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University Physics Volume 1

E | Mathematical Formulas

University Physics Volume 1E | Mathematical Formulas
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  1. Preface
  2. Unit 1. Mechanics
    1. 1 Units and Measurement
      1. Introduction
      2. 1.1 The Scope and Scale of Physics
      3. 1.2 Units and Standards
      4. 1.3 Unit Conversion
      5. 1.4 Dimensional Analysis
      6. 1.5 Estimates and Fermi Calculations
      7. 1.6 Significant Figures
      8. 1.7 Solving Problems in Physics
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Vectors
      1. Introduction
      2. 2.1 Scalars and Vectors
      3. 2.2 Coordinate Systems and Components of a Vector
      4. 2.3 Algebra of Vectors
      5. 2.4 Products of Vectors
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 Motion Along a Straight Line
      1. Introduction
      2. 3.1 Position, Displacement, and Average Velocity
      3. 3.2 Instantaneous Velocity and Speed
      4. 3.3 Average and Instantaneous Acceleration
      5. 3.4 Motion with Constant Acceleration
      6. 3.5 Free Fall
      7. 3.6 Finding Velocity and Displacement from Acceleration
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Motion in Two and Three Dimensions
      1. Introduction
      2. 4.1 Displacement and Velocity Vectors
      3. 4.2 Acceleration Vector
      4. 4.3 Projectile Motion
      5. 4.4 Uniform Circular Motion
      6. 4.5 Relative Motion in One and Two Dimensions
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 5 Newton's Laws of Motion
      1. Introduction
      2. 5.1 Forces
      3. 5.2 Newton's First Law
      4. 5.3 Newton's Second Law
      5. 5.4 Mass and Weight
      6. 5.5 Newton’s Third Law
      7. 5.6 Common Forces
      8. 5.7 Drawing Free-Body Diagrams
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 6 Applications of Newton's Laws
      1. Introduction
      2. 6.1 Solving Problems with Newton’s Laws
      3. 6.2 Friction
      4. 6.3 Centripetal Force
      5. 6.4 Drag Force and Terminal Speed
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 7 Work and Kinetic Energy
      1. Introduction
      2. 7.1 Work
      3. 7.2 Kinetic Energy
      4. 7.3 Work-Energy Theorem
      5. 7.4 Power
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 8 Potential Energy and Conservation of Energy
      1. Introduction
      2. 8.1 Potential Energy of a System
      3. 8.2 Conservative and Non-Conservative Forces
      4. 8.3 Conservation of Energy
      5. 8.4 Potential Energy Diagrams and Stability
      6. 8.5 Sources of Energy
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    9. 9 Linear Momentum and Collisions
      1. Introduction
      2. 9.1 Linear Momentum
      3. 9.2 Impulse and Collisions
      4. 9.3 Conservation of Linear Momentum
      5. 9.4 Types of Collisions
      6. 9.5 Collisions in Multiple Dimensions
      7. 9.6 Center of Mass
      8. 9.7 Rocket Propulsion
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 10 Fixed-Axis Rotation
      1. Introduction
      2. 10.1 Rotational Variables
      3. 10.2 Rotation with Constant Angular Acceleration
      4. 10.3 Relating Angular and Translational Quantities
      5. 10.4 Moment of Inertia and Rotational Kinetic Energy
      6. 10.5 Calculating Moments of Inertia
      7. 10.6 Torque
      8. 10.7 Newton’s Second Law for Rotation
      9. 10.8 Work and Power for Rotational Motion
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 11 Angular Momentum
      1. Introduction
      2. 11.1 Rolling Motion
      3. 11.2 Angular Momentum
      4. 11.3 Conservation of Angular Momentum
      5. 11.4 Precession of a Gyroscope
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 12 Static Equilibrium and Elasticity
      1. Introduction
      2. 12.1 Conditions for Static Equilibrium
      3. 12.2 Examples of Static Equilibrium
      4. 12.3 Stress, Strain, and Elastic Modulus
      5. 12.4 Elasticity and Plasticity
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    13. 13 Gravitation
      1. Introduction
      2. 13.1 Newton's Law of Universal Gravitation
      3. 13.2 Gravitation Near Earth's Surface
      4. 13.3 Gravitational Potential Energy and Total Energy
      5. 13.4 Satellite Orbits and Energy
      6. 13.5 Kepler's Laws of Planetary Motion
      7. 13.6 Tidal Forces
      8. 13.7 Einstein's Theory of Gravity
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    14. 14 Fluid Mechanics
      1. Introduction
      2. 14.1 Fluids, Density, and Pressure
      3. 14.2 Measuring Pressure
      4. 14.3 Pascal's Principle and Hydraulics
      5. 14.4 Archimedes’ Principle and Buoyancy
      6. 14.5 Fluid Dynamics
      7. 14.6 Bernoulli’s Equation
      8. 14.7 Viscosity and Turbulence
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Waves and Acoustics
    1. 15 Oscillations
      1. Introduction
      2. 15.1 Simple Harmonic Motion
      3. 15.2 Energy in Simple Harmonic Motion
      4. 15.3 Comparing Simple Harmonic Motion and Circular Motion
      5. 15.4 Pendulums
      6. 15.5 Damped Oscillations
      7. 15.6 Forced Oscillations
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 16 Waves
      1. Introduction
      2. 16.1 Traveling Waves
      3. 16.2 Mathematics of Waves
      4. 16.3 Wave Speed on a Stretched String
      5. 16.4 Energy and Power of a Wave
      6. 16.5 Interference of Waves
      7. 16.6 Standing Waves and Resonance
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 17 Sound
      1. Introduction
      2. 17.1 Sound Waves
      3. 17.2 Speed of Sound
      4. 17.3 Sound Intensity
      5. 17.4 Normal Modes of a Standing Sound Wave
      6. 17.5 Sources of Musical Sound
      7. 17.6 Beats
      8. 17.7 The Doppler Effect
      9. 17.8 Shock Waves
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
    17. Chapter 17
  12. Index

