Dear Student,

Congratulations on your results; we very much look forward to meeting you again in October.

We would like you to complete the vacation work (Page 17 of this PDF) and bring it to the tutorial

organization meeting which we expect to arrange for Friday of 0 week, after your departmental

induction.

The problem set indicates a source textbook. Additional resources can be found among the FLAP

(Flexible Learning Approach to Physics) modules from the Open University. The mathematics

modules can be found in two books, Basic Mathematics for the Physical Sciences and Further

Mathematics for the Physical Sciences.

FLAP modules for "Motion in one dimension", "Work and energy", and "Simple harmonic motion"

are available on request by contacting Jeffrey Tseng at jeff.tseng@physics.ox.ac.uk.

We are very much looking forward to having you here soon.

Sincerely,

The St Edmund Hall Physics Tutors

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Notes for Incoming Freshers

2018-2019

These notes have been produced by the University of Oxford Department of Physics. Other documents which may usefully be read alongside this one are :-

1) the leaflet “Physics at Oxford”, produced by the Physics Department and is available at http://www2.physics.ox.ac.uk/study-here/undergraduates. The leaflet gives information about all years of the three-year BA and the four-year MPhys courses; Information about the courses is also available on the web at http://www2.physics.ox.ac.uk/students

2) the booklet “Oxford Physics: The PJCC Freshers’ Guide 2018-19, produced by undergraduate physics

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NDenys Wilkinson Building

Dennis Sciama Lecture Theatre(via the Level 4 entrance)

Physics Teaching Laboratories(entrance down the steps)

Clarendon LaboratoryBeecroft Building

Lindemann and MartinWood Lecture TheatresTeaching Faculty Office (CL001)

Disabled Students can access the Lindemann and the Dennis Sciama Lecture Theatres by lifts from ground floors. The Denys Wilkinson Building has toilet facilities for disabled students. The Martin Wood Lecture Theatre has access for disabled students, a reserved area within the theatre and toilet facilities.

The Department is able to make provision for disabled students and those with other special needs. If you think you may need any special requirements, it would be very helpful to us if you could contact the Assistant Head of Teaching (Academic) about these before you arrive.

Teaching and Learning in Physics

1

LecturesThe basic form of instruction is the Lecture i.e. one person, using blackboard (whiteboard)/OHP/ slides/PowerPoint presentations/ demonstrations etc., explaining his or her advertised subject to all students in the University wishing to attend (around 180 physicists in any one year). The programme of lectures is worked out so as to cover the syllabus for the relevant papers. The lectures are frequently supported by printed notes, problem sheets, and/or other handouts. We regard attendance as generally essential, though it is not in fact compulsory. You will be going to about 4 Maths lectures per week to begin with, and maybe 3 or 4 Physics (or other) lectures. The courses cover a lot of material without repetition. Each lecture builds on the one before and it is important not to miss any if you can help it.

You will need to learn to take good lecture notes, and supplement them with your own private study, using textbooks recommended by lecturers and tutors. Lectures are intended to provide a broad overview of a topic, and cannot possibly cover every detail. It is also useful to approach a dif-ficult question from two or more different points of view.

Tutorials and ClassesLectures are backed up by Tutorials and Classes, which are compulsory. Whereas lectures are or-ganised by the Physics Department, tutorials and classes (in years 1 to 3) are the responsibility of the Colleges. This doesn’t mean that there are two separate groups of staff, one in the Department and the other in the Colleges: all the Physics Depart-ment’s academic staff are attached to Colleges. At the Department they do research and lecturing, and they also tutor for their Colleges. In tutorials and classes you have direct access to people who are actively involved in research.

A tutorial usually consists of a meeting between two students and one tutor for one hour, but some-times there can be one, or three students present

