Mode

Place data in ascending order.Mode = most frequently occurring value(1.4)

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.

Mean (1.1a) (1.1b)µ = population mean = sample meanΣxi = sum of all data values (x1, x2, x3, …)N = size of populationn = size of sample

Standard Deviation (Population)(1.5a)

(Sample)(1.5b)σ = population standard deviations = sample standard deviationxi = individual data value ( x1, x2, x3, …)

= sample meanN = size of populationn = size of sample

Median

Place data in ascending order.If N is odd, median = central value(1.2)If N is even, median = mean of two central values N = size of population

Range (1.5)

Range = xmax - xmin (1.3)

xmax = maximum data valuexmin = minimum data value2.0 Probability

Independent Events

P (A and B and C) = PAPBPC(2.3)

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 trialPA = probability of event A

Binomial Probability (order doesn’t matter)

(2.2)

Pk = binomial probability of k successes in n trialsp = probability of a successq = 1 – p = probability of failurek = number of successesn = number of trials

Conditional Probability

(2.5)

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

Version 2.1

2012 Engineering Formula Sheet

© 2012 Project Lead The Way, Inc. Formula Sheet – Page 1

1.0 Statistics

EDDBE CIMIED POE DE CEA AE

Version 2.1

© 2012 Project Lead The Way, Inc. Formula Sheet – Page 2

1

Rectangle

Perimeter = 2a + 2b (3.9)Area = ab (3.10)

Ellipse

(3.8)

2b

2aCircle

(3.1) (3.2)

Triangle (3.6)

Area = ½ bh (3.11)a2 = b2 + c2 – 2bc·cos∠A (3.12)b2 = a2 + c2 – 2ac·cos∠B (3.13)c2 = a2 + b2 – 2ab·cos∠C (3.14)

h

b

ac

A

B

C

Parallelogram

Area = bh(3.3)h

b

Regular Polygons

(3.15)

n = number of sides

f

sRight Triangle

c2 = a2 + b2 (3.4) (3.5) (3.6) (3.7)

b

c

θTrapezoid

Area = ½(a + b)h(3.16)

Cube

Volume = s3(4.1)Surface Area = 6s2(4.2)

s

s s

Sphere

Volume = π r3 (4.8)Surface Area = 4 π r2 (4.9)

r

h

dw

Rectangular Prism

Volume = wdh(4.3)Surface Area = 2(wd + wh + dh) (4.4)

h

rCylinder

Volume = π r2 h(4.10)

Surface Area = 2 π r h+2 π r2(4.11)

EDDBE CIMIED POE DE CEA AE

Irregular Prism

Volume = Ah(4.12)

A = area of base

h

5.0 Constantsg = 9.8 m/s2 = 32.27 ft/s2G = 6.67 x 10-11 m3/kg·s2 π = 3.14159

Pyramid

(4.7)A = area of base

h

Right Circular Cone

(4.5) (4.6)

r

h

Version 2.1

© 2012 Project Lead The Way, Inc. Formula Sheet – Page 3

3.0 Plane Geometry

4.0 Solid Geometryb

h

a

a

2

7.0 Defined Units

1 J= 1 N·m1 N= 1 kg·m / s21 Pa = 1 N / m21 V= 1 W / A1 W= 1 J / s1 W= 1 V / A1 Hz= 1 s-11 F= 1 A·s / V1 H= 1 V·s / V

Mass and Weight

M = VDm(9.1)W = mg(9.2)W = VDw(9.3)V = volumeDm = mass densitym = massDw = weight densityg = acceleration due to gravity

Equations of Static Equilibrium

ΣFx = 0 ΣFy = 0 ΣMP = 0 (9.8) Fx = force in the x-directionFy = force in the y-directionMP = moment about point P

EDDBE CIMIED POE DE CEA AEElectricity

Ohm’s LawV = IR (9.32) P = IV (9.33)

RT (series) = R1 + R2+ ··· + Rn (9.34)

(9.35)Kirchhoff’s Current LawIT = I1 + I2 + ··· + In or (9.36)Kirchhoff’s Voltage Law VT = V1 + V2 + ··· + Vn or (9.37)V = voltageVT = total voltageI = currentIT = total currentR = resistanceRT = total resistanceP = power

Version 2.1

Numbers Less Than One Numbers Greater Than OnePower of 10 Prefix Abbreviation Power of 10 Prefix Abbreviation

