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.

Monday, April 13, 2015

The American problem

The fundamental tension in America is that the red states want to be Sparta and the blue states want to be Athens, and the only thing that unites them in common cause is fear of Persia.

Friday, April 10, 2015

On the intractability of free will

[Author's note: the thoughts here originated as a Letter to the Editors at New Scientist, in response to a somewhat throwaway remark, in an article about randomness, that chance may be essential to the existence of free will. I felt that the point deserved delving into in more depth. New Scientist did publish the letter, but as is typical edited it down for publication -- in particular, many of my adverbs did not survive. Consequently, I wanted to share the full text here.]

Randomness may be necessary to "admit free will" in an otherwise-mechanical universe ("Chance", New Scientist 14 March 2015, p.28ff) but by itself it is not sufficient. It's hard to argue scientifically about the existence of free will in the absence of a rigorous definition of what it is, but we can say something about what it does. And at a minimum, it's existence requires that the outputs of my brain -- my actions -- are not completely determined by the inputs plus initial state. This is, of course, an astonishing proposition that is contrary to any other known physical system or law. Even if we introduce randomness, we merely allow a range of outcomes distributed probabilistically, but we still have no element of intentionality or purpose, the other essential ingredient in free will. Randomness alone would make us no more free than tumbling dice.

One intriguing possibility, however, is that whatever free will actually is, randomness provides a means for it to influence the brain without apparently violating known physical law; a curtain behind which it can hide. Imagine that free will is able to influence apparently-random outcomes deep in the brain, to achieve a desired output, but is also constrained by the need to appear random over the long term. The brain would be like a rigged casino where the roulette wheel comes up red or black at the casino's own choosing, but it must still ensure that the two come up equally over the long term if it is not to be caught breaking the rules. Correspondingly, we might speculate that free will rigs the brain game by influencing individual apparently-random quantum outcomes, which chaotic systems in turn amplify to macroscopic scale, but is limited in the long run because the overall outcomes must match our probabilistic quantum expectations.

Intriguingly, existing psychological experiments are consistent with this model of free will. For example, we know that behaviors that are usually considered exercises of free will such as "paying attention" or "resistance to temptation" are limited and can be exhausted, requiring time to recharge, even though they don't seem to be associated with anything as obvious as depletion of specific neurotransmitters or saturation of synapses. Yet this is exactly what we expect if free will can only influence a limited number of outcomes while staying hidden within known physical laws.

Whatever free will turns out to be -- assuming it exists at all -- understanding it will take at least as great a conceptual leap as that from classical mechanics to quantum theory. And perhaps it is only the reality of chance that connects these three views of reality into a consistent, scientifically explicable universe.

Wednesday, April 01, 2015

Athens and Sparta: A Parable About Open and Closed Source Software

Among all the city-states of Classical Greece, the most famous are certainly Athens and Sparta. Sometimes allies, often enemies, despite their shared language and culture, these two could not have been more different. So in the rivalry between Athens and Sparta, who ultimately emerged the winner?

In the 5th century BCE, the dominant city-state was Sparta. It was hierarchical, authoritarian and ruled by tyrannical kings and aristocrats. It’s greatest cultural values were discipline and conformity, and the kings of this highly militaristic state were also its generals. Sparta was incredibly effective at concentrating its resources to conquer a chosen goal – the phrase “the tip of the spear” could have been invented for them. As a result, Spartans were feared in battle across the Greek world, and Sparta was able to impose its military will on its neighbors.

But then, Athens began to rise to prominence and oppose the hegemony of Sparta. It became a famous center of creativity in the arts, learning and philosophy, home to Plato's Academy and Aristotle's Lyceum. Athens also gave the ancient world Socrates, Pericles, Sophocles and many more philosophers, writers and politicians. Its schools and forums were often lively, open-air marketplaces for competing ideas. It thrived on chaos. Even more remarkable were its experiments in democracy that included a unique combination of direct and representative democracy: everybody was expected to participate in and contribute to Athenian civic life. In stark contrast to Sparta’s general-kings, Athens elected its generals according to the needs of each war.

For a century, Athens and Sparta were in almost constant conflict for dominance of the Greek world, pausing occasionally and briefly to unite against a common enemy. Finally, in 404BCE, Athens was defeated for good and fell under Spartan rule. So did this mean that Sparta had won? Not exactly: Sparta’s dominance was short-lived. Neither Athens nor Sparta ever fully recovered from the costs and destruction of their wars, which impoverished most of the Greek world and ushered in the end of Greek pre-eminence.

So if both Sparta and Athens lost, who won? While Sparta and Athens were exhausting themselves in civil war, far to the west a small village called Rome was growing into a regional power. Rome was something strange and new: it borrowed many ideas from the Greeks, but had no real artistic culture of it’s own. Its sculpture, painting and poetry were second-rate derivations, sometimes even direct copies, of the works of the Greeks. It contributed no significant advances in mathematics or science, and barely anything to philosophy. Even the gods that the Romans claimed to worship were obvious imitations of the Greek pantheon. And yet, the Romans were exceptional engineers, great builders and implementers of others’ ideas. While the Greeks declined, Rome conquered a vast empire, convincing native populations almost everywhere that it was in their best interests to assimilate into Roman ways.

In the end, neither Sparta nor Athens won: both lost to Rome.