and the meeting could go on for a bit longer. A class is also a small group, often consisting of just the students in a given year in one college taking the subject, though occasionally classes may be held with another college.During the first year your time will be divided roughly equally between maths and physics with small group teaching in college, a mix of tutorials and classes - meeting roughly once a week for each subject. For each tutorial or class, work will be as-signed in advance often using material prepared by the lecturer with suggested additional reading from your tutor. Much of the work is structured around problem solving but you may be asked to write short essays from time to time. You will usually be expected to hand in the work to the tutor the day before the tutorial or class.Tutorial and class work will often be organised to be as near as possible in step with the lectures, so as to reinforce them. However, from time to time you’ll be asked to work on some topics in tutorials before they have been covered in lectures — because, for example, your tutor may think that the development of the subject is better that way. Whatever the reason, the exercise of having to work something out on your own, with the help of books, can be very valuable. Problem sets usu-ally begin with elementary topics which can be answered directly from the lecture notes, but then proceed to more challenging questions. Develop-ing the ability to solve problems independently is just as important as learning facts and standard techniques.Although you will only have something like two “contact hours” with your College tutors per week, your progress will be followed very closely on a week to week basis. A lot will depend on how you use these precious contact hours. You must prepare for them by identifying as precisely as you can the areas which you need help with, either in the lectures or tutorial work. You will soon find that they are an essentially friendly encounter (provided you are working properly!), during which you are encouraged to ask questions and contribute to the discussion.

Students sometimes have difficulty in adjusting to the less structured pattern of University education, having been accustomed to regular classroom hours and “homework” assignments at school. We think it may be helpful, therefore, to give you some advance idea of the way we operate — especially as the first few days are extremely hectic, as you will discover. The whole of the first term can fly by very quickly indeed!

2

PracticalsThe whole of physics depends on experimental observations, and learning how to make these reli-ably and quantitatively is an essential part of your education. Practical work is therefore compulsory. It is a University requirement that you do the proper amount of practical work, which averages to about one whole day per week for most of the first year. You keep a record of your practical work in ‘log-books’; some of your practicals have to be written up in detail and are marked. A termly progress report on your lab work is passed to your College tutors. Practicals are done in pairs and all arrange-ments will be explained when you arrive here.

VacationsOne very important point to understand is the role of “vacations” in the University year. At Oxford the teaching terms are very short — they add up to only about 25 weeks in one year. Vacations have to include holiday time; and everyone recognises that for many students they also have to include money-earning time.

Nevertheless, it is absolutely essential that you set aside significant amounts of time during each vacation for academic work. The course assumes that you will do this. You must go over your lecture notes, revising the material and supplementing it with information gained from tutorials and from your own reading. In addition to consolidating the previous terms work, you should also try to prepare for the next term’s courses. Your tutors will often set you vacation reading and specific vacation work.

SummaryThus the general pattern of the course is:• attendance at lectures and practicals, including

taking good notes• furthering your understanding of the lectures

by self-study of the lecture material, including reading of textbooks dealing with the same topics, as recommended by your tutors

• problem-solving and essay writing for tutorials• deepening of your understanding in the light

of the tutorial work and discussion• consolidation and preparation during vacations.

This is the ‘foundation’ year, on which the sub-sequent years of the BA and MPhys courses both depend. At the end of the year you will sit the Preliminary Examination in Physics (‘Prelims’). This is the qualifying exam and must be passed before you can continue your studies at Oxford (a resit is possible in September at the discretion of your college). Prelims consists of three com-ponents: (i) four compulsory written papers, (ii) a shorter paper chosen from a range of topics and (iii) evidence of successful completion of practical work during the year.

The four compulsory papers are: (a) CP1: Physics 1 (b) CP2: Physics 2(c) CP3: Mathematical Methods 1 (d) CP4: Mathematics Methods 2

The first year course in Physics

The Short Option topics are likely to include: (a) S01: Functions of a Complex Variable; (b) S02: Astrophysics: From Planets to the Cosmos; (c) S03: Quantum Ideas.

Full details are given in the Physics Undergraduate Course Handbook (available at the start of each year), but synopses and reading lists for the first terms work will be sent to you by your college.

Almost all students pass Prelims and continue to study physics. It is possible, with the consent of your college tutors, to change to a related subject (for example Earth Sciences or Materials) at this point provided you have passed Prelims.

Learning resources

Monitoring your progress

3

Books and LibrariesThe lecturers for each of the courses issue lists of books they recommend. These will usually specify one core text, or several alternatives, together with some others for further study or reference. Your College tutors will also have suggestions, both in general and also specifically for the subjects in which they are tutoring you. All the recommended books (some in multiple copies) should be in your College library, from which you can borrow them. If your College library doesn’t have one of the recommended books, or a book you feel you need, tell your tutor or the College Librarian. Most tu-tors, however, will expect you to buy a few books yourself: these will typically be ones that you will be using repeatedly for tutorials, perhaps for more than one year (for example “Mathematical Methods for Physics and Engineering” by Riley, Hobson and Bence).

The ultimate collection of books and journals in Oxford is in the University Library, known as the Bodleian Library, which includes a special sub-library for science called the Radcliffe Science Library (RSL). The Radcliffe Science Library is the main science and medicine research library.

