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Basic Skills in Physics and Engineering Science

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Basic Skills in Physics and Engineering Science Basic skills in physics and engineering science a guide to critical thinking about modelling Göran Grimvall KTH, Stockholm 2011 © Göran Grimvall Preface Information explosion, interdisciplinary work, lifelong learning and globalization are concepts, which mean that we must be prepared to take in and use new knowledge. In science and technology this knowledge often has a numerical aspect; in the form of data, as explicit mathematical relations, or disguised as computer models whose structure may even be unknown to the user. This challenge is made particularly acute through the Internet. Even if our internet source is reliable – how can we be sure that we are interpreting and using the information correctly? This text is meant to be a guide in this new world. Many of the presented ‘skills’ are what a practicing scientist or engineer acquires during a long life, often without being explicitly aware of it. It is like learning to speak your mother tongue, without attending a formal language course. A quicker way is offered here. Many of the ideas described in these notes are given a more popular presentation in the author's books Brainteaser Physics (2007) and Quantify! – A Crash Course in Smart Thinking (2011). Some aspects are treated in more depth than elsewhere, for instance estimation methods, while dimensional analysis and scaling have been extensively covered in many books. The main feature of the present work is a systematic approach, with examples from very many fields – not only science and engineering but also referring to, e.g., sports, economics, literature, historical aspects and daily life. The text is divided into two parts. Part I contains a number of short ‘essays’ with themes of general interest, illustrating some of the most important ‘skills’. They can be read in any order. Part II gives a systematic approach. That part is also rich in illustrating examples, many of which could have been extended into essays and may be read as such. This text is used in some courses at the Royal Institute of Technology (KTH), Stockholm. Comments of all kinds are very welcome, <[email protected]>. The 2011 edition has a few corrections made in the 2010 edition. Stockholm, August 2011 Göran Grimvall 2 Contents Part 1, Essays Estimations Are we alone? Cover the Earth with paper The energetic bicyclist Socrates and two Nobel laureates Make it simple – but not too simple The leaking water barrel Intuition, hot sauna and black cold snow Method of extreme cases Man overboard Tax rates, golf, and autobahn The coffee problem Modelling complex phenomena A loud party What time is it? Unable to see the wood for the trees Rolling friction and bicycle tires Engineering world Fiction meets school physics and engineering reality Physics vs. engineering Scaling in biology Obesity and the size of fish Gulliver Goethe, trees, and limits to growth Exact or uncertain? Is the square-law exact? Dangerous scaling The final point Part 2, Rules and methods 1. Units 1.1 General about unit systems 1.2 SI units 1.3 SI derived units with special names 1.4 Additional units to the SI 1.5 Other non-SI units 1.6 Chain rule for conversion of units 3 2. Powers of ten 2.1 SI prefixes 2.2 Named powers of ten 2.3 Prefixes in computer science 2.4 First digit in tables 3. Estimations 3.1 Order of magnitude 3.2 Simple scaling from known facts 3.3 Product of estimated factors 3.4 Estimation of an inequality 3.5 Sum of many contributions of different magnitudes 3.6 ‘Egocentric’ arguments 3.7 The ’1 %-rule’ 3.8 Rules of thumb; cost per unit 3.9 Rephrasing the problem as one related to economy 3.10 Rephrasing the problem as one related to energy 3.11 Double counting 3.12 Small changes 4. Dimensional analysis Nomenclature Equations are dimensionally homogeneous Check of formulas Direct ansatz of product of factors Buckingham’s Π-theorem 5 Laws, rules and principles 5.1 The laws of Nature 5.2 Results derived mathematically from fundamental laws 5.3 Ideal models as limiting cases 5.4 Linear response 5.5 Complex phenomena with one dominating contribution 5.6 Fitting complex phenomena through empirical parameters 5.7 ‘Laws’ that give a trend 5.8 Buckingham’s theorem and neglect of effects 6. Modelling in practice 6.1 Make it simple, but not too simple 6.2 Check of expressions in extreme limits 6.3 Check of trends in extreme limits 6.4 Fit to series expansion 6.5 Ignoring second-order effects at extrema 6.6 Phenomenological correction to ideal case 7. Quantification 7.1 Logarithmic scales 7.2 Figures of merit 7.3 Sum