PLTW Engineering Formula Sheet 2016 (v16.1)
1.0 Statistics
Mode
Mean
xi
μ=
(1.1a)
x=
Place data in ascending order.
Mode = most frequently occurring value
xi
N
n
µ = population mean
x = sample mean
Σxi = sum of all data values (x1, x2, x3, …)
N = size of population
n = size of sample
If two values occur with maximum frequency the data
set is bimodal.
If three or more values occur with maximum
frequency the data set is multi-modal.
Standard Deviation
Median
σ=
Place data in ascending order.
If N is odd, median = central value
If N is even, median = mean of two central values
(1.2)
N = size of population
Range (1.5)
Range = xmax - xmin
(1.4)
(1.1b)
(1.3)
xmax = maximum data value
xmin = minimum data value
s=
xi - μ
2
N
xi - x
(Population)
(1.5a)
(Sample)
(1.5b)
2
n ‒1
σ = population standard deviation
s = sample standard deviation
xi = individual data value ( x1, x2, x3, …)
𝜇 = population mean
x = sample mean
N = size of population
n = size of sample
2.0 Probability
Independent Events
P (A and B and C) = PAPBPC
Frequency
fx =
nx
(2.1)
n
fx = relative frequency of outcome x
nx = number of events with outcome x
n = total number of events
n!(pk )(qn-k )
k!(n-k)!
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
P (A or B) = PA + PB
(2.4)
P (A or B) = probability of either mutually exclusive
event A or B occurring in a trial
PA = probability of event A
Binomial Probability
(order doesn’t matter)
Pk =
(2.2)
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
Conditional Probability
𝑃 𝐴𝐷 =𝑃
𝑃 𝐴 ∙𝑃 𝐷 𝐴
𝐴 ∙𝑃
𝐷𝐴
(2.5)
+𝑃 ~𝐴 ∙𝑃 𝐷 ~𝐴
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
IED POE
© 2016 Project Lead The Way, Inc.
PLTW Engineering Formula Sheet 2016
(2.3)
DE
CEA
AE
ES/BE
CIM EDD
1
v16.1
3.0 Plane Geometry
Ellipse
Rectangle
2b
Perimeter = 2a + 2b (3.9)
Circle
Area = π a b (3.8)
2a
Area = ab
(3.10)
Circumference =2 π r (3.1)
Area = π r2
(3.2)
B
Triangle (3.6)
Parallelogram
Area = ½ bh
h
Area = bh
(3.11)
a2 = b2 + c2 – 2bc·cos∠A
b2 = a2 + c2 – 2ac·cos∠B
c2 = a2 + b2 – 2ab·cos∠C
(3.3)
b
Right Triangle
Regular Polygons
c2
Area = n
a2
+
sin θ =
a
=
b2
(3.4)
cos θ =
b
tan θ = ab
(3.12)
(3.13)
A
C
b
(3.14)
ns2
4tan(
c
h
(3.15)
180
)
n
n = number of sides
θ
(3.6)
c
2
=
c
a
(3.5)
c
s(12 f)
a
b
(3.7)
a
h
Trapezoid
Area = ½(a + b)h
h
h
(3.16)
b
h
4.0 Solid Geometry
Cube
Sphere
Volume = s3
Surface Area =
s
(4.1)
6s2
4
(4.2)
s
Volume = π r3
(4.8)
Surface Area = 4 π r2
(4.9)
3
s
Rectangular Prism
Cylinder
Volume = wdh
r
h
(4.3)
Surface Area = 2(wd + wh + dh) (4.4)
d
w
Volume = π r2 h
(4.10)
Surface Area = 2 π r h+2 π r2
(4.11)
h
Right Circular Cone
Volume =
h
πr2 h
3
Surface Area = π r2 + π r
Irregular Prism
(4.5)
r2 +h2 (4.6)
r
Volume = Ah
h
(4.12)
A = area of base
Pyramid
Volume =
Ah
3
(4.7)
A = area of base
© 2016 Project Lead The Way, Inc.
