Classical Dynamics
                        Jock McOrist
                         Lecture notes
                    Module MAT1036 Classical Dynamics
                     Department of Mathematics
                       University of Surrey
                   Guildford GU2 7XH, United Kingdom
                     c
                Copyright  2018 by Jock McOrist. All rights reserved.
                   E-mail address: j.mcorist@surrey.ac.uk
                Abstract
                Classical dynamics, as laid down by Newton, describes the behaviour of a huge range of everyday
                phenomena to an incredible degree of precision. It is a monumental achievement of human
                intellect. In this course we will use Newton’s laws, from first principles, to solve a wide array of
                classic problems from projectile motion with friction to how the Coriolis force influences flushing
                toilets.
                    Some helpful hints for doing well in this course. Many of you will have take mechanics at A-
                level. This course will di↵erent. Instead of rote learning formulae and using a library of examples
                as in A-levels, you are now going to solve problems from first principles. You will need to use
                your knowledge of vectors (cf. MAT1031, MAT1005) and Newton’s laws to translate a problem
                often described in words into a mathematical equation, typically an ordinary di↵erential equation
                (ODE).YouwillthenbeexpectedtosolvetheODEusingtechniquesthatyoulearntinSemester
                1 courses e.g. MAT1030.
                    This will also be a di↵erent course from what you have already taken in semester 1 and are
                concurrently doing in semester 2. It is a problem solving course, and memorising formulae, nor
                memorising answers to examples questions, will not help you do well. What will help you do
                well is working through problems by yourself without the solutions in front of you. Problem
                solving is a critical skill in your development as a mathematician, not to mention being highly
                valued by employers, so I encourage you to practice solving problems as much as possible.
                    There will be on average 3 lectures per week, and you will have fortnightly seminars in
                which you can hone your problem solving skills. There are four unassessed courseworks and
                one class test.  These will be marked and returned to you. The coursework and tests are
                important feedback for you, telling you how you are doing in the course, what your strengths
                andweaknessesare, andwhattypeofquestionstoexpectintheexam. Allimportantinformation
                will be disseminated in the usual channels via SurreyLearn. There is a syllabus on SurreyLearn,
                and I encourage to look at it. It describes the course content, what you are expected to know,
                and all of the assessments.
                    The lecture notes are structured to closely follow what we do in class, but it is important
                to attend lectures.   It is where all important information is disseminated pertaining to the
                course, including any details relating to any course tests, exams and courseworks. There are 47
                pages of worked examples throughout the notes as well as exercises at the end of each chapter,
                with solutions periodically made available throughout the semester. Key concepts are boldfaced
                when they are introduced. The fortnightly seminars have many questions on the coursework. I
                recommend working through all the worked examples, four courseworks and chapter exercises as
                a minimum. If you want to work through more questions and examples, look through the many
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       wonderful textbooks in the library. Some good example include French and Ebison, Introduction
       to classical mechanics while the modern classic from Richard Feynman (one of the greatest 20th
       century physicists) is volume 1 of the Feynman Lectures in Physics.
         Any questions on the concepts, problems, worked examples or their solutions or if you think
       you’ve found any error, inaccuracies or other bugs then please see me after lectures, during oce
       hours(05AA04,timelistedontheinformationsheet)orpleaseemailmej.mcorist@surrey.ac.uk.
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                 Contents
                 1.   Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       1
                      1.1.   What is classical dynamics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         1
                      1.2.   Coordinate systems and references frames . . . . . . . . . . . . . . . . . . . . .            1
                      1.3.   Displacement and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          3
                      1.4.   Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      6
                      1.5.   Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      8
                      1.6.   Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     9
                      1.7.   Problem solving strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        9
                      1.8.   Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    11
                 2.   Forces    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   13
                      2.1.   Mass, momentum and forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          13
                      2.2.   Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        13
                             2.2.1.  Vector equations and signs . . . . . . . . . . . . . . . . . . . . . . . . . .       15
                      2.3.   Gravity    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   15
                      2.4.   Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       19
                             2.4.1.  Translated reference frames . . . . . . . . . . . . . . . . . . . . . . . . .        20
                             2.4.2.  Rotated reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . .       20
                      2.5.   Normal forces and friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       24
                      2.6.   Air Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     29
                      2.7.   Springs and Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        30
                      2.8.   Damped Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . .            34
                      2.9.   Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    36
                 3.   Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        38
                      3.1.   Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    42
                 4.   Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         43
                      4.1.   Vector Calculus Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       43
                      4.2.   Work and Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          47
                      4.3.   Conservative Forces and Potential Energy . . . . . . . . . . . . . . . . . . . . .           50
                      4.4.   Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       53
                      4.5.   The electric force and vector fields      . . . . . . . . . . . . . . . . . . . . . . . . .   58
                      4.6.   Newton’s gravitational force . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       60
                      4.7.   The simple harmonic oscillator revisited . . . . . . . . . . . . . . . . . . . . . .         60
                      4.8.   Whythe universe is a simple harmonic oscillator . . . . . . . . . . . . . . . . .            61
                      4.9.   Dissipative Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     64
                      4.10. Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       65
                      4.11. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     66
                 5.   Systems of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      70
                      5.1.   Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           70
                      5.2.   Collisions   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   73
                      5.3.   Elastic collisions   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   74
                      5.4.   Inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   77
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