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### Engineering formula sheet

• 1. Engineering Formula Sheet Statistics Mode Mean Place data in ascending order. Mode = most frequently occurring value ∑ xi n µ= µ = mean value Σxi = sum of all data values (x1, x2, x3, …) n = number of data values 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 σ=ඨ 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. ∑(xi - µ)2 n n = number of data values σ = standard deviation xi = individual data value ( x1, x2, x3, …) ߤ = mean value n = number of data values Range Range = xmax - xmin xmax = maximum data value xmin = minimum data value Probability Independent Events P (A and B and C) = PAPBPC Frequency fx = nx n Px = fx fa 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 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) Pk = n!(pk )(qn-k ) k!(n-k)! 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 or B) = PA + PB ܲ(‫= )ܦ|ܣ‬ ܲ(‫)ܣ|ܦ(ܲ ∙ )ܣ‬ ܲ(‫)ܣ~|ܦ(ܲ ∙ )ܣ~(ܲ + )ܣ|ܦ(ܲ ∙ )ܣ‬ 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
• 2. Plane Geometry Ellipse Rectangle 2b Area = π a b Circle Perimeter = 2a + 2b Area = ab 2a Circumference =2 π r Area = π r2 B Triangle Parallelogram Area = ½ 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 h Area = bh 2 b 2 cos θ = a c a c c h 2 A C b s s(ଵ f) Area = n ଶ 2 2 c =a +b sin θ = 2 Regular Polygons Right Triangle 2 a b f n = number of sides θ c b a tan θ = b a h Trapezoid Area = ½(a + b)h h h b h Solid Geometry Cube Sphere 3 Volume = s 2 Surface Area = 6s s s ସ r 3 Volume π r ଷ 2 Surface Area = 4 π r s Rectangular Prism Cylinder r h Volume = wdh Surface Area = 2(wd + wh + dh) d w h 2 Volume = π r h 2 Surface Area = 2 π r h+2 π r Right Circular Cone πr2 h Volume = 3 Surface Area = π r ඥr2 +h2 h Irregular Prism r h Volume = Ah A = area of base Pyramid Volume = Ah 3 A = area of base PLTW, Inc. h 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
• 3. Conversions Mass Force Area 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 = 3,6000,000 J Pressure Length 1m 1 km 1 in. 1 mi 1 yd 1 atm Volume = 3.28 ft = 0.621 mi = 2.54 cm = 5280 ft = 3 ft 1L 1mL = 0.264 gal 3 = 0.0353 ft = 33.8 fl oz 3 = 1 cm = 1 cc 1psi = 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 Time Temperature Change 1K = 1 ºC = 1.8 ºF = 1.8 ºR 1d 1h 1 min 1 yr = 24 h = 60 min = 60 s = 365 d Power 1W = 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 d c m µ n p f a z y Numbers Greater Than One Power of 10 Prefix Abbreviation 101 102 103 106 109 1012 1015 1018 1021 1024 Temperature TK = TC + 273 Mass and Weight M = VDm W = mg W = VDw V = volume Dm = mass density m = mass Dw = weight density g = acceleration due to gravity PLTW, Inc. TR = TF + 460 TF - 32 TC = 180 100 TK = temperature in Kelvin TC = temperature in Celsius TR = temperature in Rankin TF = temperature in Fahrenheit Engineering Formulas decahectokiloMegaGigaTeraPetaExaZettaYotta- da h k M G T P E Z Y Force F = ma F = force m = mass a = acceleration Equations of Static Equilibrium 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
• 4. Equations (Continued) Electricity Fluid Mechanics Energy: Work W = F∙d P= V1 W = work F = force d = distance F A V2 T1 P1 T1 Power T2 P2 = T2 RT (series) = R1 + R2+ ··· + Rn (Charles’ Law) RT (parallel) = (Guy-Lussanc’s Law) 1 1 1 1 + + ∙∙∙ +R R1 R2 n Kirchhoff’s Current Law Q = Av P = power E = energy W = work t = time τ = torque rpm = revolutions per minute Efficiency Pout ∙100% Pin Pout = useful power output Pin = total power input IT = I1 + I2 + ··· + In n or IT = ∑k=1 Ik A1v1 = A2v2 E W = t t τ∙rpm P= 5252 Kirchhoff’s Voltage Law Horsepower = P = absolute pressure F = Force A = Area V = volume T = absolute