This document provides information on various topics in math, science, and engineering. It begins with plane and solid geometry concepts like polygons, circles, triangles, and polyhedra. It then covers trigonometry, analytic geometry, calculus, differential equations, matrices and more. Example formulas are given for area, volume, sine, cosine, logarithms, and other calculations. Engineering concepts like vectors, friction, centroids, and moments of inertia are also summarized. The document contains a comprehensive review of formulas and principles across multiple STEM disciplines.

1. MATH – ECON – ENGG SCIENCE
PLANE GEOMETRY
Polygons
Number of diagonal: Nd = nC2 – n
Interior angle: ( ⁄ )
Area of n-side polygon:
side b:
( )
inscribed in a circle:
( )
circumscribed in a circle:
Circles
Arc length:
Sector Area:
Segment Area: ( )
Circle Theorems
Inscribed angle (a); Tangent & chord (b):
̂
Intersecting chords (c):
( ̂ ̂ )
( )( ) ( )( )
Intersecting secants (d):
(̂ ̂ )
( )( ) ( )( )
Tangent and Secant (e):
(̂ ̂ )
( )( ) ( )
θ
θ
θ
θ θ
A
B
A
B
C
D
x
A
B
A
B
C
D A
B
C
(a) (b) (c)
(d) (e)
Triangles
√ ( )( )( )
( )⁄
Inscribed in a circle:
⁄
Circumscribes a circle:
Circle tangent to side a:
( )
Parallelogram
Rhombus:
Trapezoid
( )
Trapezium
√( )( )( )( )
s: semiperimeter
θ: average of opposite angles
Cyclic Quadrilateral
Bramaguptha’s Formula:
√( )( )( )( )
√( )( )( )
Ptolemy’s theorem:
= sum of prod. of opposite sides
Ellipse
√( )
Parabolic segment
2
3
A ab

Plane Area
 
2 2
1 1
2
, ,
1
2
2
1
1
2
x y
curve curve
x y
x
curve top curve bottom
x
A y dx x dy
A y y dx
A R d



 
 

 


SOLID GEOMETRY
Prism
Truncated Prism
Prismatoid
( )
Regular Polyhedron
Regular
Polyhedron
F A V E
(F+V-2)
Volume
Tetrahedron 4 4 4 6 =
2
12
3
Hexahedron 6 6 8 12 = 3
Octahedron 8 8 6 12 =
2
3
3
Dodecahedron 12 12 20 30 = 7.66 3
Icosahedron 20 20 12 30 = 2.18 3
Cylinder
( )
Pyramid
Frustum of a Pyramid
( √ )
Cone
Frustum of Cone
( )
( )
Sphere
Spherical:
Wedge:
Lune:
Segment: ( )
( )
Zone:
Cone:
Pyramid:
( ) ( )
Torus
Ellipsoid
Oblate Spheroid minor axis
Prolate Spheroid major axis
Paraboloid
Two bases:
[ ]
Hyperboloid
[ ]
Conoid
Volume
Circular Disk:
2
1
x
curve
x
V y dx

 
Cylindrical Shell:
2
1
2
x
curve
x
V xy dx

 
Circular Ring:  
2
2 2
1
y
y
V R r dy

 

Propositions of Pappus
First Proposition: 2
A R S

 
Second Theorem: 2
V R A

 
Length of an Arc
 
2
2
1
1 '
x
x
S y dx
 

TRIGONOMETRY
SOH CAH TOA
CHO SHA CAO
( )
( )
( )
√ √
( )
( )
Sine Law
Cosine Law
Tangent Law
( )
( )
SPHERICAL TRIGONOMETRY
A
B C
a
b
c Ac
cc
B
a
b
180 < A + B + C < 540
Napier’s Rule I: Sin-Tan-Ad
Napier’s Rule II: Sin-Cos-Op
Sine Law
Cosine Laws: “SPAN”
O
H
A
θ
T C
A
S

2. Spherical Defect, d:
( )
ANALYTIC GEOMETRY
Division of Line Segment
Angle of Inclination
Angle bet. 2 intersecting lines
( )
Distance bet. 2 parallel lines
√
Distance bet. line and a point
√
Area of n-sided polygon
* +
Conic Sections
Ellipse:
Parabola:
Hyperbola:
Diameter of the conics:
Differential, y=x/m
Polar Coordinates
;
;
Cylindrical Coordinates
Spherical Coordinates
√
DIFFERENTIAL CALCULUS
 
 
 
   
 
   
 
 
 
 
 
 
 
 
1
1
2
1
2
2
1
2
2
0 log
1
ln
1
log log
1
sin
sin cos
1
1
cos sin cos
1
1
tan sec tan
1
/
x x
a
n n
x x
a a
d d
c a e a
dx dx
d d
x nx x
dx dx x
d d
e e x e
dx dx x
d
d x
x x dx x
dx
d d
x x x
dx dx x
d d
x x x
dx dx x
d dv du
uv u v
dx dx dx
du dv
v u
d dx dx
u v
dx v




 
 
 



   

 

 


