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Murray Hill N.J. 07974
Copyright © 1991 AT&T All Rights Reserved
Welcome back from Summer Vacation! We thought it would be
nice to begin the new school year with some elementary "back
of the envelope" calculations. Scientists use these simple
techniques all the time as a cross check on more exact theory and
experiment. Just as customers expect a lunch bill at MacDonald's
to be between $1 and $10, scientists have developed enough intuition
to seemingly "guess" the right answer, or at least know
when to go back in the lab and try again. Here are a couple of examples
drawn from a wide range of disciplines. As you will see, a good
estimate can lead to new insights about the world around us...
Most of you probably put many extra miles
on the family car this summer, and you might have wondered how long
you could go before replacing the tires. This calculation is simple
A tire is about two feet in diameter, so one revolution
corresponds to about six linear feet along the ground ( remember,
we are trying to estimate the answer, so the difference between
pi and 3, or a Chevy and a Toyota, is unimportant here. If you think
you have guessed a little high on one number, guess a little lower
on the next). This corresponds to about 1000 revolutions per mile.
A typical tire lasts for 50,000 miles. So, 50000 miles * 1000 revolutions/
mile = 5*10 7 revolutions. After this many rotations,
one centimeter (1/2") of tread has worn off, or about 2A per
revolution. Microscopically, rubber consists of long hydrocarbon
chains tangled together like spagetti, and 2 A is just about a chain
From a simple, back of the envelope calculation, we deduced
that one molecular layer of rubber rips off on every spin of the
Tires aren't the only objects to get extra wear in the summer.
Clothes, beach towels, and so on get washed more than normal; how
fast are they dissolving away? Well, most laundry machines hold
about 20 pounds of clothes. If you save the lint in the wash water
and the lint in the filter on the dryer, you'll find about 1 oz.
of lint is produced each wash (remember to dry the lint before weighing,
and don't count the fuzz from old kleenex). In other words, a few
tenths of one percent of your clothes disappear each wash. In one
or two hundred loads, the clothes would be half their original thickness.
This is about four years of normal use, at which point cotton underwear
typically is worn out. An interesting insight, but are these results
The problem lies with the model, not the numbers themselves.
A new cotton towel initially sheds lint quickly, and then more slowly
over the course of time. On the other hand, polyester swim trunks
hardly wear at all. So, measuring a whole load of laundry may not
indicate the true behavior of any particular garment in the basket.
Your class might want to perform an experiment to see how different
kinds of fabric really dissolve away. Buy some new clothes, weigh
them before and after a large number of washes, and plot the results.
We'll be glad to publish your results and observations.
This "calculation" requires no numbers at all.
Three hundred and fifty years ago, tradition holds that Galileo
Galilei performed an experiment at the Tower of Pisa (the tower
began leaning four centuries earlier during construction, so perhaps
we should call it the Leaned Tower of Pisa). At the time, people
mistakenly thought heavier objects fell to earth more quickly than
lighter objects. Indeed, this is a reasonable observation when the
action of air resistance is taken into account. After all, feathers
are observed to fall slower than rocks. Galileo's insight was to
realize that two forces were at work; air resistance which slows
down objects with large cross sections, and gravity which accelerates
all objects the same way, independent of mass or area. Without this
insight, it was impossible to correctly calculate the movement of
the planets, or to develop the Laws of Motion.
Supposedly, Galileo dropped a large and a small cannonball
from the tower. The balls were dense enough to be sure air resistance
was negligible, and according to the story both hit the ground at
the same time. A nice tale, but historians doubt this event ever
Instead, Galileo probably performed what scientists today
call a "thought experiment", often referred to by the
German phrase "Gedankenexperiment". The purpose of the
experiment is to decide between two different hypothesis: all objects
fall at the same rate (hypothesis "A"), or the heavier
the object, the faster it falls (hypothesis "B"). Here
is how a Gedankenexperiment works:
Imagine you are dropping a ball of clay. It falls at some
speed. Now imagine you pull the ball apart, and then place it back
together with only a hairline gap. Clearly, it is substantially
the same object and must fall at the same rate. Now imagine you
pull the clay into two balls, connected by a thin string of clay
(see drawings above
Hypothesis "B" predicts the new object (consisting of
two connected balls) falls as fast as the original sphere. However,
once a minor change is made by cutting the interconnecting string,
hypothesis "B" predicts a dramatic change. All of a sudden,
there are two smaller balls, which are individually lighter than
the original lump of clay, and therefore must fall more slowly.
