EMF Equations
E.m.f. induced in a straight conductor
ε = B L v
E.m.f. induced between the center and the rim of a spinning
disc
ε = B πr
2
f
E.m.f. induced in a rotating coil
Ε = N A B w sin
wt
Quantity
Equation
Self-induction
dt
dI
L
/
ε
=
Formulas and Conversions
- 56 -
Quantity
Equation
N
φ
= L I
Energy stored in an inductor:
2
2
1
LI
U =
Transformers:
P
S
P
S
N
N
V
V
=
The L R (d.c.) circuit:
)
1(
/ L
Rt
e
R
E
I
=
When a great load (or smaller
resistance) is connected to
the secondary coil, the flux in
the core decreases. The
e.m.f., ε
p
, in the primary coil
falls.
V
p
p = I R;
R
V
I
p
P
ε
=
Kirchoff's laws
Kirchoff's first law (Junction Theorem)
At a junction, the total current entering the junction is equal to the total
current leaving the junction.
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
Equation
Power
P =
W
t
VI
=
Electric current
I =
q
t
Work
W = qV
Ohm's Law
V = IR
Resistances in Series
R
R
R
T
1
2
=
+
Κ
Resistances in Parallel
1
R
R
R
T
1
2
=
+
1
1
Κ
Formulas and Conversions
- 57 -
Magnetic flux
Φ = BA
Electromagnetic
induction
Emf
= −N
(
t
Φ
Φ
2
1
)
emf = vB
l
Magnetic force
F I B
= l
Transformer turns ratio
Vs
=
Ns
Vp Np
Electromagnetic spectrum
Wavelength
λ (m)
10
2
10
1 10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
-9
10
-10
10
-11
Area of
Spectrum
f(Hz)
10
6
10
7
10
8
10
9
10
10
10
11
10
12
10
13
10
14
10
15
10
16
10
17
10
18
10
19
10
20
Frequency
Note: 1. Shaded areas represent regions of overlap.
2. Gamma rays and X-rays occupy a common region.
5.2 Applied Mechanics
5.2.1 Newton's laws of motion
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
.
Newton's second law of motion
F =
m v - m u
Δ t
;
microwaves
ultraviolet
X-rays
gamma rays
vis
ibl
e
Formulas and Conversions
- 58 -
F = m a
Impulse = force · time = change of momentum
F t = m v – m u
Newton's third law of motion
When two objects interact, they exert equal and opposite forces on one another.
"Third-law pair" of forces act on two different bodies.
Universal Law
F = Gm
s
m
p
/d
2
m
s
is the mass of the sun.
m
p
is the mass of the planet.
The Universal law and the second law must be consistent
Newton's Laws of Motion and Their Applications
Physical Quantity
Equations
Average velocity
v
s
t
v + u
2
av
= =
Acceleration
a =
v-u
t
Momentum
p = mv
Force
F = ma
Weight
weight = mg
Work done
W = Fs
Kinetic energy
E
mv
k
2
=
1
2
Gravitational potential energy
E
mgh
p
=
Equations of motion
a =
v u
t
s = ut + at
v
u
as
1
2
2
2
2
=
+
;
;
2
Centripetal acceleration
a =
v
r
2
Centripetal force
F= ma =
mv
r
2
Formulas and Conversions
- 59 -
Physical Quantity
Equations
Newton's Law of Universal
Gravitation
F= G
m m
r
1
2
2
Gravitational field strength
g = G
M
r
2
Physical Quantity
Equations
Moment of a force
M = rF
Principle of
moments
∑ =
M 0
Stress
Stress
F
A
=
Strain
Strain =
Δ l
l
Young's Modulus
Y
F/ A
=
Δ l l/
Scalar: a property described by a magnitude only
Vector: a property described by a magnitude and a direction
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
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 vacuum equals 3 x 10
8
m/s
Acceleration: vector property equal to change in velocity time.