More details about the Radcliffe Science Library and lending services can be found at http://www.ouls.ox.ac.uk/rsl and http://www.ouls.ox.ac.uk/rsl/services/lending_services. Arrangements for getting access to these libraries (which requires a “University card”) will be explained to you by your College.

ComputersAll physics students have accounts on the Physics practical course computers, which enables them to book practicals, as well as use the computers (we have introductory computer courses for absolute beginners — don’t panic!). There is a University-wide network, which enables students to work on their data while back in College. Undergraduates may also have an account on the university com-puting system. The colleges all have computing facilities for their undergraduates and most college rooms have internet connections for those who bring their own machines.

All new users will be asked to sign an agreement or ‘undertaking’ to abide by the University Rules on the use of computers.

Basically, there are two ways in which this is done — one by the College, the other by the University. As mentioned above, your College tutors will be monitoring your progress week by week, in tuto-rials. They will also review the reports on your lab work. At the end of each term, they will make reports on your progress. In addition, they may set you a College (not University) exam called a “Collection” at the start of each term, on the material covered in the previous term, or on the vacation reading, or both. In this way both you and your tutors can see what has been achieved and what needs more work. In some Colleges you can be awarded a “progress prize”, and/or a prize for good work in Collections. If progress has not been satisfactory, your tutors will discuss with you what action needs to be taken.

The University assesses your progress via Univer-sity Examinations. These are the same for every-one, whichever College they belong to (the College Collections vary from College to College). At the end of your first year you take the Prelims exam as explained earlier. This is a pass/fail exam, not di-vided into First Class, Second Class, etc., although it is possible to get a Distinction. For those who fail one or more papers, there are resits in September. If you fail any resit paper, you generally have to leave (this is a rule common to all University first year courses).

4

Preparation for your first term

Oxford is quite a large University by UK standards, with over 10,000 undergraduates and more than 4,000 postgraduate students. The Physics Depart-ment is one of the largest in the country, with about 180 undergraduates entering the Oxford Physics course each year. It could be easy to feel a bit lost among such large numbers of students, but it is here that the strength of the “College system” is most apparent. There are 30 undergraduate Colleges, and you are entering one of them. It will be the focus of much of your life at Oxford. All subjects are distributed roughly equally amongst the colleges so that you should soon get to know well the ap-proximately 6 physicists and 100 freshers in your own college.

Most importantly, you will be seeing your Col-lege tutors each week for tutorials. Their main responsibility is on the academic side, of course, but they are often the first point of contact if you have a personal problem of any kind. A tutor may be able to offer help and guidance directly, or — most likely — will know whom to advise you to see. Student health and welfare are primarily College responsibilities; tutors, chaplains, and other confidential advisers make up a sympathetic and effective network of support for students. In addition, the University has a Counselling Service available to help students, and the Student Union has officers working actively to promote student health and welfare.

Student support and guidance

Monitoring our performanceThe Physics Department is concerned with ensur-ing the high quality of all aspects of the physics courses. One of the ways we do this is through the Physics Joint Consultative Committee (PJCC), which consists of elected undergraduate members who meet twice a term to discuss, with academic staff representatives, both academic and adminis-trative matters concerning the physics courses. The Department values the advice it receives from this Committee, the vitality of which is strongly de-pendent on student enthusiasm and involvement.

Students assess their lectures and practicals each term using on-line lecture and practical feedback forms. This feedback is an important source of information for the Department’s own Academic Committee, which organises the lectures and is in charge of the Physics courses. There are also sug-gestion boxes in all the physics lecture theatres, where students can put comments on lectures, and in the reception area of the practical course for comments on the practical course.

Having emphasised the importance of vacation study, we strongly recommend that you consider doing some preparation for your first term during this vacation!

We think there are two areas where some prepara-tory work is particularly necessary: Mathematics and Mechanics. As regards the first, you’ll find that the physics course here is more mathemati-cally demanding than your A-Level course was. In fact, approximately half the tutorial work in your first year will be devoted to maths. The reason for this is that most of the physics covered in the following years of the course makes essential use

of mathematical techniques that go well beyond even Further Maths at A-Levels — and they are introduced in the first year. We design the course so that it can be successfully tackled by students who have not done double maths (Mathematica and Further Mathematics) at A-level (or a compa-rable course), but single Maths people may find the first year pretty tough. Even if you’ve done double Maths, you’ll begin to meet new material quite soon in the first term, so it’s important not to think you know it all: you can soon get left behind. We want your maths to be “up and running” when you get here!

to introduce students, right at the start, to math-ematically-based problem solving, and why we think some preparatory work on solving mechanics problems will be useful for you.

