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The Mars Climate Orbiter is prepared for its mission. The laws of physics are the same everywhere, even on Mars, so the probe could be designed based on the laws of physics as discovered on earth. There is unfortunately another reason why this spacecraft is relevant to the topics of this chapter: it was destroyed attempting to enter Mars' atmosphere because engineers at Lockheed Martin forgot to convert data on engine thrusts from pounds into the metric unit of force (newtons) before giving the information to NASA. Conversions are important!
If you drop your shoe and a coin side by side, they hit the ground at the same time. Why doesn't the shoe get there first, since gravity is pulling harder on it? How does the lens of your eye work, and why do your eye's muscles need to squash its lens into different shapes in order to focus on objects nearby or far away? These are the kinds of questions that physics tries to answer about the behavior of light and matter, the two things that the universe is made of.
a / Science is a cycle of theory and experiment.
b / A satirical drawing of an alchemist's laboratory. H. Cock, after a drawing by Peter Brueghel the Elder (16th century).
Until very recently in history, no progress was made in answering questions like these. Worse than that, the wrong answers written by thinkers like the ancient Greek physicist Aristotle were accepted without question for thousands of years. Why is it that scientific knowledge has progressed more since the Renaissance than it had in all the preceding millennia since the beginning of recorded history? Undoubtedly the industrial revolution is part of the answer. Building its centerpiece, the steam engine, required improved techniques for precise construction and measurement. (Early on, it was considered a major advance when English machine shops learned to build pistons and cylinders that fit together with a gap narrower than the thickness of a penny.) But even before the industrial revolution, the pace of discovery had picked up, mainly because of the introduction of the modern scientific method. Although it evolved over time, most scientists today would agree on something like the following list of the basic principles of the scientific method:
(1) Science is a cycle of theory and experiment. Scientific theories are created to explain the results of experiments that were created under certain conditions. A successful theory will also make new predictions about new experiments under new conditions. Eventually, though, it always seems to happen that a new experiment comes along, showing that under certain conditions the theory is not a good approximation or is not valid at all. The ball is then back in the theorists' court. If an experiment disagrees with the current theory, the theory has to be changed, not the experiment.
(2) Theories should both predict and explain. The requirement of predictive power means that a theory is only meaningful if it predicts something that can be checked against experimental measurements that the theorist did not already have at hand. That is, a theory should be testable. Explanatory value means that many phenomena should be accounted for with few basic principles. If you answer every “why” question with “because that's the way it is,” then your theory has no explanatory value. Collecting lots of data without being able to find any basic underlying principles is not science.
(3) Experiments should be reproducible. An experiment should be treated with suspicion if it only works for one person, or only in one part of the world. Anyone with the necessary skills and equipment should be able to get the same results from the same experiment. This implies that science transcends national and ethnic boundaries; you can be sure that nobody is doing actual science who claims that their work is “Aryan, not Jewish,” “Marxist, not bourgeois,” or “Christian, not atheistic.” An experiment cannot be reproduced if it is secret, so science is necessarily a public enterprise.
As an example of the cycle of theory and experiment, a vital step toward modern chemistry was the experimental observation that the chemical elements could not be transformed into each other, e.g., lead could not be turned into gold. This led to the theory that chemical reactions consisted of rearrangements of the elements in different combinations, without any change in the identities of the elements themselves. The theory worked for hundreds of years, and was confirmed experimentally over a wide range of pressures and temperatures and with many combinations of elements. Only in the twentieth century did we learn that one element could be trans-formed into one another under the conditions of extremely high pressure and temperature existing in a nuclear bomb or inside a star. That observation didn't completely invalidate the original theory of the immutability of the elements, but it showed that it was only an approximation, valid at ordinary temperatures and pressures.
A psychic conducts seances in which the spirits of the dead speak to the participants. He says he has special psychic powers not possessed by other people, which allow him to “channel” the communications with the spirits. What part of the scientific method is being violated here?
(answer in the back of the PDF version of the book)The scientific method as described here is an idealization, and should not be understood as a set procedure for doing science. Scientists have as many weaknesses and character flaws as any other group, and it is very common for scientists to try to discredit other people's experiments when the results run contrary to their own favored point of view. Successful science also has more to do with luck, intuition, and creativity than most people realize, and the restrictions of the scientific method do not stifle individuality and self-expression any more than the fugue and sonata forms stifled Bach and Haydn. There is a recent tendency among social scientists to go even further and to deny that the scientific method even exists, claiming that science is no more than an arbitrary social system that determines what ideas to accept based on an in-group's criteria. I think that's going too far. If science is an arbitrary social ritual, it would seem difficult to explain its effectiveness in building such useful items as airplanes, CD players, and sewers. If alchemy and astrology were no less scientific in their methods than chemistry and astronomy, what was it that kept them from producing anything useful?
withintro{Consider whether or not the scientific method is being applied in the following examples. If the scientific method is not being applied, are the people whose actions are being described performing a useful human activity, albeit an unscientific one? }
Acupuncture is a traditional medical technique of Asian origin in which small needles are inserted in the patient's body to relieve pain. Many doctors trained in the west consider acupuncture unworthy of experimental study because if it had therapeutic effects, such effects could not be explained by their theories of the nervous system. Who is being more scientific, the western or eastern practitioners?
Goethe, a German poet, is less well known for his theory of color. He published a book on the subject, in which he argued that scientific apparatus for measuring and quantifying color, such as prisms, lenses and colored filters, could not give us full insight into the ultimate meaning of color, for instance the cold feeling evoked by blue and green or the heroic sentiments inspired by red. Was his work scientific?
A child asks why things fall down, and an adult answers “because of gravity.” The ancient Greek philosopher Aristotle explained that rocks fell because it was their nature to seek out their natural place, in contact with the earth. Are these explanations scientific?
Buddhism is partly a psychological explanation of human suffering, and psychology is of course a science. The Buddha could be said to have engaged in a cycle of theory and experiment, since he worked by trial and error, and even late in his life he asked his followers to challenge his ideas. Buddhism could also be considered reproducible, since the Buddha told his followers they could find enlightenment for themselves if they followed a certain course of study and discipline. Is Buddhism a scientific pursuit?
c / This telescope picture shows two images of the same distant object, an exotic, very luminous object called a quasar. This is interpreted as evidence that a massive, dark object, possibly a black hole, happens to be between us and it. Light rays that would otherwise have missed the earth on either side have been bent by the dark object's gravity so that they reach us. The actual direction to the quasar is presumably in the center of the image, but the light along that central line doesn't get to us because it is absorbed by the dark object. The quasar is known by its catalog number, MG1131+0456, or more informally as Einstein's Ring.
d / Reductionism.
Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the things which compose it...nothing would be uncertain, and the future as the past would be laid out before its eyes. -- Pierre Simon de Laplace
Physics is the use of the scientific method to find out the basic principles governing light and matter, and to discover the implications of those laws. Part of what distinguishes the modern outlook from the ancient mind-set is the assumption that there are rules by which the universe functions, and that those laws can be at least partially understood by humans. From the Age of Reason through the nineteenth century, many scientists began to be convinced that the laws of nature not only could be known but, as claimed by Laplace, those laws could in principle be used to predict everything about the universe's future if complete information was available about the present state of all light and matter. In subsequent sections, I'll describe two general types of limitations on prediction using the laws of physics, which were only recognized in the twentieth century.
Matter can be defined as anything that is affected by gravity, i.e., that has weight or would have weight if it was near the Earth or another star or planet massive enough to produce measurable gravity. Light can be defined as anything that can travel from one place to another through empty space and can influence matter, but has no weight. For example, sunlight can influence your body by heating it or by damaging your DNA and giving you skin cancer. The physicist's definition of light includes a variety of phenomena that are not visible to the eye, including radio waves, microwaves, x-rays, and gamma rays. These are the “colors” of light that do not happen to fall within the narrow violet-to-red range of the rainbow that we can see.
At the turn of the 20th century, a strange new phenomenon was discovered in vacuum tubes: mysterious rays of unknown origin and nature. These rays are the same as the ones that shoot from the back of your TV's picture tube and hit the front to make the picture. Physicists in 1895 didn't have the faintest idea what the rays were, so they simply named them “cathode rays,” after the name for the electrical contact from which they sprang. A fierce debate raged, complete with nationalistic overtones, over whether the rays were a form of light or of matter. What would they have had to do in order to settle the issue?
(answer in the back of the PDF version of the book)Many physical phenomena are not themselves light or matter, but are properties of light or matter or interactions between light and matter. For instance, motion is a property of all light and some matter, but it is not itself light or matter. The pressure that keeps a bicycle tire blown up is an interaction between the air and the tire. Pressure is not a form of matter in and of itself. It is as much a property of the tire as of the air. Analogously, sisterhood and employment are relationships among people but are not people themselves.
Some things that appear weightless actually do have weight, and so qualify as matter. Air has weight, and is thus a form of matter even though a cubic inch of air weighs less than a grain of sand. A helium balloon has weight, but is kept from falling by the force of the surrounding more dense air, which pushes up on it. Astronauts in orbit around the Earth have weight, and are falling along a curved arc, but they are moving so fast that the curved arc of their fall is broad enough to carry them all the way around the Earth in a circle. They perceive themselves as being weightless because their space capsule is falling along with them, and the floor therefore does not push up on their feet.
