crowell physics selection

8/8/2019 Crowell Physics Selection

http://slidepdf.com/reader/full/crowell-physics-selection 1/141

Benjamin Crowell

www.lightandmatter.com



a / Amoebas this size a

seldom encountered.

Life would be very different if y

were the size of an insect.

Chapter 1

Scaling and

Order-of-Magnitude

Estimates

1.1 Introduction

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 naturedisplay no single cells that are not just a meter tall, but a meterwide, and a meter thick as well? Believe it or not, these are questionsthat can be answered fairly easily without knowing much more aboutphysics than you already do. The only mathematical technique youreally need is the humble conversion, applied to area and volume.

Area and volumeArea can be defined by saying that we can copy the shape of

interest onto graph paper with 1 cm × 1 cm squares and count thenumber of squares inside. Fractions of squares can be estimated byeye. We then say the area equals the number of squares, in units of square cm. Although this might seem less “pure” than computingareas using formulae like A = πr2 for a circle or A = wh/2 for atriangle, 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 analgebra symbol standing for a number that can be literally multipliedby itself. But it is advantageous to write the units of area that wayand treat the units as if they were algebra symbols. For instance,

if you have a rectangle with an area of 6m2 and a width of 2 m,then calculating its length as (6 m2)/(2 m) = 3 m gives a resultthat makes sense both numerically and in terms of units. Thisalgebra-style treatment of the units also ensures that our methodsof converting units work out correctly. For instance, if we acceptthe fraction

100 cm

1 m

as a valid way of writing the number one, then one times one equalsone, so we should also say that one can be represented by

100 cm

1 m ×

100 cm

1 m ,

which is the same as10000 cm2

1 m2.

That means the conversion factor from square meters to square cen-timeters is a factor of 104, i.e., a square meter has 104 square cen-timeters in it.

All of the above can be easily applied to volume as well, usingone-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 amillion 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 beincorrect. The examples shown in figure b aim to make the correctanswer more believable, using the traditional U.S. units of feet andyards. (One foot is 12 inches, and one yard is three feet.)

b / Visualizing conversions ofarea and volume using traditionalU.S. units.

self-check A

Based on figure b, convince yourself that there are 9 ft2 in a square yard,

44 Chapter 1 Scaling and Order-of-Magnitude Estimates



and 27 ft3 in a cubic yard, then demonstrate the same thing symbolically

(i.e., with the method using fractions that equal one). Answer, p.

272

Solved problem: converting mm2 to cm2 page 61, problem 10

Solved problem: scaling a liter page 62, problem 19

Discussion Question

A 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 de-rive the conversion factor more accurately using the symbolic method.

c / Galileo Galilei (1564-1642) was a Renaissance Italian who brought thescientific method to bear on physics, creating the modern version of thescience. Coming from a noble but very poor family, Galileo had to dropout of medical school at the University of Pisa when he ran out of money.Eventually becoming a lecturer in mathematics at the same school, he

began a career as a notorious troublemaker by writing a burlesque ridi-culing the university’s regulations — he was forced to resign, but found anew teaching position at Padua. He invented the pendulum clock, inves-tigated the motion of falling bodies, and discovered the moons of Jupiter.The thrust of his life’s work was to discredit Aristotle’s physics by con-fronting it with contradictory experiments, a program which paved the wayfor Newton’s discovery of the relationship between force and motion. Inchapter 3 we’ll come to the story of Galileo’s ultimate fate at the hands ofthe Church.

1.2 Scaling of Area and Volume

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 thequestions I posed about sizes of living things? Well, imagine thatyou are shrunk like Alice in Wonderland to the size of an insect.One way of thinking about the change of scale is that what usedto 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 beingthe same as 100 cm2, then there would be no particular reasonwhy nature should behave any differently on your new, reducedscale. But nature does behave differently now that you’re small.For instance, you will find that you can walk on water, and jumpto many times your own height. The physicist Galileo Galilei hadthe basic insight that the scaling of area and volume determineshow natural phenomena behave differently on different scales. He

Section 1.2 Scaling of Area and Volume



g / Galileo discusses planmade of wood, but the concemay be easier to imagine wclay. All three clay rods in tfigure were originally the samshape. The medium-sized owas twice the height, twice tlength, and twice the width the small one, and similarly t

large one was twice as big the medium one in all its linedimensions. The big one hfour times the linear dimensioof the small one, 16 times tcross-sectional area when cperpendicular to the page, a64 times the volume. That meathat the big one has 64 times tweight to support, but only times the strength compared the smallest one.

“out of proportion to its size?” Galileo hasn’t given operationaldefinitions of things like “strength,” i.e., definitions that spell outhow to measure them numerically.

Also, a cat is shaped differently from a horse — an enlargedphotograph of a cat would not be mistaken for a horse, even if thephoto-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 wholeargument would be a lot more convincing if we could do some iso-lation of variables, a scientific term that means to change only onething at a time, isolating it from the other variables that might havean effect. If size is the variable whose effect we’re interested in see-ing, then we don’t really want to compare things that are differentin size but also different in other ways.

SALVIATI: . . . we asked the reason why [shipbuilders] em-

ployed 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 thedanger of the ship parting under its own heavy weight, a dan-

ger 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 woodenplank. The variables involved can then be narrowed down to thetype of wood, the width, the thickness, and the length. He alsogives an operational definition of what it means for the plank tohave a certain strength “in proportion to its size,” by introducingthe concept of a plank that is the longest one that would not snap

under its own weight if supported at one end. If you increasedits length by the slightest amount, without increasing its width orthickness, it would break. He says that if one plank is the sameshape as another but a different size, appearing like a reduced orenlarged photograph of the other, then the planks would be strong“in proportion to their sizes” if both were just barely able to supporttheir own weight.

h / 1. This plank is as long as

can be without collapsing undits own weight. If it was a hudredth of an inch longer, it wocollapse. 2. This plank is maout of the same kind of wood. Itwice as thick, twice as long, atwice as wide. It will collapse uder its own weight.




Also, Galileo is doing something that would be frowned on inmodern science: he is mixing experiments whose results he has ac-tually observed (building boats of different sizes), with experimentsthat he could not possibly have done (dropping an ant from theheight of the moon). He now relates how he has done actual ex-periments with such planks, and found that, according to this op-

erational definition, they are not strong in proportion to their sizes.The larger one breaks. He makes sure to tell the reader how impor-tant the result is, via Sagredo’s astonished response:

SAGREDO: My brain already reels. My mind, like a cloud

momentarily illuminated by a lightning flash, is for an instantfilled 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 impli-cations because it points out a general principle, that nature actsdifferently on different scales.

To finish the discussion, Galileo gives an explanation. He saysthat the strength of a plank (defined as, say, the weight of the heav-iest boulder you could put on the end without breaking it) is pro-portional to its cross-sectional area, that is, the surface area of thefresh wood that would be exposed if you sawed through it in themiddle. Its weight, however, is proportional to its volume.1

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, thatthese quantities don’t act the way most people naively expect. Youmight think that the volume and area of the longer plank would bothbe doubled compared to the shorter plank, so they would increasein proportion to each other, and the longer plank would be equallyable to support its weight. You would be wrong, but Galileo knowsthat this is a common misconception, so he has Salviati address thepoint specifically:

SALVIATI: . . . 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, amountsto 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

1Galileo makes a slightly more complicated argument, taking into accountthe effect of leverage (torque). The result I’m referring to comes out the sameregardless of this effect.




i / The area of a shape proportional to the square of linear dimensions, even if tshape is irregular.

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 vol-

ume 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, thatI 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 theyare 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 toits weight, and is therefore able to support its own weight while thelarge one breaks.

Scaling of area and volume for irregularly shaped objects

You probably are not going to believe Galileo’s claim that thishas deep implications for all of nature unless you can be convincedthat the same is true for any shape. Every drawing you’ve seen sofar has been of squares, rectangles, and rectangular solids. Clearlythe reasoning about sawing things up into smaller pieces would notprove anything about, say, an egg, which cannot be cut up into eightsmaller egg-shaped objects with half the length.

Is it always true that something half the size has one quarterthe surface area and one eighth the volume, even if it has an irreg-ular shape? Take the example of a child’s violin. Violins are madefor small children in smaller size to accomodate their small bodies.

Figure i shows a full-size violin, along with two violins made withhalf and 3/4 of the normal length.2 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-sizeviolin. In the 3/4-size violin, its height and width are both smallerby a factor of 3/4, so the area of the corresponding, smaller squarebecomes 3/4×3/4 = 9/16 of the original area, not 3/4 of the originalarea. Similarly, the corresponding square on the smallest violin hashalf the height and half the width of the original one, so its area is1/4 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. Theentire square scales down the same as a square in the interior, andin each violin the same fraction (about 70%) of the square is full, sothe contribution of this part to the total area scales down just the

2The customary terms “half-size” and “3/4-size” actually don’t describe thesizes in any accurate way. They’re really just standard, arbitrary marketinglabels.




j / The muffin comes out ofthe oven too hot to eat. Breakingit up into four pieces increasesits surface area while keepingthe total volume the same. Itcools faster because of thegreater surface-to-volume ratio.In general, smaller things havegreater surface-to-volume ratios,but in this example there is noeasy way to compute the effect

exactly, because the small piecesaren’t the same shape as theoriginal muffin.

same.

Since any small square region or any small region covering partof a square scales down like a square object, the entire surface areaof an irregularly shaped object changes in the same manner as thesurface area of a square: scaling it down by 3/4 reduces the area bya factor of 9/16, and so on.

In general, we can see that any time there are two objects withthe same shape, but different linear dimensions (i.e., one looks like areduced photo of the other), the ratio of their areas equals the ratioof the squares of their linear dimensions:

A1

A2

=

L1

L2

2

.

Note that it doesn’t matter where we choose to measure the linearsize, L, of an object. In the case of the violins, for instance, it couldhave 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 allthe parts are assumed to shrink or expand in the same manner, theratio L1/L2 is independent of the choice of measurement.

It is also important to realize that it is completely unnecessaryto have a formula for the area of a violin. It is only possible toderive simple formulas for the areas of certain shapes like circles,rectangles, triangles and so on, but that is no impediment to thetype of reasoning we are using.

Sometimes it is inconvenient to write all the equations in termsof ratios, especially when more than two objects are being compared.

A more compact way of rewriting the previous equation is

A ∝ L2 .

The symbol “∝” means “is proportional to.” Scientists and engi-neers 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 vol-ume. Volume goes like length cubed:

V ∝ L3 .

If different objects are made of the same material with the samedensity, ρ = m/V , then their masses, m = ρV , are proportionalto L3, and so are their weights. (The symbol for density is ρ, thelower-case Greek letter “rho.”)

An important point is that all of the above reasoning aboutscaling only applies to objects that are the same shape. For instance,a piece of paper is larger than a pencil, but has a much greatersurface-to-volume ratio.




k / Example 1. The big triagle has four times more area ththe little one.

l / A tricky way of solving eample 1, explained in solution #

One of the first things I learned as a teacher was that studentswere not very original about their mistakes. Every group of studentstends to come up with the same goofs as the previous class. Thefollowing are some examples of correct and incorrect reasoning aboutproportionality.

Scaling of the area of a triangle example 1

In figure k, the larger triangle has sides twice as long. Howmany times greater is its area?

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 ofthe 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

A1 = b 1h 1/2

A2 = b 2h 2/2

= (2b 1)(2h 1)/2

= 2b 1h 1

A2/A1 = (2b 1h 1)/(b 1h 1/2)

= 4

(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 asimple 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 theratios of the linear sizes of the triangles is as specified, so let’s

just say b 1 = 1.00 m and b 2 = 2.00 m. The heights are then also

h 1 = 1.00 m and h 2 = 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 cal-culation 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, sothe bigger triangle has 2.00 times more area. (This solution is

incorrect because no comparison has been made with the smaller

triangle.)




n / Example 3. The 48-point“S” has 1.78 times more areathan the 36-point “S.”

m / Example 2. The big spherehas 125 times more volume thanthe little one.

Scaling of the volume of a sphere example 2

In figure m, the larger sphere has a radius that is five times

greater. How many times greater is its volume?

Correct solution #1: Volume scales like the third power of the

linear size, so the larger sphere has a volume that is 125 timesgreater (53 = 125).

Correct solution #2: The volume of a sphere is V = (4/3)π r 3, so

V 1 =4

3π r 31

V 2 =4

3π r 32

=4

3π (5r 1)3

=500

3π r 31

V 2/V 1 = 500

3

π r 31 /4

3

π r 31 = 125

Incorrect solution: The volume of a sphere is V = (4/3)π r 3, so

V 1 =4

3π r 31

V 2 =4

3π r 32

=4

3π · 5r 31

=20

3π r 31

V 2/V 1 =

203

π r 31/

43π r 31

= 5

(The solution is incorrect because (5r 1)3 is not the same as 5r 31 .)

Scaling of a more complex shape example 3

The first letter “S” in figure n is in a 36-point font, the second in48-point. How many times more ink is required to make the larger

“S”? (Points are a unit of length used in typography.)

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.)

Solved problem: a telescope gathers light page 61, problem 11

Solved problem: distance from an earthquake page 61, problem 12

Discussion Questions

A A toy fire engine is 1/30 the size of the real one, but is constructedfrom the same metal with the same proportions. How many times smalleris its weight? How many times less red paint would be needed to paintit?

B Galileo spends a lot of time in his dialog discussing what reallyhappens when things break. He discusses everything in terms of Aristo-tle’s now-discredited explanation that things are hard to break, becauseif something breaks, there has to be a gap between the two halves withnothing in between, at least initially. Nature, according to Aristotle, “ab-hors 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 whyit is hard to break things an experimentally testable statement? If so, howcould it be tested experimentally?

1.3 Scaling Applied To Biology

Organisms of different sizes with the same shape

The left-hand panel in figure o shows the approximate valid-ity of the proportionality m ∝ L3 for cockroaches (redrawn fromMcMahon and Bonner). The scatter of the points around the curveindicates that some cockroaches are proportioned slightly differentlyfrom others, but in general the data seem well described by m ∝ L3.That means that the largest cockroaches the experimenter couldraise (is there a 4-H prize?) had roughly the same shape as thesmallest ones.

Another relationship that should exist for animals of differentsizes shaped in the same way is that between surface area andbody mass. If all the animals have the same average density, thenbody mass should be proportional to the cube of the animal’s lin-ear size, m ∝ L3, while surface area should vary proportionately toL2. Therefore, the animals’ surface areas should be proportional to

m2/3. As shown in the right-hand panel of figure o, this relationshipappears to hold quite well for the dwarf siren, a type of salamander.Notice how the curve bends over, meaning that the surface area doesnot increase as quickly as body mass, e.g., a salamander with eighttimes more body mass will have only four times more surface area.

This behavior of the ratio of surface area to mass (or, equiv-

Section 1.3 Scaling Applied To Biology



o / Geometrical scaling of animals.

alently, the ratio of surface area to volume) has important conse-quences for mammals, which must maintain a constant body tem-perature. It would make sense for the rate of heat loss through theanimal’s skin to be proportional to its surface area, so we shouldexpect small animals, having large ratios of surface area to volume,to need to produce a great deal of heat in comparison to their size toavoid dying from low body temperature. This expectation is borneout by the data of the left-hand panel of figure p, showing the rateof oxygen consumption of guinea pigs as a function of their bodymass. Neither an animal’s heat production nor its surface area isconvenient to measure, but in order to produce heat, the animal

must metabolize oxygen, so oxygen consumption is a good indicatorof the rate of heat production. Since surface area is proportional tom2/3, the proportionality of the rate of oxygen consumption to m2/3

is consistent with the idea that the animal needs to produce heat at arate in proportion to its surface area. Although the smaller animalsmetabolize less oxygen and produce less heat in absolute terms, theamount of food and oxygen they must consume is greater in propor-tion to their own mass. The Etruscan pigmy shrew, weighing in at




q / Galileo’s original drawinshowing how larger animabones must be greater in diaeter compared to their lengt

p / Scaling of animals’ bodies related to metabolic rate and skeletal strength.

2 grams as an adult, is at about the lower size limit for mammals.It must eat continually, consuming many times its body weight eachday to survive.

Changes in shape to accommodate changes in size

Large mammals, such as elephants, have a small ratio of surfacearea to volume, and have problems getting rid of their heat fastenough. An elephant cannot simply eat small enough amounts tokeep from producing excessive heat, because cells need to have acertain minimum metabolic rate to run their internal machinery.Hence the elephant’s large ears, which add to its surface area and

help it to cool itself. Previously, we have seen several examplesof data within a given species that were consistent with a fixedshape, scaled up and down in the cases of individual specimens. Theelephant’s ears are an example of a change in shape necessitated bya change in scale.

Large animals also must be able to support their own weight.Returning to the example of the strengths of planks of differentsizes, we can see that if the strength of the plank depends on area

Section 1.3 Scaling Applied To Biology



while its weight depends on volume, then the ratio of strength toweight goes as follows:

strength/weight ∝ A/V ∝ 1/L .

Thus, the ability of objects to support their own weights decreasesinversely in proportion to their linear dimensions. If an object is to

be just barely able to support its own weight, then a larger versionwill have to be proportioned differently, with a different shape.

Since the data on the cockroaches seemed to be consistent withroughly similar shapes within the species, it appears that the abil-ity to support its own weight was not the tightest design constraintthat Nature was working under when she designed them. For largeanimals, structural strength is important. Galileo was the first toquantify this reasoning and to explain why, for instance, a large an-imal must have bones that are thicker in proportion to their length.Consider a roughly cylindrical bone such as a leg bone or a vertebra.The length of the bone, L, is dictated by the overall linear size of theanimal, since the animal’s skeleton must reach the animal’s wholelength. We expect the animal’s mass to scale as L3, so the strengthof the bone must also scale as L3. Strength is proportional to cross-sectional area, as with the wooden planks, so if the diameter of thebone is d, then

d2 ∝ L3

or

d ∝ L3/2 .

If the shape stayed the same regardless of size, then all linear di-mensions, including d and L, would be proportional to one another.If our reasoning holds, then the fact that d is proportional to L3/2,not L, implies a change in proportions of the bone. As shown in theright-hand panel of figure p, the vertebrae of African Bovidae followthe rule d ∝ L3/2 fairly well. The vertebrae of the giant eland areas chunky as a coffee mug, while those of a Gunther’s dik-dik are asslender as the cap of a pen.


A Single-celled animals must passively absorb nutrients and oxygen

from their surroundings, unlike humans who have lungs to pump air in andout and a heart to distribute the oxygenated blood throughout their bodies.Even the cells composing the bodies of multicellular animals must absorboxygen from a nearby capillary through their surfaces. Based on thesefacts, explain why cells are always microscopic in size.

B The reasoning of the previous question would seem to be contra-dicted by the fact that human nerve cells in the spinal cord can be asmuch as a meter long, although their widths are still very small. Why isthis possible?




1.4 Order-of-Magnitude Estimates

It is the mark of an instructed mind to rest satisfied with the

degree of precision that the nature of the subject permits andnot 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 thatCaptain Kirk would ask Mr. Spock, “Spock, we’re in a pretty badsituation. 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, hecould not have estimated the odds with six significant figures of accuracy, but nevertheless one of the hallmarks of a person with agood education in science is the ability to make estimates that arelikely 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 nomore than a factor of ten in either direction. Since things that differby a factor of ten are said to differ by one order of magnitude, suchan 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

odds of survival ∼ 100 to one .

The tilde can also be used in front of an individual number to em-phasize 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 iscompletely unfamiliar to most college students. Some of the typicalmental steps can be illustrated in the following example.

Cost of transporting tomatoes example 4

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 totalcost 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 esti-mating area or volume, so it turns out the estimate of 5000 tomatoes

Section 1.4 Order-of-Magnitude Estimates



r / Consider a spherical cow.

fitting in the truck is way off. That’s why people have a hard timeat those contests where you are supposed to estimate the number of jellybeans in a big jar. Another example is that most people thinktheir families use about 10 gallons of water per day, but in realitythe average is about 300 gallons per day. When estimating areaor volume, you are much better off estimating linear dimensions,

and computing volume from the linear dimensions. Here’s a bettersolution:

Better solution: As in the previous solution, say the cost of thetrip 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 knowany formula for the volume of a tomato shape, but since this is justan 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 roughestimate, let’s round that to 10−4m3. We can find the total number

of tomatoes by dividing the volume of the bin by the volume of onetomato: 10 m3/10−4 m3 = 105 tomatoes. The transportation costper tomato is $2000/105 tomatoes=$0.02/tomato. That means thattransportation really doesn’t contribute very much to the cost of atomato.

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 howmany m2 of leather could be produced from a herd of ten thousandcattle. 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 areaas 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 threeequals about ten, we can guess that a cow has a surface area of about10 m2, so the herd as a whole might yield 105 m2 of leather.

The following list summarizes the strategies for getting a goodorder-of-magnitude estimate.

1. Don’t even attempt more than one significant figure of preci-sion.

2. Don’t guess area, volume, or mass directly. Guess linear di-mensions and get area, volume, or mass from them.

3. When dealing with areas or volumes of objects with complexshapes, idealize them as if they were some simpler shape, acube or a sphere, for example.

4. Check your final answer to see if it is reasonable. If you esti-mate that a herd of ten thousand cattle would yield 0.01 m2




a / Rotation.

b / Simultaneous rotation amotion through space.

c / One person might say that ttipping chair was only rotatinga circle about its point of contawith the floor, but another coudescribe it as having both rotatiand motion through space.

Chapter 2

Velocity and RelativeMotion

2.1 Types of Motion

If you had to think consciously in order to move your body, youwould be severely disabled. Even walking, which we consider tobe no great feat, requires an intricate series of motions that yourcerebrum would be utterly incapable of coordinating. The task of putting one foot in front of the other is controlled by the more prim-

itive parts of your brain, the ones that have not changed much sincethe mammals and reptiles went their separate evolutionary ways.The thinking part of your brain limits itself to general directivessuch as “walk faster,” or “don’t step on her toes,” rather than mi-cromanaging every contraction and relaxation of the hundred or somuscles of your hips, legs, and feet.

Physics is all about the conscious understanding of motion, butwe’re obviously not immediately prepared to understand the mostcomplicated types of motion. Instead, we’ll use the divide-and-conquer technique. We’ll first classify the various types of motion,and then begin our campaign with an attack on the simplest cases.

To make it clear what we are and are not ready to consider, we needto examine and define carefully what types of motion can exist.

Rigid-body motion distinguished from motion that changes

an object’s shape

Nobody, with the possible exception of Fred Astaire, can simplyglide forward without bending their joints. Walking is thus an ex-ample in which there is both a general motion of the whole objectand a change in the shape of the object. Another example is themotion of a jiggling water balloon as it flies through the air. We arenot presently attempting a mathematical description of the way in

which the shape of an object changes. Motion without a change inshape is called rigid-body motion. (The word “body” is often usedin physics as a synonym for “object.”)

Center-of-mass motion as opposed to rotation

A ballerina leaps into the air and spins around once before land-ing. We feel intuitively that her rigid-body motion while her feetare off the ground consists of two kinds of motion going on simul-



e / No matter what point youhang the pear from, the string

lines up with the pear’s centerof mass. The center of masscan therefore be defined as theintersection of all the lines madeby hanging the pear in this way.Note that the X in the figure

should not be interpreted asimplying that the center of massis on the surface — it is actuallyinside the pear.

f / The circus performers hangwith the ropes passing throughtheir centers of mass.

taneously: a rotation and a motion of her body as a whole throughspace, along an arc. It is not immediately obvious, however, whatis the most useful way to define the distinction between rotationand motion through space. Imagine that you attempt to balance achair and it falls over. One person might say that the only motionwas a rotation about the chair’s point of contact with the floor, but

another might say that there was both rotation and motion downand to the side.

d / The leaping dancer’s motion is complicated, but the motion ofher center of mass is simple.

It turns out that there is one particularly natural and useful wayto make a clear definition, but it requires a brief digression. Everyobject has a balance point, referred to in physics as the center of

mass. For a two-dimensional ob ject such as a cardboard cutout, thecenter of mass is the point at which you could hang the object froma string and make it balance. In the case of the ballerina (who islikely to be three-dimensional unless her diet is particularly severe),it might be a point either inside or outside her body, dependingon how she holds her arms. Even if it is not practical to attach astring to the balance point itself, the center of mass can be defined

as shown in figure e.Why is the center of mass concept relevant to the question of

classifying rotational motion as opposed to motion through space?As illustrated in figures d and g, it turns out that the motion of anobject’s center of mass is nearly always far simpler than the motionof any other part of the object. The ballerina’s body is a large objectwith a complex shape. We might expect that her motion would bemuch more complicated than the motion of a small, simply-shaped

70 Chapter 2 Velocity and Relative Motion



h / An improperly balancwheel has a center of mass this not at its geometric centWhen you get a new tire, t

mechanic clamps little weightsthe rim to balance the wheel.

i / This toy was intentionadesigned so that the mushroo

shaped piece of metal on twould throw off the center mass. When you wind it up, tmushroom spins, but the cenof mass doesn’t want to movso the rest of the toy tends counter the mushroom’s motiocausing the whole thing to jum

around.

object, say a marble, thrown up at the same angle as the angle atwhich she leapt. But it turns out that the motion of the ballerina’scenter of mass is exactly the same as the motion of the marble. Thatis, the motion of the center of mass is the same as the motion theballerina would have if all her mass was concentrated at a point. Byrestricting our attention to the motion of the center of mass, we can

therefore simplify things greatly.

g / The same leaping dancer, viewed from above. Her center ofmass traces a straight line, but a point away from her center of mass,such as her elbow, traces the much more complicated path shown by thedots.

We can now replace the ambiguous idea of “motion as a wholethrough space” with the more useful and better defined conceptof “center-of-mass motion.” The motion of any rigid body can becleanly split into rotation and center-of-mass motion. By this defini-tion, the tipping chair does have both rotational and center-of-massmotion. Concentrating on the center of mass motion allows us tomake a simplified model of the motion, as if a complicated objectlike a human body was just a marble or a point-like particle. Sciencereally never deals with reality; it deals with models of reality.

Note that the word “center” in “center of mass” is not meant

to imply that the center of mass must lie at the geometrical centerof an object. A car wheel that has not been balanced properly hasa center of mass that does not coincide with its geometrical center.An object such as the human body does not even have an obviousgeometrical center.

It can be helpful to think of the center of mass as the averagelocation of all the mass in the object. With this interpretation,we can see for example that raising your arms above your headraises your center of mass, since the higher position of the arms’mass raises the average. We won’t be concerned right now withcalculating centers of mass mathematically; the relevant equations

are in chapter 4 of Conservation Laws.

Ballerinas and professional basketball players can create an illu-sion of flying horizontally through the air because our brains intu-itively expect them to have rigid-body motion, but the body doesnot stay rigid while executing a grand jete or a slam dunk. The legsare low at the beginning and end of the jump, but come up higher at

Section 2.1 Types of Motion



j / A fixed point on the dancer’s body follows a trajectory that is flat-ter than what we expect, creating an illusion of flight.

the middle. Regardless of what the limbs do, the center of mass will

follow the same arc, but the low position of the legs at the beginningand end means that the torso is higher compared to the center of mass, while in the middle of the jump it is lower compared to thecenter of mass. Our eye follows the motion of the torso and triesto interpret it as the center-of-mass motion of a rigid body. Butsince the torso follows a path that is flatter than we expect, thisattempted interpretation fails, and we experience an illusion thatthe person is flying horizontally.

k / Example 1.

