Math Formulas with Problems, Study notes of Mathematics

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ALGEBRA 1 LOGARITHM x b bNNx =→= log Properties 1log log loglog loglog logloglog loglog)log( = = = −=      += a b xx xnx yx y x yxxy a b n REMAINDER AND FACTOR THEOREMS Given: )( )( rx xf − Remainder Theorem: Remainder = f(r) Factor Theorem: Remainder = zero QUADRATIC EQUATIONS A ACBBRoot CBxAx 2 4 0 2 2 −±− = =++ Sum of the roots = - B/A Products of roots = C/A MIXTURE PROBLEMS Quantity Analysis: A + B = C Composition Analysis: Ax + By = Cz WORK PROBLEMS Rate of doing work = 1/ time Rate x time = 1 (for a complete job) Combined rate = sum of individual rates Man-hours (is always assumed constant) 2 22 1 11 .. ))(ker( .. ))(ker( workofquantity timesWor workofquantity timesWor = ALGEBRA 2 UNIFORM MOTION PROBLEMS VtS = Traveling with the wind or downstream: 21 VVVtotal += Traveling against the wind or upstream: 21 VVVtotal −= DIGIT AND NUMBER PROBLEMS →++ uth 10100 2-digit number where: h = hundred’s digit t = ten’s digit u = unit’s digit CLOCK PROBLEMS where: x = distance traveled by the minute hand in minutes x/12 = distance traveled by the hour hand in minutes PDF created with pdfFactory trial version www.pdffactory.com PROGRESSION PROBLEMS a1 = first term an = nth term am = any term before an d = common difference S = sum of all “n” terms ARITHMETIC PROGRESSION (AP) - difference of any 2 no.’s is constant - calcu function: LINEAR (LIN) ])1(2[ 2 )( 2 ,... )( 1 1 2312 dnanS aanS etcaaaad dmnaa n mn −+= += −=−= −+= GEOMETRIC PROGRESSION (GP) - RATIO of any 2 adj, terms is always constant - Calcu function: EXPONENTIAL (EXP) ∞=<→ − = <→ − − = >→ − − = == = − nr r a S r r raS r r raS a a a ar raa n mn mn n &1 1 1 1 )1( 1 1 )1( 1 1 1 2 3 1 2 HARMONIC PROGRESSION (HP) - a sequence of number in which their reciprocals form an AP - calcu function: LINEAR (LIN) Mean – middle term or terms between two terms in the progression. COIN PROBLEMS Penny = 1 centavo coin Nickel = 5 centavo coin Dime = 10 centavo coin Quarter = 25 centavo coin Half-Dollar = 50 centavo coin DIOPHANTINE EQUATIONS If the number of equations is less than the number of unknowns, then the equations are called “Diophantine Equations”. ALGEBRA 3 Fundamental Principle: “If one event can occur in m different ways, and after it has occurred in any one of these ways, a second event can occur in n different ways, and then the number of ways the two events can occur in succession is mn different ways” PERMUTATION Permutation of n objects taken r at a time )!( ! rn nnPr − = Permutation of n objects taken n at a time !nnPn = Permutation of n objects with q,r,s, etc. objects are alike !...!! ! srq nP = Permutation of n objects arrange in a circle )!1( −= nP nth term Common difference Sum of ALL terms Sum of ALL terms Sum of ALL terms, r >1 Sum of ALL terms, r < 1 nth term ratio Sum of ALL terms, r < 1 , n = ∞ PDF created with pdfFactory trial version www.pdffactory.com 3. Given three sides 2 ))()(( cbas csbsassArea ++ = −−−= 4. Triangle inscribed in a circle r abcArea 4 = 5. Triangle circumscribing a circle rsArea = 6. Triangle escribed in a circle )( asrArea −= QUADRILATERALS 1. Given diagonals and included angle θsin 2 1 21ddArea = 2. Given four sides and sum of opposite angles 2 22 cos))()()(( 2 dcbas DBCA abcddscsbsasArea +++ = + = + = −−−−−= θ θ 3. Cyclic quadrilateral – is a quadrilateral inscribed in a circle )(4 ))()(( 2 ))()()(( Area bcadbdaccdab r