Radius of Curvature
 
3/2
2
1 '
''
y
R
y
 

 

L’hôpital’s Rule
 
 
 
 
 
 
'
Lim Lim ... Lim
'
n
n
x a x a x a
f x f x f x
g x g x g x
  
  
INTEGRAL CALCULUS
 
 
 
 
 
 
   
     
1
0
ln
1
sin cos
cos sin
tan ln sec
x x
n
n
b a
a b
b c b
a a c
dx C e dx e
dx
x
x
x dx
x
n
x dx x
x dx x
x dx x
u dv uv v du
f x dx f x dx
f x dx f x dx f x dx

 



 


 
 
 
 





 
 
  
Trigonometric Substitution
For sin cos
m n
u u du
 :
If m or n is odd,
If m and n are odd,
For tan or sec
n n
u du u du
  :
DIFFERENTIAL EQUATION
Variable Separable
Homogeneous DE
( ) ( )
degree of M = degree of N
Sol’n: y = vx or x = vy
Exact DE
( ) ( )
⁄ ⁄
Sol’n: Integrate Mdx and Ndy,
Equate to solve g(y) or h(x)
Unexact DE
⁄ ⁄
( ) ( )
∫ * + ∫ * +
Linear Differential Equation
( ) ( )
∫ ( )
∫ ( ) ∫ ( )
( ) ( )
∫ ( )
∫ ( ) ∫ ( )
Bernoulli’s Diff. Equation
( ) ( )
( )∫ ( )
∫ ( ) ( )∫ ( )
2ND order LDE
Sol’n: Solve for roots.
(a) Real and distinct
(b) Real and repeated
(c) Complex,
( )
Non-homogeneous LDE
( ) ( )
Sol’n:
yp by MUC,
Substitute and solve for coefficients.
yp by MVP,
Substitute u to coefficients of yc.
Solve for u’, integrate, and substitute.
LOGARITHM
( )
( ) ( )
COMPLEX NUMBERS
( )
( )
( )
( )
MATRIX AND DETERMINANTS
Minor, Mij
12
1 2 3
4 6
4 5 6
7 9
7 8 9
M  
Cofactor
( )
Adjoint Matrix
( )
Pivotal Method
 
     
     
2 3
4 1 3
4 3 2 1 3 0
2 0 1 1
10 5 2 6 3 0
10 6 5
x


  
  
 
Inverse Matrix, A-1
Transpose
Form Adjoint Matrix
Divide by determinant
Conics
Eccentricity
/
e f d

Discriminant
2
4
B AC

Hyperbola > 1 > 0
Parabola = 1 = 0
Ellipse < 1 < 0 (A ≠ C)
Circle = 0 < 0 (A = C)
Folium of Descartes 3 3
2 0
x y axy
  
Lemniscate of Bernoulli 2 2
cos2
r a 

Four-leaved Rose sin2
r a 

Cardiod  
1 cos
r a 
 
Limacon cos2
r b a 
 
Cycloid
 
 
sin
1 sin
x a
y a
 

 
 
𝑓( ) 𝑝
0 + 1 + 2
2
+
cos ; sin cos + sin
cos ;
sin
[ 0 + 1 + + ] cos
+[ 0 + 1 + + ] sin
x
y
z
r
z
θ
P(r,θ,z)
x
y
z
r
ϕ
θ
P(r,θ,ϕ)

3. ALGEBRA
Binomial Expansion
term with yr:
n r r
n r
C x y

Arithmetic Progression
( )
( )
Geometric Progression
( )
√
Other Sequences:
M-gonal Numbers
[ ( )( )]
Pyramidal Number (triangle base)
( )( )
Pyramidal Number (square base)
( )( )
Pyramidal Number (rectangle base)
( )( )
Work Problems
Unit work * time = 1
Total man-time = Σ each man-time
( ) ( )
Mixture Problems
Quantity: A + B = C
Composition: Ax + By = Cz
Permutation: order
( )
Alike things:
Ring:
( )
Combination: group
( )
taken 1 or 2 or n
PROBABILITY
Complementary
Joint
Non-mutual exclusive
Conditional
Independent
Repeated Trials
r n r
n r
P C p q 

p: success, f: failure
STATISTICS
Median: middle of arranged set
Mode: most frequent value
Mean: ̅, average
Variance
Population:
( ̅)
Sample:
( ̅)
Standard Deviation:
Relative Variability: SD/mean
Z-score
ECONOMICS
Simple Interest
( )
Ordinary: 360 days
Exact: 365/366 days
Compound Interest
( )
( )
Nominal rate of Interest
Effective rate
( )
Continuous Compounding
Discount
Discount:
Rate of discount:
( )
Rate of discount vs interest
Annuity:
Ordinary: *
( )
+
*
( )
+
Annuity Due
Deferred Annuity
*
( )
+( )
Perpetuity
Depreciation:
Straight Line, SLD
Annual depreciation, d
Total depreciation, Dn
( )
Book Value, Cn
Sinking Fund
Annual depreciation, d
(
( )
) (
( )
)
Total depreciation, Dn
*
( )
+
Book Value, Cn
Declining Balance
Depreciation at nth year
( )
√ √
Book value
* + ( )
Scrap value
( )
Total depreciation
Double Declining Balance, DDB
Same with Declining Balance but
Sum of Years Digit, SYD
Depreciation at nth year
( )
Total depreciation
( )
Service Output Method
( )
Working Hours Method
( )
Bonds
*
( )
+ ( )
Rate of Return, ROR
Pay out Period
Total Investment Salvage Value
Payout
Period Net Annual Cash Flow