PLTW Engineering Formula Sheet 2016
h
5.0 Constants
g = 9.8 m/s2 = 32.17 ft/s2
G = 6.67 x 10-11 m3/kg·s2
π = 3.14159
IED POE
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v16.1
6.0 Conversions
Pressure (6.8)
Mass/Weight (6.1)
Area (6.4)
1 kg
1 slug
1 ton
1 lb
1 acre = 4047 m2
= 43,560 ft2
= 0.00156 mi2
= 2.205 lbm
= 32.2 lbm
= 2000 lb
= 16 oz
1m
1 km
1 in.
1 mi
1 yd
= 3.28 ft
= 0.621 mi
= 2.54 cm
= 5280 ft
= 3 ft
1mL
= 0.264 gal
= 0.0353 ft3
= 33.8 fl oz
= 1 cm3 = 1 cc
= 24 h
= 60 min
= 60 s
= 365 d
1psi
1W
*Use equation in section
9.0 to convert
1 hp
= Δ 1 ºC
= Δ 1.8 ºF
= Δ 1.8 ºR
7.0 Defined Units
= 3.412 Btu/h
= 0.00134 hp
= 14.34 cal/min
= 0.7376 ft·lbf/s
= 550 ft∙lb/sec
1J
1N
1 Pa
1V
1W
1W
1 Hz
1F
1H
1J
= 0.239 cal
= 9.48 x 10-4 Btu
= 0.7376 ft·lbf
1kW h = 3,600,000 J
= 0.225 lb
= 1,000 lb
= 2π rad/sec
= 60 rpm
Energy (6.10)
Force (6.7)
1N
1 kip
1 Hz
= 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
Power (6.9)
Temperature Unit
Equivalents (6.6)
Δ1 K
Time (6.3)
1d
1h
1 min
1 yr
Volume (6.5)
1L
Length (6.2)
1 atm
Rotational Speed (6.11)
= 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
8.0 SI Prefixes
Numbers Less Than One
Numbers Greater Than One
Power
of 10
Decimal
Equivalent
Prefix
Abbreviation
Power
of 10
10-1
0.1
deci-
d
101
10-2
0.01
centi-
c
102
m
103
µ
106
nano-
n
109
pico-
p
1012
10-3
10-6
10-9
0.001
0.000001
0.000000001
10-12
millimicro-
Whole Number
Prefix
Abbreviation
10
deca-
da
100
hecto-
h
1000
kilo-
k
1,000,000
Mega-
M
1,000,000,000
Giga-
G
Tera-
T
Equivalent
10-15
femto-
f
1015
Peta-
P
10-18
atto-
a
1018
Exa-
E
z
1021
Zetta-
Z
y
1024
Yotta-
Y
10-21
zepto-
10-24
yocto-
9.0 Equations
Mass and Weight
Temperature
TK = TC + 273
m = VDm
(9.1)
TR = TF + 460
W = mg
(9.2)
TF =
W = VDw
(9.3)
V = volume
Dm = mass density
m = mass
Dw = weight density
W = weight
g = acceleration due to gravity
© 2016 Project Lead The Way, Inc.
PLTW Engineering Formula Sheet 2016
TC =
9
5
Tc + 32
TF - 32
1.8
Force and Moment
F = ma
M = Fd (9.7b)
(9.7a)
(9.4)
(9.5)
(9.6a)
F = force
m = mass
a = acceleration
M = moment
d= perpendicular distance
(9.6b)
Equations of Static Equilibrium
TK = temperature in Kelvin
TC = temperature in Celsius
TR = temperature in Rankin
TF = temperature in Fahrenheit
ΣFx = 0
ΣFy = 0
ΣMP = 0 (9.8)
Fx = force in the x-direction
Fy = force in the y-direction
MP = moment about point P
IED POE
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ES/BE
CIM EDD
3
v16.1
9.0 Equations (Continued)
W = F∥ ∙ d
(9.9)
W = work
F∥ = force parallel to direction of
displacement
d = displacement
t
=
p=
V1
T1
p1
T1
F
W
(9.10)
t
P=τω
(9.11)
P = power
E = energy
W = work
t = time
τ = torque
ω = angular velocity
A
V2
=T
(Charles’ Law)
2
(9.17)
p
= T2 (Gay-Lussanc’s Law)
2
Efficiency (%) =
Pout
Pin
∙100% (9.12)
Pout = useful power output
Pin = total power input
(9.13)
U = potential energy
m =mass
g = acceleration due to gravity
h = height
Energy: Kinetic
K = 12 mv2
(9.14)
K = kinetic energy
m = mass
v = velocity
Energy: Thermal
∆Q = mc∆T
(9.15)
∆Q = change in thermal energy
m = mass
c = specific heat
∆T = change in temperature
© 2016 Project Lead The Way, Inc.
