Mechanics Using Python: An Introductory Guide

Mechanics Using

Python An Introductory Guide

Mechanics Using

Python An Introductory Guide

By

Aayushman Dutta

Mechanics Using Python: An Introductory Guide

Aayushman Dutta

ISBN - 9789361521874

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Preface

Welcome to Elementary Mechanics Using Python, a comprehensive guide designed to provide a fundamental understanding of classical mechanics through the lens of Python programming. This book is tailored for undergraduate students in the United States seeking a rigorous exploration of mechanics with a hands-on, computational approach.

In the realm of introductory physics courses, the integration of programming languages has become increasingly pivotal. Python, renowned for its simplicity and versatility, serves as an ideal companion in unraveling the intricacies of elementary mechanics. This book bridges the gap between theoretical concepts and practical implementation, fostering a holistic comprehension of classical mechanics.

The primary objective is to empower students to not only grasp the theoretical foundations of mechanics but also to cultivate proficiency in applying computational methods to solve real-world problems. The narrative unfolds systematically, commencing with the essential principles of kinematics and dynamics, progressing through Newtonian mechanics, and culminating in advanced topics such as oscillations and celestial mechanics.

By immersing ourselves in Python, we harness its capabilities to simulate physical phenomena, visualize dynamic systems, and solve complex problems numerically. Each chapter presents a synergy of theoretical discourse and Python programming, facilitating a symbiotic learning experience. Extensive examples and exercises are interwoven throughout the text, providing ample opportunities for readers to reinforce their understanding and hone their programming skills.

Whether you are embarking on your first journey into the realm of mechanics or seeking to fortify your understanding with computational prowess, this book is crafted to serve as a valuable resource. Embrace the synergy of theory and practice as we delve into the captivating realm of Elementary Mechanics Using Python.

Table of Contents

1. Introduction to Mechanics1

1.1 Overview of Mechanics1

1.2 Role of Mathematics in Mechanics2

1.3 Using Python for Mechanics Problems2

2. Units and Measurement4

2.1 Fundamental units4

2.2 Derived units5

2.3 Unit Conversions6

2.4 Uncertainty and Significant Figures8

3. Vectors and Coordinate

Systems10

3.1 Scalars and Vectors10

3.2 Vector Coordinate Systems 11

3.3 Coordinate Systems12

3.4 Vector Components 14

4. Kinematics16

4.1 Displacement, Velocity, and Acceleration16

4.2 Motion Along a Straight Line 17

4.3 Projectile Motion18

4.4 Circular Motion19

5. Newton’s Laws of Motion21

5.1 Newton’s First Law21

5.2 Newton’s Second Law 22

5.3 Newton’s Third Law23

5.4 Applying Newton’s Laws24

6. Friction26

6.1 Types of Friction26

6.2 Laws of Dry Friction 27

6.3 Friction Coefficients27

6.4 Friction Examples29

7. Work and Energy31

7.1 Work 31

7.2 Kinetic Energy32

7.3 Potential energy32

7.4 Conservation of energy34

8. Momentum37

8.1 Impulse and momentum37

8.2 Conservation of momentum38

8.3 Collisions40

8.4 Center of mass42

9. Circular and Rotational

Motion45

9.1 Centripetal force45

9.2 Angular variables 47

9.3 Rotational kinetic energy47

9.4 Rotational inertia48

9.5 Torque 50

10. Gravitation52

10.1 Newton’s law of gravitation52

10.2 Gravitational potential energy53

10.3 Orbits and satellites54

10.4 Kepler’s Laws55

11. Oscillations58

11.1 Simple harmonic motion 58

11.2 Pendulums59

11.3 Mass-spring systems59

11.4 Damped Oscillations60

12. Waves63

12.1 Types of waves63

12.2 Wave properties64

12.3 Superposition and interference65

12.4 Standing waves 67

13. Fluids69

13.1 Density and Pressure69

13.2 Archimedes’ Principle70

13.3 Fluid Dynamics 70

13.4 Bernoulli’s Principle71

14. Thermodynamics73

14.1 Temperature Scales73

14.2 Thermal Expansion74

14.3 Heat and Internal Energy75

14.4 First Law of Thermodynamics 75

15. Electric Forces

and Fields77

15.1 Electric Charge77

15.2 Coulomb’s Law77

15.3 Electric Field 78

15.4 Gauss’s Law79

16. Electric Potential81

16.1 Electric Potential Energy81

16.2 Electric Potential 82

16.3 Capacitors 83

16.4 Dielectrics84

17. DC Circuits86

17.1 Current and Resistance86

17.2 DC Circuit Analysis 87

17.3 Kirchhoff’s Laws88

17.4 RC Circuits89

18. Magnetism91

18.1 Magnetic Fields91

18.2 Magnetic Forces92

18.3 Magnetic Materials94

18.4 Electromagnetic Induction94

19. Geometric Optics96

19.1 Reflection96

19.2 Refraction97

19.3 Mirrors and Lenses99

19.4 Optical Instruments101

20. Physical Optics103

20.1 Light as a Wave103

20.2 Diffraction and Interference104

20.3 Polarization106

20.4 Color and Dispersion107

Glossary110

Index112

CHAPTER 1 Introduction to Mechanics

Mechanics is the branch of physics concerned with the motion of objects and the forces acting on them. It seeks to describe motion quantitatively using mathematics. This chapter provides an overview of mechanics, the role of mathematics, and using Python to solve mechanics problems.