temperature Q = flow rate v = flow velocity U = potential energy m =mass g = acceleration due to gravity h = height t ‫܌‬ t 1 2 Q ∆t (where acceleration = 0) U= 1 k = R L kA∆T L a= vf ି vi t P= X= vi sin(2θ) -g A1v1 = A2v2 Pnet = σAe(T2 4 -T1 4 ) 2 d = d0 + v0t + ½at K = mv2 v = v0 + 2a(d – d0) K = kinetic energy m = mass v = velocity τ = dFsinθ Energy: Thermal Q =mc∆T Q = thermal energy m = mass c = specific heat ∆T = change in temperature Thermodynamics P= v = v0 + at Energy: Kinetic V = voltage VT = total voltage I = current IT = total current R = resistance RT = total resistance P = power (where acceleration = 0) d v= VT = V1 + V2 + ··· + Vn n or VT = ∑k=1 Vk P = Q′ = AU∆T Mechanics Energy: Potential U = mgh QP 1714 absolute pressure = gauge pressure + atmospheric pressure s= PLTW, Inc. V = IR P = IV P1V1 = P2V2 (Boyle’s Law) P= Efficiency (%) = = Ohm’s Law 2 2 s = speed v = velocity a = acceleration X = range t = time d = distance g = acceleration due to gravity d = distance 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 ∆t = change in time R = resistance to heat flow ( R-value) k = thermal conductivity v = velocity Pnet = net power radiated σ = 5.6696 x 10 -8 W m2 ∙K 4 e = emissivity constant T , T = temperature at time 1, time 2 POE 4 DE 4
• 5. Section Properties Rectangle Centroid Moment of Inertia h Ixx x x 3 bh = 12 b Ixx = moment of inertia of a rectangular section about x-x axis ഥ x= ∑ Ai ഥ and y = b 2 ഥ= and y h 2 Right Triangle Centroid ഥ= x b 3 ഥ= and y h 3 Semi-circle Centroid Complex Shapes Centroid ∑ xi Ai ഥ= x ഥ x = r and y = ഥ ∑ yi A i ∑ Ai ഥ= x x-distance to the centroid ത 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 ഥ= x x-distance to the centroid ത y = y-distance to the centroid Structural Analysis Material Properties Beam Formulas Stress (axial) Reaction F σ= A Moment Deflection σ = stress F = axial force A = cross-sectional area Reaction Moment Strain (axial) Deflection ϵ= δ L0 Reaction ϵ = strain L0 = original length δ = change in length Deflection Moment PL Mmax = 4 max = ߪ(F2 -F1 )L0 (ߜଶ − ߜଵ )A E = modulus of elasticity σ = stress ε = strain A = cross-sectional area F = axial force δ = deformation PLTW, Inc. ωL 2 ωL2 Mmax = (at center) 8 5ωL4 384EI (at center) RA = RB = P Mmax = Pa (between loads) Pa = 24EIቀ3L2 -4a2ቁ max Pb Moment E= 3 RA = RB = RA = Mmax = Deflection 2 (at point of load) PL max = 48EI (at point of load) Reaction Modulus of Elasticity σ E= ε P RA = RB = ୫ୟ୶ L and RB = Pab L (at center) Pa L (at Point of Load) = ౌaౘ(aశమౘ)ඥయa(aశమౘ) మళు౅ (at x = ට a(aାଶୠ) Deformation: Axial when a > b ) Truss Analysis FL0 δ = AE ଷ, 2J = M + R δ = deformation F = axial force L0 = original length A = cross-sectional area E = modulus of elasticity Engineering Formulas J = number of joints M =number of members R = number of reaction forces POE 5 AE 4 CEA 4
• 6. Simple Machines Inclined Plane Mechanical Advantage (MA) DE IMA= DR % Efficiency= ൬ FR AMA= FE AMA ൰ 100 IMA IMA= L (slope) H Wedge IMA = Ideal Mechanical Advantage AMA = Actual Mechanical Advantage DE = Effort Distance DR = Resistance Distance FE = Effort Force FR = Resistance Force IMA= L (⊥ to height) H Lever Screw 1st Class IMA = C Pitch Pitch = 2nd Class 1 TPI 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 Nout dout ωin τout GR= = = = Nin din ωout τin Effort at Axle dout ωin τout = = (pulleys) din ωout τin Compound Gears B D GRTOTAL = ቀ ቁ ቀ ቁ A C Effort at Wheel Pulley Systems IMA = Total number of strands of a single string supporting the resistance IMA = DE (string pulled) DR (resistance lifted) PLTW, Inc. 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 τin = Torque - driver τout = Torque - driven Engineering Formulas POE 6