September 1991
AT&T Bell Labs 600 Mountain Ave.
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...

Tire wear

     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 and illuminating.

       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 width.

     From a simple, back of the envelope calculation, we deduced that one molecular layer of rubber rips off on every spin of the wheel!

Laundry

     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 correct?

     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.

Gravity

     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 took place.

     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.

Breath of life

     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)

Do 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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