Breakeven
Benefit to Cost Ratio
Annual Equivalent Cost
   
0
1 1 1 1
L
n n
C C
C
i i
i i

 
   
   
   
   
   
Benefit to Cost Ratio
⁄
ENGINEERING SCIENCE
Vectors
x y z
A A A A
  
i j k
Dot Product
cos
A B A B 
 
Cross Product
n
a sin
A B A B 
 
Friction
Sliding Block
Rolling Friction
Belting Friction
Cable
Parabolic: uniformly dist. horizontally
⁄
√( ⁄ )
Catenary: uniformly dist. along length
( )

4. Centroid (1ST Moment)
S x x dS
Ax x dA
V x x dV






Moment Of Inertia (2ND Moment)
 
2
2
y
I x dA A x
  

Polar Moment of Inertia
z x y
J I I
 
Mass moment of Inertia
2
m
I r dm
 
Thin Plate
2
m
I t r dA

 
Parallel Axis Theorem
2
0
x x
I I Ar
 
2
0
x x
I I mr
 
Dynamics (Kinematics)
Uniform Accel. Motion (Free fall, a=-g)
 
2 2 2
1
2
0 0
1
2
0 0
2
f
f f
V V a x x V t at
V V at x V V t
     
    
Projectile Motion
   
 
2
1
2
0 0
2 2
0 0
sin cos
sin sin 2
2
y V t gt x V t
V V
h R
g g
 
 
   
 
Rotational Kinematics
same with linear but replace
s , v , a
  
  
Linear and Angular Relations
s , v , a
r r r
  
  
Dynamics (Kinetics)
Newton’s Law of Motion
1st Law: 0
F
 
2nd Law: F ma
 
3rd Law: F R

Newton’s Law of Universal Gravitation
1 2
2
11 2 2
G
G 6.67 10 N-m /kg
m m
F
s


 
D’Alembert’s Principle
REF 0
REF
F
ma
  

Circular Motion
Centripetal Force
2 2
n n
n
F ma
a V r r

 
Centrifugal Force
t t
t
F ma
a V t r

 
2 2
Total accel n t
a a
 
Conical Pendulum
2
cos
tan
2 /
n
T W
F V
W gr
t h g




 

Banking of Highway
 
2
tan
V
gr
 
 
Centroidal Rotation
2
2
1
2
r
M I
I mk
KE I


 


Work-Energy Theorem
net
W KE
 
Impulse-Momentum Theorem
   
0 0
f f
F t P
F t t m v v
  
  
Momentum
   
before impact after impact
2 1 2 1
' '
P P
e V V V V
  
  
Perfectly elastic: e = 1
Inelastic collision: 0<e<1
Perfectly inelastic: e = 0
Special Case:
Bounce: 2 1
e h h

Thrown at angle: 2 1
tan cot
e  

Angular Impulse
J F r t
  
Angular Momentum
0
H P r I 
  
OTHER GEOMETRIC PROPERTIES
TRIANGLE
3 3
0
12 36 3
b x
bh bh h
I I y
  
RECTANGLE
3 3
0
3 12 2
b x
bh bh h
I I y
  
CIRCLE
4 4
0 0
4 2
x y
r r
I I J
 
  
SEMI CIRCLE
4 4
8 8
4
3
x y
r r
I I
r
y
 

 

QUARTER CIRCLE
4 4
16 16
4 4
3 3
x y
r r
I I
r r
x y
 
 
 
 
QUARTER CIRCLE
3 3
16 16
4 4
3 3
x y
ab a b
I I
a b
x y
 
 
 
 
SECTOR
4
0
4
0
1 sin 2
4 2
1 sin 2
4 2
2 sin
3
x
y
I r
I r
r
x






 
 
 
 
 
 
 
 