In SI the basic unit is m/s
2
Formulas and Conversions
- 60 -
In Imperial ft/s
2
Conversion:
2
2
28
.3
1
s
ft
s
m
=
Acceleration due to gravity, g is 9.81 m/s
2
5.2.2 Linear Velocity and Acceleration
Quantity
Equations
If u initial velocity and v final velocity,
then displacement s,
⎛ +
=
2
u
v
s
If t is the elapsed time
2
2
1
at
ut
s
+
=
If a is the acceleration
as
u
v
2
2
2
+
=
Angular Velocity and Acceleration
Quantity
Equations
t×
+
=
2
2
1
ω
ω
θ
θ angular displacement
•ω angular velocity (radians/s);
ω
1
= initial, ω
2
= final
2
1
2
1
t
t
α
ω
θ
+
=
α angular acceleration
2
)
αθ
ω
ω
2
2
1
2
2
+
=
Linear displacement
s = r θ
Linear velocity
v = r ω
Linear, or tangential
acceleration
aT = r α
Tangential, Centripetal and Total Acceleration
Quantity
Equations
Formulas and Conversions
- 61 -
Tangential acceleration aT is due to angular acceleration
α
aT = rα
Centripetal (Centrifugal) acceleration ac is due to change
in direction only
ac = v
2
/r = r ω
2
Total acceleration, a, of a rotating point experiencing
angular acceleration is the vector sum of aT and ac
a = aT + ac
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/s
2
In Imperial, the mass of an object (rarely used), in slugs, can be calculated from the
known weight in pounds
g
weight
m =
2
2.
32
s
ft
g =
Torque Equation
T = I α where T is the acceleration torque in Nm, I is the moment of inertia in kg m
2
and
α is the angular acceleration in radians/s
2
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 = F s
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 ft-lb
Energy
Energy is the ability to do work, the units are the same as for work; J, kJ, and ft-lb
Formulas and Conversions
- 62 -
Kinetic Energy
2
2
2
1
ω
mk
E
R
=
Where k is radius of gyration, ω is angular velocity in rad/s
Kinetic Energy of Rotation
2
2
1
ω
I
Er =
Where I = mk
2
is the moment of inertia
5.2.4 Centripetal (Centrifugal) Force
r
mv
F
c
2
=
Where r is the radius
Where ω is angular velocity in rad/s
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 ft-lb
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 10
6
ft-lb
Formulas and Conversions
- 63 -
Power
A scalar quantity, equal to the rate of doing work
In SI the unit is the Watt W (or kW)
s
J
W 1
1 =
In Imperial, the units are:
Mechanical Power – (ft – lb) / s, horsepower h.p.
Thermal Power – Btu / s
Electrical Power - W, kW, or h.p.
Conversions
..
1
746
p
h
W =
s
lb
ft
p
h
= 550
..
1
s
Btu
kW
948
.0
1
=
Pressure
