Engineering Formula Sheet

Statistics Mode

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

∑ x If two values occur at the maximum frequency the data set is bimodal. If three or more values occur at the maximum µ = mean value frequency the data set is multi-modal. Σxi = sum of all data values (x1, x2, x3, … n = number of data values Median

Place data in ascending order. Standard Deviation If n is odd, median = central value

If n is even, median = mean of two central values

∑(x ) √ n = number of data values

σ = standard deviation Range

x = individual data value ( x , x , x , … i 1 2 3 Range = xmax - xmin

n = number of data values xmax = maximum data value x = minimum data value min Probability Independent Events

Frequency P (A and B and C) = PAPBPC

P (A and B and C) = probability of independent x x events A and B and C occurring in sequence

PA = probability of event A x x

Mutually Exclusive Events

fx = relative frequency of outcome x nx = number of events with outcome x P (A or B) = PA + PB n = total number of events P = probability of outcome x P (A or B) = probability of either mutually exclusive x event A or B occurring in a trial fa = frequency of all events P = probability of event A A Σxi = sum of all data values (x1, x2, x3, … Binomial Probability (order doesn’t matter) 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 P (A|D) = probability of event A given event D P(A) = probability of event A occurring n = number of trials P(~A) = probability of event A not occurring P(D| ~A) = probability of event D given event A did not occur

PLTW, Inc. Engineering Formulas IED POE DE CEA AE BE CIM EDD 1

Plane Geometry Ellipse Rectangle 2b

Perimeter = 2a + 2b Circle 2a Area = ab

Triangle B

Parallelogram c Area = ½ bh a h h 2 2 2 a = b + c – 2bc·cos∠A Area = bh 2 2 2 A b = a + c – 2ac·cos∠B C b b c2 = a2 + b2 – 2ab·cos∠C

Right Triangle Regular Polygons s

f 2 2 2 c = a + b

c a n = number of sides

θ

b Trapezoid a

h Area = ½(a + b)h h h b Solid Geometry h

Cube Sphere s Volume = s3 r Surface Area = 6s2 Volume r3 s s Surface Area = 4 r2

Rectangular Prism Cylinder h r Volume = wdh 2 h Surface Area = 2(wd + wh + dh) Volume = r h w d 2 Surface Area = 2 r h+2 r

Right Circular Cone

h Irregular Prism

r h √ Volume = Ah

A = area of base Pyramid

h Constants

A = area of base g = 9.8 m/s2 = 32.27 ft/s2 G = 6.67 x 10-11 m3/kg·s2 π = 3.14159

PLTW, Inc. Engineering Formulas IED POE DE CEA AE BE CIM EDD 2

Conversions

Mass Area Force Energy

2 1 kg = 2.205 lbm 1 acre = 4047 m 1 N = 0.225 lbf 1 J = 0.239 cal 2 -4 1 slug = 32.2 lbm = 43,560 ft 1 kip = 1,000 lbf = 9.48 x 10 Btu 2 1 ton = 2000 lbm = 0.00156 mi = 0.7376 ft·lbf Pressure 1kW h = 3,6000,000 J

Length Volume 1 atm = 1.01325 bar

= 33.9 ft H2O 1 m = 3.28 ft 1L = 0.264 gal = 29.92 in. Hg Defined Units 1 km = 0.621 mi = 0.0353 ft3 = 760 mm Hg 1 in. = 2.54 cm = 33.8 fl oz = 101,325 Pa 1 mi = 5280 ft 1mL = 1 cm3 = 1 cc 1 J = 1 N·m = 14.7 psi 2 1 yd = 3 ft 1 N = 1 kg·m / s 1psi = 2.31 ft of H2O 1 Pa = 1 N / m2 Time 1 V = 1 W / A Temperature Change 1 d = 24 h Power 1 W = 1 J / s

1 K = 1 ºC 1 h = 60 min 1 W = 1 V / A 1 W = 3.412 Btu/h -1 1 min = 60 s 1 Hz = 1 s = 1.8 ºF = 0.00134 hp 1 yr = 365 d 1 F = 1 A·s / V = 1.8 ºR = 14.34 cal/min 1 H = 1 V·s / V = 0.7376 ft lb /s · f

SI Prefixes Numbers Less Than One Numbers Greater Than One Power of 10 Prefix Abbreviation Power of 10 Prefix Abbreviation 10-1 deci- d 101 deca- da 10-2 centi- c 102 hecto- h 10-3 milli- m 103 kilo- k 10-6 micro- µ 106 Mega- M 10-9 nano- n 109 Giga- G 10-12 pico- p 1012 Tera- T 10-15 femto- f 1015 Peta- P 10-18 atto- a 1018 Exa- E 10-21 zepto- z 1021 Zetta- Z 10-24 yocto- y 1024 Yotta- Y

