Future 2 tech.

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

radian

rad

-

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

Farad

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

Links, (Surveyor's)

Chains

0.01

Links, (Surveyor's)

Inches

7.92

Links, (Surveyor's)

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

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

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

Lead

11.4

Magnesium

1.74

Manganese

8.0

Mercury

13.6

Lead

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)

F (Farads)

Micro

10

-6

0.001

μ

μF (Micro

farads)

Nano

10

-9

0.000001

n

nF (Nano

farads)

Pico

10

-12

0.000000000001

p

pF (Pico

farads)

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

Avogadro's number

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

(μ

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

Faraday constant

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

)

B. Quadratic Equation

•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

ad

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

ad

c

d

b

a

d

c

b

a

=

×

=

Radical Combinations

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)

(Length)(Breadth)

= 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

smaller radius

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

Degrees versus Radians

•A circle in degree contains 360 degrees

•A circle in radians contains 2π radians

Sine, Cosine and Tangent

sin

opposite

hypotenus

θ

=

cos

adjacent

hypotenus

θ

=

tan

opposite

adjacent

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

adjacent

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

jθ

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

jθ

= 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

jθ

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

[F] (Farads)

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

Farads

(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

Flux-linkage =

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

-ε

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

radio frequencies

infrared radiation

ultraviolet

radiation

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

(radians)

•ω angular velocity (radians/s);

ω

1

= initial, ω

2

= final

2

1

2

1

t

t

α

ω

θ

+

=

α angular acceleration

(radians/s

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

•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

/ρ

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

Heat added

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

Lead

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

1¼

42.2

35.1

3.56

9.653

×

10

-4

1½

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

)

2½

73.0

62.7

5.16

3.090

×

10

-3

3

88.9

77.9

5.49

4.768

×

10

-3

3½

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