dcbas dscsbsasArea +++ = +++ = −−−−= →+= bdacdd 21 Ptolemy’s Theorem 4. Quadrilateral circumscribing in a circle rsArea = 2 dcbas abcdArea +++ = = THEOREMS IN CIRCLES PDF created with pdfFactory trial version www.pdffactory.com SIMILAR TRIANGLES 2222 2 1      =     =     =     = h H c C b B a A A A SOLID GEOMETRY POLYGONS 3 sides – Triangle 4 sides – Quadrilateral/Tetragon/Quadrangle 5 sides – Pentagon 6 sides – Hexagon 7 sides – Heptagon/Septagon 8 sides – Octagon 9 sides – Nonagon/Enneagon 10 sides – Decagon 11 sides – Undecagon 12 sides – Dodecagon 15 sides – Quidecagon/ Pentadecagon 16 sides – Hexadecagon 20 sides – Icosagon 1000 sides – Chillagon Let: n = number of sides θ = interior angle α = exterior angle Sum of interior angles: S = n θ = (n – 2) 180° Value of each interior angle n n )180)(2( °− =θ Value of each exterior angle n ° =−°= 360180 θα Sum of exterior angles: S = n α = 360° Number of diagonal lines (N): )3( 2 −= nnN Area of a regular polygon inscribed in a circle of radius r       °= n nrArea 360sin 2 1 2 Area of a regular polygon circumscribing a circle of radius r       ° = n nrArea 180tan2 Area of a regular polygon having each side measuring x unit length       ° = n nxArea 180cot 4 1 2 PLANE GEOMETRIC FIGURES CIRCLES rdnceCircumfere rdA ππ π π 2 4 2 2 == == Sector of a Circle ° == ° = == 180 360 2 1 2 1 (deg) )( (deg) 2 2 θπ θ θπ θ r rs r A rrsA rad Segment of a Circle A segment = A sector – A triangle ELLIPSE A = π a b PARABOLIC SEGMENT bhA 3 2 = PDF created with pdfFactory trial version www.pdffactory.com TRAPEZOID hbaA )( 2 1 += PARALLELOGRAM θ α sin 2 1 sin 21ddA bhA abA = = = RHOMBUS αsin 2 1 2 21 aA ahddA = == SOLIDS WITH PLANE SURFACE Lateral Area = (No. of Faces) (Area of 1 Face) Polyhedron – a solid bounded by planes. The bounding planes are referred to as the faces and the intersections of the faces are called the edges. The intersections of the edges are called vertices. PRISM V = Bh A(lateral) = PL A(surface) = A(lateral) + 2B where: P = perimeter of the base L = slant height B = base area Truncated Prism       ∑ = heightsofnumber heightsBV PYRAMID BAA AA BhV lateralsurface faceslateral += ∑= = )()( )( 3 1 Frustum of a Pyramid )( 3 2121 AAAAhV ++= A1 = area of the lower base A2 = area of the upper base PRISMATOID )4( 6 21 mAAAhV ++= Am = area of the middle section REGULAR POLYHEDRON a solid bounded by planes whose faces are congruent regular polygons. There are five regular polyhedrons namely: A. Tetrahedron B. Hexahedron (Cube) C. Octahedron D. Dodecahedron E. Icosahedron PDF created with pdfFactory trial version www.pdffactory.com STRAIGHT LINES General Equation Ax + By + C = 0 Point-slope form y – y1 = m(x – x1) Two-point form )( 1 12 12 1 xx xx yyyy − − − =− Slope and y-intercept form y = mx + b Intercept form 1=+ b y a x Slope of the line, Ax + By + C = 0 B Am −= Angle between two lines       + − = − 21 121 1 tan mm mm θ Note: Angle θ is measured in a counterclockwise direction. m2 is the slope of the terminal side while m1 is the slope of the initial side. Distance of point (x1,y1) from the line Ax + By + C = 0; 22 11 BA CByAxd +± ++ = Note: The denominator is given the sign of B Distance between two parallel lines 22 21 BA CCd + − = Slope relations between parallel lines: m1 = m2 Line 1 → Ax + By + C1 = 0 Line 2 → Ax + By + C2 = 0 Slope relations between perpendicular lines: m1m2 = –1 Line 1 → Ax + By + C1 = 0 Line 2 → Bx – Ay + C2 = 0 PLANE AREAS BY COORDINATES 1321 1321 ,,....,, ,,....,, 2 1 yyyyy xxxxxA n n= Note: The points must be arranged in a counter clockwise order. LOCUS OF A MOVING POINT The curve traced by a moving point as it moves in a plane is called the locus of the point. SPACE COORDINATE SYSTEM Length of radius vector r: 222 zyxr ++= Distance between two points P1(x1,y1,z1) and P2(x2,y2,z2) 2 12 2 12 2 12 )()()( zzyyxxd −+−+−= PDF created with pdfFactory trial version www.pdffactory.com ANALYTIC GEOMETRY 2 CONIC SECTIONS a two-dimensional curve produced by slicing a plane through a three-dimensional right circular conical surface Ways of determining a Conic Section 1. By Cutting Plane 2. Eccentricity 3. By Discrimination 4. By Equation General Equation of a Conic Section: Ax2 + Cy2 + Dx + Ey + F = 0 ** Cutting plane Eccentricity Circle Parallel to base e → 0 Parabola Parallel to element e = 1.0 Ellipse none e < 1.0 Hyperbola Parallel to axis e > 1.0 Discriminant Equation** Circle B2 - 4AC < 0, A = C A = C Parabola B2 - 4AC = 0 A ≠ C same sign Ellipse B2 - 4AC < 0, A ≠ C Sign of A opp. of B Hyperbola B2 - 4AC > 0 A or C = 0 CIRCLE A locus of a moving point which moves so that its distance from a fixed point called the center is constant. Standard Equation: (x – h)2 + (y – k)2 = r2 General Equation: x2 + y2 + Dx + Ey + F = 0 Center at (h,k): A Ek A Dh 2 ; 2 −=−= Radius of the circle: A Fkhr −+= 222 or FEDr 4 2 1 22 −+= PARABOLA a locus of a moving point which moves so that it’s always equidistant from a fixed point called focus and a fixed line called directrix. where: a = distance from focus to vertex = distance from directrix to vertex AXIS HORIZONTAL: Cy2 + Dx + Ey + F = 0 Coordinates of vertex (h,k): C Ek 2 −= substitute k to solve for h Length of Latus Rectum: C DLR = PDF created with pdfFactory trial version www.pdffactory.com AXIS VERTICAL: Ax2 + Dx + Ey + F = 0 Coordinates of vertex (h,k): A Dh 2 −= substitute h to solve for k Length of Latus Rectum: A ELR = STANDARD EQUATIONS: Opening to the right: (y – k)2 = 4a(x – h) Opening to the left: (y – k)2 = –4a(x – h) Opening upward: (x – h) 2 = 4a(y – k) Opening downward: (x – h) 2 = –4a(y – k) Latus Rectum (LR) a chord drawn to the axis of symmetry of the curve. LR= 4a for a parabola Eccentricity (e) the ratio of the distance of the moving point from the focus (fixed point) to its distance from the directrix (fixed line). e = 1 for a parabola ELLIPSE a locus of a moving point which moves so that the sum of its distances from two fixed points called the foci is constant and is equal to the length of its major axis. d = distance of the center to the directrix STANDARD EQUATIONS: Major axis is horizontal: 1)()( 2 2 2 2 = − + − b ky a hx Major axis is vertical: 1)()( 2 2 2 2 = − + − a ky b hx General Equation of an Ellipse: Ax2 + Cy2 + Dx + Ey + F = 0 Coordinates of the center: C Ek A Dh 2 ; 2 −=−= If A > C, then: a2 = A; b2 = C If A < C, then: a2 = C; b2 = A KEY FORMULAS FOR ELLIPSE Length of major axis: 2a Length of minor axis: 2b Distance of focus to center: 22 bac −= PDF created with pdfFactory trial version www.pdffactory.com AREA OF SPHERICAL TRIANGLE ° = 180 2ERA π R = radius of the sphere E = spherical excess in degrees, E = A + B + C – 180° TERRESTRIAL SPHERE Radius of the Earth = 3959 statute miles Prime meridian (Longitude = 0°) Equator (Latitude = 0°) Latitude = 0° to 90° Longitude = 0° to +180° (eastward) = 0° to –180° (westward) 1 min. on great circle arc = 1 nautical mile 1 nautical mile = 6080 feet = 