We have collected a set of problems in Mathemat-ics and in Mechanics, which your College Tutor may send you. If you work through these problems, you will be able to identify which topics you are shaky on, or haven’t done at all. When you get here in October, your Tutor can discuss the situation with you, and if there are gaps to be plugged, you can be given appropriate material to work on. Your Tutor may have sent you these problems already or may be going to send them — or perhaps an alternative assignment. If in doubt, just write to the Senior Physics Tutor of your College.

On the physics side, the first subject you’ll study is Mechanics. Our students seem to be finding this more difficult than previously. We think that the reason for this may be that the mechanics course quickly begins to make use of a significant amount of maths (e.g. differential equations, and complex numbers). Although students can cope quite well with the maths on its own, what they seem to find hard is combining the maths with the physics. This involves, first taking a problem posed in ordinary language and “setting it up” in precise mathemati-cal terms. Then mathematical techniques are used to obtain a solution which has finally to be “trans-lated back” into physical terms. This process may not be very familiar to you from A-Level, but it is going to be very important in your course at Oxford — and it is a skill highly prized by employers! This is why we want to use the mechanics course

5

InductionAn Induction Session is arranged by the Department of Physics on Friday of week 0. All freshers reading physics and, physics and philosophy are required to attend. The session consists of short presentations introducing the Department and covering important aspects of the course with a tour of the practical laboratories. Students are given their practical course documentation at this session.

Your College will arrange various events during the week before lectures and tutorials start. This week is called “week 0”, or “zeroth week”. The eight weeks of term proper are “week 1”, “week 2”, etc. These events in week 0 aim to familiarise you with life at Oxford. There are also some University-based events, notably the “Freshers’ Fair” at which you will be invited to join all kinds of clubs and societies.

Issued by the Physics Teaching Faculty, July 2018

Interact, like, network… be part of the Physics community!

University of Oxford Physics @OxfordPhysics Oxford Physics

Physics Community

              

Tel: +44 (0)1865 272407        carrie.leonard‐mcintyre@physics.ox.ac.uk                   www.physics.ox.ac.uk 

 

Induction Programme for Undergraduate Physics Students 2018 

Friday 5 October 2018 (0th Week)   

To keep  the numbers manageable, students will be split by College  into  two groups; please check below which group you are in.    All Physics and Philosophy students should join group B. Physics and Philosophy students do not visit the teaching laboratories and will have their own event organised.  

Group A   Balliol, Brasenose, Exeter, Jesus, Magdalen, Mansfield, Merton, Queen’s, Pembroke, St Catherine’s, St John’s, St Edmund Hall, Wadham, Worcester.    Group B   Christ Church, Corpus Christi, Hertford, Keble, Lady Margaret Hall, Lincoln, New College, Oriel, St Anne’s, St Hilda’s, St Hugh’s, St Peter’s, Somerville, Trinity, University. 

 Supervisor (organisation, time‐keeping etc.): Carrie Leonard‐McIntyre  

Group A          

Group A will start in the Dennis Sciama Lecture Theatre     

Time  Venue  Activity  Staff involved 

14:15  Dennis Sciama Lecture Theatre  Introduction to the Practical Course  Dr Jenny Barnes 

followed by          

   Teaching Laboratories  Tour of the labs  Technicians 

15:20     Group A leaves for Clarendon Laboratory  Hannah Glanville 

15:30  Martin Wood Lecture Theatre  Introduction Oxford Physics Course  Prof Ian Shipsey & Prof Hans Kraus 

15:55  Martin Wood Lecture Theatre  (IoP) Institute of Physics  IoP Rep (Olivia Keenan) 

16:05  Martin Wood Lecture Theatre  The First Year Course  Prof Arzhang Ardavan 

16:25  Martin Wood Lecture Theatre  PJCC (student body)   PJCC (tbc) 

   Martin Wood Lecture Theatre  Women in Physics  tbc 

16:30  Session ends       

Group B          

Group B will start in the Martin Wood Lecture Theatre     

Time  Venue  Activity  Staff involved 

14:15  Martin Wood Lecture Theatre  Introduction Oxford Physics Course  Prof Ian Shipsey & Prof Hans Kraus 

