A story about a physics exam…

October 15, 2008 By Paul Wagorn
Having an idea

Sometimes it’s easy to get caught within the bounds of where an accepted solution “should” come from.  We need to look a the tools that we are given,  and get creative not just with the solution, but with the tools themselves.

The following concerns a question in a physics degree exam at the University of Copenhagen.  The story goes like this:

“Describe how to determine the height of a skyscraper with a barometer.”

One student replied: “You tie a long piece of string to the neck of the barometer, then lower the barometer from the roof of the skyscraper to the ground. The length of the string plus the length of the barometer will equal the height of the building.”

This highly original answer so incensed the examiner that the student was failed immediately. The student appealed on the grounds that his answer was indisputably correct, and the university appointed an independent arbiter to decide the case.

The arbiter judged that the answer was indeed correct, but did not display any noticeable knowledge of physics. To resolve the problem it was decided to call the student in and allow him six minutes in which to provide a verbal answer which showed at least a minimal familiarity with the basic principles of physics.

For five minutes the student sat in silence, forehead creased in thought.

The arbiter reminded him that time was running out, to which the student replied that he had several extremely relevant answers, but couldn’t make up his mind which to use. On being advised to hurry up the student replied as follows:

Measuring a building
Measuring a building

“Firstly, you could take the barometer up to the roof of the skyscraper, drop it over the edge, and measure the time it takes to reach the ground. The height of the building can then be worked out from the formula H = 0.5g x t squared. But bad luck on the barometer.” “Or if the sun is shining you could measure the height of the barometer, then set it on end and measure the length of its shadow. Then you measure the length of the skyscraper’s shadow, and thereafter it is a simple matter of proportional arithmetic to work out the height of the skyscraper.” “But if you wanted to be highly scientific about it, you could tie a short piece of string to the barometer and swing it like a pendulum, first at ground level and then on the roof of the skyscraper. The height is worked out by the difference in the gravitational restoring force T = 2 pi sq root (l/g).”

“…or if the skyscraper has an outside emergency staircase, it would be easier to walk up it and mark off the height of the skyscraper in barometer lengths, then add them up.” “If you merely wanted to be boring and orthodox about it, of course, you could use the barometer to measure the air pressure on the roof of the skyscraper and on the ground, and convert the difference in millibars into feet to give the height of the building.”

“But since we are constantly being exhorted to exercise independence of mind and apply scientific methods, undoubtedly the best way would be to knock on the janitor’s door and say to him ‘If you would like a nice new barometer, I will give you this one if you tell me the height of this skyscraper’.”

Niels Bohr
Niels Bohr

The student was Niels Bohr, who was known for proposing the model of the atom wherein the electrons were much like the planets in our solar system orbiting around the nucleus. Niels Bohr went on to be the first Dane to win the Nobel prize for Physics and even had an element named after him.

One of his sons also subsequently won a Nobel Prize.

Now disappointingly, this story is not true. It was first printed by Reader’s Digest in 1958 and has been reprinted in various forms (and with different people other than Niels Bohr).  That being said, it still demonstrates one concept quite clearly:

Sometimes the “correct” answer is not always the “right” answer.

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Reader Comments

There is an invention using cardboard to measure height. Its using 2 similar triangles principle, one big imaginery triangle on landscape, the other is small triangle drawing on cardboard. Those triangles are having same angles, ratio of triangles sides are equal to drawing scale. If one of triangle side lenght known, the rest can be calculated by measuring small triangle on cardboard. Please check to:


You can draw scale map by using the same triangles method, the produced map is good enough to check landscape distance, area, and landscape planning, etc.. And its simple enough to play treasure hunt game.

