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Engineering Formula Sheet
Statistics

Mode

Mean

Place data in ascending order.
Mode = most frequently occurring value

∑ xi
n

µ=

µ = mean value
Σxi = sum of all data values (x1, x2, x3, …)
n = number of data values

Median
Place data in ascending order.
If n is odd, median = central value
If n is even, median = mean of two central values

Standard Deviation

σ=ඨ

If two values occur at the maximum frequency the
data set is bimodal.
If three or more values occur at the maximum
frequency the data set is multi-modal.

∑(xi - µ)2
n

n = number of data values

σ = standard deviation
xi = individual data value ( x1, x2, x3, …)
ߤ = mean value
n = number of data values

Range
Range = xmax - xmin
xmax = maximum data value
xmin = minimum data value

Probability
Independent Events
P (A and B and C) = PAPBPC

Frequency
fx =

nx
n

Px =

fx
fa

P (A and B and C) = probability of independent
events A and B and C occurring in sequence
PA = probability of event A
Mutually Exclusive Events

fx = relative frequency of outcome x
nx = number of events with outcome x
n = total number of events
Px = probability of outcome x
fa = frequency of all events
Binomial Probability (order doesn’t matter)
Pk =

n!(pk )(qn-k )
k!(n-k)!

P (A or B) = probability of either mutually exclusive
event A or B occurring in a trial
PA = probability of event A
Σxi = sum of all data values (x1, x2, x3, …)
n = number of data values
Conditional Probability

Pk = binomial probability of k successes in n trials
p = probability of a success
q = 1 – p = probability of failure
k = number of successes
n = number of trials

PLTW, Inc.

P (A or B) = PA + PB

ܲ(‫= )ܦ|ܣ‬

ܲ(‫)ܣ|ܦ(ܲ ∙ )ܣ‬
ܲ(‫)ܣ~|ܦ(ܲ ∙ )ܣ~(ܲ + )ܣ|ܦ(ܲ ∙ )ܣ‬

P (A|D) = probability of event A given event D
P(A) = probability of event A occurring
P(~A) = probability of event A not occurring
P(D|~A) = probability of event D given event A did not occur

Engineering Formulas

IED POE

DE

CEA

AE

BE

CIM EDD

1
Plane Geometry

Ellipse

Rectangle

2b

Area = π a b

Circle

Perimeter = 2a + 2b
Area = ab

2a

Circumference =2 π r
Area = π r2
B

Triangle
Parallelogram

Area = ½ bh

a = b + c – 2bc·cos∠A
2
2
2
b = a + c – 2ac·cos∠B
2
2
2
c = a + b – 2ab·cos∠C

h

Area = bh

2

b

2

cos θ =

a

c

a

c

c

h

2

A

C

b

s

s(ଵ f)
Area = n ଶ
2

2

c =a +b
sin θ =

2

Regular Polygons

Right Triangle
2

a

b

f

n = number of sides
θ

c

b

a

tan θ = b

a
h

Trapezoid
Area = ½(a + b)h

h
h
b
h

Solid Geometry
Cube

Sphere
3

Volume = s
2
Surface Area = 6s

s
s

ସ

r

3

Volume π r
ଷ
2
Surface Area = 4 π r

s

Rectangular Prism
Cylinder

r

h
Volume = wdh
Surface Area = 2(wd + wh + dh)

d

w

h

2

Volume = π r h
2
Surface Area = 2 π r h+2 π r

Right Circular Cone
πr2 h
Volume =
3
Surface Area = π r ඥr2 +h2

h

Irregular Prism
r

h

Volume = Ah
A = area of base

Pyramid
Volume =

Ah
3

A = area of base

PLTW, Inc.

h

Constants
2

g = 9.8 m/s = 32.27 ft/s
-11
3
2
G = 6.67 x 10 m /kg·s
π = 3.14159

Engineering Formulas

IED POE

DE

2

CEA

AE

BE

CIM EDD

2
Conversions
Mass

Force

Area
2

1 acre = 4047 m
2
= 43,560 ft
2
= 0.00156 mi

1 kg
= 2.205 lbm
1 slug = 32.2 lbm
1 ton = 2000 lbm

1N
1 kip

Energy
= 0.225 lbf
= 1,000 lbf

1J

= 0.239 cal
-4
= 9.48 x 10 Btu
= 0.7376 ft·lbf
1kW h = 3,6000,000 J

Pressure
Length
1m
1 km
1 in.
1 mi
1 yd

1 atm

Volume
= 3.28 ft
= 0.621 mi
= 2.54 cm
= 5280 ft
= 3 ft

1L

1mL

= 0.264 gal
3
= 0.0353 ft
= 33.8 fl oz
3
= 1 cm = 1 cc
1psi

= 1.01325 bar
= 33.9 ft H2O
= 29.92 in. Hg
= 760 mm Hg
= 101,325 Pa
= 14.7 psi
= 2.31 ft of H2O

