ALL FORMULAS

Formulas Kyle Michael Sy April 4, 2017 12th Update July 2019 Update 1 (7/1/2019) DISCLAIMER (Must Read) This document is solely for review purposes only. Distribution of this copy is solely due to aid the students in their studies relevant to this document. This document is not recommended to be used as an instrument/medium for teaching on official lectures and/or classes; since the author couldn’t guarantee 100% accuracy of the document as some minor mistakes could’ve been made during the production of this document. Editing this document by removing, changing, or adding anything is strictly prohibited as this is my own work and it took me a really long time to make this 80-page document. If you 1. Want me to add more equations; 2. Want me to change anything wrong in the document; 3. Have any other comments/suggestions; 4. Simply just want to thank me; Just contact me via: Gmail – sykylemichael@gmail.com Messenger – Kyle Michael Sy 3 Changelog August 2018 1: 8/18 1. Added conversion table. 2. Added Statistics. 3. Added Table of Contents. 4. Added Disclaimer. 5. Added first page. September 2018 2: 9/11 1. Added changelog. 2. Added temperature to the conversion table. 3: 9/18 1. Added Propositional Calculus and Logical Equivalence. 2. Renamed Conversions section to Tables. 3. Added truth table to Tables. January 2019 4: 1/15 1. Previous formulas for velocity, acceleration, and UAM moved to new section called Rectilinear Motion. 2. Renamed Universally Accelerated Motion to Constant Linear Acceleration. 3. Added Rotational Motion. February 2019 5: 2/7 1. Fixed formulas for the derivative. 2. Added derivatives of inverse trigonometric functions and hyperbolic functions. 6: 2/12 1. Rearranged sections alphabetically. 2. Table of contents condensed. 3. Added Surveying section. Data correction, traverse adjustment, and area. 7: 2/14 1. Corrected formula of derivative of a logarithm to a base a. March 2019 8: 3/21 1. Added a lot of surveying formulas. So much that I can’t name all of them. Kyle Michael Sy 12th Update 4 June 2019 9: 6/4 1. Added more integral formulas. 2. Changed margin to narrow (0.5 in.) to accommodate more space. 3. Changed integral variables from x to u. 10: 6/24 1. 2. 3. 4. Changed link to bit.ly/allformulas for easier access. Changed 1 to any possible constant a in integral of inverse trig functions. Added trigonometric integrals with 5 cases. Added more trigonometric identities necessary for trigonometric integrals. 11: 6/25 1. Changed a minor mistake in trigonometric integrals double-angle identity. July 2019 12: 7/1 1. 2. 3. 4. Changed CALCULUS to MATHEMATICS. Added Wallis Formula. Added Case IV and V for Integration of Powers of Trigonometric Functions. Added Integration thru Trigonometric Substitution. Kyle Michael Sy 12th Update 5 Clickable Table of Contents – for PDF and Word DISCLAIMER (Must Read) ------------------------------------------------------------------------ 2 Changelog--------------------------------------------------------------------------------------------- 3 August 2018 September 2018 January 2019 February 2019 March 2019 June 2019 MATHEMATICS 11 Variables and Symbols ---------------------------------------------------------------------------- 12 Trigonometric Identities ------------------------------------------------------------------------- 12 Reciprocal Identities Pythagorean Identities Negative Identities Co-Function Identities Sum and Difference Double-Angle Identities Half-Angle Identities Limits Involving Trigonometric Functions Differentiation -------------------------------------------------------------------------------------- 14 Basic Formulas Trigonometric Functions Inverse Trigonometric Functions Logarithmic Functions Hyperbolic Functions Integration ------------------------------------------------------------------------------------------- 17 Simple Integration Formulas Substitution Methods Trigonometric Functions Integration of Powers of Trigonometric Functions Definite Integrals Kyle Michael Sy 12th Update 6 Propositional Calculus ---------------------------------------------------------------------------- 21 Logical Equivalences Basic and Derived Argument Forms CHEMISTRY 23 Chemical Kinetics ---------------------------------------------------------------------------------- 24 Rate Law Integrated Rate Law Half Life Temperature and Reaction Rate Chemical Equilibrium ----------------------------------------------------------------------------- 26 Equilibrium Constant Equilibrium Constants in Terms of P Reaction Quotient Acids and Bases ------------------------------------------------------------------------------------- 27 Autoionization of Water pH Scale pOH Scale Concentration Constant Acid Ionization Constant Per Cent Ionization Additional Aqueous Equilibria ----------------------------------------------------------------- 28 Henderson-Hasselbalch Equation Modified Henderson-Hasselbalch Equation Solubility Product Constant ELECTROMAGNETISM 29 Electric Field ----------------------------------------------------------------------------------------- 30 General Formulas Charge Densities Other Equations for Electric Field Electric Flux Electric Potential Energy ------------------------------------------------------------------------- 32 Electric Potential ----------------------------------------------------------------------------------- 32 Kyle Michael Sy 12th Update 7 Electrical Work ------------------------------------------------------------------------------------- 