Posted by Maruza on December 20, 2011

Yes people want predictable results according to accepted theory. This turns something that would be enjoyable, the search for knowledge, into something unpleasant since truth is not its goal. Rather, our modern system demands adherence to dogma despite the evidence opposing the accepted idea. Thus, something fun is turned into something painful. Physics is indeed fun and helps describe the world through numerical analysis. However, if it is not corrected it will not fulfill its promise. Information that criticizes the legitimacy of the constancy of speed of light indicates that Relativity might be wrong. Paul Davies of an American University indicated that that might just be the case.
Posted by mike strauss on July 30, 2010

So many classes and schools are simply teaching/preaching/forcing conformity to create employees who will behave in a predictable, dependable, if not creative, fashion. You have to learn how to do that to become a teacher.

Students who can't tolerate such environments are punished by sending them to schools that are even worse. The lucky ones are discovered by some iconoclast who moves them to an environment that rewards all the behaviors described above. But there are so few who are so lucky.
Posted by Larry on January 5, 2010

After reading all the comments, i would say that they are interesting and at the same time hilarious. I am a high school senior and have an early admit from Dartmouth College. I am planning to major in Physics. I don't think American Universities stifle creativity and independent thought. My elder brother graduated from the University of Connecticut with a degree in Mathematics. Some faculties do want narrow-minded answers but that's prevalent everywhere, may be even @ Harvard or MIT. By the way, i came across these excellent physics flash cards. Its also a great initiative by the FunnelBrain team. Amazing!!!
Posted by Peter Kaminsky on November 24, 2009

The really strange thing is that some of the "wrong" answers will give more accurate results than the right one!
Posted by Roger on November 7, 2008

I don't know if this counts, but I once had to do an experiment in a group where the teacher threw away our results on the grounds that they were wrong. She then insisted that we all do the lab write-up anyway, including a table of results and a conclusion explaining those results.
No one else in the group bothered to do it but I was annoyed so I drew the table and left it blank, wrote a conclusion that was two pages long which was worded so that it could've applied to any set of results, taken from any experiment, with even the slightest variation within the set.

I was not failed for being insolent, I was not failed for being wrong.

I was failed on the grounds that the teacher didn't believe that most of the words I had used existed.

The teacher was a graduate of Harvard and teaching at the International School Of Geneva.
Just goes to show.
Posted by G.C. on November 6, 2008

I did something similar to this in my 1st year college calculus class. I was given a question to prove that point x was equidistant from points a and b. I used a geometry proof:

1. Construct a line AB between points a and b.
2. Construct a line perpendicular to AB, bisecting AB at point c, such that AC = BC
3. If point x falls on the new line, it is equidistant from a and b (as are all points on this new line) by side-angle-side (AC = BC, ACX = BCX, and CX = CX, therefore AX = BX)

It was an introductory question to get some idea how much calculus we knew prior to taking the course. I, of course, knew nothing of calculus, and what little of geometry I remembered was from 7 years prior.

My teacher, though, was very accepting. I have to thank her for that.
Posted by Dan on October 16, 2008

CGE wrote:
"participate in current research"

Well, in math, that is as simple as reading a few papers. In an experimental science it's different. However, few undergrads doing "research" for profs are doing more than being cheap hands. They don't really have the background to
participate in current physics research at any higher level.

It is different in graduate school--it's sort of difficult to do high energy physics on the experimental level without access to a high end accelerator etc.
Posted by Penny on October 16, 2008

We are kindred spirits.

I agree that at the highest level, it could be helpful to ask professors questions.
I didn't because I enjoyed working the stuff out on my own and I was independent--perhaps to a fault.

And, I was going to post the same things you did in response to Robert, but didn't want to start a flame war.

As to nanotech, solar energy etc., these areas require lots of old physics--one
reason I chose them. Such things as the domination by viscous forces at a certain scale, thermodynamics etc. Similarly for the nuclear physics experiments that I mentioned. Building Geiger Detectors teaches principles of electron flow, and building cloud chambers teaches principles of thermodynamics.

My high school physics course ( 1960's ) New York City, included derivations of
PV=nRT from the assumption of molecular chaos, a derivation of the Bohr atom, special relativity, derivation of Kepler's laws etc. Today, that would
( sadly) be considered remarkable--but back then it was standard.

