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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 β1 σ = population standard deviation s = sample standard deviation xi = individual data value ( x1, x2, x3, …) π = 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 π π΄π· =π π π΄ βπ π· π΄ π΄ βπ π·π΄ (2.5) +π ~π΄ βπ π· ~π΄ 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 DE CEA AE ES/BE CIM EDD 2 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 ftβlb/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 = Fdο (9.7b) (9.7a) (9.4) (9.5) (9.6a) F = force m = mass a = acceleration M = moment dο= 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 DE CEA AE ES/BE CIM EDD 3 v16.1 9.0 Equations (Continued) W = Fβ₯ β d (9.9) W = work Fβ₯ = 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 β100% (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 βQ = mcβT (9.15) βQ = change in thermal energy m = mass c = specific heat βT = 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 + + βββ + 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= βd 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′ = AUβT P = Q' = βt 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 βd = 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) βQ βt (9.39) k =L (9.40) kAβT (9.41) L A1v1 = A2v2 (9.42) Pnet = σAe(T2 4 -T1 4 ) (9.43) k= PL (9.44) AβT P = rate of heat transfer Q = thermal energy A = area of thermal conductivity U = coefficient of heat conductivity (U-factor) βT = change in temperature βt = 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 m2βK4 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 πΏ2 −πΏ1 )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 πin = torque - driver πout = 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) π£ 15.0 Storm Water Runoff Storm Water Drainage Q = CfCiA C1 A1 + C2 A2 + βββ A1 + A2 + βββ qnet = qallowable - pfooting (14.5) pfooting = tfooting β150 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 ππ = allowable shear strength πΊ 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 ππ = allowable bending strength πΊ π 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′ = AUβT U= 1 R (17.1) (17.2) Q = thermal energy A = area of thermal conductivity U = coefficient of heat conductivity (U-factor) βT = 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 = β100% (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 βt (24.6) Fnet = Favg - Fg (24.7) R (24.8) E=U+K=− a= π£π βt 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 β μ 3 = 2π a2 GM GMm (24.14) (24.15) r2 π = 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 π= 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 × π 2 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