Einstein predicted as a consequence of his theory of relativity that light would after all be affected by gravity, although the effect would be extremely weak under normal conditions. His prediction was borne out by observations of the bending of light rays from stars as they passed close to the sun on their way to the Earth. Einstein's theory also implied the existence of black holes, stars so massive and compact that their intense gravity would not even allow light to escape. (These days there is strong evidence that black holes exist.)
Einstein's interpretation was that light doesn't really have mass, but that energy is affected by gravity just like mass is. The energy in a light beam is equivalent to a certain amount of mass, given by the famous equation E=mc2, where c is the speed of light. Because the speed of light is such a big number, a large amount of energy is equivalent to only a very small amount of mass, so the gravitational force on a light ray can be ignored for most practical purposes.
There is however a more satisfactory and fundamental distinction between light and matter, which should be understandable to you if you have had a chemistry course. In chemistry, one learns that electrons obey the Pauli exclusion principle, which forbids more than one electron from occupying the same orbital if they have the same spin. The Pauli exclusion principle is obeyed by the subatomic particles of which matter is composed, but disobeyed by the particles, called photons, of which a beam of light is made.
Einstein's theory of relativity is discussed more fully in book 6 of this series.
The boundary between physics and the other sciences is not always clear. For instance, chemists study atoms and molecules, which are what matter is built from, and there are some scientists who would be equally willing to call themselves physical chemists or chemical physicists. It might seem that the distinction between physics and biology would be clearer, since physics seems to deal with inanimate objects. In fact, almost all physicists would agree that the basic laws of physics that apply to molecules in a test tube work equally well for the combination of molecules that constitutes a bacterium. (Some might believe that something more happens in the minds of humans, or even those of cats and dogs.) What differentiates physics from biology is that many of the scientific theories that describe living things, while ultimately resulting from the fundamental laws of physics, cannot be rigorously derived from physical principles.
To avoid having to study everything at once, scientists isolate the things they are trying to study. For instance, a physicist who wants to study the motion of a rotating gyroscope would probably prefer that it be isolated from vibrations and air currents. Even in biology, where field work is indispensable for understanding how living things relate to their entire environment, it is interesting to note the vital historical role played by Darwin's study of the Galápagos Islands, which were conveniently isolated from the rest of the world. Any part of the universe that is considered apart from the rest can be called a “system.”
Physics has had some of its greatest successes by carrying this process of isolation to extremes, subdividing the universe into smaller and smaller parts. Matter can be divided into atoms, and the behavior of individual atoms can be studied. Atoms can be split apart into their constituent neutrons, protons and electrons. Protons and neutrons appear to be made out of even smaller particles called quarks, and there have even been some claims of experimental evidence that quarks have smaller parts inside them. This method of splitting things into smaller and smaller parts and studying how those parts influence each other is called reductionism. The hope is that the seemingly complex rules governing the larger units can be better understood in terms of simpler rules governing the smaller units. To appreciate what reductionism has done for science, it is only necessary to examine a 19th-century chemistry textbook. At that time, the existence of atoms was still doubted by some, electrons were not even suspected to exist, and almost nothing was understood of what basic rules governed the way atoms interacted with each other in chemical reactions. Students had to memorize long lists of chemicals and their reactions, and there was no way to understand any of it systematically. Today, the student only needs to remember a small set of rules about how atoms interact, for instance that atoms of one element cannot be converted into another via chemical reactions, or that atoms from the right side of the periodic table tend to form strong bonds with atoms from the left side.
I've suggested replacing the ordinary dictionary definition of light with a more technical, more precise one that involves weightlessness. It's still possible, though, that the stuff a lightbulb makes, ordinarily called “light,” does have some small amount of weight. Suggest an experiment to attempt to measure whether it does.
Heat is weightless (i.e., an object becomes no heavier when heated), and can travel across an empty room from the fireplace to your skin, where it influences you by heating you. Should heat therefore be considered a form of light by our definition? Why or why not?
Similarly, should sound be considered a form of light?
For as knowledges are now delivered, there is a kind of contract of error between the deliverer and the receiver; for he that delivereth knowledge desireth to deliver it in such a form as may be best believed, and not as may be best examined; and he that receiveth knowledge desireth rather present satisfaction than expectant inquiry. -- Francis Bacon
Many students approach a science course with the idea that they can succeed by memorizing the formulas, so that when a problem is assigned on the homework or an exam, they will be able to plug numbers in to the formula and get a numerical result on their calculator. Wrong! That's not what learning science is about! There is a big difference between memorizing formulas and understanding concepts. To start with, different formulas may apply in different situations. One equation might represent a definition, which is always true. Another might be a very specific equation for the speed of an object sliding down an inclined plane, which would not be true if the object was a rock drifting down to the bottom of the ocean. If you don't work to understand physics on a conceptual level, you won't know which formulas can be used when.
Most students taking college science courses for the first time also have very little experience with interpreting the meaning of an equation. Consider the equation w=A/h relating the width of a rectangle to its height and area. A student who has not developed skill at interpretation might view this as yet another equation to memorize and plug in to when needed. A slightly more savvy student might realize that it is simply the familiar formula A=wh in a different form. When asked whether a rectangle would have a greater or smaller width than another with the same area but a smaller height, the unsophisticated student might be at a loss, not having any numbers to plug in on a calculator. The more experienced student would know how to reason about an equation involving division --- if h is smaller, and A stays the same, then w must be bigger. Often, students fail to recognize a sequence of equations as a derivation leading to a final result, so they think all the intermediate steps are equally important formulas that they should memorize.
When learning any subject at all, it is important to become as actively involved as possible, rather than trying to read through all the information quickly without thinking about it. It is a good idea to read and think about the questions posed at the end of each section of these notes as you encounter them, so that you know you have understood what you were reading.
Many students' difficulties in physics boil down mainly to difficulties with math. Suppose you feel confident that you have enough mathematical preparation to succeed in this course, but you are having trouble with a few specific things. In some areas, the brief review given in this chapter may be sufficient, but in other areas it probably will not. Once you identify the areas of math in which you are having problems, get help in those areas. Don't limp along through the whole course with a vague feeling of dread about something like scientific notation. The problem will not go away if you ignore it. The same applies to essential mathematical skills that you are learning in this course for the first time, such as vector addition.
Sometimes students tell me they keep trying to understand a certain topic in the book, and it just doesn't make sense. The worst thing you can possibly do in that situation is to keep on staring at the same page. Every textbook explains certain things badly --- even mine! --- so the best thing to do in this situation is to look at a different book. Instead of college textbooks aimed at the same mathematical level as the course you're taking, you may in some cases find that high school books or books at a lower math level give clearer explanations.
Finally, when reviewing for an exam, don't simply read back over the text and your lecture notes. Instead, try to use an active method of reviewing, for instance by discussing some of the discussion questions with another student, or doing homework problems you hadn't done the first time.
Calculus was invented by a physicist, Isaac Newton, because he needed it as a tool for calculating velocity and acceleration; in your introductory calculus course, velocity and acceleration were probably presented as some of the first applications.
If an object's position as a function of time is given by the function x(t), then its velocity and acceleration are given by the first and second derivatives with respect to time,
The notation relates in a logical way to the units of the quantities. Velocity has units of m/s, and that makes sense because dx is interpreted as an infinitesimally small distance, with units of meters, and dt as an infinitesimally small time, with units of seconds. The seemingly weird and inconsistent placement of the superscripted twos in the notation for the acceleration is likewise meant to suggest the units: something on top with units of meters, and something on the bottom with units of seconds squared.
Velocity and acceleration have completely different physical interpretations. Velocity is a matter of opinion. Right now as you sit in a chair and read this book, you could say that your velocity was zero, but an observer watching the Earth rotate would say that you had a velocity of hundreds of miles an hour. Acceleration represents a change in velocity, and it's not a matter of opinion. Accelerations produce physical effects, and don't occur unless there's a force to cause them. For example, gravitational forces on Earth cause falling objects to have an acceleration of 9.8 m/s2.
◊ The diver starts at rest, and has an acceleration of 9.8 m/s2. We need to find a connection between the distance she travels and time it takes. In other words, we're looking for information about the function x(t), given information about the acceleration. To go from acceleration to position, we need to integrate twice:
![= int left(at+v_zu{o}right) dt qquad text{[$v_zu{o}$ is a constant of integration.]}](math/eq_08939f58.png)
![= int at :dt qquad text{[$v_zu{o}$ is zero because she's dropping from rest.]}](math/eq_9a2841b5.png)
![= frac{1}{2}at^2+x_zu{o} qquad text{[$x_zu{o}$ is a constant of integration.]}](math/eq_4e1c7095.png)
![= frac{1}{2}at^2 qquad text{[$x_zu{o}$ can be zero if we define it that way.]}](math/eq_9a5bf565.png)
Note some of the good problem-solving habits demonstrated here. We solve the problem symbolically, and only plug in numbers at the very end, once all the algebra and calculus are done. One should also make a habit, after finding a symbolic result, of checking whether the dependence on the variables make sense. A greater value of t in this expression would lead to a greater value for x; that makes sense, because if you want more time in the air, you're going to have to jump from higher up. A greater acceleration also leads to a greater height; this also makes sense, because the stronger gravity is, the more height you'll need in order to stay in the air for a given amount of time. Now we plug in numbers.