PLTW Engineering Formula Sheet 2016
P = IV
(9.33)
RT (series) = R1 + R2+ ··· + Rn (9.34)
1
RT (parallel) =
(9.19)
Kirchhoff’s Current Law
1 1
1
+ + ∙∙∙ +
R1 R2
Rn
Q = Av
(9.20)
A1v1 = A2v2
(9.21)
IT = I1 + I2 + ··· + In
n
or IT = k=1 Ik
P = Qp
(9.22)
Kirchhoff’s Voltage Law
VT = V1 + V2 + ··· + Vn
n
or VT = k=1 Vk
p = absolute pressure
F = force
A = area
V = volume
T = absolute temperature
Q = flow rate
v = flow velocity
P = power
V = voltage
VT = total voltage
I = current
IT = total current
R = resistance
RT = total resistance
P = power
Mechanics
Thermodynamics
s=
d
v=
∆d
Energy: Potential
U = mgh
(9.32)
(9.18)
absolute pressure = gauge pressure
+ atmospheric pressure (9.23)
Efficiency
V = IR
(9.16)
p1V1 = p2V2 (Boyle’s Law)
Power
E
Ohm’s Law
Fluid Mechanics
Energy: Work
P=
Electricity
a=
X=
(9.24)
t
P = Q′ = AU∆T
P = Q' =
∆t
vf − vi
t
vi 2 sin(2θ)
-g
(9.25)
U=
(9.26)
P=
(9.27)
v = vi + at
(9.28)
d = di + vit + ½at2
(9.29)
v2 = vi2 + 2a(d – di)
(9.30)
τ = dFsinθ
(9.31)
s = average speed
v = average velocity
v = velocity
vi = initial velocity (t =0)
a = acceleration
X = range
t = time
∆d = change in displacement
d = distance
di = initial distance (t=0)
g = acceleration due to gravity
θ = angle
τ = torque
F = force
1
R
(9.35)
(9.36)
(9.37)
(9.38)
∆Q
∆t
(9.39)
k
=L
(9.40)
kA∆T
(9.41)
L
A1v1 = A2v2
(9.42)
Pnet = σAe(T2 4 -T1 4 )
(9.43)
k=
PL
(9.44)
A∆T
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
W
σ = 5.6696 x 10-8 m2∙K4
e = emissivity constant
L = thickness
T1, T2 = temperature at time 1, time 2
POE 4 DE 4
AE 4
CIM 4
v16.1
10.0 Section Properties
y
Moment of Inertia
Ixx =
h
bh3
x
x=
(10.1)
12
x=
Ai
and y =
2
h
and y =
(10.3)
2
x
b
3
and y =
h
y
(10.4)
3
x
y
Semi-circle Centroid
Complex Shapes Centroid
x=
b
Right Triangle Centroid
b
Ixx = moment of inertia of a rectangular section
about x axis
xi Ai
y
Rectangle Centroid
x = r and y =
yi Ai
(10.2)
Ai
x = x-distance to the centroid
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
(10.5)
x
x = x-distance to the centroid
y = y-distance to the centroid
12.0 Structural Analysis
11.0 Material
Properties
Beam Formulas
Reaction
Stress (axial)
F
σ=
A
Moment
(11.1)
σ = stress
F = axial force
A = cross-sectional area
Reaction
RA = RB =
Moment
Reaction
(11.2)
ε = strain
L0 = original length
δ = change in length
E=
ε
(F2 -F1 )L0
𝛿2 −𝛿1 )A
2
ωL
Mmax =
(12.4)
2
(at center)
(12.5)
Δmax = 384EI (at center)
(12.6)
8
5ωL4
RA = RB = P
(12.7)
Mmax = Pa
(12.8)
Deflection
Pa
Δmax = 24EI
(3L2 -4a2 )
(12.9)
Moment
Deflection
(11.3)
RA =
Pb
L
Mmax =
Δmax =
and RB =
Pab
L
Pa
L
(at Point of Load) (12.11)
Pab a+2b 3a a+2b
27EI
(at x =
(12.10)
a a+2b
3,
(12.12)
when a>b )
E = modulus of elasticity
I = moment of inertia
(11.4)
Deformation: Axial
E = modulus of elasticity
σ = stress
ε = strain
A = cross-sectional area
F = axial force
δ = deformation
ωL
(12.3)
(at center)
Modulus of Elasticity
σ
(at point of load)
(12.2)
Moment
Reaction
E=
(at point of load)
4
PL3
48EI
Δmax =
Deflection
δ
L0
Mmax =
(12.1)
2
PL
Deflection
Strain (axial)
ε=
P
RA = R B =
δ=
FL0
AE
Truss Analysis
(12.13)
δ = deformation
F = axial force
L0 = original length
A = cross-sectional area
E = modulus of elasticity
© 2016 Project Lead The Way, Inc.