ELLIPSE
3 3
0 0
4 4
x y
ab ab
I I
 
 
PARABOLA 1
3 3
21 5
3 3
4 10
x y
bh b h
I I
x b y h
 
 
PARABOLA 2
3 3
2 2
7 15
3 3
8 5
x y
bh b h
I I
x b y h
 
 
SPHERE: SOLID HOLLOW
2
2
5
I mr
 2
2
3
I mr

CYLINDER: SOLID HOLLOW
2
1
2
I mr
  
2 2
1
2
I m R r
 
ROD, CENTER ONE END
2
1
12
I mL
 2
1
3
I mL

HOLLOW, at end
 
2 2 2
1
3 3 4
12
I m R r L
  
CONE:
RECT. PLATE thru CENTER
 
2 2
1
12
I m a b
 
2
3
10
I mr

x
cg
y
cg
y
x
cg
y
θ
θ
x
y
cg
a
b
b
x
y
cg
h
b
x
y
cg
h

5. CONVERSIONS
10^X PREFIX 10^X PREFIX
18
15
12
9
6
3
2
1
Exa
Peta
Tera
Giga
Mega
Kilo
Hecto
Deka
-1
-2
-3
-6
-9
-12
-15
-18
deci
centi
milli
micro
nano
pico
femto
atto
DISTANCE/SPEED/ACCEL
1 in = 1000 mil
1 ft = .3048 m = 3 hands
1 yd = 3 ft
1 fathom = 6 ft
1 chain = 66 ft
1 furlong = 660 ft
1 mile = 5280 ft
1 n. mile = 6080 ft = 1/60 degree
1 knot = 1 naut. mile/hr
1 m/s = 3.6 kph
1 lightyear = 9.46 x 1012 m
1 parsec = 3.084 x 1013 m
1 Angstrom = 10-10 m
9.81 m/s2 = 32.2 ft/s2
AREA
1 acre = 1 furlong x 1 chain
1 are = 100 m2
1 hectare = 10000 m2
VOLUME/FLOW RATE
1 gal = 3.785 L = 0.1337 ft3
1 bbl = 42 gal
1 m3 = 1000 L
1 ganta = 8 chupas = 3 L
MASS
1 kg = 2.2 lbm
1 lbm = 16 oz
1 slug = 32.2 lbm
1 tonne = 1 MT = 1000 kg
1 short ton = 2000 lbm
1 long ton = 2240 lbm
DENSITY/CONCENTRATION
1 kg/L = 62.4 lbm/ft3
1 ppm = 1 mg/L or 1 mg/kg
FORCE
1 N = 100 000 dynes
1 kgf = 9.81 N
1 lbf = 4.448 N
PRESSURE
1 atm = 101.325 kPa
= 14.7 psi
= 29.92 inHg = 760 mmHg
= 760 torr
1 bar = 100 kPa
1 MPa = 1 N/mm2
ENERGY
1 Btu = 1055 J
= 252 cal
= 778 ft-lbf
1 kcal = 4.187 kJ
1 J = 107 erg
1 chu = 1.8 Btu
1 eV = 1.602 x 10-19 J
POWER
1 hp = 0.746 kW
= 550 ft.lbf/s
= 2545 Btu/h
1 metric hp= 736 W
1 kW = 3412 Btu/h
1 TOR = 3.516 kW
= 12 000 Btu/hr
1 BoHP = 35 322 kJ/hr
TEMPERATURE
F = 1.8C + 32
R = F + 460
K = C + 273
R = 1.8 K
F = 1.8 C
DYNAMIC VISCOSITY
1 poise = 0.1 Pa-s
KINEMATIC VISCOSITY
1 stoke = 1 cm2/s
ANGLE
1 rev = 360
= 2π rad
= 400 grad
= 400 gons
= 6400 mils
CONSTANTS
GENERAL
̅ = 8.3143 J/mol . K
= 1545 lbf-ft/lbm.mol.R
= 0.0821 L-atm/mol-K
c = 3 x 108 m/s
NA = 6.02 x 1023 /mole
ς = 5.67 x 10-8 W/m2K4
Solar Constant = 1353 W/m2
Radius of Earth: 6.38 x 106 m
Earth Escape V: 11.2 km/s
Human Heat: 225 Btu/hr
WATER/ICE/LIQUIDS
Cp = 4.186 kJ/kg.K
Lf = 334 kJ/kg
= 144 Btu/lbm
Lv = 2257 kJ/kg
= 97 0 Btu/lbm
E = 2.1 x 106 kPa
Surface tension, ς
@ 0C ς = 0.076 N/m
@ 100C ς = 0.059 N/m
Cp of ice = 0.5(Cp water)
Liquids:
SGmercury = 13.55
SGsea water = 1.03
AIR/GASES
k = 1.4 or 1.3 (hot)
Cp = 1 kJ/kg-K = 0.24 Btu/lbm.R
Cv = 0.7186 kJ/kg
R = 0.287 kJ/kg.K
= 53.34 lbf-ft/lbm.R
ρ = 1.2 kg/m3
Latent hv = 2442 kJ/kg
Specific heat ratio:
He, noble gases k = 1.667
Carbon dioxide k = 1.287
Nitrogen k = 1.399
STEEL
E = 30 x 106 psi
G = 12 x 106 psi
α = 12 x 10-6 /C
ρ = 7860 kg/m3
OTHERS
Molecular Weights:
H(1),He(4), C(12),N(14),O(16)
S(32), Air(29)
OTHERS
1 clo = 0.880 [Btu/h· ft²·°F]-1
1 board ft = 1 ft x 1 ft x 1 in