33 Capacitors and Capacitance --------------------------------------------------------------------- 34 Spherical Capacitor Cylindrical Capacitor Capacitance in a Circuit Energy Stored in a Capacitor Dielectrics Induced Charge and Polarization Charging Capacitor Discharging Capacitor Ohm’s Law -------------------------------------------------------------------------------------------- 35 Resistance and Resistivity ----------------------------------------------------------------------- 35 Electrical Power ------------------------------------------------------------------------------------ 36 Current ------------------------------------------------------------------------------------------------ 36 Next Section ------------------------------------------------------------------------------------------ 36 GEOMETRY 37 Variables and Symbols ---------------------------------------------------------------------------- 38 Surface Area ----------------------------------------------------------------------------------------- 38 3-Dimensional Objects 2-Dimensional Objects Volume ------------------------------------------------------------------------------------------------ 39 MECHANICS 40 Variables and Symbols ---------------------------------------------------------------------------- 41 Rectilinear Motion --------------------------------------------------------------------------------- 41 Constant Linear Acceleration Velocity Acceleration Uniform Circular Acceleration Rotational Motion of a Rigid Body ------------------------------------------------------------- 42 Constant Angular Acceleration Velocity Kyle Michael Sy 12th Update 8 Acceleration Energy Moment of Inertia Parallel Axis Theorem Projectile --------------------------------------------------------------------------------------------- 44 X-Component Y-Component Force --------------------------------------------------------------------------------------------------- 45 General Formulas Friction Charge Electric Field Work and Energy ----------------------------------------------------------------------------------- 46 Momentum ------------------------------------------------------------------------------------------- 46 STATISTICS 47 Descriptive Statistics ------------------------------------------------------------------------------ 48 Measures of Center Measures of Spread Measure of Relative Position Measure of Skewness Measure of Kurtosis Sample Size------------------------------------------------------------------------------------------- 49 Point Estimation ------------------------------------------------------------------------------------ 49 Point Estimator for μ1-μ2 Other Point Estimators Interval Estimation -------------------------------------------------------------------------------- 49 Confidence Interval for μ1-μ2 Confidence Interval for p1-p2 Confidence Interval for μ Confidence Interval for p Discrete Probability Distribution-------------------------------------------------------------- 50 Binomial Distribution Kyle Michael Sy 12th Update 9 Hypergeometric Distribution Poisson Distribution Geometric Probability Distribution Negative Binomial Probability Distribution Continuous Probability Distribution --------------------------------------------------------- 52 Normal Probability Distribution Hypothesis Testing -------------------------------------------------------------------------------- 52 SURVEYING 53 Data Correction ------------------------------------------------------------------------------------- 54 Tape Correction Temperature Correction Tension Correction Sag Correction Normal Tension Traverse Adjustment ------------------------------------------------------------------------------ 55 Compass Rule Transit Rule Area ---------------------------------------------------------------------------------------------------- 56 Area by Triangle Double Meridian Distance (DMD) Double Parallel Distance Trapezoidal Rule Simpson’s One-third Rule Coordinate Method Leveling ----------------------------------------------------------------------------------------------- 57 Curvature and Refraction Reciprocal Leveling Differential Leveling Trigonometric Leveling Stadia Leveling Simple Curve ----------------------------------------------------------------------------------------- 63 Degree of Curve (D) Kyle Michael Sy 12th Update 10 Tangent Distance (T) Long Chord (LC) Subchord (SC) Length of Curve (Lc) External Distance (E) Middle Ordinate Stationing of Point of Curvature Stationing of Point of Tangency Stationing of Point of Intersection Compound Curve ----------------------------------------------------------------------------------- 67 If Common Tangent is not Parallel to the Long Chord If Common Tangent is Parallel to Long Chord Spiral Curve ------------------------------------------------------------------------------------------ 69 Elements of a Spiral Curve Properties of Spiral Curves Formulas Earthworks Engineering ------------------------------------------------------------------------- 72 Volume Computation CONSTANTS 74 TABLES 76 Mass Length Volume Temperature Truth Table Kyle Michael Sy 12th Update 11 MATHEMATICS April 4, 2017 Kyle Michael Sy 12th Update 12 Variables and Symbols 1. Theta (θ) - Angle 5. e - Natural number 2. u - Function 6. a - Any positive integer 3. A - Angle A 7. C - Arbitrary Constant 4. B - Angle B 8. k - Constant Trigonometric Identities Reciprocal Identities Sine 𝑠𝑖𝑛 𝜃 = Tangent 1 𝑐𝑠𝑐 𝜃 Cosine 𝑐𝑜𝑠 𝜃 = 1 𝑠𝑒𝑐 𝜃 Cotangent 𝑡𝑎𝑛 𝜃 = 1 𝑐𝑜𝑡 𝜃 𝑡𝑎𝑛 𝜃 = 