PLTW Engineering Formula Sheet 2016
2J = M + R
(12.14)
J = number of joints
M =number of members
R = number of reaction forces
POE 5 AE 5 CEA 4
v16.1
13.0 Simple Machines
Inclined Plane
Mechanical Advantage (MA)
IMA=
DE
(13.1)
DR
% Efficiency= (
AMA
IMA
) 100
AMA=
FR
FE
(13.2)
L
IMA=
(13.6)
H
(13.3)
IMA = ideal mechanical advantage
AMA = actual mechanical advantage
DE = effort distance
DR = resistance distance
FE = effort force
FR = resistance force
Wedge
L
IMA=
(13.7)
H
Lever
1st
Class
Screw
IMA =
2nd
Class
C
Pitch
Pitch =
(13.8)
1
(13.9)
TPI
C = circumference
r = radius
Pitch = distance between
threads
TPI = threads per inch
3rd
Class
Compound Machines
MATOTAL = (MA1) (MA2) (MA3) . . .
(13.10)
Wheel and Axle
Gears; Sprockets with Chains; and Pulleys
with Belts Ratios
Effort at Axle
GR =
dout
din
=
Nout
Nin
=
dout
=
τout
ωin
ωout
din
τin
=
ωin
ωout
=
(pulleys)
τout
τin
(13.11)
(13.12)
Compound Gears
Effort at Wheel
GRTOTAL = (
B
D
) (C)
A
(13.13)
Pulley Systems
IMA = total number of strands of a single string
supporting the resistance
(13.4)
IMA =
DE (string pulled)
DR (resistance lifted)
© 2016 Project Lead The Way, Inc.
PLTW Engineering Formula Sheet 2016
(13.5)
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
POE 6
v16.1
14.0 Structural Design
Steel Beam Design: Shear
Va ≤
Vn
Ωv
Vn = 0.6FyAw
Steel Beam Design: Moment
Ma ≤
Ωb
(14.3)
(14.2)
Mn = FyZx
(14.4)
𝑣
15.0 Storm Water Runoff
Storm Water Drainage
Q = CfCiA
C1 A1 + C2 A2 + ∙∙∙
A1 + A2 + ∙∙∙
qnet = qallowable - pfooting
(14.5)
pfooting = tfooting ∙150 lb3
(14.6)
ft
q=
Va = internal shear force
Vn = nominal shear strength
Ωv = 1.5 = factor of safety for shear
Fy = yield stress
Aw = area of web
𝑉𝑛
= allowable shear strength
𝛺
Cc =
Mn
(14.1)
Spread Footing Design
(15.1)
(15.2)
Q = peak storm water runoff rate
Cf = runoff coefficient adjustment
factor
C = runoff coefficient
i = rainfall intensity (in./h)
A = drainage area (acres)
(ft3/s)
Runoff Coefficient
Adjustment Factor
Return
Period
Cf
1, 2, 5, 10 1.0
25
1.1
50
1.2
100
1.25
© 2016 Project Lead The Way, Inc.
