Statics Appendix A

APPENDIX A

A1 SELECTED TOPICS IN MATHEMATICS

A1.1 Algebra

1. Quadratic equation

2. Logarithms

bx = y, x = logb y

Natural logarithms

b = e = 2.718 282

ex = y, x = loge y = ln y

log (ab) = log a + log b

log (a/b) = log a − log b

log (1/n) = − log n

log an = n log a

log 1 = 0

log10

x = 0.4343 ln x

3. Determinants 2nd order

3rd order

A1.2 Analytic Geometry

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Figure A1.2.1 Straight line

Figure A1.2.2 Circle

Figure A1.2.3 Parabola

Figure A1.2.4 Ellipse

A1.3 Trigonometry

1. Definitions


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Figure A1.3.1

2. Signs in the four quadrants

Figure A1.3.2

3. Miscellaneous relations

sin2 θ + cos2 θ = 1

1 + tan2θ = sec2θ

1 + cot2 θ = csc2 θ

sin 2θ = 2 sin θ cos θ

cos 2θ = cos2 θ − sin2 θ

sin(a ± b) = sin a cos b ± cos a sin b

cos(a ± b) = cos a cos b sin a sinb


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4. Law of sines

Figure A1.3.3

5. Law of cosines

c2 = a2 + b2 − 2ab cos C

c2 = a2 + b2 + 2ab cos D

A1.4 The Cross Product (Vector Product)

The cross product or vector product of two vectors (in the case at hand, r and F, in that order) is, by definition, a third vectorthat is perpendicular to the plane defined by the two vectors. Call this third vector M (for moment). We can write the crossproduct in terms of the r (the position vector) and F (the force vector) in terms of their respective components as

(5.6B)

By applying the distributive and associative laws of vector multiplication to 5.6B and noting that

(A1.4a)

we can simplify 5.6B as

(5.7)

The cross product of the position vector and the force vector, as represented in 5.7, can be written in matrix form as:

(5.12)

The form in 5.12 specifies that the determinate of the matrix be taken. This determinate consisting of three rows and threecolumns may be evaluated by repeating the first and second columns and forming products along each diagonal line. Thesum of the products obtained along the colored lines is then subtracted from the sum of the products obtained along the blacklines; this is identical to the expression on the righthand side of 5.7.


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(A1.4b)

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.


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A2 PHYSICAL QUANTITIES

A2.1 Physical Properties

Density kg/m3 Specific weight lb/ft3 Density kg/m3 Specific weight lb/ft3

Air (20°C, 1 atm) 1.2062 0.08 Mercury 13570 847.13

Aluminum 2690 167.93 Nickel 8800 549.36

Bone 500 31.21 Nylon 1150 71.79

Brick 2000 124.85 Oil (av.) 900 56.18

Concrete (av.) 2400 149.82 Oxygen (0°C, 1 atm) 1.43 0.09

Copper 8910 556.22 Platinum 21500 1342.18

Cork 250 15.61 Polyethylene 1000 62.43

Diamond 3300 206.01 Silver 10500 655.48

Earth (dry, av.) 1280 79.91 Steel 7830 488.80

(wet, av.) 1760 109.87 Styrofoam 100 6.24

Glass 2590 161.69 Titanium 3080 192.28

Gold 19300 1204.84 Tungsten 19300 1204.84

Helium (0°C, 1 atm) 0.178 0.01 Uranium 18700 1167.38

Hydrogen (0°C, 1 atm) 0.09 0.01 Water (fresh) 1000 62.43

Ice 900 56.18 (seawater) 1030 64.30

Iron (cast) 7210 450.10 Wood (Balsa wood) 120 7.49

Lead 11370 709.79 (hard oak) 800 49.94

Mammals (most) 1000 62.43 (soft pine) 480 29.96

A2.2 Solar System Constants

BODY MEAN DISTANCE TO SUNkm (mi)

PERIOD OR ORBITsolar days

MEAN DIAMETER km(mi)

MASS RELATIVE TOEARTH

Sun — — 1 392 000 333 000

(865 000)

Moon 384 398* 27.32 3 476 0.0123

(238 854)* (2 160)

Mercury 57.3 × 106 87.97 5 000 0.054

(35.6 × 106) (3 100)

Venus 108 × 106 224.70 12 400 0.815

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BODY MEAN DISTANCE TO SUNkm (mi)

PERIOD OR ORBITsolar days

MEAN DIAMETER km(mi)

MASS RELATIVE TOEARTH

(67.2 × 106) (7 700)