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 𝑐𝑜𝑡 𝜃 = 1 𝑡𝑎𝑛 𝜃 𝑐𝑜𝑡 𝜃 = 𝑐𝑜𝑠 𝜃 𝑡𝑎𝑛 𝜃 Pythagorean Identities 𝑐𝑜𝑠 2 𝑥 + 𝑠𝑖𝑛2 𝑥 = 1 1 + 𝑡𝑎𝑛2 𝑥 = 𝑠𝑒𝑐 2 𝑥 1 + 𝑐𝑜𝑡 2 𝑥 = 𝑐𝑠𝑐 2 𝑥 Negative Identities Sine 𝑠𝑖𝑛(−𝜃) = − 𝑠𝑖𝑛 𝜃 Cosine 𝑐𝑜𝑠(−𝜃) = 𝑐𝑜𝑠 𝜃 Tangent 𝑡𝑎𝑛(−𝜃) = − 𝑡𝑎𝑛 𝜃 Co-Function Identities Sine 𝑠𝑖𝑛(90° − 𝜃) = 𝑐𝑜𝑠 𝜃 Secant 𝑠𝑒𝑐(90° − 𝜃) = 𝑐𝑠𝑐 𝜃 Tangent 𝑡𝑎𝑛(90° − 𝜃) = 𝑐𝑜𝑡 𝜃 Cosine 𝑐𝑜𝑠(90° − 𝜃) = 𝑠𝑖𝑛 𝜃 Kyle Michael Sy 12th Update 13 Cotangent 𝑐𝑜𝑡(90° − 𝜃) = 𝑡𝑎𝑛 𝜃 Cosecant 𝑐𝑠𝑐(90° − 𝜃) = 𝑠𝑒𝑐 𝜃 Sum and Difference Sine 𝑠𝑖𝑛(𝐴 + 𝐵) = 𝑠𝑖𝑛 𝐴 𝑐𝑜𝑠 𝐵 + 𝑐𝑜𝑠 𝐴 𝑠𝑖𝑛 𝐵 𝑠𝑖𝑛(𝐴 − 𝐵) = 𝑠𝑖𝑛 𝐴 𝑐𝑜𝑠 𝐵 − 𝑐𝑜𝑠 𝐴 𝑠𝑖𝑛 𝐵 Cosine 𝑐𝑜𝑠(𝐴 + 𝐵) = 𝑐𝑜𝑠 𝐴 𝑐𝑜𝑠 𝐵 − 𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 𝐵 𝑐𝑜𝑠(𝐴 − 𝐵) = 𝑐𝑜𝑠 𝐴 𝑐𝑜𝑠 𝐵 + 𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 𝐵 Tangent 𝑡𝑎𝑛(𝐴 + 𝐵) = 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵 1 − 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵 𝑡𝑎𝑛(𝐴 − 𝐵) = 𝑡𝑎𝑛 𝐴 − 𝑡𝑎𝑛 𝐵 1 + 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵 Double-Angle Identities Sine 𝑐𝑜𝑠(2𝐴) = 1 − 2 𝑠𝑖𝑛2 𝐴 sin(2A) = 2 sin A cos A Cosine 𝑐𝑜𝑠(2𝐴) = 𝑐𝑜𝑠 2 𝐴 − 𝑠𝑖𝑛2 𝐴 Tangent 𝑡𝑎𝑛(2𝐴) = 2 𝑡𝑎𝑛 𝐴 1 − 𝑡𝑎𝑛2 𝐴 𝑐𝑜𝑠(2𝐴) = 2 𝑐𝑜𝑠 2 𝐴 − 1 Half-Angle Identities Kyle Michael Sy 12th Update 14 Sine 𝑠𝑖𝑛 𝐴 1 − 𝑐𝑜𝑠 𝐴 = ±√ 2 2 Cosine 𝑐𝑜𝑠 Tangent 𝐴 𝑠𝑖𝑛 𝐴 𝑡𝑎𝑛 = 2 1 + 𝑐𝑜𝑠 𝐴 𝑡𝑎𝑛 𝐴 1 − 𝑐𝑜𝑠 𝐴 = 2 𝑠𝑖𝑛 𝐴 𝐴 1 + 𝑐𝑜𝑠 𝐴 = ±√ 2 2 Limits Involving Trigonometric Functions 𝑠𝑖𝑛 𝑥 =1 𝑥→0 𝑥 𝑙𝑖𝑚 1 − 𝑐𝑜𝑠 𝑥 =0 𝑥→0 𝑥 𝑙𝑖𝑚 Differentiation Basic Formulas Derivative of a Constant 𝑑𝑦 𝑑(𝑐) = =0 𝑑𝑥 𝑑𝑥 Notes to remember: 𝒅𝒚 𝒅𝒙 can be rewritten as 𝑦′ u is a function of x Derivative of 𝒙 With Respect to 𝒙 𝑑𝑦 𝑑𝑥 = =1 𝑑𝑥 𝑑𝑥 Derivative of a Constant Multiplied by 𝒇(𝒙) 𝑑 𝑑[𝑓(𝑥)] [𝑐 ⋅ 𝑓(𝑥)] = 𝑐 ⋅ 𝑑𝑥 𝑑𝑥 Derivative of a Sum and Difference of a Function 𝑑 𝑑[𝑓(𝑥)] 𝑑[𝑔(𝑥)] [𝑓(𝑥) ± 𝑔(𝑥)] = ± 𝑑𝑥 𝑑𝑥 𝑑𝑥 Kyle Michael Sy 12th Update 15 Derivative of the Product of Two Functions 𝑑 𝑑[𝑔(𝑥)] 𝑑[𝑓(𝑥)] [𝑓(𝑥) ⋅ 𝑔(𝑥)] = 𝑓(𝑥) ⋅ + 𝑔(𝑥) ⋅ 𝑑𝑥 𝑑𝑥 𝑑𝑥 Derivative of the Quotient of Two Functions 𝑑[𝑔(𝑥)] 𝑑[𝑓(𝑥)] 𝑔(𝑥) ⋅ − 𝑓(𝑥) ⋅ 𝑑 𝑓(𝑥) 𝑑𝑥 𝑑𝑥 [ ]= 2 𝑑𝑥 𝑔(𝑥) [𝑔(𝑥)] General Power Formula 𝑑𝑦 = 𝑛 ⋅ 𝑢𝑛−1 𝑑𝑥 Trigonometric Functions Derivative of Sine 𝑑(𝑠𝑖𝑛 𝑢) 𝑑𝑢 = 𝑐𝑜𝑠 𝑢 ⋅ 𝑑𝑥 𝑑𝑥 Derivative of Cotangent 𝑑(𝑐𝑜𝑡 𝑢) 𝑑𝑢 = −𝑐𝑠𝑐 2 𝑢 ⋅ 𝑑𝑥 𝑑𝑥 Derivative of Cosine 𝑑(𝑐𝑜𝑠 𝑢) 𝑑𝑢 = − 𝑠𝑖𝑛 𝑢 ⋅ 𝑑𝑥 𝑑𝑥 Derivative of Secant 𝑑(𝑠𝑒𝑐 𝑢) 𝑑𝑢 = 𝑠𝑒𝑐 𝑢 ⋅ 𝑡𝑎𝑛 𝑢 ⋅ 𝑑𝑥 𝑑𝑥 Derivative of Tangent 𝑑(𝑡𝑎𝑛 𝑢) 𝑑𝑢 = 𝑠𝑒𝑐 2 𝑢 ⋅ 𝑑𝑥 𝑑𝑥 Derivative of Cosecant 𝑑(𝑐𝑠𝑐 𝑢) 𝑑𝑢 = − 𝑐𝑠𝑐 𝑢 ⋅ 𝑐𝑜𝑡 𝑢 ⋅ 𝑑𝑥 𝑑𝑥 Inverse Trigonometric Functions Derivative of Arcsine 𝑑 1 𝑑𝑢 [𝑠𝑖𝑛−1 𝑢] = ∙ 𝑑𝑥 √1 − 𝑢2 𝑑𝑥 Kyle Michael Sy Derivative of Arccosine 𝑑 1 𝑑𝑢 [𝑐𝑜𝑠 −1 𝑢] = − ∙ 𝑑𝑥 √1 − 𝑢2 𝑑𝑥 12th Update 16 Derivative of Arctangent 𝑑 1 𝑑𝑢 [𝑡𝑎𝑛−1 𝑢] = ∙ 𝑑𝑥 1 + 𝑢2 𝑑𝑥 Derivative of Arcsecant 𝑑 1 𝑑𝑢 [𝑠𝑒𝑐 −1 𝑢] = ∙ 𝑑𝑥 |𝑢|√𝑢2 − 1 𝑑𝑥 Derivative of Arccotangent 𝑑 1 𝑑𝑢 [𝑐𝑜𝑡 −1 𝑢] = − ∙ 𝑑𝑥 1 + 𝑢2 𝑑𝑥 Derivative of Arccosecant 𝑑 1 𝑑𝑢 [𝑐𝑠𝑐 −1 𝑢] = − ∙ 𝑑𝑥 |𝑢|√𝑢2 − 1 𝑑𝑥 Logarithmic Functions Derivative of the Logarithm of 𝒖 to the Base 𝒂 𝑑(𝑙𝑜𝑔𝑎 𝑢) 1 𝑑𝑢 = ⋅ 𝑑𝑥 𝑢 ⋅ 𝑙𝑛 𝑎 𝑑𝑥 Derivative of the Logarithm of 𝒖 to the Base 𝒆 𝑑(𝑙𝑜𝑔𝑒 𝑢) 1 𝑑𝑢 = ⋅ 𝑑𝑥 𝑢 𝑑𝑥 𝑑(𝑙𝑛 𝑢) 1 𝑑𝑢 = ⋅ 𝑑𝑥 𝑢 𝑑𝑥 Derivative of 𝒂 Raised to 𝒖 𝑑(𝑎𝑢 ) 𝑑𝑢 = 𝑎𝑢 ⋅ 𝑙𝑛 𝑎 ⋅ 𝑑𝑥 𝑑𝑥 Derivative of 𝒆 Raised to 𝒖 𝑑(𝑒 𝑢 ) 𝑑𝑢 = 𝑒𝑢 ⋅ 𝑑𝑥 𝑑𝑥 Derivative of Hyperbolic Sine 𝑑 [𝑠𝑖𝑛ℎ 𝑥] = 𝑐𝑜𝑠ℎ 𝑥 𝑑𝑥 Derivative of Hyperbolic Cosine 𝑑 [𝑐𝑜𝑠ℎ 𝑥] = 𝑠𝑖𝑛ℎ 𝑥 𝑑𝑥 Hyperbolic Functions Kyle Michael Sy 12th Update 17 Integration Simple Integration Formulas Notes to remember: Integral of 1 u is a function ∫ 𝑑𝑢 = 𝑢 + 𝐶 C is the constant of integration Integral of a Function Multiplied by a Constant C is an arbitrary constant ∫ 𝑘 ⋅ 𝑓(𝑥) = 𝑘 ⋅ ∫ 𝑓(𝑥) Integral of the Sum and Difference of Two Functions ∫ 𝑓(𝑥) ± 𝑔(𝑥) = ∫ 𝑓(𝑥) ± ∫ 𝑔(𝑥) Integral of the Function 𝒖 1 Wherein: 𝑛 = 1 ∫ ⋅ 𝑑𝑢 = 𝑙𝑛 |𝑢| + 𝐶 𝑢 ∫ 𝑢−1 ⋅ 𝑑𝑢 = 𝑙𝑛 |𝑢| + 𝐶 Integral of 𝒆 raised to u Integral of the Natural Logarithm ∫ 𝑒 𝑢 ⋅ 𝑑𝑢 = 𝑒 𝑢 + 𝐶 ∫ 𝑙𝑛 𝑢 ⋅ 𝑑𝑢 = 𝑢 ⋅ 𝑙𝑛(𝑢) − 𝑢 + 𝐶 Integral of a constant raised to u 𝑎𝑢 𝑢 ∫ 𝑎 ∙ 𝑑𝑢 = +𝐶 𝑙𝑛 𝑎 General Formula 𝑢𝑛+1 𝑛 ∫ 𝑢 ⋅ 𝑑𝑢 = +𝐶 𝑛+1 Kyle Michael Sy Wherein: 𝑛 ≠ 1 12th Update 18 Substitution Methods Trigonometric Substitution If √𝑎2 − 𝑢2 occurs in the integrand, let 𝑢 = 𝑎 sin 𝜃 If √𝑎2 + 𝑢2 occurs in the integrand, let 𝑢 = 𝑎 tan 𝜃 If √𝑢2 − 𝑎2 occurs in the integrand, let 𝑢 = 𝑎 sec 𝜃 Kyle Michael Sy 12th Update 19 Trigonometric Functions Integral of Sine ∫ 𝑠𝑖𝑛 𝑢 ⋅ 𝑑𝑢 = −𝑐𝑜𝑠 𝑢 + 𝐶 Integral of Cosine ∫ 𝑐𝑜𝑠 𝑢 ⋅ 𝑑𝑢 = 𝑠𝑖𝑛 𝑢 + 𝐶 Integral of Tangent ∫ 𝑡𝑎𝑛 𝑢 ⋅ 𝑑𝑢 = 𝑙𝑛|𝑠𝑒𝑐 𝑢| + 𝐶 Integral of Cotangent ∫ 𝑐𝑜𝑡 𝑢 ∙ 𝑑𝑢 = 𝑙𝑛|𝑠𝑖𝑛 𝑢| + 𝐶 Integral of 𝐬𝐞𝐜 𝒖 ⋅ 𝐭𝐚𝐧 𝒖 ∫ 𝑠𝑒𝑐 𝑢 ⋅ 𝑡𝑎𝑛 𝑢 ⋅ 𝑑𝑢 = 𝑠𝑒𝑐 𝑢 + 𝐶 Integral of 𝐜𝐬𝐜 𝒖 ⋅ 𝐜𝐨𝐭 𝒖 ∫ 𝑐𝑠𝑐 𝑢 ⋅ 𝑐𝑜𝑡 𝑢 ⋅ 𝑑𝑢 = − 𝑐𝑠𝑐 𝑢 + 𝐶 Integral of ∫ 𝟏 √𝒂𝟐 −𝒖𝟐 1 √𝑎2 − 𝑢2 ∙ 𝑑𝑢 = 𝑠𝑖𝑛−1 𝑢 +𝐶 𝑎 𝟏 Integral of 𝟐 𝟐 𝒂 +𝒖 1 1 𝑢 −1 ∫ 2 ∙ 𝑑𝑢 = 𝑡𝑎𝑛 +𝐶 𝑎 + 𝑢2 𝑎 𝑎 Integral of Secant ∫ 𝑠𝑒𝑐 𝑢 ∙ 𝑑𝑢 = 𝑙𝑛|𝑠𝑒𝑐 𝑢 + 𝑡𝑎𝑛 𝑢| + 𝐶 Integral of Cosecant ∫ 𝑐𝑠𝑐 𝑢 ∙ 𝑑𝑢 = 𝑙𝑛|𝑐𝑠𝑐 𝑢 − 𝑐𝑜𝑡 𝑢| + 𝐶 Integral of 𝐬𝐞𝐜 𝒖 𝟐 ∫ 𝑠𝑒𝑐 2 𝑢 ⋅ 𝑑𝑢 = 𝑡𝑎𝑛 𝑢 + 𝐶 Integral of 𝐜𝐬𝐜 𝟐 𝒖 Integral of ∫ 𝟏 |𝒖|√𝒖𝟐 −𝒂𝟐 1 |𝑢|√𝑢2 − 𝑎2 ∙ 𝑑𝑢 = 1 𝑢 𝑠𝑒𝑐 −1 + 𝐶 𝑎 𝑎 Integral of Hyperbolic Sine ∫ 𝑠𝑖𝑛ℎ 𝑢 ∙ 𝑑𝑢 = 𝑐𝑜𝑠ℎ 𝑢 + 𝐶 Integral of Hyperbolic Cosine ∫ 𝑐𝑜𝑠ℎ 𝑢 ∙ 𝑑𝑢 = 𝑠𝑖𝑛ℎ 𝑢 + 𝐶 ∫ 𝑐𝑠𝑐 2 𝑢 ⋅ 𝑑𝑢 = −𝑐𝑜𝑡 𝑢 + 𝐶 Kyle Michael Sy 12th Update 20 Integration of Powers of Trigonometric Functions Case 1 ∫ 𝑠𝑖𝑛𝑚 𝜃 𝑐𝑜𝑠 𝑛 𝜃 𝑑𝜃 Case 2 ∫ 𝑠𝑖𝑛𝑚 𝜃 𝑐𝑜𝑠 𝑛 𝜃 𝑑𝜃 Wherein: m or n is an odd integer > 1 Use identity: Pythagorean identities: sin2 𝜃 + cos 2 𝜃 = 1 Wherein: m or n is a positive even integer Use identity: Double-angle identities: cos 2 𝜃 = sin2 𝜃 = 1+cos 2𝜃 2 1−cos 2𝜃 2 sin 𝜃 cos 𝜃 = Case 3 ∫ 𝑡𝑎𝑛𝑚 𝜃 𝑠𝑒𝑐 𝑛 𝜃 𝑑𝜃 ∫ 𝑐𝑜𝑡 𝑚 𝜃 𝑐𝑠𝑐 𝑛 𝜃 𝑑𝜃 sin 2𝜃 2 Wherein: n is an even integer > 2 Use identity: Pythagorean identities: sec 2 𝜃 = tan2 𝜃 + 1 csc 2 𝜃 = cot 2 𝜃 + 1 Case 4 ∫ 𝑡𝑎𝑛𝑚 𝜃 𝑠𝑒𝑐 𝑛 𝜃 𝑑𝜃 ∫ 𝑐𝑜𝑡 𝑚 𝜃 𝑐𝑠𝑐 𝑛 𝜃 𝑑𝜃 Wherein: m and n are odd integers > 1 Use identity: Pythagorean identities: sec 2 𝜃 = tan2 𝜃 + 1 csc 2 𝜃 = cot 2 𝜃 + 1 Case 5 ∫ 𝑠𝑖𝑛 𝐴𝑥 𝑐𝑜𝑠 𝐵𝑥 𝑑𝑥 Use identity: Sum and difference identities: 1 sin 𝐴𝑥 cos 𝐵𝑥 = [sin(𝐴 − 𝐵) + sin(𝐴 + 𝐵)] 2 ∫ 𝑐𝑜𝑠 𝐴𝑥 𝑐𝑜𝑠 𝐵𝑥 𝑑𝑥 1 cos 𝐴𝑥 cos 𝐵𝑥 = [cos(𝐴 − 𝐵) + cos(𝐴 + 𝐵)] 2 1 ∫ 𝑠𝑖𝑛 𝑎𝑥 𝑠𝑖𝑛 𝑏𝑥 𝑑𝑥 Kyle Michael Sy sin 𝐴𝑥 sin 𝐵𝑥 = [cos(𝐴 − 𝐵) − cos(𝐴 + 𝐵)] 2 12th Update 21 Definite Integrals Wallis Formula 2 2 (𝑚 (𝑛 [ − 1)(𝑚 − 3) … or] [ − 1)(𝑛 − 3) … or] 𝜋 2 1 1 ∙𝛼 ∫ sin𝑚 𝜃 cos n 𝜃 𝑑𝜃 = 2 0 (𝑚 + 𝑛)(𝑚 + 𝑛 − 2)(𝑚 + 𝑛 − 4) … or 1 𝜋 Wherein: 𝛼 = if m and n are both even 2 𝛼 = 1 if otherwise Propositional Calculus Logical Equivalences Identity Law 𝑝∧𝑇 ⇔𝑝 𝑝∨𝐹 ⇔𝑝 Domination Law 𝑝∨𝑇 ⇔𝑇 𝑝∧𝐹 ⇔𝐹 Idempotent Law 𝑝∧𝑝⇔𝑝 Double Negation ¬(¬𝑝) ⇔ 𝑝 Commutative Law 𝑝∧𝑞 ⇔𝑞∧𝑝 Kyle Michael Sy 𝑝∨𝑞 ⇔𝑞∨𝑝 Associative Law (𝑝 ∧ 𝑞) ∧ 𝑟 ⇔ 𝑝 ∧ (𝑞 ∧ 𝑟) (𝑝 ∨ 𝑞) ∨ 𝑟 ⇔ 𝑝 ∨ (𝑞 ∨ 𝑟) Distributive Law 𝑝 ∨ (𝑞 ∧ 𝑟) ⇔ (𝑝 ∨ 𝑞) ∧ (𝑝 ∨ 𝑟) 𝑝 ∧ (𝑞 ∨ 𝑟) ⇔ (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟) De Morgan’s Law ¬(𝑝 ∧ 𝑞) ⇔ ¬𝑝 ∨ ¬𝑞 ¬(𝑝 ∨ 𝑞) ⇔ ¬𝑝 ∧ ¬𝑞 Absorption Law 𝑝 ∨ (𝑝 ∧ 𝑞) ⇔ 𝑝 12th Update 22 𝑝 ∧ (𝑝 ∨ 𝑞) ⇔ 𝑝 Negation Law 𝑝 ∨ ¬𝑝 ⇔ 𝑇 𝑝 ∧ ¬𝑝 ⇔ 𝐹 Basic and Derived Argument Forms Modus Ponens ((𝑝 →) ∧ 𝑝) ⇔ 𝑞 Modus Tollens ((𝑝 →) ∧ −𝑞) ⇔ −𝑝 Hypothetical Syllogism Disjunctive Syllogism Constructive Dilemma Destructive Dilemma Bi-directional Dilemma Simplification Conjunction Addition Composition De Morgan’s Theorem Commutative Associative Double Negation Transposition Material Implication Material Equivalence Exportation Importation Tautology Kyle Michael Sy 12th Update 23 CHEMISTRY August 3, 2017 Kyle Michael Sy 12th Update 24 Chemical Kinetics Rate Law Reaction Rate ∆[𝐶𝑣 + ] 𝑟𝑎𝑡𝑒 = ∆𝑡 Wherein: ∆[Cv + ] change in concentration of Cv+ ∆𝑡 change in time Overall Rate of the Reaction For any general reaction: 𝑎𝐴 + 𝑏𝐵 → 𝑐𝐶 + 𝑑𝐷 The overall rate of the reaction is: 𝑟𝑎𝑡𝑒 = − 1 ∆[𝐴] 1 ∆[𝐵] 1 ∆[𝐶] 1 ∆[𝐷] =− =+ =+ 𝑎 ∆𝑡 𝑏 ∆𝑡 𝑐 ∆𝑡 𝑑 ∆𝑡 Reactants decrease with time. Thus the negative sign. Products increase with time. Thus the positive sign. Rate Law 𝑟𝑎𝑡𝑒 = 𝑘[𝐴]𝑚 [𝐵]𝑛 … [𝐴]𝑚 [𝐵]𝑛 … 𝑟𝑎𝑡𝑒 Wherein: 𝑘 is the rate constant 𝑘= 𝑚, 𝑛 is the order for 𝑎 and 𝑏, respectively 𝑚 + 𝑛 + ⋯ is the overall order of the reaction Integrated Rate Law First-Order Reaction 𝑙𝑛[𝐴]𝑡 = −𝑘𝑡 + 𝑙𝑛[𝐴]0 Kyle Michael Sy 12th Update 25 Half Life First-Order Reaction 0.693 𝑡1 = 𝑘 2 Temperature and Reaction Rate Arrhenius Equation 𝐸𝑎 𝑘 = 𝐴 ⋅ 𝑒 −𝑅⋅𝑇 Wherein: 𝐸𝑎 is the activation energy 𝑅 is the gas constant (8.3145 J K-1 mol-1) 𝑇 is a kelvin unit 𝐴 is the frequency factor Determining Activation Energy 𝐸𝑎 1 𝑙𝑛 𝑘 = (− ) ( ) + 𝑙𝑛 𝐴 𝑅 𝑇 Kyle Michael Sy 12th Update 26 Chemical Equilibrium Equilibrium Constant General Reaction 𝑎𝐴 + 𝑏𝐵 → 𝑐𝐶 + 𝑑𝐷 𝑘𝑓𝑜𝑟𝑤𝑎𝑟𝑑 [𝐶]𝑐 [𝐷]𝑑 𝐾𝑐 = = [𝐴]𝑎 [𝐵]𝑏 𝑘𝑟𝑒𝑣𝑒𝑟𝑠𝑒 Combined Reaction 𝐾𝑐 = 𝐾𝑐 (𝑠𝑡𝑒𝑝 1) ⋅ 𝐾𝑐 (𝑠𝑡𝑒𝑝 2) Equilibrium Constants in Terms of P Gas-Phase Reaction 𝑎𝐴(𝑔) + 𝑏𝐵(𝑔) → 𝑐𝐶(𝑔) + 𝑑𝐷(𝑔) 𝑃𝐶𝑐 ⋅ 𝑃𝐶𝑑 𝐾𝑝 = 𝑎 𝑏 𝑃𝐴 ⋅ 𝑃𝐵 Wherein: 𝐾𝑝 is pressure-based General 𝐾𝑝 = 𝐾𝑐 ⋅ (𝑅 ⋅ 𝑇)∆𝑛𝑔𝑎𝑠 ∆𝑛𝑔𝑎𝑠 = 𝑛𝑔𝑎𝑠𝑒𝑜𝑢𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠 − 𝑛𝑔𝑎𝑠𝑒𝑜𝑢𝑠 𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠 = (𝑐 + 𝑑) − (𝑎 + 𝑏) Reaction Quotient [C]c [D]d Q= [A]a [B]b Kyle Michael Sy Note: 𝐾𝑐 = 𝑄 whenever a system is at equilibrium 12th Update 27 Acids and Bases Autoionization of Water General 𝐾𝑤 = [𝐻3 𝑂+ ][𝑂𝐻− ] At Room Temperature (25° Celsius) 𝐾𝑤 = 1.14 × 10−14 pH Scale Note: A pH > 7.00 is more basic A pH < 7.00 is more acidic 𝑝𝐻 = −𝑙𝑜𝑔10 [𝐻3 𝑂+ ] pOH Scale 𝑝𝑂𝐻 = −𝑙𝑜𝑔10 [𝑂𝐻 − ] Concentration Constant 𝑝𝐾𝑤 = 𝑝𝐻 + 𝑝𝑂𝐻 = 14.00 Acid Ionization Constant When an acid ionizes in water: 𝐻𝐴 (𝑎𝑞) + 𝐻2 𝑂(𝑙) → 𝐻3 𝑂+ (𝑎𝑞) + 𝐴− (𝑎𝑞) The acid ionization constant is used to report the degree of ionization: [𝐴− ] ⋅ [𝐻3 𝑂+ ] 𝐾𝑎 = [𝐻𝐴] Per Cent Ionization Note: Strong acids have large 𝐾𝑎 values Weak acids have small 𝐾𝑎 values 𝑥 % 𝑖𝑜𝑛𝑖𝑧𝑒𝑑 = ( ) ⋅ 100% 0.100 Kyle Michael Sy 12th Update 28 Additional Aqueous Equilibria Henderson-Hasselbalch Equation [𝐴− ] 𝑝𝐻 = 𝑝𝐾𝑎 + 𝑙𝑜𝑔 [𝐻𝐴] Wherein: 𝑝𝐻 = 𝑝𝐾𝑎 when [𝐻𝐴] = [𝐴− ] Modified Henderson-Hasselbalch Equation Buffer + Acid (A) 𝑝𝐻 = 𝑝𝐾𝑎 + 𝑙𝑜𝑔 𝑏−𝐴 𝑎+𝐴 𝑝𝐻 = 𝑝𝐾𝑎 + 𝑙𝑜𝑔 𝑏𝑛𝑜. 𝑚𝑜𝑙𝑒𝑠 − 𝐴𝑛𝑜. 𝑚𝑜𝑙𝑒𝑠 𝑎𝑛𝑜. 𝑚𝑜𝑙𝑒𝑠 + 𝐴𝑛𝑜. 𝑚𝑜𝑙𝑒𝑠 Buffer + Base (B) 𝑝𝐻 = 𝑝𝐾𝑎 + 𝑙𝑜𝑔 𝑏+𝐵 𝑎+𝐵 𝑝𝐻 = 𝑝𝐾𝑎 + 𝑙𝑜𝑔 𝑏𝑛𝑜. 𝑚𝑜𝑙𝑒𝑠 − 𝐵𝑛𝑜. 𝑚𝑜𝑙𝑒𝑠 𝑎𝑛𝑜. 𝑚𝑜𝑙𝑒𝑠 + 𝐵𝑛𝑜. 𝑚𝑜𝑙𝑒𝑠 Solubility Product Constant 𝐴𝑔𝐶𝑙 (𝑠) ↔ 𝐴𝑔+ (𝑎𝑞) + 𝐶𝑙 − (𝑎𝑞) 𝐾𝑠𝑝 = [𝐴𝑔+ ][𝐶𝑙− ] Kyle Michael Sy 12th Update 29 ELECTROMAGNETISM Kyle Michael Sy 12th Update 30 Electric Field General Formulas 𝐹𝑒 = 𝐸𝑞 𝐸=𝑘 𝑞 𝑟2 𝑎= Charge Densities Surface Line 𝑞𝐸 𝑚 Volume 𝜆= 𝑄 𝑙 𝜎= 𝑄 𝐴 𝜌= 𝑄 𝑉 𝜆= 𝑑𝑄 𝑑𝑙 𝜎= 𝑑𝑄 𝑑𝐴 𝜌= 𝑑𝑄 𝑑𝑉 𝑑𝑄 = 𝜆𝑑𝑙 𝑑𝑄 = 𝜎𝑑𝐴 𝑑𝑄 = 𝜌𝑑𝑉 Other Equations for Electric Field Ring with Uniform Charge 𝑞𝑥 𝐸 = 𝑘𝑒 3 (𝑥 2 + 𝑎2 )2 Rod 𝐸= 𝑘𝑒 𝑄 𝑎(𝑙 + 𝑎) Disk with Uniform Charge 𝑥 𝐸 = 2𝜋𝑘𝑒 𝜎 (1 − ) √𝑥 2 + 𝑟 2 Kyle Michael Sy Infinite Plane Disk 𝜎 𝐸= 2𝜀0 Electric Field at the Surface of a Charged Conductor 𝜎 𝐸= 𝜀0 Electric Field at the Center between Two Dipoles 𝜎 𝐸= 𝜀0 12th Update 31 Electric Flux General Formula 𝛷 = 𝐸𝐴 Gaussian Sphere (r = a) 𝑄 𝐸 = 𝑘𝑒 2 𝑎 𝛷 = 𝐸𝐴 𝑐𝑜𝑠 𝜃 Gauss’s Law a 𝛷 = ∮ 𝐸 ⋅ 𝑑𝐴 = 𝑞𝑖𝑛 𝜀0 Gaussian Sphere (r > a) 𝑞𝑖𝑛 𝐸= 𝜀0 r Conducting Sphere (r < R) 𝐸=0 a R r r Gaussian Sphere (r < a) 𝑄𝑟 𝐸 = 𝑘𝑒 3 𝑎 Conducting Sphere (r > R) 𝑞 𝐸 = 𝑘𝑒 2 𝑟 R a r Kyle Michael Sy r 12th Update 32 Sphere inside a Conducting Spherical Shell (r<a) 𝐸 = 𝑘𝑒 𝑄𝑟 𝑎2 Sphere inside a Conducting Spherical Shell (r<c, r<b) 𝑄 𝐸 = 𝑘𝑒 2 𝑟 c c b a r b a r Sphere inside a Conducting