PLTW Engineering Formula Sheet 2016
Ma = internal bending moment
Mn = nominal moment strength
Ωb = 1.67 = factor of safety for
bending moment
Fy = yield stress
Zx = plastic section modulus about
neutral axis
𝑀𝑛
= allowable bending strength
𝛺
𝑏
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
concrete)
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
condominiumsIndustrial
Light
0.5—0.8
Heavy
0.6—0.9
P
A
(14.7)
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
16.0 Water Supply
Hazen-Williams Formula
hf =
1.85
10.44LQ
1.85 4.8655
C
d
(16.1)
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
dynamic head = static head
– head loss (16.2)
static head = change in elevation
between source and
discharge
17.0 Heat Loss/Gain
Heat Loss/Gain
Q′ = AU∆T
U=
1
R
(17.1)
(17.2)
Q = thermal energy
A = area of thermal conductivity
U = coefficient of heat
conductivity (U-factor)
∆T = change in temperature
R = resistance to heat flow (Rvalue)
CEA 5
v16.1
18.0 Hazen-Williams Constants
19.0 Equivalent Length of (Generic) Fittings
Pipe Size
Screwed Fittings
1/4
3/8
1/2
3/4
1
1¼
1½
2
2½
3
4
Elbows
Regular 90 degree
Long radius 90 degree
Regular 45 degree
2.3
1.5
0.3
3.1
2.0
0.5
3.6
2.2
0.7
4.4
2.3
0.9
5.2
2.7
1.3
6.6
3.2
1.7
7.4
3.4
2.1
8.5
3.6
2.7
9.3
3.6
3.2
11.0
4.0
4.0
13.0
4.6
5.5
Tees
Line Flow
Branch Flow
0.8
2.4
1.2
3.5
1.7
4.2
2.4
5.3
3.2
6.6
4.6
8.7
5.6
9.9
7.7
12.0
9.3
13.0
12.0
17.0
17.0
21.0
Regular 180 degree
Globe
Gate
Angle
Swing Check
2.3
21.0
0.3
12.8
7.2
3.1
22.0
0.5
15.0
7.3
3.6
22.0
0.6
15.0
8.0
4.4
24.0
0.7
15.0
8.8
5.2
29.0
0.8
17.0
11.0
6.6
37.0
1.1
18.0
13.0
7.4
42.0
1.2
18.0
15.0
8.5
54.0
1.5
18.0
19.0
9.3
62.0
1.7
18.0
22.0
11.0
79.0
1.9
18.0
27.0
13.0
110.0
2.5
18.0
38.0
4.6
5.0
6.6
7.7
18.0
20.0
27.0
29.0
34.0
42.0
Return
Bends
Valves
Strainer
Flanged Fittings
Elbows
Tees
Return
Bends
Valves
Pipe Size
1/2
3/4
1
1¼
1½
2
2½
3
4
5
6
8
10
12
14
16
18
Regular 90 degree
Long radius 90
degree 45 degree
Regular
Line Flow
Branch Flow
Regular 180 degree
0.9
1.1
0.5
0.7
2.0
0.9
1.2
1.3
0.6
0.8
2.6
1.2
1.6
1.6
0.8
1.0
3.3
1.6
2.1
2.0
1.1
1.3
4.4
2.1
2.4
2.3
1.3
1.5
5.2
2.4
3.1
2.7
1.7
1.8
6.6
3.1
3.6
2.7
2.0
1.9
7.5
3.6
4.4
3.4
2.5
2.2
9.4
4.4
5.9
4.2
3.5
2.8
12.0
5.9
7.3
5.0
4.5
3.3
15.0
7.3
8.9
5.7
5.6
3.8
18.0
8.9
12.0
7.0
7.7
4.7
24.0
12.0
14.0
8.0
9.0
5.2
30.0
14.0
17.0
9.0
11.0
6.0
34.0
17.0
18.0
9.4
13.0
6.4
37.0
18.0
21.0
10.0
15.0
7.2
43.0
21.0
23.0
11.0
16.0
7.6
47.0
23.0
Long radius 180
degree
Globe
Gate
Angle
Swing Check
1.1
38.0
1.3
40.0
1.6
45.0
2.0
54.0
2.3
59.0
18.0
10.0
18.0
12.0
4.2
120.0
2.9
38.0
38.0
5.0
150.0
3.1
50.0
50.0
5.7
190.0.
