# Handbook of Formulae and Constant by Afirk - HTML preview

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### TOPIC PAGE

Basic Units (distance, area, volume, mass, density) ............................2

Mathematical Formulae .......................................................................5

Applied Mechanics .............................................................................10

Thermodynamics.................................................................................21

Fluid Mechanics..................................................................................28

Electricity............................................................................................30

Periodic Table .....................................................................................34

Names in the Metric System

VALUE EXPONENT SYMBOL PREFIX
1 000 000 000 000 1012 T tera 1 000 000 000 109 G giga 1 000 000 106 M mega 1 000 103 k kilo

100 102 h hecto 10 101 da deca 0.1 10-1 d deci 0.01 10-2 c centi 0.001 10-3 m milli 0.000 001 10-6 µ micro

0.000 000 001 10-9 n nano 0.000 000 000 001 10-12 p pico

Conversion Chart for Metric Units

To
To To To Metre, To To To Milli- Centi- Deci- Gram, Deca- HectoKilo- Litre

Kilo- x 106 x 105 x 104 x 103 x 102 x 101
Hectox 105 x 104 x 103 x 102 x 101 x 10-1
Deca- x 104 x 103 x 102 x 101 x 10-1 x 10-2
Metre, 3
x 102 x 101 x 10-1 x 10-2 x 10-3Gram, x 10
Litre
Deci- x 102 x 101 x 10-1 x 10-2 x 10-3 x 10-4 Centi- x 101 x 10-1 x 10-2 x 10-3 x 10-4 x 10-5Milli- x 10-1 x 10-2 x 10-3 x 10-4 x 10-5 x 10-6

### BASIC UNITS SI IMPERIAL DISTANCE

1 metre (1 m) = 10 decimetres (10 dm) 12 in. = 1 ft
= 100 centimetres (100 cm) 3 ft = 1 yd
= 1000 millimetres (1000 mm) 5280 ft = 1 mile

1760 yd = 1 mile 1 decametre (1 dam) = 10 m
1 hectometre (1 hm) = 100 m

1 kilometre (1 km) = 1000 m

Conversions:

1 in. = 25.4 mm 1 ft = 30.48 cm

1 mile = 1.61 km 1 yd = 0.914 m 1 m = 3.28 ft

Area

1 sq metre (1 m2) = 10 000 cm2 1 ft2 = 144 in.2 = 1 000 000 mm2 1 yd2 = 9 ft2 1 sq mile = 640 acre = 1 section 1 sq hectometre (1 hm2) = 10 000 m2

= 1 hectare (1 ha)

1 sq km (1 km2) = 1 000 000 m2

Conversions:

2 = 6.45 cm2 = 645 mm2 2 = 10.8 ft21 m 1 acre = 0.405 ha

1 sq mile = 2.59 km2

### SI IMPERIAL

Volume

1 m3 = 1 000 000 cm3 1 ft3 = 1728 in.3 = 1 x 109 mm3 1 yd3 = 27 ft3

1 dm3 = 1 litre 1(liquid) U.S. gallon = 231 in.3

1 litre = 1000 cm

3 = 4 (liquid) quarts

1 mL = 1 cm3 1 U.S. barrel (bbl) = 42 U.S. gal.

1 m3 = 1000 litres 1 imperial gallon = 1.2 U.S. gal.

Conversions:

3 31 in. 3 31 m 1 litre = 61 in.3

1 U.S.gal = 3.78 litres

1 U.S. bbl = 159 litres

1 litre/s = 15.9 U.S. gal/min

Mass and Weight

1 kilogram (1 kg) = 1000 grams 1000 kg = 1 tonne 2000 lb = 1 ton (short) 1 long ton = 2240 lb

Conversions:

1 kg (on Earth) results in a weight of 2.2 lb

Density

mass

density

=

mass weight density =weight volume volume

ρ

=
m › kg žρ = w › lb ž V œm3Ÿ  V œft3 Ÿ

Conversions:

(on Earth) a mass density of 1 kg3 results in a weight density of 0.0623 lb3m ftSI Imperial

### RELATIVE DENSITY

In SI R.D. is a comparison of mass density to a standard. For solids and liquids the standard is fresh water.
water.
In Imperial the corresponding quantity is specific gravity; for solids and liquids a comparison of weight density to that of

Conversions:

In both systems the same numbers hold for R.D. as for S.G. since these are equivalent ratios.

