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 17th and 18th 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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