6. MACHINE DESIGN & SHOP PRACTICE
STRESSES
Axial Stress St =
Shear Stress Ss =
Torsion Ss = = = ( )
Bearing Stress Sb =
Bending Stress Sf = =
Sf = = ( )
Thermal Stress ST = ( )
δ = ( )
Design Stress Sd = =
Modulus of Elasticity S =
Modulus of Rigidity G = ( )
Combined Stresses
S =
Stmax = √
Ssmax = √
Stmax = [ ]
Ssmax =
Stmax = *( ) √( ) +
Ssmax = √( )
Variable Stresses
Ductile Materials
=
Brittle Materials
=
SHAFTINGS
Power Transmission P =
Line Shaft P =
Short Shaft P =
*units in hp, inches, rpm
Diameter D = √
Power
*kW,N-mm,rpm P =
*hp,lbf-in,rpm P =
With shock factors
Stmax = [( ) √( ) ( ) ]
Ssmax = √( ) ( )
Vertical Shear SV =
Angular Deformation θ =
KEYS
Shearing Stress SS =
Compressive Stress SC =
Same Material L =1.18D
SPLINES
Shearing SS =
Compression SC =
Total Torque T =
Total Capacity TC =
COUPLING
Shearing of Bolt T =
Compression of Bolt T =
THREADED MEMBERS
Stresses
Valiance SW =
Faires Sd =
Applied Load
Valiance Fa =
Faires Fe =
Bolt Constant, C
Bronze c = 10 000
Carbon Steel c = 5 000
Alloy Steel c = 1 500
Working Strength of Bolt
Ws = [ ]
Bolt Spacing Z =
Bolt Circle Diameter Dbc =
Depth Tap
Brittle h =
Valiance (Steel) h =
Faires (S, WI) h = D
Initial Torque
Valiance T =
Faires
Lubricated T =
As received T =
Initial Tension Fi =
Power Screw
Collar friction TC = ( )
Raising & Lowering
Square Tf = ( )
ACME Tf = * +
Trapezoid Tf = * +
American Tf = * +
Total Torque T =
Efficiency e = =
Friction angle β = ( )
Linear Velocity V = NL
Lead Angle λ = ( )
Lead L = P single
L = 2P double
L = 3P triple
Outside D Do = ⁄
Handbook
Screw D d = *
( )
+
⁄
Trms Power HP = ( )
Shaft D D = ( ) or ( )
*diam. (inch); L (ft); rpm; hp
PRESSURE VESSEL
Thin walled Cylinder
Tangential St =
Longitudinal SL =
Thin walled Sphere (t >0.1ri)
Tangential St =
Thick walled
Thickness t = [√ ]
Axial Sa =
Max. Tensile Stmax=
( )
Max. Shear Ssmax=
Eqv. Max.T. Stmax=
Critical Pressure Thin Tubes
Stainless Steel Tubes
t/do < 0.025 Pcr = ( )
t/do > 0.03 Pcr = ( )
Lap-welded Steel Tubes
t/do > 0.03 Pcr = ( )
Brass Tubes
t/do < 0.025 Pcr = ( )
t/do > 0.03 Pcr = ( )
Short Tube
Collapsing/Critical Pressure
Pcr = ( )
Crushing Stress
Sc =
RIVETS AND WELDED JOINT
Rivet
St =
FS =
e =
Weld
Ave. Shear Ss =
FS FS =
Max. Shear Ssmax =
Max. Tensile Ssmax =
BEARINGS
Bearling Pressure F =
Max. Contact Stresses
Balls Ssmax = 0.31 Smax
Cylinders Ssmax = 0.31 Smax
Life in million revs
Balls L = ( )
Cylinders L = ( )
Compressive Breaking Load
FC =
Carbon steel k = 100,000
Alloy Steel k = 125,000
FS FS = 10
Maximum Load Fmax =
Diam. Clearance Cd =
SPRING
End Type Actual n Solid L Free L
Ground n
Plain n ( )
Squared &
Ground
n + 2 ( )
Squared n + 2 ( )
Spring Index c = =
Whal Factor k =
Stresses