Defined Units
1J
1N
1 Pa
1V
1W
1W
1 Hz
1F
1H

Time
Temperature Change

1K

= 1 ºC
= 1.8 ºF
= 1.8 ºR

1d
1h
1 min
1 yr

= 24 h
= 60 min
= 60 s
= 365 d

Power
1W

= 3.412 Btu/h
= 0.00134 hp
= 14.34 cal/min
= 0.7376 ft·lbf/s

= 1 N·m
= 1 kg·m / s2
= 1 N / m2
=1W/A
=1J/s
=1V/A
= 1 s-1
= 1 A·s / V
= 1 V·s / V

SI Prefixes
Numbers Less Than One
Power of 10
Prefix
Abbreviation
10-1
10-2
10-3
10-6
10-9
10-12
10-15
10-18
10-21
10-24

decicentimillimicronanopicofemtoattozeptoyocto-

Equations

d
c
m
µ
n
p
f
a
z
y

Numbers Greater Than One
Power of 10
Prefix
Abbreviation
101
102
103
106
109
1012
1015
1018
1021
1024

Temperature
TK = TC + 273

Mass and Weight
M = VDm
W = mg
W = VDw
V = volume
Dm = mass density
m = mass
Dw = weight density
g = acceleration due to gravity

PLTW, Inc.

TR = TF + 460
TF - 32
TC
=
180
100
TK = temperature in Kelvin
TC = temperature in Celsius
TR = temperature in Rankin
TF = temperature in Fahrenheit

Engineering Formulas

decahectokiloMegaGigaTeraPetaExaZettaYotta-

da
h
k
M
G
T
P
E
Z
Y

Force
F = ma
F = force
m = mass
a = acceleration
Equations of Static Equilibrium
ΣFx = 0

ΣFy = 0

ΣMP = 0

Fx = force in the x-direction
Fy = force in the y-direction
MP = moment about point P

IED POE

DE

CEA

AE

BE

CIM EDD

3
Equations (Continued)

Electricity
Fluid Mechanics

Energy: Work
W = F∙d

P=
V1

W = work
F = force
d = distance

F
A
V2

T1
P1
T1

Power

T2
P2

=

T2

RT (series) = R1 + R2+ ··· + Rn
(Charles’ Law)
RT (parallel) =
(Guy-Lussanc’s Law)

1
1 1
1
+ + ∙∙∙ +R
R1 R2
n

Kirchhoff’s Current Law

Q = Av

P = power
E = energy
W = work
t = time
τ = torque
rpm = revolutions per minute

Efficiency
Pout
∙100%
Pin

Pout = useful power output
Pin = total power input

IT = I1 + I2 + ··· + In
n
or IT = ∑k=1 Ik

A1v1 = A2v2

E W
=
t
t
τ∙rpm
P=
5252

Kirchhoff’s Voltage Law

Horsepower =

P = absolute pressure
F = Force
A = Area
V = volume
T = absolute temperature
Q = flow rate
v = flow velocity

U = potential energy
m =mass
g = acceleration due to gravity
h = height

t
‫܌‬
t

1
2

Q
∆t

(where acceleration = 0)

U=

1 k
=
R L
kA∆T
L

a=

vf ି vi
t

P=

X=

vi sin(2θ)
-g

A1v1 = A2v2
Pnet = σAe(T2 4 -T1 4 )
2

d = d0 + v0t + ½at

K = mv2

v = v0 + 2a(d – d0)

K = kinetic energy
m = mass
v = velocity

τ = dFsinθ

Energy: Thermal
Q =mc∆T
Q = thermal energy
m = mass
c = specific heat
∆T = change in temperature

Thermodynamics

P=

v = v0 + at
Energy: Kinetic

V = voltage
VT = total voltage
I = current
IT = total current
R = resistance
RT = total resistance
P = power

(where acceleration = 0)

d

v=

VT = V1 + V2 + ··· + Vn
n
or VT = ∑k=1 Vk

P = Q′ = AU∆T

Mechanics

Energy: Potential
U = mgh

QP
1714

absolute pressure = gauge pressure
+ atmospheric pressure

s=

PLTW, Inc.

V = IR
P = IV

P1V1 = P2V2 (Boyle’s Law)

P=

Efficiency (%) =

=

Ohm’s Law

2

2

s = speed
v = velocity
a = acceleration
X = range
t = time
d = distance
g = acceleration due to gravity
d = distance
θ = angle
τ = torque
F = force

Engineering Formulas

P = rate of heat transfer
Q = thermal energy
A = Area of thermal conductivity
U = coefficient of heat conductivity
(U-factor)
∆T = change in temperature
∆t = change in time
R = resistance to heat flow ( R-value)
k = thermal conductivity
v = velocity
Pnet = net power radiated
σ = 5.6696 x 10

