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Elementary Mechanics Using Python.pdf
Preface
Contents
1 Introduction
1.1 Physics
1.2 Mechanics
1.3 Integrating Numerical Methods
1.4 Problems and Exercises
1.5 How to Learn Physics
1.5.1 Advice for How to Succeed
1.6 How to Use This Book
2 Getting Started with Programming
2.1 A Python Calculator
2.2 Scripts and Functions
2.3 Plotting Data-Sets
2.4 Plotting a Function
2.5 Random Numbers
2.6 Conditions
2.7 Reading Real Data
2.7.1 Example: Plot of Function and Derivative
3 Units and Measurement
3.1 Standardized Units
3.2 Changing Units
3.3 Uncertainty and Significant Digits
3.4 Numerical Representation
4 Motion in One Dimension
4.1 Description of Motion
4.1.1 Example: Motion of a Falling Tennis Ball
4.2 Calculation of Motion
4.2.1 Example: Modeling the Motion of a Falling Tennis Ball
5 Forces in One Dimension
5.1 What Is a Force?
5.2 Identifying Forces
5.3 Newton's Second Law of Motion
5.3.1 Example: Acceleration and Forces on a Lunar Lander
5.4 Force Models
5.5 Force Model: Gravitational Force
5.6 Force Model: Viscous Force
5.6.1 Example: Falling Raindrops
5.7 Force Model: Spring Force
5.7.1 Example: Motion of a Hanging Block
5.8 Newton's First Law
5.9 Newton's Third Law
5.9.1 Example: Weight in an Elevator
6 Motion in Two and Three Dimensions
6.1 Vectors
6.2 Description of Motion
6.2.1 Example: Mars Express
6.3 Calculation of Motion
6.3.1 Example: Feather in the Wind
6.4 Frames of Reference
6.4.1 Example: Motion of a Boat on a Flowing River
7 Forces in Two and Three Dimensions
7.1 Identifying Forces
7.2 Newton's Second Law
7.3 Force Model---Constant Gravity
7.3.1 Example: Motion of a Ball with Gravity
7.4 Force Model---Viscous Force
7.4.1 Example: Path Through a Tornado
7.5 Force Model---Spring Force
7.5.1 Example: Motion of a Bouncing Ball with Air Resistance
7.6 Force Model---Central Force
7.6.1 Example: Comet Trajectory
8 Constrained Motion
8.1 Linear Motion
8.2 Curved Motion
8.2.1 Example: Acceleration of a Matchbox Car
8.2.2 Example: Acceleration of a Rotating Rod
8.2.3 Example: Normal Acceleration in Circular Motion
9 Forces and Constrained Motion
9.1 Linear Constraints
9.1.1 Example: A Bead in the Wind
9.2 Force Model---Friction
9.2.1 Example: Static Friction Forces
9.2.2 Example: Dynamic Friction of a Block Sliding up a Hill
9.2.3 Example: Oscillations During an Earthquake
9.3 Circular Motion
9.3.1 Example: A Car Driving Through a Curve
9.3.2 Example: Pendulum with Air Resistance
10 Work
10.1 Integration Methods
10.2 Work-Energy Theorem
10.3 Work Done by One-Dimensional Force Models
10.3.1 Example: Jumping from the Roof
10.3.2 Example: Stopping in a Cushion
10.4 Work Done in Two- and Three-Dimensional Motions
10.4.1 Example: Work of Gravity
10.4.2 Example: Roller-Coaster Motion
10.4.3 Example: Work on a Block Sliding Down a Plane
10.5 Power
10.5.1 Example: Power Exerted When Climbing the Stairs
10.5.2 Example: Power of Small Bacterium
11 Energy
11.1 Motivating Examples
11.2 Potential Energy in One Dimension
11.2.1 Example: Falling Faster
11.2.2 Example: Roller-Coaster Motion
11.2.3 Example: Pendulum
11.2.4 Example: Spring Cannon
11.3 Energy Diagrams
11.3.1 Example: Energy Diagram for the Vertical Bow-Shot
11.3.2 Example: Atomic Motion Along a Surface
11.4 The Energy Principle
11.4.1 Example: Lift and Release
11.4.2 Example: Sliding Block
11.5 Potential Energy in Three Dimensions
11.5.1 Example: Constant Gravity in Three Dimensions
11.5.2 Example: Gravity in Three Dimensions
11.5.3 Example: Non-conservative Force Field
11.6 Energy Conservation as a Test of Numerical Solutions
12 Momentum, Impulse, and Collisions
