Classical Mechanics with Numerical Methods

Timo Budarz, Nov 17, 2017

A book on introductory mechanics and numerical methods for lower division undergraduate students in engineering or physical sciences. Edit 08/24 Added a section on multiplying vectors and the nonexistence of a vector inverse Edit 08/24 Added history of quaternions to the vectors chapter Edit 10/24/23 Rewrote introduction to Work and Energy chapter and added section on notation Edit 9/10/22. Edited fuel economy section Edit 9/10/22. Added kinematics examples. Edit 9/27/20. Added clarification of train transport efficiency. Edit 9/19/19. Clarification of gravitational fields in fundamental forces section. Edit 9/10/19. Graphic on box with static friction fixed. Edit 4/19/19. Section added on drag force derivation and other small details were modified. "Beach" by Pexels is in the Public Domain, CC0

Table of Contents

  1. Introduction
    1. Introduction
    2. The Scope of Physics
    3. Science is Testable
    4. Units
    5. PROBLEMS
  2. Calculus and Vectors
    1. Why Vectors?
    2. Vectors and Complex Numbers
    3. Vector Properties
    4. Three Dimensional Vectors
    5. Direction Cosines
    6. Basic Operations with Vectors
    7. Derivatives and Integrals of Vectors
    8. The Meanings of Functions
    9. Miscellaneous Comments
    10. Problems
  3. Kinematics
    1. When Left Alone
    2. Position
    3. Velocity
    4. Acceleration
    5. Eliminating Time
    6. Drag-Free Projectile Motion
    7. Time-Varying Acceleration Example
    8. PROBLEMS
    9. Lab Exercise
  4. Dynamics
    1. Background
    2. Fundamental Interactions
    3. Contact Interactions
    4. Newton's Laws of Motion
    5. Momentum
    6. Application of Newton's Second Law
    7. Motion on an Incline
    8. Friction on Moving Surfaces
    9. GeoGebra Exercise - Universal Gravitation
    10. PROBLEMS
    11. Lab Exercise - Slope and Friction of a Cart
  5. Dynamics with Variable Forces
    1. Motivation
    2. Differential Equations of Motion
    3. Air Drag
    4. Free Fall with Drag
    5. Propulsion and Machines
    6. Vehicle Resistive Forces
    7. Solving Differential Equations of Motion
    8. PROBLEMS
    9. Lab Exercise - Drag Coefficient Measurement
  6. Harmonic Motion
    1. Elasticity
    2. Hooke's Law
    3. Stress and Strain
    4. Simple Harmonic Motion
    5. Numerical Solutions to Harmonic Motion Problems
    6. The Simple Pendulum
    7. Phase Space
    8. PROBLEMS
    9. Lab Exercise
  7. Circular Motion
    1. Introduction to Circular Motion
    2. Circular Motion Derivation
    3. Circular Orbits
    4. Kepler's Laws of Planetary Motion
    5. Some Examples
    6. Banked Track with Friction
    7. PROBLEMS
    8. Laboratory Exercise
  8. Effective Gravitation
    1. We Can't Feel Gravity
    2. Further Examples
    3. Earth's Effective Gravitation
    4. Coriolis Force
    5. PROBLEMS
    6. Experiment - Conical Pendulum
  9. Work and Energy
    1. Kinetic Energy
    2. Work
    3. Parametric Path Integrals
    4. Simpler Cases
    5. Power
    6. Fuel Economy
    7. PROBLEMS
  10. Potential Energy
    1. What you Already Know
    2. Potential Energy Defined
    3. Potential Energy for Universal Gravitation
    4. Energy Conservation
    5. Elastic Potential Energy
    6. Dissipative Forces
    7. PROBLEMS
  11. Collisions
    1. Momentum
    2. Momentum Conservation
    3. Center of Mass
    4. Coefficient of Restitution and Elasticity
    5. Relative Velocity
    6. Collisions
    7. Some Potential Simplifications
    8. PROBLEMS
  12. Rigid Body Motion
    1. Angular Kinematics
    2. Preview of Rotational Motion
    3. Moment of Inertia
    4. Parallel Axis Theorem
    5. Torque
    6. Angular Momentum
    7. Precession
    8. Problems
  13. Lab Tools and Concepts
    1. Precision and Accuracy
    2. Estimating Error
    3. Units
    4. PROBLEMS
    5. Lab Exercise

1.

Introduction

Introduction and Motivation

  • 1. Introduction
  • 2. The Scope of Physics
  • 3. Science is Testable
  • 4. Units
  • 5. PROBLEMS

1.1.

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1.2.

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1.3.

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1.4.

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1.5.

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2.

Calculus and Vectors

  • 1. Why Vectors?
  • 2. Vectors and Complex Numbers
  • 3. Vector Properties
  • 4. Three Dimensional Vectors
  • 5. Direction Cosines
  • 6. Basic Operations with Vectors
  • 7. Derivatives and Integrals of Vectors
  • 8. The Meanings of Functions
  • 9. Miscellaneous Comments
  • 10. Problems

2.1.

