Download PLTW Engineering Formulas: Probability, Statistics, and Geometry and more Lecture notes Engineering in PDF only on Docsity!
PLTW, Inc. Engineering Formulas
Engineering Formula Sheet
Probability
Pk =
n!(p k^ )(qn-k^ ) k!(n-k)!
Binomial Probability (order doesn’t matter)
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
Independent Events
P (A and B and C) = P (^) A PB PC
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) = P (^) A + 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 (x 1 , x 2 , x 3 , …) n = number of data values
fx =
n (^) x n
Px =
fx fa
Frequency
fx = relative frequency of outcome x n (^) x = number of events with outcome x n = total number of events Px = probability of outcome x fa = frequency of all events
Conditional Probability
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
Mode
Place data in ascending order. Mode = most frequently occurring value
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
n
Mean
μ = mean value Σxi = sum of all data values (x 1 , x 2 , x 3 , …) n = number of data values
∑ሺxi - μሻ^2
n
ߤ = mean value
Standard Deviation
σ = standard deviation xi = individual data value ( x 1 , x 2 , x 3 , …)
n = number of data values
Range
Range = xmax - xmin
xmax = maximum data value xmin = minimum data value
Median
Place data in ascending order. If n is odd, median = central value If n is even, median = mean of two central values
n = number of data values
Statistics
IED POE DE CEA AE BE CIM EDD 1
PLTW, Inc. Engineering Formulas
Plane Geometry
Triangle
Area = ½ bh
a 2 = b 2 + c 2 – 2bc∙ cos ∠A
b 2 = a 2 + c 2 – 2ac∙ cos ∠B
c 2 = a 2 + b 2 – 2ab∙ cos ∠C
h
b
a
c
A
B
C
Area = π a b
Ellipse 2b
2a
Area = n
s(ଵଶ f) 2
Regular Polygons
n = number of sides
f
s
Rectangle
Perimeter = 2a + 2b Area = ab Circumference = 2 π r Area = π r^2
Circle
Parallelogram
Area = bh
h
b
Volume =
Ah 3
Pyramid
A = area of base
h
Solid Geometry
Sphere
Volume
ସ ଷ π^ r
3
Surface Area = 4 π r 2
r
h
w d
Rectangular Prism
Volume = wdh
Surface Area = 2(wd + wh + dh) h
r
Cylinder
Volume = π r^2 h Surface Area = 2 π r h+2 π r^2
Cube
Volume = s 3 Surface Area = 6s^2
s
s s
Volume =
πr 2 h 3 Surface Area = π r ඥr 2 +h 2
Right Circular Cone
r
h
Trapezoid
Area = ½(a + b)h
b
h
a
Constants
g = 9.8 m/s^2 = 32.27 ft/s 2 G = 6.67 x 10-11^ m^3 /kg∙s 2
h
Irregular Prism
Volume = Ah
A = area of base
a
tan θ = ab
Right Triangle
c 2 = a 2 + b 2
sin θ =
a c
cos θ =
b c b
c
θ
IED POE DE CEA AE BE CIM EDD 2
PLTW, Inc. Engineering Formulas
Section Properties
xഥ= x-distance to the centroid yത^ = y-distance to the centroid
Complex Shapes Centroid
xഥ =
∑ xi A (^) i ∑ A (^) i^ and^ yഥ^ =^
∑ (^) yi A (^) i ∑ A (^) i
xi = x distance to centroid of shape i yi = y distance to centroid of shape i Ai = Area of shape i
Material Properties
σ =
F
A
Stress (axial)
σ = stress F = axial force A = cross-sectional area
ϵ = δ L (^0)
Strain (axial)
ϵ = strain L 0 = original length δ = change in length
E =
σ ε
E =
(F 2 -F 1 )L 0
ߜሺଶ ߜ െଵ)A
Modulus of Elasticity
E = modulus of elasticity σ = stress ε = strain A = cross-sectional area F = axial force δ = deformation
Structural Analysis
Truss Analysis
2J = M + R
J = number of joints M =number of members R = number of reaction forces
Beam Formulas
Reaction R (^) A = R (^) B =
P 2 Moment Mmax =
PL 4 (at point of load) Deflection ∆max = PL
3 48EI (at point of load)
Reaction R (^) A = R (^) B =
ωL 2 Moment M (^) max =
ωL 2 8 (at center) Deflection ∆max = 5 ωL
4 384EI (at center) Reaction R (^) A = R (^) B = P
Moment M (^) max = Pa (between loads)
Deflection ∆max = 24EIP a ቀ3L 2 -4 a^2 ቁ (at center)
Reaction R (^) A =
Pb L and^ R^ B^ =^
Pa L Moment M (^) max =
Pab L (at Point of Load) Deflection Δ୫ୟ୶ ൌౌ aౘሺaశమౘሻඥయమళుaሺaశమౘሻ
(at x ൌ ටaሺaାଶୠሻ ଷ,
when a b )
xഥ^ = r and yഥ=
4r
3 π
xഥ= x-distance to the centroid yത^ = y-distance to the centroid
Rectangle Centroid
xഥ=
b 2 and^ yഥ=^
h 2
Right Triangle Centroid
xഥ=
b 3 and^ y
ഥ= h 3
Semi-circle Centroid
δ =
FL 0
AE
Deformation: Axial
δ = deformation F = axial force L 0 = original length A = cross-sectional area E = modulus of elasticity
POE 5 AE 4 CEA 4
Ixx =
bh 3 12
Moment of Inertia
Ixx = moment of inertia of a rectangular section about x-x axis
x
h
b
x
PLTW, Inc. Engineering Formulas
Aerospace Equations
ቆPs +
ρv^2 2
1
= ቆPs +
ρv^2 2
2
Bernoulli’s Law
PS = static pressure v = velocity ρ = density
K =
1 2 mv
2
U =
െ GMm R
E = U + K = െ
GMm 2R
Energy
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
C D =
2D
Aρv^2
R (^) e =
ρvl μ
C L =
2L
Aρv^2
M = Fd
Forces of Flight
C (^) L = coefficient of lift C (^) D = coefficient of drag L = lift D = drag A = wing area ρ = density R (^) e = 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)
b 2 a 2
T = 2π
a
య మ
√μ^
= 2π
a
య మ
√GM
F =
GMm r 2
Orbital Mechanics
݁ = 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
F (^) N = W൫vj - vo൯
I = Fave∆t
Fnet = Favg - Fg
a = vf∆t
Propulsion
FN = net thrust W = air mass flow vo = flight velocity vj = jet velocity I = total impulse Fave = average thrust force ∆t = change in time (thrust duration) Fnet = net force Favg = average force Fg = force of gravity vf = final velocity a = acceleration ∆t = change in time (thrust duration)
NOTE: Fave and Favg are
easily confused.
T = 15.04 - 0.00649h
p = 101.29 ቈ
ሺT + 273.1ሻ
ρ =
p 0.2869ሺT + 273.1ሻ
Atmosphere Parameters
T = temperature h = height p = pressure ρ = density
AE 5