1852 meters 1 statute mile = 5280 feet = 1760 yards 1 statute mile = 8 furlongs = 80 chains Derivatives dx du u u dx d dx duuuu dx d dx duuuu dx d dx duuu dx d dx duuu dx d dx duuu dx d dx duuu dx d u dx du u dx d u dx due u dx d dx duee dx d dx duaaa dx d u dx duc u c dx d u dx du u dx d dx dunuu dx d v dx dvu dx duv v u dx d dx duv dx dvuuv dx d dx dv dx duvu dx d dx dC a a uu uu nn 2 1 2 2 2 1 2 1 1)(sin cotcsc)(csc tansec)(sec csc)(cot sec)(tan sin)(cos cos)(sin )(ln log )(ln )( ln)( 2 )( )( )( 0 − = −= = −= = −= = = = = = − =      = = − =      += +=+ = − − PDF created with pdfFactory trial version www.pdffactory.com dx du uu uh dx d dx du uu uh dx d dx du u u dx d dx du u u dx d dx du u u dx d dx du u u dx d dx duuhuhu dx d dx duuhuhu dx d dx duuhu dx d dx duuhu dx d dx duuu dx d dx duuu dx d dx du uu u dx d dx du uu u dx d dx du u u dx d dx du u u dx d dx du u u dx d 2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 1 2 1 2 1 2 1 2 1 1 1)(csc 1 1)(sec 1 1)(sinh 1 1)(tanh 1 1)(cosh 1 1)(sinh cothcsc)(csc tanhsec)(sec csc)(coth sec)(tanh sinh)(cosh cosh)(sinh 1 1)(csc 1 1)(sec 1 1)(cot 1 1)(tan 1 1)(cos + − = − − = − − = − = − = + = −= −= −= = = = − − = − = + − = + = − − = − − − − − − − − − − − DIFFERENTIAL CALCULUS LIMITS Indeterminate Forms ∞∞∞∞∞ ∞ ∞ 1,,0,-,)(0)(,, 0 0 00 L’Hospital’s Rule ..... )(" )(" )(' )(' )( )( xg xfLim xg xfLim xg xfLim axaxax →→→ == Shortcuts Input equation in the calculator TIP 1: if x → 1, substitute x = 0.999999 TIP 2: if x → ∞ , substitute x = 999999 TIP 3: if Trigonometric, convert to RADIANS then do tips 1 & 2 MAXIMA AND MINIMA Slope (pt.) Y’ Y” Concavity MAX 0 (-) dec down MIN 0 (+) inc up INFLECTION - No change - HIGHER DERIVATIVES nth derivative of xn !)( nx dx d n n n = nth derivative of xe n Xn n n enxxe dx d )()( += PDF created with pdfFactory trial version www.pdffactory.com TIME RATE the rate of change of the variable with respect to time dt dx + = increasing rate dt dx − = decreasing rate APPROXIMATION AND ERRORS If “dx” is the error in the measurement of a quantity x, then “dx/x” is called the RELATIVE ERROR. RADIUS OF CURVATURE " ])'(1[ 2 3 2 y yR + = INTEGRAL CALCULUS 1 Cuuduu Cuuduu Cuudu Cuudu Cuudu Cuudu Cedue C a adua Cu u du nC n uduu duugduufduuguf Cauadu Cudu uu u u n n +−= += +−= += += +−= += += += ≠+ + = +=+ += += ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫∫ ∫ ∫ + csccotcsc sectansec cotcsc tansec sincos cossin ln ln )1....(.......... 1 )()()]()([ 2 2 1 ∫∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ −= >+= − <+= − +−= + += − += + += += +−= +−= +−= += += += +      −= − += − += + += − +−= ++= += += − − − − − − − − − vduuvudv auC a u aua du auC a u aua du C u a aauu du C a u au du C a u au du Cuudu Cuudu Chuuduhu Chuuduhu Cuuduh Cuuduh Cuudu Cuudu C a u uau du C a u aauu du C a u aua du C a u ua du Cuuudu Cuuudu Cuudu Cuudu ..............coth1 ..............tanh1 sinh1 cosh sinh sinhlncoth coshlntanh csccothcsc sectanhsec cothcsc tanhsec sinhcosh coshsinh 1cos 2 sec1 tan1 sin cotcsclncsc tanseclnsec sinlncot seclntan 1 22 1 22 1 22 1 22 1 22 2 2 1 2 1 22 1 22 1 22 PDF created with pdfFactory trial version www.pdffactory.com MOMENT OF INERTIA Moment of Inertia about the x- axis: ∫= 2 1 2 x x x dAyI Moment of Inertia about the y- axis: ∫= 2 1 2 y y y dAxI Parallel Axis Theorem The moment of inertia of an area with respect to any coplanar line equals the moment of inertia of the area with respect to the parallel centroidal line plus the area times the square of the distance between the lines. 