The center of mass as an average example 1

Explain how we know that the center of mass of each object is

at the location shown in figure k.

The center of mass is a sort of average, so the height of the

centers of mass in 1 and 2 has to be midway between the two

squares, because that height is the average of the heights of thetwo squares. Example 3 is a combination of examples 1 and

2, so we can find its center of mass by averaging the horizontal

positions of their centers of mass. In example 4, each square

has been skewed a little, but just as much mass has been moved

up as down, so the average vertical position of the mass hasn’t

changed. Example 5 is clearly not all that different from example

4, the main difference being a slight clockwise rotation, so just as




l / The high-jumper’s bopasses over the bar, but center of mass passes under it

m / Self-check B.

in example 4, the center of mass must be hanging in empty space,

where there isn’t actually any mass. Horizontally, the center of

mass must be between the heels and toes, or else it wouldn’t be

possible to stand without tipping over.

Another interesting example from the sports world is the high jump, in which the jumper’s curved body passes over the bar, but

the center of mass passes under the bar! Here the jumper lowers hislegs and upper body at the peak of the jump in order to bring hiswaist higher compared to the center of mass.

Later in this course, we’ll find that there are more fundamentalreasons (based on Newton’s laws of motion) why the center of massbehaves in such a simple way compared to the other parts of anobject. We’re also postponing any discussion of numerical methodsfor finding an object’s center of mass. Until later in the course, wewill only deal with the motion of objects’ centers of mass.

Center-of-mass motion in one dimension

In addition to restricting our study of motion to center-of-massmotion, we will begin by considering only cases in which the centerof mass moves along a straight line. This will include cases suchas objects falling straight down, or a car that speeds up and slowsdown but does not turn.

Note that even though we are not explicitly studying the morecomplex aspects of motion, we can still analyze the center-of-massmotion while ignoring other types of motion that might be occurringsimultaneously . For instance, if a cat is falling out of a tree andis initially upside-down, it goes through a series of contortions thatbring its feet under it. This is definitely not an example of rigid-

body motion, but we can still analyze the motion of the cat’s centerof mass just as we would for a dropping rock.

self-check A

Consider a person running, a person pedaling on a bicycle, a person

coasting on a bicycle, and a person coasting on ice skates. In which

cases is the center-of-mass motion one-dimensional? Which cases are

examples of rigid-body motion? Answer, p. 272

self-check B

The figure shows a gymnast holding onto the inside of a big wheel.

From inside the wheel, how could he make it roll one way or the other?

Answer, p. 272

2.2 Describing Distance and Time

Center-of-mass motion in one dimension is particularly easy to dealwith because all the information about it can be encapsulated in twovariables: x, the position of the center of mass relative to the origin,and t, which measures a point in time. For instance, if someone

Section 2.2 Describing Distance and Time



supplied you with a sufficiently detailed table of x and t values, youwould know pretty much all there was to know about the motion of the object’s center of mass.

A point in time as opposed to duration

In ordinary speech, we use the word “time” in two different

senses, which are to be distinguished in physics. It can be used,as in “a short time” or “our time here on earth,” to mean a lengthor duration of time, or it can be used to indicate a clock reading, asin “I didn’t know what time it was,” or “now’s the time.” In sym-bols, t is ordinarily used to mean a point in time, while ∆t signifiesan interval or duration in time. The capital Greek letter delta, ∆,means “the change in...,” i.e. a duration in time is the change ordifference between one clock reading and another. The notation ∆tdoes not signify the product of two numbers, ∆ and t, but ratherone single number, ∆t. If a matinee begins at a point in time t = 1o’clock and ends at t = 3 o’clock, the duration of the movie was thechange in t,

∆t = 3 hours − 1 hour = 2 hours .

To avoid the use of negative numbers for ∆t, we write the clockreading “after” to the left of the minus sign, and the clock reading“before” to the right of the minus sign. A more specific definitionof the delta notation is therefore that delta stands for “after minusbefore.”

Even though our definition of the delta notation guarantees that∆t is positive, there is no reason why t can’t be negative. If tcould not be negative, what would have happened one second beforet = 0? That doesn’t mean that time “goes backward” in the sense

that adults can shrink into infants and retreat into the womb. It just means that we have to pick a reference point and call it t = 0,and then times before that are represented by negative values of t.An example is that a year like 2007 A.D. can be thought of as apositive t value, while one like 370 B.C. is negative. Similarly, whenyou hear a countdown for a rocket launch, the phrase “t minus tenseconds” is a way of saying t = −10 s, where t = 0 is the time of blastoff, and t > 0 refers to times after launch.

Although a point in time can be thought of as a clock reading, itis usually a good idea to avoid doing computations with expressionssuch as “2:35” that are combinations of hours and minutes. Times

can instead be expressed entirely in terms of a single unit, such ashours. Fractions of an hour can be represented by decimals ratherthan minutes, and similarly if a problem is being worked in termsof minutes, decimals can be used instead of seconds.

self-check C Of the following phrases, which refer to points in time, which refer totime intervals, and which refer to time in the abstract rather than as a

measurable number?




(1) “The time has come.”

(2) “Time waits for no man.”

(3) “The whole time, he had spit on his chin.” Answer, p. 272

Position as opposed to change in position

As with time, a distinction should be made between a pointin space, symbolized as a coordinate x, and a change in position,symbolized as ∆x.

As with t, x can be negative. If a train is moving down thetracks, not only do you have the freedom to choose any point alongthe tracks and call it x = 0, but it’s also up to you to decide whichside of the x = 0 point is positive x and which side is negative x.

Since we’ve defined the delta notation to mean “after minusbefore,” it is possible that ∆x will be negative, unlike ∆t which isguaranteed to be positive. Suppose we are describing the motionof a train on tracks linking Tucson and Chicago. As shown in the

figure, it is entirely up to you to decide which way is positive.

n / Two equally valid ways of dscribing the motion of a train froTucson to Chicago. In examplethe train has a positive ∆x agoes from Enid to Joplin. Inthe same train going forwardthe same direction has a negat

∆x .

Note that in addition to x and ∆x, there is a third quantity wecould define, which would be like an odometer reading, or actualdistance traveled. If you drive 10 miles, make a U-turn, and driveback 10 miles, then your ∆x is zero, but your car’s odometer readinghas increased by 20 miles. However important the odometer readingis to car owners and used car dealers, it is not very important inphysics, and there is not even a standard name or notation for it.

The change in position, ∆x, is more useful because it is so mucheasier to calculate: to compute ∆x, we only need to know the be-ginning and ending positions of the object, not all the informationabout how it got from one position to the other.

self-check D

A ball falls vertically, hits the floor, bounces to a height of one meter,

falls, and hits the floor again. Is the ∆x between the two impacts equal

to zero, one, or two meters? Answer, p. 273

Section 2.2 Describing Distance and Time



o / Motion with constant ve-locity.

p / Motion that decreases x is represented with negativevalues of ∆x and v .

q / Motion with changing ve-locity.

Frames of reference

The example above shows that there are two arbitrary choicesyou have to make in order to define a position variable, x. You haveto decide where to put x = 0, and also which direction will be posi-tive. This is referred to as choosing a coordinate system or choosinga frame of reference. (The two terms are nearly synonymous, but

the first focuses more on the actual x variable, while the second ismore of a general way of referring to one’s point of view.) As long asyou are consistent, any frame is equally valid. You just don’t wantto change coordinate systems in the middle of a calculation.

Have you ever been sitting in a train in a station when suddenlyyou notice that the station is moving backward? Most people woulddescribe the situation by saying that you just failed to notice thatthe train was moving — it only seemed like the station was moving.But this shows that there is yet a third arbitrary choice that goesinto choosing a coordinate system: valid frames of reference candiffer from each other by moving relative to one another. It might

seem strange that anyone would bother with a coordinate systemthat was moving relative to the earth, but for instance the frame of reference moving along with a train might be far more convenientfor describing things happening inside the train.

2.3 Graphs of Motion; Velocity

Motion with constant velocity

In example o, an object is moving at constant speed in one di-rection. We can tell this because every two seconds, its position

changes by five meters.In algebra notation, we’d say that the graph of x vs. t shows

the same change in position, ∆x = 5.0 m, over each interval of ∆t = 2.0 s. The object’s velocity or speed is obtained by calculatingv = ∆x/∆t = (5.0 m)/(2.0 s) = 2.5 m/s. In graphical terms, thevelocity can be interpreted as the slope of the line. Since the graphis a straight line, it wouldn’t have mattered if we’d taken a longertime interval and calculated v = ∆x/∆t = (10.0 m)/(4.0 s). Theanswer would still have been the same, 2.5 m/s.

Note that when we divide a number that has units of meters byanother number that has units of seconds, we get units of meters

per second, which can be written m/s. This is another case wherewe treat units as if they were algebra symbols, even though they’renot.

In example p, the object is moving in the opposite direction: astime progresses, its x coordinate decreases. Recalling the definitionof the ∆ notation as “after minus before,” we find that ∆t is stillpositive, but ∆x must be negative. The slope of the line is therefore




r / The velocity at any givmoment is defined as the slo

of the tangent line through trelevant point on the graph.

s / Example: finding the v

locity at the point indicated wthe dot.

t / Reversing the direction motion.

negative, and we say that the object has a negative velocity, v =∆x/∆t = (−5.0 m)/(2.0 s) = −2.5 m/s. We’ve already seen thatthe plus and minus signs of ∆x values have the interpretation of telling us which direction the object moved. Since ∆t is alwayspositive, dividing by ∆t doesn’t change the plus or minus sign, andthe plus and minus signs of velocities are to be interpreted in the

same way. In graphical terms, a positive slope characterizes a linethat goes up as we go to the right, and a negative slope tells us thatthe line went down as we went to the right.

Solved problem: light-years page 89, problem 4

Motion with changing velocity

Now what about a graph like figure q? This might be a graphof a car’s motion as the driver cruises down the freeway, then slowsdown to look at a car crash by the side of the road, and then speedsup again, disappointed that there is nothing dramatic going on suchas flames or babies trapped in their car seats. (Note that we are

still talking about one-dimensional motion. Just because the graphis curvy doesn’t mean that the car’s path is curvy. The graph is notlike a map, and the horizontal direction of the graph represents thepassing of time, not distance.)

Example q is similar to example o in that the object moves atotal of 25.0 m in a period of 10.0 s, but it is no longer true that itmakes the same amount of progress every second. There is no way tocharacterize the entire graph by a certain velocity or slope, becausethe velocity is different at every moment. It would be incorrect tosay that because the car covered 25.0 m in 10.0 s, its velocity was2.5 m/s. It moved faster than that at the beginning and end, but

slower in the middle. There may have been certain instants at whichthe car was indeed going 2.5 m/s, but the speedometer swept pastthat value without “sticking,” just as it swung through various othervalues of speed. (I definitely want my next car to have a speedometercalibrated in m/s and showing both negative and positive values.)

We assume that our speedometer tells us what is happening tothe speed of our car at every instant, but how can we define speedmathematically in a case like this? We can’t just define it as theslope of the curvy graph, because a curve doesn’t have a singlewell-defined slope as does a line. A mathematical definition thatcorresponded to the speedometer reading would have to be one that

attached a different velocity value to a single point on the curve,i.e., a single instant in time, rather than to the entire graph. If wewish to define the speed at one instant such as the one marked witha dot, the best way to proceed is illustrated in r, where we havedrawn the line through that point called the tangent line, the linethat “hugs the curve.” We can then adopt the following definitionof velocity:

Section 2.3 Graphs of Motion; Velocity



definition of velocity

The velocity of an object at any given moment is the slope of thetangent line through the relevant point on its x − t graph.

One interpretation of this definition is that the velocity tells us

how many meters the object would have traveled in one second, if it had continued moving at the same speed for at least one second.To some people the graphical nature of this definition seems “in-accurate” or “not mathematical.” The equation by itself, however,is only valid if the velocity is constant, and so cannot serve as ageneral definition.

The slope of the tangent line example 2

What is the velocity at the point shown with a dot on the graph?

First we draw the tangent line through that point. To find the

slope of the tangent line, we need to pick two points on it. Theo-

retically, the slope should come out the same regardless of whichtwo points we pick, but in practical terms we’ll be able to measure

more accurately if we pick two points fairly far apart, such as the

two white diamonds. To save work, we pick points that are directly

above labeled points on the t axis, so that ∆t = 4.0 s is easy to

read off. One diamond lines up with x ≈ 17.5 m, the other with

x ≈ 26.5 m, so ∆x = 9.0 m. The velocity is ∆x /∆t = 2.2 m/s.

Conventions about graphing

The placement of t on the horizontal axis and x on the uprightaxis may seem like an arbitrary convention, or may even have dis-turbed you, since your algebra teacher always told you that x goeson the horizontal axis and y goes on the upright axis. There is areason for doing it this way, however. In example s, we have anobject that reverses its direction of motion twice. It can only bein one place at any given time, but there can be more than onetime when it is at a given place. For instance, this object passedthrough x = 17 m on three separate occasions, but there is no wayit could have been in more than one place at t = 5.0 s. Resurrectingsome terminology you learned in your trigonometry course, we saythat x is a function of t, but t is not a function of x. In situationssuch as this, there is a useful convention that the graph should beoriented so that any vertical line passes through the curve at only

one point. Putting the x axis across the page and t upright wouldhave violated this convention. To people who are used to interpret-ing graphs, a graph that violates this convention is as annoying asfingernails scratching on a chalkboard. We say that this is a graphof “x versus t.” If the axes were the other way around, it wouldbe a graph of “t versus x.” I remember the “versus” terminologyby visualizing the labels on the x and t axes and remembering thatwhen you read, you go from left to right and from top to bottom.




Discussion question G.


A Park is running slowly in gym class, but then he notices Jennawatching him, so he speeds up to try to impress her. Which of the graphscould represent his motion?

B The figure shows a sequence of positions for two racing tractors.Compare the tractors’ velocities as the race progresses. When do theyhave the same velocity? [Based on a question by Lillian McDermott.]

C If an object had a straight-line motion graph with ∆x =0 and ∆t = 0,what would be true about its velocity? What would this look like on agraph? What about ∆t =0 and ∆x = 0?

D If an object has a wavy motion graph like the one in figure t onthe previous page, which are the points at which the object reverses its

direction? What is true about the object’s velocity at these points?

E Discuss anything unusual about the following three graphs.

F I have been using the term “velocity” and avoiding the more commonEnglish word “speed,” because introductory physics texts typically define

them to mean different things. They use the word “speed,” and the symbol“s ” to mean the absolute value of the velocity, s = |v |. Although I’vechosen not to emphasize this distinction in technical vocabulary, thereare clearly two different concepts here. Can you think of an example ofa graph of x -versus-t in which the object has constant speed, but notconstant velocity?

G For the graph shown in the figure, describe how the object’s velocitychanges.

H Two physicists duck out of a boring scientific conference. On the

Section 2.3 Graphs of Motion; Velocity



street, they witness an accident in which a pedestrian is injured by a hit-and-run driver. A criminal trial results, and they must testify. In her testi-mony, Dr. Transverz Waive says, “The car was moving along pretty fast,I’d say the velocity was +40 mi/hr. They saw the old lady too late, and eventhough they slammed on the brakes they still hit her before they stopped.Then they made a U turn and headed off at a velocity of about -20 mi/hr,I’d say.” Dr. Longitud N.L. Vibrasheun says, “He was really going too fast,

maybe his velocity was -35 or -40 mi/hr. After he hit Mrs. Hapless, heturned around and left at a velocity of, oh, I’d guess maybe +20 or +25mi/hr.” Is their testimony contradictory? Explain.

2.4 The Principle of Inertia

Physical effects relate only to a change in velocity

Consider two statements of a kind that was at one time madewith the utmost seriousness:

People like Galileo and Copernicus who say the earth is ro-

tating must be crazy. We know the earth can’t be moving.

Why, if the earth was really turning once every day, then ourwhole city would have to be moving hundreds of leagues in

an hour. That’s impossible! Buildings would shake on their

foundations. Gale-force winds would knock us over. Trees

would fall down. The Mediterranean would come sweeping

across the east coasts of Spain and Italy. And furthermore,

what force would be making the world turn?

All this talk of passenger trains moving at forty miles an hour

is sheer hogwash! At that speed, the air in a passenger com-

partment would all be forced against the back wall. People in

the front of the car would suffocate, and people at the back

would die because in such concentrated air, they wouldn’t be

able to expel a breath.

Some of the effects predicted in the first quote are clearly justbased on a lack of experience with rapid motion that is smooth andfree of vibration. But there is a deeper principle involved. In eachcase, the speaker is assuming that the mere fact of motion musthave dramatic physical effects. More subtly, they also believe that aforce is needed to keep an object in motion: the first person thinksa force would be needed to maintain the earth’s rotation, and thesecond apparently thinks of the rear wall as pushing on the air to

keep it moving.Common modern knowledge and experience tell us that these

people’s predictions must have somehow been based on incorrectreasoning, but it is not immediately obvious where the fundamentalflaw lies. It’s one of those things a four-year-old could infuriateyou by demanding a clear explanation of. One way of getting atthe fundamental principle involved is to consider how the modernconcept of the universe differs from the popular conception at the




v / Why does Aristotle lo

so sad? Has he realized thhis entire system of physics wrong?

w / The earth spins. Peopin Shanghai say they’re at reand people in Los Angeles amoving. Angelenos say the samabout the Shanghainese.

x / The jets are at rest. TEmpire State Building is moving

time of the Italian Renaissance. To us, the word “earth” impliesa planet, one of the nine planets of our solar system, a small ballof rock and dirt that is of no significance to anyone in the universeexcept for members of our species, who happen to live on it. ToGalileo’s contemporaries, however, the earth was the biggest, mostsolid, most important thing in all of creation, not to be compared

with the wandering lights in the sky known as planets. To us, theearth is just another object, and when we talk loosely about “howfast” an object such as a car “is going,” we really mean the car-object’s velocity relative to the earth-object.

u / This Air Force doctor volunteered to ride a rocket sled as amedical experiment. The obvious effects on his head and face are notbecause of the sled’s speed but because of its rapid changes in speed:

increasing in 2 and 3, and decreasing in 5 and 6. In 4 his speed isgreatest, but because his speed is not increasing or decreasing verymuch at this moment, there is little effect on him.

Motion is relative

According to our modern world-view, it really isn’t that reason-able to expect that a special force should be required to make theair in the train have a certain velocity relative to our planet. Afterall, the “moving” air in the “moving” train might just happen tohave zero velocity relative to some other planet we don’t even knowabout. Aristotle claimed that things “naturally” wanted to be atrest, lying on the surface of the earth. But experiment after exper-iment has shown that there is really nothing so special about being

Section 2.4 The Principle of Inertia



Discussion question A.

Discussion question B.

Discussion question D.

at rest relative to the earth. For instance, if a mattress falls out of the back of a truck on the freeway, the reason it rapidly comes torest with respect to the planet is simply because of friction forcesexerted by the asphalt, which happens to be attached to the planet.

Galileo’s insights are summarized as follows:

The principle of inertiaNo force is required to maintain motion with constant velocity ina straight line, and absolute motion does not cause any observablephysical effects.

There are many examples of situations that seem to disprove theprinciple of inertia, but these all result from forgetting that frictionis a force. For instance, it seems that a force is needed to keep asailboat in motion. If the wind stops, the sailboat stops too. Butthe wind’s force is not the only force on the boat; there is also africtional force from the water. If the sailboat is cruising and the

wind suddenly disappears, the backward frictional force still exists,and since it is no longer being counteracted by the wind’s forwardforce, the boat stops. To disprove the principle of inertia, we wouldhave to find an example where a moving object slowed down eventhough no forces whatsoever were acting on it.

self-check E What is incorrect about the following supposed counterexamples to theprinciple of inertia?

(1) When astronauts blast off in a rocket, their huge velocity does causea physical effect on their bodies — they get pressed back into theirseats, the flesh on their faces gets distorted, and they have a hard time

lifting their arms.(2) When you’re driving in a convertible with the top down, the wind in

your face is an observable physical effect of your absolute motion.

Answer, p. 273

Solved problem: a bug on a wheel page 89, problem 7


A A passenger on a cruise ship finds, while the ship is docked, thathe can leap off of the upper deck and just barely make it into the poolon the lower deck. If the ship leaves dock and is cruising rapidly, will thisadrenaline junkie still be able to make it?

B You are a passenger in the open basket hanging under a heliumballoon. The balloon is being carried along by the wind at a constantvelocity. If you are holding a flag in your hand, will the flag wave? If so,which way? [Based on a question from PSSC Physics.]

C Aristotle stated that all objects naturally wanted to come to rest, withthe unspoken implication that “rest” would be interpreted relative to thesurface of the earth. Suppose we go back in time and transport Aristotleto the moon. Aristotle knew, as we do, that the moon circles the earth; he

said it didn’t fall down because, like everything else in the heavens, it was




made out of some special substance whose “natural” behavior was to goin circles around the earth. We land, put him in a space suit, and kickhim out the door. What would he expect his fate to be in this situation? Ifintelligent creatures inhabited the moon, and one of them independentlycame up with the equivalent of Aristotelian physics, what would they thinkabout objects coming to rest?

D The glass is sitting on a level table in a train’s dining car, but thesurface of the water is tilted. What can you infer about the motion of thetrain?

2.5 Addition of Velocities

Addition of velocities to describe relative motion

Since absolute motion cannot be unambiguously measured, theonly way to describe motion unambiguously is to describe the motionof one object relative to another. Symbolically, we can write vPQfor the velocity of object P relative to object Q.

Velocities measured with respect to different reference points canbe compared by addition. In the figure below, the ball’s velocityrelative to the couch equals the ball’s velocity relative to the truckplus the truck’s velocity relative to the couch:

vBC = vBT + vTC

= 5 cm/s + 10 cm/s

= 15 cm/s

The same equation can be used for any combination of threeobjects, just by substituting the relevant subscripts for B, T, and

C. Just remember to write the equation so that the velocities beingadded have the same subscript twice in a row. In this example, if you read off the subscripts going from left to right, you get BC . . . =. . . BTTC. The fact that the two “inside” subscripts on the right arethe same means that the equation has been set up correctly. Noticehow subscripts on the left look just like the subscripts on the right,but with the two T’s eliminated.

Negative velocities in relative motion

My discussion of how to interpret positive and negative signs of

velocity may have left you wondering why we should bother. Whynot just make velocity positive by definition? The original reasonwhy negative numbers were invented was that bookkeepers decidedit would be convenient to use the negative number concept for pay-ments to distinguish them from receipts. It was just plain easier thanwriting receipts in black and payments in red ink. After adding upyour month’s positive receipts and negative payments, you either gota positive number, indicating profit, or a negative number, showing

Section 2.5 Addition of Velocities



y / These two highly competent physicists disagree on absolute ve-locities, but they would agree on relative velocities. Purple Dinoconsiders the couch to be at rest, while Green Dino thinks of the truck asbeing at rest. They agree, however, that the truck’s velocity relative to thecouch is v T C = 10 cm/s, the ball’s velocity relative to the truck is v BT = 5

cm/s, and the ball’s velocity relative to the couch is v BC = v BT + v T C = 15cm/s.

a loss. You could then show that total with a high-tech “+” or “−”sign, instead of looking around for the appropriate bottle of ink.

Nowadays we use positive and negative numbers for all kinds

of things, but in every case the point is that it makes sense toadd and subtract those things according to the rules you learnedin grade school, such as “minus a minus makes a plus, why this istrue we need not discuss.” Adding velocities has the significanceof comparing relative motion, and with this interpretation negativeand positive velocities can be used within a consistent framework.For example, the truck’s velocity relative to the couch equals thetruck’s velocity relative to the ball plus the ball’s velocity relativeto the couch:

vTC = vTB + vBC

= −5 cm/s + 15 cm/s

= 10 cm/s

If we didn’t have the technology of negative numbers, we would havehad to remember a complicated set of rules for adding velocities: (1)if the two objects are both moving forward, you add, (2) if one ismoving forward and one is moving backward, you subtract, but (3)if they’re both moving backward, you add. What a pain that wouldhave been.




z / Graphs of x and v verst for a car accelerating away froa traffic light, and then stoppifor another red light.

Solved problem: two dimensions page 90, problem 10


A Interpret the general rule v AB = −v BA in words.

B Wa-Chuen slips away from her father at the mall and walks up thedown escalator, so that she stays in one place. Write this in terms ofsymbols.

2.6 Graphs of Velocity Versus Time

Since changes in velocity play such a prominent role in physics, weneed a better way to look at changes in velocity than by laboriouslydrawing tangent lines on x-versus-t graphs. A good method is todraw a graph of velocity versus time. The examples on the left showthe x− t and v − t graphs that might be produced by a car startingfrom a traffic light, speeding up, cruising for a while at constantspeed, and finally slowing down for a stop sign. If you have an airfreshener hanging from your rear-view mirror, then you will see an

effect on the air freshener during the beginning and ending periodswhen the velocity is changing, but it will not be tilted during theperiod of constant velocity represented by the flat plateau in themiddle of the v − t graph.

Students often mix up the things being represented on these twotypes of graphs. For instance, many students looking at the topgraph say that the car is speeding up the whole time, since “thegraph is becoming greater.” What is getting greater throughout thegraph is x, not v.

Similarly, many students would look at the bottom graph andthink it showed the car backing up, because “it’s going backwardsat the end.” But what is decreasing at the end is v, not x. Havingboth the x − t and v − t graphs in front of you like this is oftenconvenient, because one graph may be easier to interpret than theother for a particular purpose. Stacking them like this means thatcorresponding points on the two graphs’ time axes are lined up witheach other vertically. However, one thing that is a little counter-intuitive about the arrangement is that in a situation like this oneinvolving a car, one is tempted to visualize the landscape stretchingalong the horizontal axis of one of the graphs. The horizontal axes,however, represent time, not position. The correct way to visualizethe landscape is by mentally rotating the horizon 90 degrees coun-

terclockwise and imagining it stretching along the upright axis of thex-t graph, which is the only axis that represents different positionsin space.

Section 2.6 Graphs of Velocity Versus Time



2.7

Applications of Calculus

The integral symbol,

, in the heading for this section indicates thatit is meant to be read by students in calculus-based physics. Stu-dents in an algebra-based physics course should skip these sections.The calculus-related sections in this book are meant to be usableby students who are taking calculus concurrently, so at this earlypoint in the physics course I do not assume you know any calculusyet. This section is therefore not much more than a quick preview of calculus, to help you relate what you’re learning in the two courses.

Newton was the first person to figure out the tangent-line defi-nition of velocity for cases where the x − t graph is nonlinear. Be-fore Newton, nobody had conceptualized the description of motionin terms of x − t and v − t graphs. In addition to the graphicaltechniques discussed in this chapter, Newton also invented a set of symbolic techniques called calculus. If you have an equation for xin terms of t, calculus allows you, for instance, to find an equation

for v in terms of t. In calculus terms, we say that the function v(t)is the derivative of the function x(t). In other words, the derivativeof a function is a new function that tells how rapidly the originalfunction was changing. We now use neither Newton’s name for histechnique (he called it “the method of fluxions”) nor his notation.The more commonly used notation is due to Newton’s German con-temporary Leibnitz, whom the English accused of plagiarizing thecalculus from Newton. In the Leibnitz notation, we write

v =dx

dtto indicate that the function v(t) equals the slope of the tangent lineof the graph of x(t) at every time t. The Leibnitz notation is meantto evoke the delta notation, but with a very small time interval.Because the dx and dt are thought of as very small ∆x’s and ∆t’s,i.e., very small differences, the part of calculus that has to do withderivatives is called differential calculus.