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2.2.

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2.3.

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2.4.

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2.5.

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2.6.

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2.7.

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2.8.

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2.9.

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2.10.

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3.

Kinematics

  • 1. When Left Alone
  • 2. Position
  • 3. Velocity
  • 4. Acceleration
  • 5. Eliminating Time
  • 6. Drag-Free Projectile Motion
  • 7. Time-Varying Acceleration Example
  • 8. PROBLEMS
  • 9. Lab Exercise

3.1.

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3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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3.8.

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3.9.

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4.

Dynamics

  • 1. Background
  • 2. Fundamental Interactions
  • 3. Contact Interactions
  • 4. Newton's Laws of Motion
  • 5. Momentum
  • 6. Application of Newton's Second Law
  • 7. Motion on an Incline
  • 8. Friction on Moving Surfaces
  • 9. GeoGebra Exercise - Universal Gravitation
  • 10. PROBLEMS
  • 11. Lab Exercise - Slope and Friction of a Cart

4.1.

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4.2.

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4.3.

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4.4.

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4.5.

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4.6.

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4.7.

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4.8.

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4.9.

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4.10.

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4.11.

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5.

Dynamics with Variable Forces

  • 1. Motivation
  • 2. Differential Equations of Motion
  • 3. Air Drag
  • 4. Free Fall with Drag
  • 5. Propulsion and Machines
  • 6. Vehicle Resistive Forces
  • 7. Solving Differential Equations of Motion
  • 8. PROBLEMS
  • 9. Lab Exercise - Drag Coefficient Measurement

5.1.

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5.2.

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5.3.

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5.4.

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5.5.

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5.6.

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5.7.

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5.8.

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5.9.

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6.

Harmonic Motion

Springs, elasticity and the resulting harmonic motion are discussed in this chapter.

  • 1. Elasticity
  • 2. Hooke's Law
  • 3. Stress and Strain
  • 4. Simple Harmonic Motion
  • 5. Numerical Solutions to Harmonic Motion Problems
  • 6. The Simple Pendulum
  • 7. Phase Space
  • 8. PROBLEMS
  • 9. Lab Exercise

6.1.

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6.2.

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6.3.

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6.4.

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6.5.

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6.6.

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6.7.

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6.8.

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6.9.

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7.

Circular Motion

  • 1. Introduction to Circular Motion
  • 2. Circular Motion Derivation
  • 3. Circular Orbits
  • 4. Kepler's Laws of Planetary Motion
  • 5. Some Examples
  • 6. Banked Track with Friction
  • 7. PROBLEMS
  • 8. Laboratory Exercise

7.1.

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7.2.

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7.3.

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7.4.

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7.5.

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7.6.

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7.7.

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7.8.

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8.

Effective Gravitation

  • 1. We Can't Feel Gravity
  • 2. Further Examples
  • 3. Earth's Effective Gravitation
  • 4. Coriolis Force
  • 5. PROBLEMS
  • 6. Experiment - Conical Pendulum

8.1.

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8.2.

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8.3.

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8.4.

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8.5.

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8.6.

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9.

Work and Energy

  • 1. Kinetic Energy
  • 2. Work
  • 3. Parametric Path Integrals
  • 4. Simpler Cases
  • 5. Power
  • 6. Fuel Economy
  • 7. PROBLEMS

9.1.

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9.2.

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9.3.

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9.4.

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9.5.

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9.6.

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9.7.

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10.

Potential Energy

  • 1. What you Already Know
  • 2. Potential Energy Defined
  • 3. Potential Energy for Universal Gravitation
  • 4. Energy Conservation
  • 5. Elastic Potential Energy
  • 6. Dissipative Forces
  • 7. PROBLEMS

10.1.

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10.2.

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10.3.

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10.4.

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10.5.

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10.6.

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10.7.

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11.

Collisions

  • 1. Momentum
  • 2. Momentum Conservation
  • 3. Center of Mass
  • 4. Coefficient of Restitution and Elasticity
  • 5. Relative Velocity
  • 6. Collisions
  • 7. Some Potential Simplifications
  • 8. PROBLEMS

11.1.

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11.2.

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11.3.

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11.4.

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11.5.

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11.6.

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11.7.

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11.8.

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12.

Rigid Body Motion

  • 1. Angular Kinematics
  • 2. Preview of Rotational Motion
  • 3. Moment of Inertia
  • 4. Parallel Axis Theorem
  • 5. Torque
  • 6. Angular Momentum
  • 7. Precession
  • 8. Problems

12.1.

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12.2.

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12.3.

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12.4.

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12.5.

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12.6.

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12.7.

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12.8.

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13.

Lab Tools and Concepts

  • 1. Precision and Accuracy
  • 2. Estimating Error
  • 3. Units
  • 4. PROBLEMS
  • 5. Lab Exercise

13.1.

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13.2.

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13.3.

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13.4.

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13.5.

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