A vector quantity, force per unit area
In SI the basic units of pressure are pascals Pa and kPa
2
1
1
m
N
Pa =
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 H
2
O = 9.81 kPa
Other pressure unit conversions:
Formulas and Conversions
- 64 -
•1 bar = 14.5 psi = 100 kPa
•1 kg/cm
2
= 98.1 kPa = 14.2 psi = 0.981 bar
•1 atmosphere (atm) = 101.3 kPa = 14.7 psi
Simple Harmonic Motion
Velocity of P =
s
m
x
R
2
2
ω
5.2.5 Stress, Strain And Modulus Of Elasticity
Young's modulus and the breaking stress for selected materials
Material
Young modulus
x 10
11
Pa
Breaking stress
x 10
8
Pa
Aluminium
0.70
2.4
Copper
1.16
4.9
Brass
0.90
4.7
Iron (wrought)
1.93
3.0
Mild steel
2.10
11.0
Glass
0.55
10
Tungsten
4.10
20
Bone
0.17
1.8
5.3 Thermodynamics
5.3.1 Laws of Thermodynamics
•W = PΔV
•ΔU = Q – W
•W= nRT lnV
f
/V
i
•Q = CnΔT
•C
v
= 3/2R
•C
p
= 5/2R
•C
p
/C
v
= γ= 5/3
•e = 1 – Qc/Q
h
= W/Q
h
•e
c
= 1 – T
c
/T
h
•COP = Q
c
/W (refrigerators)
•COP = Q
h
/W (heat pumps)
•Wmax= (1-T
c
/T
h
)Q
h
•ΔS = Q/T
Formulas and Conversions
- 65 -
5.3.2 Momentum
•p = mv
•∑F = Δp/Δt
5.3.3 Impulse
I = F
av
t = mv
f
– mv
i
5.3.4 Elastic and Inelastic collision
•m
i
v
1i
+ m
2
v
2i
= m
1
v
1f
+ m
2
v
2f
•(½) m
i
v
1i
2
+ (½) m
2
v
2i
2
= ½ m
1
v
1f
2
+ ½ m
2
v
2f
2
•m
i
v
1i
+ m
2
v
2i
= (m
1
+ m
2
)v
f
5.3.5 Center of Mass
•x
cm
= ∑mx/M
•V
cm
= ∑mv/M
•A
cm
= ∑ma/M
•MA
cm
= F
net
5.3.6 Angular Motion
•s = rθ
•v
t
= rω
•a
t
= rα
•a
c
= v
t
2
/r = rω
2
•ω = 2π/T
•1 rev = 2π rad = 360
o
For constant α
•ω = ω
o
+ αt
•ω
2
= ω
o
2
+2αθ
•θ = ω
o
t + ½αt
2
•θ = (ω
o
+ ω)·t/2
•I = ∑mr
2
•KE
R
= ½Iω
2
•τ = rF
•∑τ = Iα
•W
R
= τθ
•L = Iω
•∑τ = Iα
•W
R
= τθ
•L = Iω
•L
i
= L
f
Formulas and Conversions
- 66 -
5.3.7 Conditions of Equilibrium
•∑ F
x
= 0
•∑ F
y
= 0
•∑τ = 0
(any axis)
5.3.8 Gravity
•F = Gm
1
m
2
/r
2
•T = 2π / √r
3
/GM
s
•G = 6.67 x 10
-11
N-m
2
/kg
2
•g = GM
E
/ R
2
E
•PE = - Gm
1
m
2
/ r
•v
e
= √2GM
E
/ R
E
•v
s
= √GM
E
/ r
•M
E
= 5.97 x 10
24
kg
•R
E
= 6.37 x 10
6
m
5.3.9 Vibrations & Waves
•F = -kx
•PE
s
= ½kx
2
•x = Acosθ = Acos(ωt)
•v = -Aωsin(ωt)
•a = -Aω
2
cos(ωt)
•ω = √k / m
•f = 1 / T
•T = 2π√m / k
•E = ½kA
2
•T = 2π√L / g
•v
max
= Aω
•a
max
= Aω
2
•v = λ f
v = √F
T
•μ = m/L
•I = P/A
•β = 10log(I/I
o
)
•I
o
= 1 x 10
-12
W/m
2
•f
'
= f[(1 ± v
0
/v)/(1 μ v
s
/v)]
•Surface area of the sphere = 4πr
2
•Speed of sound waves = 343 m/s
5.3.10 Standing Waves
•f
n
= nf
1
•f
n
= nv/2L (air column, string fixed both ends) n = 1,2,3,4…….