3.2
63.0
63.0
7.0
260.0
3.2
90.0
90.0
8.0
310.0
3.2
120.0
120.0
9.0
390.0
3.2
140.0
140.0
11.0
17.0
7.2
3.4
94.0
2.8
285.0
27.0
10.0
15.0
5.3
2.9
77.0
2.7
22.0
21.0
9.4
15.0
3.8
2.7
70.0
2.6
21.0
17.0
3.2
160.0
3.2
190.0
3.2
210.0
© 2016 Project Lead The Way, Inc.
PLTW Engineering Formula Sheet 2016
CEA 6
v16.1
20.0 555 Timer Design
quaons
T = 0.693 (RA + 2RB)C
f =
1
21.A Boolean Algebra
(20.1)
(20.2)
T
(RA+ RB)
duty-cycle =
∙100%
(RA+2RB)
(20.3)
T = period
f = frequency
RA = resistance A
RB = resistance B
C = capacitance
21.B Resistor Color Code
Boolean Theorems
Consensus Theorems
X• 0 = 0
(21.1)
X + XY = X + Y
(21.16)
X•1 = X
(21.2)
X + XY = X + Y
(21.17)
X• X =X
(21.3)
X + XY = X + Y
(21.18)
X • X=0
(21.4)
X + XY = X + Y
(21.19)
X+0=X
(21.5)
X+1=1
(21.6)
X+X=X
(21.7)
XY = X + Y
(21.20)
X+X=1
(21.8)
X+Y = X • Y
(21.21)
X=X
(21.9)
Commutative Law
DeMorgan’s Theorems
X•Y = Y•X
(21.10)
X+Y = Y+X
(21.11)
Associative Law
X(YZ) = (XY)Z
(21.12)
X + (Y + Z) = (X + Y) + Z
(21.13)
Distributive Law
21.C Capacitor
Code
X(Y+Z) = XY + XZ
(21.14)
(X+Y)(W+Z) = XW+XZ+YW+YZ
(21.15)
23.0 G&M Codes
22.0 Speeds and Feeds
N=
CS(12in.
)
ft
πd
fm = ft·nt·N
(22.1)
(22.2)
Plunge Rate = ½·fm
N = spindle speed (rpm)
CS = cutting speed (ft/min)
d = diameter (in.)
fm = feed rate (in./min)
ft = feed (in./tooth/rev)
nt = number of teeth
© 2016 Project Lead The Way, Inc.
PLTW Engineering Formula Sheet 2016
G00 = Rapid Traverse
G01 = Straight Line Interpolation
G02 = Circular Interpolation (clockwise)
G03 = Circular Interpolation (c-clockwise)
G04 = Dwell (wait)
G05 = Pause for user intervention
G20 = Inch programming units
G21 = Millimeter programming units
G80 = Canned cycle cancel
G81 = Drilling cycle
G82 = Drilling cycle with dwell
G90 = Absolute Coordinates
G91 = Relative Coordinates
M00 = Pause
M01 = Optional stop
M02 = End of program
M03 = Spindle on
M05 = Spindle off
M06 = Tool change
M08 = Accessory # 1 on
M09 = Accessory # 1 off
M10 = Accessory # 2 on
M11 = Accessory # 2 off
M30 = Program end and reset
M47 = Rewind
DE 5
(23.1)
(23.2)
(23.3)
(23.4)
(23.5)
(23.6)
(23.7)
(23.8)
(23.9)
(23.10)
(23.11)
(23.12)
(23.13)
(23.14)
(23.15)
(23.16)
(23.17)
(23.18)
(23.19)
(23.20)
(23.21)
(23.22)
(23.23)
(23.24)
(23.25)
CIM 5
v16.1
24.0 Aerospace
Equations
Forces of Flight
CD =
2D
(24.1)
Aρv2
R e=
ρvl
CL =
2L
(24.2)
μ
(24.3)
Aρv2
M = Fd
(24.4)
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
μ = fluid viscosity
F = force
m = mass
g = acceleration due to gravity
M = moment
d = moment arm (distance from
datum perpendicular to F)
(Ps +
(Ps +
K = 12 mv2
F N = W vj - vo
(24.5)
I = Fave ∆t
(24.6)
Fnet = Favg - Fg
(24.7)
R
(24.8)
E=U+K=−
a=
𝑣𝑓
∆t
U=
ρv2
)
2 2
PS = static pressure
v = velocity
ρ = density
T = 2π
(24.17)
T + 273.1 5.256
288.08
]
p
(24.18)
(24.19)
0.2869 T + 273.1
# of transformants
μg of DNA
∙
μ
3
= 2π
a2
GM
GMm
(24.14)
(24.15)
r2
𝑒 = eccentricity
b = semi-minor axis
a =semi-major axis
T = orbital period
a = semi-major axis
μ = 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
final volume at recovery
volume plated (mL)
# of moles of CO2
# of moles consumed in experiment
=
# of moles of glucose produced in formula # of moles of glucose produced in experiment
distance the substance travels
distance the solvent travels
F=
a2
(25.1)
Transformation Efficiency (# Transformants/μg) =
Rf =
(24.12)
(24.13)
a2
3
p = 101.29 [
(24.11)
2R
b2
𝑒= 1-
25.0 Environmental Sustainability
GDP2 - GDP1
GDP1
GMm
Orbital Mechanics
T = temperature
h = height
p = pressure
ρ = density
Economic Growth =
(24.10)
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
NOTE: Fave and Favg are
easily confused.
