Newtonian Mechanics for Undergraduates

Vijay Tymms

• Publisher: WSPC (UK)
• Year: 2015
• eBook ISBN: 9781786340108
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

Newtonian mechanics is a cornerstone topic in physics. Regardless of the path an aspiring physicist takes, an intimate and intuitive understanding of how objects behave within Newton's law of motion is essential.

Yet the transition from high school physics to university level physics can be — and should be — difficult. The aim of this book is to teach Newtonian mechanics suitable for the first two years of university study. Using carefully chosen and detailed examples to expose areas of frequent misunderstanding, the first two thirds of the book introduces material familiar to high school students from the ground up, with a more mature point of view. The final third of the book contains new material, introducing detailed sections on the rotation of rigid objects and providing an insight into subtleties that can be troubling to the first-time learner. Tabletop physics demonstrations are suggested to assist in understanding the worked examples.

As a teacher and lecturer of physics with experience at both high school and university level, Professor Vijay Tymms offers a lucid and sensitive presentation of Newtonian mechanics to help make the step from high school to university as smooth as possible.

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Newtonian mechanics is a cornerstone topic in physics. Regardless of the path an aspiring physicist takes, an intimate and intuitive understanding of how objects behave within Newton's law of motion is essential.

Yet the transition from high school physics to university level physics can be — and should be — difficult. The aim of this book is to teach Newtonian mechanics suitable for the first two years of university study. Using carefully chosen and detailed examples to expose areas of frequent misunderstanding, the first two thirds of the book introduces material familiar to high school students from the ground up, with a more mature point of view. The final third of the book contains new material, introducing detailed sections on the rotation of rigid objects and providing an insight into subtleties that can be troubling to the first-time learner. Tabletop physics demonstrations are suggested to assist in understanding the worked examples.

As a teacher and lecturer of physics with experience at both high school and university level, Professor Vijay Tymms offers a lucid and sensitive presentation of Newtonian mechanics to help make the step from high school to university as smooth as possible.

Request Inspection Copy

Readership: Undergraduate physics students, high school students looking to study physics or applied physics at university level, high school physics teachers and first year undergraduate physics lecturers.
Key Features:

• Focuses on Newtonian mechanics taught at undergraduate level
• Features dedicated sections on toppling, rolling and sliding that are more detailed than most mechanics textbooks

1

Overview

1.1.Introduction

I have been teaching physics for 16 years, starting with secondary school teaching, then later university lecturing where I taught the first year mechanics lecture course at Imperial College for four years from 2010–2014. Teaching this course has been one of the most enjoyable parts of my career thus far, giving me an opportunity to rein-spect some of the most fundamental concepts in the discipline for delivery to a demanding (though appreciative) audience, complete with multiple demonstrations plus interesting problems and puzzles. During these years I developed and refined a set of comprehensive course notes tailored for the students I was teaching. This textbook is an adaptation of the notes, altered to appeal to a broader audience.

1.2.Why This Book is Needed

School syllabuses are in a state of constant flux. The breadth and depth of core physics and mathematics curricula taught in schools varies a little from year to year and a lot from generation to generation. So while well-established subjects in physics remain the same, the level of knowledge and understanding of students that enter university to study the discipline varies. This means that lecturers have to constantly update their courses to suit their target audiences and make the transition from A-level to degree as smooth as possible.

Although there are already many mechanics textbooks out there, there is a need for producing up-to-date reference material to match the level of development of the target audience. Essentially, textbooks quickly become out of date and there will always be a need for new ones. This particular one is designed to be in line with the level of physics and mathematics that contemporary school leavers ready to start a physics or physics-related degree will have.

1.3.Who Will Benefit From This Book?

The lecture course that led to the creation of this book was designed specifically for first year physics undergraduates at Imperial College and as such the direct target audience of this textbook are students making the transition from school to university.

The book should also appeal to advanced A-level students unsatisfied with the level they have reached, and especially those who are considering studying physics or physics-related subjects beyond school. It contains some A-level material that is delivered at university level of presentation and should strengthen such students’ understanding while also providing a smooth introduction to subtopics beyond the syllabus.

A-level physics teachers and first year university lecturers should also find the book useful; as well as the basic subject matter, in-depth examples and problems, there are also suggestions as to basic demonstrations that can easily be recreated in the classroom at minimal expense.

1.4.Assumed Prior Knowledge

Regarding mathematics, all the content that can feature in a standard A-level mathematics syllabus is assumed knowledge throughout the text. Differential and integral calculus plus logarithms and exponents are used from the outset, with a gentle introduction to good practice in the use of integral calculus outlined early on. Knowledge of vectors is also essential, with scalar products being used from Chapter 5 and vector products from Chapter 14.

Regarding physics content, this course text goes from the ground up — i.e. in terms of classical mechanics everything starts from the beginning, though occasionally in examples and discussions some other subdisciplines of physics are invoked with a GCSE level of knowledge being assumed.

1.5.Structure and Topics

Readers of this text no doubt come from a broad range of backgrounds and have covered a wide range of high school-level syllabuses. This means that some readers will have a lot more knowledge of mechanics than others already. This course will start at a relatively basic level from the beginning and become quite advanced by the end. This means that all students will find some of the course to be revision, but it will vary from person to person. No one will find the whole of the course to be revision, and readers’ understanding of familiar concepts will be further enhanced by the course in all cases.

