Order-of-Magnitude PhysicsUnderstanding the World with Dimensional Analysis,Educated Guesswork, and White LiesSanjoy MahajanUniversity of CambridgeSterl PhinneyCalifornia Institute of TechnologyPeter GoldreichInstitute for Advanced StudyCopyr ightc 1995–2006Send comments to [email protected] of 2006-03-20 23:51:19 [rev 22987c8b4860]iiContents1 Wetting your feet 11.1 Armored cars 11.2 Cost of lighting Pasadena, Californi a 91.3 Pasadena’s budget 111.4 Diaper pro d uction 121.5 Meteorite impacts 141.6 What you have learned 161.7 Exercises 172 Some financial math 182.1 Rule of 72 182.2 Mortgages: A first approximation 192.3 Realistic mortgages 212.4 Short-term limit 222.5 Long-term limit 232.6 What you have learned 242.7 Exercises 25Bibliography 262006-03-20 23:51:19 [rev 22987c8b4860]11 Wetting your feetMost technical education emph asizes exact answers. If you are a physi-cist, you s olve for the energy levels of the hydrogen atom to six dec-imal places. If you are a chemist, you measure reaction rates andconcentrations to two or three decimal places. In this book, you learncomplementary skills. You learn that an approximate answer is notmerely good enough; it’s often more useful than an exact answer.When you approach an unfamiliar problem, you want to learn firstthe main ideas and the important principles, because these ideas andprinciples structure your understanding of the problem. It is easier torefine this understanding than to create the refined analysis in onestep.The adjective in the title of the book, order of magnitude,reflects our emphasis on ap proximation. An order of magnitude is afactor of 10. To be ‘within an order of magnitude’, or to estimate aquantity ‘to order of magnitude’, means that your estimate is roughlywithin a factor of 10 on either side. This chapter introduces the artof determining such approximations.Writer’s block is broken by writing; estimator’s block is broken byestimating. So we begin our study of approximation using everydayexamples, such as estimating budgets or annual production of diapers .These warmups flex your estimation muscles, which may have laindormant thr ough many years of traditional education.Everyday estimations provide practice for our later problems, andalso provid e a method to sanity check information that you see. Sup-pose that a newspaper article says that the annual cost of healthcare in the United States will soon surpass $1 trillion. Whenever youread any such claim, you should automatically think: Does this num-ber seem reasonable? Is it far too small, or far too large? You needmethods for such estimations, methods that we develop in severalexamples. We dedicate the first example to physicists who need em-ployment outside of physics.1.1 Armored carsHow much money is there in a fully loaded Brinks armored car?The amount of money depends on the size of the car, the d en om-ination of the bills, the volume of each bill, the amount of air be-tween the bills, and many other factors. The question, at first glance,seems vague. One important skill that you will learn from this text,2006-03-20 23:51:19 [rev 22987c8b4860]1. Wetting your feet 2by practice and example, is what assumptions to make. Because wedo not need an exact answer, any reasonable set of assumptions willdo. Getting started is more important than dotting every i; make anassumption—any assumption—and begin. You can correct the grosslies after you have got a feeling for the problem, and have learnedwhich assumptions are most critical. If you keep silent, rather thantell a gross lie, you never discover anything.Let’s begin with our equality conventions, in ascending order ofprecision. We use ∝ for proportionalities, where the units on the leftand right sides of the ∝ do not match; for example, Newton’s secondlaw could read F ∝ m. We use ∼ f or dimensionally correct relations(the units do match), which are often accurate to, say, a factor of 5in either direction. An example iskinetic energy ∼ Mv2. (1.1)Like the ∝ s ign, the ∼ sign indicates that we’ve left out a constant;with ∼, the constant is dimensionless. We use ≈ to emphasize th at therelation is accurate to, say, 20 or 30 percent. Sometimes, ∼ relationsare also that accurate; the context will make the distinction.Now we return to the armored car. How much money does itcontain? Before you try a sy s tematic method, take a guess. Make it aneducated guess if you have some knowledge (perhaps you work for aninsurance company, and you happened to write the insurance policythat the armored-car company bought); make it an uneducated guessif you have no knowledge. Then, after you get a more reliable estimate,compare it to your guess: The wonderful learning machine that is yourbrain magically improves your guesses for the next problem. You trainyour intuition, and, as we see at the end of this example, you aid yourmemory. As a pure guess, let’s say that the armored car contains$1 million.Now we introduce a systematic method. A general method in manyestimations is to break the problem into pieces that we can handle:We divide and conquer. The amount of money is large by every-day standards; the largeness suggests that we break the problem intosmaller chunks, which we can estimate more reliably. If we know thevolume V of the car, and the volume v of a us bill, then we can countthe bills inside the car by dividing the two volumes, N ∼ V/v. Afterwe count the bills, we can worry abou t the denominations (divide andconquer again). [We do not want to say that N ≈ V/v. Our volumeestimates may be in error easily by 30 or 40 percent, or only a frac-tion of the storage space may be occupied by bills. We do not wantto commit ourselves. We h ave divided the problem into two simplersubproblems: determining the volume of the car, and determining thevolume of a bill. What is the volume of an armored car? The stor-age space in an armored car has a funny shape, with ledges, corners,nooks, and crann ies; no s imple formula would tell us the volume, even2006-03-20 23:51:19 [rev 22987c8b4860]1. Wetting your feet 32 m2 m2 mFigure 1.1. Interior of a Br i nks ar-mored car. The actual shape is irreg-ular, but to order of magnitude, theinterior is a cube. A person can prob-ably lie down or stand up with roomto spare, so we estimate the volume asV ∼ 2 m × 2 m × 2 m ∼ 10 m3.1. ‘I seen my opportunitie s and I took’em.’—George Washington Plunkitt, ofTammany Hall, quoted