7. Round Wire S =
Square Wire S =
Rect. Wire S =
( )
Deflections
Round Wire δ =
Square Wire δ =
Rect. Wire δ = ( )
Stress (Torsion) S =
Deflection (Torsion)
Helical round δ =
Spiral round δ =
Spiral rect. δ =
*a-moment arm; L-wire length
Stresses (Leaf)
Single S =
Multiple S = ( )
Deflections (Leaf)
Single δ =
Multiple δ =
( )
Length of Wire L =
Free Length FL =
=
Impact Load ( ) = ( )
Spring Rate k =
Spring System
Series k = [ ( ⁄ )]
Parallel k =
FLYWHEEL
Total Weight Wf = WA+WH +WR
Rim weight WR = =
( )
Punch hole Energy E =
Punching Force
Steel round F = ( )
Steel square F = ( )
Brass rect. F = ( )
*units in tons, inches
Hoop Stress S =
Coef. Of Fluctuation Cf =
BRAKES
Band Brake
Tension Ratio =
Torque T = ( )
Max. unit pressure Pmax =
Max. stress Smax =
Actuating Force Fa =
( )
Differential Brake
Actuating Force Fa,cw =
( ) ( )
Block Brake
Braking Torque T =
= ( )
Brake Shoe
Heat dissipated
in brakes H =
for lowering brakes H =
Temperature rise tr =
cast iron C = 0.13 Btu/lb.F
cast steel C = 0.116 Btu/lb.F
Spot Brake
Braking torque capacity
T =
= ( )
CLUTCH
Plate/Disk Clutch
Uniform Pressure T = * ( )+
Fa = ( )
Uniform Wear T = * +
Fa = ( )
Cone Clutch
Torque T =
Axial Force Fa =
Fa = ( )
Block Clutch
Torque T =
Radial Force Fr =
= ( )
Engagement Force Fe = ( )
Max. Pressure Pmax = ( )
Expanding ring clutch
T =
Band Clutch (same with band brake)
Centrifugal Clutch
Torque T = ( )
Radial spring force
Radial spring S =
Garter spring S =
( )
*units in lb, inches, rpm
BELTS
Belt tension ratio =
Centrifugal Force FC=
Effective Belt pull = F1 – F2
Angle of Contact
Open θ = ( )
Crossed θ = ( )
Power transmitted P = ( )
Belt cross-section A =
( )
( )
Belt length
Open L = ( )
( )
Crossed L = ( )
( )
Belt Speed V = ( )
= ( )
V-belts
Tension Ratio =
WIRE ROPES
Bending Load Fb =
Weight of rope Wr =
Total Tension Ft = ( ) ( )
Ultimate Strength for plow steel
6 x 7; 6 x 19; 6 x 37 Fu =
*units in lbf, inches
Factor of Safety FS =
POWER CHAIN
Pitch Diameter Di = ( ⁄ )
Outside Diam. D0i = * ( )+
Chain Length L =
( )
GEAR (SPUR)
Diametral Pitch Pd =
Circular Pitch PC =
Addendum a =
Dedendum
14.5 and 22.5 d =
20 and 25 d =
Outside D Do =
Root D Drp =
Drg =
Whole depth W =
Working depth Wr =
Clearance c = =
Tooth thickness t =
Backlash B =
Face width b =
Base circle D Db = D
Center distance
external C = ( )⁄
internal C = ( )⁄
BF Strength Fs =
Dynamic Load Fd =
( )
√
Intermittent Service
Commercial cut (Vm  2000 fpm)
Fd = * +
Carefully cut (2000  Vm  4000 fpm)
Fd = * +
Precision cut (Vm > 4000 fpm)
Fd = [
√
]
Failure based on fatigue Nsf =
Uniform load w/o shock Nsf = 1.0 to 1.25
Medium shock Nsf = 1.25 to 1.5
Moderately heavy shock Nsf = 1.5 to 1.75
Heavy shock Nsf = 1.75 to 2.0
Failure based on wear Fw  Fd
Wear Load Fw =
GEARS (HELICAL)
Radial Force Fr =
Tangential Force Ft = ⁄
Axial Force Fa =
Normal Pressure Angle
ϕn = ( )
Normal Diametral Pitch Pdn =
Normal Circular Pitch Pcn =
Axial Pitch Pa = =
Lead
single helix L = Pa
double helix L = 2Pa
triple helix L = 3Pa
multiple helix L = nPa
BF Strength Fs =
Dynamic Load
Fd =
( )
√
Wear Load Fw =
Formative no. of teeth Nev =
GEARS (WORM)
Diametral Pitch Pd = ⁄
Lead L =
Lead Angle λ =
Pitch line velocity Vw =

8. Vg =
Worm Force Fw =
Separating Force
FS = * +
Tangential Force on worm
FG = * +
Efficiency of the worm gear
e = * +
Face width b =
Worm OD Dwo =
Worm Diameter Dw =
Teeth BFS Fs =
Dynamic Load Fd = * +
Worm Load Fw =
Thermal Capacity Q = ( )
GEARS (BEVEL)
Cutting or Root angle ω =
Face angle β =
Pitch angle γp = ( ⁄ )
γg = ( ⁄ )
Face width b 
Length of cone L = √
Strength Fs = * +
Dynamic Load Fd = * +
MACHINE SHOP
Time =
RPM(speed) =
Feed (in/min) = ( )
OTHERS
Petrox Formula Tf = , N.m