I found it boring, as I had read much more physics on my own, but it was
a very good background for most students.
Posted by Penny on October 16, 2008

All I can say to this is lol. This is basically the exact experience that I have had with the post-secondary institutions within the United States. I can honestly say that for the majority that professors and the institution wish to stifle creativity and independent thought. The narrow minded answers they want completely disregard the plethora of other choices that would reveal the correct answer. Whether or not the story is true, this is the attitude of the American educational institute, for those that think outside the box continue to. Revolutionist, non-conventionalist, and the blatant disregard for normalcy are the fundamentals to continuing our species in the area of science.
Posted by web design company on October 16, 2008

In response to Mr. Oetting, I personally think that most students choosing to take an AP Physics course are going to be taking it to learn physics, not to learn how to study. I'm not convinced that making physics secondary is wise or appropriate; there are other ways to learn how to study. One way, for example, might be to simply structure and teach the course as if it were university-level.

As for contemporary work, as Penny noted, nanotech is interesting, though I'm a bit biased. However, I think that focusing on recent developments can be unfortunate in a low level physics course. Most recent physics work is built so heavily upon older theories that understanding it well isn't possible without a good grasp of basic physics. Thus, your students will only be able to hear about things that sound impressive, without being able to really understand them, or do any theoretical work from basic principles. That gives a very distorted view of physics, and makes real physics all the more shocking for undergraduates when they reach it. Quite a bit of very old physics is also very beautiful and interesting, and could probably be taught to bright 12th grade students. I'm particularly fond of classical mechanics, Noether's theorem, Lagrangians, and so on; the math is wonderful, and neat, and the applications are amazing in their range. Statistical mechanics, too, is wonderful, with absurdly simple calculations being able to make startlingly accurate approximations. Don't assume that older areas of physics won't be interesting to students simply because they are old; many of them, if taught correctly, can be far more interesting, because the students would be learning real physics, from first principles, rather than something that is necessarily just memorization. Memorization is not physics; one should be able to go quite far with only a handful of equations.
Posted by CGE on October 15, 2008

Penny, it seems your story is remarkably similar to mine. It does seem that junior high school is far more concerned with pointless procedure than actual teaching, and also that the teachers and administration there can be extremely hostile to the brightest students. In one junior high, in a program for gifted students, instead of teaching at a higher level, some teachers decided to just give us so much work at the same level that we couldn't possibly finish it. When I discussed leaving and going to university early with some of my friends, some of our teachers actually added a section into their curriculum forcing us to repeatedly listen to all the reasons why we shouldn't do so, and how it would destroy our lives; it didn't keep a around dozen of us from leaving. I must admit, though, that I'd never thought of sending them a triumphant letter now.

However, I'm not sure I'd entirely agree with your view of universities, and I do believe they serve a useful purpose even for bright students. While things like MIT's materials and the variety of very useful books are all nice, the ability to discuss things with professors, participate in current research, and generally be around colleagues can all add to one's understanding and make things more enjoyable and productive. I often ignored many lectures, and learned things from books like Landau's series instead, but being able to ask questions and discuss things with professors was quite helpful, especially at higher levels.

I do think that these things are very dependent on the university and the department, so at some universities, such procedural nonsense might be strictly enforced. At my university, when I thought that I knew classical mechanics well enough that the undergraduate level course would be pointless, I was told that I could take the graduate level course instead, and had quite a bit of fun learning far more interesting aspects of nonlinear and chaotic systems, complex canonical transforms, and so on. When I noted that taking a programming course, required for physics majors, would be boring for me, and pointed to the code for my research, I was exempted from the requirement entirely, without the slightest argument. My department actually encouraged students to do things differently and skip useless work, and respected them more for doing so: one professor I worked with was held in particular awe among the faculty in the department because, when he was an undergraduate at the university, he had decided to skip *all* of the undergraduate physics courses, and proceeded to only take graduate-level courses.
Posted by CGE on October 15, 2008

Ben, The length of the barometer is not needed. Simply provide the answer in units of 1 barometer length.
Posted by bob on October 15, 2008

Dear Robert,
There are lots of interesting and elementary physics problems that arise in
"nanotech". You might give a few to your students. That's contemporary.