-8

W
m2 ∙K

4

e = emissivity constant
T , T = temperature at time 1, time 2

POE 4 DE 4
Section Properties
Rectangle Centroid

Moment of Inertia
h
Ixx

x

x

3

bh
=
12

b

Ixx = moment of inertia of a rectangular section
about x-x axis

ഥ
x=

∑ Ai

ഥ
and y =

b
2

ഥ=
and y

h
2

Right Triangle Centroid
ഥ=
x

b
3

ഥ=
and y

h
3

Semi-circle Centroid

Complex Shapes Centroid
∑ xi Ai

ഥ=
x

ഥ
x = r and y =
ഥ

∑ yi A i
∑ Ai

ഥ=
x x-distance to the centroid
ത
y = y-distance to the centroid
xi = x distance to centroid of shape i
yi = y distance to centroid of shape i
Ai = Area of shape i

4r
3π

ഥ=
x x-distance to the centroid
ത
y = y-distance to the centroid

Structural Analysis
Material Properties
Beam Formulas
Stress (axial)

Reaction

F
σ=
A

Moment
Deflection

σ = stress
F = axial force
A = cross-sectional area

Reaction
Moment

Strain (axial)

Deflection

ϵ= δ
L0

Reaction

ϵ = strain
L0 = original length
δ = change in length

Deflection

Moment

PL

Mmax =

4

max

=

ߪ(F2 -F1 )L0
(ߜଶ − ߜଵ )A

E = modulus of elasticity
σ = stress
ε = strain
A = cross-sectional area
F = axial force
δ = deformation

PLTW, Inc.

ωL
2

ωL2

Mmax =

(at center)

8

5ωL4
384EI

(at center)

RA = RB = P
Mmax = Pa (between loads)
Pa
= 24EIቀ3L2 -4a2ቁ

max

Pb

Moment

E=

3

RA = RB =

RA =

Mmax =

Deflection

2

(at point of load)

PL
max = 48EI (at point of load)

Reaction

Modulus of Elasticity
σ
E=
ε

P

RA = RB =

୫ୟ୶

L

and RB =
Pab
L

(at center)

Pa
L

(at Point of Load)

= ౌaౘ(aశమౘ)ඥయa(aశమౘ)
మళు౅

(at x = ට

a(aାଶୠ)

Deformation: Axial

when a > b )

Truss Analysis

FL0
δ = AE

ଷ,

2J = M + R

δ = deformation
F = axial force
L0 = original length
A = cross-sectional area
E = modulus of elasticity

Engineering Formulas

J = number of joints
M =number of members
R = number of reaction forces

POE 5 AE 4 CEA 4
Simple Machines
Inclined Plane
Mechanical Advantage (MA)
DE
IMA=
DR
% Efficiency= ൬

FR
AMA=
FE

AMA
൰ 100
IMA

IMA=

L (slope)
H

Wedge

IMA = Ideal Mechanical Advantage
AMA = Actual Mechanical Advantage
DE = Effort Distance
DR = Resistance Distance
FE = Effort Force
FR = Resistance Force

IMA=

L (⊥ to height)
H

Lever
Screw
1st
Class

IMA =

C
Pitch

Pitch =
2nd
Class

1
TPI
C = Circumference
r = radius
Pitch = distance between
threads
TPI = Threads Per Inch

3rd
Class

Compound Machines
MATOTAL = (MA1) (MA2) (MA3) . . .

Wheel and Axle

Gears; Sprockets with Chains; and Pulleys with
Belts Ratios
Nout dout ωin τout
GR=
=
=
=
Nin din ωout τin

Effort at Axle

dout ωin τout
=
=
(pulleys)
din ωout τin
Compound Gears
B
D
GRTOTAL = ቀ ቁ ቀ ቁ
A
C

Effort at Wheel

Pulley Systems
IMA = Total number of strands of a single string
supporting the resistance
IMA =

DE (string pulled)
DR (resistance lifted)

PLTW, Inc.

GR = Gear Ratio
ωin = Angular Velocity - driver
ωout = Angular Velocity - driven
Nin = Number of Teeth - driver
Nout = Number of Teeth - driven
din = Diameter - driver
dout = Diameter - driven
τin = Torque - driver
τout = Torque - driven

Engineering Formulas

POE 6
Structural Design
Steel Beam Design: Shear
Va =

Steel Beam Design: Moment

Vn

Ma =

v

Mn
b

Vn = 0.6FyAw

Mn = FyZx

Va = allowable shear strength
Vn = nominal shear strength
v = 1.5 = factor of safety for shear
Fy = yield stress
Aw = area of web

Ma = allowable bending moment
Mn = nominal moment strength
b = 1.67 = factor of safety for
bending moment
Fy = yield stress
Zx = plastic section modulus about
neutral axis

Storm Water Runoff
Storm Water Drainage
Q = CfCiA
Cc =

C1 A1 + C2 A2 + ∙∙∙
A1 + A2 + ∙∙∙
3

Q = peak storm water runoff rate (ft /s)
Cf = runoff coefficient adjustment
factor
C = runoff coefficient
i = rainfall intensity (in./h)
A = drainage area (acres)
Runoff Coefficient
Adjustment Factor
Return
Period
Cf
1, 2, 5, 10 1.0
25
1.1
50
1.2
100
1.25