14:40  Martin Wood Lecture Theatre  The First Year Course  Prof Arzhang Ardavan 

15:00  Martin Wood Lecture Theatre  PJCC (student body)   PJCC (tbc) 

   Martin Wood Lecture Theatre  Women in Physics  tbc 

Assistant Head of Teaching (Academic) Carrie Leonard-McIntyre

15:05  Martin Wood Lecture Theatre  (IoP) Institute of Physics  IoP Rep (Olivia Keenan) 

15:20   

Group B leaves for the Denys Wilkinson Building 

Hannah Glanville 

 15:20   

Physics and Philosophy students leave for their event 

tbc 

15:30  Dennis Sciama Lecture Theatre  Introduction to the Practical Course  Dr Jenny Barnes  

followed by          

   Teaching Laboratories  Tour of the labs  Technicians 

16:30  Session ends       

   

 

Introduction to Computing in the Oxford Physics Course

Being able to write a computer program is an important skill for all physicists. In physics, computing is

mainly required for the following types of task:

1. Analysis of experimental data

2. Solving numerical problems such as differential equations

3. Controlling scientific instruments and acquiring data from them

We aim to teach you all these skills during your time in Oxford. In the first year we focus on the first two

aspects. The third aspect is generally learnt by doing experiments in the teaching labs, where you will

experience some data acquisition software during first year practicals. You will spend more time writing

your own code for data acquisition and control in the second and third year lab experiments.

Physicists use many different computing packages, and our philosophy is to expose you to a range of

different programming environments during your time in Oxford, so that, like professional physicists,

you learn to choose which you prefer for different tasks. In the first year we teach you two

programming languages, RStudio (which uses an underlying language called R) and Matlab. Your first lab

classes will be introductory exercises in both languages.

Many modern computing languages are interchangeable in the sense they can all be used to carry out

the tasks above, although there are a few differences between languages, which you will learn about in

the computing lectures. In the first year, we encourage you to use RStudio for data analysis, whereas

Matlab is used to train you mainly in solving numerical problems.

Here we provide some introductory resources that use free and friendly web-based tools to get you

used to the basics of computing. We also provide resources for a third language, Python, which we do

not formally teach, but is easy to learn, and popular with scientists due to its versatility. Python and

R/RStudio are open source which means they are free to download and use, whereas Matlab needs a

license and cannot be downloaded in its full form before you arrive in Oxford. All three systems are

essentially independent of operating system i.e. it will not make a difference if you try something on a

Mac at home and use a Windows machine in Oxford, or vice versa.

We strongly recommend that you do EITHER

BOTH the online introductions to Matlab and R/RStudio. Then download R/RStudio and do the

short data analysis exercises provided.

OR the CodeAcademy Python course

Physics and Philosophy student? You will need to do an exercise in R when you start lab work in Oxford

in the second year, so best to focus on this. However, if you are keen to learn to code, we encourage

you to do the Python course which should equip you well for anything you might need to do later on.

Matlab

An online introduction to Matlab, Matlab Onramp, is available here:

https://matlabacademy.mathworks.com/

The course is web-based (you must register with the provider) and uses an interface similar to Matlab. It

covers the basics of programming in Matlab and develops many of the skills you will need for the first

year computing course. The course providers suggest that the course will take about two hours. You can

work through the sections in any order.

If you do want to try out the full version of Matlab before you come to Oxford, a free 30-day trial can be

downloaded.

https://uk.mathworks.com/programs/trials/trial_request.html

RStudio (R)

The recommended introductory course is called Try R:

http://tryr.codeschool.com/

Like the Matlab course, this uses a web interface that is similar to R, but does not require you to

download the full package. It provides a simple introduction to many of the key concepts, starting at a

very basic level. The sections must be worked through in order, but they are all short.

R can be downloaded from here:

https://cran.r-project.org/

The R interface is relatively simple, and very similar to the Try R environment, with a window to enter

commands (called the “command line”) and with plots appearing in an adjacent window. Once you have

completed TryR online, you should download and install R and do the data analysis exercises that we

provide (at the end of this document for now, but will need a link) in the command line.

In the Physics Teaching Labs we use RStudio, which is a more “friendly” interface for programming in R,

and provides more “point and click” functionality. The underlying language is exactly the same. RStudio

can be downloaded here (note that you will need to download R first)

https://www.rstudio.com/products/rstudio/download/

Python

CodeAcademy run a good introductory Python course:

https://www.codecademy.com/

This online course is longer (13 hours) and goes further than the others. It also provides more general

background to programming than the Matlab and R courses, though all are equally good at introducing

their language. For this reason, if you take this course you will be well prepared to start our computing

course.