ρ=
colonies/mL = # colonies/dilution
− GMm
m3
T = 15.04 - 0.00649h
( 23.16)
(24.9)
G = 6.67 × 10−11 kg × 𝑠2
FN = net thrust
W = air mass flow
vo = flight velocity
vj = jet velocity
I = total impulse
Fave = average thrust force
Δt = change in time (thrust
duration)
Fnet = net force
Favg = average force
Fg = force of gravity
vf = final velocity
a = acceleration
Δt = change in time (thrust
duration)
Atmosphere Parameters
Bernoulli’s Law
ρv2
) =
2 1
Energy
Propulsion
(25.2)
(25.3)
(25.4)
(25.5)
GDP = gross domestic product
Rf = retention factor
© 2016 Project Lead The Way, Inc.
PLTW Engineering Formula Sheet 2016
ES 4
AE 6
v16.1
26.0 USCS Soil Classification Chart
Visual
Examination. Is
soil highly organic,
coarse grained, or
50% or more retained on the No. 200 sieve
fine grained?
%G > %S
Coarse
Grained
Run sieve
analysis
Well
graded
GW
Poorly
graded
GP
Fine
grained
Run LL and
PL on
minus No.
40 sieve
material
Sand (S)
Between
5% and
12% pass
No. 200
sieve
More than
12% pass
No. 200
seive
Dual
symbol
based on
gradation
and
plasticity
Run LL and
PL on minus
No. 40 sieve
material
GW-GM
GP-GM
GW-GC
GP-GC
More than 50% passes the No. 200 sieve
%S > %G
Gravel
(G)
Less
than 5%
pass No.
200
sieve
Highly Organic Soils (Pt)
Color, odor, very high moisture
content, particles of plant life,
fibrous.
Below
line or
shaded
area on
Plasticity
Chart
GM
Limits
plot in
shaded
area on
Plasticity
Chart
GM-GC
Less than
5% pass
No. 200
sieve
Well
grade
d
Poorly
grade
d
Above
line and
shaded
area on
Plasticity
Chart
GC
SW
SP
Between
5% and
12% pass
No. 200
sieve
More than
12% pass
No. 200
sieve
Dual
symbol
based
on
gradatio
n and
plasticity
Run LL
and PL on
minus No.
40 sieve
material
SW-SM
SP-SM
SW-SC
SP-SC
Below
line or
shaded
area on
Plasticity
Chart
Limits
plot in
shaded
area on
Plasticity
Chart
LL ≥ 50
LL < 50
Below
line on
Plasticity
Chart or
PI < 4
Above
line and
shaded
area on
Plasticity
Chart
Limits plot
in shaded
area of
Plasticity
Chart
Above
line on
Plasticity
Chart
and PI>
7
Color or
odor
Color or
odor
Above line
on
Plasticity
Chart
Below
line on
Plasticity
Chart
Inorganic
Inorganic
Organi
c
Organic
MH
ML
SM
SM-SC
SC
OL
ML-CL
CL
OH
CH
CEA 7
© 2016 Project Lead The Way, Inc.
PLTW Engineering Formula Sheet 2016