Wednesday, April 15, 2015

Physics Problem: barometers and building height

There's a physics problem that typically gets presented to children when they learn about pressure (air, water, and often mercury) and barometers. And it seems to me that the problem is broken. The question is: how could you use an ordinary barometer to measure the height of a tall building? And the expected answer is to measure the air pressure at the top and bottom, and then knowing the weight of air, compute the elevation change.

But let's check the feasibility of this with a quick back-of-the-envelope calculation. For any realistic earthbound circumstance, we can assume that the pressure drops linearly with altitude; see the chart below. For reference, Denver, Colorado is at about 1600m, and the highest town in Great Britain is under 500m; we won't find much in the way of tall buildings above an altitude of 4000m.

("Atmospheric Pressure vs. Altitude" by Geek.not.nerd - Own work. Licensed under CC0 via Wikimedia Commons)

So what would our barometer tell us? Since we're just investigating the feasibility here, we're going to round things a little to make the math easy. Don't worry, no truths were harmed during the making of this calculation:
  • For every 1000m of altitude gained, the pressure drops ~10kPa, the chart tells us (at least, over the range we are concerned with).
  • Thus for each meter, the pressure drops ~10Pa; that's 0.1hPa. (1 hectopascal or 1hPa = 100Pa, and is the modern unit equivalent to millibars, commonly used in meteorology.)
A typical household digital barometer can detect a change of +/- 0.5hPa (50Pa), and an analog "certified precision" $600 aneroid barometer is only accurate to 1hPa (100Pa). So with one of these instruments, we can hope to measure building height to, at best, an accuracy of 5 to 10m -- maybe good enough to estimate the number of stories, but not the height.

Fortunately there are two other possible ways to use a barometer to determine a building's height:
  1. Drop the barometer off the top of the building and time how long it takes to hit the ground below. For a building in the range of 100m to 200m, timing accurate to 0.01s would give a precision of around 0.5m, so we will probably want to use some kind of electronic timing device rather than a hand-operated stopwatch. (With manual timing we could reasonably only count on a timing accuracy of 1/10s, which by coincidence converts to about the same height accuracy as the digital barometer.) For example, we could have synchronized clocks at top and bottom, an electromagnet release that records the start time and a sound-activated circuit to record the stop time. For a heavy barometer, we can ignore air resistance.
  2. Find the building custodian and say to him "If you can tell me how tall this building is, I will give you this lovely barometer". This is definitely my favorite solution.
NOTE TO STUDENTS: do not use this answer in any test unless you are very sure about the sense of humor of your teacher.

1 comment:

Carl Zetie said...

Incidentally, the last answer above is apocryphally attributed to Neils Bohr, who supposedly was bored by the obvious answer. He also supposedly proposed the solution "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", which the examiners rejected because it demonstrated no knowledge of physics.