12.1 Motivating Example---Meteor Impact
12.2 Translational Momentum
12.3 Impulse and Change in Momentum
12.3.1 Example: Ball Colliding with Wall
12.3.2 Example: Hitting a Tennis Ball
12.4 Isolated Systems and Conservation of Momentum
12.5 Collisions
12.5.1 Example: Ballistic Pendulum
12.5.2 Example: Super-Ball
12.6 Modeling and Visualization of Collisions
12.7 Rocket Equation
12.7.1 Example: Adding Mass to a Railway Car
12.7.2 Example: Rocket with Diminishing Mass
13 Multiparticle Systems
13.1 Motion of a Multiparticle System
13.2 The Center of Mass
13.2.1 Example: Points on a Line
13.2.2 Example: Center of Mass of Object with Hole
13.2.3 Example: Center of Mass by Integration
13.2.4 Example: Center of Mass from Image Analysis
13.3 Newton's Second Law for Particle Systems
13.3.1 Example: Ballistic Motion with an Explosion
13.4 Motion in the Center of Mass System
13.5 Energy Partitioning
13.5.1 Example: Bouncing Dumbbell
13.6 Energy Principle for Multi-particle Systems
14 Rotational Motion
14.1 Rotational State---Angle of Rotation
14.2 Angular Velocity
14.3 Angular Acceleration
14.3.1 Example: Oscillating Antenna
14.4 Comparing Linear and Rotational Motion
14.5 Solving for the Rotational Motion
14.5.1 Example: Revolutions of an Accelerating Disc
14.5.2 Example: Angular Velocities of Two Objects in Contact
14.6 Rotational Motion in Three Dimensions
14.6.1 Example: Velocity and Acceleration of a Conical Pendulum
15 Rotation of Rigid Bodies
15.1 Rigid Bodies
15.2 Kinetic Energy of a Rotating Rigid Body
15.3 Calculating the Moment of Inertia
15.3.1 Example: Moment of Inertia of Two-Particle System
15.3.2 Example: Moment of Inertia of a Plate
15.4 Conservation of Energy for Rigid Bodies
15.4.1 Example: Rotating Rod
15.5 Relating Rotational and Translational Motion
15.5.1 Example: Weight and Spinning Wheel
15.5.2 Example: Rolling Down a Hill
16 Dynamics of Rigid Bodies
16.1 Motivating Example---Spinning a Wheel
16.2 Newton's Second Law for Rotational Motion
16.2.1 Example: Torque and Vector Decomposition
16.2.2 Example: Pulling at a Wheel
16.2.3 Example: Blowing at a Pendulum
16.3 Rotational Motion Around a Moving Center of Mass
16.3.1 Example: Kicking a Ball
16.3.2 Example: Rolling down an Inclined Plane
16.3.3 Example: Bouncing Rod
16.4 Collisions and Conservation Laws
16.4.1 Example: Block on a Frictionless Table
16.4.2 Example: Changing Your Angular Velocity
16.4.3 Example: Conservation of Rotational Momentum
16.4.4 Example: Ballistic Pendulum
16.4.5 Example: Rotating Rod
16.5 General Rotational Motion
Appendix A Proofs
Appendix B Solutions
Index
Undergraduate Lecture Notes in Physics
Anders Malthe-Sørenssen
Elementary
Mechanics
Using Python
A Modern Course Combining Analytical
and Numerical Techniques
Undergraduate Lecture Notes in Physics
Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics
throughout pure and applied physics. Each title in the series is suitable as a basis for
undergraduate instruction, typically containing practice problems, worked examples, chapter
summaries, and suggestions for further reading.
ULNP titles must provide at least one of the following:
An exceptionally clear and concise treatment of a standard undergraduate subject.
A solid undergraduate-level introduction to a graduate, advanced, or non-standard subject.
A novel perspective or an unusual approach to teaching a subject.
ULNP especially encourages new, original, and idiosyncratic approaches to physics teaching
at the undergraduate level.
The purpose of ULNP is to provide intriguing, absorbing books that will continue to be the
reader’s preferred reference throughout their academic career.