RELATIVE DENSITY (SPECIFIC GRAVITY) OF VARIOUS SUBSTANCES

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 Carbon (graphite)........2.3 Carbon (charcoal) .......1.8 Sand (dry) ...................1.42 Silicon.........................2.6 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 Wood (lignum-vitae) ..1.3
Lead ..........................11.4 Wood (oak).................0.7-1.0 Magnesium .................1.74 Wood (pine)................0.56 Manganese..................8.0 Wood (teak) ................0.8 Mercury ....................13.6 Zinc.............................7.0

Greek Alphabet

Alpha α Beta β Gamma γ Delta Epsilon ε Zeta ζ Eta η Theta θ Pi Omega &, ω Iota ι Rho ρ Kappa κ Sigma Σ, σ Lambda λ Tau τ Mu µ Upsilon υ Nu ν Phi Φ, φ Xi ξ Kai χ Omicron Ο Psi ψ

MATHEMATICAL FORMULAE
Algebra
1. Expansion Formulae

(x + y)2 = x2 + 2xy + y2

(x - y)2 = x2 - 2xy + y2

x2 - y2 = (x - y) (x + y)

(x + y)3 = x3 + 3x2y + 3xy2 + y3

x3 + y3 = (x + y) (x2 - xy + y2)

(x - y)3 = x3 - 3x2y + 3xy2 - y3

x3 - y3 = (x - y) (x2 + xy + y2)

If ax + bx + c = 0,

-b± b2−4acThen x = 2a

Trigonometry
1. Basic Ratios

Sin

=

y , cos =x , tan =y h h x

2. Pythagoras' Law

x2 + y2 = h2

3. Trigonometric Function Values

Sin is positive from 0° to 90° and positive from 90° to 180°

Cos is positive from 0° to 90° and negative from 90° to 180°

Tan is positive from 0° to 90° and negative from 90° to 180°

4. Solution of Triangles
a. Sine Law

a= b= c Sin A Sin B Sin C

b. Cosine Law

c2 = a2 + b2 - 2 ab Cos C

a2 = b2 + c2 - 2 bc Cos A

b2 = a2 + c2 - 2 ac Cos B

Geometry
1. Areas of Triangles
a. All Triangles

Area

=
base x perpendicular height 2

Area

=
bcSin A= ab Sin C= acSin B 2 2 2

and,

Area= s(s-a)(s-b) (s-c)

where, s is half the sum of the sides, or s = a + b c 2

b. Equilateral Triangles

Area = 0.433 x side2

2. Circumference of a Circle

C = πd

3. Area of a Circle

A =

π

r

2 = circumference x r= πd2 = 0.7854d2 2 4

4. Area of a Sector of a Circle

A =
arcx r 2

A =θ° x πr2 (θ = angle in degrees) 360

A =

θ

°

r2

2

5. Area of a Segment of a Circle

A = area of sector – area of triangle

Also approximate area =4 h2 d -0.6083 h

6. Ellipse

A = π Dd4

Approx. circumference =

### ()

π2

7. Area of Trapezoid

A = › +bžhœ Ÿ

8. Area of Hexagon

A = 2.6s2 where s is the length of one side

9. Area of Octagon

A = 4.83s2 where s is the length of one side

10. Sphere

Total surface area A =4πr2

Surface area of segment As = πdh

Volume V =

4 πr3 3

Volume of segment

V

s

=

π

h2

3 (3r – h)

Vs =π h 2+3a2) where a = radius of segment base6 (h

11. Volume of a Cylinder

V = πd2L where L is cylinder length 4

12. Pyramid

Volume

V = 1 base area x perpendicular height

3

Volume of frustum

VF = h (A +a + Aa) where h is the perpendicular height, A and a are areas as shown 3

13. Cone

Area of curved surface of cone:

A =
πDL 2

Area of curved surface of frustum

A

F

=
π +d)L 2

Volume of cone:

V= base area

× perpendicular height 3

Volume of frustum:

= perpendicular height×π (R2 + r2+Rr)VF 3

APPLIED MECHANICS
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 ms , in Imperial ft Other common units are kmh , mi

m ftConversions: 1 =3.28

s s

km mi1 =0.621

h h

Speed of sound in dry air is 331 ms at 0°C and increases by about 0.61 ms 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

s

2

m , in Imperial

ft s2

Conversion

: 1

s

2

m = 3.28

ft s2

Acceleration due to gravity, symbol "g", is 9.81 m or 32.2 ft s2 s2

### LINEAR VELOCITY AND ACCELERATION

u initial velocity v final velocity t elapsed time s displacement a acceleration
v=u+at

s= v+ut

2

s = ut + 12 at2

v2=u2+2as

Angular Velocity and Acceleration

ω angular velocity (radians/s); ω1 = initial, ω2 = final α angular acceleration (radians/s2)

ω2 = ω1 + α t

θ = ω1 + ω2 x t 2

θ = ω1 t + ½ α t2

ω22 = ω12 + 2 α θ

linear displacement, s = r θ linear velocity, v = r ω linear, or tangential acceleration, aT = r