Note that when we put in the numbers,
we check that the units work out correctly, 
. We should
also check that the result makes sense: 4.9 meters is pretty high, but not unreasonable.
The notation dq in calculus represents an infinitesimally small change in the variable q. The corresponding notation for a finite change in a variable is Δ q. For example, if q represents the value of a certain stock on the stock market, and the value falls from qo=5 dollars initially to qf=3 dollars finally, then Δ q=-2 dollars. When we study linear functions, whose slopes are constant, the derivative is synonymous with the slope of the line, and dy/dx is the same thing as Δ y/Δ x, the rise over the run.
Under conditions of constant acceleration, we can relate velocity and time,
or, as in the example 1, position and time,
It can also be handy to have a relation involving velocity and position, eliminating time. Straightforward algebra gives
where vf is the final velocity, vo the initial velocity, and Δ x the distance traveled.
◊ Solved problem: Dropping a rock on Mars — problem 17
◊ Solved problem: The Dodge Viper — problem 19
The introductory part of a book like this is hard to write, because every student arrives at this starting point with a different preparation. One student may have grown up outside the U.S. and so may be completely comfortable with the metric system, but may have had an algebra course in which the instructor passed too quickly over scientific notation. Another student may have already taken vector calculus, but may have never learned the metric system. The following self-evaluation is a checklist to help you figure out what you need to study to be prepared for the rest of the course.
If you disagree with this statement… | you should study this section: |
I am familiar with the basic metric units of meters, kilograms, and seconds, and the most common metric prefixes: milli- (m), kilo- (k), and centi- (c). | subsection ?? Basic of the Metric System |
I am familiar with these less common metric prefixes: mega- (M), micro- μ), and nano- (n). | subsection ?? Less Common Metric Prefixes |
I am comfortable with scientific notation. | subsection ?? Scientific Notation |
I can confidently do metric conversions. | subsection ?? Conversions |
I understand the purpose and use of significant figures. | subsection ?? Significant Figures |
It wouldn't hurt you to skim the sections you think you already know about, and to do the self-checks in those sections.
e / The original definition of the meter.
f / A duplicate of the Paris kilogram, maintained at the Danish National Metrology Institute.
Units were not standardized until fairly recently in history, so when the physicist Isaac Newton gave the result of an experiment with a pendulum, he had to specify not just that the string was 37 7/8 inches long but that it was “37 7/8 London inches long.” The inch as defined in Yorkshire would have been different. Even after the British Empire standardized its units, it was still very inconvenient to do calculations involving money, volume, distance, time, or weight, because of all the odd conversion factors, like 16 ounces in a pound, and 5280 feet in a mile. Through the nineteenth century, schoolchildren squandered most of their mathematical education in preparing to do calculations such as making change when a customer in a shop offered a one-crown note for a book costing two pounds, thirteen shillings and tuppence. The dollar has always been decimal, and British money went decimal decades ago, but the United States is still saddled with the antiquated system of feet, inches, pounds, ounces and so on.
Every country in the world besides the U.S. uses a system of units known in English as the “metric system.1” This system is entirely decimal, thanks to the same eminently logical people who brought about the French Revolution. In deference to France, the system's official name is the Système International, or SI, meaning International System. (The phrase “SI system” is therefore redundant.)
The wonderful thing about the SI is that people who live in countries more modern than ours do not need to memorize how many ounces there are in a pound, how many cups in a pint, how many feet in a mile, etc. The whole system works with a single, consistent set of Greek and Latin prefixes that modify the basic units. Each prefix stands for a power of ten, and has an abbreviation that can be combined with the symbol for the unit. For instance, the meter is a unit of distance. The prefix kilo- stands for 103, so a kilometer, 1 km, is a thousand meters.
The basic units of the metric system are the meter for distance, the second for time, and the gram for mass.
The following are the most common metric prefixes. You should memorize them.
| prefix | meaning | example | ||
| kilo- | k | 103 | 60 kg | = a person’s mass |
| centi- | c | 10 − 2 | 28 cm | = height of a piece of paper |
| milli- | m | 10 − 3 | 1 ms | = time for one vibration of a guitar string playing the note D |
The prefix centi-, meaning 10-2, is only used in the centimeter; a hundredth of a gram would not be written as 1 cg but as 10 mg. The centi- prefix can be easily remembered because a cent is 10-2 dollars. The official SI abbreviation for seconds is “s” (not “sec”) and grams are “g” (not “gm”).
When I stated briefly above that the second was a unit of time, it may not have occurred to you that this was not much of a definition. We can make a dictionary-style definition of a term like “time,” or give a general description like Isaac Newton's: “Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external...” Newton's characterization sounds impressive, but physicists today would consider it useless as a definition of time. Today, the physical sciences are based on operational definitions, which means definitions that spell out the actual steps (operations) required to measure something numerically.
In an era when our toasters, pens, and coffee pots tell us the time, it is far from obvious to most people what is the fundamental operational definition of time. Until recently, the hour, minute, and second were defined operationally in terms of the time required for the earth to rotate about its axis. Unfortunately, the Earth's rotation is slowing down slightly, and by 1967 this was becoming an issue in scientific experiments requiring precise time measurements. The second was therefore redefined as the time required for a certain number of vibrations of the light waves emitted by a cesium atoms in a lamp constructed like a familiar neon sign but with the neon replaced by cesium. The new definition not only promises to stay constant indefinitely, but for scientists is a more convenient way of calibrating a clock than having to carry out astronomical measurements.
What is a possible operational definition of how strong a person is?
(answer in the back of the PDF version of the book)The French originally defined the meter as 10-7 times the distance from the equator to the north pole, as measured through Paris (of course). Even if the definition was operational, the operation of traveling to the north pole and laying a surveying chain behind you was not one that most working scientists wanted to carry out. Fairly soon, a standard was created in the form of a metal bar with two scratches on it. This was replaced by an atomic standard in 1960, and finally in 1983 by the current definition, which is that the meter is the distance traveled by light in a vacuum over a period of (1/299792458) seconds.
The third base unit of the SI is the kilogram, a unit of mass. Mass is intended to be a measure of the amount of a substance, but that is not an operational definition. Bathroom scales work by measuring our planet's gravitational attraction for the object being weighed, but using that type of scale to define mass operationally would be undesirable because gravity varies in strength from place to place on the earth.
There's a surprising amount of disagreement among physics textbooks about how mass should be defined, but here's how it's actually handled by the few working physicists who specialize in ultra-high-precision measurements. They maintain a physical object in Paris, which is the standard kilogram, a cylinder made of platinum-iridium alloy. Duplicates are checked against this mother of all kilograms by putting the original and the copy on the two opposite pans of a balance. Although this method of comparison depends on gravity, the problems associated with differences in gravity in different geographical locations are bypassed, because the two objects are being compared in the same place. The duplicates can then be removed from the Parisian kilogram shrine and transported elsewhere in the world. It would be desirable to replace this at some point with a universally accessible atomic standard rather than one based on a specific artifact, but as of 2010 the technology for automated counting of large numbers of atoms has not gotten good enough to make that work with the desired precision.
Just about anything you want to measure can be measured with some combination of meters, kilograms, and seconds. Speed can be measured in m/s, volume in m3, and density in kg/m3. Part of what makes the SI great is this basic simplicity. No more funny units like a cord of wood, a bolt of cloth, or a jigger of whiskey. No more liquid and dry measure. Just a simple, consistent set of units. The SI measures put together from meters, kilograms, and seconds make up the mks system. For example, the mks unit of speed is m/s, not km/hr.
A useful technique for finding mistakes in one's algebra is to analyze the units associated with the variables.
for the volume of a
cone, where A is the area of its base, and h is its height.
He wants to find an equation that will tell him how tall a conical
tent has to be in order to have a certain volume, given its radius.
His algebra goes like this:
◊ Line 4 is supposed to be an equation for the height, so the units of the expression on the right-hand side had better equal meters. The pi and the 3 are unitless, so we can ignore them. In terms of units, line 4 becomes
This is false, so there must be a mistake in the algebra. The units of lines 1, 2, and 3 check out, so the mistake must be in the step from line 3 to line 4. In fact the result should have been
Now the units check: m = m3/m2.
Isaac Newton wrote, “... the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time... It may be that there is no such thing as an equable motion, whereby time may be accurately measured. All motions may be accelerated or retarded...” Newton was right. Even the modern definition of the second in terms of light emitted by cesium atoms is subject to variation. For instance, magnetic fields could cause the cesium atoms to emit light with a slightly different rate of vibration. What makes us think, though, that a pendulum clock is more accurate than a sundial, or that a cesium atom is a more accurate timekeeper than a pendulum clock? That is, how can one test experimentally how the accuracies of different time standards compare?
g / This is a mnemonic to help you remember the most important metric prefixes. The word “little” is to remind you that the list starts with the prefixes used for small quantities and builds upward. The exponent changes by 3, except that of course that we do not need a special prefix for 100, which equals one.