Differential calculus consists of three things:

• The concept and definition of the derivative, which is coveredin this book, but which will be discussed more formally in yourmath course.

• The Leibnitz notation described above, which you’ll need to

get more comfortable with in your math course.

• A set of rules that allows you to find an equation for the deriva-tive of a given function. For instance, if you happened to havea situation where the position of an object was given by theequation x = 2t7, you would be able to use those rules tofind dx/dt = 14t6. This bag of tricks is covered in your mathcourse.




location g (m/s2)asteroid Vesta (surface) 0.3Earth’s moon (surface) 1.6Mars (surface) 3.7Earth (surface) 9.8Jupiter (cloud-tops) 26

Sun (visible surface) 270typical neutron star (surface) 1012

black hole (center) infinite according to some theo-ries, on the order of 1052 accord-ing to others

A typical neutron star is not so different in size from a large asteroid,but is orders of magnitude more massive, so the mass of a bodydefinitely correlates with the g it creates. On the other hand, aneutron star has about the same mass as our Sun, so why is its gbillions of times greater? If you had the misfortune of being on thesurface of a neutron star, you’d be within a few thousand miles of all

its mass, whereas on the surface of the Sun, you’d still be millionsof miles from most of its mass.


A What is wrong with the following definitions of g ?

(1) “g is gravity.”

(2) “g is the speed of a falling object.”

(3) “g is how hard gravity pulls on things.”

B When advertisers specify how much acceleration a car is capableof, they do not give an acceleration as defined in physics. Instead, theyusually specify how many seconds are required for the car to go from rest

to 60 miles/hour. Suppose we use the notation “a ” for the acceleration asdefined in physics, and “a car ad” for the quantity used in advertisements forcars. In the US’s non-metric system of units, what would be the units ofa and a car ad? How would the use and interpretation of large and small,positive and negative values be different for a as opposed to acar ad?

C Two people stand on the edge of a cliff. As they lean over the edge,one person throws a rock down, while the other throws one straight upwith an exactly opposite initial velocity. Compare the speeds of the rockson impact at the bottom of the cliff.

3.3 Positive and Negative Acceleration

Gravity always pulls down, but that does not mean it always speedsthings up. If you throw a ball straight up, gravity will first slowit down to v = 0 and then begin increasing its speed. When Itook physics in high school, I got the impression that positive signsof acceleration indicated speeding up, while negative accelerationsrepresented slowing down, i.e., deceleration. Such a definition wouldbe inconvenient, however, because we would then have to say thatthe same downward tug of gravity could produce either a positive

98 Chapter 3 Acceleration and Free Fall



i / The ball’s acceleration stathe same — on the way up, at ttop, and on the way back dowIt’s always negative.

or a negative acceleration. As we will see in the following example,such a definition also would not be the same as the slope of the v− tgraph

Let’s study the example of the rising and falling ball. In the ex-ample of the person falling from a bridge, I assumed positive velocityvalues without calling attention to it, which meant I was assuming

a coordinate system whose x axis pointed down. In this example,where the ball is reversing direction, it is not possible to avoid neg-ative velocities by a tricky choice of axis, so let’s make the morenatural choice of an axis pointing up. The ball’s velocity will ini-tially be a positive number, because it is heading up, in the samedirection as the x axis, but on the way back down, it will be a neg-ative number. As shown in the figure, the v − t graph does not doanything special at the top of the ball’s flight, where v equals 0. Itsslope is always negative. In the left half of the graph, there is anegative slope because the positive velocity is getting closer to zero.On the right side, the negative slope is due to a negative velocity

that is getting farther from zero, so we say that the ball is speedingup, but its velocity is decreasing!

To summarize, what makes the most sense is to stick with theoriginal definition of acceleration as the slope of the v − t graph,∆v/∆t. By this definition, it just isn’t necessarily true that thingsspeeding up have positive acceleration while things slowing downhave negative acceleration. The word “deceleration” is not usedmuch by physicists, and the word “acceleration” is used unblush-ingly to refer to slowing down as well as speeding up: “There was ared light, and we accelerated to a stop.”

Numerical calculation of a negative acceleration example 4

In figure i, what happens if you calculate the acceleration be-tween t = 1.0 and 1.5 s?

Reading from the graph, it looks like the velocity is about −1 m/s

at t = 1.0 s, and around −6 m/s at t = 1.5 s. The acceleration,

figured between these two points, is

a =∆v

∆t =

(−6 m/s)− (−1 m/s)

(1.5 s) − (1.0 s)= −10 m/s2 .

Even though the ball is speeding up, it has a negative accelera-

tion.

Another way of convincing you that this way of handling the plusand minus signs makes sense is to think of a device that measuresacceleration. After all, physics is supposed to use operational defini-tions, ones that relate to the results you get with actual measuringdevices. Consider an air freshener hanging from the rear-view mirrorof your car. When you speed up, the air freshener swings backward.Suppose we define this as a positive reading. When you slow down,the air freshener swings forward, so we’ll call this a negative reading

Section 3.3 Positive and Negative Acceleration



on our accelerometer. But what if you put the car in reverse andstart speeding up backwards? Even though you’re speeding up, theaccelerometer responds in the same way as it did when you weregoing forward and slowing down. There are four possible cases:

motion of car accelerometerswings

slope of v-t graph

directionof force

acting oncar

forward, speeding up backward + forwardforward, slowing down forward − backwardbackward, speeding up forward − backwardbackward, slowing down b ackward + forward

Note the consistency of the three right-hand columns — nature istrying to tell us that this is the right system of classification, notthe left-hand column.

Because the positive and negative signs of acceleration depend

on the choice of a coordinate system, the acceleration of an objectunder the influence of gravity can be either positive or negative.Rather than having to write things like “g = 9.8 m/s2 or −9.8 m/s2”every time we want to discuss g’s numerical value, we simply defineg as the absolute value of the acceleration of objects moving underthe influence of gravity. We consistently let g = 9.8 m/s2, but wemay have either a = g or a = −g, depending on our choice of acoordinate system.

Acceleration with a change in direction of motion example 5

A person kicks a ball, which rolls up a sloping street, comes to

a halt, and rolls back down again. The ball has constant accel-

eration. The ball is initially moving at a velocity of 4.0 m/s, andafter 10.0 s it has returned to where it started. At the end, it has

sped back up to the same speed it had initially, but in the opposite

direction. What was its acceleration?

By giving a positive number for the initial velocity, the statement

of the question implies a coordinate axis that points up the slope

of the hill. The “same” speed in the opposite direction should

therefore be represented by a negative number, -4.0 m/s. The

acceleration is

a = ∆v /∆t

= (v f −

v o)/10.0 s= [(−4.0 m/s)− (4.0 m/s)]/10.0s

= −0.80 m/s2 .

The acceleration was no different during the upward part of the

roll than on the downward part of the roll.

Incorrect solution: Acceleration is ∆v /∆t, and at the end it’s not

moving any faster or slower than when it started, so ∆v=0 and




Discussion question C.

a = 0.

The velocity does change, from a positive number to a negative

number.

Discussion question B.


A A child repeatedly jumps up and down on a trampoline. Discuss thesign and magnitude of his acceleration, including both the time when he is

in the air and the time when his feet are in contact with the trampoline.

B The figure shows a refugee from a Picasso painting blowing on a

rolling water bottle. In some cases the person’s blowing is speeding the

bottle up, but in others it is slowing it down. The arrow inside the bottleshows which direction it is going, and a coordinate system is shown at thebottom of each figure. In each case, figure out the plus or minus signs ofthe velocity and acceleration. It may be helpful to draw a v − t graph ineach case.

C Sally is on an amusement park ride which begins with her chair being

hoisted straight up a tower at a constant speed of 60 miles/hour. Despitestern warnings from her father that he’ll take her home the next time shemisbehaves, she decides that as a scientific experiment she really needsto release her corndog over the side as she’s on the way up. She doesnot throw it. She simply sticks it out of the car, lets it go, and watches itagainst the background of the sky, with no trees or buildings as referencepoints. What does the corndog’s motion look like as observed by Sally?

Does its speed ever appear to her to be zero? What acceleration doesshe observe it to have: is it ever positive? negative? zero? What wouldher enraged father answer if asked for a similar description of its motion

as it appears to him, standing on the ground?

D Can an object maintain a constant acceleration, but meanwhile

reverse the direction of its velocity?

E Can an object have a velocity that is positive and increasing at the

same time that its acceleration is decreasing?

Section 3.3 Positive and Negative Acceleration 1



k / Example 6.

3.4 Varying Acceleration

So far we have only been discussing examples of motion for whichthe v − t graph is linear. If we wish to generalize our definition tov-t graphs that are more complex curves, the best way to proceedis similar to how we defined velocity for curved x− t graphs:

definition of accelerationThe acceleration of an object at any instant is the slope of the tangent line passing through its v-versus-t graph at therelevant point.

A skydiver example 6

The graphs in figure k show the results of a fairly realistic com-

puter simulation of the motion of a skydiver, including the effects

of air friction. The x axis has been chosen pointing down, so x

is increasing as she falls. Find (a) the skydiver’s acceleration at

t = 3.0 s, and also (b) at t = 7.0 s.

The solution is shown in figure l. I’ve added tangent lines at the

two points in question.

(a) To find the slope of the tangent line, I pick two points on the

line (not necessarily on the actual curve): (3.0 s, 28m/s) and

(5.0 s, 42 m/s). The slope of the tangent line is (42 m/s−28 m/s)/(5.0 s−

3.0 s) = 7.0 m/s2.

(b) Two points on this tangent line are (7.0 s,47 m/s) and (9.0 s, 52 m/s).

The slope of the tangent line is (52 m/s−47 m/s)/(9.0 s−7.0 s) =

2.5 m/s2.

Physically, what’s happening is that at t = 3.0 s, the skydiver is

not yet going very fast, so air friction is not yet very strong. She

therefore has an acceleration almost as great as g . At t = 7.0 s,she is moving almost twice as fast (about 100 miles per hour), and

air friction is extremely strong, resulting in a significant departure

from the idealized case of no air friction.

In example 6, the x−t graph was not even used in the solution of the problem, since the definition of acceleration refers to the slopeof the v − t graph. It is possible, however, to interpret an x − tgraph to find out something about the acceleration. An object withzero acceleration, i.e., constant velocity, has an x− t graph that is a

straight line. A straight line has no curvature. A change in velocityrequires a change in the slope of the x− t graph, which means thatit is a curve rather than a line. Thus acceleration relates to thecurvature of the x− t graph. Figure m shows some examples.




l / The solution to example 6.

In example 6, the x − t graph was more strongly curved at thebeginning, and became nearly straight at the end. If the x− t graphis nearly straight, then its slope, the velocity, is nearly constant, andthe acceleration is therefore small. We can thus interpret the accel-eration as representing the curvature of the x − t graph, as shownin figure m. If the “cup” of the curve points up, the acceleration is

positive, and if it points down, the acceleration is negative.

m / Acceleration relates to the curvature of the x − t graph.

Section 3.4 Varying Acceleration 1



o / How position, velocity, and

acceleration are related.

Since the relationship between a and v is analogous to the rela-tionship between v and x, we can also make graphs of accelerationas a function of time, as shown in figure n.

n / Examples of graphs of x , v , and a versus t . 1. A object in freefall, with no friction. 2. A continuation of example 6, the skydiver.

Solved problem: Drawing a v − t graph. page 117, problem 14

Solved problem: Drawing v − t and a− t graphs. page 118, problem 20

Figure o summarizes the relationships among the three types of graphs.


A Describe in words how the changes in the a − t graph in figure n/2

relate to the behavior of the v − t graph.




B Explain how each set of graphs contains inconsistencies, and fixthem.

C In each case, pick a coordinate system and draw x − t , v − t , anda − t graphs. Picking a coordinate system means picking where you wantx = 0 to be, and also picking a direction for the positive x axis.

(1) An ocean liner is cruising in a straight line at constant speed.

(2) You drop a ball. Draw two different sets of graphs (a total of 6), withone set’s positive x axis pointing in the opposite direction compared to theother’s.

(3) You’re driving down the street looking for a house you’ve never beento before. You realize you’ve passed the address, so you slow down, put

the car in reverse, back up, and stop in front of the house.

3.5 The Area Under the Velocity-Time Graph

A natural question to ask about falling objects is how fast they fall,but Galileo showed that the question has no answer. The physicallaw that he discovered connects a cause (the attraction of the planetEarth’s mass) to an effect, but the effect is predicted in terms of anacceleration rather than a velocity. In fact, no physical law predictsa definite velocity as a result of a specific phenomenon, becausevelocity cannot be measured in absolute terms, and only changes in

velocity relate directly to physical phenomena.The unfortunate thing about this situation is that the definitions

of velocity and acceleration are stated in terms of the tangent-linetechnique, which lets you go from x to v to a, but not the otherway around. Without a technique to go backwards from a to v to x,we cannot say anything quantitative, for instance, about the x − tgraph of a falling object. Such a technique does exist, and I used itto make the x− t graphs in all the examples above.

Section 3.5 The Area Under the Velocity-Time Graph 1



p / The area under the v − t graph gives ∆x .

First let’s concentrate on how to get x information out of a v− tgraph. In example p/1, an object moves at a speed of 20 m/s fora period of 4.0 s. The distance covered is ∆x = v∆t = (20 m/s) ×(4.0 s) = 80 m. Notice that the quantities being multiplied are thewidth and the height of the shaded rectangle — or, strictly speaking,the time represented by its width and the velocity represented by

its height. The distance of ∆x = 80 m thus corresponds to the areaof the shaded part of the graph.

The next step in sophistication is an example like p/2, where theobject moves at a constant speed of 10 m/s for two seconds, thenfor two seconds at a different constant speed of 20 m/s. The shadedregion can be split into a small rectangle on the left, with an arearepresenting ∆x = 20 m, and a taller one on the right, correspondingto another 40 m of motion. The total distance is thus 60 m, whichcorresponds to the total area under the graph.

An example like p/3 is now just a trivial generalization; thereis simply a large number of skinny rectangular areas to add up.

But notice that graph p/3 is quite a good approximation to thesmooth curve p/4. Even though we have no formula for the area of a funny shape like p/4, we can approximate its area by dividing it upinto smaller areas like rectangles, whose area is easier to calculate.If someone hands you a graph like p/4 and asks you to find thearea under it, the simplest approach is just to count up the littlerectangles on the underlying graph paper, making rough estimatesof fractional rectangles as you go along.

That’s what I’ve done in figure q. Each rectangle on the graph

paper is 1.0 s wide and 2 m/s tall, so it represents 2 m. Adding upall the numbers gives ∆x = 41 m. If you needed better accuracy,you could use graph paper with smaller rectangles.

It’s important to realize that this technique gives you ∆x, notx. The v − t graph has no information about where the object waswhen it started.

The following are important points to keep in mind when apply-ing this technique:

• If the range of v values on your graph does not extend down

to zero, then you will get the wrong answer unless you com-pensate by adding in the area that is not shown.

• As in the example, one rectangle on the graph paper does notnecessarily correspond to one meter of distance.

• Negative velocity values represent motion in the opposite di-rection, so area under the t axis should be subtracted, i.e.,counted as “negative area.”




q / An example using estimatof fractions of a rectangle.

• Since the result is a ∆x value, it only tells you xafter−xbefore,which may be less than the actual distance traveled. For in-stance, the object could come back to its original position atthe end, which would correspond to ∆x=0, even though it had

actually moved a nonzero distance.

Finally, note that one can find ∆v from an a − t graph usingan entirely analogous method. Each rectangle on the a − t graphrepresents a certain amount of velocity change.

Discussion Question

A Roughly what would a pendulum’s v − t graph look like? What wouldhappen when you applied the area-under-the-curve technique to find thependulum’s ∆x for a time period covering many swings?

3.6 Algebraic Results for Constant

Acceleration

Although the area-under-the-curve technique can be applied to anygraph, no matter how complicated, it may be laborious to carry out,and if fractions of rectangles must be estimated the result will onlybe approximate. In the special case of motion with constant accel-eration, it is possible to find a convenient shortcut which produces

Section 3.6 Algebraic Results for Constant Acceleration 1



r / The shaded area tells ushow far an object moves whileaccelerating at a constant rate.

exact results. When the acceleration is constant, the v − t graphis a straight line, as shown in the figure. The area under the curvecan be divided into a triangle plus a rectangle, both of whose areascan be calculated exactly: A = bh for a rectangle and A = bh/2for a triangle. The height of the rectangle is the initial velocity, vo,and the height of the triangle is the change in velocity from begin-

ning to end, ∆v. The object’s ∆x is therefore given by the equation∆x = vo∆t + ∆v∆t/2. This can be simplified a little by using thedefinition of acceleration, a = ∆v/∆t, to eliminate ∆v, giving

∆x = vo∆t +1

2a∆t2 . [motion with

constant acceleration]

Since this is a second-order polynomial in ∆t, the graph of ∆x versus∆t is a parabola, and the same is true of a graph of x versus t —the two graphs differ only by shifting along the two axes. AlthoughI have derived the equation using a figure that shows a positive vo,positive a, and so on, it still turns out to be true regardless of whatplus and minus signs are involved.

Another useful equation can be derived if one wants to relatethe change in velocity to the distance traveled. This is useful, forinstance, for finding the distance needed by a car to come to a stop.For simplicity, we start by deriving the equation for the special caseof vo = 0, in which the final velocity vf is a synonym for ∆v. Sincevelocity and distance are the variables of interest, not time, we takethe equation ∆x = 1

2a∆t2 and use ∆t = ∆v/a to eliminate ∆t. This

gives ∆x = (∆v)2/2a, which can be rewritten as

v2f = 2a∆x . [motion with constant acceleration, vo = 0]

For the more general case where , we skip the tedious algebra leading

to the more general equation,

v2f = v2o

+ 2a∆x . [motion with constant acceleration]




a / Johannes Kepler found mathematical description of t

motion of the planets, which lto Newton’s theory of gravity.

Gravity is the only really important force on the cosmic scale. This false-

color representation of saturn’s rings was made from an image sent back

by the Voyager 2 space probe. The rings are composed of innumerable

tiny ice particles orbiting in circles under the influence of saturn’s gravity.

Chapter 10

Gravity

Cruise your radio dial today and try to find any popular song thatwould have been imaginable without Louis Armstrong. By introduc-ing solo improvisation into jazz, Armstrong took apart the jigsawpuzzle of popular music and fit the pieces back together in a dif-ferent way. In the same way, Newton reassembled our view of theuniverse. Consider the titles of some recent physics books writtenfor the general reader: The God Particle, Dreams of a Final The-ory. When the subatomic particle called the neutrino was recentlyproven for the first time to have mass, specialists in cosmology be-

gan discussing seriously what effect this would have on calculationsof the ultimate fate of the universe: would the neutrinos’ mass causeenough extra gravitational attraction to make the universe eventu-ally stop expanding and fall back together? Without Newton, suchattempts at universal understanding would not merely have seemeda little pretentious, they simply would not have occurred to anyone.

This chapter is about Newton’s theory of gravity, which he usedto explain the motion of the planets as they orbited the sun. Whereas

2



b / Tycho Brahe made his nameas an astronomer by showing thatthe bright new star, today calleda supernova, that appeared in

the skies in 1572 was far beyondthe Earth’s atmosphere. This,along with Galileo’s discovery ofsunspots, showed that contraryto Aristotle, the heavens werenot perfect and unchanging.Brahe’s fame as an astronomerbrought him patronage from KingFrederick II, allowing him to carryout his historic high-precision

measurements of the planets’motions. A contradictory charac-ter, Brahe enjoyed lecturing other

nobles about the evils of dueling,but had lost his own nose in ayouthful duel and had it replacedwith a prosthesis made of analloy of gold and silver. Willing toendure scandal in order to marrya peasant, he nevertheless used

the feudal powers given to him bythe king to impose harsh forcedlabor on the inhabitants of hisparishes. The result of their work,an Italian-style palace with anobservatory on top, surely ranks

as one of the most luxuriousscience labs ever built. He diedof a ruptured bladder after fallingfrom a wagon on the way home

from a party — in those days, itwas considered rude to leave thedinner table to relieve oneself.

this book has concentrated on Newton’s laws of motion, leavinggravity as a dessert, Newton tosses off the laws of motion in thefirst 20 pages of the Principia Mathematica and then spends thenext 130 discussing the motion of the planets. Clearly he saw thisas the crucial scientific focus of his work. Why? Because in it heshowed that the same laws of motion applied to the heavens as to

the earth, and that the gravitational force that made an apple fallwas the same as the force that kept the earth’s motion from carryingit away from the sun. What was radical about Newton was not hislaws of motion but his concept of a universal science of physics.

10.1 Kepler’s Laws

Newton wouldn’t have been able to figure out why the planetsmove the way they do if it hadn’t been for the astronomer TychoBrahe (1546-1601) and his protege Johannes Kepler (1571-1630),who together came up with the first simple and accurate descriptionof how the planets actually do move. The difficulty of their task issuggested by figure c, which shows how the relatively simple orbitalmotions of the earth and Mars combine so that as seen from earthMars appears to be staggering in loops like a drunken sailor.

c / As the Earth and Mars revolve around the sun at different rates,the combined effect of their motions makes Mars appear to trace astrange, looped path across the background of the distant stars.

Brahe, the last of the great naked-eye astronomers, collected ex-

234 Chapter 10 Gravity



tensive data on the motions of the planets over a period of manyyears, taking the giant step from the previous observations’ accuracyof about 10 minutes of arc (10/60 of a degree) to an unprecedented1 minute. The quality of his work is all the more remarkable consid-ering that his observatory consisted of four giant brass protractorsmounted upright in his castle in Denmark. Four different observers

would simultaneously measure the position of a planet in order tocheck for mistakes and reduce random errors.

With Brahe’s death, it fell to his former assistant Kepler to tryto make some sense out of the volumes of data. Kepler, in con-tradiction to his late boss, had formed a prejudice, a correct oneas it turned out, in favor of the theory that the earth and planetsrevolved around the sun, rather than the earth staying fixed andeverything rotating about it. Although motion is relative, it is not just a matter of opinion what circles what. The earth’s rotationand revolution about the sun make it a noninertial reference frame,which causes detectable violations of Newton’s laws when one at-

tempts to describe sufficiently precise experiments in the earth-fixedframe. Although such direct experiments were not carried out untilthe 19th century, what convinced everyone of the sun-centered sys-tem in the 17th century was that Kepler was able to come up witha surprisingly simple set of mathematical and geometrical rules fordescribing the planets’ motion using the sun-centered assumption.After 900 pages of calculations and many false starts and dead-endideas, Kepler finally synthesized the data into the following threelaws:

Kepler’s elliptical orbit law

The planets orbit the sun in elliptical orbits with the sun at

one focus.

Kepler’s equal-area law

The line connecting a planet to the sun sweeps out equal areasin equal amounts of time.

Kepler’s law of periods

The time required for a planet to orbit the sun, called itsperiod, is proportional to the long axis of the ellipse raised tothe 3/2 power. The constant of proportionality is the samefor all the planets.

Although the planets’ orbits are ellipses rather than circles, most

are very close to being circular. The earth’s orbit, for instance, isonly flattened by 1.7% relative to a circle. In the special case of aplanet in a circular orbit, the two foci (plural of “focus”) coincideat the center of the circle, and Kepler’s elliptical orbit law thus saysthat the circle is centered on the sun. The equal-area law impliesthat a planet in a circular orbit moves around the sun with constantspeed. For a circular orbit, the law of periods then amounts to astatement that the time for one orbit is proportional to r3/2, where

Section 10.1 Kepler’s Laws 2



d / An ellipse is a circle thathas been distorted by shrinking

and stretching along perpendicu-lar axes.

e / An ellipse can be con-structed by tying a string to twopins and drawing like this with thepencil stretching the string taut.Each pin constitutes one focus ofthe ellipse.

f / If the time interval takenby the planet to move from P to Qis equal to the time interval fromR to S, then according to Kepler’sequal-area law, the two shadedareas are equal. The planet

is moving faster during intervalRS than it did during PQ, whichNewton later determined was dueto the sun’s gravitational forceaccelerating it. The equal-arealaw predicts exactly how much itwill speed up.

r is the radius. If all the planets were moving in their orbits at thesame speed, then the time for one orbit would simply depend onthe circumference of the circle, so it would only be proportional tor to the first power. The more drastic dependence on r3/2 meansthat the outer planets must be moving more slowly than the innerplanets.

10.2 Newton’s Law of Gravity

The sun’s force on the planets obeys an inverse square law.

Kepler’s laws were a beautifully simple explanation of what theplanets did, but they didn’t address why they moved as they did.Did the sun exert a force that pulled a planet toward the center of its orbit, or, as suggested by Descartes, were the planets circulatingin a whirlpool of some unknown liquid? Kepler, working in theAristotelian tradition, hypothesized not just an inward force exertedby the sun on the planet, but also a second force in the direction

of motion to keep the planet from slowing down. Some speculatedthat the sun attracted the planets magnetically.

Once Newton had formulated his laws of motion and taughtthem to some of his friends, they began trying to connect themto Kepler’s laws. It was clear now that an inward force would beneeded to bend the planets’ paths. This force was presumably anattraction between the sun and each planet. (Although the sun doesaccelerate in response to the attractions of the planets, its mass is sogreat that the effect had never been detected by the prenewtonianastronomers.) Since the outer planets were moving slowly alongmore gently curving paths than the inner planets, their accelerations

were apparently less. This could be explained if the sun’s force wasdetermined by distance, becoming weaker for the farther planets.Physicists were also familiar with the noncontact forces of electricityand magnetism, and knew that they fell off rapidly with distance,so this made sense.

In the approximation of a circular orbit, the magnitude of thesun’s force on the planet would have to be

[1] F = ma = mv2/r .

Now although this equation has the magnitude, v, of the velocityvector in it, what Newton expected was that there would be a more

fundamental underlying equation for the force of the sun on a planet,and that that equation would involve the distance, r, from the sunto the object, but not the object’s speed, v — motion doesn’t makeobjects lighter or heavier.

self-check A

If eq. [1] really was generally applicable, what would happen to an

object released at rest in some empty region of the solar system?

Answer, p. 274




g / The moon’s acceleratiis 602 = 3600 times smaller th

the apple’s.

Equation [1] was thus a useful piece of information which couldbe related to the data on the planets simply because the planetshappened to be going in nearly circular orbits, but Newton wantedto combine it with other equations and eliminate v algebraically inorder to find a deeper truth.

To eliminate v, Newton used the equation

[2] v =circumference

T =

2πr

T .

Of course this equation would also only be valid for planets in nearlycircular orbits. Plugging this into eq. [1] to eliminate v gives

[3] F =4π2mr

T 2.

This unfortunately has the side-effect of bringing in the period, T ,which we expect on similar physical grounds will not occur in thefinal answer. That’s where the circular-orbit case, T ∝ r3/2, of

Kepler’s law of periods comes in. Using it to eliminate T gives aresult that depends only on the mass of the planet and its distancefrom the sun:

F ∝ m/r2 . [force of the sun on a planet of mass

m at a distance r from the sun; same

proportionality constant for all the planets]

(Since Kepler’s law of periods is only a proportionality, the finalresult is a proportionality rather than an equation, and there is thisno point in hanging on to the factor of 4π2.)