1.1 Overview of Mechanics

Mechanics can be divided into two main areas:

Kinematics - The study of motion without regard to the forces causing it. Kinematics describes motion in terms of variables like position, velocity, acceleration and time without considering causes.

Dynamics - The study of forces and how they affect motion. Dynamics analyzes the forces causing observed kinematic motion using concepts like mass, force, momentum and energy.

Mechanics utilizes core mathematical concepts like calculus, vectors, differential equations and coordinate systems. Mechanics provides mathematical models that approximate the real physical world. Models are evaluated based on predictiveness and experimentally validated.

Key subfields of mechanics include:

- Statistics - Analysis of forces in non-moving rigid bodies.

- Dynamics - Analysis of forces causing acceleration in moving bodies.

- Kinematics - Description of motion without force causes.

- Kinetics - Relationship between motion and its underlying forces.

- Newtonian mechanics - Mechanics modeled using Newton’s laws of motion.

- Lagrangian mechanics - Mechanics modeled using Lagrangians and Hamiltonians.

- Quantum mechanics - Behavior of subatomic particles described by quantum theory.

- Relativistic mechanics - Models that account for effects from special and general relativity.

- Continuum mechanics - Models describing materials as continuous mass rather than discrete particles.

Mechanics has broad applications in fields like engineering, physics, robotics, aerospace, biomechanics, chemistry, seismology, and astronomy.

Fig. 1.1 Overview of Mechanics

https://images.app.goo.gl/LU8GcAZepFxRNz4k9

1.2 Role of Mathematics in Mechanics

Mathematics enables mechanics theories to be expressed quantitatively and generate testable predictions. Key mathematical concepts used include:

Calculus - Derivatives describe instantaneous rates of change like velocity and acceleration. Integrals describe accumulated change like position.

Vectors - Vectors represent quantities like force and velocity that have both magnitude and direction. Vector math is extensively used.

Differential Equations - Relate a function to its derivatives. Used to equate rates of change to other variables.

Coordinate Systems - Orthogonal coordinate systems like Cartesian, polar and spherical simplify analysis of motion by aligning reference frames with key variables.

Linear Algebra - Matrix transformations and Eigenanalysis provide compact representations for multi-dimensional systems.

Numerical Methods - Techniques like Euler’s method approximate solutions to differential equations through discrete iterations.

Transforms - Fourier transforms decompose complex signals into underlying frequencies for easier analysis.

Probability/Statistics - Handle inherent randomness and uncertainty in measurements and phenomena.

By applying these mathematical concepts, the continuous dynamics of the real world can be translated into tractable discrete models. These models generate testable hypotheses. Their accuracy is refined over time based on experimental data.

1.3 Using Python for Mechanics Problems

The Python programming language provides an excellent environment for implementing mechanics models and analyzing data. Key features of Python include:

- A simple, readable syntax enabling clear expression of mathematical concepts.

- Vast libraries like NumPy and SciPy for fast array math and scientific computing.

- Matplotlib and Seaborn for flexible 2D and 3D data visualization.

- SymPy for symbolic math manipulation.

- Pandas for convenient data wrangling and analysis.

- Powerful IDEs like Jupyter Notebook for interactive development.

- Integration with tools like CAD, FEA and CFD.

For mechanics, Python allows complex mathematical models to be efficiently coded up and simulated. Numerical techniques like Euler’s method for solving differential equations can be straightforwardly implemented.

Key Python libraries used in mechanics include:

- NumPy - Provides N-dimensional arrays and linear algebra capabilities. Used widely.

- SciPy - Collection of scientific computing algorithms spanning optimization, integration, interpolation, etc.

- SymPy - Symbolic math library for manipulating expressions and deriving equations.

- Matplotlib - Flexible 2D/3D plotting with control over all chart elements.

- Seaborn - Built on Matplotlib, provides preset statistical plot styles.

- Pandas - Data analysis toolkit enabling data cleaning, manipulation and statistics.

Python allows quick iteration and testing of mathematical models against data. It can integrate with domain-specific tools while offering high programmer productivity. These characteristics make Python well-suited for applied mechanics.

Fig. 1.2 NumPy

https://images.app.goo.gl/5SEgHaMo58kKZGpg8

Conclusion

Mechanics aims to mathematically describe the motion of objects and the forces driving it. Kinematics focuses on the motion itself while dynamics considers the forces causing kinematic motion. Key subfields range from statics to quantum mechanics.

Mathematics like calculus, linear algebra, and differential equations are fundamental to expressing mechanical theories quantitatively. These allow continuous real-world phenomena to be translated into solvable discrete models.

Python provides an excellent programming environment for mechanics applications due to its balance of usability, libraries for scientific computing, and easy data