The following are three metric prefixes which, while less common than the ones discussed previously, are well worth memorizing.
| prefix | meaning | example | ||
| mega- | M 106 | 6.4 Mm | = radius of the earth | |
| micro- | μ 10 − 6 | 10mumunit | = size of a white blood cell | |
| nano- | n 10 − 9 | 0.154 nm | = distance between carbon nuclei in an ethane molecule | |
Note that the abbreviation for micro is the Greek letter mu, μ --- a common mistake is to confuse it with m (milli) or M (mega).
There are other prefixes even less common, used for
extremely large and small quantities. For instance,
1 femtometer=10-15 m is a convenient unit of distance in
nuclear physics, and 1 gigabyte=109 bytes is used for
computers' hard disks. The international committee that
makes decisions about the SI has recently even added some
new prefixes that sound like jokes, e.g.,
is about half the mass of a proton. In the
immediate future, however, you're unlikely to see prefixes
like “yocto-” and “zepto-” used except perhaps in trivia
contests at science-fiction conventions or other geekfests.
Suppose you could slow down time so that according to your perception, a beam of light would move across a room at the speed of a slow walk. If you perceived a nanosecond as if it was a second, how would you perceive a microsecond?
(answer in the back of the PDF version of the book)Most of the interesting phenomena in our universe are not on the human scale. It would take about 1,000,000,000,000,000,000,000 bacteria to equal the mass of a human body. When the physicist Thomas Young discovered that light was a wave, it was back in the bad old days before scientific notation, and he was obliged to write that the time required for one vibration of the wave was 1/500 of a millionth of a millionth of a second. Scientific notation is a less awkward way to write very large and very small numbers such as these. Here's a quick review.
Scientific notation means writing a number in terms of a product of something from 1 to 10 and something else that is a power of ten. For instance,
Each number is ten times bigger than the previous one.
Since 101 is ten times smaller than 102 , it makes sense to use the notation 100 to stand for one, the number that is in turn ten times smaller than 101 . Continuing on, we can write 10-1 to stand for 0.1, the number ten times smaller than 100 . Negative exponents are used for small numbers:
A common source of confusion is the notation used on the displays of many calculators. Examples:
| 3.2×106 | (written notation) |
| 3.2E+6 | (notation on some calculators) |
| 3.26 | (notation on some other calculators) |
The last example is particularly unfortunate, because 3.26 really stands for the number 3.2 × 3.2 × 3.2 × 3.2 × 3.2 × 3.2= 1074, a totally different number from 3.2 × 106=3200000. The calculator notation should never be used in writing. It's just a way for the manufacturer to save money by making a simpler display.
A student learns that 104 bacteria, standing in line to register for classes at Paramecium Community College, would form a queue of this size:
\includegraphics{../share/intro/figs/sc-bacteria-queue-1}
The student concludes that 102 bacteria would form a line of this length:
\includegraphics{../share/intro/figs/sc-bacteria-queue-2}
Why is the student incorrect?
(answer in the back of the PDF version of the book)Conversions are one of the three essential mathematical skills, summarized on pp.848-850, that you need for success in this course.
I suggest you avoid memorizing lots of conversion factors between SI units and U.S. units, but two that do come in handy are:
{}1 inch = 2.54 cm
An object with a weight on Earth of 2.2 pounds-force has a mass of 1 kg.
The first one is the present definition of the inch, so it's exact. The second one is not exact, but is good enough for most purposes. (U.S. units of force and mass are confusing, so it's a good thing they're not used in science. In U.S. units, the unit of force is the pound-force, and the best unit to use for mass is the slug, which is about 14.6 kg.)
More important than memorizing conversion factors is understanding the right method for doing conversions. Even within the SI, you may need to convert, say, from grams to kilograms. Different people have different ways of thinking about conversions, but the method I'll describe here is systematic and easy to understand. The idea is that if 1 kg and 1000 g represent the same mass, then we can consider a fraction like
to be a way of expressing the number one. This may bother you. For instance, if you type 1000/1 into your calculator, you will get 1000, not one. Again, different people have different ways of thinking about it, but the justification is that it helps us to do conversions, and it works! Now if we want to convert 0.7 kg to units of grams, we can multiply kg by the number one:
If you're willing to treat symbols such as “kg” as if they were variables as used in algebra (which they're really not), you can then cancel the kg on top with the kg on the bottom, resulting in
To convert grams to kilograms, you would simply flip the fraction upside down.
One advantage of this method is that it can easily be applied to a series of conversions. For instance, to convert one year to units of seconds,
A common mistake is to write the conversion fraction incorrectly. For instance the fraction
does not equal one, because 103 kg is the mass of a car, and 1 g is the mass of a raisin. One correct way of setting up the conversion factor would be
You can usually detect such a mistake if you take the time to check your answer and see if it is reasonable.
If common sense doesn't rule out either a positive or a negative exponent, here's another way to make sure you get it right. There are big prefixes and small prefixes:
| big prefixes: | k M |
| small prefixes: | m μ n |
(It's not hard to keep straight which are which, since “mega” and “micro” are evocative, and it's easy to remember that a kilometer is bigger than a meter and a millimeter is smaller.) In the example above, we want the top of the fraction to be the same as the bottom. Since k is a big prefix, we need to compensate by putting a small number like 10-3 in front of it, not a big number like 103.
◊ Solved problem: a simple conversion — problem 6
◊ Solved problem: the geometric mean — problem 8
Each of the following conversions contains an error. In each case, explain what the error is.
(a) 
(b) 
(c) “Nano” is 10-9, so there are 10-9 nm in a meter.
(d) “Micro” is 10-6, so 1 kg is 106 μg.
An engineer is designing a car engine, and has been told that the diameter of the pistons (which are being designed by someone else) is 5 cm. He knows that 0.02 cm of clearance is required for a piston of this size, so he designs the cylinder to have an inside diameter of 5.04 cm. Luckily, his supervisor catches his mistake before the car goes into production. She explains his error to him, and mentally puts him in the “do not promote” category.
What was his mistake? The person who told him the pistons were 5 cm in diameter was wise to the ways of significant figures, as was his boss, who explained to him that he needed to go back and get a more accurate number for the diameter of the pistons. That person said “5 cm” rather than “5.00 cm” specifically to avoid creating the impression that the number was extremely accurate. In reality, the pistons' diameter was 5.13 cm. They would never have fit in the 5.04-cm cylinders.
The number of digits of accuracy in a number is referred to as the number of significant figures, or “sig figs” for short. As in the example above, sig figs provide a way of showing the accuracy of a number. In most cases, the result of a calculation involving several pieces of data can be no more accurate than the least accurate piece of data. In other words, “garbage in, garbage out.” Since the 5 cm diameter of the pistons was not very accurate, the result of the engineer's calculation, 5.04 cm, was really not as accurate as he thought. In general, your result should not have more than the number of sig figs in the least accurate piece of data you started with. The calculation above should have been done as follows:
The fact that the final result only has one significant figure then alerts you to the fact that the result is not very accurate, and would not be appropriate for use in designing the engine.
Note that the leading zeroes in the number 0.04 do not count as significant figures, because they are only placeholders. On the other hand, a number such as 50 cm is ambiguous --- the zero could be intended as a significant figure, or it might just be there as a placeholder. The ambiguity involving trailing zeroes can be avoided by using scientific notation, in which 5×101 cm would imply one sig fig of accuracy, while 5.0×101 cm would imply two sig figs.
The following quote is taken from an editorial by Norimitsu Onishi in the New York Times, August 18, 2002.
Consider Nigeria. Everyone agrees it is Africa's most populous nation. But what is its population? The United Nations says 114 million; the State Department, 120 million. The World Bank says 126.9 million, while the Central Intelligence Agency puts it at 126,635,626.
What should bother you about this?
(answer in the back of the PDF version of the book)Dealing correctly with significant figures can save you time! Often, students copy down numbers from their calculators with eight significant figures of precision, then type them back in for a later calculation. That's a waste of time, unless your original data had that kind of incredible precision.
The rules about significant figures are only rules of thumb, and are not a substitute for careful thinking. For instance, $20.00 + $0.05 is $20.05. It need not and should not be rounded off to $20. In general, the sig fig rules work best for multiplication and division, and we also apply them when doing a complicated calculation that involves many types of operations. For simple addition and subtraction, it makes more sense to maintain a fixed number of digits after the decimal point.
When in doubt, don't use the sig fig rules at all. Instead, intentionally change one piece of your initial data by the maximum amount by which you think it could have been off, and recalculate the final result. The digits on the end that are completely reshuffled are the ones that are meaningless, and should be omitted.
How many significant figures are there in each of the following measurements?
(1) 9.937 m
(2) 4.0 s
(3) 0.0000000000000037 kg
(answer in the back of the PDF version of the book)Why can't an insect be the size of a dog? Some skinny stretched-out cells in your spinal cord are a meter tall --- why does nature display no single cells that are not just a meter tall, but a meter wide, and a meter thick as well? Believe it or not, these are questions that can be answered fairly easily without knowing much more about physics than you already do. The only mathematical technique you really need is the humble conversion, applied to area and volume.