Quadratic formula

If ax2+bx+c=0,ax2+bx+c=0, then x=b±b24ac2ax=b±b24ac2a

Triangle of base bb and height hh Area =12bh=12bh
Circle of radius rr Circumference =2πr=2πr Area =πr2=πr2
Sphere of radius rr Surface area =4πr2=4πr2 Volume =43πr3=43πr3
Cylinder of radius rr and height hh Area of curved surface =2πrh=2πrh Volume =πr2h=πr2h
Table E1 Geometry

Trigonometry

Trigonometric Identities

  1. sinθ=1/cscθsinθ=1/cscθ
  2. cosθ=1/secθcosθ=1/secθ
  3. tanθ=1/cotθtanθ=1/cotθ
  4. sin(900θ)=cosθsin(900θ)=cosθ
  5. cos(900θ)=sinθcos(900θ)=sinθ
  6. tan(900θ)=cotθtan(900θ)=cotθ
  7. sin2θ+cos2θ=1sin2θ+cos2θ=1
  8. sec2θtan2θ=1sec2θtan2θ=1
  9. tanθ=sinθ/cosθtanθ=sinθ/cosθ
  10. sin(α±β)=sinαcosβ±cosαsinβsin(α±β)=sinαcosβ±cosαsinβ
  11. cos(α±β)=cosαcosβsinαsinβcos(α±β)=cosαcosβsinαsinβ
  12. tan(α±β)=tanα±tanβ1tanαtanβtan(α±β)=tanα±tanβ1tanαtanβ
  13. sin2θ=2sinθcosθsin2θ=2sinθcosθ
  14. cos2θ=cos2θsin2θ=2cos2θ1=12sin2θcos2θ=cos2θsin2θ=2cos2θ1=12sin2θ
  15. sinα+sinβ=2sin12(α+β)cos12(αβ)sinα+sinβ=2sin12(α+β)cos12(αβ)
  16. cosα+cosβ=2cos12(α+β)cos12(αβ)cosα+cosβ=2cos12(α+β)cos12(αβ)

Triangles

  1. Law of sines: asinα=bsinβ=csinγasinα=bsinβ=csinγ
  2. Law of cosines: c2=a2+b22abcosγc2=a2+b22abcosγ
    Figure shows a triangle with three dissimilar sides labeled a, b and c. All three angles of the triangle are acute angles. The angle between b and c is alpha, the angle between a and c is beta and the angle between a and b is gamma.
  3. Pythagorean theorem: a2+b2=c2a2+b2=c2
    Figure shows a right triangle. Its three sides are labeled a, b and c with c being the hypotenuse. The angle between a and c is labeled theta.

