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 =

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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