As an example, the “twin planets” Uranus and Neptune havenearly the same mass, but Neptune is about twice as far from thesun as Uranus, so the sun’s gravitational force on Neptune is aboutfour times smaller.

self-check B

Fill in the steps leading from equation [3] to F ∝ m /r 2. Answer, p.

275

The forces between heavenly bodies are the same type of

force as terrestrial gravity.

OK, but what kind of force was it? It probably wasn’t magnetic,

since magnetic forces have nothing to do with mass. Then cameNewton’s great insight. Lying under an apple tree and looking upat the moon in the sky, he saw an apple fall. Might not the earthalso attract the moon with the same kind of gravitational force?The moon orbits the earth in the same way that the planets orbitthe sun, so maybe the earth’s force on the falling apple, the earth’sforce on the moon, and the sun’s force on a planet were all the sametype of force.

Section 10.2 Newton’s Law of Gravity 2



There was an easy way to test this hypothesis numerically. If itwas true, then we would expect the gravitational forces exerted bythe earth to follow the same F ∝ m/r2 rule as the forces exerted bythe sun, but with a different constant of proportionality appropriateto the earth’s gravitational strength. The issue arises now of how todefine the distance, r, between the earth and the apple. An apple

in England is closer to some parts of the earth than to others, butsuppose we take r to be the distance from the center of the earth tothe apple, i.e., the radius of the earth. (The issue of how to measurer did not arise in the analysis of the planets’ motions because thesun and planets are so small compared to the distances separatingthem.) Calling the proportionality constant k, we have

F earth on apple = k mapple/r2earth

F earth on moon = k mmoon/d2earth-moon .

Newton’s second law says a = F/m, so

aapple = k / r 2earth

amoon = k / d2earth-moon .

The Greek astronomer Hipparchus had already found 2000 yearsbefore that the distance from the earth to the moon was about 60times the radius of the earth, so if Newton’s hypothesis was right,the acceleration of the moon would have to be 60 2 = 3600 times lessthan the acceleration of the falling apple.

Applying a = v2/r to the acceleration of the moon yielded anacceleration that was indeed 3600 times smaller than 9.8 m/s2, and

Newton was convinced he had unlocked the secret of the mysteriousforce that kept the moon and planets in their orbits.

Newton’s law of gravity

The proportionality F ∝ m/r2 for the gravitational force on anobject of mass m only has a consistent proportionality constant forvarious objects if they are being acted on by the gravity of the sameobject. Clearly the sun’s gravitational strength is far greater thanthe earth’s, since the planets all orbit the sun and do not exhibitany very large accelerations caused by the earth (or by one another).What property of the sun gives it its great gravitational strength?Its great volume? Its great mass? Its great temperature? Newton

reasoned that if the force was proportional to the mass of the objectbeing acted on, then it would also make sense if the determiningfactor in the gravitational strength of the object exerting the forcewas its own mass. Assuming there were no other factors affectingthe gravitational force, then the only other thing needed to makequantitative predictions of gravitational forces would be a propor-tionality constant. Newton called that proportionality constant G,so here is the complete form of the law of gravity he hypothesized.




j / The conic sections are thecurves made by cutting thesurface of an infinite cone with aplane.

k / An imaginary cannon ableto shoot cannonballs at very highspeeds is placed on top of an

imaginary, very tall mountainthat reaches up above the at-mosphere. Depending on thespeed at which the ball is fired,it may end up in a tightly curvedelliptical orbit, 1, a circular orbit,

2, a bigger elliptical orbit, 3, or anearly straight hyperbolic orbit, 4.

The proportionality to 1/r2 in Newton’s law of gravity was notentirely unexpected. Proportionalities to 1/r2 are found in manyother phenomena in which some effect spreads out from a point.For instance, the intensity of the light from a candle is proportionalto 1/r2, because at a distance r from the candle, the light has to

be spread out over the surface of an imaginary sphere of area 4πr2.The same is true for the intensity of sound from a firecracker, or theintensity of gamma radiation emitted by the Chernobyl reactor. It’simportant, however, to realize that this is only an analogy. Forcedoes not travel through space as sound or light does, and force isnot a substance that can be spread thicker or thinner like butter ontoast.

Although several of Newton’s contemporaries had speculatedthat the force of gravity might be proportional to 1/r2, none of them, even the ones who had learned Newton’s laws of motion, hadhad any luck proving that the resulting orbits would be ellipses, as

Kepler had found empirically. Newton did succeed in proving thatelliptical orbits would result from a 1/r2 force, but we postpone theproof until the end of the next volume of the textbook because itcan be accomplished much more easily using the concepts of energyand angular momentum.

Newton also predicted that orbits in the shape of hyperbolasshould be possible, and he was right. Some comets, for instance,orbit the sun in very elongated ellipses, but others pass throughthe solar system on hyperbolic paths, never to return. Just as thetrajectory of a faster baseball pitch is flatter than that of a moreslowly thrown ball, so the curvature of a planet’s orbit depends on

its speed. A spacecraft can be launched at relatively low speed,resulting in a circular orbit about the earth, or it can be launchedat a higher speed, giving a more gently curved ellipse that reachesfarther from the earth, or it can be launched at a very high speedwhich puts it in an even less curved hyperbolic orbit. As you govery far out on a hyperbola, it approaches a straight line, i.e., itscurvature eventually becomes nearly zero.

Newton also was able to prove that Kepler’s second law (sweep-ing out equal areas in equal time intervals) was a logical consequenceof his law of gravity. Newton’s version of the proof is moderatelycomplicated, but the proof becomes trivial once you understand the

concept of angular momentum, which will be covered later in thecourse. The proof will therefore be deferred until section 5.7 of book2.

self-check C

Which of Kepler’s laws would it make sense to apply to hyperbolic or-

bits? Answer, p.

275




a / This Global PositioniSystem (GPS) system, runnion a smartphone attached tobike’s handlebar, depends Einstein’s theory of relativTime flows at a different rataboard a GPS satellite thandoes on the bike, and the G

software has to take this inaccount.

b / The atomic clock has own ticket and its own seat.

Chapter 1

Relativity1.1 Time Is Not Absolute

When Einstein first began to develop the theory of relativity, around1905, the only real-world observations he could draw on were am-biguous and indirect. Today, the evidence is part of everyday life.For example, every time you use a GPS receiver, a, you’re usingEinstein’s theory of relativity. Somewhere between 1905 and today,technology became good enough to allow conceptually simple exper-iments that students in the earth 20th century could only discuss in

terms like “Imagine that we could. . . ” A good jumping-on point is1971. In that year, J.C. Hafele and R.E. Keating of the U.S. NavalObservatory brought atomic clocks aboard commercial airliners, b,and went around the world, once from east to west and once fromwest to east. (The clocks had their own tickets, and occupied theirown seats.) Hafele and Keating observed that there was a discrep-ancy between the times measured by the traveling clocks and thetimes measured by similar clocks that stayed at the lab in Wash-ington. The east-going clock lost time, ending up off by −59 ± 10nanoseconds, while the west-going one gained 273 ± 7 ns.

The correspondence principle

This establishes that time doesn’t work the way Newton believedit did when he wrote that “Absolute, true, and mathematical time,of itself, and from its own nature flows equably without regard toanything external. . . ” We are used to thinking of time as absoluteand universal, so it is disturbing to find that it can flow at a differ-ent rate for observers in different frames of reference. Nevertheless,the effect that Hafele and Keating observed were small. This makessense: Newton’s laws have already been thoroughly tested by ex-periments under a wide variety of conditions, so a new theory likerelativity must agree with Newton’s to a good approximation, withinthe Newtonian theory’s realm of applicability. This requirement of

backward-compatibility is known as the correspondence principle.

Causality

It’s also reassuring that the effects on time were small comparedto the three-day lengths of the plane trips. There was therefore noopportunity for paradoxical scenarios such as one in which the east-going experimenter arrived back in Washington before he left andthen convinced himself not to take the trip. A theory that maintains



d / All three clocks are mov-ing to the east. Even though thewest-going plane is moving to thewest relative to the air, the airis moving to the east due to the

earth’s rotation.

c / Newton’s laws do not dis-tinguish past from future. Thefootball could travel in eitherdirection while obeying Newton’slaws.

this kind of orderly relationship between cause and effect is said tosatisfy causality.

Causality is like a water-hungry front-yard lawn in Los Angeles:we know we want it, but it’s not easy to explain why. Even in plainold Newtonian physics, there is no clear distinction between pastand future. In figure c, number 18 throws the football to number

25, and the ball obeys Newton’s laws of motion. If we took a videoof the pass and played it backward, we would see the ball flying from25 to 18, and Newton’s laws would still be satisfied. Nevertheless,we have a strong psychological impression that there is a forwardarrow of time. I can remember what the stock market did last year,but I can’t remember what it will do next year. Joan of Arc’s mil-itary victories against England caused the English to burn her atthe stake; it’s hard to accept that Newton’s laws provide an equallygood description of a process in which her execution in 1431 causedher to win a battle in 1429. There is no consensus at this pointamong physicists on the origin and significance of time’s arrow, and

for our present purposes we don’t need to solve this mystery. In-stead, we merely note the empirical fact that, regardless of whatcausality really means and where it really comes from, its behavioris consistent. Specifically, experiments show that if an observer in acertain frame of reference observes that event A causes event B, thenobservers in other frames agree that A causes B, not the other wayaround. This is merely a generalization about a large body of ex-perimental results, not a logically necessary assumption. If Keatinghad gone around the world and arrived back in Washington beforehe left, it would have disproved this statement about causality.

Time distortion arising from motion and gravity

Hafele and Keating were testing specific quantitative predictionsof relativity, and they verified them to within their experiment’serror bars. Let’s work backward instead, and inspect the empiricalresults for clues as to how time works.

The two traveling clocks experienced effects in opposite direc-tions, and this suggests that the rate at which time flows dependson the motion of the observer. The east-going clock was moving inthe same direction as the earth’s rotation, so its velocity relative tothe earth’s center was greater than that of the clock that remainedin Washington, while the west-going clock’s velocity was correspond-ingly reduced. The fact that the east-going clock fell behind, andthe west-going one got ahead, shows that the effect of motion is tomake time go more slowly. This effect of motion on time was pre-dicted by Einstein in his original 1905 paper on relativity, writtenwhen he was 26.

If this had been the only effect in the Hafele-Keating experiment,then we would have expected to see effects on the two flying clocksthat were equal in size. Making up some simple numbers to keep the

14 Chapter 1 Relativity



e / The correspondence prciple requires that the relativis

distortion of time become smfor small velocities.

arithmetic transparent, suppose that the earth rotates from west toeast at 1000 km/hr, and that the planes fly at 300 km/hr. Then thespeed of the clock on the ground is 1000 km/hr, the speed of theclock on the east-going plane is 1300 km/hr, and that of the west-going clock 700 km/hr. Since the speeds of 700, 1000, and 1300km/hr have equal spacing on either side of 1000, we would expect

the discrepancies of the moving clocks relative to the one in the labto be equal in size but opposite in sign.

In fact, the two effects are unequal in size: −59 ns and 273 ns.This implies that there is a second effect involved, simply due to theplanes’ being up in the air. Relativity predicts that time’s rate of flow also changes with height in a gravitational field. Einstein didn’tfigure out how to incorporate gravity into relativity until 1915, aftermuch frustration and many false starts. The simpler version of thetheory without gravity is known as special relativity, the full versionas general relativity. We’ll restrict ourselves to special relativity inthis book, and that means that what we want to focus on right now

is the distortion of time due to motion, not gravity.We can now see in more detail how to apply the correspondence

principle. The behavior of the three clocks in the Hafele-Keatingexperiment shows that the amount of time distortion increases asthe speed of the clock’s motion increases. Newton lived in an erawhen the fastest mode of transportation was a galloping horse, andthe best pendulum clocks would accumulate errors of perhaps aminute over the course of several days. A horse is much slowerthan a jet plane, so the distortion of time would have had a relativesize of only ∼ 10−15 — much smaller than the clocks were capableof detecting. At the speed of a passenger jet, the effect is about

10−12

, and state-of-the-art atomic clocks in 1971 were capable of measuring that. A GPS satellite travels much faster than a jet air-plane, and the effect on the satellite turns out to be ∼ 10−10. Thegeneral idea here is that all physical laws are approximations, andapproximations aren’t simply right or wrong in different situations.Approximations are better or worse in different situations, and thequestion is whether a particular approximation is good enough in agiven situation to serve a particular purpose. The faster the motion,the worse the Newtonian approximation of absolute time. Whetherthe approximation is good enough depends on what you’re tryingto accomplish. The correspondence principle says that the approxi-mation must have been good enough to explain all the experimentsdone in the centuries before Einstein came up with relativity.

By the way, don’t get an inflated idea of the importance of theHafele-Keating experiment. Special relativity had already been con-firmed by a vast and varied body of experiments decades before 1971.The only reason I’m giving such a prominent role to this experiment,which was actually more important as a test of general relativity, isthat it is conceptually very direct.

Section 1.1 Time Is Not Absolute



f / Two events are given aspoints on a graph of positionversus time. Joan of Arc helps torestore Charles VII to the throne.At a later time and a differentposition, Joan of Arc is sentenced

to death.

g / A change of units distortsan x -t graph. This graph depictsexactly the same events as figure

f. The only change is that the x and t coordinates are measuredusing different units, so the grid iscompressed in t and expanded inx .

h / A convention we’ll use torepresent a distortion of time andspace.

1.2 Distortion of Space and Time

The Lorentz transformation

Relativity says that when two observers are in different frames of reference, each observer considers the other one’s perception of timeto be distorted. We’ll also see that something similar happens to

their observations of distances, so both space and time are distorted.What exactly is this distortion? How do we even conceptualize it?

The idea isn’t really as radical as it might seem at first. Wecan visualize the structure of space and time using a graph withposition and time on its axes. These graphs are familiar by now,but we’re going to look at them in a slightly different way. Before, weused them to describe the motion of objects. The grid underlyingthe graph was merely the stage on which the actors played theirparts. Now the background comes to the foreground: it’s time andspace themselves that we’re studying. We don’t necessarily needto have a line or a curve drawn on top of the grid to represent a

particular object. We may, for example, just want to talk aboutevents, depicted as points on the graph as in figure f. A distortionof the Cartesian grid underlying the graph can arise for perfectlyordinary reasons that Isaac Newton would have readily accepted.For example, we can simply change the units used to measure timeand position, as in figure g.

We’re going to have quite a few examples of this type, so I’lladopt the convention shown in figure h for depicting them. Figureh summarizes the relationship between figures f and g in a morecompact form. The gray rectangle represents the original coordinategrid of figure f, while the grid of black lines represents the new

version from figure g. Omitting the grid from the original grayrectangle makes the graphs easier to decode visually.

Our goal of unraveling the mysteries of special relativity amountsto nothing more than finding out how to draw a diagram like hin the case where the two different sets of coordinates representmeasurements of time and space made by two different observers,each in motion relative to the other. Galileo and Newton thoughtthey knew the answer to this question, but their answer turnedout to be only approximately right. To avoid repeating the samemistakes, we need to clearly spell out what we think are the basicproperties of time and space that will be a reliable foundation for

our reasoning. I want to emphasize that there is no purely logicalway of deciding on this list of properties. The ones I’ll list are simplya summary of the patterns observed in the results from a large bodyof experiments. Furthermore, some of them are only approximate.For example, property 1 below is only a good approximation whenthe gravitational field is weak, so it is a property that applies tospecial relativity, not to general relativity.




i / A Galilean version of trelationship between two framof reference. As in all sugraphs in this chapter, the origin

coordinates, represented by tgray rectangle, have a time ax

that goes to the right, andposition axis that goes straigup.

Experiments show that:

1. No point in time or space has properties that make it differentfrom any other point.

2. Likewise, all directions in space have the same properties.

3. Motion is relative, i.e., all inertial frames of reference areequally valid.

4. Causality holds, in the sense described on page 13.

5. Time depends on the state of motion of the observer.

Most of these are not very subversive. Properties 1 and 2 dateback to the time when Galileo and Newton started applying thesame universal laws of motion to the solar system and to the earth;this contradicted Aristotle, who believed that, for example, a rockwould naturally want to move in a certain special direction (down)in order to reach a certain special location (the earth’s surface).Property 3 is the reason that Einstein called his theory “relativity,”but Galileo and Newton believed exactly the same thing to be true,as dramatized by Galileo’s run-in with the Church over the questionof whether the earth could really be in motion around the sun.Property 4 would probably surprise most people only because itasserts in such a weak and specialized way something that they feeldeeply must be true. The only really strange item on the list is 5,but the Hafele-Keating experiment forces it upon us.

If it were not for property 5, we could imagine that figure iwould give the correct transformation between frames of referencein motion relative to one another. Let’s say that observer 1, whosegrid coincides with the gray rectangle, is a hitch-hiker standing bythe side of a road. Event A is a raindrop hitting his head, andevent B is another raindrop hitting his head. He says that A and Boccur at the same location in space. Observer 2 is a motorist whodrives by without stopping; to him, the passenger compartment of his car is at rest, while the asphalt slides by underneath. He saysthat A and B occur at different points in space, because during thetime between the first raindrop and the second, the hitch-hiker hasmoved backward. On the other hand, observer 2 says that events Aand C occur in the same place, while the hitch-hiker disagrees. The

slope of the grid-lines is simply the velocity of the relative motionof each observer relative to the other.

Figure i has familiar, comforting, and eminently sensible behav-ior, but it also happens to be wrong, because it violates property5. The distortion of the coordinate grid has only moved the verticallines up and down, so both observers agree that events like B andC are simultaneous. If this was really the way things worked, thenall observers could synchronize all their clocks with one another for

Section 1.2 Distortion of Space and Time



j / A transformation that leads todisagreements about whethertwo events occur at the same

time and place. This is not justa matter of opinion. Either thearrow hit the bull’s-eye or it didn’t.

k / A nonlinear transformation.

once and for all, and the clocks would never get out of sync. Thiscontradicts the results of the Hafele-Keating experiment, in whichall three clocks were initially synchronized in Washington, but laterwent out of sync because of their different states of motion.

It might seem as though we still had a huge amount of wiggleroom available for the correct form of the distortion. It turns out,

however, that properties 1-5 are sufficient to prove that there is onlyone answer, which is the one found by Einstein in 1905. To see whythis is, let’s work by a process of elimination.

Figure j shows a transformation that might seem at first glanceto be as good a candidate as any other, but it violates property3, that motion is relative, for the following reason. In observer 2’sframe of reference, some of the grid lines cross one another. Thismeans that observers 1 and 2 disagree on whether or not certainevents are the same. For instance, suppose that event A marks thearrival of an arrow at the bull’s-eye of a target, and event B is thelocation and time when the bull’s-eye is punctured. Events A and B

occur at the same location and at the same time. If one observer saysthat A and B coincide, but another says that they don’t, we havea direct contradiction. Since the two frames of reference in figure j give contradictory results, one of them is right and one is wrong.This violates property 3, because all inertial frames of reference aresupposed to be equally valid. To avoid problems like this, we clearlyneed to make sure that none of the grid lines ever cross one another.

The next type of transformation we want to kill off is shown infigure k, in which the grid lines curve, but never cross one another.The trouble with this one is that it violates property 1, the unifor-mity of time and space. The transformation is unusually “twisty”

at A, whereas at B it’s much more smooth. This can’t be correct,because the transformation is only supposed to depend on the rela-tive state of motion of the two frames of reference, and that giveninformation doesn’t single out a special role for any particular pointin spacetime. If, for example, we had one frame of reference rotating

relative to the other, then there would be something special aboutthe axis of rotation. But we’re only talking about inertial frames of reference here, as specified in property 3, so we can’t have rotation;each frame of reference has to be moving in a straight line at con-stant speed. For frames related in this way, there is nothing thatcould single out an event like A for special treatment compared to

B, so transformation k violates property 1.The examples in figures j and k show that the transformation

we’re looking for must be linear, meaning that it must transformlines into lines, and furthermore that it has to take parallel lines toparallel lines. Einstein wrote in his 1905 paper that “. . . on accountof the property of homogeneity [property 1] which we ascribe to time




m / In the units that are moconvenient for relativity, the tranformation has symmetry abou

45-degree diagonal line.

and space, the [transformation] must be linear.”1 Applying this toour diagrams, the original gray rectangle, which is a special typeof parallelogram containing right angles, must be transformed intoanother parallelogram. There are three types of transformations,figure l, that have this property. Case I is the Galilean transforma-tion of figure i on page 17, which we’ve already ruled out.

l / Three types of transformations that preserve parallelism. Theirdistinguishing feature is what they do to simultaneity, as shown by what

happens to the left edge of the original rectangle. In I, the left edgeremains vertical, so simultaneous events remain simultaneous. In II, theleft edge turns counterclockwise. In III, it turns clockwise.

Case II can also be discarded. Here every point on the grid ro-tates counterclockwise. What physical parameter would determinethe amount of rotation? The only thing that could be relevant wouldbe v, the relative velocity of the motion of the two frames of referencewith respect to one another. But if the angle of rotation was pro-portional to v, then for large enough velocities the grid would haveleft and right reversed, and this would violate property 4, causality:

one observer would say that event A caused a later event B, butanother observer would say that B came first and caused A.

The only remaining possibility is case III, which I’ve redrawn infigure m with a couple of changes. This is the one that Einsteinpredicted in 1905. The transformation is known as the Lorentztransformation, after Hendrik Lorentz (1853-1928), who partiallyanticipated Einstein’s work, without arriving at the correct inter-pretation. The distortion is a kind of smooshing and stretching, assuggested by the hands. Also, we’ve already seen in figures f-h onpage 16 that we’re free to stretch or compress everything as much aswe like in the horizontal and vertical directions, because this simply

corresponds to choosing different units of measurement for time anddistance. In figure m I’ve chosen units that give the whole drawinga convenient symmetry about a 45-degree diagonal line. Ordinarilyit wouldn’t make sense to talk about a 45-degree angle on a graphwhose axes had different units. But in relativity, the symmetric ap-pearance of the transformation tells us that space and time ought

1A. Einstein, “On the Electrodynamics of Moving Bodies,” Annalen der Physik 17 (1905), p. 891, tr. Saha and Bose, 1920




n / Interpretation of the Lorentztransformation. The slope in-dicated in the figure gives therelative velocity of the two framesof reference. Events A and B thatwere simultaneous in frame 1are not simultaneous in frame 2,

where event A occurs to the rightof the t = 0 line represented bythe left edge of the grid, but eventB occurs to its left.

to be treated on the same footing, and measured in the same units.

As in our discussion of the Galilean transformation, slopes areinterpreted as velocities, and the slope of the near-horizontal lines infigure n is interpreted as the relative velocity of the two observers.The difference between the Galilean version and the relativistic oneis that now there is smooshing happening from the other side as

well. Lines that were vertical in the original grid, representing si-multaneous events, now slant over to the right. This tells us that, asrequired by property 5, different observers do not agree on whetherevents that occur in different places are simultaneous. The Hafele-Keating experiment tells us that this non-simultaneity effect is fairlysmall, even when the velocity is as big as that of a passenger jet,and this is what we would have anticipated by the correspondenceprinciple. The way that this is expressed in the graph is that if wepick the time unit to be the second, then the distance unit turns outto be hundreds of thousands of miles. In these units, the velocityof a passenger jet is an extremely small number, so the slope v in

figure n is extremely small, and the amount of distortion is tiny —it would be much too small to see on this scale.

The only thing left to determine about the Lorentz transforma-tion is the size of the transformed parallelogram relative to the sizeof the original one. Although the drawing of the hands in figure mmay suggest that the grid deforms like a framework made of rigidcoat-hanger wire, that is not the case. If you look carefully at thefigure, you’ll see that the edges of the smooshed parallelogram areactually a little longer than the edges of the original rectangle. Infact what stays the same is not lengths but areas. The proof of thisfact is straightforward, but a little lengthy, so I’ve relegated it to

page 130, following it with some remarks that may be of interestto teachers. Oversimplifying a little, the basic idea of the proof isthat it wouldn’t make sense if the area was increased by the Lorentztransformation, because then area would have to be decreased bya Lorentz transformation corresponding to motion in the oppositedirection, and this would violate property 2 on page 17, which statesthat all directions in space have the same properties.

The γ factor

With a little algebra and geometry (homework problem 7, page39), one can use the equal-area property to show that the factor γ

(Greek letter gamma) defined in figure o is given by the equation

γ =1√

1− v2.

If you’ve had good training in physics, the first thing you probablythink when you look at this equation is that it must be nonsense,




o / The γ factor.

because its units don’t make sense. How can we take somethingwith units of velocity squared, and subtract it from a unitless 1?But remember that this is expressed in our special relativistic units,in which the same units are used for distance and time. In thissystem, velocities are always unitless. This sort of thing happensfrequently in physics. For instance, before James Joule discovered

conservation of energy, nobody knew that heat and mechanical en-ergy were different forms of the same thing, so instead of measuringthem both in units of joules as we would do now, they measuredheat in one unit (such as calories) and mechanical energy in another(such as foot-pounds). In ordinary metric units, we just need anextra convension factor c, and the equation becomes

γ =

1 1−

v

c

2 .

Here’s why we care about γ . Figure o defines it as the ratio of twotimes: the time between two events as expressed in one coordinatesystem, and the time between the same two events as measured inthe other one. The interpretation is:

Time dilationA clock runs fastest in the frame of reference of an an observerwho is at rest relative to the clock. An observer in motionrelative to the clock at speed v perceives the clock as runningmore slowly by a factor of γ .

Since the Lorentz transformation treats time and distance entirelysymmetrically, we could just as well have defined γ using the uprightx axis in figure o, and we therefore have a similar interpretation interms of space:

Length contraction

A meter-stick appears longest to an observer who is at restrelative to it. An observer moving relative to the meter-stickat v observes the stick to be shortened by a factor of γ .

self-check A

What is γ when v = 0? What does this mean? Answer, p. 135




q / Muons accelerated to nearly c

undergo radioactive decay much

more slowly than they wouldaccording to an observer at restwith respect to the muons. Thefirst two data-points (unfilledcircles) were subject to largesystematic errors.

p / Apparatus used for the testof relativistic time dilation de-scribed in example 1. The promi-

nent black and white blocks arelarge magnets surrounding a cir-

cular pipe with a vacuum inside.(c) 1974 by CERN.

Large time dilation example 1

The time dilation effect in the Hafele-Keating experiment was very

small. If we want to see a large time dilation effect, we can’t doit with something the size of the atomic clocks they used; the ki-

netic energy would be greater than the total megatonnage of all

the world’s nuclear arsenals. We can, however, accelerate sub-

atomic particles to speeds at which γ is large. For experimental

particle physicists, relativity is something you do all day before

heading home and stopping off at the store for milk. An early, low-

precision experiment of this kind was performed by Rossi and Hall

in 1941, using naturally occurring cosmic rays. Figure p shows a

1974 experiment2 of a similar type which verified the time dilation

predicted by relativity to a precision of about one part per thou-

sand.Particles called muons (named after the Greek letter µ , “myoo”)

were produced by an accelerator at CERN, near Geneva. A muon

is essentially a heavier version of the electron. Muons undergo

radioactive decay, lasting an average of only 2.197 µ s before they

evaporate into an electron and two neutrinos. The 1974 experi-

ment was actually built in order to measure the magnetic proper-

ties of muons, but it produced a high-precision test of time dilation

as a byproduct. Because muons have the same electric charge

as electrons, they can be trapped using magnetic fields. Muons

were injected into the ring shown in figure p, circling around it un-

til they underwent radioactive decay. At the speed at which thesemuons were traveling, they had γ = 29.33, so on the average they

lasted 29.33 times longer than the normal lifetime. In other words,

they were like tiny alarm clocks that self-destructed at a randomly

selected time. Figure q shows the number of radioactive decays

counted, as a function of the time elapsed after a given stream of

muons was injected into the storage ring. The two dashed lines

2Bailey at al., Nucl. Phys. B150(1979) 1




s / In the garage’s frame reference, 1, the bus is movinand can fit in the garage. In tbus’s frame of reference, tgarage is moving, and can’t hothe bus.

show the rates of decay predicted with and without relativity. The

relativistic line is the one that agrees with experiment.

