Fermi Problems Physics Estimation
Physicists Often Use Order of Magnitude or Power of Ten Estimates
Sep 9, 2008
A good early step for solving difficult physics (and other) problems is estimating the order of magnitude (or power of ten) of the answer. These estimates are also called back of the envelope calculations or Fermi problems. Physicist, Enrico Fermi, often asked his students these types of questions. For order of magnitude estimates, one might estimate whether the answer is about one, ten, one hundred, or one thousand. One does not worry whether the answer is ten, twenty, or thirty. One does not even think about the second significant digit.
Why Make Order of Magnitude Estimates
The estimated answer provides a good quick check. This check is especially useful for complex problems; simply ignore the complexities to get a quick estimate. After accounting for the complexities, if an answer is the same order of magnitude as the estimate, it may be ok. Otherwise, look for errors in the solution.
Estimate are also useful when the numbers used in the calculation are poorly known. In such cases a calculation to many significant figures is unjustified. After the input numbers for the calculation are better determined, more precise calculations may be justified.
Estimates can provide insight into problems. A scientists who does an estimate for a difficult problem will have a better understanding of what quantities are needed to solve the problem more accurately. Then it is possible to design the experiments or calculations needed to get accurate values for these quantities.
How Many Piano Tuners Are There in Chicago?
According to physics folklore, this question was one of Enrico Fermi's favorite estimation problems. If Fermi had really cared how many piano tuners there were in Chicago, consulting the Chicago business listings would have easily given an accurate answer. The point however was exercising his students' problem solving abilities.
To answer this question first assume that the number of piano tuners approximately equals the demand for piano tuners. Next estimate the demand.
Don't look up accurate census figures. Chicago has a few million people. If one guesses that roughly 10% of the population owns pianos, there are a few hundred thousand pianos in Chicago. For this type of estimate, don't poll Chicago's citizens on their musical instruments. Simply choose between the orders of magnitude. One percent is too small and one hundred percent is too large, so ten percent is close enough.
Pianos belonging to the Chicago Symphony must be tuned very often. Pianos in the basement from piano lessons in the distant past may never be tuned. A guessed rough average might be that pianos are tuned yearly. So the Chicago piano tuners must collectively tune a few hundred thousand pianos per year.
A piano tuner might tune a several (between one and ten) pianos a day and work a few hundred days per year. Several multiplied by a few hundred gives a thousand to a few thousand, so a piano tuner could reasonably expect to tune a few thousand pianos per year. Hence about a hundred piano tuners are needed to collectively tune a few hundred thousand pianos per year. In this context, a hundred means there are somewhere between about ten and about a thousand piano tuners in Chicago.
For real scientific problems, a physicist might make such an estimate and gain insight into the problem. Next the physicist would find more accurate values of the necessary numbers and take into account various complicating factors to refine the calculation.
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