Sci: The art of estimation

 All tools can have surprising uses, and I once used a plumb bob to make my electricity supplier replace the power/light pole outside my house. I phoned them to report that the pole was in danger of falling and causing injury, but they ignored me. The second time I called, I told the person answering that the pole was 5.05º out of plumb, and that I was monitoring it, expecting the angle to increase.

The offending light pole and my high-tech equipment.

The operator was clearly suspicious of my claimed level of accuracy (as I would be, in the same position), and wanted to know how I measured it. “A plumb bob pinned, 1500 millimetres up the pole was 132 millimetres out at the base. That’s a sine of 0.088, and there’s my angle,” I said. The lesson: like tools, numbers also have surprising uses, and one of the best tricks is when you get to use numbers as a cattle prod—and common folk are alarmed by even simple science. The pole was replaced a week later, which was what I had estimated.

A very unlucky rabbit.

To a mathematician or physicist, the notion of an immortal rabbit is quite acceptable for calculation purposes. As a school student, my English teacher encouraged me to psychoanalyse Macbeth, even when I protested that Freud hadn’t been invented when Shakespeare was writing. Ever a historically-minded cuss, I argued that it would be more relevant to look at the political situation in London, with a Scot (James I) sitting on the throne. Exasperated, he urged the class to ignore me, to engage instead in the willing suspension of disbelief.

In the same way, a spherical horse or spherical cow can be a useful starting point to explore ideas, to get a first approximation that can be extended. That brings us to the claim about the bumblebee that was shown not to be able to fly: this is often trotted out as evidence that scientists are thick, but there is a little more to the story than that.

In 1934, a French entomologist called Antoine Magnan tried to apply an engineer’s equation to bumblebees, and showed how, according to that equation, designed for aircraft that did not flap their wings, the bee could not generate enough lift to take off.

There is a great deal of folklore wrapped around this “event” and who actually was involved, but it appears that the equation was worked out by André Saint-Lagué. While the incident is often dressed up as “a scientist proving that bumblebees can’t fly”, all that Magnan really showed was that the equation was inadequate to describe the flight of the bumblebee.

He had shown that you can’t apply that particular equation to bumblebees, rather than proving that spherical bumblebees can’t fly, even if real ones can, flapping their wings at 130 times a second, moving happily along at 3 metres/sec, 11 km/h. Like Zeno’s paradox, Magnan’s calculation merely showed that there was a faulty assumption in there somewhere. This minor paradox showed that the mathematical model was flawed.

Safely out of the English classroom and into the lab, we heard of the marvels that could be done with simple apparatus. The muzzle velocity of a bullet could be measured with just a block of wood, a piece of string, a protractor and a measuring tape.

Our physics teacher, as equally at home with fiction as our English teacher, explained how, in the days of gunpowder and muzzle-loading firearms, slight variations in the ingredients or their amounts and proportions, could make a lot of difference. In the 17^th and 18^th centuries, Britain and France were always at war, and better gunpowder could make all the difference between winning and losing. The very best saltpetre, an essential part of gunpowder, came from India.

The most obvious measure of powder quality was the speed at which a cannon ball or musket ball left the barrel of the gun, or in physics-speak, the muzzle velocity. The idea was quite simple. You suspended a large block of wood and fired a bullet at it from close range.

The bullet lodged in the block, and all the energy of the bullet was transferred to the block, which would swing like a pendulum. Then the researcher only had to measure the swing angle and calculate the height the block reached.

This gadget even has a name: it is the ballistic pendulum, and it was invented by Benjamin Robins in 1742. From the swing, or so we were told, it is an elementary calculation to estimate the energy and hence the velocity of the bullet. Unfortunately, this explanation ignores the 800-pound spherical horse that is rolling around the room. (OK, it could have been a spherical elephant, but that’s a different joke.)

Some of the energy would go into deforming the bullet and the wood, some would be wasted as friction, and to do any calculations, we would have to assume that the bullet stopped instantaneously (which is about as likely as a girder with negligible mass).