An example of length contraction example 2

Figure r shows an artist’s rendering of the length contraction for

the collision of two gold nuclei at relativistic speeds in the RHIC

accelerator in Long Island, New York, which went on line in 2000.

The gold nuclei would appear nearly spherical (or just slightlylengthened like an American football) in frames moving along with

them, but in the laboratory’s frame, they both appear drastically

foreshortened as they approach the point of collision. The later

pictures show the nuclei merging to form a hot soup, in which

experimenters hope to observe a new form of matter.

r / Colliding nuclei show relativtic length contraction.

The garage paradox example 3

One of the most famous of all the so-called relativity paradoxes

has to do with our incorrect feeling that simultaneity is well de-

fined. The idea is that one could take a schoolbus and drive it at

relativistic speeds into a garage of ordinary size, in which it nor-

mally would not fit. Because of the length contraction, the bus

would supposedly fit in the garage. The paradox arises when we

shut the door and then quickly slam on the brakes of the bus.

An observer in the garage’s frame of reference will claim that the

bus fit in the garage because of its contracted length. The driver,

however, will perceive the garage as being contracted and thus

even less able to contain the bus. The paradox is resolved when

we recognize that the concept of fitting the bus in the garage “all

at once” contains a hidden assumption, the assumption that it

makes sense to ask whether the front and back of the bus can

simultaneously be in the garage. Observers in different framesof reference moving at high relative speeds do not necessarily

agree on whether things happen simultaneously. The person in

the garage’s frame can shut the door at an instant he perceives to

be simultaneous with the front bumper’s arrival at the back wall of

the garage, but the driver would not agree about the simultaneity

of these two events, and would perceive the door as having shut

long after she plowed through the back wall.




t / A proof that causality im-

poses a universal speed limit. In

the original frame of reference,represented by the square, eventA happens a little before event B.In the new frame, shown by theparallelogram, A happens aftert = 0, but B happens before t = 0;that is, B happens before A. Thetime ordering of the two events

has been reversed. This can onlyhappen because events A and Bare very close together in timeand fairly far apart in space. Theline segment connecting A and

B has a slope greater than 1,meaning that if we wanted to bepresent at both events, we wouldhave to travel at a speed greaterthan c (which equals 1 in the

units used on this graph). You willfind that if you pick any two pointsfor which the slope of the linesegment connecting them is lessthan 1, you can never get them tostraddle the new t = 0 line in thisfunny, time-reversed way. Since

different observers disagree onthe time order of events like Aand B, causality requires that

information never travel fromA to B or from B to A; if it did,then we would have time-travelparadoxes. The conclusion is thatc is the maximum speed of causeand effect in relativity.

The universal speed c

Let’s think a little more about the role of the 45-degree diagonalin the Lorentz transformation. Slopes on these graphs are inter-preted as velocities. This line has a slope of 1 in relativistic units,but that slope corresponds to c in ordinary metric units. We al-ready know that the relativistic distance unit must be extremely

large compared to the relativistic time unit, so c must be extremelylarge. Now note what happens when we perform a Lorentz transfor-mation: this particular line gets stretched, but the new version of the line lies right on top of the old one, and its slope stays the same.In other words, if one observer says that something has a velocityequal to c, every other observer will agree on that velocity as well.(The same thing happens with −c.)

Velocities don’t simply add and subtract.

This is counterintuitive, since we expect velocities to add andsubtract in relative motion. If a dog is running away from me at 5

m/s relative to the sidewalk, and I run after it at 3 m/s, the dog’svelocity in my frame of reference is 2 m/s. According to everythingwe have learned about motion, the dog must have different speedsin the two frames: 5 m/s in the sidewalk’s frame and 2 m/s inmine. But velocities are measured by dividing a distance by a time,and both distance and time are distorted by relativistic effects, sowe actually shouldn’t expect the ordinary arithmetic addition of velocities to hold in relativity; it’s an approximation that’s valid atvelocities that are small compared to c.

A universal speed limit

For example, suppose Janet takes a trip in a spaceship, andaccelerates until she is moving at 0.6c relative to the earth. Shethen launches a space probe in the forward direction at a speedrelative to her ship of 0.6c. We might think that the probe was thenmoving at a velocity of 1.2c, but in fact the answer is still less thanc (problem 1, page 38). This is an example of a more general factabout relativity, which is that c represents a universal speed limit.This is required by causality, as shown in figure t.

Light travels at c.

Now consider a beam of light. We’re used to talking casuallyabout the “speed of light,” but what does that really mean? Motion

is relative, so normally if we want to talk about a velocity, we haveto specify what it’s measured relative to. A sound wave has a certainspeed relative to the air, and a water wave has its own speed relativeto the water. If we want to measure the speed of an ocean wave, forexample, we should make sure to measure it in a frame of referenceat rest relative to the water. But light isn’t a vibration of a physicalmedium; it can propagate through the near-perfect vacuum of outerspace, as when rays of sunlight travel to earth. This seems like a




paradox: light is supposed to have a specific speed, but there is noway to decide what frame of reference to measure it in. The wayout of the paradox is that light must travel at a velocity equal to c.Since all observers agree on a velocity of c, regardless of their frameof reference, everything is consistent.

The Michelson-Morley experiment

The constancy of the speed of light had in fact already beenobserved when Einstein was an 8-year-old boy, but because nobodycould figure out how to interpret it, the result was largely ignored.In 1887 Michelson and Morley set up a clever apparatus to measureany difference in the speed of light beams traveling east-west andnorth-south. The motion of the earth around the sun at 110,000km/hour (about 0.01% of the speed of light) is to our west during theday. Michelson and Morley believed that light was a vibration of amysterious medium called the ether, so they expected that the speedof light would be a fixed value relative to the ether. As the earthmoved through the ether, they thought they would observe an effecton the velocity of light along an east-west line. For instance, if theyreleased a beam of light in a westward direction during the day, theyexpected that it would move away from them at less than the normalspeed because the earth was chasing it through the ether. They weresurprised when they found that the expected 0.01% change in thespeed of light did not occur.

u / The Michelson-Morley expe

ment, shown in photographs, adrawings from the original 18paper. 1. A simplified draing of the apparatus. A beamlight from the source, s, is ptially reflected and partially tramitted by the half-silvered mirh1. The two half-intensity parts

the beam are reflected by the mrors at a and b, reunited, and o

served in the telescope, t. If tearth’s surface was supposedbe moving through the ether, ththe times taken by the two ligwaves to pass through the moing ether would be unequal, athe resulting time lag would detectable by observing the intference between the waves whthey were reunited. 2. In the r

apparatus, the light beams wereflected multiple times. The fective length of each arm wincreased to 11 meters, whgreatly improved its sensitivitythe small expected differencethe speed of light. 3. In earlier version of the experimethey had run into problems wits “extreme sensitiveness to

bration,” which was “so great tit was impossible to see the terference fringes except at br

intervals . . . even at two o’cloin the morning.” They therefomounted the whole thing onmassive stone floating in a poomercury, which also made it posible to rotate it easily. 4. A phoof the apparatus.




Discussion question B


A A person in a spaceship moving at 99.99999999% of the speedof light relative to Earth shines a flashlight forward through dusty air, sothe beam is visible. What does she see? What would it look like to anobserver on Earth?

B A question that students often struggle with is whether time and

space can really be distorted, or whether it just seems that way. Comparewith optical illusions or magic tricks. How could you verify, for instance,that the lines in the figure are actually parallel? Are relativistic effects thesame, or not?

C On a spaceship moving at relativistic speeds, would a lecture seemeven longer and more boring than normal?

D Mechanical clocks can be affected by motion. For example, it was

a significant technological achievement to build a clock that could sailaboard a ship and still keep accurate time, allowing longitude to be deter-mined. How is this similar to or different from relativistic time dilation?

E Figure r from page 23, depicting the collision of two nuclei at theRHIC accelerator, is reproduced below. What would the shapes of the twonuclei look like to a microscopic observer riding on the left-hand nucleus?To an observer riding on the right-hand one? Can they agree on what ishappening? If not, why not — after all, shouldn’t they see the same thing

if they both compare the two nuclei side-by-side at the same instant intime?

v / Discussion question E: colliding nuclei show relativistic lengthcontraction.




In recent decades, a huge hole in the ozone layer has spread out from

Antarctica.

Chapter 3Light as a Particle

The only thing that interferes with my learning is my educa-

tion.

Albert Einstein

Radioactivity is random, but do the laws of physics exhibit ran-domness in other contexts besides radioactivity? Yes. Radioactive

decay was just a good playpen to get us started with concepts of randomness, because all atoms of a given isotope are identical. Bystocking the playpen with an unlimited supply of identical atom-toys, nature helped us to realize that their future behavior could bedifferent regardless of their original identicality. We are now readyto leave the playpen, and see how randomness fits into the structureof physics at the most fundamental level.

The laws of physics describe light and matter, and the quantumrevolution rewrote both descriptions. Radioactivity was a good ex-ample of matter’s behaving in a way that was inconsistent withclassical physics, but if we want to get under the hood and under-

stand how nonclassical things happen, it will be easier to focus onlight rather than matter. A radioactive atom such as uranium-235is after all an extremely complex system, consisting of 92 protons,143 neutrons, and 92 electrons. Light, however, can be a simple sinewave.

However successful the classical wave theory of light had been— allowing the creation of radio and radar, for example — it still



failed to describe many important phenomena. An example thatis currently of great interest is the way the ozone layer protects usfrom the dangerous short-wavelength ultraviolet part of the sun’sspectrum. In the classical description, light is a wave. When a wavepasses into and back out of a medium, its frequency is unchanged,and although its wavelength is altered while it is in the medium,

it returns to its original value when the wave reemerges. Luckilyfor us, this is not at all what ultraviolet light does when it passesthrough the ozone layer, or the layer would offer no protection atall!

3.1 Evidence for Light As a Particle

a / Images made by a digital cam-era. In each successive image,the dim spot of light has been

made even dimmer.

For a long time, physicists tried to explain away the problemswith the classical theory of light as arising from an imperfect under-standing of atoms and the interaction of light with individual atomsand molecules. The ozone paradox, for example, could have beenattributed to the incorrect assumption that the ozone layer was a

smooth, continuous substance, when in reality it was made of indi-vidual ozone molecules. It wasn’t until 1905 that Albert Einsteinthrew down the gauntlet, proposing that the problem had nothing todo with the details of light’s interaction with atoms and everythingto do with the fundamental nature of light itself.

In those days the data were sketchy, the ideas vague, and theexperiments difficult to interpret; it took a genius like Einstein to cutthrough the thicket of confusion and find a simple solution. Today,however, we can get right to the heart of the matter with a piece of ordinary consumer electronics, the digital camera. Instead of film, adigital camera has a computer chip with its surface divided up into a

grid of light-sensitive squares, called “pixels.” Compared to a grainof the silver compound used to make regular photographic film, adigital camera pixel is activated by an amount of light energy ordersof magnitude smaller. We can learn something new about light byusing a digital camera to detect smaller and smaller amounts of light, as shown in figures a/1 through a/3. Figure 1 is fake, but 2and 3 are real digital-camera images made by Prof. Lyman Pageof Princeton University as a classroom demonstration. Figure 1 is

68 Chapter 3 Light as a Particle



b / A water wave is partiaabsorbed.

c / A stream of bullets is ptially absorbed.

what we would see if we used the digital camera to take a picture of a fairly dim source of light. In figures 2 and 3, the intensity of thelight was drastically reduced by inserting semitransparent absorberslike the tinted plastic used in sunglasses. Going from 1 to 2 to 3,more and more light energy is being thrown away by the absorbers.

The results are dramatically different from what we would expect

based on the wave theory of light. If light was a wave and nothingbut a wave, b, then the absorbers would simply cut down the wave’samplitude across the whole wavefront. The digital camera’s entirechip would be illuminated uniformly, and weakening the wave withan absorber would just mean that every pixel would take a long timeto soak up enough energy to register a signal.

But figures a/2 and a/3 show that some pixels take strong hitswhile others pick up no energy at all. Instead of the wave picture,the image that is naturally evoked by the data is something morelike a hail of bullets from a machine gun, c. Each “bullet” of lightapparently carries only a tiny amount of energy, which is why de-

tecting them individually requires a sensitive digital camera ratherthan an eye or a piece of film.

Although Einstein was interpreting different observations, thisis the conclusion he reached in his 1905 paper: that the pure wavetheory of light is an oversimplification, and that the energy of a beamof light comes in finite chunks rather than being spread smoothlythroughout a region of space.

d / Einstein and Seurat: twseparated at birth? Detail frSeine Grande Jatte by GeorgSeurat, 1886.

We now think of these chunks as particles of light, and call them

Section 3.1 Evidence for Light As a Particle



e / Apparatus for observingthe photoelectric effect. A beamof light strikes a capacitor plate

inside a vacuum tube, and elec-trons are ejected (black arrows).

“photons,” although Einstein avoided the word “particle,” and theword “photon” was invented later. Regardless of words, the trou-ble was that waves and particles seemed like inconsistent categories.The reaction to Einstein’s paper could be kindly described as vig-orously skeptical. Even twenty years later, Einstein wrote, “Thereare therefore now two theories of light, both indispensable, and —

as one must admit today despite twenty years of tremendous efforton the part of theoretical physicists — without any logical connec-tion.” In the remainder of this chapter we will learn how the seemingparadox was eventually resolved.


A Suppose someone rebuts the digital camera data in figure a, claim-ing that the random pattern of dots occurs not because of anything fun-damental about the nature of light but simply because the camera’s pixelsare not all exactly the same — some are just more sensitive than others.How could we test this interpretation?

B Discuss how the correspondence principle applies to the observa-

tions and concepts discussed in this section.

3.2 How Much Light Is One Photon?

The photoelectric effect

We have seen evidence that light energy comes in little chunks,so the next question to be asked is naturally how much energy isin one chunk. The most straightforward experimental avenue foraddressing this question is a phenomenon known as the photoelec-tric effect. The photoelectric effect occurs when a photon strikesthe surface of a solid object and knocks out an electron. It occurs

continually all around you. It is happening right now at the surfaceof your skin and on the paper or computer screen from which youare reading these words. It does not ordinarily lead to any observ-able electrical effect, however, because on the average, free electronsare wandering back in just as frequently as they are being ejected.(If an object did somehow lose a significant number of electrons,its growing net positive charge would begin attracting the electronsback more and more strongly.)

Figure e shows a practical method for detecting the photoelec-tric effect. Two very clean parallel metal plates (the electrodes of acapacitor) are sealed inside a vacuum tube, and only one plate is ex-

posed to light. Because there is a good vacuum between the plates,any ejected electron that happens to be headed in the right direc-tion will almost certainly reach the other capacitor plate withoutcolliding with any air molecules.

The illuminated (bottom) plate is left with a net positive charge,and the unilluminated (top) plate acquires a negative charge fromthe electrons deposited on it. There is thus an electric field between




f / The hamster in her hamsball is like an electron emergifrom the metal (tiled kitchen flointo the surrounding vacuu(wood floor). The wood floor

higher than the tiled floor, so

she rolls up the step, the hamswill lose a certain amount kinetic energy, analogous to EIf her kinetic energy is too smashe won’t even make it up tstep.

the plates, and it is because of this field that the electrons’ paths arecurved, as shown in the diagram. However, since vacuum is a goodinsulator, any electrons that reach the top plate are prevented fromresponding to the electrical attraction by jumping back across thegap. Instead they are forced to make their way around the circuit,passing through an ammeter. The ammeter measures the strength

of the photoelectric effect.

An unexpected dependence on frequency

The photoelectric effect was discovered serendipitously by Hein-rich Hertz in 1887, as he was experimenting with radio waves. Hewas not particularly interested in the phenomenon, but he did noticethat the effect was produced strongly by ultraviolet light and moreweakly by lower frequencies. Light whose frequency was lower than acertain critical value did not eject any electrons at all.1 This depen-dence on frequency didn’t make any sense in terms of the classicalwave theory of light. A light wave consists of electric and magneticfields. The stronger the fields, i.e., the greater the wave’s ampli-tude, the greater the forces that would be exerted on electrons thatfound themselves bathed in the light. It should have been amplitude(brightness) that was relevant, not frequency. The dependence onfrequency not only proves that the wave model of light needs mod-ifying, but with the proper interpretation it allows us to determinehow much energy is in one photon, and it also leads to a connec-tion between the wave and particle models that we need in order toreconcile them.

To make any progress, we need to consider the physical processby which a photon would eject an electron from the metal electrode.A metal contains electrons that are free to move around. Ordinarily,

in the interior of the metal, such an electron feels attractive forcesfrom atoms in every direction around it. The forces cancel out. Butif the electron happens to find itself at the surface of the metal,the attraction from the interior side is not balanced out by anyattraction from outside. In popping out through the surface theelectron therefore loses some amount of energy E s, which dependson the type of metal used.

Suppose a photon strikes an electron, annihilating itself and giv-ing up all its energy to the electron.2 The electron will (1) losekinetic energy through collisions with other electrons as it plowsthrough the metal on its way to the surface; (2) lose an amount of

kinetic energy equal to E s as it emerges through the surface; and(3) lose more energy on its way across the gap between the plates,

1In fact this was all prior to Thomson’s discovery of the electron, so Hertz

would not have described the effect in terms of electrons — we are discussing

everything with the benefit of hindsight.2We now know that this is what always happens in the photoelectric effect,

although it had not yet been established in 1905 whether or not the photon was

completely annihilated.

Section 3.2 How Much Light Is One Photon?



g / A different way of study-ing the photoelectric effect.

h / The quantity E s + e ∆V in-dicates the energy of one photon.It is found to be proportional tothe frequency of the light.

due to the electric field between the plates. Even if the electronhappens to be right at the surface of the metal when it absorbs thephoton, and even if the electric field between the plates has not yetbuilt up very much, E s is the bare minimum amount of energy thatthe electron must receive from the photon if it is to contribute toa measurable current. The reason for using very clean electrodes is

to minimize E s and make it have a definite value characteristic of the metal surface, not a mixture of values due to the various typesof dirt and crud that are present in tiny amounts on all surfaces ineveryday life.

We can now interpret the frequency dependence of the photo-electric effect in a simple way: apparently the amount of energypossessed by a photon is related to its frequency. A low-frequencyred or infrared photon has an energy less than E s, so a beam of them will not produce any current. A high-frequency blue or violetphoton, on the other hand, packs enough of a punch to allow anelectron to get out of the electrode. At frequencies higher than the

minimum, the photoelectric current continues to increase with thefrequency of the light because of effects (1) and (3).

Numerical relationship between energy and frequency

Prompted by Einstein’s photon paper, Robert Millikan (whomwe encountered in book 4 of this series) figured out how to use thephotoelectric effect to probe precisely the link between frequencyand photon energy. Rather than going into the historical details of Millikan’s actual experiments (a lengthy experimental program thatoccupied a large part of his professional career) we will describe asimple version, shown in figure g, that is used sometimes in college

laboratory courses. The idea is simply to illuminate one plate of the vacuum tube with light of a single wavelength and monitor thevoltage difference between the two plates as they charge up. Sincethe resistance of a voltmeter is very high (much higher than theresistance of an ammeter), we can assume to a good approximationthat electrons reaching the top plate are stuck there permanently,so the voltage will keep on increasing for as long as electrons aremaking it across the vacuum tube.

At a moment when the voltage difference has a reached a value∆V, the minimum energy required by an electron to make it out of the bottom plate and across the gap to the other plate is E s + e∆V.As ∆V increases, we eventually reach a point at which E s + e∆V equals the energy of one photon. No more electrons can cross thegap, and the reading on the voltmeter stops rising. The quantityE s+e∆V now tells us the energy of one photon. If we determine thisenergy for a variety of frequencies, h, we find the following simplerelationship between the energy of a photon and the frequency of the light:

E = hf ,




where h is a constant with a numerical value of 6.63 × 10−34 J ·s.Note how the equation brings the wave and particle models of lightunder the same roof: the left side is the energy of one particle of light, while the right side is the frequency of the same light, inter-preted as a wave. The constant h is known as Planck’s constant (seehistorical note on page 73).

self-check AHow would you extract h from the graph in figure h? What if you didn’t

even know E s in advance, and could only graph e ∆V versus f ?

Answer, p. 135

Since the energy of a photon is hf , a beam of light can only haveenergies of hf , 2hf , 3hf , etc. Its energy is quantized — there is nosuch thing as a fraction of a photon. Quantum physics gets its namefrom the fact that it quantizes things like energy, momentum, andangular momentum that had previously been thought to be smooth,continuous and infinitely divisible.

Historical NoteWhat I’m presenting in this chapter is a simplified explanation of howthe photon could have been discovered. The actual history is morecomplex. Max Planck (1858-1947) began the photon saga with a the-oretical investigation of the spectrum of light emitted by a hot, glowingobject. He introduced quantization of the energy of light waves, in multi-ples of hf , purely as a mathematical trick that happened to produce theright results. Planck did not believe that his procedure could have anyphysical significance. In his 1905 paper Einstein took Planck’s quantiza-tion as a description of reality, and applied it to various theoretical and

experimental puzzles, including the photoelectric effect. Millikan thensubjected Einstein’s ideas to a series of rigorous experimental tests. Al-though his results matched Einstein’s predictions perfectly, Millikan was

skeptical about photons, and his papers conspicuously omit any refer-ence to them. Only in his autobiography did Millikan rewrite history andclaim that he had given experimental proof for photons.

Number of photons emitted by a lightbulb per second example 1

Roughly how many photons are emitted by a 100-W lightbulb in

1 second?

People tend to remember wavelengths rather than frequencies

for visible light. The bulb emits photons with a range of frequen-

cies and wavelengths, but let’s take 600 nm as a typical wave-

length for purposes of estimation. The energy of a single photon

Section 3.2 How Much Light Is One Photon?



is

E photon = hf

=hc

λ

A power of 100 W means 100 joules per second, so the number

of photons is100 J

E photon =

100 J

hc /λ≈ 3 × 1020 .

Momentum of a photon example 2

According to the theory of relativity, the momentum of a beam

of light is given by p = E /c (see homework problem 11 on page

40). Apply this to find the momentum of a single photon in terms

of its frequency, and in terms of its wavelength.

Combining the equations p = E /c and E = hf , we find

p = E c

=hf

c .

To reexpress this in terms of wavelength, we use c = f λ:

p =hf

f λ

=h

λ

The second form turns out to be simpler.


A The photoelectric effect only ever ejects a very tiny percentage ofthe electrons available near the surface of an object. How well does thisagree with the wave model of light, and how well with the particle model?Consider the two different distance scales involved: the wavelength of thelight, and the size of an atom, which is on the order of 10−10 or 10−9 m.

B What is the significance of the fact that Planck’s constant is numeri-cally very small? How would our everyday experience of light be differentif it was not so small?

C How would the experiments described above be affected if a singleelectron was likely to get hit by more than one photon?

D Draw some representative trajectories of electrons for ∆V = 0, ∆V less than the maximum value, and ∆V greater than the maximum value.

E Explain based on the photon theory of light why ultraviolet light wouldbe more likely than visible or infrared light to cause cancer by damagingDNA molecules. How does this relate to discussion question C?

F Does E = hf imply that a photon changes its energy when it passesfrom one transparent material into another substance with a different in-

dex of refraction?




j / Bullets pass through a doubslit.

k / A water wave passes throu

a double slit.

3.3 Wave-Particle Duality

i / Wave interference pattephotographed by Prof. LymPage with a digital camera. Las

light with a single well-defin

wavelength passed through series of absorbers to cut doits intensity, then through a setslits to produce interference, afinally into a digital camera ch(A triple slit was actually usbut for conceptual simplicity discuss the results in the mtext as if it was a double slit.)

panel 2 the intensity has bereduced relative to 1, and evmore so for panel 3.

How can light be both a particle and a wave? We are nowready to resolve this seeming contradiction. Often in science whensomething seems paradoxical, it’s because we either don’t define ourterms carefully, or don’t test our ideas against any specific real-worldsituation. Let’s define particles and waves as follows:

• Waves exhibit superposition, and specifically interference phe-nomena.

• Particles can only exist in whole numbers, not fractions

As a real-world check on our philosophizing, there is one partic-ular experiment that works perfectly. We set up a double-slit inter-ference experiment that we know will produce a diffraction patternif light is an honest-to-goodness wave, but we detect the light witha detector that is capable of sensing individual photons, e.g., a dig-ital camera. To make it possible to pick out individual dots fromindividual photons, we must use filters to cut down the intensity of the light to a very low level, just as in the photos by Prof. Page insection 3.1. The whole thing is sealed inside a light-tight box. Theresults are shown in figure i. (In fact, the similar figures in section3.1 are simply cutouts from these figures.)

Neither the pure wave theory nor the pure particle theory canexplain the results. If light was only a particle and not a wave, therewould be no interference effect. The result of the experiment wouldbe like firing a hail of bullets through a double slit, j. Only twospots directly behind the slits would be hit.

If, on the other hand, light was only a wave and not a particle,we would get the same kind of diffraction pattern that would happen

Section 3.3 Wave-Particle Duality



l / A single photon can gothrough both slits.

with a water wave, k. There would be no discrete dots in the photo,only a diffraction pattern that shaded smoothly between light anddark.

Applying the definitions to this experiment, light must be botha particle and a wave. It is a wave because it exhibits interferenceeffects. At the same time, the fact that the photographs contain

discrete dots is a direct demonstration that light refuses to be splitinto units of less than a single photon. There can only be wholenumbers of photons: four photons in figure i/3, for example.

A wrong interpretation: photons interfering with each other

One possible interpretation of wave-particle duality that occurredto physicists early in the game was that perhaps the interference ef-fects came from photons interacting with each other. By analogy, awater wave consists of moving water molecules, and interference of water waves results ultimately from all the mutual pushes and pullsof the molecules. This interpretation was conclusively disproved by

G.I. Taylor, a student at Cambridge. The demonstration by Prof.Page that we’ve just been discussing is essentially a modernizedversion of Taylor’s work. Taylor reasoned that if interference effectscame from photons interacting with each other, a bare minimum of two photons would have to be present at the same time to produceinterference. By making the light source extremely dim, we can bevirtually certain that there are never two photons in the box at thesame time. In figure i, the intensity of the light has been cut downso much by the absorbers that if it was in the open, the averageseparation between photons would be on the order of a kilometer!At any given moment, the number of photons in the box is most

likely to be zero. It is virtually certain that there were never twophotons in the box at once.

The concept of a photon’s path is undefined.