Notes

We regret that we are not able to provide IT support before you attend the introductory

lectures at the start of Michaelmas Term, however there are many web resources available

beyond the ones we have listed here.

We do not recommend any specific operating systems or types of computer. We have Windows,

Mac and Linux systems available for undergraduate use, and our students buy many different

types of personal computer.

Introductory Data Analysis with R

Before starting, you should have completed the TryR online course, and downloaded R to your own

computer. Most of the concepts are discussed in our bridging material on Data Analysis and Statistics,

so it is also a good idea to look at this beforehand.

These exercises use a dataset from the famous Michelson-Morley experiment, in which the speed of

light in air was measured in 1879. The dataset is conveniently already stored in R, where it is called

“morley”. To look at it, you can just type

morley

into the command line. Because the dataset contains 100 numbers and a header, you probably won’t be

able to see it all. A more useful way to quickly inspect the data is to try

head(morley)

which only shows the first few lines of data. The first column gives the row number, and the other three

columns are labelled. Each row represents a speed (labelled Speed) recorded in a particular “run” (Run),

and in this case the data are divided into five groups of 20, each labelled an “experiment” (Expt). The

speeds are in units of kms-1 after subtracting 299 000 kms-1. You can see the speed data by typing:

morley$Speed plot(morley$Speed)

One way to obtain a visual overview of data is to produce a histogram. In a histogram, rectangles are

used to represent frequency, with the area of each rectangle proportional to the frequency. A histogram

of the Michelson Morley data can be generated with:

hist(morley$Speed)

If you want to edit the histogram bins, colour and/or labels, these can be added by expanding the

hist() command as follows:

hist(morley$Speed,breaks=25,col="darkgray",xlab="speed - 299 000 km/s")

The concepts of mean and median are probably familiar from school maths, and are also mentioned in

our Data Analysis bridging material. They can be displayed (remembering to include the offset

subtracted earlier), with the following commands:

mean(morley$Speed) + 299000 median(morley$Speed) + 299000

R includes functions that make it very simple to work out the standard deviation and variance of a data

set (again, these are explained fully in Data Analysis). To save time typing, we can define a variable

called Speed before calculating its variance and standard deviation:

Speed = morley$Speed var(Speed) sd(Speed)

Finally, the summary command is a useful way to get an overview of a dataset. This lists the minimum,

maximum, median, mean, first and third quartiles. (If you don’t understand any of these quantities,

check Data Analysis again.)

summary(Speed)

Note that the summary(morley) command also works (try it), although it is not particularly helpful for

this dataset. To continue working out some of the quantities in Data Analysis, how would we calculate

the standard error in the speed measurement? There is no single command for standard error, but it is

easy to calculate, using the length function to work out the number of data points

N=length(Speed) sd(Speed)/sqrt(N)

Reference

This material is mainly taken from Scientific Inference: Learning with Data by Simon Vaughan (CUP,

2014)

http://www.star.le.ac.uk/sav2/stats.html

Vacation Work: Maths & Mechanics

The Oxford Physics course contains a lot of maths and it is important toensure that you are as far as possible up to speed before arriving in Oxford.This is particularly important if you have not studied Further Maths orequivalent. This document gives some advice on preparation before arrival,and a set of problems which you should attempt over the summer.

Maths Textbooks

The recommended maths text for the course is Mathematical Methods forPhysics and Engineering by K. F. Riley, M. P. Hobson and S. J. Bence (3rdedition, Cambridge University Press). This is a very large book, but willcover all the maths you need for the compulsory parts of the physics course.You are advised, if possible, to obtain a copy before coming up to Oxford.

You may initially find this text very tough. A useful transition text isFurther Mathematics for the Physical Sciences by Tinker and Lambourne,which is divided into a number of modules with self assessment tests so thatyou can work out which chapters you need to read.

Introductory Problems

We will assume that you are familiar with Chapter 1 of Riley, Hobson andBence on arrival. This chapter covers the solution of polynomials, trigono-metric identities, coordinate geometry, partial fractions, the binomial expan-sion, and methods of proof. The problems listed below are adapted from theend of this chapter. You may find these problems quite tricky at first, butwith thought they should be possible. If you find them easy then look atsome other problems in Chapter 1.

