FERMI PROBLEMS

Every day in lecture, I will give out a Fermi problem. The ability to do Fermi problems is one of the most important traits of a good scientist, and many who took this class years ago have come back and told me that it was one of the more valuable skills they learned in the class.

At first, one often thinks that a Fermi problem is impossible to answer, but simple logical reasoning can lead one to an approximate answer. Each step in a Fermi problem will be uncertain by a factor of two or more, but sometimes the uncertainty is in one direction, sometimes in the other, and the uncertainties do tend to balance. The hope is to get the answer within a factor of two or so—a factor of ten is often ok..

The classic Fermi problem was asked by Fermi during graduate oral exams. His most famous question was “How many piano tuners are in Chicago”. The logical steps are as follows: What is the population of Chicago? I have no idea, but it’s more than a million and less than 10 million, so I’ll guess 2 million. How many families? Some are big, some are small---I’ll guess 4 people per family, or 500,000 families. How many pianos? I suspect that more than 1% of families have pianos, and certainly less than 20% have them, so I’ll guess 10% have pianos, so there are 50,000 pianos. How many tunings does a piano need per year? Certainly at least one every couple of years, and less than twice a year, so I’ll guess once a year—50,000 tunings. How many pianos can a tuner tune in a year? Well, it takes a couple of hours, so maybe 2-3 per day, or 10-15 per week, for 50 weeks is around 500-700, so I’ll guess 600 tunings per tuner per year. For 600 tunings of 50,000 pianos, you need around 80 tuners. The estimate is 80 tuners. (Note: in 2004, I counted---there were 60). Note that each estimate could easily be off by a factor of a few, but the final answer is usually fairly close.

Over the term, some Fermi problems will be very silly (like piano tuners in Chicago), and some will be extremely relevant for medicine (involving epidemiological questions and public health issues). By the end of the semester, you will understand why this is such an important skill.

Although you can talk to friends about your ideas about how to do Fermi problems, you should work out the numbers on your own. Also, you should NOT look up anything on the internet—the point of these problems is to figure things out on your own, and any additional knowledge needed will be given with the problem. Finally, no Fermi problem solution should EVER have more than two significant figures---if they have many more, you’ll lose half the credit.

Fermi problems will be turned in at the table by the back door every Friday (not counting the first Friday) before class. During class Friday, I’ll give you my personal answers to the problems, and show you my rationale. There are no right answers---all that is important is for your reasoning to be somewhat logical. Your answers should explain your reasoning—no credit for just an answer. They won’t be turned back to you in problem session---when finished they’ll be put into a box just outside my office.

Following are numbers that may prove useful in doing Fermi problems:

Population of the US = 3 x 10⁸

Population of the World = 7 x 10⁹

Number of objects/mole (Avogadro’s number) = 6 x 10²³

Mass of one mole of hydrogen = 1 gram

Size of an atom = 10^-10 meters

Radius of the Earth = 6400 km

Earth-Sun distance – 1.5 x 10¹¹ meters

1 m/sec = 2 miles/hour

1 year = 3 x 10⁷ seconds

Energy of a typical chemical reaction = 1.5 eV

1 Joule = 6 x 10¹⁸ eV

Density of water or the human body = 1000 kg per cubic meter

Density of air = 1.3 kg per cubic meter

Density of solid = a few times that of water