•f
n
= nv/4L (open at one end) n = 1,3,5,7………
5.3.11 Beats
Formulas and Conversions
- 67 -
f
beats
= | f
1
– f
2
|
•Fluids
•ρ = m/V
•P = F/A
•P
2
= P
1
+ ρgh
•P
atm
= 1.01 x 10
5
Pa = 14.7 lb/in
2
F
B
= ρ
f
Vg = W
f
(weight of the displaced fluid)
•ρ
o
f
= V
f
/V
o
(floating object)
ρ
water
= 1000 kg/m
3
• W
a
=W-F
B
Equation of Continuity: Av = constant
Bernoulli's equation: P + ½ ρv
2
+ ρgy = 0
5.3.12 Temperature and Heat
•T
F
= (9/5) T
C
+32
T
C
= 5/9(T
F
-32)
•∆T
F
= (9/5) ∆T
C
•T= T
C
+273.15
•ρ= m/v
•∆L = αL
o
ΔT
•ΔA = γA
o
ΔT
•ΔV = βV
o
ΔT β=3α
•Q = mcΔT
•Q = mL
•1 kcal = 4186 J
•Heat Loss = Heat Gain
•Q = (kAΔT)t/L,
•H = Q/t =(kAΔT)/L
•Q = eσT
4
At
•P = Q/t
•P = σAeT
4
•P
net
= σAe(T
4
-T
S
4
)
•σ = 5.67 × 10
-8
W/m
2
K
4
5.3.13 Ideal Gases
•PV = nRT
•R = 8.31 J/mol K
•PV = NkT
•N
A
= 6.02 × 10
23
molecules/mol
•k = 1.38 × 10
-23
J/K
•M=N
A
m
•(KE)
av
=(1/2mv
2
)
av
= 3/2kT
•U= 3/2NkT = 3/2nRT
Formulas and Conversions
- 68 -
5.3.14 Elastic Deformation
•P = F/A
•Y = FL
o
/A∆L
•S = Fh/A∆x
•B = –V
o
∆F / A∆V
•Volume of the sphere = 4πr
3
/3
•1 atm = 1.01 × 10
5
Pa
5.3.15 Temperature Scales
•°C = 5/9 (°F – 32)
•°F = (9/5) °C + 32
•°R = °F + 460 (R Rankine)
•K = °C + 273 (K Kelvin)
5.3.16 Sensible Heat Equation
•Q=mcΔT
•M=mass
•C=specific heat
•ΔT=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
P
1
V
1
= P
2
V
2
Where P is absolute pressure and V is volume
Charles' Law
When gas pressure is constant,
.
const
T
V
=
or
2
2
1
1
T
V
T
V
=
where V is volume and T is absolute temperature
Formulas and Conversions
- 69 -
Gay-Lussac's Law
When gas volume is constant,
.
const
T
P
=
or
2
2
1
1
T
P
T
P
=
where P is absolute pressure and T is absolute temperature
General Gas Law
.
2
2
2
1
1
1
const
T
V
P
T
V
P
=
=
P V = m R T where P = absolute pressure (kPa)
V = volume (m
3
)
T = absolute temp (K)
m = mass (kg)
R = characteristic constant (kJ/kgK)
Also
PV = nRoT where P = absolute pressure (kPa)
V = volume (m
3
)
T = absolute temperature K
N = the number of kmoles of gas
Ro = the universal gas constant 8.314 kJ/kmol/K
5.3.19 Specific Heats Of Gases
GAS
Specific Heat
at Constant
Pressure
kJ/kgK or
kJ/kg
o
C
Specific Heat
at Constant
Volume
kJ/kgK or
kJ/kg
o
C
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
1.051
0.751
1.40
Formulas and Conversions
- 70 -
GAS
Specific Heat
at Constant
Pressure
kJ/kgK or
kJ/kg
o
C
Specific Heat
at Constant
Volume
kJ/kgK or
kJ/kg
o
C
Ratio of
Specific
γ= cp / cv
Monoxide
Helium
5.234
3.153
1.66
Hydrogen
14.235
10.096
1.41
Hydrogen
Sulphide
1.105
0.85
1.30
Methane
2.177
1.675
1.30
Nitrogen
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
1
2
1
T
T
T
=
η
where T
1
and T
2
are absolute temperatures of heat source and sink
Air Standard Efficiencies
Spark Ignition Gas and Oil Engines (Constant Volume Cycle)
)1
(
1
1
=
γ
η
v
r
r
v
= compression ratio
γ = specific heat (constant pressure) / Specific heat (constant volume)
Diesel Cycle
)1
(
)1
1
1
=
R
r
R
v
γ
γ
η
γ
Where r = ratio of compression
R = ratio of cut-off volume to clearance volume
High Speed Diesel (Dual-Combustion) Cycle
Formulas and Conversions
- 71 -
[
]
)1
(
)1
(
1
1
1
+
=
β
γ
β
η
γ
γ
k
k
r
k
v
Where r
v
= 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)
⎛ −
=
γ
γ
η
1
1
1
p
r
where r
p
= pressure ratio = compressor discharge pressure / compressor intake pressure
5.3.21 Heat Transfer by Conduction
Material
Coefficient of Thermal
Conductivity
W/m °C
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
Felt
0.038
Glass, fibre
0.04
Plastic, cellular
0.04
Wood
0.15
Formulas and Conversions
- 72 -
5.3.22 Thermal Expansion of Solids
Increase in length = L α (T
2
– T
1
)
Where L = original length
α = coefficient of linear expansion
(T
2
– T
1
) = rise in temperature
Increase in volume = V β (T
2
– T
1
)
Where V = original volume
β = coefficient of volumetric expansion
(T
2
– T
1
) = 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 =
S
O
H
C
3.
