LADY MARGARET HALL Oxford OX2 6QA
15 August 2012
Dear Physics Student, Preliminary reading for the Physics course
I am enclosing some notes on Physics at LMH, to give you some information about what to expect when you arrive here. I am also enclosing some suggestions suggestions for preliminary reading, reading, and a set of problems in Maths and Mechanics, for you to tackle before you you arrive in Oxford. This gives you a chance to consolidate consolidat e some of your A Level course (or other school work) and start getting acquainted with the material you will be covering covering next year. year. The idea is to try to make the transition transition from school work to university work as smooth as possible! Another purpose of these problems is to give us an idea of what things you have covered at school, so first try the questions on material material that is familiar to you, omitting the others. Please bring your your solutions when you come up in October, and we shall go through them in tutorials. If you do have time to read up any unfamiliar material now, and answer the questions on it, then feel free to do so. But don’t spend too long struggling struggling with these questions, questions, or become discouraged, discouraged, if you get get badly stuck. Most of this material will be covered during the first-year first-year course. I look forward to seeing you in October. Meanwhile, please don’t hesitate to contact me if you have any queries.
Yours sincerely,
David Andrews
(Senior Physics Tutor, Lady Margaret Hall)
[email protected]
Physics at Lady Margaret Hall These notes are intended to give you some information on the course of study you will be starting in October. They should be read in conjunction with the Physics Department’s Notes for Incoming Freshers, which you will receive separately. There are currently three Physics Tutors at LMH, Dr Todd Huffman, Dr Gianluca Gregori and Professor David Andrews, all of whom are Fellows or Lecturers of the College and members of the University’s Physics Department. They will give many of your tutorials during the first two years (though Dr Huffman will be away on research leave for 2012-13), with some being given by other tutors such as Professor Niranjan Thatte. You will normally have two one-hour tutorials per week (together with one other student). Some topics may be taught in larger classes. For each tutorial you will usually be set a selection of problems, mostly based on material that has recently been covered in lectures. Your solutions should be handed in before tutorials, so that the tutor can look through them in advance. During the tutorial the tutor will go through your work with you, correcting any errors and discussing any difficulties that you may have encountered. The tutor will also answer questions relating to the material covered in lectures, mention wider implications and connections with other parts of physics, and provide other relevant background. Your tutors monitor your progress week by week; at the end of each term, they write reports on your work. If the reports highlight any particular weaknesses, your tutors will discuss with you what measures you need to take. Tutorials give you the opportunity to discuss physics in detail on a regular one-to-one basis with your tutors. On the other hand, lectures (given in the Physics Department to the 180 first-year physicists from all the colleges) are the primary means of presenting the bulk of the material on the syllabus. It is very important that you should attend all relevant lecture courses. 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. Your tutors will be happy to advise on this. At the beginning of Hilary and Trinity Terms (in January and April), there are college examinations (called ‘Collections’) on work covered during the previous terms: t hese give you some exam practice, and are a further way of monitoring your progress. During the first year, approximately half of the tutorial work is devoted to physics and half to mathematics. The reason for the large proportion of mathematics is that it is an essential tool for most of the physics material covered in the course: virtually all of the mathematics that is done is used in some aspect of the physics course. (However, the proportion of mathematics is much lower in the later years.) In addition to lectures and tutorial work, you will spend about one day per week doing practical work in the Physics Department. You keep a record of your practical work in ‘daybooks’, which are checked by the Lab Demonstrators. Some of your practicals have to be written up in detail and marked. A termly progress report on your lab work is passed to your college tutors. You must complete a total of 15 days’ credit during the first year. At the end of the first year, in June, you take the Preliminary Examination (‘Prelims’). This consists of four compulsory papers, plus one short option paper.
Since the Oxford term is only eight weeks in length, life is pretty hectic, and you will have to work conscientiously to keep up to schedule! Revision and consolidation during each vacation - going through the previous term’s lecture notes and tutorial work, and making sure that you fully understand them (with help from textbooks, where necessary) - are an essential supplement to the term-time coursework. The Lady Margaret Hall library is well stocked with Physics books, which you can borrow for extensive periods. It includes multiple copies of the most popular and important texts, so it is not necessary for you to purchase lots of expensive books for your own use if you do not wish to do so. If you find that a book you need is not in the library catalogue (or there are not enough copies) please ask your tutors, who will be happy to arrange for one to be ordered, if it i s in print.