Water Supply
Hazen-Williams Formula
hf =

10.44LQ
C

1.85

1.85 4.8655

d

hf = head loss due to friction (ft of H2O)
L = length of pipe (ft)
Q = water flow rate (gpm)
C = Hazen-Williams constant
d = diameter of pipe (in.)
Dynamic Head

Rational Method Runoff Coefficients
Categorized by Surface
Forested
0.059—0.2
Asphalt
0.7—0.95
Brick
0.7—0.85
Concrete
0.8—0.95
Shingle roof
0.75—0.95
Lawns, well drained (sandy soil)
Up to 2% slope
0.05—0.1
2% to 7% slope
0.10—0.15
Over 7% slope
0.15—0.2
Lawns, poor drainage (clay soil)
Up to 2% slope
0.13—0.17
2% to 7% slope
0.18—0.22
Over 7% slope
0.25—0.35
Driveways,
0.75—0.85
walkways
Categorized by Use
Farmland
0.05—0.3
Pasture
0.05—0.3
Unimproved
0.1—0.3
Parks
0.1—0.25
Cemeteries
0.1—0.25
Railroad yard
0.2—0.40
Playgrounds
0.2—0.35
(except asphalt or Districts
Business
Neighborhood
0.5—0.7
City (downtown)
0.7—0.95
Residential
Single-family
0.3—0.5
Multi-plexes,
0.4—0.6
detached
Multi-plexes,
0.6—0.75
attached
Suburban
0.25—0.4
Apartments,
0.5—0.7
condominiums
Industrial
Light
0.5—0.8
Heavy
0.6—0.9

Spread Footing Design
qnet = qallowable - pfooting
pfooting = tfooting ∙150
q=

lb
2

ft

P
A

qnet = net allowable soil
bearing pressure
qallowable = total allowable soil
bearing pressure
pfooting = soil bearing pressure
due to footing weight
tfooting = thickness of footing
q = soil bearing pressure
P = column load applied
A = area of footing

dynamic head = static head – head loss

PLTW, Inc.

Engineering Formulas

CEA 5
PLTW, Inc.

Engineering Formulas

CEA 6

Equivalent Length of (Generic) Fittings

Hazen-Williams Constants
555 Timer Design Equations
T = 0.693 (RA + 2RB)C
f =

1
T

duty-cycle =

(RA + RB )
∙100%
(RA +2RB )

T = period
f = frequency
RA = resistance A
RB = resistance B
C = capacitance

Boolean Algebra
Boolean Theorems

Commutative Law

Consensus Theorems

X• 0 = 0

X•Y = Y•X

ഥ
X + XY = X + Y

X•1 = X

X+Y = Y+X

ഥഥ
ഥ
X + XY = X + Y

X• X =X

Associative Law

ഥ
X • X=0

X(YZ) = (XY)Z

X+0=X

ഥ
തത
X + XY =തത + Y
X
ഥ
ഥ ഥ ഥ
X + XY = X + Y

X + (Y + Z) = (X + Y) + Z
DeMorgan’s Theorems

X+1=1
X+X=X

Distributive Law

തതതതത= X + ഥ
XY ഥ Y

X+ഥ =1
X

X(Y+Z) = XY + XZ

തതതതതതത= ഥ • Y
X+Y X ഥ

ന
X=X

(X+Y)(W+Z) = XW+XZ+YW+YZ

Speeds and Feeds
N=

CSቀ12in.ቁ
ft
πd

fm = ft·nt·N
Plunge Rate = ½·fm
N = spindle speed (rpm)
CS = cutting speed (in./min)
d = diameter (in.)
fm = feed rate (in./min)
ft = feed (in./tooth)
nt = number of teeth

PLTW, Inc.

Engineering Formulas

DE 5

CIM 4
Aerospace Equations

R e=
CL =

Orbital Mechanics

I = Fave ∆t

F N = W൫vj - vo ൯

2D
Aρv2

Fnet = Favg - Fg

݁ =ඨ1 -

ρvl

a = vf ∆t

T = 2π

Forces of Flight
CD =

Propulsion

2L
Aρv2

M = Fd
CL = coefficient of lift
CD = coefficient of drag
L = lift
D = drag
A = wing area
ρ = density
Re = Reynolds number
v = velocity
l = length of fluid travel
= fluid viscosity
F = force
m = mass
g = acceleration due to gravity
M = moment
d = moment arm (distance from
datum perpendicular to F)

FN = net thrust
W = air mass flow
vo = flight velocity
vj = jet velocity
I = total impulse
Fave = average thrust force
t = change in time (thrust
duration)
Fnet = net force
Favg = average force
Fg = force of gravity
vf = final velocity
a = acceleration
t = change in time (thrust
duration)

NOTE: Fave and Favg are
easily confused.