Area can be defined by saying that we can copy the shape of interest onto graph paper with 1 cm × 1 cm squares and count the number of squares inside. Fractions of squares can be estimated by eye. We then say the area equals the number of squares, in units of square cm. Although this might seem less “pure” than computing areas using formulae like A=π r2 for a circle or A=wh/2 for a triangle, those formulae are not useful as definitions of area because they cannot be applied to irregularly shaped areas.
Units of square cm are more commonly written as cm2 in science. Of course, the unit of measurement symbolized by “cm” is not an algebra symbol standing for a number that can be literally multiplied by itself. But it is advantageous to write the units of area that way and treat the units as if they were algebra symbols. For instance, if you have a rectangle with an area of 6 m2 and a width of 2 m, then calculating its length as (6 m2)/(2 m)=3 m gives a result that makes sense both numerically and in terms of units. This algebra-style treatment of the units also ensures that our methods of converting units work out correctly. For instance, if we accept the fraction
as a valid way of writing the number one, then one times one equals one, so we should also say that one can be represented by
which is the same as
That means the conversion factor from square meters to square centimeters is a factor of 104, i.e., a square meter has 104 square centimeters in it.
All of the above can be easily applied to volume as well, using one-cubic-centimeter blocks instead of squares on graph paper.
To many people, it seems hard to believe that a square meter equals 10000 square centimeters, or that a cubic meter equals a million cubic centimeters --- they think it would make more sense if there were 100 cm2 in 1 m2, and 100 cm3 in 1 m3, but that would be incorrect. The examples shown in figure b aim to make the correct answer more believable, using the traditional U.S. units of feet and yards. (One foot is 12 inches, and one yard is three feet.)
b / Visualizing conversions of area and volume using traditional U.S. units.
Based on figure b, convince yourself that there are 9 ft2 in a square yard, and 27 ft3 in a cubic yard, then demonstrate the same thing symbolically (i.e., with the method using fractions that equal one).
(answer in the back of the PDF version of the book)◊ Solved problem: converting mm2 to cm2 — problem 31
◊ Solved problem: scaling a liter — problem 40
How many square centimeters are there in a square inch? (1 inch = 2.54 cm) First find an approximate answer by making a drawing, then derive the conversion factor more accurately using the symbolic method.
c / Galileo Galilei (1564-1642).
d / The small boat holds up just fine.
e / A larger boat built with the same proportions as the small one will collapse under its own weight.
f / A boat this large needs to have timbers that are thicker compared to its size.
h / Galileo discusses planks made of wood, but the concept may be easier to imagine with clay. All three clay rods in the figure were originally the same shape. The medium-sized one was twice the height, twice the length, and twice the width of the small one, and similarly the large one was twice as big as the medium one in all its linear dimensions. The big one has four times the linear dimensions of the small one, 16 times the cross-sectional area when cut perpendicular to the page, and 64 times the volume. That means that the big one has 64 times the weight to support, but only 16 times the strength compared to the smallest one.
i / The area of a shape is proportional to the square of its linear dimensions, even if the shape is irregular.
j / The muffin comes out of the oven too hot to eat. Breaking it up into four pieces increases its surface area while keeping the total volume the same. It cools faster because of the greater surface-to-volume ratio. In general, smaller things have greater surface-to-volume ratios, but in this example there is no easy way to compute the effect exactly, because the small pieces aren't the same shape as the original muffin.
k / Example 3. The big triangle has four times more area than the little one.
l / A tricky way of solving example 3, explained in solution #2.
m / Example 4. The big sphere has 125 times more volume than the little one.
n / Example 5. The 48-point “S” has 1.78 times more area than the 36-point “S.”
Great fleas have lesser fleas
Upon their backs to bite 'em.
And lesser fleas have lesser still,
And so ad infinitum. -- Jonathan Swift
Now how do these conversions of area and volume relate to the questions I posed about sizes of living things? Well, imagine that you are shrunk like Alice in Wonderland to the size of an insect. One way of thinking about the change of scale is that what used to look like a centimeter now looks like perhaps a meter to you, because you're so much smaller. If area and volume scaled according to most people's intuitive, incorrect expectations, with 1 m2 being the same as 100 cm2, then there would be no particular reason why nature should behave any differently on your new, reduced scale. But nature does behave differently now that you're small. For instance, you will find that you can walk on water, and jump to many times your own height. The physicist Galileo Galilei had the basic insight that the scaling of area and volume determines how natural phenomena behave differently on different scales. He first reasoned about mechanical structures, but later extended his insights to living things, taking the then-radical point of view that at the fundamental level, a living organism should follow the same laws of nature as a machine. We will follow his lead by first discussing machines and then living things.
One of the world's most famous pieces of scientific writing is Galileo's Dialogues Concerning the Two New Sciences. Galileo was an entertaining writer who wanted to explain things clearly to laypeople, and he livened up his work by casting it in the form of a dialogue among three people. Salviati is really Galileo's alter ego. Simplicio is the stupid character, and one of the reasons Galileo got in trouble with the Church was that there were rumors that Simplicio represented the Pope. Sagredo is the earnest and intelligent student, with whom the reader is supposed to identify. (The following excerpts are from the 1914 translation by Crew and de Salvio.)
Yes, that is what I mean; and I refer especially to his last assertion which I have always regarded as false...; namely, that in speaking of these and other similar machines one cannot argue from the small to the large, because many devices which succeed on a small scale do not work on a large scale. Now, since mechanics has its foundations in geometry, where mere size [ is unimportant], I do not see that the properties of circles, triangles, cylinders, cones and other solid figures will change with their size. If, therefore, a large machine be constructed in such a way that its parts bear to one another the same ratio as in a smaller one, and if the smaller is sufficiently strong for the purpose for which it is designed, I do not see why the larger should not be able to withstand any severe and destructive tests to which it may be subjected.
Salviati contradicts Sagredo:
... Please observe, gentlemen, how facts which at first seem improbable will, even on scant explanation, drop the cloak which has hidden them and stand forth in naked and simple beauty. Who does not know that a horse falling from a height of three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury? Equally harmless would be the fall of a grasshopper from a tower or the fall of an ant from the distance of the moon.
The point Galileo is making here is that small things are sturdier in proportion to their size. There are a lot of objections that could be raised, however. After all, what does it really mean for something to be “strong”, to be “strong in proportion to its size,” or to be strong “out of proportion to its size?” Galileo hasn't given operational definitions of things like “strength,” i.e., definitions that spell out how to measure them numerically.
Also, a cat is shaped differently from a horse --- an enlarged photograph of a cat would not be mistaken for a horse, even if the photo-doctoring experts at the National Inquirer made it look like a person was riding on its back. A grasshopper is not even a mammal, and it has an exoskeleton instead of an internal skeleton. The whole argument would be a lot more convincing if we could do some isolation of variables, a scientific term that means to change only one thing at a time, isolating it from the other variables that might have an effect. If size is the variable whose effect we're interested in seeing, then we don't really want to compare things that are different in size but also different in other ways.
... we asked the reason why [shipbuilders] employed stocks, scaffolding, and bracing of larger dimensions for launching a big vessel than they do for a small one; and [an old man] answered that they did this in order to avoid the danger of the ship parting under its own heavy weight, a danger to which small boats are not subject?
After this entertaining but not scientifically rigorous beginning, Galileo starts to do something worthwhile by modern standards. He simplifies everything by considering the strength of a wooden plank. The variables involved can then be narrowed down to the type of wood, the width, the thickness, and the length. He also gives an operational definition of what it means for the plank to have a certain strength “in proportion to its size,” by introducing the concept of a plank that is the longest one that would not snap under its own weight if supported at one end. If you increased its length by the slightest amount, without increasing its width or thickness, it would break. He says that if one plank is the same shape as another but a different size, appearing like a reduced or enlarged photograph of the other, then the planks would be strong “in proportion to their sizes” if both were just barely able to support their own weight.
g / 1. This plank is as long as it can be without collapsing under its own weight. If it was a hundredth of an inch longer, it would collapse. 2. This plank is made out of the same kind of wood. It is twice as thick, twice as long, and twice as wide. It will collapse under its own weight.
Also, Galileo is doing something that would be frowned on in modern science: he is mixing experiments whose results he has actually observed (building boats of different sizes), with experiments that he could not possibly have done (dropping an ant from the height of the moon). He now relates how he has done actual experiments with such planks, and found that, according to this operational definition, they are not strong in proportion to their sizes. The larger one breaks. He makes sure to tell the reader how important the result is, via Sagredo's astonished response:
My brain already reels. My mind, like a cloud momentarily illuminated by a lightning flash, is for an instant filled with an unusual light, which now beckons to me and which now suddenly mingles and obscures strange, crude ideas. From what you have said it appears to me impossible to build two similar structures of the same material, but of different sizes and have them proportionately strong.
In other words, this specific experiment, using things like wooden planks that have no intrinsic scientific interest, has very wide implications because it points out a general principle, that nature acts differently on different scales.