10-1 deci- d 101 deca- da10-2 centi- c 102 hecto- h10-3 milli- m 103 kilo- k10-6 micro- µ 106 Mega- M10-9 nano- n 109 Giga- G10-12 pico- p 1012 Tera- T10-15 femto- f 1015 Peta- P10-18 atto- a 1018 Exa- E10-21 zepto- z 1021 Zetta- Z10-24 yocto- y 1024 Yotta- Y

© 2012 Project Lead The Way, Inc. Formula Sheet – Page 4

8.0 SI Prefixes

9.0 Equations Force and Moment

F = ma (9.7a) M = Fd (9.7b)

9.0 Equations (Continued)

Temperature

TK = TC + 273 (9.4)

3

6.0 Conversions

Mass/Weight (6.1)

1 kg = 2.205 lbm

1 slug = 32.2 lbm

1 ton = 2000 lbm

Force (6.7)

1 N = 0.225 lb

1 kip = 1,000 lb

Length (6.2)

1 m = 3.28 ft

1 km = 0.621 mi

1 in. = 2.54 cm

1 mi = 5280 ft

Area (6.4)

1 acre = 4047 m2

= 43,560 ft2

= 0.00156

Volume (6.5)

1L = 0.264 gal

= 0.0353 ft3

= 33.8 fl oz

Pressure (6.8)

1 atm = 1.01325 bar

= 33.9 ft H2O

= 29.92 in. Hg

= 760 mm Hg

= 101,325 Pa

= 14.7 psi

Power (6.9)

1 W = 3.412 Btu/h

= 0.00134 hp

= 14.34 cal/min

= 0.7376 ft·lbf/s

Energy (6.10)

1 J = 0.239 cal

= 9.48 x 10-4 Btu

= 0.7376 ft·lbf

Temperature Unit Equivalents (6.6)

1 K = 1 ºC= 1.8 ºF= 1.8 ºR

See below for

Time (6.3)

1 d = 24 h

1 h = 60 min

1 min = 60 s

Fluid Mechanics

(9.16) (Charles’ Law) (9.17) (Gay-Lussanc’s Law) (9.18)p1V1 = p2V2 (Boyle’s Law) (9.19)Q = Av(9.20)

A1v1 = A2v2(9.21)

(9.22)absolute pressure = gauge pressure + atmospheric pressure (9.23) p = absolute pressureF = forceA = areaV = volumeT = absolute temperatureQ = flow ratev = flow velocityP = power

Energy: Work

(9.9)W = workF∥ = force parallel to direction of displacementd = displacement

Power

(9.10)(9.11)P = powerE = energyW = workt = timeτ = torqueω = angular velocity

Efficiency

(9.12)

Pout = useful power outputPin = total power input

Mechanics

(9.24)(9.25) (9.26)(9.27)v = vi + at(9.28)d = di + vit + ½at2(9.29)v2 = vi2 + 2a(d – di)(9.30)τ = dFsinθ(9.31)

v = velocity vi = initial velocity (t =0)a = accelerationX = ranget = time∆d = change in displacementd = distancedi = initial distance (t=0)g = acceleration due to gravityθ = angleτ = torqueF = force

Energy: Potential

(9.13)U = potential energym =massg = acceleration due to gravityh = height

Energy: Kinetic

(9.14)K = kinetic energym = massv = velocity

POE 4 DE 4

Energy: Thermal

(9.15)∆Q = change in thermal energy m = massc = specific heat∆T = change in temperature

Version 2.1

© 2012 Project Lead The Way, Inc. Formula Sheet – Page 5

Force and Moment

F = ma (9.7a) M = Fd (9.7b)

Temperature

TK = TC + 273 (9.4)

Thermodynamics

P = Q′ = AU∆T (9.38)

P = Q' = ∆Q∆t (9.39)

U = 1R

=kL (9.40)

P = kA∆TL (9.41)

A1v1 = A2v2 (9.42)

Pnet= σ Ae( T24 - T1

4 ) (9.43)

k = PLA∆T (9.44)

P = rate of heat transferQ = thermal energy A = area of thermal conductivityU = coefficient of heat conductivity

(U-factor)∆T = change in temperature∆t = change in timeR = resistance to heat flow ( R-value)k = thermal conductivity

CIM 4

x

x

y

x

y

y

xh

b

Complex Shapes Centroid and (10.2)

xi = x distance to centroid of shape iyi = y distance to centroid of shape iAi = Area of shape i