1

K = 2 mv2
U=

− GMm
R

E=U+K=−

√

య

= 2π

aమ

√GM

݁ = eccentricity
b = semi-minor axis
a =semi-major axis
T = orbital period
a = semi-major axis
= gravitational parameter
F = force of gravity between two
bodies
G = universal gravitation constant
M =mass of central body
m = mass of orbiting object
r = distance between center of two
objects
Bernoulli’s Law
ρv2
ρv2
ቇ = ቆPs +
ቇ
2 1
2 2

PS = static pressure
v = velocity
ρ = density
GMm
2R

K = kinetic energy
m =mass
v = velocity
U = gravitational potential energy
G = universal gravitation constant
M =mass of central body
m = mass of orbiting object
R = Distance center main body to
center of orbiting object
E = Total Energy of an orbit

PLTW, Inc.

య

aమ

GMm
F=
r2

ቆPs +

Energy

b2
a2

Engineering Formulas

Atmosphere Parameters
T = 15.04 - 0.00649h
p = 101.29 ቈ
ρ=

(T + 273.1)
቉
288.08

5.256

p
0.2869(T + 273.1)

T = temperature
h = height
p = pressure
ρ = density

AE 5

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Engineering formula sheet

  • 1. Engineering Formula Sheet Statistics Mode Mean Place data in ascending order. Mode = most frequently occurring value ∑ xi n µ= µ = mean value Σxi = sum of all data values (x1, x2, x3, …) n = number of data values Median Place data in ascending order. If n is odd, median = central value If n is even, median = mean of two central values Standard Deviation σ=ඨ If two values occur at the maximum frequency the data set is bimodal. If three or more values occur at the maximum frequency the data set is multi-modal. ∑(xi - µ)2 n n = number of data values σ = standard deviation xi = individual data value ( x1, x2, x3, …) ߤ = mean value n = number of data values Range Range = xmax - xmin xmax = maximum data value xmin = minimum data value Probability Independent Events P (A and B and C) = PAPBPC Frequency fx = nx n Px = fx fa P (A and B and C) = probability of independent events A and B and C occurring in sequence PA = probability of event A Mutually Exclusive Events fx = relative frequency of outcome x nx = number of events with outcome x n = total number of events Px = probability of outcome x fa = frequency of all events Binomial Probability (order doesn’t matter) Pk = n!(pk )(qn-k ) k!(n-k)! P (A or B) = probability of either mutually exclusive event A or B occurring in a trial PA = probability of event A Σxi = sum of all data values (x1, x2, x3, …) n = number of data values Conditional Probability Pk = binomial probability of k successes in n trials p = probability of a success q = 1 – p = probability of failure k = number of successes n = number of trials PLTW, Inc. P (A or B) = PA + PB ܲ(‫= )ܦ|ܣ‬ ܲ(‫)ܣ|ܦ(ܲ ∙ )ܣ‬ ܲ(‫)ܣ~|ܦ(ܲ ∙ )ܣ~(ܲ + )ܣ|ܦ(ܲ ∙ )ܣ‬ P (A|D) = probability of event A given event D P(A) = probability of event A occurring P(~A) = probability of event A not occurring P(D|~A) = probability of event D given event A did not occur Engineering Formulas IED POE DE CEA AE BE CIM EDD 1
  • 2. Plane Geometry Ellipse Rectangle 2b Area = π a b Circle Perimeter = 2a + 2b Area = ab 2a Circumference =2 π r Area = π r2 B Triangle Parallelogram Area = ½ bh a = b + c – 2bc·cos∠A 2 2 2 b = a + c – 2ac·cos∠B 2 2 2 c = a + b – 2ab·cos∠C h Area = bh 2 b 2 cos θ = a c a c c h 2 A C b s s(ଵ f) Area = n ଶ 2 2 c =a +b sin θ = 2 Regular Polygons Right Triangle 2 a b f n = number of sides θ c b a tan θ = b a h Trapezoid Area = ½(a + b)h h h b h Solid Geometry Cube Sphere 3 Volume = s 2 Surface Area = 6s s s ସ r 3 Volume π r ଷ 2 Surface Area = 4 π r s Rectangular Prism Cylinder r h Volume = wdh Surface Area = 2(wd + wh + dh) d w h 2 Volume = π r h 2 Surface Area = 2 π r h+2 π r Right Circular Cone πr2 h Volume = 3 Surface Area = π r ඥr2 +h2 h Irregular Prism r h Volume = Ah A = area of base Pyramid Volume = Ah 3 A = area of base PLTW, Inc. h Constants 2 g = 9.8 m/s = 32.27 ft/s -11 3 2 G = 6.67 x 10 m /kg·s π = 3.14159 Engineering Formulas IED POE DE 2 CEA AE BE CIM EDD 2