2AdIxI ox == Moment of Inertia for Common Geometric Figures Square 3 3bhI x = 12 3bhI xo = Triangle 12 3bhI x = 36 3bhI xo = Circle 4 4rI xo π = Half-Circle 8 4rI x π = Quarter-Circle 16 4rI x π = Ellipse 4 3abI x π = 4 3baI y π = FLUID PRESSURE ∫= == dAhwF AhAhwF γ F = force exerted by the fluid on one side of the area h = distance of the c.g. to the surface of liquid w = specific weight of the liquid (γ) A = vertical plane area Specific Weight: Volume Weight =γ γwater = 9.81 kN/m2 SI γwater = 45 lbf/ft2 cgs PDF created with pdfFactory trial version www.pdffactory.com MECHANICS 1 VECTORS Dot or Scalar product θcosQPQP =• zzyyxx QPQPQPQP ++=• Cross or Vector product θsinQPQP =× zyx zyx QQQ PPP kji QP =× EQUILIBRIUM OF COPLANAR FORCE SYSTEM Conditions to attain Equilibrium: ∑ ∑ ∑ = = = − − 0 0 0 int)( )( )( po axisy axisx M F F Friction Ff = μN tanφ = μ φ = angle of friction if no forces are applied except for the weight, φ = θ CABLES PARABOLIC CABLES the load of the cable of distributed horizontally along the span of the cable. Uneven elevation of supports 22 22 22 11 2 2 2 1 2 1 )( )( 22 HwxT HwxT d wx d wxH += += == Even elevation of supports 10> d L 2 2 2 2 8 HwLT d wLH +     = = 3 42 5 32 3 8 L d L dLS −+= L = span of cable d = sag of cable T = tension of cable at support H = tension at lowest point of cable w = load per unit length of span S = total length of cable PDF created with pdfFactory trial version www.pdffactory.com CATENARY the load of the cable is distributed along the entire length of the cable. Uneven elevation of supports 21 22 2 11 1 22 2 2 2 22 1 2 1 22 11 ln ln xxSpan c yScx c yScx cSy cSy wcH wyT wyT +=       + =       + = += += = = = Total length of cable = S1 + S2 Even elevation of supports xSpan c yScx cSy wcH wyT 2 ln 222 =       + = += = = Total length of cable = 2S MECHANICS 2 RECTILINEAR MOTION Constant Velocity S = Vt Constant Acceleration: Horizontal Motion aSVV atVV attVS 2 2 1 2 0 2 0 2 0 ±= ±= ±= + (sign) = body is speeding up – (sign) = body is slowing down Constant Acceleration: Vertical Motion gHVV gtVV gttVH 2 2 1 2 0 2 0 2 0 ±= ±= −=± + (sign) = body is moving down – (sign) = body is moving up Values of g, SI (m/s2) English (ft/s2) general 9.81 32.2 estimate 9.8 32 exact 9.806 32.16 PDF created with pdfFactory trial version www.pdffactory.com LAW OF CONSERVATION OF MOMENTUM “In every process where the velocity is changed, the momentum lost by one body or set of bodies is equal to the momentum gain by another body or set of bodies” Momentum lost = Momentum gained ' 22 ' 112211 VmVmVmVm +=+ m1 = mass of the first body m2 = mass of the second body V1 = velocity of mass 1 before the impact V2 = velocity of mass 2 before the impact V1’ = velocity of mass 1 after the impact V2’ = velocity of mass 2 after the impact Coefficient of Restitution (e) 21 ' 1 ' 2 VV VVe − − = Type of collision e Kinetic Energy ELASTIC 100% conserved 10 >< e INELASTIC Not 100% conserved 0=e PERFECTLY INELASTIC Max Kinetic Energy Lost 1=e Special Cases d r h he = βθ tancot=e Work, Energy and Power Work SFW ⋅= Force Distance Work Newton (N) Meter Joule Dyne Centimeter ft-lbf Pound (lbf) Foot erg Potential Energy WhmghPE == Kinetic Energy 2 2 1 mVKElinear = 2 2 1 ωIKErotational = → V = rω I = mass moment of inertia ω = angular velocity Mass moment of inertia of rotational INERTIA for common geometric figures: Solid sphere: 2 5 2 mrI = Thin-walled hollow