Tangential, Centripetal and Total Acceleration

Tangential acceleration aT is due to angular acceleration α

aT = rα

Centripetal (Centrifugal) acceleration ac is due to change in direction only

ac = v2/r = r ω2

Total acceleration, a, of a rotating point experiencing angular acceleration is the vector sum of aT and ac

a = aT + ac

### 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 m2s

In Imperial the unit of force is the pound lb

Conversion: 9.81 N = 2.2 lb

Weight

The gravitational force of attraction between a mass, m, and the mass of the Earth

In SI weight can be calculated from

Weight = F = mg , where g = 9.81 m/s2

In Imperial, the mass of an object (rarely used), in slugs, can be calculated from the known weight in pounds

m= Weightg g=32.2ft2s

Newton's Second Law of Motion

An unbalanced force F will cause an object of mass m to accelerate a, according to:

F = ma (Imperial F = wg a, where w is weight)

Torque Equation

T = I α where T is the acceleration torque in Nm, I is the moment of inertia in kg m2 and α is the angular acceleration in radians/s2

Momentum

Vector quantity, symbol p,

p = mv (Imperial p = wg v, where w is weight)

in SI unit is kg ms

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

Kinetic Energy

Energy due to motion

Ek = 12mv2

In Imperial this is usually expressed as Ek = w2gv2 where w is weight

Kinetic Energy of Rotation

ER =1 mk2ω2 where k is radius of gyration, ω is angular velocity in rad/s 2

or ER =12 where I = mk2 is the moment of inertia 2

### CENTRIPETAL (CENTRIFUGAL) FORCE

F

C

=

mv2r where r is the radius or

FC = m ω2 r where ω is angular velocity in rad/s

Potential Energy

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 106 ft-lb

Power

A scalar quantity, equal to the rate of doing work

In SI the unit is the Watt W (or kW)

1W=1Js

In Imperial, the units are:

Mechanical Power - ft–lb, horsepower h.p. s

Thermal Power - Btu

s

Electrical Power - W, kW, or h.p.

Conversions: 746 W = 1 h.p.

1 h.p. = 550 ft–lbs 1 kW = 0.948 Btus

Pressure

A vector quantity, force per unit area

In SI the basic units of pressure are pascals Pa and kPa

1Pa=1 N2m

In Imperial, the basic unit is the pound per square inch, psi

Atmospheric Pressure

At sea level atmospheric pressure equals 101.3 kPa or 14.7 psi

Pressure Conversions

1 psi = 6.895 kPa

Pressure may be expressed in standard units, or in units of static fluid head, in both SI and Imperial systems

Common equivalencies are:

1 kPa = 0.294 in. mercury = 7.5 mm mercury
1 kPa = 4.02 in. water = 102 mm water
1 psi = 2.03 in. mercury = 51.7 mm mercury
1 psi = 27.7 in. water = 703 mm water
1 m H2O = 9.81 kPa

Other pressure unit conversions:

1 bar = 14.5 psi = 100 kPa
1 kg/cm2 = 98.1 kPa = 14.2 psi = 0.981 bar
1 atmosphere (atm) = 101.3 kPa = 14.7 psi

Simple Harmonic Motion

Velocity of P =

ω

R

2 -x2 m s

Acceleration of P = ω2 x m/s2

The period or time of a complete oscillation = seconds ω General formula for the period of S.H.M.

T = 2

π
displacement acceleration

Simple Pendulum

T = 2πL T = period or time in seconds for a double swing g

L = length in metres

The Conical Pendulum

R/H = tan θ= Fc/W = ω2 R/g

Lifting Machines

W = load lifted, F = force applied

M.A. =

V.R. (velocity ratio) =

M.A.

η = efficiency = V.R.

1. Lifting Blocks

V.R. = number of rope strands supporting the load block

2. Wheel & Differential Axle

2πR Velocity ratio = 2π(r -r1) 2

2R 2 R

=

r -r1

2D Velocity ratio = (d -d1)

3. Inclined Plane

length

V.R. = height

4. Screw Jack

circumferenceof leverageV.R. = pitch of thread

Indicated Power

I.P. = Pm A L N where I.P. is power in W, Pm is mean or "average" effective pressure in Pa, A is piston area in m2, L is length of stroke in m and N is number of power strokes per second

Brake Power

B.P. = Tω velocity in radian/second where B.P. is brake power in W, T is torque in Nm and ω is angular STRESS, STRAIN and MODULUS OF ELASTICITY

Direct stress = load= P area A

Direct strain = extension original length= L Modulus of elasticity

E=
direct stress= P/A= PL direct strain /L A

force Shear stress = area under shear

x

Shear strain = L Modulus of rigidity

G=
shear stress shear strain

General Torsion Equation (Shafts of circular cross-section) T

J =

τr = Gθ

L 1. For Solid Shaft T = torque or twisting moment in newton metres π

r

4

=

π

d

4 J = polar second moment of area of cross-section

J =2 32 about shaft axis.
τ = shear stress at outer fibres in pascals r = radius of shaft in metres