  • 3. Conversions Mass Force Area 2 1 acre = 4047 m 2 = 43,560 ft 2 = 0.00156 mi 1 kg = 2.205 lbm 1 slug = 32.2 lbm 1 ton = 2000 lbm 1N 1 kip Energy = 0.225 lbf = 1,000 lbf 1J = 0.239 cal -4 = 9.48 x 10 Btu = 0.7376 ft·lbf 1kW h = 3,6000,000 J Pressure Length 1m 1 km 1 in. 1 mi 1 yd 1 atm Volume = 3.28 ft = 0.621 mi = 2.54 cm = 5280 ft = 3 ft 1L 1mL = 0.264 gal 3 = 0.0353 ft = 33.8 fl oz 3 = 1 cm = 1 cc 1psi = 1.01325 bar = 33.9 ft H2O = 29.92 in. Hg = 760 mm Hg = 101,325 Pa = 14.7 psi = 2.31 ft of H2O Defined Units 1J 1N 1 Pa 1V 1W 1W 1 Hz 1F 1H Time Temperature Change 1K = 1 ºC = 1.8 ºF = 1.8 ºR 1d 1h 1 min 1 yr = 24 h = 60 min = 60 s = 365 d Power 1W = 3.412 Btu/h = 0.00134 hp = 14.34 cal/min = 0.7376 ft·lbf/s = 1 N·m = 1 kg·m / s2 = 1 N / m2 =1W/A =1J/s =1V/A = 1 s-1 = 1 A·s / V = 1 V·s / V SI Prefixes Numbers Less Than One Power of 10 Prefix Abbreviation 10-1 10-2 10-3 10-6 10-9 10-12 10-15 10-18 10-21 10-24 decicentimillimicronanopicofemtoattozeptoyocto- Equations d c m µ n p f a z y Numbers Greater Than One Power of 10 Prefix Abbreviation 101 102 103 106 109 1012 1015 1018 1021 1024 Temperature TK = TC + 273 Mass and Weight M = VDm W = mg W = VDw V = volume Dm = mass density m = mass Dw = weight density g = acceleration due to gravity PLTW, Inc. TR = TF + 460 TF - 32 TC = 180 100 TK = temperature in Kelvin TC = temperature in Celsius TR = temperature in Rankin TF = temperature in Fahrenheit Engineering Formulas decahectokiloMegaGigaTeraPetaExaZettaYotta- da h k M G T P E Z Y Force F = ma F = force m = mass a = acceleration Equations of Static Equilibrium ΣFx = 0 ΣFy = 0 ΣMP = 0 Fx = force in the x-direction Fy = force in the y-direction MP = moment about point P IED POE DE CEA AE BE CIM EDD 3
  • 4. Equations (Continued) Electricity Fluid Mechanics Energy: Work W = F∙d P= V1 W = work F = force d = distance F A V2 T1 P1 T1 Power T2 P2 = T2 RT (series) = R1 + R2+ ··· + Rn (Charles’ Law) RT (parallel) = (Guy-Lussanc’s Law) 1 1 1 1 + + ∙∙∙ +R R1 R2 n Kirchhoff’s Current Law Q = Av P = power E = energy W = work t = time τ = torque rpm = revolutions per minute Efficiency Pout ∙100% Pin Pout = useful power output Pin = total power input IT = I1 + I2 + ··· + In n or IT = ∑k=1 Ik A1v1 = A2v2 E W = t t τ∙rpm P= 5252 Kirchhoff’s Voltage Law Horsepower = P = absolute pressure F = Force A = Area V = volume T = absolute temperature Q = flow rate v = flow velocity U = potential energy m =mass g = acceleration due to gravity h = height t ‫܌‬ t 1 2 Q ∆t (where acceleration = 0) U= 1 k = R L kA∆T L a= vf ି vi t P= X= vi sin(2θ) -g A1v1 = A2v2 Pnet = σAe(T2 4 -T1 4 ) 2 d = d0 + v0t + ½at K = mv2 v = v0 + 2a(d – d0) K = kinetic energy m = mass v = velocity τ = dFsinθ Energy: Thermal Q =mc∆T Q = thermal energy m = mass c = specific heat ∆T = change in temperature Thermodynamics P= v = v0 + at Energy: Kinetic V = voltage VT = total voltage I = current IT = total current R = resistance RT = total resistance P = power (where acceleration = 0) d v= VT = V1 + V2 + ··· + Vn n or VT = ∑k=1 Vk P = Q′ = AU∆T Mechanics Energy: Potential U = mgh QP 1714 absolute pressure = gauge pressure + atmospheric pressure s= PLTW, Inc. V = IR P = IV P1V1 = P2V2 (Boyle’s Law) P= Efficiency (%) = = Ohm’s Law 2 2 s = speed v = velocity a = acceleration X = range t = time d = distance g = acceleration due to gravity d = distance θ = angle τ = torque F = force Engineering Formulas P = rate of heat transfer Q = thermal energy A = Area of thermal conductivity U = coefficient of heat conductivity (U-factor) ∆T = change in temperature ∆t = change in time R = resistance to heat flow ( R-value) k = thermal conductivity v = velocity Pnet = net power radiated σ = 5.6696 x 10 -8 W m2 ∙K 4 e = emissivity constant T , T = temperature at time 1, time 2 POE 4 DE 4