To finish the discussion, Galileo gives an explanation. He says that the strength of a plank (defined as, say, the weight of the heaviest boulder you could put on the end without breaking it) is proportional to its cross-sectional area, that is, the surface area of the fresh wood that would be exposed if you sawed through it in the middle. Its weight, however, is proportional to its volume.2
How do the volume and cross-sectional area of the longer plank compare with those of the shorter plank? We have already seen, while discussing conversions of the units of area and volume, that these quantities don't act the way most people naively expect. You might think that the volume and area of the longer plank would both be doubled compared to the shorter plank, so they would increase in proportion to each other, and the longer plank would be equally able to support its weight. You would be wrong, but Galileo knows that this is a common misconception, so he has Salviati address the point specifically:
... Take, for example, a cube two inches on a side so that each face has an area of four square inches and the total area, i.e., the sum of the six faces, amounts to twenty-four square inches; now imagine this cube to be sawed through three times [with cuts in three perpendicular planes] so as to divide it into eight smaller cubes, each one inch on the side, each face one inch square, and the total surface of each cube six square inches instead of twenty-four in the case of the larger cube. It is evident therefore, that the surface of the little cube is only one-fourth that of the larger, namely, the ratio of six to twenty-four; but the volume of the solid cube itself is only one-eighth; the volume, and hence also the weight, diminishes therefore much more rapidly than the surface... You see, therefore, Simplicio, that I was not mistaken when ... I said that the surface of a small solid is comparatively greater than that of a large one.
The same reasoning applies to the planks. Even though they are not cubes, the large one could be sawed into eight small ones, each with half the length, half the thickness, and half the width. The small plank, therefore, has more surface area in proportion to its weight, and is therefore able to support its own weight while the large one breaks.
You probably are not going to believe Galileo's claim that this has deep implications for all of nature unless you can be convinced that the same is true for any shape. Every drawing you've seen so far has been of squares, rectangles, and rectangular solids. Clearly the reasoning about sawing things up into smaller pieces would not prove anything about, say, an egg, which cannot be cut up into eight smaller egg-shaped objects with half the length.
Is it always true that something half the size has one quarter the surface area and one eighth the volume, even if it has an irregular shape? Take the example of a child's violin. Violins are made for small children in smaller size to accomodate their small bodies. Figure i shows a full-size violin, along with two violins made with half and 3/4 of the normal length.3 Let's study the surface area of the front panels of the three violins.
Consider the square in the interior of the panel of the full-size violin. In the 3/4-size violin, its height and width are both smaller by a factor of 3/4, so the area of the corresponding, smaller square becomes 3/4×3/4=9/16 of the original area, not 3/4 of the original area. Similarly, the corresponding square on the smallest violin has half the height and half the width of the original one, so its area is 1/4 the original area, not half.
The same reasoning works for parts of the panel near the edge, such as the part that only partially fills in the other square. The entire square scales down the same as a square in the interior, and in each violin the same fraction (about 70%) of the square is full, so the contribution of this part to the total area scales down just the same.
Since any small square region or any small region covering part of a square scales down like a square object, the entire surface area of an irregularly shaped object changes in the same manner as the surface area of a square: scaling it down by 3/4 reduces the area by a factor of 9/16, and so on.
In general, we can see that any time there are two objects with the same shape, but different linear dimensions (i.e., one looks like a reduced photo of the other), the ratio of their areas equals the ratio of the squares of their linear dimensions:
Note that it doesn't matter where we choose to measure the linear size, L, of an object. In the case of the violins, for instance, it could have been measured vertically, horizontally, diagonally, or even from the bottom of the left f-hole to the middle of the right f-hole. We just have to measure it in a consistent way on each violin. Since all the parts are assumed to shrink or expand in the same manner, the ratio L1/L2 is independent of the choice of measurement.
It is also important to realize that it is completely unnecessary to have a formula for the area of a violin. It is only possible to derive simple formulas for the areas of certain shapes like circles, rectangles, triangles and so on, but that is no impediment to the type of reasoning we are using.
Sometimes it is inconvenient to write all the equations in terms of ratios, especially when more than two objects are being compared. A more compact way of rewriting the previous equation is
The symbol “∝” means “is proportional to.” Scientists and engineers often speak about such relationships verbally using the phrases “scales like” or “goes like,” for instance “area goes like length squared.”
All of the above reasoning works just as well in the case of volume. Volume goes like length cubed:
When a car or truck travels over a road, there is wear and tear on the road surface, which incurs a cost. Studies show that the cost C per kilometer of travel is related to the weight per axle w by C ∝ w4. Translate this into a statement about ratios.
(answer in the back of the PDF version of the book)If different objects are made of the same material with the same density, ρ =m/V, then their masses, m=ρ V, are proportional to L3. (The symbol for density is ρ, the lower-case Greek letter “rho.”)
An important point is that all of the above reasoning about scaling only applies to objects that are the same shape. For instance, a piece of paper is larger than a pencil, but has a much greater surface-to-volume ratio.
Correct solution #1: Area scales in proportion to the square of the linear dimensions, so the larger triangle has four times more area (22=4).
Correct solution #2: You could cut the larger triangle into four of the smaller size, as shown in fig. (b), so its area is four times greater. (This solution is correct, but it would not work for a shape like a circle, which can't be cut up into smaller circles.)
Correct solution #3: The area of a triangle is given by
A=bh/2, where b is the base and h is the height. The areas of the triangles are
(Although this solution is correct, it is a lot more work than solution #1, and it can only be used in this case because a triangle is a simple geometric shape, and we happen to know a formula for its area.)
Correct solution #4: The area of a triangle is A= bh/2. The comparison of the areas will come out the same as long as the ratios of the linear sizes of the triangles is as specified, so let's just say b1=1.00 m and b2=2.00 m. The heights are then also h1=1.00 m and h2=2.00 m, giving areas A1=0.50 m2 and A2=2.00 m2, so A2/A1=4.00.
(The solution is correct, but it wouldn't work with a shape for whose area we don't have a formula. Also, the numerical calculation might make the answer of 4.00 appear inexact, whereas solution #1 makes it clear that it is exactly 4.)
Incorrect solution: The area of a triangle is A=bh/2, and if you plug in b=2.00 m and h=2.00 m, you get A=2.00 m2, so the bigger triangle has 2.00 times more area. (This solution is incorrect because no comparison has been made with the smaller triangle.)
Correct solution #1: Volume scales like the third power of the linear size, so the larger sphere has a volume that is 125 times greater (53=125).
Correct solution #2: The volume of a sphere is V=(4/3)π r3, so
Incorrect solution: The volume of a sphere is V=(4/3)π r3, so
(The solution is incorrect because (5r1)3 is not the same as 5r13.)
Correct solution: The amount of ink depends on the area to be covered with ink, and area is proportional to the square of the linear dimensions, so the amount of ink required for the second “S” is greater by a factor of (48/36)2=1.78.
Incorrect solution: The length of the curve of the second “S” is longer by a factor of 48/36=1.33, so 1.33 times more ink is required.
(The solution is wrong because it assumes incorrectly that the width of the curve is the same in both cases. Actually both the width and the length of the curve are greater by a factor of 48/36, so the area is greater by a factor of (48/36)2=1.78.)
Reasoning about ratios and proportionalities is one of the three essential mathematical skills, summarized on pp.848-850, that you need for success in this course.
◊ Solved problem: a telescope gathers light — problem 32
◊ Solved problem: distance from an earthquake — problem 33
A toy fire engine is 1/30 the size of the real one, but is constructed from the same metal with the same proportions. How many times smaller is its weight? How many times less red paint would be needed to paint it?
Galileo spends a lot of time in his dialog discussing what really happens when things break. He discusses everything in terms of Aristotle's now-discredited explanation that things are hard to break, because if something breaks, there has to be a gap between the two halves with nothing in between, at least initially. Nature, according to Aristotle, “abhors a vacuum,” i.e., nature doesn't “like” empty space to exist. Of course, air will rush into the gap immediately, but at the very moment of breaking, Aristotle imagined a vacuum in the gap. Is Aristotle's explanation of why it is hard to break things an experimentally testable statement? If so, how could it be tested experimentally?
o / Can you guess how many jelly beans are in the jar? If you try to guess directly, you will almost certainly underestimate. The right way to do it is to estimate the linear dimensions, then get the volume indirectly. See problem 44, p. 53.
p / Consider a spherical cow.
It is the mark of an instructed mind to rest satisfied with the degree of precision that the nature of the subject permits and not to seek an exactness where only an approximation of the truth is possible. -- Aristotle
It is a common misconception that science must be exact. For instance, in the Star Trek TV series, it would often happen that Captain Kirk would ask Mr. Spock, “Spock, we're in a pretty bad situation. What do you think are our chances of getting out of here?” The scientific Mr. Spock would answer with something like, “Captain, I estimate the odds as 237.345 to one.” In reality, he could not have estimated the odds with six significant figures of accuracy, but nevertheless one of the hallmarks of a person with a good education in science is the ability to make estimates that are likely to be at least somewhere in the right ballpark. In many such situations, it is often only necessary to get an answer that is off by no more than a factor of ten in either direction. Since things that differ by a factor of ten are said to differ by one order of magnitude, such an estimate is called an order-of-magnitude estimate. The tilde, ∼, is used to indicate that things are only of the same order of magnitude, but not exactly equal, as in
The tilde can also be used in front of an individual number to emphasize that the number is only of the right order of magnitude.