An important thing to note about this textbook is that none of the material is redundant for any student taking a physics degree — every topic, subtopic, equation and example is relevant to the journey towards understanding physics. Whether you are aiming to excel at theory, experiment or computation, whether your interest lies in quantum field theory, atmospheric physics or cosmology, all of the material within these pages is relevant and will be beneficial on your route to becoming an expert.

The textbook follows a reasonably traditional route though is split into shorter chapters than most, which are of variable length. It starts with an overview, defining and categorising the most important quantities in the discipline, i.e. displacement, velocity, acceleration, force and mass. The next four chapters expand on this and bring in Newton’s three laws of motion. Chapter 6 then briefly brings in linear momentum before introducing work and energy for the first time, leading to a much more in-depth view of momentum and potential energy. The second half of the book starts with motion on a curved path (not necessarily circular motion, though that is very much part of it) leading to simple harmonic motion and gravitation. Chapters 13 to 18 deal with the dynamics of rotating objects, ending with a brief look at gyroscopic motion and precession.

Feedback for the Author

I am happy to hear from any readers so please contact me via the publishers if you have any comments, queries or criticisms.

Vijay Tymms, April 2015

2

Introductory Concepts

This chapter introduces the five most important quantities in classical mechanics, namely displacement, velocity, acceleration, force, and mass. They are the most important quantities because it is impossible to make progress in the subject without first having an appreciation of what these quantities mean and how they relate to each other. The chapter provides definitions of the quantities at a level suitable to undergraduates and provides a discussion on what they physically mean.

The chapter also provides an introduction to scalar and vector quantities, SI prefixes and highlights a sensible approach to introducing new quantities and units that will be used throughout the text.

2.1.Quantities, Units, and Coordinate Systems

2.1.1.Scalar and Vector Quantities

In physics most measured quantities can be expressed by either:

(1)	A magnitude only. These are known as scalar quantities or simply scalars. Examples of scalar quantities are time, mass, energy, power and density. Mathematical operations on scalar quantities are familiar, i.e. they add, subtract, multiply, and divide like normal numbers. Scalar quantities do not usually require any special notation to denote that they are scalars.
(2)	A magnitude and a direction. These are known as vector quantities or simply vectors. Examples of vector quantities encountered in classical mechanics are force, acceleration, velocity, momentum and torque.

2.1.2.When Vectors Will Be Used and What Knowledge Will Be Assumed

Mathematical operations on vector quantities are less familiar and more complicated than with scalar quantities but you will be expected to know some of them to get through this book. Vector addition and splitting vectors into components are absolutely essential and will appear from Chapter 5 onwards. Vector dot products will be required from a similar stage, and vector cross products will be used in detail from Chapter 14 onwards. You should already have some familiarity with some of these topics from high school mathematics and you should refer to your favourite mathematics textbook or other resource if unsure. Vector calculus will not be used in this book as it is seldom seen at school and is usually first met midway through the first year of university degrees in physics and applied mathematics in the UK. On occasions where a knowledge of vector calculus could be utilised to enhance understanding, notes are written in the text with optional further reading cited in order to allow the interested student to pursue the topic further.

2.1.3.Vector Notation in Print and in Handwriting

When writing vector quantities, it is essential to use some sort of special notation to denote the vector. In this textbook they will usually be denoted in bold.

In handwriting, the two most common and generally unambiguous conventions are either to draw an arrow above

or a tilde or straight line below the symbol

. Just as inclusion of these accoutrements implies a quantity is a vector, an omission means the quantity is a scalar, even if missed by accident. One must therefore be careful, especially if the handwritten work is going to be studied by someone else.

That said, there are occasions where a vector quantity can effectively be treated as a scalar (for example, when dealing with velocity along a straight line and the use of + and − symbols is sufficient to denote a direction); provided the preamble to such work states matters clearly enough in these situations then a vector notation can be omitted.

2.1.4.Knowing When a Quantity is Scalar or Vector

When dealing with quantities in physics, it is usually important to know whether the quantity is a scalar or a vector. In this book this will be stated whenever a new quantity is introduced. If unsure, a good question to ask yourself is whether directions are required when adding parts of the same quantity together. For example, if two times in seconds are added together it does not make sense to ask what directions they have (you do not add five seconds north to seven seconds west) but if two displacements in metres are added then directions must be considered. Sometimes this will not be as easy as it sounds but asking oneself this quest...

Table of contents

• Cover
• Title
• Copyright
• Contents
• 1. Overview
• 2. Introductory Concepts
• 3. 1D Motion
• 4. Newton’s First and Second Laws of Motion
• 5. Types of Force and Free Body Diagrams
• 6. Newton’s Third Law of Motion
• 7. Linear Momentum
• 8. Work, Energy and Power
• 9. Potential Energy
• 10. Collisions and Rockets
• 11. Motion on a Curved Path
• 12. Simple Harmonic Motion
• 13. Gravitation
• 14. Rotational Analogues
• 15. Equilibrium and Balance
• 16. Unbalanced Objects
• 17. Rolling and Sliding
• 18. Angular Momentum
• 19. Angular Momentum, Gyroscopes, and Precession
• Bibliography
• Index