If a single photon can demonstrate double-slit interference, thenwhich slit did it pass through? The unavoidable answer must be thatit passes through both! This might not seem so strange if we thinkof the photon as a wave, but it is highly counterintuitive if we tryto visualize it as a particle. The moral is that we should not thinkin terms of the path of a photon. Like the fully human and fullydivine Jesus of Christian theology, a photon is supposed to be 100%wave and 100% particle. If a photon had a well defined path, then it

would not demonstrate wave superposition and interference effects,contradicting its wave nature. (In the next chapter we will discussthe Heisenberg uncertainty principle, which gives a numerical wayof approaching this issue.)




Another wrong interpretation: the pilot wave hypothesis

A second possible explanation of wave-particle duality was takenseriously in the early history of quantum mechanics. What if thephoton particle is like a surfer riding on top of its accompanyingwave? As the wave travels along, the particle is pushed, or “piloted”by it. Imagining the particle and the wave as two separate entities

allows us to avoid the seemingly paradoxical idea that a photon isboth at once. The wave happily does its wave tricks, like super-position and interference, and the particle acts like a respectableparticle, resolutely refusing to be in two different places at once. If the wave, for instance, undergoes destructive interference, becomingnearly zero in a particular region of space, then the particle simplyis not guided into that region.

The problem with the pilot wave interpretation is that the onlyway it can be experimentally tested or verified is if someone managesto detach the particle from the wave, and show that there really aretwo entities involved, not just one. Part of the scientific method is

that hypotheses are supposed to be experimentally testable. Sincenobody has ever managed to separate the wavelike part of a photonfrom the particle part, the interpretation is not useful or meaningfulin a scientific sense.

The probability interpretation

The correct interpretation of wave-particle duality is suggestedby the random nature of the experiment we’ve been discussing: eventhough every photon wave/particle is prepared and released in thesame way, the location at which it is eventually detected by thedigital camera is different every time. The idea of the probability

interpretation of wave-particle duality is that the location of thephoton-particle is random, but the probability that it is in a certainlocation is higher where the photon-wave’s amplitude is greater.

More specifically, the probability distribution of the particle mustbe proportional to the square of the wave’s amplitude,

(probability distribution) ∝ (amplitude)2 .

This follows from the correspondence principle and from the factthat a wave’s energy density is proportional to the square of its am-plitude. If we run the double-slit experiment for a long enough time,the pattern of dots fills in and becomes very smooth as would have

been expected in classical physics. To preserve the correspondencebetween classical and quantum physics, the amount of energy de-posited in a given region of the picture over the long run must beproportional to the square of the wave’s amplitude. The amount of energy deposited in a certain area depends on the number of pho-




fields are nearly constant throughout it. We then have

P =

1

8πk |E|2 + 1

2µo|B|2

v

hf .

We can simplify this formidable looking expression by recogniz-

ing that in an electromagnetic wave, |E| and |B| are related by|E| = c |B|. With some algebra, it turns out that the electric and

magnetic fields each contribute half the total energy, so we can

simplify this to

P = 2

1

8πk |E|2

v

hf

=v

4π khf |E|2 .

As advertised, the probability is proportional to the square of the

wave’s amplitude.


A In example 3 on page 78, about the carrot in the microwave oven,show that it would be nonsensical to have probability be proportional tothe field itself, rather than the square of the field.

B Einstein did not try to reconcile the wave and particle theories oflight, and did not say much about their apparent inconsistency. Einsteinbasically visualized a beam of light as a stream of bullets coming froma machine gun. In the photoelectric effect, a photon “bullet” would onlyhit one atom, just as a real bullet would only hit one person. Supposesomeone reading his 1905 paper wanted to interpret it by saying thatEinstein’s so-called particles of light are simply short wave-trains that only

occupy a small region of space. Comparing the wavelength of visible light

(a few hundred nm) to the size of an atom (on the order of 0.1 nm), explainwhy this poses a difficulty for reconciling the particle and wave theories.

C Can a white photon exist?

D In double-slit diffraction of photons, would you get the same patternof dots on the digital camera image if you covered one slit? Why should itmatter whether you give the photon two choices or only one?




Chapter 1

Electricity and the Atom

Where the telescope ends, the microscope begins. Which of the twohas the grander view? Victor Hugo

His father died during his mother’s pregnancy. Rejected by heras a boy, he was packed off to boarding school when she remarried.

He himself never married, but in middle age he formed an intenserelationship with a much younger man, a relationship that he ter-minated when he underwent a psychotic break. Following his earlyscientific successes, he spent the rest of his professional life mostlyin frustration over his inability to unlock the secrets of alchemy.

The man being described is Isaac Newton, but not the triumphantNewton of the standard textbook hagiography. Why dwell on thesad side of his life? To the modern science educator, Newton’s life-long obsession with alchemy may seem an embarrassment, a distrac-tion from his main achievement, the creation the modern science of mechanics. To Newton, however, his alchemical researches were nat-

urally related to his investigations of force and motion. What wasradical about Newton’s analysis of motion was its universality: itsucceeded in describing both the heavens and the earth with thesame equations, whereas previously it had been assumed that thesun, moon, stars, and planets were fundamentally different fromearthly objects. But Newton realized that if science was to describeall of nature in a unified way, it was not enough to unite the humanscale with the scale of the universe: he would not be satisfied until



he fit the microscopic universe into the picture as well.

It should not surprise us that Newton failed. Although he was afirm believer in the existence of atoms, there was no more experimen-tal evidence for their existence than there had been when the ancientGreeks first posited them on purely philosophical grounds. Alchemylabored under a tradition of secrecy and mysticism. Newton had

already almost single-handedly transformed the fuzzyheaded fieldof “natural philosophy” into something we would recognize as themodern science of physics, and it would be unjust to criticize himfor failing to change alchemy into modern chemistry as well. Thetime was not ripe. The microscope was a new invention, and it wascutting-edge science when Newton’s contemporary Hooke discoveredthat living things were made out of cells.

1.1 The Quest for the Atomic Force

Newton was not the first of the age of reason. He was the last of

the magicians. John Maynard Keynes

Nevertheless it will be instructive to pick up Newton’s train of thought and see where it leads us with the benefit of modern hind-sight. In uniting the human and cosmic scales of existence, he hadreimagined both as stages on which the actors were objects (treesand houses, planets and stars) that interacted through attractionsand repulsions. He was already convinced that the objects inhab-iting the microworld were atoms, so it remained only to determinewhat kinds of forces they exerted on each other.

His next insight was no less brilliant for his inability to bring it tofruition. He realized that the many human-scale forces — friction,sticky forces, the normal forces that keep objects from occupyingthe same space, and so on — must all simply be expressions of amore fundamental force acting between atoms. Tape sticks to paperbecause the atoms in the tape attract the atoms in the paper. Myhouse doesn’t fall to the center of the earth because its atoms repelthe atoms of the dirt under it.

Here he got stuck. It was tempting to think that the atomic forcewas a form of gravity, which he knew to be universal, fundamental,and mathematically simple. Gravity, however, is always attractive,so how could he use it to explain the existence of both attractive

and repulsive atomic forces? The gravitational force between objects of ordinary size is also extremely small, which is why we nevernotice cars and houses attracting us gravitationally. It would behard to understand how gravity could be responsible for anythingas vigorous as the beating of a heart or the explosion of gunpowder.Newton went on to write a million words of alchemical notes filledwith speculation about some other force, perhaps a “divine force” or“vegetative force” that would for example be carried by the sperm

14 Chapter 1 Electricity and the Atom



a / Four pieces of tape aprepared, 1, as described in ttext. Depending on which cobination is tested, the interactican be either repulsive, 2, attractive, 3.

to the egg.

Luckily, we now know enough to investigate a different suspectas a candidate for the atomic force: electricity. Electric forces areoften observed between objects that have been prepared by rubbing(or other surface interactions), for instance when clothes rub againsteach other in the dryer. A useful example is shown in figure a/1:

stick two pieces of tape on a tabletop, and then put two more pieceson top of them. Lift each pair from the table, and then separatethem. The two top pieces will then repel each other, a/2, as willthe two bottom pieces. A bottom piece will attract a top piece,however, a/3. Electrical forces like these are similar in certain waysto gravity, the other force that we already know to be fundamental:

• Electrical forces are universal . Although some substances,such as fur, rubber, and plastic, respond more strongly toelectrical preparation than others, all matter participates inelectrical forces to some degree. There is no such thing as a

“nonelectric” substance. Matter is both inherently gravita-tional and inherently electrical.

• Experiments show that the electrical force, like the gravita-tional force, is an inverse square force. That is, the electricalforce between two spheres is proportional to 1/r2, where r isthe center-to-center distance between them.

Furthermore, electrical forces make more sense than gravity ascandidates for the fundamental force between atoms, because wehave observed that they can be either attractive or repulsive.

1.2 Charge, Electricity and Magnetism

Charge

“Charge” is the technical term used to indicate that an objecthas been prepared so as to participate in electrical forces. This isto be distinguished from the common usage, in which the term isused indiscriminately for anything electrical. For example, althoughwe speak colloquially of “charging” a battery, you may easily verifythat a battery has no charge in the technical sense, e.g., it does notexert any electrical force on a piece of tape that has been preparedas described in the previous section.

Two types of charge

We can easily collect reams of data on electrical forces betweendifferent substances that have been charged in different ways. Wefind for example that cat fur prepared by rubbing against rabbitfur will attract glass that has been rubbed on silk. How can wemake any sense of all this information? A vast simplification isachieved by noting that there are really only two types of charge.

Section 1.2 Charge, Electricity and Magnetism



Suppose we pick cat fur rubbed on rabbit fur as a representative of type A, and glass rubbed on silk for type B. We will now find thatthere is no “type C.” Any object electrified by any method is eitherA-like, attracting things A attracts and repelling those it repels, orB-like, displaying the same attractions and repulsions as B. The twotypes, A and B, always display opposite interactions. If A displays

an attraction with some charged object, then B is guaranteed toundergo repulsion with it, and vice-versa.

The coulomb

Although there are only two types of charge, each type can comein different amounts. The metric unit of charge is the coulomb(rhymes with “drool on”), defined as follows:

One Coulomb (C) is the amount of charge such that a force of 9.0×109 N occurs between two pointlike objects with chargesof 1 C separated by a distance of 1 m.

The notation for an amount of charge is q. The numerical factor

in the definition is historical in origin, and is not worth memoriz-ing. The definition is stated for pointlike, i.e., very small, ob jects,because otherwise different parts of them would be at different dis-tances from each other.

A model of two types of charged particles

Experiments show that all the methods of rubbing or otherwisecharging objects involve two objects, and both of them end up get-ting charged. If one object acquires a certain amount of one type of charge, then the other ends up with an equal amount of the othertype. Various interpretations of this are possible, but the simplest

is that the basic building blocks of matter come in two flavors, onewith each type of charge. Rubbing objects together results in thetransfer of some of these particles from one object to the other. Inthis model, an object that has not been electrically prepared may ac-tually possesses a great deal of both types of charge, but the amountsare equal and they are distributed in the same way throughout it.Since type A repels anything that type B attracts, and vice versa,the object will make a total force of zero on any other object. Therest of this chapter fleshes out this model and discusses how thesemysterious particles can be understood as being internal parts of atoms.

Use of positive and negative signs for charge

Because the two types of charge tend to cancel out each other’sforces, it makes sense to label them using positive and negative signs,and to discuss the total charge of an object. It is entirely arbitrarywhich type of charge to call negative and which to call positive.Benjamin Franklin decided to describe the one we’ve been calling“A” as negative, but it really doesn’t matter as long as everyone is




consistent with everyone else. An object with a total charge of zero(equal amounts of both types) is referred to as electrically neutral.

self-check A

Criticize the following statement: “There are two types of charge, attrac-

tive and repulsive.” Answer, p.

205

Coulomb’s law

A large body of experimental observations can be summarizedas follows:

Coulomb’s law: The magnitude of the force acting betweenpointlike charged objects at a center-to-center distance r is givenby the equation

|F| = k|q1||q2|

r2,

where the constant k equals 9.0×109 N·m2/C2. The force is attrac-

tive if the charges are of different signs, and repulsive if they havethe same sign.

Clever modern techniques have allowed the 1/r2 form of Coulomb’slaw to be tested to incredible accuracy, showing that the exponentis in the range from 1.9999999999999998 to 2.0000000000000002.

Note that Coulomb’s law is closely analogous to Newton’s lawof gravity, where the magnitude of the force is Gm1m2/r

2, exceptthat there is only one type of mass, not two, and gravitational forcesare never repulsive. Because of this close analogy between the two

types of forces, we can recycle a great deal of our knowledge of gravitational forces. For instance, there is an electrical equivalentof the shell theorem: the electrical forces exerted externally by auniformly charged spherical shell are the same as if all the chargewas concentrated at its center, and the forces exerted internally arezero.

Conservation of charge

An even more fundamental reason for using positive and nega-tive signs for electrical charge is that experiments show that chargeis conserved according to this definition: in any closed system, thetotal amount of charge is a constant. This is why we observe that

rubbing initially uncharged substances together always has the re-sult that one gains a certain amount of one type of charge, whilethe other acquires an equal amount of the other type. Conservationof charge seems natural in our model in which matter is made of positive and negative particles. If the charge on each particle is afixed property of that type of particle, and if the particles themselvescan be neither created nor destroyed, then conservation of charge isinevitable.




b / A charged piece of tapeattracts uncharged pieces ofpaper from a distance, and theyleap up to it.

c / The paper has zero totalcharge, but it does have chargedparticles in it that can move.

Electrical forces involving neutral objects

As shown in figure b, an electrically charged object can attractobjects that are uncharged. How is this possible? The key is thateven though each piece of paper has a total charge of zero, it has atleast some charged particles in it that have some freedom to move.Suppose that the tape is positively charged, c. Mobile particles

in the paper will respond to the tape’s forces, causing one end of the paper to become negatively charged and the other to becomepositive. The attraction is between the paper and the tape is nowstronger than the repulsion, because the negatively charged end iscloser to the tape.

self-check B

What would have happened if the tape was negatively charged?

Answer, p. 205

The path ahead

We have begun to encounter complex electrical behavior that we

would never have realized was occurring just from the evidence of oureyes. Unlike the pulleys, blocks, and inclined planes of mechanics,the actors on the stage of electricity and magnetism are invisiblephenomena alien to our everyday experience. For this reason, theflavor of the second half of your physics education is dramaticallydifferent, focusing much more on experiments and techniques. Eventhough you will never actually see charge moving through a wire,you can learn to use an ammeter to measure the flow.

Students also tend to get the impression from their first semesterof physics that it is a dead science. Not so! We are about to pickup the historical trail that leads directly to the cutting-edge physics

research you read about in the newspaper. The atom-smashing ex-periments that began around 1900, which we will be studying in thischapter, were not that different from the ones of the year 2000 — just smaller, simpler, and much cheaper.

Magnetic forces

A detailed mathematical treatment of magnetism won’t comeuntil much later in this book, but we need to develop a few simpleideas about magnetism now because magnetic forces are used in theexperiments and techniques we come to next. Everyday magnetscome in two general types. Permanent magnets, such as the ones

on your refrigerator, are made of iron or substances like steel thatcontain iron atoms. (Certain other substances also work, but ironis the cheapest and most common.) The other type of magnet,an example of which is the ones that make your stereo speakersvibrate, consist of coils of wire through which electric charge flows.Both types of magnets are able to attract iron that has not been




magnetically prepared, for instance the door of the refrigerator.

A single insight makes these apparently complex phenomenamuch simpler to understand: magnetic forces are interactions be-tween moving charges, occurring in addition to the electric forces.Suppose a permanent magnet is brought near a magnet of the coiled-wire type. The coiled wire has moving charges in it because we force

charge to flow. The permanent magnet also has moving charges init, but in this case the charges that naturally swirl around inside theiron. (What makes a magnetized piece of iron different from a blockof wood is that the motion of the charge in the wood is randomrather than organized.) The moving charges in the coiled-wire mag-net exert a force on the moving charges in the permanent magnet,and vice-versa.

The mathematics of magnetism is significantly more complexthan the Coulomb force law for electricity, which is why we willwait until chapter 6 before delving deeply into it. Two simple factswill suffice for now:

(1) If a charged particle is moving in a region of space near whereother charged particles are also moving, their magnetic force on itis directly proportional to its velocity.

(2) The magnetic force on a moving charged particle is alwaysperpendicular to the direction the particle is moving.

A magnetic compass example 1

The Earth is molten inside, and like a pot of boiling water, it roils

and churns. To make a drastic oversimplification, electric charge

can get carried along with the churning motion, so the Earth con-

tains moving charge. The needle of a magnetic compass is itself

a small permanent magnet. The moving charge inside the earthinteracts magnetically with the moving charge inside the compass

needle, causing the compass needle to twist around and point

north.

A television tube example 2

A TV picture is painted by a stream of electrons coming from

the back of the tube to the front. The beam scans across the

whole surface of the tube like a reader scanning a page of a book.

Magnetic forces are used to steer the beam. As the beam comes

from the back of the tube to the front, up-down and left-right forces

are needed for steering. But magnetic forces cannot be used

to get the beam up to speed in the first place, since they can

only push perpendicular to the electrons’ direction of motion, not

forward along it.


A If the electrical attraction between two pointlike objects at a distanceof 1 m is 9×109 N, why can’t we infer that their charges are +1 and −1 C?What further observations would we need to do in order to prove this?




B An electrically charged piece of tape will be attracted to your hand.Does that allow us to tell whether the mobile charged particles in yourhand are positive or negative, or both?

1.3 Atoms

I was brought up to look at the atom as a nice, hard fellow, red orgrey in color according to taste. Rutherford

Atomism

The Greeks have been kicked around a lot in the last couple of millennia: dominated by the Romans, bullied during the crusadesby warlords going to and from the Holy Land, and occupied byTurkey until recently. It’s no wonder they prefer to remember theirsalad days, when their best thinkers came up with concepts likedemocracy and atoms. Greece is democratic again after a periodof military dictatorship, and an atom is proudly pictured on one of their coins. That’s why it hurts me to have to say that the ancient

Greek hypothesis that matter is made of atoms was pure guess-work. There was no real experimental evidence for atoms, and the18th-century revival of the atom concept by Dalton owed little tothe Greeks other than the name, which means “unsplittable.” Sub-tracting even more cruelly from Greek glory, the name was shownto be inappropriate in 1897 when physicist J.J. Thomson proved ex-perimentally that atoms had even smaller things inside them, whichcould be extracted. (Thomson called them “electrons.”) The “un-splittable” was splittable after all.

But that’s getting ahead of our story. What happened to theatom concept in the intervening two thousand years? Educated peo-

ple continued to discuss the idea, and those who were in favor of itcould often use it to give plausible explanations for various facts andphenomena. One fact that was readily explained was conservationof mass. For example, if you mix 1 kg of water with 1 kg of dirt,you get exactly 2 kg of mud, no more and no less. The same is truefor the a variety of processes such as freezing of water, fermentingbeer, or pulverizing sandstone. If you believed in atoms, conserva-tion of mass made perfect sense, because all these processes couldbe interpreted as mixing and rearranging atoms, without changingthe total number of atoms. Still, this is nothing like a proof thatatoms exist.

If atoms did exist, what types of atoms were there, and what dis-tinguished the different types from each other? Was it their sizes,their shapes, their weights, or some other quality? The chasm be-tween the ancient and modern atomisms becomes evident when weconsider the wild speculations that existed on these issues until thepresent century. The ancients decided that there were four types of atoms, earth, water, air and fire; the most popular view was that




they were distinguished by their shapes. Water atoms were spher-ical, hence water’s ability to flow smoothly. Fire atoms had sharppoints, which was why fire hurt when it touched one’s skin. (Therewas no concept of temperature until thousands of years later.) Thedrastically different modern understanding of the structure of atomswas achieved in the course of the revolutionary decade stretching

1895 to 1905. The main purpose of this chapter is to describe thosemomentous experiments.

Are you now or have you ever been an atomist?“You are what you eat.” The glib modern phrase more or less assumesthe atomic explanation of digestion. After all, digestion was pretty mys-

terious in ancient times, and premodern cultures would typically believethat eating allowed you to extract some kind of “life force” from the food.Myths abound to the effect that abstract qualities such as bravery orritual impurity can enter your body via the food you eat. In contrast tothese supernatural points of view, the ancient atomists had an entirelynaturalistic interpretation of digestion. The food was made of atoms,and when you digested it you were simply extracting some atoms from

it and rearranging them into the combinations required for your ownbody tissues. The more progressive medieval and renaissance scien-tists loved this kind of explanation. They were anxious to drive a stake

through the heart of Aristotelian physics (and its embellished, Church-friendly version, scholasticism), which in their view ascribed too manyoccult properties and “purposes” to objects. For instance, the Aris-totelian explanation for why a rock would fall to earth was that it wasits “nature” or “purpose” to come to rest on the ground.

The seemingly innocent attempt to explain digestion naturalistically,however, ended up getting the atomists in big trouble with the Church.The problem was that the Church’s most important sacrament involveseating bread and wine and thereby receiving the supernatural effect offorgiveness of sin. In connection with this ritual, the doctrine of transub-stantiation asserts that the blessing of the eucharistic bread and wineliterally transforms it into the blood and flesh of Christ. Atomism was

perceived as contradicting transubstantiation, since atomism seemedto deny that the blessing could change the nature of the atoms. Al-though the historical information given in most science textbooks aboutGalileo represents his run-in with the Inquisition as turning on the issueof whether the earth moves, some historians believe his punishmenthad more to do with the perception that his advocacy of atomism sub-verted transubstantiation. (Other issues in the complex situation wereGalileo’s confrontational style, Pope Urban’s military problems, and ru-mors that the stupid character in Galileo’s dialogues was meant to bethe Pope.) For a long time, belief in atomism served as a badge of

nonconformity for scientists, a way of asserting a preference for naturalrather than supernatural interpretations of phenomena. Galileo andNewton’s espousal of atomism was an act of rebellion, like later gener-ations’ adoption of Darwinism or Marxism.

Another conflict between scholasticism and atomism came from thequestion of what was between the atoms. If you ask modern people thisquestion, they will probably reply “nothing” or “empty space.” But Aris-totle and his scholastic successors believed that there could be no suchthing as empty space, i.e., a vacuum. That was not an unreasonable

Section 1.3 Atoms



point of view, because air tends to rush in to any space you open up,and it wasn’t until the renaissance that people figured out how to makea vacuum.

Atoms, light, and everything else

Although I tend to ridicule ancient Greek philosophers like Aris-totle, let’s take a moment to praise him for something. If you readAristotle’s writings on physics (or just skim them, which is all I’vedone), the most striking thing is how careful he is about classifyingphenomena and analyzing relationships among phenomena. The hu-man brain seems to naturally make a distinction between two typesof physical phenomena: objects and motion of objects. When aphenomenon occurs that does not immediately present itself as oneof these, there is a strong tendency to conceptualize it as one orthe other, or even to ignore its existence completely. For instance,physics teachers shudder at students’ statements that “the dynamiteexploded, and force came out of it in all directions.” In these exam-ples, the nonmaterial concept of force is being mentally categorized

as if it was a physical substance. The statement that “winding theclock stores motion in the spring” is a miscategorization of electricalenergy as a form of motion. An example of ignoring the existenceof a phenomenon altogether can be elicited by asking people whywe need lamps. The typical response that “the lamp illuminatesthe room so we can see things,” ignores the necessary role of lightcoming into our eyes from the things being illuminated.

If you ask someone to tell you briefly about atoms, the likelyresponse is that “everything is made of atoms,” but we’ve now seenthat it’s far from obvious which “everything” this statement wouldproperly refer to. For the scientists of the early 1900s who were

trying to investigate atoms, this was not a trivial issue of defini-tions. There was a new gizmo called the vacuum tube, of which theonly familiar example today is the picture tube of a TV. In shortorder, electrical tinkerers had discovered a whole flock of new phe-nomena that occurred in and around vacuum tubes, and given thempicturesque names like “x-rays,” “cathode rays,” “Hertzian waves,”and “N-rays.” These were the types of observations that ended uptelling us that we know about matter, but fierce controversies ensuedover whether these were themselves forms of matter.

Let’s bring ourselves up to the level of classification of phenom-ena employed by physicists in the year 1900. They recognized threecategories:

• Matter has mass, can have kinetic energy, and can travelthrough a vacuum, transporting its mass and kinetic energywith it. Matter is conserved, both in the sense of conservationof mass and conservation of the number of atoms of each ele-ment. Atoms can’t occupy the same space as other atoms, so




mHe

mH

= 3.97

mNe

mH

= 20.01

mSc

mH

= 44.60

d / Examples of masseatoms compared to that of gen. Note how some, but nare close to integers.

a convenient way to prove something is not a form of matteris to show that it can pass through a solid material, in whichthe atoms are packed together closely.

• Light has no mass, always has energy, and can travel through avacuum, transporting its energy with it. Two light beams canpenetrate through each other and emerge from the collision

without being weakened, deflected, or affected in any otherway. Light can penetrate certain kinds of matter, e.g., glass.

• The third category is everything that doesn’t fit the defini-tion of light or matter. This catch-all category includes, forexample, time, velocity, heat, and force.

The chemical elements

How would one find out what types of atoms there were? To-day, it doesn’t seem like it should have been very difficult to workout an experimental program to classify the types of atoms. For

each type of atom, there should be a corresponding element, i.e., apure substance made out of nothing but that type of atom. Atomsare supposed to be unsplittable, so a substance like milk could notpossibly be elemental, since churning it vigorously causes it to splitup into two separate substances: butter and whey. Similarly, rustcould not be an element, because it can be made by combining twosubstances: iron and oxygen. Despite its apparent reasonableness,no such program was carried out until the eighteenth century. Theancients presumably did not do it because observation was not uni-versally agreed on as the right way to answer questions about nature,and also because they lacked the necessary techniques or the tech-

niques were the province of laborers with low social status, such assmiths and miners. Alchemists were hindered by atomism’s repu-tation for subversiveness, and by a tendency toward mysticism andsecrecy. (The most celebrated challenge facing the alchemists, thatof converting lead into gold, is one we now know to be impossible,since lead and gold are both elements.)

By 1900, however, chemists had done a reasonably good job of finding out what the elements were. They also had determined theratios of the different atoms’ masses fairly accurately. A typicaltechnique would be to measure how many grams of sodium (Na)would combine with one gram of chlorine (Cl) to make salt (NaCl).

(This assumes you’ve already decided based on other evidence thatsalt consisted of equal numbers of Na and Cl atoms.) The masses of individual atoms, as opposed to the mass ratios, were known onlyto within a few orders of magnitude based on indirect evidence, andplenty of physicists and chemists denied that individual atoms wereanything more than convenient symbols.

The following table gives the atomic masses of all the elements,on a standard scale in which the mass of hydrogen is very close to 1.0.

Section 1.3 Atoms



The absolute calibration of the whole scale was only very roughlyknown for a long time, but was eventually tied down, with the massof a hydrogen atom being determined to be about 1.7 × 10−27 kg.