Vacation Work: Maths and Mechanics The Oxford Physics course contains a lot of maths, and it is important to ensure 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 or equivalent. This document gives some advice on preparation before arrival, and a set of problems which you should attempt over the summer. Maths Textbooks
A recommended maths text for the course is Mathematical Methods for Physics and Engineering by K. F. Riley, M. P. Hobson and S. J. Bence (3rd edition, Cambridge University Press). This is a very large book, but will cover all the maths you need for the compulsory parts of the physics course. However, there are many others that cover much the same ground, including Mathematical Methods nd in the Physical Sciences, by M L Boas (2 Edition, Wiley). The LMH library has multiple copies of both of these, so you can borrow them when you require them. When you’ve settled in to the course, you may decide it’s worth buying your own copy of one of them. You may initially find these texts very tough. A useful transition text is Further Mathematics for the Physical Sciences by Tinker and Lambourne, which is divided into a number of modules with self assessment tests so that you can work out which chapters you need to read. Introductory Problems
We will assume that you are familiar with the topics in Chapter 1 of Riley, Hobson and Bence on arrival. This chapter covers the solution of polynomials, trigonometric identities, coordinate geometry, partial fractions, the binomial expansion, and methods of proof. The problems li sted below are adapted from the end of this chapter. You may find these problems quite tricky at first, but with thought they should be possible. Calculus
Although the Oxford Physics course in principle covers calculus from scratch, in practice elementary techniques are covered very quickly, and you should ensure that you are familiar with these before arrival. This includes differentiation and integration of polynomials, trigonometric functions, and exponential and logarithmic functions; differentiation of products and quotients and use of the chain rule; implicit differentiation; integration by parts and by substitution.
Mechanics
The first “physics” subject you will study at Oxford is mechanics. Unlike school courses the Oxford course quickly begins to make use of a significant amount of maths, and many students find it hard to combine maths with physics. This involves taking a problem posed in ordinary language, setting it up in precise mathematical terms, using mathematical techniques to find a solution, and then translating this back into ordinary terms. This process may not be very familiar to you now but is going to be very important throughout your course. The mechanics problems provided can mostly be solved using techniques familiar from A-level, but it is important to begin by adopting a proper approach straight away: the way in which you solve these problems and set out your answers is more important than the answers themselves! In particular you should avoid at all costs simply plugging numbers into standard formulae using a calculator: you should solve each problem in the general case before dealing with the numbers given. Special Relativity
Later in the first term you will study Special Relativity. For preliminary reading we strongly recommend the recently-published book The Wonderful World of Relativity by our Oxford colleague Andrew Steane (Oxford University Press). This is specifically aimed at people like you, with a good A-level background in Physics and Maths.
Other reading
If you have time before you arrive in Oxford, you might also look at some of the following: Maths: It would be useful to read something about the following, more advanced, topics. Use whatever books you have access to, or look at these sections of Riley et al. (R) or Boas (B):
Complex numbers: (R §3.1-3.5, B §2.1-2.5); Determinants: (R §8.9, B §3.3); Taylor's theorem: (R §4.6, B §1.12); Partial differentiation: (R §5.1, B §4.1); Differential equations: (R § 14.1-14.2.4, B § 8.1-8.5); Vectors: (R §7.1-7.7 B §3.4-3.5).
Physics: The initial work in Physics will concentrate on mechanics and special relativity. Later in the First Year you will go on to vibrations, waves, optics, and electromagnetism, with optional courses on topics like introductory quantum mechanics, atomic physics and astrophysics.
W.J. Duffin's Electricity and magnetism (Fourth edition, McGraw-Hill), chapters 1-3 and 6-7, would be good preliminary reading on electromagnetism. The book by I.G. Main Vibrations and waves in physics (Cambridge), chapters 1-3, is also worth looking at, if you can find it. There are several other good texts with similar titles. You may enjoy dipping into R.P. Feynman's Lectures on Physics (Addison-Wesley), vol. I. And Feynman’s ‘popular’ writings, several paperbacks by P.C.W. Davies, and books like J. Gribbin’s In search of Schrödinger's cat all give interesting background on modern physics and physicists.
A. Introductory Problems 1. Consider the 3rd order polynomial g(x) = 4x3 + 3x2 − 6x − 1 (a) Show that g(x) must have at least one real root. (b) Make a table of g(x) for integer values of x between −2 and 2, and hence find one root by inspection. (c) Factorise the equation as a product of the first root and a quadratic, and hence find values of all three roots. 2. A polynomial equation can be written in terms of coefficients f (x) = a n xn + an−1 xn−1 + · · · + a1x + a0 = 0 or in terms of roots f (x) = a n(x − α1 )(x − α2 ) · · · (x − αn ) = 0. (a) By expanding these expressions show that n
α = −a k
k=1
n−1
an
n
and
α = (−1) a k
k=1
0
n
an
where Π indicates a product of terms just as Σ indicates a sum. (b) Show that these formulae give the right results for g(x) in the previous question. 3. Use double angle formulae to find an expression for cos(4θ) in terms of 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 simplify this using surds.) 3
4. Show that a sin θ + b cos θ = K sin(θ + φ) with K 2 = a 2 + b2
and φ = arctan(b/a).