Spherical Shell (b<r<c) 𝐸=0 c b a Sphere inside a Conducting Spherical Shell (r>c) 𝑞𝑖𝑛 𝐸 = 𝑘𝑒 2 𝑟 c r b a r Electric Potential Energy General Formula 𝑄𝑞 𝑈 = 𝑘𝑒 𝑟 Electric Potential Energy with Several Point Charges 𝑛 𝑈 = 𝑘𝑒 𝑞 ∑ 𝑖=1 𝑞𝑖 𝑟𝑖 Electric Potential General Formula 𝑈 𝑉= 𝑞 Kyle Michael Sy Potential Due to a Continuous Distribution of Charge 𝑑𝑞 𝑉 = 𝑘𝑒 ∫ 𝑟 12th Update 33 Electrical Work General Formula 𝐵 𝑊 = ∫ 𝐹 ⋅ 𝑑𝑙 𝑊 = ∫|𝐹| 𝑐𝑜𝑠(𝜃) ⋅ 𝑑𝑠 𝐴 Kyle Michael Sy 12th Update 34 Capacitors and Capacitance Spherical Capacitor Potential 𝑉𝑎𝑏 = ( Capacitance 𝑄 𝑟𝑏 − 𝑟𝑎 )( ) 4𝜋𝜖0 𝑟𝑎 𝑟𝑏 𝐶 = (4𝜋𝜖0 ) ( 𝑟𝑎 𝑟𝑏 ) 𝑟𝑏 − 𝑟𝑎 Cylindrical Capacitor Potential 𝑉𝑎𝑏 = ( 𝜆 𝑟0 ) (𝑙𝑛 ) 2𝜋𝜖0 𝑟 Capacitance 2𝜋𝜖0 𝑙 𝐶= 𝑟 𝑙𝑛 𝑏 𝑟𝑎 Capacitance in a Circuit Capacitance in Series 1 1 1 1 = + + +⋯ 𝐶𝑒𝑞 𝐶1 𝐶2 𝐶3 Capacitance in Parallel 𝐶𝑒𝑞 = 𝐶1 + 𝐶2 + 𝐶3 + ⋯ Energy Stored in a Capacitor Work needed to Transfer Charge from one Plate to Another 𝑄2 𝑊= 2𝐶 Work done in Charging the Capacitor 1 𝑈 = 𝐶(∆𝑉)2 2 Dielectrics Insulators 𝐴 𝐶 = 𝜖0 𝑑 𝐶= 𝑄 𝑉 𝐾= 𝐶 𝐶0 Kyle Michael Sy 12th Update 35 Induced Charge and Polarization With Dielectric 𝜎 − 𝜎𝑖 𝐸= 𝜖0 Without Dielectric 𝜎 𝐸= 𝜖0 Charging Capacitor 1 Charging Capacitor 𝜀 𝑞 𝑖= − 𝑅 𝑅𝐶 q(t) = Q max (1 − e−τ ) Instantaneous Charge 𝑞(𝑡) = 𝑄𝑚𝑎𝑥 (1 − 1 𝑒 −𝑅𝐶 ) Instantaneous Current 𝜀 −1 𝜏(𝑡) = 𝑒 𝑅𝐶 𝑅 Discharging Capacitor 𝑞(𝑡) = 𝑄𝑖 ∙ 𝑡 − 𝑅𝐶 𝑒 𝜏(𝑡) = − 𝑄𝑖 − 𝑡 ∙ 𝑒 𝑅𝐶 𝑅𝐶 Ohm’s Law General Formula 𝑉 = 𝐼𝑅 Conductivity 𝐽 = 𝜎𝐸 Resistance and Resistivity General Formula 𝑉 𝑅= 𝐼 Resistivity 1 𝜌= 𝜎 Kyle Michael Sy 1 𝑅 = 𝜌( ) 𝐴 𝑙 𝑅 = 𝜌( ) 𝑎 12th Update 36 Resistance of a Hollow Cylinder of Silicon 𝜌 𝑟𝑏 𝑅= ∙ 𝑙𝑛 ( ) 2𝜋𝑙 𝑟𝑎 Resistance at a Given Temperature 𝑅 = 𝑅0 [1 + 𝛼(𝑇 − 𝑇0 )] Internal Resistance 𝜀 − 𝑣𝑎𝑏 𝑅= 𝐼 Temperature Coefficient for Resistivity 1 ∆𝜌 𝛼= ∙ 𝜌 ∆𝑇 Electrical Power General Formula 𝑊 𝑃= 𝑡 𝑃 = 𝐼𝑉𝑎𝑏 Power Output of a Source 𝑃 = 𝜀𝐼 − 𝐼 2 𝑟 Power Input to a Source 𝑃 = 𝜀𝐼 + 𝐼 2 𝑅 Power Input to a Pure Resistance 2 𝑉𝑎𝑏 𝑃= 𝑅 Current General Formula 𝑞 𝐼= 𝑡 Drift Velocity 𝐼 = 𝑛𝑞𝑉𝑑 𝐴 Vector Current Density 𝑛𝑞 2 𝐸 𝐽=( )𝜏 𝑚 𝐽= 𝐼 𝐴 Next Section Kyle Michael Sy 12th Update 37 GEOMETRY January 23, 2018 Kyle Michael Sy 12th Update 38 Variables and Symbols 1. A - Area 6. b - Base 2. l - Length 7. s - Slope length 3. w - Width 8. r - Radius 4. h - Height 9. d - Diagonal length 5. a - Side length Surface Area 3-Dimensional Objects Cuboid 𝐴 = 2(𝑙𝑤 + 𝑤ℎ + ℎ𝑙) Right Prism 𝐴 = 𝑏ℎ + 2𝑙𝑠 + 𝑙𝑏 Cube 𝐴 = 6𝑎2 Cylinder 𝐴 = 2𝜋𝑟(𝑟 + ℎ) Right Pyramid 2 𝑤 2 𝑙 𝐴 = 𝑙𝑤 + 𝑙 √( ) + ℎ2 + 𝑤 √( ) + ℎ2 2 2 Sphere 𝐴 = 4𝜋𝑟 2 Cone 𝐴 = 𝜋𝑟(𝑙 + 𝑟) 2-Dimensional Objects Square 𝐴 = 𝑙2 Parallelogram 𝐴 = 𝑏ℎ Rectangle 𝐴=𝑙×𝑤 Circle 𝐴 = 𝜋𝑟 2 Sector Triangle 1 A = bh 2 Kyle Michael Sy θ A=( ) πr 2 360 Trapezoid 1 𝐴 = (𝑎 + 𝑏)ℎ 2 Rhombus 1 𝐴 = 𝑑1 𝑑2 2 12th Update 39 Volume Cuboid 𝑉 = 𝑙𝑏ℎ Cylinder 𝑉 = 𝜋𝑟 2 ℎ Cube 𝑉 = 𝑎3 Right Pyramid 𝑙𝑤ℎ 𝑉= 3 Right Prism 1 𝑉 = 𝑏𝑙ℎ 2 Kyle Michael Sy Cone 1 𝑉 = 𝜋𝑟 2 ℎ 3 Sphere 4 𝑉 = 𝜋𝑟 3 3 12th Update 40 MECHANICS Kyle Michael Sy 12th Update 41 Variables and Symbols v - Velocity Fg - Gravitational force vx/vy Fe - Electric force -Velocity at a given point/height Fs - Static friction force vox/voy Fk - Kinetic friction force - Initial velocity d - Distance μs - Coefficient of static friction t - Time μk - Coefficient of kinetic friction r - Radius G - Gravitational constant a - Acceleration k - Coulomb’s constant ax - Acceleration at a point q - Electric charge x - Displacement ε0 - Permittivity of free space xo - Starting point E - Electric field g - Gravitational acceleration W - Work m - Mass PE/U F - Force KE- Kinetic energy Fn - Normal force p - Momentum - Potential energy Rectilinear Motion Constant Linear Acceleration Equation 1 𝑣𝑥 = 𝑣𝑜𝑥 + 𝑎𝑥 𝑡 Equation 3 𝑣𝑥 2 = 𝑣𝑜𝑥 2 + 2𝑎𝑥 ⋅ (𝑥 − 𝑥𝑜 ) Equation 2 Equation 4 1 𝑥 = 𝑥𝑜 + 𝑣𝑜𝑥 𝑡 + 𝑎𝑥 𝑡 2 2 𝑥 − 𝑥𝑜 = ( 𝑣𝑜𝑥 + 𝑣𝑥 )⋅𝑡 2 Velocity Kyle Michael Sy 12th Update 42 Velocity of an Object Traversing a Circular Path 2𝜋𝑟 𝑣= 𝑡 General Equation 𝑑 𝑣= 𝑡 Acceleration Instantaneous Acceleration dv dx = ⋅( ) dt dt d2 x = 2 dt Average Acceleration 𝛥𝑣 𝑎= 𝛥𝑡 Uniform Circular Acceleration Circular Acceleration 𝑣2 𝑎= 𝑟 a= 4π2 r t Rotational Motion of a Rigid Body Angular Coordinate 𝑠 𝜃= 𝑟 Constant Angular Acceleration Equation 1 𝜔𝑧 = 𝜔0𝑧 + 𝛼𝑧 𝑡 Equation 3 2 𝜔𝑧2 = 𝜔0𝑧 + 2𝛼𝑧 (𝜃 − 𝜃0 ) Equation 2 1 𝜃 = 𝜃0 + 𝜔0𝑧 𝑡 + 𝛼𝑧 𝑡 2 2 Equation 4 1 𝜃 − 𝜃0 = (𝜔0𝑧 + 𝜔𝑧 )𝑡 2 Velocity Kyle Michael Sy 12th Update 43 Instantaneous Angular Velocity 𝛥𝜃 𝑑𝜃 𝜔𝑧 = 𝑙𝑖𝑚 = 𝛥𝑡→0 𝛥𝑡 𝑑𝑡 Average Angular Velocity 𝛥𝜃 𝜔𝑎𝑣−𝑧 = 𝛥𝑡 Linear Speed of a Point 𝑣 = 𝑟𝜔 Acceleration Average Angular Acceleration 𝜔2𝑧 − 𝜔1𝑧 𝛥𝜔𝑧 𝛼𝑎𝑣−𝑧 = = 𝑡2 − 𝑡1 𝛥𝑡 [Linear] Tangential Acceleration 𝑑𝑣 𝑑𝜔 𝑎𝑡𝑎𝑛 = =𝑟 = 𝑟𝛼 𝑑𝑡 𝑑𝑡 Instantaneous Angular Acceleration 𝛥𝜔𝑧 𝑑𝜔𝑧 𝛼𝑧 = 𝑙𝑖𝑚 = 𝛥𝑡→0 𝛥𝑡 𝑑𝑡 [Linear] Centripetal Acceleration 𝑣2 𝑎𝑟𝑎𝑑 = = 𝜔2 𝑟 𝑟 Energy Rotational Kinetic Energy 1 1 𝐾 = (𝑚1 𝑟12 + 𝑚2 𝑟22 + ⋯ + 𝑚𝑖 𝑟𝑖2 )𝜔2 = (∑ 𝑚𝑖 𝑟𝑖2 ) 𝜔2 2 2 𝑖 1 𝐾 = 𝐼𝜔2 2 Gravitational Potential Energy for an Extended Body 𝑈 = 𝑀𝑔𝑦𝑐𝑚 𝑈 = (𝑚1 𝑦1 + 𝑚2 𝑦2 + ⋯ + 𝑚𝑖 𝑦𝑖 )𝑔 Moment of Inertia Standard Formula 𝐼 = 𝑚1 𝑟12 + 𝑚2 𝑟22 + ⋯ + 𝑚𝑖 𝑟𝑖2 = ∑ 𝑚𝑖 𝑟𝑖2 𝑖 Kyle Michael Sy 12th Update 44 Slender Rod, Axis through Center 1 𝐼= 𝑀𝐿2 12 Hollow Cylinder 1 𝐼 = 𝑀(𝑅12 + 𝑅22 ) 2 Slender Rod, Axis through one end 1 𝐼 = 𝑀𝐿2 3 Solid Cylinder 1 𝐼 = 𝑀𝑅2 2 Rectangular Plane, Axis through Center 1 𝐼= 𝑀(𝑎2 + 𝑏 2 ) 12 Thin-walled Hollow Cylinder 𝐼 = 𝑀𝑅2 Thin Rectangular Plane, Axis along Edge 1 𝐼 = 𝑀𝑎2 3 Solid Sphere 2 𝐼 = 𝑀𝑅2 5 Thin-walled Hollow Sphere 2 𝐼 = 𝑀𝑅2 3 Parallel Axis Theorem 𝐼𝑃 = 𝐼𝑐𝑚 + 𝑀𝑑 2 Projectile X-Component Position on the x-axis 𝑥 = 𝑥𝑜 + 𝑣𝑥𝑜 