Engineering Formula Sheet Statistics
Mode
Mean
Place data in ascending order. Mode = most frequently occurring value
∑x µ = mean value Σxi = sum of all data values (x1, x2, x3, … n = number of data values
∑(x
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
√
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.
)
n = number of data values
σ = standard deviation xi = individual data value ( x1, x2, x3, …
Range
n = number of data values
xmax = maximum data value xmin = minimum data value
Range = xmax - xmin
Probability Independent Events P (A and B and C) = PAPBPC
Frequency
P (A and B and C) = probability of independent events A and B and C occurring in sequence PA = probability of event A
x
x
x
x
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)
P (A or B) = PA + PB 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|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
Circle
Perimeter = 2a + 2b Area = ab
2a
B
Triangle Parallelogram h
Area = 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
C
2
c
h
2
A b
s
Regular Polygons
Right Triangle 2
a
2
b
2
Area = ½ bh
f
2
c =a +b
c
a
n = number of sides θ
b
a h
Trapezoid Area = ½(a + b)h
h h b h
Solid Geometry Cube
Sphere
s
3
Volume = s 2 Surface Area = 6s
r
3
s
Volume = r Surface Area = 4
s
r
2
Rectangular Prism Cylinder
r
h Volume = wdh Surface Area = 2(wd + wh + dh)
d
w
h
2
Volume = r h Surface Area = 2
r h+2
r
2
Right Circular Cone h
Irregular Prism r
√
h
Volume = Ah A = area of base
Pyramid
h A = area of base
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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
Area
Force 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 = 3,600,000 J
Pressure Length
Volume
1m 1 km 1 in. 1 mi 1 yd
= 3.28 ft = 0.621 mi = 2.54 cm = 5280 ft = 3 ft
1L
1mL
1 atm = 0.264 gal 3 = 0.0353 ft = 33.8 fl oz 3 = 1 cm = 1 cc 1psi
Temperature Unit Equivalents
Time 1d 1h 1 min 1 yr
1K
= 24 h = 60 min = 60 s = 365 d
= 1 ºC = 1.8 ºF = 1.8 ºR
= 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
Power 1W
See below for temperature calculation
= 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 Mass and Weight
Numbers Greater Than One Power of 10 Prefix Abbreviation 101 102 103 106 109 1012 1015 1018 1021 1024
d c m µ n p f a z y Temperature TK = TC + 273
M = VDm
TR = TF + 460
W = mg
TF =
PLTW, Inc.
da h k M G T P E Z Y
Force F = ma F = force m = mass a = acceleration
Tc + 32
W = VDw V = volume Dm = mass density m = mass Dw = weight density g = acceleration due to gravity
decahectokiloMegaGigaTeraPetaExaZettaYotta-
Equations of Static Equilibrium TK = temperature in Kelvin TC = temperature in Celsius TR = temperature in Rankin TF = temperature in Fahrenheit
Engineering Formulas
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) Energy: Work
Electricity Ohm’s Law
Fluid Mechanics
V = IR P = IV
W = work F = force parallel to direction of displacement d = displacement Power
Efficiency y Pout = useful power output Pin = total power input
’ L
(Gay-L p1V1 = p2V2
P = power E = energy W = work t = time τ = torque rpm = revolutions per minute
RT (series) = R1 + R2+ ··· + Rn
’L
B y ’ L
Kirchhoff’s Current Law
Q = Av
IT = I1 + I2 + ··· + In ∑ or
A1v1 = A2v2
Kirchhoff’s Voltage Law
VT = V1 + V2 + ··· + Vn ∑ or absolute pressure = gauge pressure + atmospheric pressure
p = absolute pressure F = Force A = Area V = volume T = absolute temperature Q = flow rate v = flow velocity
V = voltage VT = total voltage I = current IT = total current R = resistance RT = total resistance P = power Thermodynamics ′
Mechanics
∆T
∆ ̅
Energy: Potential g
L
̅
U = potential energy m =mass g = acceleration due to gravity h = height Energy: Kinetic
L A1v1 = A2v2 g v = v0 + at d = d0 + v0t + ½at 2
2
2
v = v0 + 2a(d – d0) K = kinetic energy m = mass v = velocity Energy: Thermal
Q = thermal energy m = mass c = specific heat ∆T = change in temperature
PLTW, Inc.
τ = dFsinθ ̅ g ̅ g y v = velocity a = acceleration X = range t = time ∆d = change in displacement d = distance g = acceleration due to gravity 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 g R = resistance to heat flow ( R-value) k = thermal conductivity v = velocity Pnet = net power radiated = 5.6696 x 10
-8
e = emissivity constant L = thickness T1, T2 = temperature at time 1, time 2
v = flow velocity
POE 4 DE 4
Section Properties Moment of Inertia
Rectangle Centroid h
x
x xx
b
Ixx = moment of inertia of a rectangular section about x-x axis
∑x ∑
and y̅
and y̅
Right Triangle Centroid x̅
and y̅
Semi-circle Centroid
Complex Shapes Centroid
x̅
x̅
x̅
∑y
y̅
∑
x̅ x y̅ y xi = x distance to centroid of shape i yi = y distance to centroid of shape i Ai = Area of shape i
x̅ x y̅ y
Structural Analysis Material Properties Beam Formulas Reaction
Stress (axial)
B L
Moment Deflection = stress F = axial force A = cross-sectional area
L
x
B L
Moment
L
x
Reaction
= strain L0 = original length δ = change in length
Moment
x
Deflection
x
x
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and
Deformation: Axial δ
L
δ = deformation F = axial force L0 = original length A = cross-sectional area E = modulus of elasticity
Engineering Formulas
√
) (at center)
B
L
(at Point of Load)
L
Deflection (at
E = modulus of elasticity = stress = strain A = cross-sectional area F = axial force δ = deformation
( L L
Moment
(
(at center)
B
Reaction
Modulus of Elasticity
(at center)
x
Deflection
L
(at point of load) L
Reaction
Strain (axial)
(at point of load)
x
(
)√
(
)
(
)
)
Truss Analysis 2J = M + R 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)
y (
L
) Wedge
IMA = Ideal Mechanical Advantage AMA = Actual Mechanical Advantage DE = Effort Distance DR = Resistance Distance FE = Effort Force FR = Resistance Force
L
g
Lever Screw 1st Class
IMA =
Pitch = 2nd Class
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
Effort at Axle (
)
Compound Gears B GRTOTAL = ( ) (
Effort at Wheel
Pulley Systems IMA = Total number of strands of a single string supporting the resistance IMA =
PLTW, Inc.
g
)
GR = Gear Ratio in = Angular Velocity - driver out = Angular Velocity - driven Nin = Number of Teeth - driver Nout = Number of Teeth - driven din = Diameter - driver dout = Diameter - driven in = Torque - driver out = Torque - driven
Engineering Formulas
POE 6