9
)
8
(
144
7.
33
2
2
+
+
C is the mass of carbon per kg of fuel
H
2
is the mass of hydrogen per kg of fuel
O
2
is the mass of oxygen per kg of fuel
S is the mass of sulphur per kg of fuel
Theoretical Air Required to Burn Fuel
Air (kg per kg of fuel) =
23
100
)
(8
3
8
2
2
+
+
S
O
H
C
Air Supplied from Analysis of Flue Gases
Air in kg per kg of fuel =
C
CO
CO
N
×
+
)
(
33
2
2
Boiler Formulae
Equivalent evaporation
kg
kj
h
h
m
s
/
2257
)
(
2
1
=
Factor of evaporation
kg
kj
h
h
/
2257
)
(
2
1
=
Boiler Efficiency
)
(
)
(
2
1
alue
calorificv
mf
h
h
m
s
×
Where
m
s
= mass flow rate of steam
h
1
= enthalpy of steam produced in boiler
h
2
= enthalpy of feedwater to boiler
Formulas and Conversions
- 73 -
m
f
= mass flow rate of fuel
Formulas and Conversions
- 74 -
P-V-T Relationships
Name of
process
Value
of n
P-V
T-P
T-V
Work done
Change in
Internal
Energy
Change in
Enthalpy
Change in
Entropy
Constant
Volume
V=Constant
--
2
1
2
1
P
P
T
T
=
--
(
)
1
2
T
T
mc
v
0
(
)
1
2
T
T
mc
v
(
)
1
2
T
T
mc
p
1
2
log
T
T
mc
e
v
Constant
pressure
P=Pressure
0
--
--
2
1
2
1
V
V
T
T
=
(
)
1
2
T
T
mc
p
P(V
2
-V
1
)
(
)
1
2
T
T
mc
v
(
)
1
2
T
T
mc
p
1
2
log
T
T
mc
e
n
Isothermal
T=Constant
1
1
2
2
1
V
V
P
P
=
--
--
2
1
log
P
P
mRT
e
2
1
log
P
P
mRT
e
0
0
2
1
log
P
P
mR
e
Isentropic
S=Constant
γ
γ
=
1
2
2
1
V
V
P
P
γ
γ
l
P
P
T
T
=
2
1
2
1
1
1
2
2
1
=
γ
V
V
T
T
0
(
)
2
1
T
T
mc
v
(
)
1
2
T
T
mc
v
(
)
1
2
T
T
mc
p
0
Polytropic
PV
n
=
Constant
n
n
V
V
P
P
=
1
2
2
1
n
l
n
P
P
T
T
=
2
1
2
1
1
1
2
2
1
=
n
V
V
T
T
(
)
1
2
T
T
mc
n
(
)
2
1
1
T
T
n
mR
(
)
1
2
T
T
mc
v
(
)
1
2
T
T
mc
p
1
2
log
T
T
mc
e
n
Thermodynamic Equations for perfect gases
*Can be used for reversible adiabatic processes
c
v
= Specific heat at constant volume, kJ/kgK
c
p
= Specific heat at constant pressure, kJ/kgK
Formulas and Conversions
- 75 -
c
m
= Specific heat for polytropic process =
kgK
kJ
n
n
c
v
/
1
γ
H = Enthalpy, kJ
γ = Isentropic Exponent, c
p
/c
v
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, m
3
m = Mass of gas, kg
Formulas and Conversions
- 76 -
Specific Heat and Linear
Expansion of Solids
Mean Specific Heat between 0
C
o
and 100
C
o
kJ/kgK or kJ/kg
C
o
Coefficient of Linear Expansion
between 0
C
o
and 100
C
o
(multiply by 10
-6
)
Aluminum