On the other hand, hypothesis "A" predicts the rate of
fall to be the same before and after breaking the string. Similarly,
if we formed the ball of clay into a string of pearls, and then
cut the string, hypothesis "A" predicts their fall to
continue unabated, but hypothesis "B" practically calls
for the pearls to stop moving! Clearly, "B" leads to unintuitive
and nonsensical behavior under some conditions. Thus, by a process
of incremental reasoning and some demand over the consistency of
nature, we can understand why gravity must act equally on all masses.
Galileo, though a great experimentalist, probably solved the mystery
of the masses through a similar line of reasoning.
You may have heard the claim that every breath you take contains
air inhaled by Isaac Newton, or for that matter, any other person
in history (Atoms, it appears, do not discriminate between scientist
or soldier). Is this a true observation?
To find out, we need to calculate the fraction of the atmosphere
a person breathes in a lifetime, and then compare that fraction
to the number of atoms in a breath. For example, if Isaac breathed
one in every thousand atoms in the atmosphere, and there were a
million atoms in each breath, you and Ike would have a good chance
of sharing more than a passing interest in physics.
Each time you breathe you inhale about one liter of air (a
liter is 1000 cm3, or 10-3 m3).
The average person takes ~10 breaths/ minute, which over a lifetime
corresponds to 300 million breaths. At 10-3 m3/
breath, this is 300,000 m3 of air. The atmosphere, of
course, is much larger. You can estimate its volume from the thickness
of the atmosphere (about 10 miles before the density has dropped
precipitously) and the earth's diameter (8000 miles). Working out
the numbers, the earth is covered by about 1019 m3,
so a person in a lifetime consumes 1 part in 1013 of
all the available air. That doesn't sound like much, but there are
a lot of atoms in a breath of air. In fact, there are about 1025
atoms in 1 m3, so you are probably recycling billions
of atoms from Sir Newton (or indeed, simultaneously billions with
anyone and everyone you can think of).
Similarly, a car "breathes" air as well. A car's engine
is often rated in liters, but unlike a person, it breathes with
every r.p.m, about 1000/minute. So even though a car is off most
of the day, it breathes about as much as a person. Consequently,
your lungs have as much in common with a V-8 as they do with Sir
Isaac Newton. Fortunately, there are mechanisms in place in the
environment to cleanse and recycle the atmosphere, but we do breathe
the sins of our past.
(As an aside, how can you and your class measure the thickness
of the atmosphere? The diameter of the earth? The number of atoms
in a breath of air? Answers next issue)
fish have salivary glands?
Salivary glands produce a watery liquid for a variety
of purposes. Saliva helps food slide from the mouth into the stomach,
increases contact between the food and taste buds, aids in clearing
starchy residue from the mouth, and begins the digestion process.
Although the most well developed glands are found in mammals,
many other vertebrates and invertebrates have salivary glands. Fish
and other aquatic animals clearly do not lack opportunities to add
water to their meals; hence most aquatic animals are devoid of "true"
salivary glands. However, some form of lubrication is still necessary
to assist swallowing even in water, and this is provided by mucous
glands along the tongue and roof of mouth (Mucous secretion is present
in all animals.)
Nature has adapted salivary glands to solve many interesting
challenges. In mammals, birds and some frogs, saliva contains the
enzyme pytalin, that begins the digestion of starch into sugar.
This is why when you chew on a simple starch, like rice or potato,
it begins to taste sweet after a few minutes. Meat eaters, such
as lions, have no need for this enzyme and its presence in humans
(along with the shape of our teeth) is thought to indicate our evolution
from plant eating (or at least omnivorous) ancestors. In some lower
vertebrates such as the lamprey, an anticoagulant is secreted from
a type of salivary gland to prevent blood (the lamprey's sole food)
from clotting. Fish embryos have a gland which secretes an enzyme
to aid in hatching by breaking down its eggshell. The venom glands
of snakes are modified salivary glands, producing some of the most
dangerous substances in the world.
Perhaps the most spectacular (or, depending on your point
of view, gross) use of modified salivary glands is found in certain
kinds of beetles. These beetles hunt frogs. Once caught, the frog
is killed and a chemical that dissolves all the frog's internal
organs is injected through a tube. The beetle, in effect, turns
the frog's skin into a bag, drinking its meal with a straw! If this
sounds like something out of science fiction or a horror film, the
inverse is true. Many of Hollywood's most spectacular beasts are
based on the shape and behavior of lesser known insects and animals.
You might want to look through an advanced book on insects from
the library, or visit the Natural History Museum in New York, to
get a closer insight into the true sources of Hollywood's imagination.
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