Calculus

Although the Oxford Physics course in principle covers calculus from scratch,in practice elementary techniques are covered very quickly, and you should

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ensure that you are familiar with these before arrival. This includes differ-entiation and integration of polynomials, trigonometric functions, and expo-nential and logarithmic functions; differentiation of products and quotientsand use of the chain rule; implicit differentiation; integration by parts andby substitution.

Mechanics

The first “physics” subject you will study at Oxford is mechanics. Unlikeschool courses the Oxford course quickly begins to make use of a significantamount of maths, and many students find it hard to combine maths withphysics. This involves taking a problem posed in ordinary language, settingit up in precise mathematical terms, using mathematical techniques to finda solution, and then translating this back into ordinary terms. This processmay not be very familiar to you now but is going to be very importantthroughout your course.

The mechanics problems provided can mostly be solved using techniquesfamiliar from A-level, but it is important to begin by adopting a proper ap-proach straight away: the way in which you solve these problems and set outyour answers is more important than the answers themselves! In particularyou should avoid at all costs simply plugging numbers into standard formulaeusing a calculator: you should solve each problem in the general case beforedealing with the numbers given.

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A. Introductory Problems

1. A polynomial equation can be written in terms of coefficients

f(x) = anxn + an−1x

n−1 + · · ·+ a1x+ a0 = 0

or in terms of roots

f(x) = an(x− α1)(x− α2) · · · (x− αn) = 0.

(a) By expanding these expressions show that

n∑k=1

αk = −an−1

anand

n∏k=1

αk = (−1)na0an

where Π indicates a product of terms just as Σ indicates a sum.

(b) Consider the polynomial g(x) = 4x3 + 3x2 − 6x− 1. Show thatx = 1 is a root, and find the sum and product of the other tworoots.

2. Use double angle formulae to find an expression for cos(4θ) in termsof sin(θ); hence prove that s = sin(π/8) is one of the four roots of

8s4 − 8s2 + 1 = 0

and use your knowledge of the sin function to show that

sin(π/8) =

√2−√

2

4.

Find an expression for cos(π/8) and show that tan(π/8) =√

2− 1.(Hint: first find the “obvious” form for tan(π/8) and then simplifythis using surds.)

3. Show thata sin θ + b cos θ = K sin(θ + φ)

withK2 = a2 + b2 and φ = arctan(b/a).

(Hint: it is much easier to show this “the other way round” byexpanding K sin(θ + φ) using an angle sum formula.)

4. Show that the equation

f(x, y) = x2 + y2 + 6x+ 8y = 0

represents a circle, and find its centre and radius.

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5. Express the following as partial fractions:

(a)2x+ 1

x2 + 3x− 10(b)

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x2 − 3x(c)

x2 + x− 1

x2 + x− 2

(d)2x

(x+ 1)(x− 1)2(e)

2 + 4x

(x+ 2)(x2 + 2)

(In case (c) you should start by dividing to find the quotient andremainder; in cases (d) and (e) you will need three constants.)

6. Use a binomial expansion to evaluate (a) (1 + x)5, (b) 1/(1 + x), and(c) 1/

√1 + x up to second order. Use the last result to find a value of

1/√

4.2 to three decimal places.

7. Prove by induction that

(a)∑nr=1 r = 1

2n(n+ 1)

(b)∑nr=1 r

3 = 14n2(n+ 1)2

8. The sum of a geometric progression can be written as

Sn = 1 + r + r2 + · · ·+ rn

Evaluate rSn and use this to show that Sn = (1− rn+1)/(1− r).

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B. Calculus

Differentiation

1. Differentiate each of the following functions:

(a) f(x) = x2 sin(x) + loge(x)

(b) f(x) = 1/ sin(x)

(c) f(x) = 2x

2. Find the first two derivatives of the following functions:

(a) F (x) = 3 sinx+ 4 cosx

(b) y = loge x x > 0

3. Let x = a(θ − sin θ) and y = a(1− cos θ), where a is a constant; finddydx

in terms of θ. Then find d2y/dx2 in terms of θ.

(Hint: let f(θ) stand for your expression for dydx

in terms of θ; thend2ydx2

= ddx

( dydx

) = ddxf(θ) = df

dθdθdx

.)