𝑡 Vertically Launched Projectile 𝑣𝑥𝑜 = 𝑣𝑜 𝑐𝑜𝑠 𝜃 Time 𝑇 = 2(𝑡𝑚𝑎𝑥 𝐻 ) Kyle Michael Sy 12th Update 45 Y-Component Time General Equations 1 𝑦 = 𝑦𝑜 + 𝑣𝑦𝑜 𝑡 + 𝑔𝑡 2 2 𝑡𝑚𝑎𝑥 𝐻 = 𝑣𝑦 2 = 𝑣𝑦𝑜 𝑡 + 2𝑔∆𝑦 𝑣𝑦 − 𝑣𝑦𝑜 𝑔 Vertically Launched Projectile 𝑣𝑦𝑜 = 𝑣𝑜 𝑠𝑖𝑛 𝜃 𝑣𝑦 = 𝑣𝑦𝑜 + 𝑔𝑡 Force General Formulas Force 𝐹 = 𝑚𝑎 Weight 𝑤 = 𝑚𝑔 Centripetal Force 𝑚𝑣 2 𝐹= 𝑟 Friction Static Friction 𝐹𝑠,𝑚𝑎𝑥 𝐹𝑛 = 𝜇𝑠 Kinetic Friction 𝐹𝑘 𝐹𝑛 = 𝜇𝑘 Charge Newton’s Universal Law of Gravitation 𝑚1 𝑚2 𝐹𝑔 = 𝐺 𝑟2 Coulomb’s Law |𝑞1 𝑞2 | 𝐹𝑒 = 𝑘 𝑟2 𝑘= 1 4𝜋𝜀0 Electric Field General Formula 𝐹𝑒 = 𝐸𝑞 Kyle Michael Sy 12th Update 46 Work and Energy General Formula 𝑊 = 𝐹𝑑 𝑊 = 𝐹𝑑 𝑐𝑜𝑠 𝜃 Kinetic Energy 1 𝐾 = 𝑚𝑣 2 2 Potential Energy 𝑈 = 𝑚𝑔ℎ Mechanical Energy 𝑀𝐸 = 𝐾 + 𝑈 Momentum General Formulas 𝑝 = 𝑚𝑣 ∆𝑝 = 𝐹∆𝑡 Kyle Michael Sy 12th Update 47 STATISTICS August 9, 2018 Kyle Michael Sy 12th Update 48 Descriptive Statistics Measures of Center Mean of a Sample ∑𝑛𝑖=1 𝑥𝑖 ̅ 𝑋= 𝑛 Mean of a Population ∑𝑁 𝑖=1 𝑥𝑖 𝜇̅ = 𝑁 Range 𝑅𝑎𝑛𝑔𝑒 = 𝑚𝑎𝑥 − 𝑚𝑖𝑛 Standard Deviation Measures of Spread Variance 𝑛 ∑𝑖=1(𝑋𝑖 − 𝑋̅)2 𝑠2 = 𝑛−1 𝑛 ∑𝑖=1(𝑋𝑖 − 𝑋̅)2 √ 𝑠= 𝑛−1 Coefficient of Variation 𝑠 𝑐𝑣 = ( ) ∙ 100% 𝑋̅ Measure of Relative Position *Section under construction* *Still Googling the formulas* Measure of Skewness Skewness (Pearson’s Second Skewness Coefficient) 3(𝑋̅ − 𝑀𝑒) 𝑆𝑘 = 𝑠 Measure of Kurtosis *Still Googling the formulas, hoping you don’t need this yet 😊* Kyle Michael Sy 12th Update 49 Sample Size Sample Size for p (Proportion) 𝑍𝛼2 ∙ 𝑝𝑞 𝑛= 2 2 𝑒 Sample Size for μ (Mean) 𝑍𝛼 ∙ 𝜎 2 𝑛=( 2 ) 𝑒 Point Estimation Point Estimator for μ1-μ2 Related Samples ∑𝑛𝑖=1 𝑑𝑖 ̅ 𝑑= 𝑛 Independent Samples 𝑥1 − 𝑥2 Other Point Estimators p1-p2 𝑝1 − 𝑝2 Interval Estimation Mean (μ) Proportion (p) 𝑥 𝑝= 𝑛 𝑛 𝑥̅ = ∑ 𝑋𝑖 𝑖=1 Confidence Interval for μ1-μ2 𝝈𝟐𝟏 and 𝝈𝟐𝟐 Known [(𝑥̅1 − 𝑥̅2 ) − 𝑒, (𝑥̅1 − 𝑥̅2 ) + 𝑒] Wherein: 𝑒 = 𝑍𝑎 √ 2 𝜎12 𝑛1 + 𝜎22 𝑛2 𝝈𝟐𝟏 and 𝝈𝟐𝟐 Unknown, and 𝒏𝟏 , 𝒏𝟐 Large 2 2 [(𝑥̅1 − 𝑥̅2 ) − 𝑒, (𝑥̅1 − 𝑥̅2 ) + 𝑒] Wherein: 𝑒 = 𝑍𝑎 √ 𝑠1 + 𝑠2 2 𝑛1 𝑛2 𝝈𝟐𝟏 and 𝝈𝟐𝟐 Unknown but Assumed Equal 1 [(𝑥̅1 − 𝑥̅2 ) − 𝑒, (𝑥̅1 − 𝑥̅2 ) + 𝑒] Wherein: 𝑒 = 𝑡𝑎(𝑛 +𝑛 −2) √𝑠𝑝2 ( + 1 2 𝑛1 2 𝑠𝑝2 = 𝝈𝟐𝟏 and Unknown and Assumed Unequal [(𝑥̅1 − 𝑥̅2 ) − 𝑒, (𝑥̅1 − 𝑥̅2 ) + 𝑒] 𝝈𝟐𝟐 Kyle Michael Sy 1 𝑛2 ) (𝑛1 −1)𝑠12 +(𝑛2 −1)𝑠22 𝑛1 +𝑛2 −2 12th Update Wherein: 𝑒 = 𝑡𝑎(𝑣) √ 2 2 𝑣= Related Samples [𝑑̅ − 𝑒, 𝑑̅ + 𝑒] Wherein: 𝑒 = 𝑡𝑎,𝑣 2 𝑆𝑑 𝑠12 𝑛1 + 𝑠22 50 𝑛2 2 2 𝑠 𝑠 ( 1+ 2) 𝑛1 𝑛2 2 2 𝑠2 𝑠2 (𝑛1 ) (𝑛2 ) 1 2 + 𝑛1 −1 𝑛2 −1 √𝑛 v = degrees of freedom = n-1 Confidence Interval for p1-p2 Sufficiently Large 𝒏𝟏 and 𝒏𝟐 [(𝑝1 − 𝑝2 ) − 𝑒, (𝑝1 − 𝑝2 ) + 𝑒] Wherein: 𝑒 = 𝑍𝑎 √ 2 𝑝̂1 𝑞̂1 𝑛1 + 𝑝̂2 𝑞̂2 𝑛2 Confidence Interval for μ σ Known [𝑥̅ − 𝑒, 𝑥̅ + 𝑒] Wherein: 𝑒 = 𝑍𝑎 2 σ Unknown and n Large [𝑥̅ − 𝑒, 𝑥̅ + 𝑒] Wherein: 𝑒 = 𝑍𝑎 2 𝜎 √𝑛 𝑠 √𝑛 σ Unknown and n Small Wherein: 𝑒 = 𝑡𝑎,𝑣 [𝑥̅ − 𝑒, 𝑥̅ + 𝑒] 2 𝑠 √𝑛 v = degrees of freedom = n-1 Confidence Interval for p Sufficiently Large n 𝑝̂𝑞̂ [𝑝 − 𝑒, 𝑝 + 𝑒] Wherein: 𝑒 = 𝑍𝑎 √ 2 𝑛 Discrete Probability Distribution Expected Value 𝑛 𝐸(𝑋) = 𝜇 = ∑ 𝑥𝑖 𝑃(𝑋 = 𝑥𝑖 ) 𝑖=1 Variance 𝑛 𝑉𝑎𝑟(𝑥) = 𝜎 2 = ∑(𝑥𝑖 − 𝜇)2 𝑃(𝑋 = 𝑥𝑖 ) 𝑖=1 Kyle Michael Sy 12th Update 51 Binomial Distribution Function 𝑃(𝑋 = 𝑥) = 𝐶𝑥𝑛 𝑝 𝑥 (1 − 𝑝)𝑛−𝑥 Variance 𝑉𝑎𝑟(𝑋) = 𝑛 ∙ 𝑝 ∙ 𝑞 Expected Value 𝐸(𝑋) = 𝑛 ∙ 𝑝 Standard Deviation 𝑆𝑑(𝑋) = √𝑛 ∙ 𝑝 ∙ 𝑞 Hypergeometric Distribution Function 𝑁−𝑘 𝐶𝑥𝑘 𝐶𝑛−𝑥 𝑃(𝑋 = 𝑥) = 𝐶𝑛𝑁 Expected Value 𝑘 𝐸(𝑥) = 𝑛 ( ) 𝑁 Variance 𝑘 𝑘 𝑁−𝑛 𝑉𝑎𝑟(𝑥) = 𝑛 ( ) (1 − ) ( ) 𝑁 𝑁 𝑁−1 Poisson Distribution Function 𝑒 −𝜆 ∙ 𝜆𝑥 𝑃(𝑋 = 𝑥) = 𝑥! Expected Value 𝐸(𝑋) = 𝜆 Variance 𝑉𝑎𝑟(𝑋) = 𝜆 Standard Deviation 𝑆𝑑(𝑋) = √𝜆 Geometric Probability Distribution Function 𝑃(𝑋 = 𝑥) = (𝑞 𝑥−1 )(𝑝) Variance 𝑉𝑎𝑟(𝑋) = 𝑞 𝑝2 Expected Value 1 𝐸(𝑋) = 𝑝 Kyle Michael Sy 12th Update 52 Negative Binomial Probability Distribution Function 𝑥−1 𝑃(𝑋 = 𝑥) = (𝐶𝑟−1 )(𝑞 𝑥−𝑟 )(𝑝𝑟 ) Variance 𝑉𝑎𝑟(𝑋) = 𝑟𝑞 𝑝2 Expected Value 𝑟 𝐸(𝑋) = 𝑝 Continuous Probability Distribution Normal Probability Distribution Hypothesis Testing Kyle Michael Sy 12th Update 53 SURVEYING February 9, 2019 Kyle Michael Sy 12th Update 54 Data Correction Tape Correction Correction per Tape Length 𝐶𝑑 = 𝑇𝐿 − 𝑁𝐿 Wherein: TL is the tape length NL is the nominal length Total Correction to be Applied 𝑀𝐿 Wherein: ML is the measured length 𝐶𝑙 = 𝐶𝑑 ( ) 𝑁𝐿 NL is the nominal length Corrected Length 𝐶𝐿 = 𝑀𝐿 ± 𝐶𝑙 𝐶𝑙 is the total correction to be applied CL is the corrected length Temperature Correction 𝐶𝑡 = 𝛼𝐿(𝑇 − 𝑇0 ) Wherein: L is the measured length. T is the observed temperature of the tape T0 is the temperature at which the tape was standardized α= 0.0000116 °𝐶 OR 𝛼 = 0.00000645 °𝐹 Tension Correction 𝐶𝑝 = 𝐿(𝑃 − 𝑃0 ) 𝑎𝐸 Wherein: L is the measured length P is the applied tension P0 is the standardized tension for the tape a is the cross-sectional area E is the elastic modulus of the steel Kyle Michael Sy 12th Update 55 Sag Correction 𝑤 2 𝐿3 𝐶𝑠 = 24𝑃2 Wherein: L is the distance between supports 2 𝐶𝑠 = 𝑊 𝐿 24𝑃2 w is the weight of the tape W is the total weight of tape between supports P is the applied tension Normal Tension 𝑎𝐸 𝑃𝑛 = 0.204 ∙ 𝑊√ 𝑃𝑛 − 𝑃0 Traverse Adjustment Compass Rule Latitude Correction 𝑑 𝑐𝑙 = 𝐶𝐿 ( ) 𝐷 Departure Correction 𝑑 𝑐𝑑 = 𝐶𝐷 ( ) 𝐷 Wherein: 𝑑 is the length of any course D is the perimeter of the traverse CL is the total closure in latitude CD is the total closure in departure Transit Rule Latitude Correction 𝐿𝑎𝑡 𝑐𝑙 = 𝐶𝐿 ( ) ∑𝑁𝐿 − ∑𝑆𝐿 Departure Correction 𝐷𝑒𝑝 𝑐𝑑 = 𝐶𝐷 ( ) ∑𝐸𝐷 − ∑𝑊𝐷 Kyle Michael Sy Wherein: Lat is the latitude of a given length Dep is the departure of a given length CL is the total closure in latitude CD is the total closure in departure 12th Update 56 Area Area by Triangle Known