  • 5. Section Properties Rectangle Centroid Moment of Inertia h Ixx x x 3 bh = 12 b Ixx = moment of inertia of a rectangular section about x-x axis ഥ x= ∑ Ai ഥ and y = b 2 ഥ= and y h 2 Right Triangle Centroid ഥ= x b 3 ഥ= and y h 3 Semi-circle Centroid Complex Shapes Centroid ∑ xi Ai ഥ= x ഥ x = r and y = ഥ ∑ yi A i ∑ Ai ഥ= x x-distance to the centroid ത y = y-distance to the centroid xi = x distance to centroid of shape i yi = y distance to centroid of shape i Ai = Area of shape i 4r 3π ഥ= x x-distance to the centroid ത y = y-distance to the centroid Structural Analysis Material Properties Beam Formulas Stress (axial) Reaction F σ= A Moment Deflection σ = stress F = axial force A = cross-sectional area Reaction Moment Strain (axial) Deflection ϵ= δ L0 Reaction ϵ = strain L0 = original length δ = change in length Deflection Moment PL Mmax = 4 max = ߪ(F2 -F1 )L0 (ߜଶ − ߜଵ )A E = modulus of elasticity σ = stress ε = strain A = cross-sectional area F = axial force δ = deformation PLTW, Inc. ωL 2 ωL2 Mmax = (at center) 8 5ωL4 384EI (at center) RA = RB = P Mmax = Pa (between loads) Pa = 24EIቀ3L2 -4a2ቁ max Pb Moment E= 3 RA = RB = RA = Mmax = Deflection 2 (at point of load) PL max = 48EI (at point of load) Reaction Modulus of Elasticity σ E= ε P RA = RB = ୫ୟ୶ L and RB = Pab L (at center) Pa L (at Point of Load) = ౌaౘ(aశమౘ)ඥయa(aశమౘ) మళు౅ (at x = ට a(aାଶୠ) Deformation: Axial when a > b ) Truss Analysis FL0 δ = AE ଷ, 2J = M + R δ = deformation F = axial force L0 = original length A = cross-sectional area E = modulus of elasticity Engineering Formulas J = number of joints M =number of members R = number of reaction forces POE 5 AE 4 CEA 4
  • 6. Simple Machines Inclined Plane Mechanical Advantage (MA) DE IMA= DR % Efficiency= ൬ FR AMA= FE AMA ൰ 100 IMA IMA= L (slope) H Wedge IMA = Ideal Mechanical Advantage AMA = Actual Mechanical Advantage DE = Effort Distance DR = Resistance Distance FE = Effort Force FR = Resistance Force IMA= L (⊥ to height) H Lever Screw 1st Class IMA = C Pitch Pitch = 2nd Class 1 TPI C = Circumference r = radius Pitch = distance between threads TPI = Threads Per Inch 3rd Class Compound Machines MATOTAL = (MA1) (MA2) (MA3) . . . Wheel and Axle Gears; Sprockets with Chains; and Pulleys with Belts Ratios Nout dout ωin τout GR= = = = Nin din ωout τin Effort at Axle dout ωin τout = = (pulleys) din ωout τin Compound Gears B D GRTOTAL = ቀ ቁ ቀ ቁ A C Effort at Wheel Pulley Systems IMA = Total number of strands of a single string supporting the resistance IMA = DE (string pulled) DR (resistance lifted) PLTW, Inc. GR = Gear Ratio ωin = Angular Velocity - driver ωout = Angular Velocity - driven Nin = Number of Teeth - driver Nout = Number of Teeth - driven din = Diameter - driver dout = Diameter - driven τin = Torque - driver τout = Torque - driven Engineering Formulas POE 6
  • 7. Structural Design Steel Beam Design: Shear Va = Steel Beam Design: Moment Vn Ma = v Mn b Vn = 0.6FyAw Mn = FyZx Va = allowable shear strength Vn = nominal shear strength v = 1.5 = factor of safety for shear Fy = yield stress Aw = area of web Ma = allowable bending moment Mn = nominal moment strength b = 1.67 = factor of safety for bending moment Fy = yield stress Zx = plastic section modulus about neutral axis Storm Water Runoff Storm Water Drainage Q = CfCiA Cc = C1 A1 + C2 A2 + ∙∙∙ A1 + A2 + ∙∙∙ 3 Q = peak storm water runoff rate (ft /s) Cf = runoff coefficient adjustment factor C = runoff coefficient i = rainfall intensity (in./h) A = drainage area (acres) Runoff Coefficient Adjustment Factor Return Period Cf 1, 2, 5, 10 1.0 25 1.1 50 1.2 100 1.25 Water Supply Hazen-Williams Formula hf = 10.44LQ C 1.85 1.85 4.8655 d hf = head loss due to friction (ft of H2O) L = length of pipe (ft) Q = water flow rate (gpm) C = Hazen-Williams constant d = diameter of pipe (in.) Dynamic Head Rational Method Runoff Coefficients Categorized by