0.909
23.8
Antimony
0.209
17.5
Bismuth
0.125
12.4
Brass
0.383
18.4
Carbon
0.795
7.9
Cobalt
0.402
12.3
Copper
0.388
16.5
Glass
0.896
9.0
Gold
0.130
14.2
Ice (between -20
C
o
& 0
C
o
)
2.135
50.4
Iron (cast)
0.544
10.4
Iron (wrought)
0.465
12.0
0.131
29.0
Nickel
0.452
13.0
Platinum
0.134
8.6
Silicon
0.741
7.8
Silver
0.235
19.5
Steel (mild)
0.494
12.0
Tin
0.230
26.7
Zinc
0.389
16.5
Formulas and Conversions
- 77 -
Specific Heat and Volume Expansion for Liquids
Liquid
Specific Heat
(at 20
C
o
)
KJ/kgK or kJ/kg
C
o
Coefficient of Volume Expansion
(Multiply by 10
-4
)
Alcohal
2.470
11.0
Ammonia
0.473
Benzine
1.138
12.4
Carbon Dioxide
3.643
1.82
Mercury
0.139
1.80
Olive oil
1.633
Petroleum
2.135
Gasoline
2.093
12.0
Turpentine
1.800
9.4
Water
4.183
3.7
Formulas and Conversions
- 78 -
5.4 Fluid Mechanics
5.4.1 Discharge from an Orifice
Let A = cross-sectional area of the orifice =
2
4
d
π
And Ac = cross-sectional area of the jet at the vena
conrtacta
2
4
c
d
π
Then Ac = CcA
Or
2
=
=
d
d
A
A
C
c
c
c
Where C
c
is the coefficient of contraction
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 = A
c
V
•Or
gh
AC
C
Q
v
c
2
=
Typically, values for Cd vary between 0.6 and 0.65
•Circular orifice: Q = 0.62 A √2gh
Where Q = flow (m
3
/s) A = area (m
2
) h = head (m)
• Rectangular notch: Q = 0.62 (B · H) 2/3 √2gh
Formulas and Conversions
- 79 -
Where B = breadth (m)
H = head (m above sill)
Triangular Right Angled Notch: Q = 2.635 H
5/2
Where H = head (m above sill)
5.4.2 Bernoulli's Theory
g
v
w
P
h
H
2
2
+
+
=
H = total head (meters)
w = force of gravity on 1 m
3
of fluid (N)
h = height above datum level (meters)
v = velocity of water (meters per second)
P = pressure (N/m
2
or Pa)
Loss of Head in Pipes Due to Friction
Loss of head in meters =
g
v
d
L
f
2
2
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
5.4.3 Actual pipe dimensions
Nominal
pipe size
(in)
Outside
diameter
(mm)
Inside
diameter
(mm)
Wall
thickness
(mm)
Flow area
(m
2
)
1/8
10.3
6.8
1.73
3.660
×
10
-5
1/4
13.7
9.2
2.24
6717
×
10
-5
3/8
17.1
12.5
2.31
1.236
×
10
-4
1/2
21.3
15.8
2.77
1.960
×
10
-4
3/4
26.7
20.9
2.87
3.437
×
10
-4
1
33.4
26.6
3.38
5.574
×
10
-4
42.2
35.1
3.56
9.653
×
10
-4
48.3
40.9
3.68
1.314
×
10
-3
2
60.3
52.5
3.91
2.168
×
10