9. POWER AND INDUSTRIAL PLANT
ENGINEERING
THERMODYNAMICS
Pabs =
QS = 
QL =
H =
Ideal Gases
=
=
= ( )
=
= = ̅
=
=
=
Dalton’s Law of Partial Pressure
=
Processes
Nonflow Work:
= ∫ = ( )
Steady flow Work:
= ∫ = ( )
Heat Transferred:
=
Isometric: V = C
Wn = 0
Ws = ( )
Q = U
S = ( )
Isobaric: P = C
Wn = ( )
Ws = 0
Q = H
S = ( )
Isothermal: PV = C
Wn = ( ) = ( )
Ws = Wn
U = H = 0
Q = Wn
S = ( )
Isentropic: PVk = C
Wn =
( )
Ws = k Wn
U = -Wn
H =
Q = S = 0
=
= ( ) = ( )
( )⁄
Polytropic: PVn = C
Wn = =
( )
Ws = n Wn
Q = → ( )
S = ( )
CYCLES
SVSV OTTO STST CARNOT
SPSP BRAYTON TVTV STIRLING
SPSP RANKINE SPSV DIESEL
PTPT ERICCSON SVPSV DUAL
Carnot Cycle
INTERNAL COMBUSTION ENGINE
Otto Cycle:
Compression ratio rk = =
Volume Displacement VD =
Percentage clearance c =
Clearance Volume VC =
Efficiency e =
Mean eff. Pressure, Pm =
Diesel Cycle:
Compression ratio rk =
Cut-off Ratio rc =
Expansion Ratio re =
Efficiency e = ( )
Dual Cycle
Pressure ratio rp =
Efficiency e = ( )
FLUID MECHANICS
SG =
Bulk Modulus of Elasticity
⁄
Viscosity
Surface Tension
Soap: σ = ⁄
Liquid: σ = ⁄
Capillary Action
Water: θ = 0
Mercury: θ = 140
Variation in pressure
Liquids:
Gases:
Manometer
Pressure decreases upwards
Pressure increases downwards
Buoyancy
Flow in Pipes
Continuity Eqn
Compressible:
̇ = ̇ → =
Incompressible:
= → =
Bernoulli’s Eqn
( )
Reynold’s Number
Noncircular:
;
Laminar flow:
Turbulent flow:
Friction Losses
Orifices √
Weirs
*( )
⁄
( )
⁄
+
⁄
√ ;
⁄
Lift
Drag
Stokes’ Law
( )
Velocity of Sound
√
FUELS AND COMBUSTION
API =
Baume =
SGt = [ ( )]
Qh = kJ/kg
Qh = 33820C+144212(H- )+9304S kJ/kg
Qh = 13500C + 60890H Btu/lb
Wta = ( )
Wta =
( )
Wta = ( ⁄ ) =
( ⁄ )
Waa = ( ⁄ )
= ( )
= ( )
Composition of air:
By Weight: 23% O2 77% N2
By Volume: 21% O2 79% N2
 =
( ⁄ )
( ⁄ )
Gravimetric Analysis: %G =
Volumetric Analysis: %V = =
%G = %V( )
DIESEL POWERPLANT
Piston Disp.: VD =
Piston Speed = 2 L N
Indicated Power
Pind = Pmi VD
Brake Power
Pb = =
T = Fr
Pb = Pmb VD
Friction Power
Pf = Pind - Pb
Mechanical efficiency: em =
Electrical efficiency: egen =
Thermal efficiency: eti =
etb =
etc =
Engine efficiency: eei =
eeb =
eec =

10. Volumetric efficiency: ev =
Va =
Specific fuel consumption
mi =
( )
mb =
( )
mc =
( )
Heat rate
HRi =
( )
HRb =
( )
HRc =
( )
Generator Speed
N =
Engine at High altitudes
P = ( √ )
Pact = inHg
T = R
*h in feet
GAS TURBINE POWERPLANT
Thermal efficiency
eth =
Overall efficiency
eth =
Combustor efficiency
Eh =
Net heat plant rate
NHR =
STEAM POWERPLANT
Steam Rate
SR =
Rated Boiler HP
Water tube: RBoHP =
Fire tube: RBoHP =
*A in m2
Developed Boiler HP
DBoHP =
( )
*ms in kg/hr
*h in kJ/kg
Percent Rating
Percent Rating =
ASME Evaporation Units
AEU = ( )
Factor of Evaporation
FE =
Equivalent Evaporation
EE =
Actual Specific Evap. or Boiler Economy
ASE =
Equivalent Specific Evaporation
ESE =
Boiler Efficiency
ebo =
( )
Grate Efficiency
egrate =
*mC is amount of carbon in ash
Turbine
Wt =
Wact = ( )
Pump
Wp =
Wact = ( )
GAS AND FEEDWATER LOOP
Draft Loss
D = ( ) cm H2O
*Units in SI
Friction factor, f
Air-steel: f = 0.005
Air-concrete: f = 0.007
Fluegas-steel: f = 0.014
Fluegas-concrete: f = 0.014
Fan Work
W =
Air Horsepower
HPt =
Draft per 30m chimney
D30 = ( ) √
*Brick and steel: k =2.7
HYDROELECTRIC POWERPLANT
Pwater =
Pelton
h =
Reaction (Francis and Kaplan)
h =
Peripheral Coefficient
 =
√
Specific Speed
NS = ⁄
Total efficiency
etotal = ehemev
NONCONVENTIONAL POWERPLANT
Solar Power
Qsun = Qw + PE + Qloss
MACHINE FOUNDATION
Clearance, c
Bedplate to edge: 6 in to 12 in
To ground: 6 in min
Upper width
a = w +2c
Weight of foundation
Wf = 3 to 5 times Wm
Vf = Wf / ρ
Lower width, b
=
Depth, h
h = 3.2 to 4.2 times stroke
Vf = ( )
Weight of Steel bar reinforcements
WSB =
Anchor bolts
Depth =
VARIABLE LOAD PROBLEMS
PLANT CAPACITY
PEAK LOAD
AVERAGE LOAD
KW LOAD
kW-hrs
TIME (hrs)
Reserve
Over Peak
Reserve over Peak
ROP = Plant Capacity – Peak Load
Average Load
Ave. Load =
Load Factor
LF =
Capacity Factor
CF =
Annual Capacity Factor
ACF =
Use Factor
UsF =
Demand Factor
DeF =
Diversity Factor
DiF =
Utilization Factor
UtF =
Operation Factor
OF =
Plant Factor
PF =
CHIMNEY
Densities ρair =
ρgas =
Draft head
hw = ( )
Volume Flow Rate of Flue gas
Qg =
Theoretical Velocity of Flue gas
Vt = √ ( )
Actual Velocity of Flue gas
Va = ( )
Chimney Inside Diameter, D
Qg = ( )
PIPING
GREEN Water
SILVER-GRAY Steam
VIOLET Acid/Alkali
LIGHT BLUE Air
LIGHT ORANGE Electricity
WHITE Communications
BROWN Flammable, Oil
YELLOW OCHRE Gases
BLACK Other Fluids, Drainage
SAFETY RED Fire fighting
SAFETY YELLOW Hazardous
Pipe wall thickness
Power Piping Systems:
tmin =
tnominal =
Industrial and Gas Piping Systems:
tmin =
Refrigeration Piping Systems:
tmin =