Stress (axial) (11.1)σ = stressF = axial forceA = cross-sectional area

Strain (axial) (11.2)ε = strainL0 = original lengthδ = change in length

Modulus of Elasticity (11.3) (11.4)

E = modulus of elasticityσ = stressε = strainA = cross-sectional areaF = axial forceδ = deformation

Deformation: Axial(12.13)δ = deformationF = axial forceL0 = original lengthA = cross-sectional areaE = modulus of elasticity

POE 5 CEA 4AE 4

Version 2.1

© 2012 Project Lead The Way, Inc. Formula Sheet – Page 6

Moment of Inertia

Ixx = bh3

12 (10.1)

Ixx = moment of inertia of a rectangular section

about x axis

y

10.0 Section Properties

11.0 Material 12.0 Structural Analysis

Truss Analysis2J = M + R (12.14)

J = number of jointsM =number of membersR = number of reaction forces

Beam Formulas

Reaction RA = RB = P2

(12.1)

Moment Mmax = PL4

(at point of load) (12.2)

Deflection Δmax = P L3

48EI (at point of load)

(12.3)

Reaction RA = RB = ωL2

(12.4)

Moment Mmax = ω L2

8 (at center)

(12.5)

Deflection Δmax = 5ωL4

384EI (at center)

(12.6)

Reaction RA = RB = P (12.7)

Moment Mmax = P a (12.8)

Rectangle Centroid

x = b2 and y = h

2 (10.3)

Right Triangle Centroid

x = b3 and y = h

3 (10.4)

Semi-circle Centroid

x = r and y = 4r3 π (10.5)

Gears; Sprockets with Chains; and Pulleys with Belts Ratios

Inclined Plane

Compound Machines

MATOTAL = (MA1) (MA2) (MA3) . . .(13.10)

Screw

Pulley Systems

IMA = total number of strands of a single string supporting the resistance(13.4)

IMA = (13.5)

Wheel and Axle

Wedge

Mechanical Advantage (MA)

(13.3)

IMA = ideal mechanical advantageAMA = actual mechanical advantageDE = effort distanceDR = resistance distanceFE = effort forceFR = resistance force

Version 2.1

© 2012 Project Lead The Way, Inc. Formula Sheet – Page 7

13.0 Simple Machines

POE 6

Steel Beam Design: Shear(14.1)Vn = 0.6FyAw(14.2)

Va = internal shear forceVn = nominal shear strengthΩv = 1.5 = factor of safety for shearFy = yield stressAw = area of web = allowable shear strength

Storm Water Drainage

Q = CfCiA(15.1)(15.2)Q = peak storm water runoff rate (ft3/s)Cf = runoff coefficient adjustment factorC = runoff coefficienti = rainfall intensity (in./h)A = drainage area (acres)

Hazen-Williams Formula(16.1)hf = head loss due to friction (ft of H2O)L = length of pipe (ft)Q = water flow rate (gpm)C = Hazen-Williams constantd = diameter of pipe (in.)

Dynamic Headdynamic head = static head – head loss (16.2)static head = change in elevation between source and discharge (16.3)

Version 2.1

© 2012 Project Lead The Way, Inc. Formula Sheet – Page 8

15.0 Storm Water Runoff Rational Method Runoff CoefficientsCategorized by SurfaceForested 0.059—0.2Asphalt 0.7—0.95Brick 0.7—0.85Concrete 0.8—0.95Shingle roof 0.75—0.95

Lawns, well drained (sandy soil)Up to 2% slope 0.05—0.12% to 7% slope 0.10—0.15Over 7% slope 0.15—0.2

Lawns, poor drainage (clay soil)Up to 2% slope 0.13—0.172% to 7% slope 0.18—0.22Over 7% slope 0.25—0.35Driveways, walkways

0.75—0.85Categorized by Use

Farmland 0.05—0.3Pasture 0.05—0.3Unimproved 0.1—0.3Parks 0.1—0.25Cemeteries 0.1—0.25Railroad yard 0.2—0.40Playgrounds (except asphalt or

0.2—0.35Business Districts

Neighborhood 0.5—0.7City (downtown) 0.7—0.95ResidentialSingle-family 0.3—0.5Multi-plexes, detached