Series editors
Neil Ashby
Professor Emeritus, University of Colorado, Boulder, CO, USA
William Brantley
Professor, Furman University, Greenville, SC, USA
Michael Fowler
Professor, University of Virginia, Charlottesville, VA, USA
Morten Hjorth-Jensen
Professor, University of Oslo, Oslo, Norway
Michael Inglis
Professor, SUNY Suffolk County Community College, Long Island, NY, USA
Heinz Klose
Professor Emeritus, Humboldt University Berlin, Germany
Helmy Sherif
Professor, University of Alberta, Edmonton, AB, Canada
More information about this series at http://www.springer.com/series/8917
Anders Malthe-Sørenssen
Elementary Mechanics
Using Python
A Modern Course Combining Analytical
and Numerical Techniques
123
Anders Malthe-Sørenssen
Department of Physics
University of Oslo
Oslo
Norway
ISSN 2192-4791
Undergraduate Lecture Notes in Physics
ISBN 978-3-319-19595-7
DOI 10.1007/978-3-319-19596-4
ISBN 978-3-319-19596-4
(eBook)
ISSN 2192-4805
(electronic)
Library of Congress Control Number: 2015940747
Springer Cham Heidelberg New York Dordrecht London
© Springer International Publishing Switzerland 2015
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed.
The use of general descriptive names, registered names,
in this
publication does not imply, even in the absence of a specific statement, that such names are exempt from
the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained herein or
for any errors or omissions that may have been made.
trademarks, service marks, etc.
Printed on acid-free paper
Springer International Publishing AG Switzerland is part of Springer Science+Business Media
(www.springer.com)
To Mina, Aurora and Olav.
Preface
This book was developed as a textbook for use in the course “Introduction to
Mechanics” in the Department of Physics at the University of Oslo starting 2007. In
this course we aimed at providing a seamless integration of analytical and numerical
methods when solving physics problems,
thereby allowing us to solve more
advanced and applied problems in mechanics, and providing examples that are
perceived as more relevant for the students. We could address not only the very
special cases that have analytical solutions, but could instead focus on choosing
problems that would initiate discussions and provide the students with physical
insights.
Through the processes of introducing and developing advanced problems, it also
became clear that this approach brought the students closer to the way physics is
discovered and applied. In addition, it introduced the students to a more exploratory
way of understanding phenomena and of developing their physical concepts. Well-
developed examples that also include elements of numerical computations gave the
students a feeling of discovering physical processes while also understanding how
they are results of the underlying simple physical laws. In many cases, the advanced
examples and exercises spawned interesting and rewarding discussions about the
underlying physical processes, and also forced the students to understand the
various forms of representation used to illustrate physical processes, such as motion
diagrams and energy diagrams, and use these diagrams to reason about physical
processes.
As the course, examples, and exercises were developed it also became clear that
the introduction of numerical methods in an introductory course in physics also
helped build the notion that numerical methods are no different from analytical
methods—they are part of the theoretical toolbox that any physicist is supposed to
master. Our aim became to make it as natural for our students to solve their
problems by developing a small program and discussing the results, as it was to use
a calculator.
It has been particularly rewarding to observe the way that many of the examples
and exercises trigger discussions when students discover unexpected results, in the
form of unexpected resonances in a simple model for friction or in the case of
vii
viii
Preface
Greenwood gaps in the distribution of asteroids in the solar system. The insight that
the simple laws of mechanics that they learned actually had observable conse-
quences and explanatory power was often an important insight as well as an
important reinforcer for the students. We also believe that this helps the student
build a more realistic image of how science actually is done.
In order to get most of the numerical parts of this text it is advantageous for the
students to have some prior knowledge of scientific programming, preferably with a
scripting type language such as Matlab or Python, but this is not absolutely nec-
essary. We encourage readers who are not familiar with scripting type programming
first to study Chap. 2. However, in our experience students who read the book,
study the examples, and do the exercises will already be developing programmers
by the end of the course.
This book grew out of a larger, collaborative effort at the University of Oslo.
I would like to thank Morten Hjorth-Jensen and Arnt Inge Vistnes for including me
in the physics part of the Computers in Science Education program. I also thank
Hans Petter Langtangen and Knut Mørken at the Department of Informatics for
their dedication, support, and inspiration for introducing numerical approaches in
the basic curriculum. I thank the Faculty for Mathematics and Natural Sciences for
their support used to develop exercises and examples used in this text. I would also
like to thank Arnt Inge Vistnes, Jonas van den Brinck, and Sigve Bøe Skattum for
developing some of the exercises that have been included in this book as examples
or exercises. Sigve Bøe Skattum has also provided many of the illustrations.
Oslo
March 2015
Anders Malthe-Sørenssen
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