11. HEAT TRANSFER
Conduction
Q =
( )
Fluid to Wall to Fluid
Q =
( )
Composite Pipe
Q = ( ⁄ ) ( ⁄ )
Critical radius: rc =
Radiation Q/t = σ [ ]
Perfect Black Body
Convection Q = ( )
Heat Exchangers
LMTD = ( ⁄ )
AMTD =
Reynolds Number
(inertial/viscous) Re =
Prandtl Number
(momentum/heat) NPr =
Nusselt Number
(Tgradient/overall T) NNu =
Grashof Number
(buoyancy/viscous) NGr =
COMPRESSORS
SINGLE STAGE
, c = n, k, or 1
VD =
Capacity: V1’ =
Clearance: c = VC/VD
Volumetric Efficiency
ev = = ( )
⁄
Actual:
ev = [ ( )
⁄
]
Work: WS
Polytropic and Isentropic (n=k)
*( ) +
Isothermal:
( ) ( )
Compressor Efficiency
ec =
Piston Speed: V = 2LN
Indicated Power: Pind = PmiVD
Adiabatic Compressor Efficiency
ec(adiabatic) =
MULTISTAGE COMPRESSOR
Pm = √( ) ( )
*( ) +
PUMPS
Total dynamic head:
H =
( )
( )
Power: P =
Efficiencies:
epump =
emotor =
eoverall = = epem
evol = Q/VD
Volume flow rate: Q = VA
Slip: S = VD - Q
Percent Slip: %S= S/VD
Specific Speed NS =
√
⁄
Similar pumps:
Q H P
N
D
1 2 3
5
2
3
FANS AND BLOWERS
Static Head, hs =
Total head: h = hs +
Capacity: Q = AV
Power output: Pair =
Power input: Pbrake =
Static efficiency, es = ( )
Similar Fans:
Q H P
N
D
ρ
1 2 3
5
2
3
1 1
0
REFRIGERATION
Reverse Carnot COP COP =
Refrigeration Load Q =
Vapor Compression Cycle
1-2: compression
2-3: condensation
3-4: expansion
4-1: evaporation
Volume flow rate: V1’ = mv1
Heat rejected, QR = ( )
Refrigerating Capacity, QA = ( )
Refrigerating Effect, RE =
Coefficient of Performance
COP = =
AIRCONDITIONING
Pressure
Pt = Pa + Pv
Humidity ratio
w = =
Relative Humidity
RH =
Specific Volume
υ =
Enthalpy
H =
Degree of saturation
D = = ( )
Psychometric chart
DRY-BULB TEMP
SPECIFIC
HUMIDITY
SATURATION LINE
WET BULB TEMP
DEW POINT TEMP
SPECIFIC VOLUME
REL. HUMIDITY
Air mixing
Mass:
Energy:
Moisture:
Temp:
Air conditioner
RC = ( )
Rate of moisture removal = ( )
Volume flow rate: V1’ =
Cooling tower “drawing
Range: TR =
Approach: TA = –
Cooling tower efficiency
e =
Dryer
Regain =
Moisture content =
Aircon calculation
Sensible: Qs = ( )
Latent: QL = ( )
Total: QT =
Sensible Heat Ratio: SHR =
Recirculated air: mr =
Ventilation load: QV = ( )
MACHINERY ROOM
Exhaust air, Q = ,m3/s
Free aperture, F = , m2
*G in kg
OTHERS
1 yd3 = 6 sacks cement
Turbine specific speed, ns = ⁄
rk = rcre
 =
1
2
3
4
P
h