Elementary Physics, Mechanics, and Acoustics

This appendix derives some basic results from the field of physics, particularly mechanics and acoustics, which are referenced elsewhere in this book, or which are known to be needed in practical synthesis models.

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Subsections

• Newton's Laws of Motion
  • Newton's Three Laws of Motion
  • Mass
  • Gravitational Force
  • Hooke's Law
  • Applying Newton's Laws of Motion
  
  
• Work and Energy
  • Potential Energy in a Spring
  • Kinetic Energy of a Mass
  • Mass Kinetic Energy from Virtual Work
  • Energy in the Mass-Spring Oscillator
  • Energy Conservation
  • Energy Conservation in the Mass-Spring System
  
  
• Momentum
  • Conservation of Momentum
  
  
• Rigid-Body Dynamics
  • Center of Mass
    • Linear Momentum of the Center of Mass
    • Whoops, No Angular Momentum!
  • Translational Kinetic Energy
  • Rotational Kinetic Energy
  • Mass Moment of Inertia
    • Circular Disk Rotating in Its Own Plane
    • Circular Disk Rotating About Its Diameter
  • Perpendicular Axis Theorem
  • Parallel Axis Theorem
  • Stretch Rule
  • Area Moment of Inertia
  • Radius of Gyration
    • Rectangular Cross-Section
    • Circular Cross-Section
  • Two Masses Connected by a Rod
    • Striking the Rod in the Middle
    • Striking One of the Masses
  • Angular Velocity Vector
  • Vector Cross Product
    • Cross-Product Magnitude
    • Mass Moment of Inertia as a Cross Product
    • Tangential Velocity as a Cross Product
  • Angular Momentum
    • Relation of Angular to Linear Momentum
  • Angular Momentum Vector
    • Angular Momentum Vector in Matrix Form
  • Mass Moment of Inertia Tensor
    • Simple Example
    • Example with Coupled Rotations
    • Off-Diagonal Terms in Moment of Inertia Tensor
  • Principal Axes of Rotation
    • Positive Definiteness of the Moment of Inertia Tensor
  • Rotational Kinetic Energy Revisited
  • Torque
  • Newton's Second Law for Rotations
  • Equations of Motion for Rigid Bodies
    • Body-Fixed and Space-Fixed Frames of Reference
    • Angular Motion in the Space-Fixed Frame
    • Euler's Equations for Rotations in the Body-Fixed Frame
    • Examples
  
  
• Properties of Elastic Solids
  • Young's Modulus
    • Young's Modulus as a Spring Constant
  • String Tension
  
  
• Wave Equation for the Vibrating String
  • Non-Stiff String
  • Wave Momentum
  
  
• Properties of Gases
  • Particle Velocity of a Gas
  • Volume Velocity of a Gas
  • Pressure is Confined Kinetic Energy
  • Bernoulli Equation
  • Bernoulli Effect
  • Air Jets
  • Acoustic Intensity
  • Acoustic Energy Density
  • Energy Decay through Lossy Boundaries
  • Ideal Gas Law
  • Isothermal versus Isentropic
  • Adiabatic Gas Constant
  • Heat Capacity of Ideal Gases
  • Speed of Sound in Air
  • Air Absorption
  
  
• Wave Equation in Higher Dimensions
  • Plane Waves in Air
  • Vector Wavenumber
  • Solving the 2D Wave Equation
  • 2D Boundary Conditions
  • 3D Sound

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Previous: Summary
Next: Newton's Laws of Motion

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.