Also include the Feynman article ( on the web) that inspired the field.

Similarly, solar power is a good field for them to explore--both computationally and in the lab.

They might also enjoy a unit based on cloud chambers, Geiger counters, and simple experiments. I had an " atomic energy kit" as a child--including this
stuff and a spintheroscope, and a zinc sulfide screen etc. They should build the cloud chambers and Geiger Counters.

Lots of good physics in designing and building telescopes and their use in astronomy too.
Posted by Penny on October 15, 2008

Today, with the amazing things available on the web, such as MIT's open courseware, school is even more worthless for a bright motivated kid.
Posted by Penny on October 15, 2008

I wrote:

"When I was a freshman at uni, we were given physics exams and required to
show all the arithmetic at every stage. I wouldn't do that, and did all the algebra and calculus first–and lost points on every test."

To be clear, they wanted us to not use variables and derive formulae and then substitute the numbers at the end. When I did use formulae--correctly, and lost points, I was told " You are not ready to do that."

In retrospect, I should have gone to the chairman of the department, and told him I wanted to skip undergraduate school and take my Phd qualifiers. I was only 15 and scared to death of authority figures. That is because the sort of teachers I had in elementary school and JHS as described above had battered me to an emotional fine pulp.

I had taught myself most of the physics I knew ( though I had high school physics) by reading Dover books and an early version of the Feynman lectures.
Posted by Penny on October 15, 2008

I once had a Junior High School teacher, in a special program for gifted kids, decide on the last day to grade our notebooks. I didn't have one--but, I had a perfect grade on the all the tests. He battered me for hours, gave me a very low grade, and told me " I don't care how smart you think you are, if you don't learn to keep a notebook, you will never get through high school, let alone into college." I told him to open the textbook and ask me to recite it backwards
from memory starting anywhere--but he didn't care.

In the same year, I had an arithmetic teacher who insisted on her own "method" of arithmetic, which mostly involved many cross checks for mistakes. For example to add a column of three figure numbers, one had to add up, add down, and
cast out nines, and cast out elevens. I just did all the problems mentally--quite easily and quickly and got them all correct--but with the points deducted for
not following orders--I was failing.

I offered to do an extra credit report to pass, and this was accepted and I did
the report on the special theory of relativity--with the equations derived. My teacher failed me anyway, and told my parents that: : Your daughter might be another Einstein, but she will fail in life if she doesn't learn to conform."

About ten years later I sent them both a letter on Institute for Advanced Study
stationary ( as I was a member that year) which said: " I still don't have a notebook, and I never learned to conform."

School---basically, Poison.
Posted by Penny on October 15, 2008

Well ok, here is a story of mine:

When I was a freshman at uni, we were given physics exams and required to
show all the arithmetic at every stage. I wouldn't do that, and did all the algebra and calculus first--and lost points on every test.

About three weeks into the course, I was so bored that I decided to solve all the problems using Lagrangian and Hamiltonian mechanics--and got C's instead of A's for not following procedure--even if I was displaying a graduate student's level of understanding!

This was one reason that I entered as a physics major and very quickly became a math major. The math department liked creativity, let me minor in graduate school ( as an undergrad) and this led to a very early PhD.