Surface Forested 0.059—0.2 Asphalt 0.7—0.95 Brick 0.7—0.85 Concrete 0.8—0.95 Shingle roof 0.75—0.95 Lawns, well drained (sandy soil) Up to 2% slope 0.05—0.1 2% to 7% slope 0.10—0.15 Over 7% slope 0.15—0.2 Lawns, poor drainage (clay soil) Up to 2% slope 0.13—0.17 2% to 7% slope 0.18—0.22 Over 7% slope 0.25—0.35 Driveways, 0.75—0.85 walkways Categorized by Use Farmland 0.05—0.3 Pasture 0.05—0.3 Unimproved 0.1—0.3 Parks 0.1—0.25 Cemeteries 0.1—0.25 Railroad yard 0.2—0.40 Playgrounds 0.2—0.35 (except asphalt or Districts Business Neighborhood 0.5—0.7 City (downtown) 0.7—0.95 Residential Single-family 0.3—0.5 Multi-plexes, 0.4—0.6 detached Multi-plexes, 0.6—0.75 attached Suburban 0.25—0.4 Apartments, 0.5—0.7 condominiums Industrial Light 0.5—0.8 Heavy 0.6—0.9 Spread Footing Design qnet = qallowable - pfooting pfooting = tfooting ∙150 q= lb 2 ft P A qnet = net allowable soil bearing pressure qallowable = total allowable soil bearing pressure pfooting = soil bearing pressure due to footing weight tfooting = thickness of footing q = soil bearing pressure P = column load applied A = area of footing dynamic head = static head – head loss PLTW, Inc. Engineering Formulas CEA 5
  • 8. PLTW, Inc. Engineering Formulas CEA 6 Equivalent Length of (Generic) Fittings Hazen-Williams Constants
  • 9. 555 Timer Design Equations T = 0.693 (RA + 2RB)C f = 1 T duty-cycle = (RA + RB ) ∙100% (RA +2RB ) T = period f = frequency RA = resistance A RB = resistance B C = capacitance Boolean Algebra Boolean Theorems Commutative Law Consensus Theorems X• 0 = 0 X•Y = Y•X ഥ X + XY = X + Y X•1 = X X+Y = Y+X ഥഥ ഥ X + XY = X + Y X• X =X Associative Law ഥ X • X=0 X(YZ) = (XY)Z X+0=X ഥ തത X + XY =തത + Y X ഥ ഥ ഥ ഥ X + XY = X + Y X + (Y + Z) = (X + Y) + Z DeMorgan’s Theorems X+1=1 X+X=X Distributive Law തതതതത= X + ഥ XY ഥ Y X+ഥ =1 X X(Y+Z) = XY + XZ തതതതതതത= ഥ • Y X+Y X ഥ ന X=X (X+Y)(W+Z) = XW+XZ+YW+YZ Speeds and Feeds N= CSቀ12in.ቁ ft πd fm = ft·nt·N Plunge Rate = ½·fm N = spindle speed (rpm) CS = cutting speed (in./min) d = diameter (in.) fm = feed rate (in./min) ft = feed (in./tooth) nt = number of teeth PLTW, Inc. Engineering Formulas DE 5 CIM 4
  • 10. Aerospace Equations R e= CL = Orbital Mechanics I = Fave ∆t F N = W൫vj - vo ൯ 2D Aρv2 Fnet = Favg - Fg ݁ =ඨ1 - ρvl a = vf ∆t T = 2π Forces of Flight CD = Propulsion 2L Aρv2 M = Fd CL = coefficient of lift CD = coefficient of drag L = lift D = drag A = wing area ρ = density Re = Reynolds number v = velocity l = length of fluid travel = fluid viscosity F = force m = mass g = acceleration due to gravity M = moment d = moment arm (distance from datum perpendicular to F) FN = net thrust W = air mass flow vo = flight velocity vj = jet velocity I = total impulse Fave = average thrust force t = change in time (thrust duration) Fnet = net force Favg = average force Fg = force of gravity vf = final velocity a = acceleration t = change in time (thrust duration) NOTE: Fave and Favg are easily confused. 1 K = 2 mv2 U= − GMm R E=U+K=− √ య = 2π aమ √GM ݁ = eccentricity b = semi-minor axis a =semi-major axis T = orbital period a = semi-major axis = gravitational parameter F = force of gravity between two bodies G = universal gravitation constant M =mass of central body m = mass of orbiting object r = distance between center of two objects Bernoulli’s Law ρv2 ρv2 ቇ = ቆPs + ቇ 2 1 2 2 PS = static pressure v = velocity ρ = density GMm 2R K = kinetic energy m =mass v = velocity U = gravitational potential energy G = universal gravitation constant M =mass of central body m = mass of orbiting object R = Distance center main body to center of orbiting object E = Total Energy of an orbit PLTW, Inc. య aమ GMm F= r2 ቆPs + Energy b2 a2 Engineering Formulas Atmosphere Parameters T = 15.04 - 0.00649h p = 101.29 ቈ ρ= (T + 273.1) ቉ 288.08 5.256 p 0.2869(T + 273.1) T = temperature h = height p = pressure ρ = density AE 5