Ag 107.9 Eu 152.0 Mo 95.9 Sc 45.0Al 27.0 F 19.0 N 14.0 Se 79.0Ar 39.9 Fe 55.8 Na 23.0 Si 28.1As 74.9 Ga 69.7 Nb 92.9 Sn 118.7Au 197.0 Gd 157.2 Nd 144.2 Sr 87.6B 10.8 Ge 72.6 Ne 20.2 Ta 180.9Ba 137.3 H 1.0 Ni 58.7 Tb 158.9Be 9.0 He 4.0 O 16.0 Te 127.6Bi 209.0 Hf 178.5 Os 190.2 Ti 47.9Br 79.9 Hg 200.6 P 31.0 Tl 204.4C 12.0 Ho 164.9 Pb 207.2 Tm 168.9Ca 40.1 In 114.8 Pd 106.4 U 238Ce 140.1 Ir 192.2 Pt 195.1 V 50.9Cl 35.5 K 39.1 Pr 140.9 W 183.8Co 58.9 Kr 83.8 Rb 85.5 Xe 131.3Cr 52.0 La 138.9 Re 186.2 Y 88.9Cs 132.9 Li 6.9 Rh 102.9 Yb 173.0Cu 63.5 Lu 175.0 Ru 101.1 Zn 65.4Dy 162.5 Mg 24.3 S 32.1 Zr 91.2Er 167.3 Mn 54.9 Sb 121.8

Making sense of the elements

As the information accumulated, the challenge was to find away of systematizing it; the modern scientist’s aesthetic sense rebelsagainst complication. This hodgepodge of elements was an embar-rassment. One contemporary observer, William Crookes, describedthe elements as extending “before us as stretched the wide Atlanticbefore the gaze of Columbus, mocking, taunting and murmuringstrange riddles, which no man has yet been able to solve.” It wasn’t

long before people started recognizing that many atoms’ masses werenearly integer multiples of the mass of hydrogen, the lightest ele-ment. A few excitable types began speculating that hydrogen wasthe basic building block, and that the heavier elements were madeof clusters of hydrogen. It wasn’t long, however, before their paradewas rained on by more accurate measurements, which showed thatnot all of the elements had atomic masses that were near integermultiples of hydrogen, and even the ones that were close to beinginteger multiples were off by one percent or so.

Chemistry professor Dmitri Mendeleev, preparing his lectures in1869, wanted to find some way to organize his knowledge for his stu-

dents to make it more understandable. He wrote the names of allthe elements on cards and began arranging them in different wayson his desk, trying to find an arrangement that would make sense of the muddle. The row-and-column scheme he came up with is essen-tially our modern periodic table. The columns of the modern versionrepresent groups of elements with similar chemical properties, andeach row is more massive than the one above it. Going across eachrow, this almost always resulted in placing the atoms in sequence




e / A modern periodic table. Ements in the same column hasimilar chemical properties. Tmodern atomic numbers, dcussed in section 2.3, were

known in Mendeleev’s time, sinthe table could be flipped in va

ous ways.

by weight as well. What made the system significant was its predic-tive value. There were three places where Mendeleev had to leavegaps in his checkerboard to keep chemically similar elements in the

same column. He predicted that elements would exist to fill thesegaps, and extrapolated or interpolated from other elements in thesame column to predict their numerical properties, such as masses,boiling points, and densities. Mendeleev’s professional stock sky-rocketed when his three elements (later named gallium, scandiumand germanium) were discovered and found to have very nearly theproperties he had predicted.

One thing that Mendeleev’s table made clear was that mass wasnot the basic property that distinguished atoms of different ele-ments. To make his table work, he had to deviate from orderingthe elements strictly by mass. For instance, iodine atoms are lighter

than tellurium, but Mendeleev had to put iodine after tellurium sothat it would lie in a column with chemically similar elements.

Direct proof that atoms existed

The success of the kinetic theory of heat was taken as strong evi-dence that, in addition to the motion of any object as a whole, thereis an invisible type of motion all around us: the random motion of atoms within each object. But many conservatives were not con-vinced that atoms really existed. Nobody had ever seen one, afterall. It wasn’t until generations after the kinetic theory of heat wasdeveloped that it was demonstrated conclusively that atoms really

existed and that they participated in continuous motion that neverdied out.

The smoking gun to prove atoms were more than mathematicalabstractions came when some old, obscure observations were reex-amined by an unknown Swiss patent clerk named Albert Einstein.A botanist named Brown, using a microscope that was state of theart in 1827, observed tiny grains of pollen in a drop of water on amicroscope slide, and found that they jumped around randomly for

Section 1.3 Atoms



f / A young Robert Millikan.

g / A simplified diagram ofMillikan’s apparatus.

no apparent reason. Wondering at first if the pollen he’d assumed tobe dead was actually alive, he tried looking at particles of soot, andfound that the soot particles also moved around. The same resultswould occur with any small grain or particle suspended in a liquid.The phenomenon came to be referred to as Brownian motion, andits existence was filed away as a quaint and thoroughly unimportant

fact, really just a nuisance for the microscopist.It wasn’t until 1906 that Einstein found the correct interpreta-

tion for Brown’s observation: the water molecules were in continuousrandom motion, and were colliding with the particle all the time,kicking it in random directions. After all the millennia of speculationabout atoms, at last there was solid proof. Einstein’s calculationsdispelled all doubt, since he was able to make accurate predictionsof things like the average distance traveled by the particle in a cer-tain amount of time. (Einstein received the Nobel Prize not for histheory of relativity but for his papers on Brownian motion and thephotoelectric effect.)


A How could knowledge of the size of an individual aluminum atom beused to infer an estimate of its mass, or vice versa?

B How could one test Einstein’s interpretation of Brownian motion byobserving it at different temperatures?

1.4 Quantization of Charge

Proving that atoms actually existed was a big accomplishment, butdemonstrating their existence was different from understanding theirproperties. Note that the Brown-Einstein observations had nothingat all to do with electricity, and yet we know that matter is inher-ently electrical, and we have been successful in interpreting certainelectrical phenomena in terms of mobile positively and negativelycharged particles. Are these particles atoms? Parts of atoms? Par-ticles that are entirely separate from atoms? It is perhaps prema-ture to attempt to answer these questions without any conclusiveevidence in favor of the charged-particle model of electricity.

Strong support for the charged-particle model came from a 1911experiment by physicist Robert Millikan at the University of Chicago.Consider a jet of droplets of perfume or some other liquid made byblowing it through a tiny pinhole. The droplets emerging from the

pinhole must be smaller than the pinhole, and in fact most of themare even more microscopic than that, since the turbulent flow of airtends to break them up. Millikan reasoned that the droplets wouldacquire a little bit of electric charge as they rubbed against the chan-nel through which they emerged, and if the charged-particle modelof electricity was right, the charge might be split up among so manyminuscule liquid drops that a single drop might have a total charge




amounting to an excess of only a few charged particles — perhapsan excess of one positive particle on a certain drop, or an excess of two negative ones on another.

Millikan’s ingenious apparatus, g, consisted of two metal plates,which could be electrically charged as needed. He sprayed a cloud of oil droplets into the space between the plates, and selected one drop

through a microscope for study. First, with no charge on the plates,he would determine the drop’s mass by letting it fall through theair and measuring its terminal velocity, i.e., the velocity at whichthe force of air friction canceled out the force of gravity. The forceof air drag on a slowly moving sphere had already been found byexperiment to be bvr2, where b was a constant. Setting the totalforce equal to zero when the drop is at terminal velocity gives

bvr2 −mg = 0 ,

and setting the known density of oil equal to the drop’s mass dividedby its volume gives a second equation,

ρ = m43πr3

.

Everything in these equations can be measured directly except form and r, so these are two equations in two unknowns, which can besolved in order to determine how big the drop is.

Next Millikan charged the metal plates, adjusting the amountof charge so as to exactly counteract gravity and levitate the drop.If, for instance, the drop being examined happened to have a totalcharge that was negative, then positive charge put on the top platewould attract it, pulling it up, and negative charge on the bottom

plate would repel it, pushing it up. (Theoretically only one platewould be necessary, but in practice a two-plate arrangement like thisgave electrical forces that were more uniform in strength throughoutthe space where the oil drops were.) The amount of charge on theplates required to levitate the charged drop gave Millikan a handleon the amount of charge the drop carried. The more charge thedrop had, the stronger the electrical forces on it would be, and theless charge would have to be put on the plates to do the trick. Un-fortunately, expressing this relationship using Coulomb’s law wouldhave been impractical, because it would require a perfect knowledgeof how the charge was distributed on each plate, plus the abilityto perform vector addition of all the forces being exerted on thedrop by all the charges on the plate. Instead, Millikan made use of the fact that the electrical force experienced by a pointlike chargedobject at a certain point in space is proportional to its charge,

F

q= constant .

With a given amount of charge on the plates, this constant could bedetermined for instance by discarding the oil drop, inserting between

Section 1.4 Quantization of Charge



q/(1.64

q (C) ×10−19 C)

−1.970 × 10−18 −12.02−0.987 × 10−18 −6.02−2.773 × 10−18 −16.93

h / A few samples of Millikan’sdata.

the plates a larger and more easily handled object with a knowncharge on it, and measuring the force with conventional methods.(Millikan actually used a slightly different set of techniques for de-termining the constant, but the concept is the same.) The amountof force on the actual oil drop had to equal mg, since it was justenough to levitate it, and once the calibration constant had been

determined, the charge of the drop could then be found based on itspreviously determined mass.

Table h shows a few of the results from Millikan’s 1911 paper.(Millikan took data on both negatively and positively charged drops,but in his paper he gave only a sample of his data on negativelycharged drops, so these numbers are all negative.) Even a quicklook at the data leads to the suspicion that the charges are notsimply a series of random numbers. For instance, the second chargeis almost exactly equal to half the first one. Millikan explained theobserved charges as all being integer multiples of a single number,1.64×10−19 C. In the second column, dividing by this constant gives

numbers that are essentially integers, allowing for the random errorspresent in the experiment. Millikan states in his paper that theseresults were a

. . . direct and tangible demonstration . . . of the correct-ness of the view advanced many years ago and supportedby evidence from many sources that all electrical charges,however produced, are exact multiples of one definite,elementary electrical charge, or in other words, that an

electrical charge instead of being spread uniformly overthe charged surface has a definite granular structure,consisting, in fact, of . . . specks, or atoms of electric-ity, all precisely alike, peppered over the surface of thecharged body.

In other words, he had provided direct evidence for the charged-particle model of electricity and against models in which electricitywas described as some sort of fluid. The basic charge is notated e,

and the modern value is e = 1.60 × 10−19

C. The word “quantized ”is used in physics to describe a quantity that can only have certainnumerical values, and cannot have any of the values between those.In this language, we would say that Millikan discovered that chargeis quantized. The charge e is referred to as the quantum of charge.

self-check C

Is money quantized? What is the quantum of money? Answer, p.

205




A historical note on Millikan’s fraudVery few undergraduate physics textbooks mention the well-documentedfact that although Millikan’s conclusions were correct, he was guilty ofscientific fraud. His technique was difficult and painstaking to perform,and his original notebooks, which have been preserved, show that thedata were far less perfect than he claimed in his published scientificpapers. In his publications, he stated categorically that every single

oil drop observed had had a charge that was a multiple of e , with noExceptions or omissions. But his notebooks are replete with notationssuch as “beautiful data, keep,” and “bad run, throw out.” Millikan, then,appears to have earned his Nobel Prize by advocating a correct positionwith dishonest descriptions of his data.

Why do textbook authors fail to mention Millikan’s fraud? It may bethat they think students are too unsophisticated to correctly evaluate theimplications of the fact that scientific fraud has sometimes existed andeven been rewarded by the scientific establishment. Maybe they areafraid students will reason that fudging data is OK, since Millikan gotthe Nobel Prize for it. But falsifying history in the name of encourag-ing truthfulness is more than a little ironic. English teachers don’t edit

Shakespeare’s tragedies so that the bad characters are always pun-ished and the good ones never suffer!

Another possible explanation is simply a lack of originality; it’s possi-

ble that some venerated textbook was uncritical of Millikan’s fraud, andlater authors simply followed suit. Biologist Stephen Jay Gould has writ-ten an essay tracing an example of how authors of biology textbookstend to follow a certain traditional treatment of a topic, using the gi-raffe’s neck to discuss the nonheritability of acquired traits. Yet anotherinterpretation is that scientists derive status from their popular imagesas impartial searchers after the truth, and they don’t want the public torealize how human and imperfect they can be. (Millikan himself was aneducational reformer, and wrote a series of textbooks that were of muchhigher quality than others of his era.)

Note added September 2002 Several years after I wrote this historical digression, I came across an

interesting defense of Millikan by David Goodstein (American Scientist,Jan-Feb 2001, pp. 54-60). Goodstein argues that although Millikanwrote a sentence in his paper that was a lie, Millikan is nevertheless notguilty of fraud when we take that sentence in context: Millikan statedthat he never threw out any data, and he did throw out data, but he hadgood, objective reasons for throwing out the data. The Millikan affair willprobably remain controversial among historians, but the lesson I wouldtake away is that although the episode may reduce our confidence inMillikan, it should deepen our faith in science. The correct result waseventually recognized; it might not have been in a pseudo-scientific field

like political science.

Section 1.4 Quantization of Charge



i / Cathode rays observed in

a vacuum tube.

1.5 The Electron

Cathode rays

Nineteenth-century physicists spent a lot of time trying to comeup with wild, random ways to play with electricity. The best ex-periments of this kind were the ones that made big sparks or pretty

colors of light.One such parlor trick was the cathode ray. To produce it, you

first had to hire a good glassblower and find a good vacuum pump.The glassblower would create a hollow tube and embed two pieces of metal in it, called the electrodes, which were connected to the out-side via metal wires passing through the glass. Before letting himseal up the whole tube, you would hook it up to a vacuum pump,and spend several hours huffing and puffing away at the pump’shand crank to get a good vacuum inside. Then, while you were stillpumping on the tube, the glassblower would melt the glass and sealthe whole thing shut. Finally, you would put a large amount of pos-

itive charge on one wire and a large amount of negative charge onthe other. Metals have the property of letting charge move throughthem easily, so the charge deposited on one of the wires wouldquickly spread out because of the repulsion of each part of it forevery other part. This spreading-out process would result in nearlyall the charge ending up in the electrodes, where there is more roomto spread out than there is in the wire. For obscure historical rea-sons a negative electrode is called a cathode and a positive one isan anode.

Figure i shows the light-emitting stream that was observed. If,as shown in this figure, a hole was made in the anode, the beam

would extend on through the hole until it hit the glass. Drilling ahole in the cathode, however would not result in any beam comingout on the left side, and this indicated that the stuff, whatever itwas, was coming from the cathode. The rays were therefore chris-tened “cathode rays.” (The terminology is still used today in theterm “cathode ray tube” or “CRT” for the picture tube of a TV orcomputer monitor.)

Were cathode rays a form of light, or of matter?

Were cathode rays a form of light, or matter? At first no one re-ally cared what they were, but as their scientific importance became

more apparent, the light-versus-matter issue turned into a contro-versy along nationalistic lines, with the Germans advocating lightand the English holding out for matter. The supporters of the ma-terial interpretation imagined the rays as consisting of a stream of atoms ripped from the substance of the cathode.

One of our defining characteristics of matter is that materialobjects cannot pass through each other. Experiments showed thatcathode rays could penetrate at least some small thickness of matter,




j / J.J. Thomson in the la

such as a metal foil a tenth of a millimeter thick, implying that theywere a form of light.

Other experiments, however, pointed to the contrary conclusion.Light is a wave phenomenon, and one distinguishing property of waves is demonstrated by speaking into one end of a paper towelroll. The sound waves do not emerge from the other end of the

tube as a focused beam. Instead, they begin spreading out in alldirections as soon as they emerge. This shows that waves do notnecessarily travel in straight lines. If a piece of metal foil in the shapeof a star or a cross was placed in the way of the cathode ray, thena “shadow” of the same shape would appear on the glass, showingthat the rays traveled in straight lines. This straight-line motionsuggested that they were a stream of small particles of matter.

These observations were inconclusive, so what was really neededwas a determination of whether the rays had mass and weight. Thetrouble was that cathode rays could not simply be collected in a cupand put on a scale. When the cathode ray tube is in operation, one

does not observe any loss of material from the cathode, or any crustbeing deposited on the anode.

Nobody could think of a good way to weigh cathode rays, so thenext most obvious way of settling the light/matter debate was tocheck whether the cathode rays possessed electrical charge. Lightwas known to be uncharged. If the cathode rays carried charge,they were definitely matter and not light, and they were presum-ably being made to jump the gap by the simultaneous repulsion of the negative charge in the cathode and attraction of the positivecharge in the anode. The rays would overshoot the anode becauseof their momentum. (Although electrically charged particles do notnormally leap across a gap of vacuum, very large amounts of chargewere being used, so the forces were unusually intense.)

Thomson’s experiments

Physicist J.J. Thomson at Cambridge carried out a series of definitive experiments on cathode rays around the year 1897. Byturning them slightly off course with electrical forces, k, he showedthat they were indeed electrically charged, which was strong evi-dence that they were material. Not only that, but he proved thatthey had mass, and measured the ratio of their mass to their charge,m/q. Since their mass was not zero, he concluded that they were

a form of matter, and presumably made up of a stream of micro-scopic, negatively charged particles. When Millikan published hisresults fourteen years later, it was reasonable to assume that thecharge of one such particle equaled minus one fundamental charge,q = −e, and from the combination of Thomson’s and Millikan’s re-sults one could therefore determine the mass of a single cathode rayparticle.

Section 1.5 The Electron



k / Thomson’s experiment provingcathode rays had electric charge(redrawn from his original paper).The cathode, c, and anode, A, areas in any cathode ray tube. Therays pass through a slit in the an-

ode, and a second slit, B, is inter-posed in order to make the beamthinner and eliminate rays thatwere not going straight. Chargingplates D and E shows that cath-ode rays have charge: they areattracted toward the positive plateD and repelled by the negativeplate E.

The basic technique for determining m/q was simply to measurethe angle through which the charged plates bent the beam. Theelectric force acting on a cathode ray particle while it was betweenthe plates would be proportional to its charge,

F elec = (known constant) · q .

Application of Newton’s second law, a = F/m, would allow m/qto be determined:

m

q=

known constant

a

There was just one catch. Thomson needed to know the cathoderay particles’ velocity in order to figure out their acceleration. Atthat point, however, nobody had even an educated guess as to thespeed of the cathode rays produced in a given vacuum tube. Thebeam appeared to leap across the vacuum tube practically instan-taneously, so it was no simple matter of timing it with a stopwatch!

Thomson’s clever solution was to observe the effect of both elec-tric and magnetic forces on the beam. The magnetic force exertedby a particular magnet would depend on both the cathode ray’scharge and its speed:

F mag = (known constant #2) · qv

Thomson played with the electric and magnetic forces until ei-ther one would produce an equal effect on the beam, allowing himto solve for the speed,

v =(known constant)

(known constant #2).

Knowing the speed (which was on the order of 10% of the speedof light for his setup), he was able to find the acceleration and thusthe mass-to-charge ratio m/q. Thomson’s techniques were relativelycrude (or perhaps more charitably we could say that they stretchedthe state of the art of the time), so with various methods he came




up with m/q values that ranged over about a factor of two, evenfor cathode rays extracted from a cathode made of a single mate-rial. The best modern value is m/q = 5.69 × 10−12 kg/C, which isconsistent with the low end of Thomson’s range.

The cathode ray as a subatomic particle: the electron

What was significant about Thomson’s experiment was not theactual numerical value of m/q, however, so much as the fact that,combined with Millikan’s value of the fundamental charge, it gavea mass for the cathode ray particles that was thousands of timessmaller than the mass of even the lightest atoms. Even withoutMillikan’s results, which were 14 years in the future, Thomson rec-ognized that the cathode rays’ m/q was thousands of times smallerthan the m/q ratios that had been measured for electrically chargedatoms in chemical solutions. He correctly interpreted this as evi-dence that the cathode rays were smaller building blocks — he calledthem electrons — out of which atoms themselves were formed. Thiswas an extremely radical claim, coming at a time when atoms hadnot yet been proven to exist! Even those who used the word “atom”often considered them no more than mathematical abstractions, notliteral objects. The idea of searching for structure inside of “un-splittable” atoms was seen by some as lunacy, but within ten yearsThomson’s ideas had been amply verified by many more detailedexperiments.


A Thomson started to become convinced during his experiments thatthe “cathode rays” observed coming from the cathodes of vacuum tubeswere building blocks of atoms — what we now call electrons. He thencarried out observations with cathodes made of a variety of metals, andfound that m /q was roughly the same in every case, considering his lim-ited accuracy. Given his suspicion, why did it make sense to try different

metals? How would the consistent values of m /q test his hypothesis?

B My students have frequently asked whether the m /q that Thomsonmeasured was the value for a single electron, or for the whole beam. Can

you answer this question?

C Thomson found that the m /q of an electron was thousands of times

smaller than that of charged atoms in chemical solutions. Would this implythat the electrons had more charge? Less mass? Would there be no wayto tell? Explain. Remember that Millikan’s results were still many years inthe future, so q was unknown.

D Can you guess any practical reason why Thomson couldn’t justlet one electron fly across the gap before disconnecting the battery andturning off the beam, and then measure the amount of charge depositedon the anode, thus allowing him to measure the charge of a single electrondirectly?

E Why is it not possible to determine m and q themselves, rather than just their ratio, by observing electrons’ motion in electric and magneticfields?

Section 1.5 The Electron



l / The raisin cookie model of

the atom with four units ofcharge, which we now know to beberyllium.

1.6 The Raisin Cookie Model of the Atom

Based on his experiments, Thomson proposed a picture of the atomwhich became known as the raisin cookie model. In the neutralatom, l, there are four electrons with a total charge of −4e, sittingin a sphere (the “cookie”) with a charge of +4e spread throughout it.It was known that chemical reactions could not change one element

into another, so in Thomson’s scenario, each element’s cookie spherehad a permanently fixed radius, mass, and positive charge, differentfrom those of other elements. The electrons, however, were not apermanent feature of the atom, and could be tacked on or pulled outto make charged ions. Although we now know, for instance, that aneutral atom with four electrons is the element beryllium, scientistsat the time did not know how many electrons the various neutralatoms possessed.

This model is clearly different from the one you’ve learned ingrade school or through popular culture, where the positive chargeis concentrated in a tiny nucleus at the atom’s center. An equally

important change in ideas about the atom has been the realizationthat atoms and their constituent subatomic particles behave entirelydifferently from objects on the human scale. For instance, we’ll seelater that an electron can be in more than one place at one time.The raisin cookie model was part of a long tradition of attemptsto make mechanical models of phenomena, and Thomson and hiscontemporaries never questioned the appropriateness of building amental model of an atom as a machine with little parts inside. To-day, mechanical models of atoms are still used (for instance thetinker-toy-style molecular modeling kits like the ones used by Wat-son and Crick to figure out the double helix structure of DNA), but

scientists realize that the physical objects are only aids to help ourbrains’ symbolic and visual processes think about atoms.

Although there was no clear-cut experimental evidence for manyof the details of the raisin cookie model, physicists went ahead andstarted working out its implications. For instance, suppose you hada four-electron atom. All four electrons would be repelling eachother, but they would also all be attracted toward the center of the“cookie” sphere. The result should be some kind of stable, sym-metric arrangement in which all the forces canceled out. Peoplesufficiently clever with math soon showed that the electrons in afour-electron atom should settle down at the vertices of a pyramid

with one less side than the Egyptian kind, i.e., a regular tetrahe-dron. This deduction turns out to be wrong because it was basedon incorrect features of the model, but the model also had manysuccesses, a few of which we will now discuss.

Flow of electrical charge in wires example 3

One of my former students was the son of an electrician, and

had become an electrician himself. He related to me how his




father had remained refused to believe all his life that electrons

really flowed through wires. If they had, he reasoned, the metal

would have gradually become more and more damaged, eventu-

ally crumbling to dust.

His opinion is not at all unreasonable based on the fact that elec-

trons are material particles, and that matter cannot normally pass

through matter without making a hole through it. Nineteenth-century physicists would have shared his objection to a charged-

particle model of the flow of electrical charge. In the raisin-cookie

model, however, the electrons are very low in mass, and there-

fore presumably very small in size as well. It is not surprising that

they can slip between the atoms without damaging them.

Flow of electrical charge across cell membranes example 4

Your nervous system is based on signals carried by charge mov-

ing from nerve cell to nerve cell. Your body is essentially all liquid,

and atoms in a liquid are mobile. This means that, unlike the case

of charge flowing in a solid wire, entire charged atoms can flow in

your nervous system

Emission of electrons in a cathode ray tube example 5

Why do electrons detach themselves from the cathode of a vac-

uum tube? Certainly they are encouraged to do so by the re-

pulsion of the negative charge placed on the cathode and the

attraction from the net positive charge of the anode, but these are

not strong enough to rip electrons out of atoms by main force —

if they were, then the entire apparatus would have been instantly

vaporized as every atom was simultaneously ripped apart!

The raisin cookie model leads to a simple explanation. We know

that heat is the energy of random motion of atoms. The atoms inany object are therefore violently jostling each other all the time,

and a few of these collisions are violent enough to knock electrons

out of atoms. If this occurs near the surface of a solid object, the

electron may can come loose. Ordinarily, however, this loss of

electrons is a self-limiting process; the loss of electrons leaves

the object with a net positive charge, which attracts the lost sheep

home to the fold. (For objects immersed in air rather than vacuum,

there will also be a balanced exchange of electrons between the

air and the object.)

This interpretation explains the warm and friendly yellow glow of

the vacuum tubes in an antique radio. To encourage the emissionof electrons from the vacuum tubes’ cathodes, the cathodes are

intentionally warmed up with little heater coils.


A Today many people would define an ion as an atom (or molecule)with missing electrons or extra electrons added on. How would peoplehave defined the word “ion” before the discovery of the electron?

Section 1.6 The Raisin Cookie Model of the Atom



B Since electrically neutral atoms were known to exist, there had to bepositively charged subatomic stuff to cancel out the negatively chargedelectrons in an atom. Based on the state of knowledge immediately afterthe Millikan and Thomson experiments, was it possible that the positivelycharged stuff had an unquantized amount of charge? Could it be quan-tized in units of +e? In units of +2e? In units of +5/7e?




Summary

Selected Vocabularyatom . . . . . . . the basic unit of one of the chemical elementsmolecule . . . . . a group of atoms stuck togetherelectrical force . one of the fundamental forces of nature; a non-

contact force that can be either repulsive or

attractivecharge . . . . . . a numerical rating of how strongly an object

participates in electrical forcescoulomb (C) . . . the unit of electrical chargeion . . . . . . . . . an electrically charged atom or moleculecathode ray . . . the mysterious ray that emanated from the

cathode in a vacuum tube; shown by Thomsonto be a stream of particles smaller than atoms

electron . . . . . . Thomson’s name for the particles of which acathode ray was made

quantized . . . . describes a quantity, such as money or elec-

trical charge, that can only exist in certainamounts

Notationq . . . . . . . . . . chargee . . . . . . . . . . the quantum of charge

Summary

All the forces we encounter in everyday life boil down to twobasic types: gravitational forces and electrical forces. A force suchas friction or a “sticky force” arises from electrical forces betweenindividual atoms.

Just as we use the word “mass” to describe how strongly anobject participates in gravitational forces, we use the word “charge”for the intensity of its electrical forces. There are two types of charge. Two charges of the same type repel each other, but objectswhose charges are different attract each other. Charge is measuredin units of coulombs (C).

Mobile charged particle model: A great many phenomena areeasily understood if we imagine matter as containing two types of charged particles, which are at least partially able to move around.

Positive and negative charge: Ordinary objects that have not

been specially prepared have both types of charge spread evenlythroughout them in equal amounts. The object will then tend notto exert electrical forces on any other object, since any attractiondue to one type of charge will be balanced by an equal repulsionfrom the other. (We say “tend not to” because bringing the objectnear an object with unbalanced amounts of charge could cause itscharges to separate from each other, and the force would no longercancel due to the unequal distances.) It therefore makes sense to

Summary



describe the two types of charge using positive and negative signs,so that an unprepared object will have zero total charge.