My general principle is that for the really bright students school at all levels is a complete waste of time. One can just read and follow one's one interests.
Posted by Penny on October 15, 2008

I think a point that a lot of people are missing is that, in this story, the guy could actually answer the question. In my experience as a teaching assistant, students who give creative answers like this usually can't back it up with a real answer. They think they can write down some joke and they should get marks, but don't actually know any answer to the question at all. It's not about stifling the creativity of students, it's about making sure that they can actually answer the damn questions. I don't know about you, but I don't want some jokester designing bridges and office buildings with the engineering degree he learned by being clever. I'd prefer someone who knew what they were doing, regardless of whether or not they are a dull guy.
Posted by K on October 15, 2008

The lesson: It's fine to be clever, but don't be an ass about it.
Posted by me on October 15, 2008

It's been about 25 years since I graduated from college, but this story reminds me of something similar that happened to me. On the final for a digital logic class, the prof gave a problem that went something like this:

You are hired to design the digital logic for a locking mechanism that reads an identity card, decides whether the bearer is supposed to have access to a parking lot the mechanism secures, and opens the gate if the bearer is, and doesn't if he doesn't. There are 6 types of bearers: Freshman, Sophomores, Juniors, Seniors, Professors, and Other Inferior Staff Types. There are three parking lots.

Determine how many bits it'll take to represent the 6 classes of bearers, assign them unique IDs, and determine the logic circuits necessary to decode the classes into a true/false decision for each parking lot. Use the least number of logic gates possible.

Then he gave some list of who should be allowed where, of which I remember nothing. Freshmen only could use lot A, maybe, and sophomores lot B, juniors both A and B, and of course, professors could park anywhere they like.

Now, what he wanted us to do was say you needed three bits to represent 6 classes of bearer, assign them arbitrary three bit codes, 0 through 5 probably, write out the formulas to represent the logic, minimize that, and then sketch out all the and, or, and not gates it would take to come up with the right signal to open the gate. So of course, that's not what I did.

If, instead of making the assignment of codes to bearer classes be arbitrary, you instead notice that there are three bits in the bearer code, and three parking lots, and let each bit of the bearer code represent a parking lot, and assign the codes to the bearers intelligently, then you don't need any logic circuits at all to decode the desired signal! For instance, if bit 0 represents lot A, and bit 1 lot B, and bit 2 lot C, then the freshman should be assigned bearer code 1 (001 in binary), sophomores code 2 (010), juniors 3(011), professors should be code 7 (111), etc.

This was so obviously the most elegant solution to the problem possible, and the least cost to build, so I wrote all this on the paper. This problem was worth 30 points on the final out of 100. The professor awarded me 5 points for this answer. Unfortunately, at my school, there was no one to whom to appeal, professor's grading decisions were final, and I ended up with a B- in the course. What a pinhead.
Posted by Bob on October 15, 2008

Ok, here's a true story: I was taking a Pascal class at a University in the early 80's. I was not a regular student there, but was taking the class to catch up on some prereqs so that I could enroll. Our first assignment was to generate a NxM cell maze, and then to solve the maze programatically.

I spent the first 30 minutes or so writing the program. Then, because I was curious, rather than just printing the maze and solution (as required), I started displaying the maze, and the progress of the solution to visualize the solution and see if I could come up with a better way of running the maze. After playing with it for a while and finding several optimizations I packaged it up and turned it all in. I was expecting an easy "A" on the project since I fulfilled all the requirements, and then some.

The next week I got my program listing back: "Grade: C -- too long."

What? You can't be serious. I went and talked to the TA, and he said, yes the grade was right, and that the program was way to long given the scope of the project that we were supposed to be doing. I asked him if he had looked at the code at all, or had run it. He just looked at me, as if I were pond scum, and said that he had graded the paper.

I went to complain to the professor (a well known and published researcher and author). He looked at me, looked at the program for about 10 seconds. He then said "Does it run?"

"Of course," I replied.

"Show me."

So, I popped the disk into the computer on his desk and showed him my maze and solution, all drawn out in IBM character graphics (it was an IBM XT, after all.)

He shook his head, took my paper, scratched out the grade, and wrote in "A". He then took out the class book and made a note. He then said to me, "The mid term is on and the final is on . They are both open book exams. Your grade for the class will be determined by those two tests. You don't have to turn in any assignments. However, I will warn you that if you don't know the material, having an open book will not help you."