of figures of merit 7.4 Categorization scales 7.5 Standardized measurements 4 8. Extrapolation of data 8.1 Simple power law 8.2 Sum of two power laws 8.3 Power law with non-rational exponents 8.4 Exponential law 8.5 Doubling times, the 69 and the 72 rule 8.6 Sum rule for exponential growth 8.7 The long tail 8.8 Sigmoid growth and saturation 9. Scaling and analytic extrapolation 9.1 Simple scale factor 9.2 Scaling of surface area and volume 9.3 Scaling with dimensionless ratios 9.4 Named dimensionless numbers 9.5 Reversal of trends 9.6 Instabilities 5 Is there anybody out there? Arguably the most challenging estimation problem is that which tries to find the number of intelligent civilizations in our galaxy, from which we may receive a message. A conference in 1961 at the National Radio Astronomy Observatory in Green Bank, West Virginia, was the starting point for the ‘search for extraterrestrial intelligence’, usually abbreviated SETI. The conference resulted in the famous Green Bank equation, often referred to as the Drake equation because essential ideas were formulated by Frank Drake. The equation, which seeks to quantify the number of such civilizations, reads N = R fp ne fl fi fc L. N is the number of civilizations in our galaxy (the Milky Way) whose radio emissions are now detectable; R is the rate of formation of stars in the Milky Way, expressed as an average number per time; fp is the fraction of those stars that form planetary systems; ne is the number of planets in those systems that are ecologically suitable for life forms; fl is the fraction of those planets, on which life forms actually develop; fi is the fraction of planets with life forms that then evolve to an intelligent form; fc is the fraction of planets where the intelligent life also develops technology that releases detectable signals; L is the average ‘life time’ during which such advanced civilizations release signals. At the 1961 conference it was suggested that 1 4 R = 10 a− fp = 0.5 ne = 2 fl = 1 fi = fc = 0.01 L = 10 a. These numbers give N ≈ 10, thus implying that it may be worth searching for extraterrestrial life. We now take a closer look at the values assigned to the parameters. 1 The mean rate of star formation in the Milky Way is well understood, and R = 10 a− is widely accepted to be reasonable.1 At the time of the Green Bank conference one had no information about the existence of planets around stars other than our own Sun, and fp = 0.5 was considered reasonable. Many stars are now known to be surrounded by planets.2 Furthermore, if our solar system is regarded as typical, one expects that there is often at least one planet with conditions3 suitable for the development of life. Therefore there is no reason for a change 6 in the original assumption fpne ≈ 1. Next comes the question if life actually develops on planets that are ‘suitable’ for life. It has been argued that because life was established early in the history of the Earth, and because there are life forms that evolve also under the harsh conditions of ocean depths and extreme colds, fl should be close to 1. It is difficult to estimate the probability fi that life, once it has been established, also develops to an intelligent form. Equally difficult is it to estimate the probability fc that intelligent life at some stage develops a technology that sends out signals, which we can detect. The originally suggested average life time L = 104 years was considered by some people to be much too long. One must then remember that human self annihilation, e.g., through nuclear war, was at that time much debated. Also today many think that a technologically advanced life form will not live long before it destroys itself. If technologically advanced civilizations can avoid self annihilation during a period that is as long as the existence of Homo sapiens (over 100 000 years), our data give N = 100 and the consequences suggested by the Drake equation are enormous for how we view ourselves.5,6 Finally it should be remarked that the Drake equation as presented here is meant to illustrate a technique in estimations, and its numerical results for extraterrestrial life should not be taken too seriously. Comments 1 The symbols for time units are: a for year (annum), h for hour, min for minute, s for second. The abbreviations yr, hr and sec, which are language dependent, shall not be used. 2 Planets around stars are too small to be seen from the Earth. In 1995 variations in the motion of the star 51 Pegasi were interpreted as caused by an orbiting planet. About of stars having planets have now (2010) been identified. 3 One usually assumes the presence of liquid water and an atmosphere.
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