Order-of-Magnitude Physics

Understanding the World with Dimensional Analysis, Educated Guesswork, and White Lies

Sanjoy Mahajan University of Cambridge Sterl Phinney California Institute of Technology Peter Goldreich Institute for Advanced Study

Copyright c 1995–2006 Send comments to sanjoy@mrao.cam.ac.uk Draft of 2006-03-20 23:51:19 [rev 22987c8b4860]

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Contents

1 Wetting your feet 1 1.1 Armored cars 1 1.2 Cost of lighting Pasadena, California 1.3 Pasadena’s budget 11 1.4 Diaper production 12 1.5 Meteorite impacts 14 1.6 What you have learned 16 1.7 Exercises 17 2 Some ﬁnancial math 18 2.1 Rule of 72 18 2.2 Mortgages: A ﬁrst approximation 2.3 Realistic mortgages 21 2.4 Short-term limit 22 2.5 Long-term limit 23 2.6 What you have learned 24 2.7 Exercises 25 Bibliography 26 9

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2006-03-20 23:51:19 [rev 22987c8b4860]

1

1 Wetting your feet

Most technical education emphasizes exact answers. If you are a physicist, you solve for the energy levels of the hydrogen atom to six decimal places. If you are a chemist, you measure reaction rates and concentrations to two or three decimal places. In this book, you learn complementary skills. You learn that an approximate answer is not merely good enough; it’s often more useful than an exact answer. When you approach an unfamiliar problem, you want to learn ﬁrst the main ideas and the important principles, because these ideas and principles structure your understanding of the problem. It is easier to reﬁne this understanding than to create the reﬁned analysis in one step. The adjective in the title of the book, order of magnitude, reﬂects our emphasis on approximation. An order of magnitude is a factor of 10. To be ‘within an order of magnitude’, or to estimate a quantity ‘to order of magnitude’, means that your estimate is roughly within a factor of 10 on either side. This chapter introduces the art of determining such approximations. Writer’s block is broken by writing; estimator’s block is broken...

Understanding the World with Dimensional Analysis, Educated Guesswork, and White Lies

Sanjoy Mahajan University of Cambridge Sterl Phinney California Institute of Technology Peter Goldreich Institute for Advanced Study

Copyright c 1995–2006 Send comments to sanjoy@mrao.cam.ac.uk Draft of 2006-03-20 23:51:19 [rev 22987c8b4860]

ii

Contents

1 Wetting your feet 1 1.1 Armored cars 1 1.2 Cost of lighting Pasadena, California 1.3 Pasadena’s budget 11 1.4 Diaper production 12 1.5 Meteorite impacts 14 1.6 What you have learned 16 1.7 Exercises 17 2 Some ﬁnancial math 18 2.1 Rule of 72 18 2.2 Mortgages: A ﬁrst approximation 2.3 Realistic mortgages 21 2.4 Short-term limit 22 2.5 Long-term limit 23 2.6 What you have learned 24 2.7 Exercises 25 Bibliography 26 9

19

2006-03-20 23:51:19 [rev 22987c8b4860]

1

1 Wetting your feet

Most technical education emphasizes exact answers. If you are a physicist, you solve for the energy levels of the hydrogen atom to six decimal places. If you are a chemist, you measure reaction rates and concentrations to two or three decimal places. In this book, you learn complementary skills. You learn that an approximate answer is not merely good enough; it’s often more useful than an exact answer. When you approach an unfamiliar problem, you want to learn ﬁrst the main ideas and the important principles, because these ideas and principles structure your understanding of the problem. It is easier to reﬁne this understanding than to create the reﬁned analysis in one step. The adjective in the title of the book, order of magnitude, reﬂects our emphasis on approximation. An order of magnitude is a factor of 10. To be ‘within an order of magnitude’, or to estimate a quantity ‘to order of magnitude’, means that your estimate is roughly within a factor of 10 on either side. This chapter introduces the art of determining such approximations. Writer’s block is broken by writing; estimator’s block is broken...