I ended up missing 2 points for the whole quarter. I think I forgot a semicolon somewhere on one of the code samples.
Posted by Mark on October 15, 2008

I teach an AP Physics course to 12th grade boys in a private school. I tell my students that the first thing they will learn in my course is how to study for a college level science or engineering course. The second thing they will learn is how to solve problems. Finally, they will learn some physics, as a vehicle for the first two. I think this is an instructive story, urban legend or fact, because it illustrates a certain agility in solving problems. Consequently, I plan to post this site for my students to read. Any "contemporary" thing I can do to celebrate physics and physicist is welcome! Thank you.
Posted by Robert Oetting on October 15, 2008

A Danish student wouldn't say, "convert the difference in millibars into feet", they're on the metric system. This was clearly written by an American.
Posted by Mike on October 15, 2008

Even though this is clearly apocraphyl, there are only two correct answers here - the traditional 'right' one, and perhaps the final answer. All the other answers use additional objects that are not available in the question, and so none of them could be correct. Anyone who has ever answered an exam paper knows that you have to stick to the environment given. Some might be valid if you knew the length of the barometer, but again there can be no presumption you know that.
Posted by Ben on October 15, 2008

This is stupid.

"Knowledgeable students can give knowledgeable and technically correct answers to poorly phrased test questions without dignifying the spirit of the question that's being asked." Oh. Ok.

This has nothing to do with stifling creativity, independent thought, etc. At all.

The only problem with the 'redo' was that the redo question was even vaguer than the original.
Posted by Jesse on October 15, 2008

Hi. These urban legends travel more than you think. I heard this story as a lawstudent as an example of the importance for future lawyers to think outside the box. This must have been 94 or 95. I think it was given by the (then former) president of the Amsterdam Court, Ben Asscher.
Posted by Arwin on October 15, 2008

The idea of this being an inspirational little anecdote about creativity, independent thinking, etc. makes me eyeroll. Even pretending it was true: The question was obviously not testing the student's knowledge of determining the height of buildings. It was testing the student's knowledge of the (intended) use of a barometer, interpreting the measurements it gives, and the math involved in putting those measurements to practical uses. If the question had been essentially about buildings, any of these answers would be fine. But given that the question was about barometry, the examiner would have been right to consider it incorrect. Giving answers like this is not being smart, it's just being a smartass.

Meanwhilst, if this story is circulated to say something like "Look, geniuses are bad at school, so if you're bad at school, you might actually be a genius!" Now, if some given person is bad at school, they *could* be an unappreciated genius, but most likely not. So at that this story also fails, just like the equally apocryphal claims that Einstein did poorly in school (he actually excelled in math class, etc. as any reasonable person would expect).
Posted by Adam on October 15, 2008

All I can say to this is lol. This is basically the exact experience that I have had with the post-secondary institutions within the United States. I can honestly say that for the majority that professors and the institution wish to stifle creativity and independent thought. The narrow minded answers they want completely disregard the plethora of other choices that would reveal the correct answer. Whether or not the story is true, this is the attitude of the American educational institute, for those that think outside the box continue to. Revolutionist, non-conventionalist, and the blatant disregard for normalcy are the fundamentals to continuing our species in the area of science.
Posted by NeXuS on October 15, 2008

I wrote too quickly--I didn't notice that you said the story was nonsense.
So please ignore my comment
Posted by Penny on October 15, 2008

This is nonsense. I heard this urban legend when i was a high school student and it has been told about practically every physics genius besides Bohr.

One proof that it is nonsense is that such a question would have been--in Bohr's day --in Europe, a question whose answer would have been expected of a twelve year old.
In my school days ( 1960's), it was part of the High School physics curriculum
in New York City ( on the state regents exam). In Bohr's case, there is no way
that he would have been given such a question in university.

Moreover, Bohr didn't get an undergraduate degree in physics--as they didn't exist at his university--his only exam would have been his doctoral exam--as they didn't give exams along the way--as we do today.

Posted by Penny on October 15, 2008

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