base and altitude 1 𝐴 = 𝑏ℎ 2 Two sides and included angle known/measured 1 𝐴 = 𝑎𝑏 𝑠𝑖𝑛 𝛼 2 Three sides known/measured 𝐴 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) 1 𝑠 = (𝑎 + 𝑏 + 𝑐) 2 Double Meridian Distance (DMD) Double Area 2𝐴 = 𝐷𝑀𝐷 × 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐿𝑎𝑡𝑖𝑡𝑢𝑑𝑒 Double Parallel Distance Double Area 2𝐴 = 𝐷𝑃𝐷 × 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 Area 1 𝐴 = (𝐷𝑀𝐷 × 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐿𝑎𝑡𝑖𝑡𝑢𝑑𝑒) 2 Area 1 𝐴 = (𝐷𝑃𝐷 2 × 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒) Trapezoidal Rule ℎ1 + ℎ𝑛 𝐴 = 𝑑[ + ℎ2 + ℎ3 + ⋯ + ℎ𝑛−1 ] 2 Kyle Michael Sy 12th Update 57 Simpson’s One-third Rule When n is odd 𝑑 𝐴= [(ℎ1 + ℎ𝑛 ) + 2(ℎ3 + ℎ5 + ℎ7 + ⋯ + ℎ𝑛−2 ) + 4(ℎ2 + ℎ4 + ℎ6 + ⋯ + ℎ𝑛−1 ) 3 When n is even 𝑑 𝐴= [(ℎ1 + ℎ𝑛−1 ) + 2(ℎ3 + ℎ5 + ℎ7 + ⋯ + ℎ𝑛−3 ) + 4(ℎ2 + ℎ4 + ℎ6 + ⋯ + ℎ𝑛−2 )] 3 + ℎ1 + ℎ𝑛−1 𝑑 2 Coordinate Method 𝐴= 1 𝑥1 × [𝑦 1 2 𝑥2 𝑦2 𝑥3 𝑥𝑛 … 𝑦3 𝑦1 𝑥1 𝑦1 ] Leveling Curvature and Refraction Note that K is in kilometers and h is in meters. Curvature Height ℎ𝑐 = 0.0675𝐾 2 Kyle Michael Sy Curvature and Refraction Height ℎ𝑐𝑟 = 0.0785𝐾 2 12th Update 58 Reciprocal Leveling Mean Diff. in Elev. at Left 𝐷𝐸𝐴 = 𝑎 − 𝑏 Mean Diff. in Elev. at Right 𝐷𝐸𝐵 = 𝑎′ − 𝑏′ True Mean Diff. in Elev. 𝐷𝐸𝐴 + 𝐷𝐸𝐵 𝑇𝐷𝐸 = 2 Elevation of Benchmark 2 𝐸𝑙𝑒𝑣. 𝐵𝑀2 = 𝐸𝑙𝑒𝑣. 𝐵𝑀1 ± 𝑇𝐷𝐸 Differential Leveling Height of Instrument 𝐻𝐼 = 𝐸𝑙𝑒𝑣. 𝐵𝑀𝑎 + 𝐵𝑆 Elevation of the Turning Point 𝐸𝑙𝑒𝑣. 𝑇𝑃1 = 𝐻𝐼 − 𝐹𝑆 Wherein: HI is the height of the instrument BM is the benchmark BS is the backsight FS is the foresight Trigonometric Leveling Vertical Distance 𝑉 = 𝑑 𝑡𝑎𝑛 𝛼 𝑉 = 𝑠 𝑠𝑖𝑛 𝛼 Kyle Michael Sy 12th Update 59 Upward Line of Sight 𝐷𝐸𝑎𝑏 = 𝑉 + 𝐻𝐼 Without curvature (hcr = 0) − 𝑅𝑅 + ℎ𝑐𝑟 With curvature (hcr ≠ 0) 𝐸𝑙𝑒𝑣. 𝐴 = 𝐸𝑙𝑒𝑣. 𝐵 + 𝑅𝑅 − 𝑉 − ℎ𝑐𝑟 − 𝐻𝐼 𝐸𝑙𝑒𝑣. 𝐵 = 𝐸𝑙𝑒𝑣. 𝐴 + 𝐻𝐼 + 𝑉 + ℎ𝑐𝑟 − 𝑅𝑅 Wherein: HI is the instrument height hcr is the effect of curvature and refraction RR is the rod reading V/VD is the vertical dist. from the horizontal to the line of sight Kyle Michael Sy 12th Update 60 Downward Line of Sight Without curvature (hcr = 0) With curvature (hcr ≠ 0) 𝐷𝐸𝑎𝑏 = 𝑉 − 𝐻𝐼 − 𝑅𝑅 − ℎ𝑐𝑟 𝐸𝑙𝑒𝑣. 𝐴 = 𝐸𝑙𝑒𝑣. 𝐵 + 𝑅𝑅 + 𝑉 − 𝐻𝐼 − ℎ𝑐𝑟 𝐸𝑙𝑒𝑣. 𝐵 = 𝐸𝑙𝑒𝑣. 𝐴 + 𝐻𝐼 + ℎ𝑐𝑟 − 𝑉 − 𝑅𝑅 Wherein: HI is the instrument height hcr is the effect of curvature and refraction RR is the rod reading V/VD is the vertical dist. from the horizontal to the line of sight Kyle Michael Sy 12th Update 61 Stadia Leveling Horizontal Sights 𝑑 𝑓 = 𝑠 𝑖 𝐾= 𝑓 𝑖 Wherein: c is the distance from the instrument center to the objective lens center C = 0.0m for internal, C = 0.3m for external focusing telescope 𝐶 =𝑓+𝑐 d is the distance from the focal point to the face of the rod 𝐷 =𝐶+𝑑 D is the distance from the instrument center to the face of the rod 𝐷 =𝐾×𝑆+𝐶 f is the focal length i is the spacing between stadia hairs K is the stadia constant S is the stadia intercept/interval Kyle Michael Sy 12th Update 62 Inclined Sights 𝐻 = 𝐾 × 𝑆 𝑐𝑜𝑠 2 𝛼 + 𝐶 𝑐𝑜𝑠 𝛼 𝑉 = 𝐾 × 𝑆 𝑐𝑜𝑠 𝛼 𝑠𝑖𝑛 𝛼 + 𝐶 𝑠𝑖𝑛 𝛼 1 𝑉 = 𝐾 × 𝑆 𝑠𝑖𝑛 2𝛼 + 𝐶 𝑠𝑖𝑛 𝛼 2 𝐷 = 𝐾 × 𝑆 𝑐𝑜𝑠 𝛼 + 𝐶 Wherein: D is the line of sight from the instrument to the rod C = 0.0m for internal, C = 0.3m for external focusing telescope H/HD is the horizontal distance K is the stadia constant S is the stadia intercept/interval V/VD is the vertical distance α is the angle of the inclined stadia Kyle Michael Sy 12th Update 63 Simple Curve Degree of Curve (D) Arc Basis (Metric) 20 2𝜋𝑅 = 𝐷 360° 𝐷= 1145.916 𝑅 Arc Basis (English) 5(20) 2𝜋𝑅 = 𝐷 360° 𝐷= 5(1145.916) 𝑅 Chord Basis (Metric) 𝐷 10 𝑠𝑖𝑛 = 2 𝑅 𝑅= 10 𝐷 𝑠𝑖𝑛 2 Chord Basis (English) 𝐷 50 𝑠𝑖𝑛 = 2 𝑅 𝑅= 50 𝐷 𝑠𝑖𝑛 2 Kyle Michael Sy 12th Update 64 Tangent Distance (T) 𝑡𝑎𝑛 𝛥 𝐼 𝑇 = 𝑡𝑎𝑛 = 2 2 𝑅 𝑇 = 𝑅 𝑡𝑎𝑛 𝐼 2 Long Chord (LC) 𝐿𝐶 𝐼 𝑠𝑖𝑛 = 2 2 𝑅 𝐿𝐶 = 2𝑅 𝑠𝑖𝑛 𝐼 2 Subchord (SC) 𝑆𝐶 𝜃 𝑠𝑖𝑛 = 2 2 𝑅 𝑆𝐶 = 2𝑅 𝑠𝑖𝑛 𝜃 2 Length of Curve (Lc) From Arc Definition, 𝜋 𝐿𝑐 = 𝑅𝛥 ( ) 180° Metric Lc 20 = 𝐼 𝐷 𝐼 Lc = 20 ( ) 𝐷 Kyle Michael Sy 12th Update 65 English 𝐿𝑐 100 = 𝐼 𝐷 𝐼 𝐿𝑐 = 100 ( ) 𝐷 External Distance (E) 𝐸 = 𝑂𝑃𝐼 − 𝑅 𝑂𝑃𝐼 = 𝑅 𝑠𝑒𝑐 𝐼 2 𝐼 𝐸 = 𝑅 𝑠𝑒𝑐 − 𝑅 2 𝐼 𝐸 = 𝑅(𝑠𝑒𝑐 − 1) 2 Middle Ordinate 𝑀 = 𝑅 − 𝑂𝐹 𝑀 = 𝑅 − 𝑅 𝑐𝑜𝑠 𝐼 2 𝐼 𝑀 = 𝑅(1 − 𝑐𝑜𝑠 ) 2 Kyle Michael Sy 12th Update 66 Stationing of Point of Curvature If STA PI is known 𝑆𝑇𝐴 𝑃𝐶 = 𝑆𝑇𝐴 𝑃𝐼 − 𝑇 IF STA PT is known 𝑆𝑇𝐴 𝑃𝐶 = 𝑆𝑇𝐴 𝑃𝑇 − 𝐿𝑐 Stationing of Point of Tangency If STA PC is known 𝑆𝑇𝐴 𝑃𝑇 = 𝑆𝑇𝐴 𝑃𝐶 + 𝐿𝑐 IF STA PI is known 𝑆𝑇𝐴 𝑃𝑇 = (𝑆𝑇𝐴 𝑃𝐼 − 𝑇) + 𝐿𝑐 Stationing of Point of Intersection If STA PC is known 𝑆𝑇𝐴 𝑃𝐼 = 𝑆𝑇𝐴 𝑃𝐶 + 𝑇 IF STA PT is known 𝑆𝑇𝐴 𝑃𝐼 = (𝑆𝑇𝐴 𝑃𝑇 − 𝐿𝑐 ) + 𝑇 Kyle Michael Sy 12th Update 67 Compound Curve If Common Tangent is not Parallel to the Long Chord Triangle PC-V-PT Triangle PC-PCC-PT Triangle V1-V-V2 Kyle Michael Sy 12th Update 68 If Common Tangent is Parallel to Long Chord Kyle Michael Sy 12th Update 69 Spiral Curve Elements of a Spiral Curve TS: Point of change from tangent to spiral SC: Point of change from spiral to circle CS: Point of change from circle to spiral ST: Point of change from spiral to tangent L: Spiral arc length from TS to any point on the spiral Lc: Total length of spiral from TS to SC Sc: Central angle of spiral (from TS to SC) Kyle Michael Sy 12th Update 70 S: The spiral angle from TS to any point on the spiral i: Spiral deflection angle at the TS from initial tangent to any point on the spiral D: Degree of curve of the spiral at any point, and R = its radius Dc: Degree of curve of the shifted circle to which the spiral becomes tangent at the SC, and R-c the radius of the circle I: Total central angle of the circular curve Ic: Central angle of circular