Although making order-of-magnitude estimates seems simple and natural to experienced scientists, it's a mode of reasoning that is completely unfamiliar to most college students. Some of the typical mental steps can be illustrated in the following example.
◊ Roughly what percentage of the price of a tomato comes from the cost of transporting it in a truck?
◊ The following incorrect solution illustrates one of the main ways you can go wrong in order-of-magnitude estimates.
Incorrect solution: Let's say the trucker needs to make a $400 profit on the trip. Taking into account her benefits, the cost of gas, and maintenance and payments on the truck, let's say the total cost is more like $2000. I'd guess about 5000 tomatoes would fit in the back of the truck, so the extra cost per tomato is 40 cents. That means the cost of transporting one tomato is comparable to the cost of the tomato itself. Transportation really adds a lot to the cost of produce, I guess.
The problem is that the human brain is not very good at estimating area or volume, so it turns out the estimate of 5000 tomatoes fitting in the truck is way off. That's why people have a hard time at those contests where you are supposed to estimate the number of jellybeans in a big jar. Another example is that most people think their families use about 10 gallons of water per day, but in reality the average is about 300 gallons per day. When estimating area or volume, you are much better off estimating linear dimensions, and computing volume from the linear dimensions. Here's a better solution to the problem about the tomato truck:
As in the previous solution, say the cost of the trip is $2000. The dimensions of the bin are probably 4 m × 2 m × 1 m, for a volume of 8 m3. Since the whole thing is just an order-of-magnitude estimate, let's round that off to the nearest power of ten, 10 m3. The shape of a tomato is complicated, and I don't know any formula for the volume of a tomato shape, but since this is just an estimate, let's pretend that a tomato is a cube, 0.05 m × 0.05 m × 0.05 m, for a volume of 1.25×10-4 m3. Since this is just a rough estimate, let's round that to 10-4 m3. We can find the total number of tomatoes by dividing the volume of the bin by the volume of one tomato: 10 m3/10-4 m3=105 tomatoes. The transportation cost per tomato is $$2000/10^5$ tomatoes=$0.02/tomato. That means that transportation really doesn't contribute very much to the cost of a tomato.
Approximating the shape of a tomato as a cube is an example of another general strategy for making order-of-magnitude estimates. A similar situation would occur if you were trying to estimate how many m2 of leather could be produced from a herd of ten thousand cattle. There is no point in trying to take into account the shape of the cows' bodies. A reasonable plan of attack might be to consider a spherical cow. Probably a cow has roughly the same surface area as a sphere with a radius of about 1 m, which would be 4π (1 m)2. Using the well-known facts that pi equals three, and four times three equals about ten, we can guess that a cow has a surface area of about 10 m2, so the herd as a whole might yield 105 m2 of leather.
Its torso looks like it can be approximated by a rectangular box with dimensions 10 m×5 m×3 m, giving about 2×102 m3. Living things are mostly made of water, so we assume the animal to have the density of water, 1 g/cm3, which converts to 103 kg/m3. This gives a mass of about 2×105 kg, or 200 metric tons.
The following list summarizes the strategies for getting a good order-of-magnitude estimate.
a / Problem 10.
b / Problem 12.
c / Albert Einstein, and his moustache, problem 35.
d / Problem 40.
e / Problem 42.
f / Problem 43.
1. Correct use of a calculator: (a) Calculate 
on a calculator.
[Self-check: The most common mistake results in 97555.40.] (answer check available at lightandmatter.com)
(b) Which would be more like the price of a TV, and which
would be more like the price of a house, $3.5×105 or $3.55?
2. Compute the following things. If they don't make sense
because of units, say so.
(a) 3 cm + 5 cm
(b) 1.11 m + 22 cm
(c) 120 miles + 2.0 hours
(d) 120 miles / 2.0 hours
3. Your backyard has brick walls on both ends. You measure a distance of 23.4 m from the inside of one wall to the inside of the other. Each wall is 29.4 cm thick. How far is it from the outside of one wall to the outside of the other? Pay attention to significant figures.
4. The speed of light is 3.0×108 m/s. Convert
this to furlongs per fortnight. A furlong is 220 yards, and
a fortnight is 14 days. An inch is 2.54 cm.(answer check available at lightandmatter.com)
5. Express each of the following quantities in micrograms:
(a) 10 mg, (b) 104 g, (c) 10 kg, (d) 100×103 g, (e) 1000 ng. (answer check available at lightandmatter.com)
6. (solution in the pdf version of the book) Convert 134 mg to units of kg, writing your answer in scientific notation.
7. In the last century, the average age of the onset of
puberty for girls has decreased by several years. Urban
folklore has it that this is because of hormones fed to beef
cattle, but it is more likely to be because modern girls
have more body fat on the average and possibly because of
estrogen-mimicking chemicals in the environment from the
breakdown of pesticides. A hamburger from a hormone-implanted
steer has about 0.2 ng of estrogen (about double the amount
of natural beef). A serving of peas contains about 300 ng of
estrogen. An adult woman produces about 0.5 mg of estrogen
per day (note the different unit!). (a) How many hamburgers
would a girl have to eat in one day to consume as much
estrogen as an adult woman's daily production? (b) How
many servings of peas? (answer check available at lightandmatter.com)
8. (solution in the pdf version of the book) The usual definition of the mean (average) of two numbers a and b is (a+b)/2. This is called the arithmetic mean. The geometric mean, however, is defined as (ab)1/2 (i.e., the square root of ab). For the sake of definiteness, let's say both numbers have units of mass. (a) Compute the arithmetic mean of two numbers that have units of grams. Then convert the numbers to units of kilograms and recompute their mean. Is the answer consistent? (b) Do the same for the geometric mean. (c) If a and b both have units of grams, what should we call the units of ab? Does your answer make sense when you take the square root? (d) Suppose someone proposes to you a third kind of mean, called the superduper mean, defined as (ab)1/3. Is this reasonable?
9. In an article on the SARS epidemic, the May 7, 2003 New York Times discusses conflicting estimates of the disease's incubation period (the average time that elapses from infection to the first symptoms). “The study estimated it to be 6.4 days. But other statistical calculations ... showed that the incubation period could be as long as 14.22 days.” What's wrong here?
11. The distance to the horizon is given by the expression 
, where r is the
radius of the Earth, and h is the observer's height above the Earth's surface. (This can
be proved using the Pythagorean theorem.) Show that the units of this expression make
sense. (See example 2 on p.
26 for an example of how to do this.)
Don't try to prove the result, just check its units.
12. (solution in the pdf version of the book) (a) Based on the definitions of the sine, cosine, and tangent, what units must they have? (b) A cute formula from trigonometry lets you find any angle of a triangle if you know the lengths of its sides. Using the notation shown in the figure, and letting s=(a+b+c)/2 be half the perimeter, we have
Show that the units of this equation make sense. In other words, check that the units of the right-hand side are the same as your answer to part a of the question.
13. A physics homework question asks, “If you start from rest and accelerate at 1.54 m/s2 for 3.29 s, how far do you travel by the end of that time?” A student answers as follows:
His Aunt Wanda is good with numbers, but has never taken physics. She doesn't know the formula for the distance traveled under constant acceleration over a given amount of time, but she tells her nephew his answer cannot be right. How does she know?
14. You are looking into a deep well. It is dark, and you cannot see the bottom. You want to find out how deep it is, so you drop a rock in, and you hear a splash 3.0 seconds later. How deep is the well? (answer check available at lightandmatter.com)
15. You take a trip in your spaceship to another star.
Setting off, you increase your speed at a constant
acceleration. Once you get half-way there, you start
decelerating, at the same rate, so that by the time you get
there, you have slowed down to zero speed. You see the
tourist attractions, and then head home by the same method.
(a) Find a formula for the time, T, required for the round
trip, in terms of d, the distance from our sun to the
star, and a, the magnitude of the acceleration. Note that
the acceleration is not constant over the whole trip, but
the trip can be broken up into constant-acceleration parts.
(b) The nearest star to the Earth (other than our own sun)
is Proxima Centauri, at a distance of d=4×1016 m.
Suppose you use an acceleration of a=10 m/s2, just enough
to compensate for the lack of true gravity and make you feel
comfortable. How long does the round trip take, in years?
(c) Using the same numbers for d and a, find your
maximum speed. Compare this to the speed of light, which is
3.0×108 m/s. (Later in this course, you will learn
that there are some new things going on in physics when one
gets close to the speed of light, and that it is impossible
to exceed the speed of light. For now, though, just use the
simpler ideas you've learned so far.) (answer check available at lightandmatter.com)
16. You climb half-way up a tree, and drop a rock. Then you climb to the top, and drop another rock. How many times greater is the velocity of the second rock on impact? Explain. (The answer is not two times greater.)
