Saturday, November 12, 2011

Fermi Problems, Introduction

“Fermi Problems” is one of the neatest things you will learn from taking physics: you can calculate anything, simply and efficiently. This is worth the price of admission. The only tools needed are your imagination the analytical power of your brain and a pencil and paper. You guess data, estimate sizes and make simple assumptions for your calculations. The title of one of the books recommended below pretty much gives away the whole trick: “Assume a spherical cow.”
Fermi Problems is a standard problem solving technique commonly taught in high school and college physics classes. It involves  essentially nothing more than solving word problems without having problem-relevant data.
For example how many ping-pong balls will fit into a suitcase, where the suitcase and ping-pong ball sizes are left undefined. Sometimes a hint of data can be gotten like from watching pieces of torn paper float in the wind.  The point is to concentrate on the physics of a situation and not waste time worrying about exact numerical answers (some people will say that’s the engineer’s job anyway).
In my days of yore (slide rule times) my physics professors had a standing rule that as long the answer was within half an order of magnitude it was correct; nobody cared about the numbers, just the thinking.  
This concept has even made its way into employment interviews where it’s been reported interviewees are asked to give a rough estimate, using only their brain, to some seemingly bizarre problem—like how many dump trucks are needed to haul away Mt. Fuji (hint: it’s a fairly well defined cone)—to show they are capable of rational analysis and reasonable problem solving abilities.  
Fermi problems are a good way to sharpen your intuition, but who was this Fermi guy and why did he get an eponymous genera of physics problems named after him?
Fermi was an Italian physicist, both theoretical and experimental (eponymous “Fermi level” anyone?), who was experimenting on why atoms disintegrated when bombarded with alpha particles. He emigrated to America in the late 1930s; built the first nuclear reactor; demonstrated the uranium fission chain reaction; and worked at Los Alamos. One early morning in 1945 he found himself standing on a desert in New Mexico gazing south into the darkness wondering if the first atomic bomb was going to work or not. He was a crafty applied physicist too and since there was only minimal instrumentation available to measure the yield of the “gadget” he thought of a way to make his own measurements while waiting for the official results.  The bomb goes off as planned with a big fire ball and a shockwave starts propagating away from the blast. Fermi waits until it gets to him, drops some tiny pieces of paper he had ripped up from shoulder height and watches to see how far the shock wave takes them before they hit the ground.  He estimates the distance they traveled and looks this number up on a pre-calculated table to find the yield was at least 10 kilotons of TNT.  The rest is history (the actual yield from instrumentation was 20 kilo tons, but at least Fermi knew things had gone per theory).
This story give the outlines of doing Fermi problems: collect or even just assume some simple data; use a simple model to analyze the data; critique the answer for plausibility.
Fermi problems have a serious side too. Physicist find it useful to check their work against simple models and calculations to make sure they are on the right track often even making a rough hand drawn graph of the data to make sure an experiment is performing properly.
Over the years a few books have been written by researchers to illustrate the techniques of quickly calculating results. The following is a short bibliography of some of these books.
“Used Math For the First Two Years of College Science,” Clifford E. Swartz, p.4. 1993, American Assoc. of Physics Teachers. This book also contains a splendid review of all the math one needs in college.
“Guesstimation Solving the World’s Problems on the Back of a Cocktail Napkin,” Lawrence Weinstein and John A. Adam, Princeton U. Press, 2008. 73 Fermi questions posed and answered.
“The Cosmological Milkshake A Semi-Serious Look at the Size of Things,” Robert Ehrlich, 1994. 135 somewhat off-beat, but important Fermi Questions.
“Consider a Spherical Cow A Course in Environmental Problem Solving” and “Consider a Cylindrical Cow More Adventures in Environmental Problem Solving,” John Harte, 1988 and 2001 University Science Books. These are more focused questions with deeper computations that show how Fermi Questions can be made to give real quantitative insight.
“The Flying Circus of Physics With Answers,” Jearl Walker, 1977, John Wiley & Sons. Interesting questions most demanding only an intuitive answer, so perhaps the simplest Fermi Problems, but the answers require real physical insight.
Other references and historical information can be found at:
Finally, what can go wrong? Well, plenty, if one is not careful in assuming data for the calculation. Ludwig Prandtl a founder of the science of aerodynamics was at a fancy dinner party early in the 20th century. A guest asked how it was that a bumble bee can fly. He immediately made some calculations on the back of a menu and concluded by announcing that he had just proven that it is not theoretically possible for bumble bees to fly! So one must choose input data wisely for the Fermi Problem to work! Oh, and yes, performing Prandtl’s calculation with more accurate values from biology shows it is perfectly acceptable for bumble bees to fly.
Note added Nov. 15th 2011.
It has recently come to my attention that Tom Murphy, Professor of Physics at UC San Diego, has a blog called “Do the Math,” somewhat focused on the environment, but other things as well, where he uses Fermi-type computations to get a handle on challenges in our future. This is a very interesting blog, highly recommended!
Subsequent post will illustrate a couple of Fermi Problem calculations I have made concerning measuring the earth’s axial tilt and the size of rocks in the rings of Saturn. Stay tuned!

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