Insights on the Barometer and Building Height Joke as a Physics Question

· Essay

While thinking, this afternoon, about writing something less frivolous, I suddenly found myself remembering the joke about how to measure the height of a building by using a barometer. For those who do not know it, and who do not want to visit one of the various pages describing it, attributing this to an allegedly real-life situation with the physicist Niels Bohr, let us briefly summarize:  there is an examiner who is given an unexpected answer to the question we have suggested, for which he wants to fail the student.  The student claims his answer is correct, and appeals.  He is allowed time to give an answer more in the line of what is expected on a physics examination.  I had forgotten that point, because I myself would have criticized his answers on that ground, and was going to attack the alternate answers one-by-one.

Nevertheless, even the correct answer is not correct enough.  Am I quibbling?  I do not think so.  This takes me back to a science experiment which was conducted in one of my high-school classes.  The objective was to determine the thermal conductivity of aluminum.  However, the report was not only to give the outcome, but to consider the possible errors that might have arisen.  I did think of quite a few, and for that, high marks were obtained – even if the calculated thermal conductivity was not 100% in agreement with the textbook result.

I am not going back to those high school years and try to be so thorough now, but during my afternoon reflection, there came to me ideas why the correct answer to the barometer question is unsatisfactory, and perhaps even wrong.  (I hope no one has written on this before, because I hereby state, this is wholly the result of my own musings.)

The supposedly correct answer is to measure the air pressure at the bottom of the building, and to compare the reading with that at the top of the same edifice, and (usually unstated), to make a calculation, based on these readings and the knowledge of the relationship between barometric pressure and the standard pressure of the same at sea level.

However, the student, to be more correct, would have to take into consideration some other factors.  Would the height of the building, and the time it took to get to the top allow atmospheric pressure changes at one or both levels (although it is the second of these that we must first worry about) which would cause a distortion of the reading sufficient to give an error about the structure’s height?  We believe this could easily be the case, as when we have seen an intense summer storm approaching – first: the sky is sunny, things are calm, and the weather is hot; but on the horizon we see black clouds.  Suddenly, we feel a cold wind, and the black clouds quickly approach. (This is from memory, and may only reflect the approximate sequence from sun to storm.)  I would suggest that here we have a situation where the barometric readings change with unexpected rapidity.

Let us assume that the situation is not quite as extreme as in the above example, but that it is of note.  The calculation could be refined by going back down the stairs or the elevator, and checking to see how this compares with the original reading.  If that second reading on the ground floor checks out, everything is fine.  If not, it should be necessary to go back to the top floor, and get a second reading.  This time, if the reading is the same as previously, then the second reading at street level would have been correct.  If not, the process would have to be repeated again, or maybe it would have to be determined if the barometer was being viewed correctly in the first place.  This would require an impartial observer, but as a criticism of possible defects in the reading, it could be included by the very person making the original observations.

A physicist might disagree with this next point, but I think it has some merit.  I once read, in an old textbook, Steam, by the Babcock and Wilcox Company, (and strangely, given to me out of the library of a medical practitioner) that in measuring temperature with a glass thermometer, a correction must be made for the differences between the bulb and the stem.  If I remember correctly, this was the bulb correction factor.  I submit that we might have a similar phenomenon by carrying a barometer around – especially if it is one of the larger types – in a nice wooden casing – and by the contact with the human body affecting the reading.

If the building were extremely tall, which at the time the original question was postulated, would either not have been the case (if Bohr had anything to do with it), or probably was not even contemplated: building sway might have to be taken into consideration.  At least, it might contemplate the taking of measurements on the roof, if the sway were sufficient.  To allay this problem, it must be specified that measurements be taken on a calm day.

A final point occurred to me while writing the previous paragraph – and if this is valid, it would seriously complicate the readings between the upper and lower levels of the building.  In the first place, it is not stated whether, in these tests, the pressure is taken indoors or outside.  I imagine on ground level, it could be either, especially on a sunny day.  I take it that to get the height, one has to get to the roof, so that would be outside.  However, to get there, our intrepid atmospheric temperature taker, if not already inside, would have to go there, and would be there, until getting outside onto the flat roof. (I take it that if it were glass covered, that would increase the height, and would preclude a precise calculation.)  Would the barometric pressure inside the building have been the same as outside? I have my doubts.

And as long as the answers to the original were expected to be tongue-in-cheek, we include our own now – we must be sure that in these trips up and down the building, the barometer had not been dropped, thus ensuring an inaccurate reading beyond any doubt!


May 10, 2019

Ⓒ 2019, Paul Karl Moeller

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