-3
Formulas and Conversions
- 80 -
Nominal
pipe size
(in)
Outside
diameter
(mm)
Inside
diameter
(mm)
Wall
thickness
(mm)
Flow area
(m
2
)
73.0
62.7
5.16
3.090
×
10
-3
3
88.9
77.9
5.49
4.768
×
10
-3
101.6
90.1
5.74
6.381
×
10
-3
4
114.3
102.3
6.02
8.213
×
10
-3
5
141.3
128.2
6.55
1.291
×
10
-2
6
168.3
154.1
7.11
1.864
×
10
-2
8
219.1
202.7
8.18
3.226
×
10
-2
10
273.1
254.5
9.27
5.090
×
10
-2
12
323.9
303.2
10.31
7.219
×
10
-2
14
355.6
333.4
11.10
8.729
×
10
-2
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
Formulas and Conversions
- 81 -
References
6.1 Periodic Table of Elements
A
1
8A
18
1
H
1.00
8
2A
2
3A
13
4A
14
5A
15
6A
16
7A
17
2
He
4.00
3
3
Li
6.94
1
4
Be
9.01
2
5
B
10.8
1
6
C
12.0
1
7
N
14.0
1
8
O
16.0
0
9
F
19.0
0
10
Ne
20.1
8
11
Na
22.9
9
12
Mg
24.3
1
3B
3
4B
4
5B
5
6B
6
7B
7
8B
8
8B
9
8B
10
1B
11
2B
12
13
Al
26.9
8
14
Si
28.0
9
15
P
30.9
7
16
S
32.0
7
17
Cl
35.4
5
18
Ar
39.9
5
19
K
39.1
0
20
Ca
40.0
8
21
Sc
44.9
6
22
Ti
47.9
0
23
V
50.9
4
24
Cr
52.0
0
25
Mn
54.9
4
26
Fe
55.8
5
27
Co
58.9
3
28
Ni
58.7
0
29
Cu
63.5
5
30
Zn
65.3
8
31
Ga
69.7
2
32
Ge
72.5
9
33
As
74.9
2
34
Se
78.9
6
35
Br
79.9
0
36
Kr
83.8
0
37
Rb
85.4
7
38
Sr
87.6
2
39
Y
88.9
1
40
Zr
91.2
2
41
Nb
92.9
1
42
Mo
95.9
4
43
Tc
97.9
44
Ru
101.
1
45
Rh
102.
9
46
Pd
106.
4
47
Ag
107.
9
48
Cd
112.
4
49
In
114.
8
50
Sn
118.
7
51
Sb
121.
8
52
Te
127.
6
53
I
126.
9
54
Xe
131.
3
55
Cs
132.
9
56
Ba
137.
3
57
La
138.
9
72
Hf
178.
5
73
Ta
180.
9
74
W
183.
8
75
Re
186.
2
76
Os
190.
2
77
Ir
192.
2
78
Pt
195.
1
79
Au
197.
0
80
Hg
200.
6
81
Tl
204.
4
82
Pb
207.
2
83
Bi
209.
0
84
Po
(209)
85
At
(210)
86
Rn
(222)
87
Fr
(223)
88
Ra
226.
0
89
Ac
227.
0
104
Rf
(261)
105
Db
(262)
106
Sg
(266)
107
Bh
(264)
108
Hs
(265)
109
Mt
(268)
58
Ce
140.
1
59
Pr
140.
9
60
Nd
144.
2
61
Pm
(145)
62
Sm
150.
4
63
Eu
152.
0
64
Gd
157.
3
65
Tb
158.
9
66
Dy
162.
5
67
Ho
164.
9
68
Er
167.
3
69
Tm
168.
9
70
Yb
173.
0
71
Lu
175.
0
90
Th
232.
0
91
Pa
231.
0
92
U
238.
0
93
Np
237.
0
94
Pu
(244)
95
Am
(243)
96
Cm
(247)
97
Bk
(247)
98
Cf
(251)
99
Es
(252)
100
Fm
(257)
101
Md
(258)
102
No
(259)
103
Lr
(262)
Chapter 6
Formulas and Conversions
- 82 -
6.2 Resistor Color Coding
Color
Value
Black
0
Brown
1
Red
2
Orange
3
Yellow
4
Green
5
Blue
6
Violet / Purple
7
Grey
8
White
9