Series expansions

  1. Binomial theorem: (a+b)n=an+nan1b+n(n1)an2b22!+n(n1)(n2)an3b33!+···(a+b)n=an+nan1b+n(n1)an2b22!+n(n1)(n2)an3b33!+···
  2. (1±x)n=1±nx1!+n(n1)x22!±···(x2<1)(1±x)n=1±nx1!+n(n1)x22!±···(x2<1)
  3. (1±x)n=1nx1!+n(n+1)x22!···(x2<1)(1±x)n=1nx1!+n(n+1)x22!···(x2<1)
  4. sinx=xx33!+x55!···sinx=xx33!+x55!···
  5. cosx=1x22!+x44!···cosx=1x22!+x44!···
  6. tanx=x+x33+2x515+···tanx=x+x33+2x515+···
  7. ex=1+x+x22!+···ex=1+x+x22!+···
  8. ln(1+x)=x12x2+13x3···(|x|<1)ln(1+x)=x12x2+13x3···(|x|<1)

Derivatives

  1. ddx[af(x)]=addxf(x)ddx[af(x)]=addxf(x)
  2. ddx[f(x)+g(x)]=ddxf(x)+ddxg(x)ddx[f(x)+g(x)]=ddxf(x)+ddxg(x)
  3. ddx[f(x)g(x)]=f(x)ddxg(x)+g(x)ddxf(x)ddx[f(x)g(x)]=f(x)ddxg(x)+g(x)ddxf(x)
  4. ddxf(u)=[dduf(u)]dudxddxf(u)=[dduf(u)]dudx
  5. ddxxm=mxm1ddxxm=mxm1
  6. ddxsinx=cosxddxsinx=cosx
  7. ddxcosx=sinxddxcosx=sinx
  8. ddxtanx=sec2xddxtanx=sec2x
  9. ddxcotx=csc2xddxcotx=csc2x
  10. ddxsecx=tanxsecxddxsecx=tanxsecx
  11. ddxcscx=cotxcscxddxcscx=cotxcscx
  12. ddxex=exddxex=ex
  13. ddxlnx=1xddxlnx=1x
  14. ddxsin−1x=11x2ddxsin−1x=11x2
  15. ddxcos−1x=11x2ddxcos−1x=11x2
  16. ddxtan−1x=11+x2ddxtan−1x=11+x2

Integrals

  1. af(x)dx=af(x)dxaf(x)dx=af(x)dx
  2. [f(x)+g(x)]dx=f(x)dx+g(x)dx[f(x)+g(x)]dx=f(x)dx+g(x)dx
  3. xmdx=xm+1m+1(m1)=lnx(m=−1)xmdx=xm+1m+1(m1)=lnx(m=−1)
  4. sinxdx=cosxsinxdx=cosx
  5. cosxdx=sinxcosxdx=sinx
  6. tanxdx=ln|secx|tanxdx=ln|secx|
  7. sin2axdx=x2sin2ax4asin2axdx=x2sin2ax4a
  8. cos2axdx=x2+sin2ax4acos2axdx=x2+sin2ax4a
  9. sinaxcosaxdx=cos2ax4asinaxcosaxdx=cos2ax4a
  10. eaxdx=1aeaxeaxdx=1aeax
  11. xeaxdx=eaxa2(ax1)xeaxdx=eaxa2(ax1)
  12. lnaxdx=xlnaxxlnaxdx=xlnaxx
  13. dxa2+x2=1atan−1xadxa2+x2=1atan−1xa
  14. dxa2x2=12aln|x+axa|dxa2x2=12aln|x+axa|
  15. dxa2+x2=sinh−1xadxa2+x2=sinh−1xa
  16. dxa2x2=sin−1xadxa2x2=sin−1xa
  17. a2+x2dx=x2a2+x2+a22sinh−1xaa2+x2dx=x2a2+x2+a22sinh−1xa
  18. a2x2dx=x2a2x2+a22sin−1xaa2x2dx=x2a2x2+a22sin−1xa
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