17. (solution in the pdf version of the book) If the acceleration of gravity on Mars is 1/3 that on Earth, how many times longer does it take for a rock to drop the same distance on Mars? Ignore air resistance.
18. A person is parachute jumping. During the time between when she leaps out of the plane and when she opens her chute, her altitude is given by an equation of the form
where e is the base of natural logarithms, and b, c, and
k are constants. Because of air resistance, her velocity
does not increase at a steady rate as it would for an
object falling in vacuum.
(a) What units would b, c, and k have to have for the
equation to make sense?
(b) Find the person's velocity, v, as a function of time.
[You will need to use the chain rule, and the fact that
d(ex)/dx=ex.] (answer check available at lightandmatter.com)
(c) Use your answer from part (b) to get an interpretation
of the constant c. [Hint: e-x approaches zero for
large values of x.]
(d) Find the person's acceleration, a, as a function of time.(answer check available at lightandmatter.com)
(e) Use your answer from part (b) to show that if she waits
long enough to open her chute, her acceleration will become very small.
19. (solution in the pdf version of the book) In July 1999, Popular Mechanics carried out tests to find which car sold by a major auto maker could cover a quarter mile (402 meters) in the shortest time, starting from rest. Because the distance is so short, this type of test is designed mainly to favor the car with the greatest acceleration, not the greatest maximum speed (which is irrelevant to the average person). The winner was the Dodge Viper, with a time of 12.08 s. The car's top (and presumably final) speed was 118.51 miles per hour (52.98 m/s). (a) If a car, starting from rest and moving with constant acceleration, covers a quarter mile in this time interval, what is its acceleration? (b) What would be the final speed of a car that covered a quarter mile with the constant acceleration you found in part a? (c) Based on the discrepancy between your answer in part b and the actual final speed of the Viper, what do you conclude about how its acceleration changed over time?
20. The speed required for a low-earth orbit is 7.9×103 m/s (see ch. 10). When a rocket is launched into orbit, it goes up a little at first to get above almost all of the atmosphere, but then tips over horizontally to build up to orbital speed. Suppose the horizontal acceleration is limited to 3g to keep from damaging the cargo (or hurting the crew, for a crewed flight). (a) What is the minimum distance the rocket must travel downrange before it reaches orbital speed? How much does it matter whether you take into account the initial eastward velocity due to the rotation of the earth? (b) Rather than a rocket ship, it might be advantageous to use a railgun design, in which the craft would be accelerated to orbital speeds along a railroad track. This has the advantage that it isn't necessary to lift a large mass of fuel, since the energy source is external. Based on your answer to part a, comment on the feasibility of this design for crewed launches from the earth's surface.
21. Consider the following passage from Alice in Wonderland, in which Alice has been falling for a long time down a rabbit hole:
Down, down, down. Would the fall never come to an end? “I wonder how many miles I've fallen by this time?” she said aloud. “I must be getting somewhere near the center of the earth. Let me see: that would be four thousand miles down, I think” (for, you see, Alice had learned several things of this sort in her lessons in the schoolroom, and though this was not a very good opportunity for showing off her knowledge, as there was no one to listen to her, still it was good practice to say it over)...
Alice doesn't know much physics, but let's try to calculate the amount of time it would take to fall four thousand miles, starting from rest with an acceleration of 10 m/s2. This is really only a lower limit; if there really was a hole that deep, the fall would actually take a longer time than the one you calculate, both because there is air friction and because gravity gets weaker as you get deeper (at the center of the earth, g is zero, because the earth is pulling you equally in every direction at once). (answer check available at lightandmatter.com)
22. How many cubic inches are there in a cubic foot? The answer is not 12.(answer check available at lightandmatter.com)
23. Assume a dog's brain is twice as great in diameter as a cat's, but each animal's brain cells are the same size and their brains are the same shape. In addition to being a far better companion and much nicer to come home to, how many times more brain cells does a dog have than a cat? The answer is not 2.
24. The population density of Los Angeles is about 4000 people/km2. That of San Francisco is about 6000 people/km2. How many times farther away is the average person's nearest neighbor in LA than in San Francisco? The answer is not 1.5. (answer check available at lightandmatter.com)
25. A hunting dog's nose has about 10 square inches of active
surface. How is this possible, since the dog's nose is only
about 1 in × 1 in × 1 in = 1 
After all, 10 is
greater than 1, so how can it fit?
27. In a computer memory chip, each bit of information (a 0 or a 1) is stored in a single tiny circuit etched onto the surface of a silicon chip. The circuits cover the surface of the chip like lots in a housing development. A typical chip stores 64 Mb (megabytes) of data, where a byte is 8 bits. Estimate (a) the area of each circuit, and (b) its linear size.
28. Suppose someone built a gigantic apartment building, measuring 10 km × 10 km at the base. Estimate how tall the building would have to be to have space in it for the entire world's population to live.
29. A hamburger chain advertises that it has sold 10 billion Bongo Burgers. Estimate the total mass of feed required to raise the cows used to make the burgers.
32. (solution in the pdf version of the book) Compare the light-gathering powers of a 3-cm-diameter telescope and a 30-cm telescope.
33. (solution in the pdf version of the book) One step on the Richter scale corresponds to a factor of 100 in terms of the energy absorbed by something on the surface of the Earth, e.g., a house. For instance, a 9.3-magnitude quake would release 100 times more energy than an 8.3. The energy spreads out from the epicenter as a wave, and for the sake of this problem we'll assume we're dealing with seismic waves that spread out in three dimensions, so that we can visualize them as hemispheres spreading out under the surface of the earth. If a certain 7.6-magnitude earthquake and a certain 5.6-magnitude earthquake produce the same amount of vibration where I live, compare the distances from my house to the two epicenters.
34. In Europe, a piece of paper of the standard size, called A4, is a little narrower and taller than its American counterpart. The ratio of the height to the width is the square root of 2, and this has some useful properties. For instance, if you cut an A4 sheet from left to right, you get two smaller sheets that have the same proportions. You can even buy sheets of this smaller size, and they're called A5. There is a whole series of sizes related in this way, all with the same proportions. (a) Compare an A5 sheet to an A4 in terms of area and linear size. (b) The series of paper sizes starts from an A0 sheet, which has an area of one square meter. Suppose we had a series of boxes defined in a similar way: the B0 box has a volume of one cubic meter, two B1 boxes fit exactly inside an B0 box, and so on. What would be the dimensions of a B0 box? (answer check available at lightandmatter.com)
36. According to folklore, every time you take a breath, you are inhaling some of the atoms exhaled in Caesar's last words. Is this true? If so, how many?
37. The Earth's surface is about 70% water. Mars's diameter is about half the Earth's, but it has no surface water. Compare the land areas of the two planets.(answer check available at lightandmatter.com)
38. (solution in the pdf version of the book) The traditional Martini glass is shaped like a cone with the point at the bottom. Suppose you make a Martini by pouring vermouth into the glass to a depth of 3 cm, and then adding gin to bring the depth to 6 cm. What are the proportions of gin and vermouth?
39. The central portion of a CD is taken up by the hole and some surrounding clear plastic, and this area is unavailable for storing data. The radius of the central circle is about 35% of the outer radius of the data-storing area. What percentage of the CD's area is therefore lost? (answer check available at lightandmatter.com)
40. The one-liter cube in the photo has been marked off into smaller cubes, with linear dimensions one tenth those of the big one. What is the volume of each of the small cubes?(solution in the pdf version of the book)
41. Estimate the number of man-hours required for building the Great Wall of China. (solution in the pdf version of the book)
42. (a) Using the microscope photo in the figure, estimate the mass of a one cell of the E. coli bacterium, which is one of the most common ones in the human intestine. Note the scale at the lower right corner, which is 1 μm. Each of the tubular objects in the column is one cell. (b) The feces in the human intestine are mostly bacteria (some dead, some alive), of which E. coli is a large and typical component. Estimate the number of bacteria in your intestines, and compare with the number of human cells in your body, which is believed to be roughly on the order of 1013. (c) Interpreting your result from part b, what does this tell you about the size of a typical human cell compared to the size of a typical bacterial cell?
43. The figure shows a practical, simple experiment for
determining g to high precision. Two steel balls are
suspended from electromagnets, and are released simultaneously
when the electric current is shut off. They fall through
unequal heights Δ x1 and Δ x2. A computer
records the sounds through a microphone as first one ball
and then the other strikes the floor. From this recording,
we can accurately determine the quantity T defined as
T=Δ t2-Δ t1, i.e., the time lag between the
first and second impacts. Note that since the balls do not
make any sound when they are released, we have no way of
measuring the individual times Δ t2 and Δ t1.
(a) Find an equation for g in terms of the measured
quantities T, Δ x1 and Δ x2.(answer check available at lightandmatter.com)
(b) Check the
units of your equation.
(c) Check that your equation gives
the correct result in the case where Δ x1 is very
close to zero. However, is this case realistic?
(d) What
happens when Δ x1=Δ x2? Discuss this both
mathematically and physically.
44. (solution in the pdf version of the book) Estimate the number of jellybeans in figure o on p. 44.