2. For Hollow Shaft G = modulus of rigidity in pascals θ = angle of twist in radians

J=π (r4 - r4) L = length of shaft in metres 2 1 2

d = diameter of shaft in metres

=π (d4 −d4)32 1 2

Relationship Between Bending Stress and External Bending Moment M=σ

I y = ER

1. For Rectangle

M = external bending moment in newton metres I = second moment of area in m4
σ = bending stress at outer fibres in pascals y = distance from centroid to outer fibres in metres E = modulus of elasticity in pascals R = radius of currative in metres

BD3

I =

12

2. For Solid Shaft

I=

πD4 64

THERMODYNAMICS
Temperature Scales

5 9

°C= °F =

9 5

°R = °F + 460 (R Rankine) K = °C + 273 (K Kelvin)

Sensible Heat Equation

Q = mcT

m is mass
c is specific heat T is temperature change

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

Gas Laws
1. Boyle’s Law

When gas temperature is constant

PV = constant or

P1V1 = P2V2

where P is absolute pressure and V is volume

2. Charles’ Law

When gas pressure is constant, V=constantT

or V1 = V2 , where V is volume and T is absolute temperature

T1 T2

3. Gay-Lussac's Law

When gas volume is constant, P= constantT

Or P1=P2 , where P is absolute pressure and T is absolute temperature T1 T2

4. General Gas Law

P1V1 = P2V2 = constantT1 T2

P V = m R T where P = absolute pressure (kPa)
3V = volume (m T = absolute temp (K)
m = mass (kg) R = characteristic constant (kJ/kgK)

Also

PV = nRoT where P = absolute pressure (kPa)
3V = volume (m T = absolute temperature K
N = the number of kmoles of gas

Ro = the universal gas constant 8.314 kJ/kmol/K

SPECIFIC HEATS OF GASES

Specific Heat at Specific Heat at Ratio of Specific Constant Pressure Constant Volume Heats kJ/kgK kJ/kgKγ = cp/ cv GAS or
kJ/kg
or oC kJ/kg oC

Air 1.005 0.718 1.40 Ammonia 2.060 1.561 1.32 Carbon Dioxide 0.825 0.630 1.31 Carbon Monoxide 1.051 0.751 1.40 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

Efficiency of Heat Engines

Carnot Cycle η = T1 – T2 where T1 and T2 are absolute temperatures of heat source and T1

sink

Air Standard Efficiencies
1. Spark Ignition Gas and Oil Engines (Constant Volume Cycle or Otto Cycle)

η

=

1

-

1

r

(
γ
-
1)

where rv = compression ratio =

cylinder volume clearancevolumev

γ

=
specificheat (constant pressure) specificheat (constant volume)

2. Diesel Cycle

(Rγ−1) η=1-rv γ(R-1) where r = ratio of compression -1

R = ratio of cut-off volume to clearance volume

3. High Speed Diesel (Dual-Combustion) Cycle

η

=

1

-

r

γ -1

-1

### v []

where r

v

=
cylinder volume clearancevolume

k =
absolutepressueat end of constant V heating(combustion) absolutepressueat beginningof constant Vcombustion

β

=
volumeat end of constant Pheating (combustion) clearance volume

4. Gas Turbines (Constant Pressure or Brayton Cycle)

η
=
1
-
1
› −1ž
œœγ ŸŸ

r
p

where r

= pressure ratio =
compressordischargepressure compressorintake pressure

Heat Transfer by Conduction

Q=

λAtT

d

where Q = heat transferred in joules

λ = thermal conductivity or coeficient of heat transfer in

J× m W

m2× s×°C or m×°C

A = area in m2

t = time in seconds

T = temperature difference between surfaces in°C d = thickness of layer in m

COEFFICIENTS OF THERMAL CONDUCTIVITY

Material Coefficient of Thermal Conductivity W/m °C

Air 0.025 Aluminum 206 Brass 104 Brick 0.6 Concrete 0.85 Copper 380 Cork 0.043 Felt 0.038 Glass 1.0 Glass, fibre 0.04 Iron, cast 70 Plastic, cellular 0.04 Steel 60 Wood 0.15 Wallboard, paper 0.076

Thermal Expansion of Solids

Increase in length = L α (T2 – T1 )

where L = original length

α = coefficient of linear expansion (T2 – T1 ) = rise in temperature

Increase in volume = V β (T2 – T1 )

Where V = original volume

β = coefficient of volumetric expansion (T2 – T1 ) = rise in temperature

coefficient of volumetric expansion = coefficient of linear expansion x 3 β = 3α

Chemical Heating Value of a Fuel