Earth 149.6 × 106 365.26 12 742† 1.000

(92.96 × 106) (7 918)†

Mars 227.9 × 106 686.98 6 788 0.107

(141.6 × 106) (4 218)

A2.3 Conversion Factors from U.S. Customary Units

PHYSICAL QUANTITY U.S. CUSTOMARY UNIT = SI EQUIVALENT

BASIC UNITS

Length 1 foot (ft) = 3.048(10−1) meter (m)

1 inch (in.) = 2.54(10−2) meter (m)

1 mile (U.S. statute) = 1.6093(103) meter (m)

Mass 1 slug (lb · s2/ft) = 1.4594(10) kilogram (kg)

1 pound mass (lbm) = 4.5359(10−1) kilogram (kg)

DERIVED UNITS

Acceleration 1 foot/second2 (ft/s2) = 3.048(10−1) meter/second2 (m/s2)

1 inch/second2 (in./s2) = 2.54(10−2) meter/second2 (m/s2)

Area 1 foot2 (ft2) = 9.2903(10−2) meter2 (m2)

1 inch2 (in.2) = 6.4516(10−4) meter2 (m2)

Density 1 slug/foot3 (lb · s2/ft4) = 5.1537(102) kilogram/meter3 (kg/m3)

1 pound mass/foot3 (lbm/ft3) = 1.6018(10) kilogram/meter3 (kg/m3)

Energy and Work (1 joule ≡ 1 meter-newton)

1 foot-pound (ft · lb) = 1.3558 joules (J)

1 kilowatt-hour (kW · hr) = 3.60(106) joules (J)

1 British thermal unit (Btu) = 1.0551(103) joules (J)

Force (1 newton ≡ 1 kilogram-meter/second2)

1 pound (lb) = 4.4482 newtons (N)

1 kip (1000 lb) = 4.4482(103) newtons (N)

Power (1 watt ≡ 1 joule/second)

1 foot-pound/second (ft · lb/s) = 1.3558 watt (W)

1 horsepower (hp) = 7.4570(102) watt (W)


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PHYSICAL QUANTITY U.S. CUSTOMARY UNIT = SI EQUIVALENT

Pressure and Stress (1 pascal ≡ 1 newton/meter2)

1 pound/foot2 (lb/ft2) = 4.7880(10) pascal (Pa)

1 pound/inch2 (lb/in.2) = 6.8948(103) pascal (Pa)

1 atmosphere (standard, 14.7 lb/in.2) = 1.0133(105) pascal (Pa)

Speed 1 foot/second (ft/s) = 3.048(10−1) meter/second (m/s)

1 mile/hr = 4.4704(10−1) meter/second (m/s)

1 mile/hr = 1.6093 kilometer/hour (km/hr)

Volume 1 foot3 (ft3) = 2.8317(10−2) meter3 (m3)

1 inch3 (in.3) = 1.6387(10−5) meter3 (m3)

1 gallon (U.S. liquid) = 3.7854(10−3) meter3 (m3)



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A3 PROPERTIES OF AREAS AND VOLUMES

A3.1 Areas, Centroids, and Area Moments of Inertia

Shape Area Moment of Inertia

Figure A3.1.1 Rectangle

Figure A3.1.2 Right triangle

Figure A3.1.3 Scalene triangle

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Figure A3.1.4 Circle

Figure A3.1.5 Quarter circle

Figure A3.1.6 Circular sector

Figure A3.1.7 Circular arc

Figure A3.1.8 Ellipse


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Figure A3.1.9 Quarter ellipse

Figure A3.1.10 Parabolic section

Figure A3.1.11 Parabolic spandrel

A3.2 Volumes and Centroids

Shape Shape

Figure A3.2.1 Uniform slender rod

Figure A3.2.3 Cylinder


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Shape Shape

Figure A3.2.2 Rectangular parallelopiped

Figure A3.2.4 Semicylinder

Figure A3.2.5 Semicylindrical shell

Figure A3.2.9 Semiellipsoid

Figure A3.2.6 SphereFigure A3.2.10 Cone


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Shape Shape

Figure A3.2.7 HemisphereFigure A3.2.11 Semicone

Figure A3.2.8 Hemispherical shell Figure A3.2.12 Semiconical shell

Figure A3.2.13 Orthogonal tetrahedron

Figure A3.2.17 Thin circular ring

Figure A3.2.14 Right triangular prism

Figure A3.2.18 Circular plate


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Shape Shape

Figure A3.2.15 Rectangular plate Figure A3.2.19 Quarter circular plate

Figure A3.2.16 Triangular plate

Figure A3.2.20 Elliptical plate



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Statics Appendix A