(Hint: it is much easier to show this “the other way round” by expanding K sin(θ + φ) using an angle sum formula.) 5. Show that the equation f (x, y) = x 2 + y 2 + 6x + 8y = 0 represents a circle, and find its centre and radius. 6. Express the following as partial fractions: (a)
2x + 1 x2 + 3x − 10
(b)
4 x2 − 3x
(c)
x2 + x − 1 x2 + x − 2
(In the last case you will have to start by identifying multiples of the denominator in the numerator and subtracting these from the fraction.) 7. 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. 8. Prove by induction that (a)
n 1 r=1 r = 2 n(n + 1) n 1 2 3 2 r=1 r = 4 n (n + 1)
(b)
9. The sum of a geometric progression can be written as S n = 1 + r + r 2 + · · · + r n Evaluate rS n and use this to show that S n = (1 − rn+1 )/(1 − r).
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B. Calculus Differentiation 1. Differentiate each of the following functions: (a) f (x) = x 2 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) = 3sin x + 4 cos x (b) y = loge x x > 0 3. Let x = a(θ − sin θ) and y = a(1 − cos θ), where a is a constant; find dy in terms of θ. Then find d2 y/dx2 in terms of θ. dx dy (Hint: let f (θ) stand for your expression for dx in terms of θ; then d y df dθ d dy d = dx ( dx ) = dx f (θ) = dθ .) dx dx 2
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Stationary points and graph sketching 4. Under certain circumstances, when sound travels from one medium to another, the fraction of the incident energy that is transmitted across the interface is given by E (r) = 4r/(1 + r)2 where r is the ratio of the acoustic resistances of the two media. Find any stationary points of E (r), and classify them as local maxima, minima, or points of inflection. For what value of r is d2 E/dr 2 = 0? Sketch the graph of E (r) for r > 0, marking the special values of r you have found.
Hyperbolic functions 5. The hyperbolic functions are defined in terms of the exponential function ex by ex − e−x sinh(x) = 2
ex + e−x cosh(x) = 2
tanh(x) =
sinh(x) cosh(x)
Sketch each of these functions against x. Do not use a calculator! 5
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: 2
(1 + 2x + 3x ) dx (b) [sin(2x) − cos(3x)] dx (c) (e + ) dt (d) dw (a)
1 t2
t
8. Evaluate the following definite integrals: 1/4 −1/4
cos(2πx) dx (b) (2t − 1) dt dt (c) √ x(x − ) dx (d) x dx; explain your answer with a sketch. (e) 9. Find the indefinite integral x e dx. 10. Find the indefinite integral sin x(1 + cos x) dx. (a)
3 0
2
2 (1+et )2 1 et 9 4 1 −1
1 x
3
2 −x
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11. Find the area of the region bounded by the graph of the function y = x 2 + 2 and the line y = 5 − 2x.
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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. How far has the car travelled? 2. A stone falls from rest with an acceleration of 9.8 ms−2 . How fast is it moving after it has fallen through 2 m? 3. A car is travelling at an initial velocity of 6 ms−1 . It then accelerates at 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 of 30 m. Use energy considerations to determine the magnitude of the total frictional force acting on the tyres, assuming it to be constant throughout the braking process. What is the car’s speed after the first 15 m of this skid? 5. The gravitational potential energy for a mass m at a distance R + h from 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 M is the mass of the earth. If h R show that this is approximately equal 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 the binomial theorem.] 6. Show that the minimum speed with which a body can be projected from the surface of the earth to enable it to just escape from the earth’s gravity (and reach “infinity” with zero speed) is given by vescape = 2GM/R, where G, M and R are defined as in the previous question. If a body is projected vertically with speed 1 v , how high will it get? (Give the answer in terms of R and 2 escape 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 by x(t) = A sin(ωt + φ), where A, ω and φ are constants. Obtain similar formulae for (a) the velocity, (b) the acceleration of the particle. If φ = π/6, find in terms of A the value of x at t = π/6ω. What is the value of x (in terms of A) when the acceleration is greatest in magnitude? 8. A particle of mass m moves in one dimension under the action of a force 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 differential equation for x, and verify that its solution is x = A cos(ωt + φ), where ω 2 = k/m. If the body starts from rest at the point x = A at time t = 0, find an expression for x at later times.
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