arc of Lc extending from the SC to the CS xc: Tangent offset of the SC with reference to the TS and the initial tangent x: Tangent offset yc: Tangent distance for the SC y: Tangent distance q: Distance along tangent to the point perpendicular to the PC of the shifted curve p: Offset from the initial tangent to the PC of the shifted circular curve or throw Ts: Total tangent distance = distance from PI to TS or ST Es: T otal external distance = distance from PI to midpoint of curve Rc: Radius of simple curve R: Radius of spiral at any point e: Superelevation k: Velocity of vehicle in kph v: Velocity of vehicle Kyle Michael Sy 12th Update 71 Properties of Spiral Curves At the end of the spiral adjacent to the tangent, the radius of the spiral is large; along the curve it decreases gradually until at the point where the spiral joins the circular curve, the radii of the curves are equal, hence, the radius of the spiral varies inversely proportional to the radius of the circular curve. 𝑅 𝐿𝑐 = 𝑅𝑐 𝐿 The spiral angle varies as the squares of the lengths along the spiral. 𝑆 𝐿 2 =( ) 𝑆𝑐 𝐿𝑐 The tangent offset varies as the cubes of the lengths along the spiral. 𝑥 𝐿 3 =( ) 𝑥𝑐 𝐿𝑐 The deflection angle varies as the squares of the lengths along the spiral. 𝑖 𝐿 2 =( ) 𝑖𝑐 𝐿𝑐 Formulas Superelevation 0.0079𝑘 2 ∙ 𝑊 𝑒= 𝑅 Desirable Length of Spiral 0.036𝑘 3 𝐿𝑐 = 𝑅 Radius of Spiral 1145.916𝐿𝑐 𝑅= 𝐷𝑐 𝐿 Kyle Michael Sy Spiral Angle 𝑆 = 𝐿2 /2𝑅𝑐 𝐿𝑐 Spiral Angle at the SC 𝐿𝑐 𝑆𝑐 = 2𝑅𝑐 Tangent Offset 𝐿3 𝑥= 6𝑅𝑐 𝐿𝑐 12th Update 72 Tangent Offset at the SC 𝐿2 𝑥𝑐 = 6𝑅𝑐 Distance Along Tangent at the SC Deflection Angle 1 𝑖= 𝑆 3 Angle of Intersection 𝐼 = 𝐼𝑐 + 2𝑆𝑐 Deflection Angle at the SC 1 𝑖𝑐 = 𝑆𝑐 3 𝐿3𝑐 𝑦𝑐 = 𝐿𝑐 − 40𝑅𝑐2 Length of Ghost Curve 1 𝑄 = 𝐿𝑐 2 Throw Distance Along Tangent 𝐿5 𝑦=𝐿− 40𝑅𝑐2 𝐿2𝑐 External Distance 𝐼 𝐸𝑠 = (𝑅𝑐 + 𝑝) 𝑠𝑒𝑐 − 𝑅𝑐 2 𝑥𝑐 𝐿2𝑐 𝑝= = 4 24𝑅𝑐 Tangent Distance 𝐿𝑐 𝐼 𝑇𝑠 = + (𝑅𝑐 + 𝑝) 𝑡𝑎𝑛 2 2 Earthworks Engineering Volume Computation End Area Method 𝐴1 + 𝐴2 𝑉=( )𝐿 2 Prismoidal Formula 𝐿 𝑉 = (𝐴1 + 4𝐴𝑚 + 𝐴2 ) 6 Prismoidal Correction 𝑉 = 𝑉𝐸 − 𝑉𝑐𝑝 𝑉𝑐𝑝 = 𝐿 (𝐶 − 𝐶2 )(𝐷1 − 𝐷2 ) 12 1 Volume of Regular Prism 𝑎+𝑏+𝑐+𝑑 𝑉 = 𝐴( ) 4 Assembly of Regular Prism 𝐴 𝑉 = [∑ℎ1 + 2∑ℎ2 + 3∑ℎ3 + 4∑ℎ4 ] 4 Kyle Michael Sy 12th Update 73 Truncated Prism 𝑎+𝑏+𝑐 𝑉 = 𝐴( ) 3 Kyle Michael Sy 12th Update 74 CONSTANTS January 23, 2018 Kyle Michael Sy 12th Update 75 Euler’s Number 𝑒 = 2.718 Pi Coulomb’s Constant/Electrostatic Constant 𝑁 ⋅ 𝑚2 𝑘𝑒 = 8.987 × 109 𝐶2 𝜋 = 3.142 Gravitational Acceleration 𝑚 𝑔 = 9.807 2 𝑠 Permittivity of Free Space 𝐶2 −12 𝜀0 = 8.854 × 10 𝑁 ⋅ 𝑚2 Elementary Charge 𝑞 = 1.602 × 10−19 𝐶 Gravitational Constant 𝑁 ⋅ 𝑚2 −11 𝐺 = 6.67 × 10 𝑘𝑔2 Mass of a Proton 𝑚𝑝 = 1.673 × 10−27 𝑘𝑔 Electron-volt 𝑒𝑉 = 1.602 × 10−19 𝐽 Mass of an Electron 𝑚𝑒 = 9.109 × 10−31 𝑘𝑔 Speed of Light Mass of a Neutron 𝑚𝑛 = 1.675 × 10−27 𝑘𝑔 𝑐 = 2.998 × 108 𝑚 𝑠2 Faraday’s Constant ℱ = 9.649 × 104 Kyle Michael Sy 𝐶 𝑚𝑜𝑙 12th Update 76 TABLES August 17, 2018 Kyle Michael Sy 12th Update 77 Metric Prefixes and Symbols Prefix Symbol Factor yotta Y 1,000,000,000,000,000,000,000,000 Scientific 1024 zetta Z 1,000,000,000,000,000,000,000 1021 exa E 1,000,000,000,000,000,000 1018 peta P 1,000,000,000,000,000 1015 tera T 1,000,000,000,000 1012 giga G 1,000,000,000 109 mega M 1,000,000 106 kilo k 1,000 103 hecto h 100 102 deka da 10 101 1 100 deci d 0.1 10-1 centi c 0.01 10-2 milli m 0.001 10-3 micro μ 0.000001 10-6 nano n 0.000000001 10-9 pico p 0.000000000001 10-12 femto f 0.000000000000001 10-15 atto a 0.000000000000000001 10-18 zepto z 0.000000000000000000001 10-21 yocto y 0.000000000000000000000001 10-24 Kyle Michael Sy 9th Update 78 Mass Kilogram Pound Stone Quarter Hundredweight Ton 1 2.2046 0.1575 0.0787 0.0197 0.0011 1 pound = 0.4536 1 0.0714 0.0357 0.0089 0.0004 1 stone = 6.3503 14 1 0.5 0.125 0.0063 1 quarter = 12.7006 28 2 1 0.25 0.0125 1 hundredweight = 50.8024 112 8 4 1 0.05 1,016.0469 2240 160 80 20 1 1 kilogram = 1 ton = Length Meter Inch Foot Yard Chain Furlong Mile League 1 39.3701 3.2808 1.0936 0.0497 0.0050 0.0006 0.0002 1 inch = 0.0254 1 0.0833 0.0278 0.0013 0.0001 1.6e-5 4.6e-6 1 foot = 0.3048 12 1 0.3333 0.0152 0.0015 0.0002 5.5e-5 1 yard = 0.9144 36 3 1 0.0455 0.0045 0.0006 0.0002 1 chain = 20.1168 792 66 22 1 0.1000 0.0125 0.0036 1 furlong = 201.168 7920.02 660.001 220 10 1 0.125 0.0362 1 mile = 1609.34 63360 1760 80 8.0000 1 0.2897 1 league = 5556 218740 1 fathom = 1.8288 72 6 2 0.0909 0.0091 0.0011 0.0003 1 naut. mi. = 1852 7.3e4 6.1e3 2.03e3 92.0624 9.2062 1.1508 0.3333 5.0292 198 16.5 5.5 0.25 0.025 0.0031 0.0009 1 meter = 1 rod = Kyle Michael Sy 5280 18228.3 6076.12 276.187 27.6187 3.4523 1 9th Update 79 Volume ml l fl. oz. pt qt gal in3 1 0.001 0.0338 0.0021 0.0011 0.0002 0.0610 1000 1 33.8140 2.1134 1.0567 0.2641 61.0237 1 fluid ounce = 29.5735 0.0296 1 0.0625 0.0313 0.0078 1.8047 1 pint = 473.1765 0.4732 16 1 0.5 0.125 28.875 1 quart = 946.3529 0.9464 32 2 1 0.25 57.75 1 gallon = 3785.4118 3.7854 128 8 4 1 231 16.3871 0.0164 0.5541 0.0346 0.0173 0.0043 1 1 milliliter = 1 liter = 1 in3 = Temperature Celsius Fahrenheit Kelvin °C = 1 5 ([°F] − 32) 9 [K] − 273.15 °F = 9 [°C] + 32 5 1 K= [°C] + 273.15 °R = ([°C] + 273.15) × °Ré = [°C] × 4 5 [K] × ([K] + 459.67) × 9 5 [°F] + 459.67 ([°F] − 32) × 4 9 9 5 Rankine ([°R] − 491.67) × 9 − 459.67 5 1 [K] × ([K] − 273.15) × 5 9 [°R] − 459.67 [°R] × 9 5 Réaumur [°Ré] × 5 9 1 4 5 ([°R] − 491.67) × [°Ré] × 4 9 5 4 9 + 32 4 [°Ré] × 5 + 273.15 4 [°Ré] × 9 + 491.67 4 1 Truth Table Kyle Michael Sy 9th Update

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