Equations Temperature Force

F = ma TK = TC + 273 Mass and Weight F = force T = T + 460 m = mass M = VD R F m a = acceleration

W = mg

W = VDw Equations of Static Equilibrium

V = volume ΣF = 0 ΣF = 0 ΣM = 0 TK = temperature in Kelvin x y P Dm = mass density TC = temperature in Celsius m = mass Fx = force in the x-direction TR = temperature in Rankin Fy = force in the y-direction Dw = weight density TF = temperature in Fahrenheit g = acceleration due to gravity MP = moment about point P

PLTW, Inc. Engineering Formulas IED POE DE CEA AE BE CIM EDD 3

Equations (Continued) Electricity

Ohm’s Law Energy: Work Fluid Mechanics V = IR

P = IV

W = work RT (series) = R1 + R2+ ··· + Rn

F = force ’ L d = distance

(Guy-L ’ L

Power Kirchhoff’s Current Law P V = P V B y ’ L 1 1 2 2 IT = I1 + I2 + ··· + In

Q = Av or ∑

A1v1 = A2v2 Kirchhoff’s Voltage Law

VT = V1 + V2 + ··· + Vn

or ∑ P = power E = energy W = work absolute pressure = gauge pressure V = voltage t = time + atmospheric pressure VT = total voltage

τ = torque I = current P = absolute pressure rpm = revolutions per minute I = total current F = Force T R = resistance A = Area R = total resistance V = volume T Efficiency P = power T = absolute temperature Q = flow rate y v = flow velocity Thermodynamics

′ ∆T Pout = useful power output Mechanics Pin = total power input

(where acceleration = 0) ∆

Energy: Potential (where acceleration = 0) L

U = potential energy m =mass L

g = acceleration due to gravity A v = A v h = height 1 1 2 2

v = v0 + at Energy: Kinetic d = d + v t + ½at2 P = rate of heat transfer 0 0 Q = thermal energy v2 = v 2 + 2a(d – d ) 0 0 A = Area of thermal conductivity τ = dFsinθ U = coefficient of heat conductivity K = kinetic energy (U-factor) m = mass s = speed ∆T = change in temperature v = velocity v = velocity a = acceleration R = resistance to heat flow ( R-value) Energy: Thermal X = range k = thermal conductivity t = time v = velocity d = distance Pnet = net power radiated g = acceleration due to gravity -8 = 5.6696 x 10 Q = thermal energy d = distance m = mass θ = angle e = emissivity constant c = specific heat τ = torque T1, T2 = temperature at time 1, time 2 ∆T = change in temperature F = force

v = flow velocity PLTW, Inc. Engineering Formulas POE 4 DE 4

Section Properties

Moment of Inertia Rectangle Centroid h x x x̅ and y̅

xx b Right Triangle Centroid

x̅ and y̅

Ixx = moment of inertia of a rectangular section about x-x axis Semi-circle Centroid Complex Shapes Centroid

∑ x ∑ y x̅ y̅ x̅ and y̅ ∑ ∑

x̅ x x̅ x y̅ y y̅ y x = x distance to centroid of shape i i y = y distance to centroid of shape i i A = Area of shape i i Structural Analysis Material Properties Beam Formulas

Reaction Stress (axial) B L Moment x (at point of load)

L Deflection x (at point of load)

= stress L Reaction F = axial force B A = cross-sectional area L Moment (at center) x L Strain (axial) Deflection x (at center)

Reaction B

L Moment x (between loads)

= strain Deflection ( L - ) (at center) x L0 = original length

δ = change in length Reaction and L B L

Moment (at Point of Load) x L Modulus of Elasticity ( )√ ( ) Deflection ( ) (at √ )

Deformation: Axial Truss Analysis ( L 2J = M + R

E = modulus of elasticity δ = stress J = number of joints = strain δ = deformation M =number of members A = cross-sectional area F = axial force R = number of reaction forces F = axial force L0 = original length δ = deformation A = cross-sectional area E = modulus of elasticity

PLTW, Inc. Engineering Formulas POE 5 AE 4 CEA 4

Simple Machines Inclined Plane

Mechanical Advantage (MA) L

y ( )

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

Lever

Screw 1st

IMA = Class

Pitch =

2nd C = Circumference Class 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 Effort at Wheel GRTOTAL = ( ) ( )

GR = Gear Ratio in = Angular Velocity - driver Pulley Systems out = Angular Velocity - driven Nin = Number of Teeth - driver IMA = Total number of strands of a single string Nout = Number of Teeth - driven supporting the resistance din = Diameter - driver dout = Diameter - driven in = Torque - driver IMA = = Torque - driven out