Of course, if you were simply trying to compare different grades of gunpowder, rather than measuring the muzzle velocities, the losses will be similar in each case, and they can be ignored. Whichever powder produces the biggest swing is the best, if everything else is kept constant, and in fairy physics (which is, you will recall, the name that engineers give to this sort of thinking), that always applies.

Robins’ ballistic pendulum would have shown that Indian saltpetre made the best gunpowder. He died in India in 1751, supervising the construction of forts, and a few years later, the British drove the French out of India, getting all that excellent saltpetre for their own use. (History is one of the Arts, but chemistry has to be understood as well.)

And you need to keep an open mind. When you enquire about fast animals, more often than not, you will read that the fastest animal of all is the deer botfly. This is credited with an amazing 1287 km/hr, though if you convert this to miles per hour, that comes out as a round 800 mph, a figure that smells a little bit of fudged science—and rightly so, because round numbers are always suspicious.

A 1927 article in the Journal of the New York Entomological Society, written by an entomologist called Charles Henry Tyler Townsend reported a speed like that. Townsend said these flies passed in a blur, and so must have been travelling very fast. On that ‘scientific’ basis and no other, he credited the flies with a nice round 400 yards per second.

In 1938, when Irving Langmuir, a Nobel laureate in chemistry tested the assumptions. He found that the air pressure on the fly at that speed would be more than half an atmosphere, enough to crush it. The energy needed to maintain the flight would be 370 watts, half a horsepower. Aside from anything else, the fly would use up its own weight in fuel every second.

Langmuir had been hit by these flies, and while it hurt, a fly of that weight, going at 1300 km/hr would have left a significant hole in him, rather like that of a soft bullet, and the fly would have been mashed inside the wound. Instead, the fly bounced off. He used solder to make a pellet, the size of a fly, 1 cm long and 0.5 cm wide, and tied it to a string.

He whirled the pellet around his head. Knowing the length of the string and how many circuits it made each second, he calculated the speed, and found that at 13 mph, it was a blur. At 26 mph, it was barely visible; at 43 mph, an observer could not tell which way it was going; and at 64 mph, it was invisible. He said Townsend’s blur came from a fly travelling at 25 mph (40 km/hr).

Langmuir’s results were published in Science and reported in Time magazine, but legends are tough things, even when they are debunked by Nobel Prize winners. So even today, the same old values keep emerging from the woodwork.

Post script 1: catching crooks with numbers

When one is engaged in fraud investigation, one fertile method is to do some rough estimating, because these will often show up fraud. If an operation averages eight sick days per worker each year, you can make certain assumptions. If after allowing for seasonal colds and the like, the averages or the frequencies don’t fit the estimates, a closer look is recommended.

Post script 2: a floating cork

I was once asked (never mind why) to calculate the mass of a cork ball, two metres in diameter. Getting the volume is easy: ⁴/₃ x πr³ or 4.189 cubic metres, but how much does a cubic metre of cork weigh? There are two ways to find out: one is to look it up, the other is to estimate it, by looking at a cork in water.

Estimating the density of cork.

I decided that 20% of the cork in the photo above was submerged, giving a density of 0.2, while the reference books give a value of 0.24. My estimated mass was 838 kg, while the official answer would be just over 1000 kg. As they say. near enough for government work, and this was, as it happens, government work.

Whatever: under each answer, a 2-metre cork ball was a bad idea in a children’s playground! Oops, I gave the why away, but my calculations saved somebody else’s job, and perhaps a life or two.

Post script 3: Round numbers

There is a legend that the surveyors who measured the height of Mount Everest found that the numbers they had gave a value of exactly 29,000 feet, but they decided that nobody would accept that, so they gave the value as 29,002 feet (8,839.8 m), fearing that the calculated value of 8,839.2 m would be dismissed as a rounded estimate. This remains a conjecture that most scientists would like to be true.

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