0.4—0.6Multi-plexes, attached

0.6—0.75Suburban 0.25—0.4Apartments, condominiums

0.5—0.7Industrial

Light 0.5—0.8Heavy 0.6—0.9

Runoff Coefficient Adjustment FactorReturn Period Cf1, 2, 5, 10 1.025 1.150 1.2100 1.25

16.0 Water Supply

CEA 5

Heat Loss/Gain

Q′ = AU∆T (17.1)

U = 1R (17.2)

Q = thermal energy A = area of thermal conductivityU = coefficient of heat

conductivity (U-factor)∆T = change in temperature

14.0 Structural Design

Spread Footing Design

qnet = qallowable - pfooting(14.5)

pfooting = tfooting ∙150 lbft2

(14.6)

q = PA (14.7)

qnet = net allowable soil bearing pressure

qallowable = total allowable soil bearing pressure

pfooting = soil bearing pressure

Steel Beam Design: Moment

Ma ≤ Mn

Ωb(14.3)

Mn = FyZx (14.4)

Ma = internal bending momentMn = nominal moment strengthΩb = 1.67 = factor of safety for

bending momentFy = yield stressZx = plastic section modulus

about neutral axis

17.0 Heat Loss/Gain

Version 2.1

© 2012 Project Lead The Way, Inc. Formula Sheet – Page 9

H a z e n - W i l l

1 9 . 0 E q u i v a l e n t L e n g t h o f ( G e n e r

CEA 6

T = 0.693 (RA + 2RB)C(20.1) (20.2)(20.3)T = periodf = frequencyRA = resistance ARB = resistance BC = capacitance

22.0 Speeds and Feeds

(22.1)fm = ft·nt·N (22.2)Plunge Rate = ½·fmN = spindle speed (rpm)CS = cutting speed (in./min)d = diameter (in.)fm = feed rate (in./min)ft = feed (in./tooth/rev)nt = number of teeth

Consensus Theorems (21.16) (21.17) (21.18) (21.19)

DeMorgan’s Theorems (21.20) (21.21)

Associative LawX(YZ) = (XY)Z (21.12)X + (Y + Z) = (X + Y) + Z (21.13)

Commutative LawX•Y = Y•X(21.10)X+Y = Y+X(21.11)

Distributive LawX(Y+Z) = XY + XZ (21.14)(X+Y)(W+Z) = XW+XZ+YW+YZ (21.15)

Boolean TheoremsX• 0 = 0(21.1)X•1 = X(21.2)X• X =X(21.3)(21.4)X + 0 = X(21.5)X + 1 = 1(21.6)X + X = X(21.7)(21.8)(21.9)

Version 2.1

© 2012 Project Lead The Way, Inc. Formula Sheet – Page 10

20.0 555 Timer Design

CIM 5DE 5

21.0 Boolean Algebra

Orbital Mechanics

(23.13) (23.14) (23.15) = eccentricityb = semi-minor axisa =semi-major axisT = orbital perioda = semi-major axisμ = gravitational parameterF = force of gravity between two bodiesG = universal gravitation constantM =mass of central bodym = mass of orbiting objectr = distance between center of two objects

Propulsion (23.5)(23.6)(23.7)(23.8)FN = net thrustW = air mass flowvo = flight velocityvj = jet velocityI = total impulseFave = average thrust forceΔt = change in time (thrust duration)Fnet = net forceFavg = average forceFg = force of gravityvf = final velocitya = accelerationΔt = change in time (thrust duration)

NOTE: Fave and Favg are easily confused.

Forces of Flight (23.1)(23.2)(23.3)(23.4)

CL = coefficient of liftCD = coefficient of dragL = liftD = dragA = wing areaρ = densityRe = Reynolds numberv = velocityl = length of fluid travelμ = fluid viscosityF = forcem = massg = acceleration due to gravityM = momentd = moment arm (distance from datum perpendicular to F)

Energy (23.9) (23.10) (23.11) (23.12)K = kinetic energym =massv = velocityU = gravitational potential energyG = universal gravitation constantM =mass of central bodym = mass of orbiting objectR = Distance center main body to center of orbiting objectE = Total Energy of an orbit

Atmosphere Parameters (23.17) (23.18) (23.19)T = temperatureh = heightp = pressureρ = density

Bernoulli’s Law (23.16)PS = static pressurev = velocityρ = density

Version 2.1

© 2012 Project Lead The Way, Inc. Formula Sheet – Page 11

23.0 Aerospace

AE 5