sphere: 2 3 2 mrI = Solid disk: 2 2 1 mrI = Solid Cylinder: 2 2 1 mrI = Hollow Cylinder: )( 2 1 22 innerouter rrmI −= m = mass of the body r = radius PDF created with pdfFactory trial version www.pdffactory.com POWER rate of using energy VF t WP ⋅== 1 watt = 1 Newton-m/s 1 joule/sec = 107 ergs/sec 1 hp = 550 lb-ft per second = 33000 lb-ft per min = 746 watts LAW ON CONSERVATION OF ENERGY “Energy cannot be created nor destroyed, but it can be change from one form to another” Kinetic Energy = Potential Energy WORK-ENERGY RELATIONSHIP The net work done on an object always produces a change in kinetic energy of the object. Work Done = ΔKE Positive Work – Negative Work = ΔKE Total Kinetic Energy = linear + rotation HEAT ENERGY AND CHANGE IN PHASE Sensible Heat is the heat needed to change the temperature of the body without changing its phase. Q = mcΔT Q = sensible heat m = mass c = specific heat of the substance ΔT = change in temperature Specific heat values Cwater = 1 BTU/lb–°F Cwater = 1 cal/gm–°C Cwater = 4.156 kJ/kg Cice = 50% Cwater Csteam = 48% Cwater Latent Heat is the heat needed by the body to change its phase without changing its temperature. Q = ±mL Q = heat needed to change phase m = mass L = latent heat (fusion/vaporization) (+) = heat is entering (substance melts) (–) = heat is leaving (substance freezes) Latent heat of Fusion – solid to liquid Latent heat of Vaporization – liquid to gas Values of Latent heat of Fusion and Vaporization, Lf = 144 BTU/lb Lf = 334 kJ/kg Lf ice = 80 cal/gm Lv boil = 540 cal/gm Lf = 144 BTU/lb = 334 kJ/kg Lv = 970 BTU/lb = 2257 kJ/kg 1 calorie = 4.186 Joules 1 BTU = 252 calories = 778 ft–lbf LAW OF CONSERVATION OF HEAT ENERGY When two masses of different temperatures are combined together, the heat absorbed by the lower temperature mass is equal to the heat given up by the higher temperature mass. Heat gained = Heat lost PDF created with pdfFactory trial version www.pdffactory.com THERMAL EXPANSION For most substances, the physical size increase with an increase in temperature and decrease with a decrease in temperature. ΔL = LαΔT ΔL = change in length L = original length α = coefficient of linear expansion ΔT = change in temperature ΔV = VβΔT ΔV = change in volume V = original volume β = coefficient of volume expansion ΔT = change in temperature Note: In case β is not given; β = 3α THERMODYNAMICS In thermodynamics, there are four laws of very general validity. They can be applied to systems about which one knows nothing other than the balance of energy and matter transfer. ZEROTH LAW OF THERMODYNAMICS stating that thermodynamic equilibrium is an equivalence relation. If two thermodynamic systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each other. FIRST LAW OF THERMODYNAMICS about the conservation of energy The increase in the energy of a closed system is equal to the amount of energy added to the system by heating, minus the amount lost in the form of work done by the system on its surroundings. SECOND LAW OF THERMODYNAMICS about entropy The total entropy of any isolated thermodynamic system tends to increase over time, approaching a maximum value. THIRD LAW OF THERMODYNAMICS, about absolute zero temperature As a system asymptotically approaches absolute zero of temperature all processes virtually cease and the entropy of the system asymptotically approaches a minimum value. This law is more clearly stated as: "the entropy of a perfectly crystalline body at