Stationary points and graph sketching

4. Under certain circumstances, when sound travels from one medium toanother, the fraction of the incident energy that is transmitted acrossthe interface is given by

E(r) = 4r/(1 + r)2

where r is the ratio of the acoustic resistances of the two media. Findany stationary points of E(r), and classify them as local maxima,minima, or points of inflection. For what value of r is d2E/dr2 = 0?Sketch the graph of E(r) for r > 0, marking the special values of ryou have found.

Hyperbolic functions

5. The hyperbolic functions are defined in terms of the exponentialfunction ex by

sinh(x) =ex − e−x

2cosh(x) =

ex + e−x

2tanh(x) =

sinh(x)

cosh(x)

Sketch each of these functions against x. Do not use a calculator!

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6. Find the first derivatives of sinh(x) and tanh(x) with respect to x,expressing your answers in terms of cosh(x).

Integration

7. Evaluate the following indefinite integrals: (a)∫

(1 + 2x+ 3x2) dx,(b)

∫[sin(2x)− cos(3x)] dx, (c)

∫(et + 1

t2) dt, (d)

∫dw.

8. Evaluate the following definite integrals:

(a)∫ 1/4−1/4 cos(2πx) dx

(b)∫ 30 (2t− 1)2 dt

(c)∫ 21

(1+et)2

etdt

(d)∫ 94

√x(x− 1

x) dx

(e)∫ 1−1 x

3 dx; explain your answer with a sketch.

9. Find the indefinite integral∫x2e−x dx.

10. Find the indefinite integral∫

sinx(1 + cos x)4 dx.

11. Find the area of the region bounded by the graph of the functiony = x2 + 2 and the line y = 5− 2x.

12. Although integrals are frequently thought of as areas under curves, amore general interpretation is possible and useful.

(a) Justify the standard formula for the volume of revolutionobtained by rotating the curve y = f(x) for the intervala < x < b by 2π around the x axis. (Start by slicing the volumeup into infinitely thin coaxial discs and sum the volumes ofthese.) Find the volume of Torricelli’s trumpet, defined byy = 1/x for 1 < x <∞.

(b) Consider an elastic spring of natural length lm which has atension k x2 N when it is stretched by xm. Find the work donewhen this spring is stretched to twice its natural length. (Note:this spring does not obey Hooke’s law and so standard results donot apply; start by finding the work done when the spring isstretched by an infinitesimal amount dx.)

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C. Mechanics

Motion in one dimension

1. A car accelerates uniformly from rest to 80 km per hour in 10 s. Howfar has the car travelled?

2. A stone falls from rest with an acceleration of 9.8 ms−2. How fast is itmoving after it has fallen through 2 m?

3. A car is travelling at an initial velocity of 6 ms−1. It then acceleratesat 3 ms−2 over a distance of 20 m. What is its final velocity?

Work and energy

4. The brakes on a car of mass 1000 kg travelling at a speed of 15 ms−1

are suddenly applied so that the car skids to a stop in a distance of30 m. Use energy considerations to determine the magnitude of thetotal frictional force acting on the tyres, assuming it to be constantthroughout the braking process. What is the car’s speed after thefirst 15 m of this skid?

5. The gravitational potential energy for a mass m at a distance R + hfrom the centre of the earth (where R is the radius of the earth) is−GMm/(R + h) where G is Newton’s gravitational constant and Mis the mass of the earth. If h� R show that this is approximatelyequal to a constant (independent of h) plus mgh, where g = GM/R2.[Hint: write R + h = R(1 + h/R) and expand (1 + h/R)−1 by thebinomial theorem.]

6. Show that the minimum speed with which a body can be projectedfrom the surface of the earth to enable it to just escape from theearth’s gravity (and reach “infinity” with zero speed) is given by

vescape =√

2GM/R, where G, M and R are defined as in theprevious question. If a body is projected vertically with speed12vescape, how high will it get? (Give the answer in terms of R and

neglect air resistance throughout this question.)

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Simple harmonic motion

7. The position of a particle as a function of time is given byx(t) = A sin(ωt+ φ), where A, ω and φ are constants. Obtain similarformulae for (a) the velocity, (b) the acceleration of the particle. Ifφ = π/6, find in terms of A the value of x at t = π/6ω. What is thevalue of x (in terms of A) when the acceleration is greatest inmagnitude?

8. A particle of mass m moves in one dimension under the action of aforce given by −kx where x is the displacement of the body at time t,and k is a positive constant. Using F = ma write down a differentialequation for x, and verify that its solution is x = A cos(ωt+ φ), whereω2 = k/m. If the body starts from rest at the point x = x0 at timet = 0, find an expression for x at later times.

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