PLTW, Inc. Engineering Formulas POE 6

Structural Design

Steel Beam Design: Shear Steel Beam Design: Moment Spread Footing Design

qnet = qallowable - pfooting

Vn = 0.6FyAw Mn = FyZx

M = allowable bending moment Va = allowable shear strength a M = nominal moment strength Vn = nominal shear strength n q = net allowable soil Ω = 1.67 = factor of safety for net Ωv = 1.5 = factor of safety for shear b bearing pressure bending moment Fy = yield stress qallowable = total allowable soil Fy = yield stress Aw = area of web bearing pressure Zx = plastic section modulus about pfooting = soil bearing pressure neutral axis due to footing weight Storm Water Runoff tfooting = thickness of footing q = soil bearing pressure Rational Method Runoff Coefficients P = column load applied Storm Water Drainage Categorized by Surface A = area of footing Q = CfCiA Forested 0.059—0.2 Asphalt 0.7—0.95

Brick 0.7—0.85

Concrete 0.8—0.95 Q = peak storm water runoff rate (ft3/s) Shingle roof 0.75—0.95 Cf = runoff coefficient adjustment Lawns, well drained (sandy soil) factor Up to 2% slope 0.05—0.1 C = runoff coefficient 2% to 7% slope 0.10—0.15 i = rainfall intensity (in./h) Over 7% slope 0.15—0.2 A = drainage area (acres) Lawns, poor drainage (clay soil) Up to 2% slope 0.13—0.17 Runoff Coefficient 2% to 7% slope 0.18—0.22 Adjustment Factor Over 7% slope 0.25—0.35 Return Driveways, 0.75—0.85 Period Cf walkwaysCategorized by Use 1, 2, 5, 10 1.0 Farmland 0.05—0.3 25 1.1 Pasture 0.05—0.3 50 1.2 Unimproved 0.1—0.3 100 1.25 Parks 0.1—0.25

Cemeteries 0.1—0.25 Railroad yard 0.2—0.40 Water Supply Playgrounds 0.2—0.35 (except asphaltBusiness or Districts Hazen-Williams Formula Neighborhoodconcrete) 0.5—0.7 L City (downtown) 0.7—0.95

Residential

Single-family 0.3—0.5 hf = head loss due to friction (ft of H2O) Multi-plexes, 0.4—0.6 L = length of pipe (ft) Multidetached-plexes, 0.6—0.75 Q = water flow rate (gpm) attached C = Hazen-Williams constant Suburban 0.25—0.4 d = diameter of pipe (in.) Apartments, 0.5—0.7 condominiumsIndustrial Light 0.5—0.8 Dynamic Head Heavy 0.6—0.9

dynamic head = static head – head loss

PLTW, Inc. Engineering Formulas CEA 5

Williams Constants Williams

-

Equivalent Length of (Generic) Fittings of (Generic) Length Equivalent Hazen

PLTW, Inc. Engineering Formulas CEA 6

555 Timer Design Equations

T = 0.693 (RA + 2RB)C

y y B B 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 ̅ X•1 = X X+Y = Y+X ̅ ̅ ̅ X• X =X ̅ ̅ ̅̅̅ Associative Law ̅ ̅ ̅ ̅ ̅ X(YZ) = (XY)Z X + 0 = X X + (Y + Z) = (X + Y) + Z X + 1 = 1 DeMorgan’s Theorems

X + X = X Distributive Law ̅ ̅̅ ̅̅ ̅ ̅ ̅ X(Y+Z) = XY + XZ ̅ ̅̅ ̅̅ ̅̅ ̅ • ̅ ̿ (X+Y)(W+Z) = XW+XZ+YW+YZ

Speeds and Feeds

( )

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 Propulsion Orbital Mechanics

( ) Forces of Flight √

√ √ F = net thrust N L W = air mass flow

L vo = flight velocity v = jet velocity j = eccentricity I = total impulse b = semi-minor axis F = average thrust force ave a =semi-major axis C = coefficient of lift t = change in time (thrust L T = orbital period C = coefficient of drag duration) D a = semi-major axis L = lift F = net force net gravitational parameter D = drag F = average force avg F = force of gravity between two A = wing area Fg = force of gravity bodies density vf = final velocity G = universal gravitation constant R = Reynolds number e a = acceleration M =mass of central body v = velocity t = change in time (thrust m = mass of orbiting object l = length of fluid travel duration) r = distance between center of two = fluid viscosity objects F = force NOTE: F and F are m = mass ave avg easily confused. g = acceleration due to gravity Ber oulli’s L w M = moment

d = moment arm (distance from Energy ( ) ( ) datum perpendicular to F)

P = static pressure S v = velocity y

Atmosphere Parameters 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 T = temperature E = Total Energy of an orbit h = height p = pressure density

PLTW, Inc. Engineering Formulas AE 5