absolute zero temperature is zero." PDF created with pdfFactory trial version www.pdffactory.com d. Biaxial and Triaxial Deformation x z x y ε ε ε ε µ −=−= μ = Poisson’s ratio μ = 0.25 to 0.3 for steel = 0.33 for most metals = 0.20 for concrete μmin = 0 μmax = 0.5 TORSIONAL SHEARING STRESS Torsion – refers to twisting of solid or hollow rotating shaft. Solid shaft 3 16 d T π τ = Hollow shaft )( 16 44 dD TD − = π τ τ = torsional shearing stress T = torque exerted by the shaft D = outer diameter d = inner diameter Maximum twisting angle of the shaft’s fiber: JG TL =θ θ = angular deformation (radians) T = torque L = length of the shaft G = modulus of rigidity J = polar moment of inertia of the cross 32 4dJ π = → Solid shaft 32 )( 44 dDJ − = π → Hollow shaft Gsteel = 83 GPa; Esteel = 200 GPa Power delivered by a rotating shaft: min3300 2 sec550 2 60 2 2 lbftTNP lbftTNP rpmTNP rpsTNP TP hp hp rpm rpm − = − = = = = π π π π ω T = torque N = revolutions/time HELICAL SPRINGS       += R d d PR 4 116 3π τ       + − − = mm m d PR 615.0 44 1416 3π τ where, r R d Dm meanmean == elongation, 4 364 Gd nPR =δ τ = shearing stress δ = elongation R = mean radius d = diameter of the spring wire n = number of turns G = modulus of rigidity PDF created with pdfFactory trial version www.pdffactory.com ENGINEERING ECONOMICS 1 SIMPLE INTEREST PinI = )1( inPF += P = principal amount F = future amount I = total interest earned i = rate of interest n = number of interest periods Ordinary Simple Interest 360 daysn = 12 monthsn = Exact Simple Interest →= 365 daysn ordinary year →= 366 daysn leap year COMPOUND INTEREST niPF )1( += Nominal Rate of Interest mNn m NRi =⇔= Effective Rate of Interest ( ) AnnualifequalNRER m NRER iER m m ; 11 11 ≥ −      += −+= i = rate of interest per period NR = nominal rate of interest m = number of interest periods per year n = total number of interest periods N = number pf years ER = effective rate of interest Mode of Interest m Annually 1 Semi-Annually 2 Quarterly 4 Semi-quarterly 8 Monthly 12 Semi-monthly 24 Bimonthly 6 Daily 360 Shortcut on Effective Rate ANNUITY Note: interest must be effective rate Ordinary Annuity ii iAP i iAF n n n )1( ]1)1[( ]1)1[( + −+ = −+ = A = uniform periodic amount or annuity Perpetuity or Perpetual Annuity i AP = PDF created with pdfFactory trial version www.pdffactory.com LINEAR / UNIFORM GRADIENT SERIES GA PPP +=       + − + −+ = nn n G i n ii i i GP )1()1( 1)1(       − −+ = n i i i GF n G 1)1(       −+ −= 1)1( 1 nG i n i GA Perpetual Gradient 2i GPG = UNIFORM GEOMETRIC GRADIENT       − −++ = − iq iqCP nn 1)1()1( if q ≠ i       − +−+ = iq iqCF nn )1()1( if q ≠ i q CnP + = 1 q iCnP n + + = 1 )1( if q = i 1sec −= first ondq C = initial cash flow of the geometric gradient series which occurs one period after the present q = fixed percentage or rate of increase ENGINEERING ECONOMICS 2 DEPRECIATION Straight Line Method (SLM) n CC d n− = 0 Dm = md Cm = C0 – Dm d = annual depreciation C0 = first cost Cm = book value Cn = salvage or scrap value n = life of the property Dm = total depreciation after m-years m = mth year Sinking Fund Method (SFM) 1)1( )( 0 −+ − = n n i iCCd i idD m m ]1)1[( −+ = Cm = C0 – Dm i = standard rate of interest Sum of the Years Digit (SYD) Method       + +− −= )1( )1(2)( 0 nn mnCCd nm       + +− −= )1( )12()( 0 nn mmnCCD nm 2 )1( + = nnSYD Cm = C0 – Dm SYD = sum of the years digit dm = depreciation at year m PDF created with pdfFactory trial version www.pdffactory.com