The Coulomb force law states that the magnitude of the electri-cal force between two charged particles is given by |F| = k|q1||q2|/r

2.

Conservation of charge: An even more fundamental reason for

using positive and negative signs for charge is that with this defini-tion the total charge of a closed system is a conserved quantity.

Quantization of charge: Millikan’s oil drop experiment showedthat the total charge of an object could only be an integer multipleof a basic unit of charge (e). This supported the idea the the “flow”of electrical charge was the motion of tiny particles rather than themotion of some sort of mysterious electrical fluid.

Einstein’s analysis of Brownian motion was the first definitiveproof of the existence of atoms. Thomson’s experiments with vac-uum tubes demonstrated the existence of a new type of microscopicparticle with a very small ratio of mass to charge. Thomson cor-

rectly interpreted these as building blocks of matter even smallerthan atoms: the first discovery of subatomic particles. These parti-cles are called electrons.

The above experimental evidence led to the first useful model of the interior structure of atoms, called the raisin cookie model. Inthe raisin cookie model, an atom consists of a relatively large, mas-sive, positively charged sphere with a certain number of negativelycharged electrons embedded in it.




Problem 1. Top: A realispicture of a neuron. BottoA simplified diagram of osegment of the tail (axon).

Problems

Key√ A computerized answer check is available online. A problem that requires calculus.

A difficult problem.

1 The figure shows a neuron, which is the type of cell your nervesare made of. Neurons serve to transmit sensory information to thebrain, and commands from the brain to the muscles. All this datais transmitted electrically, but even when the cell is resting and nottransmitting any information, there is a layer of negative electricalcharge on the inside of the cell membrane, and a layer of positivecharge just outside it. This charge is in the form of various ionsdissolved in the interior and exterior fluids. Why would the negativecharge remain plastered against the inside surface of the membrane,and likewise why doesn’t the positive charge wander away from theoutside surface?

2

Use the nutritional information on some packaged food tomake an order-of-magnitude estimate of the amount of chemicalenergy stored in one atom of food, in units of joules. Assume thata typical atom has a mass of 10−26 kg. This constitutes a roughestimate of the amounts of energy there are on the atomic scale.[See chapter 1 of book 1, Newtonian Physics, for help on how to doorder-of-magnitude estimates. Note that a nutritional “calorie” isreally a kilocalorie; see page 218.]

√

3 (a) Recall that the gravitational energy of two gravitationallyinteracting spheres is given by PE = −Gm1m2/r, where r is thecenter-to-center distance. What would be the analogous equation

for two electrically interacting spheres? Justify your choice of aplus or minus sign on physical grounds, considering attraction andrepulsion.

√

(b) Use this expression to estimate the energy required to pull aparta raisin-cookie atom of the one-electron type, assuming a radius of 10−10 m.

√

(c) Compare this with the result of problem 2.

4 A neon light consists of a long glass tube full of neon, withmetal caps on the ends. Positive charge is placed on one end of thetube, and negative charge on the other. The electric forces generatedcan be strong enough to strip electrons off of a certain number of

neon atoms. Assume for simplicity that only one electron is everstripped off of any neon atom. When an electron is stripped off of an atom, both the electron and the neon atom (now an ion) haveelectric charge, and they are accelerated by the forces exerted by thecharged ends of the tube. (They do not feel any significant forcesfrom the other ions and electrons within the tube, because onlya tiny minority of neon atoms ever gets ionized.) Light is finallyproduced when ions are reunited with electrons. Give a numerical

Problems



Problem 8.

Problem 6.

comparison of the magnitudes and directions of the accelerations of the electrons and ions. [You may need some data from page 218.]√

5 If you put two hydrogen atoms near each other, they will feelan attractive force, and they will pull together to form a molecule.(Molecules consisting of two hydrogen atoms are the normal form of

hydrogen gas.) How is this possible, since each is electrically neu-tral? Shouldn’t the attractive and repulsive forces all cancel outexactly? Use the raisin cookie model. (Students who have takenchemistry often try to use fancier models to explain this, but if youcan’t explain it using a simple model, you probably don’t understandthe fancy model as well as you thought you did!) It’s not so easyto prove that the force should actually be attractive rather than re-pulsive, so just concentrate on explaining why it doesn’t necessarilyhave to vanish completely.

6 The figure shows one layer of the three-dimensional structureof a salt crystal. The atoms extend much farther off in all directions,

but only a six-by-six square is shown here. The larger circles arethe chlorine ions, which have charges of −e. The smaller circlesare sodium ions, with charges of +e. The center-to-center distancebetween neighboring ions is about 0.3 nm. Real crystals are neverperfect, and the crystal shown here has two defects: a missing atomat one location, and an extra lithium atom, shown as a grey circle,inserted in one of the small gaps. If the lithium atom has a chargeof +e, what is the direction and magnitude of the total force on it?Assume there are no other defects nearby in the crystal besides thetwo shown here. [Hints: The force on the lithium ion is the vectorsum of all the forces of all the quadrillions of sodium and chlorine

atoms, which would obviously be too laborious to calculate. Nearlyall of these forces, however, are canceled by a force from an ion onthe opposite side of the lithium.]

√

7 The Earth and Moon are bound together by gravity. If, in-stead, the force of attraction were the result of each having a chargeof the same magnitude but opposite in sign, find the quantity of charge that would have to be placed on each to produce the re-quired force.

√

8 In the semifinals of an electrostatic croquet tournament,Jessica hits her positively charged ball, sending it across the playingfield, rolling to the left along the x axis. It is repelled by two other

positive charges. These two equal charges are fixed on the y axis atthe locations shown in the figure. (a) Express the force on the ballin terms of the ball’s position, x. (b) At what value of x does theball experience the greatest deceleration? Express you answer interms of b. [Based on a problem by Halliday and Resnick.]




a / Marie and Pierre Curie wethe first to purify radium in signcant quantities. Radium’s inten

radioactivity made possible texperiments that led to the moern planetary model of the atoin which electrons orbit a nuclemade of protons and neutrons.

Chapter 2

The Nucleus

2.1 Radioactivity

Becquerel’s discovery of radioactivity

How did physicists figure out that the raisin cookie model wasincorrect, and that the atom’s positive charge was concentrated ina tiny, central nucleus? The story begins with the discovery of ra-dioactivity by the French chemist Becquerel. Up until radioactivitywas discovered, all the processes of nature were thought to be basedon chemical reactions, which were rearrangements of combinationsof atoms. Atoms exert forces on each other when they are close to-

gether, so sticking or unsticking them would either release or storeelectrical energy. That energy could be converted to and from otherforms, as when a plant uses the energy in sunlight to make sugarsand carbohydrates, or when a child eats sugar, releasing the energyin the form of kinetic energy.

Becquerel discovered a process that seemed to release energyfrom an unknown new source that was not chemical. Becquerel,



b / Henri Becquerel (1852-1908).

c / Becquerel’s photographicplate. In the exposure at the

bottom of the image, he hasfound that he could absorb theradiations, casting the shadowof a Maltese cross that wasplaced between the plate and theuranium salts.

whose father and grandfather had also been physicists, spent thefirst twenty years of his professional life as a successful civil engi-neer, teaching physics on a part-time basis. He was awarded thechair of physics at the Musee d’Histoire Naturelle in Paris after thedeath of his father, who had previously occupied it. Having now asignificant amount of time to devote to physics, he began studying

the interaction of light and matter. He became interested in the phe-nomenon of phosphorescence, in which a substance absorbs energyfrom light, then releases the energy via a glow that only graduallygoes away. One of the substances he investigated was a uraniumcompound, the salt UKSO5. One day in 1896, cloudy weather in-terfered with his plan to expose this substance to sunlight in orderto observe its fluorescence. He stuck it in a drawer, coincidentally ontop of a blank photographic plate — the old-fashioned glass-backedcounterpart of the modern plastic roll of film. The plate had beencarefully wrapped, but several days later when Becquerel checked itin the darkroom before using it, he found that it was ruined, as if ithad been completely exposed to light.

History provides many examples of scientific discoveries thathappened this way: an alert and inquisitive mind decides to in-vestigate a phenomenon that most people would not have worriedabout explaining. Becquerel first determined by further experimentsthat the effect was produced by the uranium salt, despite a thickwrapping of paper around the plate that blocked out all light. Hetried a variety of compounds, and found that it was the uraniumthat did it: the effect was produced by any uranium compound, butnot by any compound that didn’t include uranium atoms. The effectcould be at least partially blocked by a sufficient thickness of metal,and he was able to produce silhouettes of coins by interposing them

between the uranium and the plate. This indicated that the effecttraveled in a straight line., so that it must have been some kind of ray rather than, e.g., the seepage of chemicals through the paper.He used the word “radiations,” since the effect radiated out fromthe uranium salt.

At this point Becquerel still believed that the uranium atomswere absorbing energy from light and then gradually releasing theenergy in the form of the mysterious rays, and this was how hepresented it in his first published lecture describing his experiments.Interesting, but not earth-shattering. But he then tried to determinehow long it took for the uranium to use up all the energy that hadsupposedly been stored in it by light, and he found that it neverseemed to become inactive, no matter how long he waited. Not onlythat, but a sample that had been exposed to intense sunlight for awhole afternoon was no more or less effective than a sample thathad always been kept inside. Was this a violation of conservationof energy? If the energy didn’t come from exposure to light, wheredid it come from?

42 Chapter 2 The Nucleus



Three kinds of “radiations”

Unable to determine the source of the energy directly, turn-of-the-century physicists instead studied the behavior of the “radia-tions” once they had been emitted. Becquerel had already shownthat the radioactivity could penetrate through cloth and paper, sothe first obvious thing to do was to investigate in more detail what

thickness of material the radioactivity could get through. They soonlearned that a certain fraction of the radioactivity’s intensity wouldbe eliminated by even a few inches of air, but the remainder wasnot eliminated by passing through more air. Apparently, then, theradioactivity was a mixture of more than one type, of which one wasblocked by air. They then found that of the part that could pene-trate air, a further fraction could be eliminated by a piece of paperor a very thin metal foil. What was left after that, however, wasa third, extremely penetrating type, some of whose intensity wouldstill remain even after passing through a brick wall. They decidedthat this showed there were three types of radioactivity, and with-

out having the faintest idea of what they really were, they made upnames for them. The least penetrating type was arbitrarily labeledα (alpha), the first letter of the Greek alphabet, and so on throughβ (beta) and finally γ (gamma) for the most penetrating type.

Radium: a more intense source of radioactivity

The measuring devices used to detect radioactivity were crude:photographic plates or even human eyeballs (radioactivity makesflashes of light in the jelly-like fluid inside the eye, which can beseen by the eyeball’s owner if it is otherwise very dark). Because theways of detecting radioactivity were so crude and insensitive, furtherprogress was hindered by the fact that the amount of radioactivityemitted by uranium was not really very great. The vital contribu-tion of physicist/chemist Marie Curie and her husband Pierre wasto discover the element radium, and to purify and isolate significantquantities it. Radium emits about a million times more radioactivityper unit mass than uranium, making it possible to do the experi-ments that were needed to learn the true nature of radioactivity.The dangers of radioactivity to human health were then unknown,and Marie died of leukemia thirty years later. (Pierre was run overand killed by a horsecart.)

Tracking down the nature of alphas, betas, and gammas

As radium was becoming available, an apprentice scientist namedErnest Rutherford arrived in England from his native New Zealandand began studying radioactivity at the Cavendish Laboratory. Theyoung colonial’s first success was to measure the mass-to-charge ra-tio of beta rays. The technique was essentially the same as the oneThomson had used to measure the mass-to-charge ratio of cathoderays by measuring their deflections in electric and magnetic fields.The only difference was that instead of the cathode of a vacuum

Section 2.1 Radioactivity



d / A simplified version ofRutherford’s 1908 experiment,showing that alpha particles were

doubly ionized helium atoms.

e / These pellets of uraniumfuel will be inserted into the metalfuel rod and used in a nuclearreactor. The pellets emit alphaand beta radiation, which the

gloves are thick enough to stop.

tube, a nugget of radium was used to supply the beta rays. Notonly was the technique the same, but so was the result. Beta rayshad the same m/q ratio as cathode rays, which suggested they wereone and the same. Nowadays, it would make sense simply to usethe term “electron,” and avoid the archaic “cathode ray” and “betaparticle,” but the old labels are still widely used, and it is unfortu-

nately necessary for physics students to memorize all three namesfor the same thing.

At first, it seemed that neither alphas or gammas could be de-flected in electric or magnetic fields, making it appear that neitherwas electrically charged. But soon Rutherford obtained a much morepowerful magnet, and was able to use it to deflect the alphas butnot the gammas. The alphas had a much larger value of m/q thanthe betas (about 4000 times greater), which was why they had beenso hard to deflect. Gammas are uncharged, and were later found tobe a form of light.

The m/q ratio of alpha particles turned out to be the same

as those of two different types of ions, He++ (a helium atom withtwo missing electrons) and H+

2 (two hydrogen atoms bonded into amolecule, with one electron missing), so it seemed likely that theywere one or the other of those. The diagram shows a simplified ver-sion of Rutherford’s ingenious experiment proving that they wereHe++ ions. The gaseous element radon, an alpha emitter, was in-troduced into one half of a double glass chamber. The glass walldividing the chamber was made extremely thin, so that some of therapidly moving alpha particles were able to penetrate it. The otherchamber, which was initially evacuated, gradually began to accu-mulate a population of alpha particles (which would quickly pick up

electrons from their surroundings and become electrically neutral).Rutherford then determined that it was helium gas that had ap-peared in the second chamber. Thus alpha particles were proved tobe He++ ions. The nucleus was yet to be discovered, but in modernterms, we would describe a He++ ion as the nucleus of a He atom.

To summarize, here are the three types of radiation emitted byradioactive elements, and their descriptions in modern terms:

α particle stopped by a few inches of air He nucleus

β particle stopped by a piece of paper electron

γ ray penetrates thick shielding a type of light

Discussion QuestionA Most sources of radioactivity emit alphas, betas, and gammas, not just one of the three. In the radon experiment, how did Rutherford knowthat he was studying the alphas?




f / Ernest Rutherford (187

1937).

g / Marsden and Rutherforapparatus.

2.2 The Planetary Model of the Atom

The stage was now set for the unexpected discovery that the posi-tively charged part of the atom was a tiny, dense lump at the atom’scenter rather than the “cookie dough” of the raisin cookie model.By 1909, Rutherford was an established professor, and had studentsworking under him. For a raw undergraduate named Marsden, he

picked a research project he thought would be tedious but straight-forward.

It was already known that although alpha particles would bestopped completely by a sheet of paper, they could pass through asufficiently thin metal foil. Marsden was to work with a gold foilonly 1000 atoms thick. (The foil was probably made by evaporatinga little gold in a vacuum chamber so that a thin layer would bedeposited on a glass microscope slide. The foil would then be liftedoff the slide by submerging the slide in water.)

Rutherford had already determined in his previous experiments

the speed of the alpha particles emitted by radium, a fantastic 1.5×107 m/s. The experimenters in Rutherford’s group visualized themas very small, very fast cannonballs penetrating the “cookie dough”part of the big gold atoms. A piece of paper has a thickness of ahundred thousand atoms or so, which would be sufficient to stopthem completely, but crashing through a thousand would only slowthem a little and turn them slightly off of their original paths.

Marsden’s supposedly ho-hum assignment was to use the appa-ratus shown in figure g to measure how often alpha particles weredeflected at various angles. A tiny lump of radium in a box emit-ted alpha particles, and a thin beam was created by blocking all

the alphas except those that happened to pass out through a tube.Typically deflected in the gold by only a small amount, they wouldreach a screen very much like the screen of a TV’s picture tube,which would make a flash of light when it was hit. Here is the firstexample we have encountered of an experiment in which a beam of particles is detected one at a time. This was possible because eachalpha particle carried so much kinetic energy; they were moving atabout the same speed as the electrons in the Thomson experiment,but had ten thousand times more mass.

Marsden sat in a dark room, watching the apparatus hour afterhour and recording the number of flashes with the screen moved to

various angles. The rate of the flashes was highest when he set thescreen at an angle close to the line of the alphas’ original path, but if he watched an area farther off to the side, he would also occasionallysee an alpha that had been deflected through a larger angle. Afterseeing a few of these, he got the crazy idea of moving the screen tosee if even larger angles ever occurred, perhaps even angles largerthan 90 degrees.

Section 2.2 The Planetary Model of the Atom



i / The planetary model ofthe atom.

h / Alpha particles being scattered by a gold nucleus. On this scale,the gold atom is the size of a car, so all the alpha particles shown hereare ones that just happened to come unusually close to the nucleus.

For these exceptional alpha particles, the forces from the electrons areunimportant, because they are so much more distant than the nucleus.

The crazy idea worked: a few alpha particles were deflectedthrough angles of up to 180 degrees, and the routine experimenthad become an epoch-making one. Rutherford said, “We have beenable to get some of the alpha particles coming backwards. It wasalmost as incredible as if you fired a 15-inch shell at a piece of tissuepaper and it came back and hit you.” Explanations were hard tocome by in the raisin cookie model. What intense electrical forcescould have caused some of the alpha particles, moving at such astro-nomical speeds, to change direction so drastically? Since each goldatom was electrically neutral, it would not exert much force on analpha particle outside it. True, if the alpha particle was very near toor inside of a particular atom, then the forces would not necessarilycancel out perfectly; if the alpha particle happened to come veryclose to a particular electron, the 1/r2 form of the Coulomb forcelaw would make for a very strong force. But Marsden and Ruther-ford knew that an alpha particle was 8000 times more massive thanan electron, and it is simply not possible for a more massive objectto rebound backwards from a collision with a less massive objectwhile conserving momentum and energy. It might be possible inprinciple for a particular alpha to follow a path that took it very

close to one electron, and then very close to another electron, and soon, with the net result of a large deflection, but careful calculationsshowed that such multiple “close encounters” with electrons wouldbe millions of times too rare to explain what was actually observed.

At this point, Rutherford and Marsden dusted off an unpop-ular and neglected model of the atom, in which all the electronsorbited around a small, positively charged core or “nucleus,” justlike the planets orbiting around the sun. All the positive charge




and nearly all the mass of the atom would be concentrated in thenucleus, rather than spread throughout the atom as in the raisincookie model. The positively charged alpha particles would be re-pelled by the gold atom’s nucleus, but most of the alphas would notcome close enough to any nucleus to have their paths drasticallyaltered. The few that did come close to a nucleus, however, could

rebound backwards from a single such encounter, since the nucleus of a heavy gold atom would be fifty times more massive than an alphaparticle. It turned out that it was not even too difficult to derive aformula giving the relative frequency of deflections through variousangles, and this calculation agreed with the data well enough (towithin 15%), considering the difficulty in getting good experimentalstatistics on the rare, very large angles.

What had started out as a tedious exercise to get a studentstarted in science had ended as a revolution in our understandingof nature. Indeed, the whole thing may sound a little too muchlike a moralistic fable of the scientific method with overtones of

the Horatio Alger genre. The skeptical reader may wonder whythe planetary model was ignored so thoroughly until Marsden andRutherford’s discovery. Is science really more of a sociological enter-prise, in which certain ideas become accepted by the establishment,and other, equally plausible explanations are arbitrarily discarded?Some social scientists are currently ruffling a lot of scientists’ feath-ers with critiques very much like this, but in this particular case,there were very sound reasons for rejecting the planetary model. Asyou’ll learn in more detail later in this course, any charged particlethat undergoes an acceleration dissipate energy in the form of light.In the planetary model, the electrons were orbiting the nucleus in

circles or ellipses, which meant they were undergoing acceleration, just like the acceleration you feel in a car going around a curve. Theyshould have dissipated energy as light, and eventually they shouldhave lost all their energy. Atoms don’t spontaneously collapse likethat, which was why the raisin cookie model, with its stationaryelectrons, was originally preferred. There were other problems aswell. In the planetary model, the one-electron atom would haveto be flat, which would be inconsistent with the success of molecu-lar modeling with spherical balls representing hydrogen and atoms.These molecular models also seemed to work best if specific sizeswere used for different atoms, but there is no obvious reason in theplanetary model why the radius of an electron’s orbit should be afixed number. In view of the conclusive Marsden-Rutherford results,however, these became fresh puzzles in atomic physics, not reasonsfor disbelieving the planetary model.

Section 2.2 The Planetary Model of the Atom



k / A modern periodic tablabeled with atomic numbeMendeleev’s original table wupside-down compared to tone.

from helium, making He+ or He++, but nobody could make He+++,presumably because the nuclear charge of helium was only +2e.Unfortunately only a few of the lightest elements could be strippedcompletely, because the more electrons were stripped off, the greater

the positive net charge remaining, and the more strongly the rest of the negatively charged electrons would be held on. The heavy ele-ments’ atomic numbers could only be roughly extrapolated from thelight elements, where the atomic number was about half the atom’smass expressed in units of the mass of a hydrogen atom. Gold, forexample, had a mass about 197 times that of hydrogen, so its atomicnumber was estimated to be about half that, or somewhere around100. We now know it to be 79.

How did we finally find out? The riddle of the nuclear chargeswas at last successfully attacked using two different techniques,which gave consistent results. One set of experiments, involvingx-rays, was performed by the young Henry Mosely, whose scientificbrilliance was soon to be sacrificed in a battle between European im-perialists over who would own the Dardanelles, during that pointlessconflict then known as the War to End All Wars, and now referredto as World War I.

l / An alpha particle has to comuch closer to the low-chargcopper nucleus in order to be dflected through the same angle

Section 2.3 Atomic Number



Since Mosely’s analysis requires several concepts with which youare not yet familiar, we will instead describe the technique usedby James Chadwick at around the same time. An added bonus of describing Chadwick’s experiments is that they presaged the impor-tant modern technique of studying collisions of subatomic particles.In grad school, I worked with a professor whose thesis adviser’s the-

sis adviser was Chadwick, and he related some interesting storiesabout the man. Chadwick was apparently a little nutty and a com-plete fanatic about science, to the extent that when he was held in aGerman prison camp during World War II, he managed to cajole hiscaptors into allowing him to scrounge up parts from broken radiosso that he could attempt to do physics experiments.

Chadwick’s experiment worked like this. Suppose you performtwo Rutherford-type alpha scattering measurements, first one with agold foil as a target as in Rutherford’s original experiment, and thenone with a copper foil. It is possible to get large angles of deflectionin both cases, but as shown in figure m, the alpha particle must

be heading almost straight for the copper nucleus to get the sameangle of deflection that would have occurred with an alpha thatwas much farther off the mark; the gold nucleus’ charge is so muchgreater than the copper’s that it exerts a strong force on the alphaparticle even from far off. The situation is very much like that of ablindfolded person playing darts. Just as it is impossible to aim analpha particle at an individual nucleus in the target, the blindfoldedperson cannot really aim the darts. Achieving a very close encounterwith the copper atom would be akin to hitting an inner circle on thedartboard. It’s much more likely that one would have the luck tohit the outer circle, which covers a greater number of square inches.

By analogy, if you measure the frequency with which alphas arescattered by copper at some particular angle, say between 19 and20 degrees, and then perform the same measurement at the sameangle with gold, you get a much higher percentage for gold than forcopper.

m / An alpha particle must beheaded for the ring on the frontof the imaginary cylindrical pipein order to produce scattering atan angle between 19 and 20 de-

grees. The area of this ringis called the “cross-section” forscattering at 19-20 ◦because it isthe cross-sectional area of a cutthrough the pipe.




In fact, the numerical ratio of the two nuclei’s charges can bederived from this same experimentally determined ratio. Using thestandard notation Z for the atomic number (charge of the nucleusdivided by e), the following equation can be proved (example 1):

Z 2gold

Z 2

copper

=number of alphas scattered by gold at 19-20 ◦

number of alphas scattered by copper at 19-20 ◦

By making such measurements for targets constructed from all theelements, one can infer the ratios of all the atomic numbers, andsince the atomic numbers of the light elements were already known,atomic numbers could be assigned to the entire periodic table. Ac-cording to Mosely, the atomic numbers of copper, silver and plat-inum were 29, 47, and 78, which corresponded well with their posi-tions on the periodic table. Chadwick’s figures for the same elementswere 29.3, 46.3, and 77.4, with error bars of about 1.5 times the fun-damental charge, so the two experiments were in good agreement.

The point here is absolutely not that you should be ready to plugnumbers into the above equation for a homework or exam question!My overall goal in this chapter is to explain how we know what weknow about atoms. An added bonus of describing Chadwick’s ex-periment is that the approach is very similar to that used in modernparticle physics experiments, and the ideas used in the analysis areclosely related to the now-ubiquitous concept of a “cross-section.”In the dartboard analogy, the cross-section would be the area of thecircular ring you have to hit. The reasoning behind the invention of the term “cross-section” can be visualized as shown in figure m. Inthis language, Rutherford’s invention of the planetary model camefrom his unexpected discovery that there was a nonzero cross-section

for alpha scattering from gold at large angles, and Chadwick con-firmed Mosely’s determinations of the atomic numbers by measuringcross-sections for alpha scattering.

Proof of the relationship between Z and scattering example 1

The equation above can be derived by the following not very rigor-

ous proof. To deflect the alpha par ticle by a certain angle requires

that it acquire a certain momentum component in the direction

perpendicular to its original momentum. Although the nucleus’s

force on the alpha particle is not constant, we can pretend that

it is approximately constant during the time when the alpha is

within a distance equal to, say, 150% of its distance of closest

approach, and that the force is zero before and after that part ofthe motion. (If we chose 120% or 200%, it shouldn’t make any

difference in the final result, because the final result is a ratio,

and the effects on the numerator and denominator should cancel

each other.) In the approximation of constant force, the change

in the alpha’s perpendicular momentum component is then equal

Section 2.3 Atomic Number



to F ∆t . The Coulomb force law says the force is proportional to

Z /r 2. Although r does change somewhat during the time interval

of interest, it’s good enough to treat it as a constant number, since

we’re only computing the ratio between the two experiments’ re-

sults. Since we are approximating the force as acting over the

time during which the distance is not too much greater than the

distance of closest approach, the time interval ∆t must be propor-tional to r , and the sideways momentum imparted to the alpha,

F ∆t , is proportional to (Z /r 2)r , or Z /r . If we’re comparing alphas

scattered at the same angle from gold and from copper, then ∆p

is the same in both cases, and the proportionality ∆p ∝ Z /r tells

us that the ones scattered from copper at that angle had to be

headed in along a line closer to the central axis by a factor equal-

ing Z gold/Z copper. If you imagine a “dartboard ring” that the alphas

have to hit, then the ring for the gold experiment has the same

proportions as the one for copper, but it is enlarged by a factor

equal to Z gold/Z copper. That is, not only is the radius of the ring

greater by that factor, but unlike the rings on a normal dartboard,

the thickness of the outer ring is also greater in proportion to its

radius. When you take a geometric shape and scale it up in size

like a photographic enlargement, its area is increased in propor-

tion to the square of the enlargement factor, so the area of the

dartboard ring in the gold experiment is greater by a factor equal

to (Z gold/Z copper)2. Since the alphas are aimed entirely randomly,

the chances of an alpha hitting the ring are in proportion to the

area of the ring, which proves the equation given above.

As an example of the modern use of scattering experiments andcross-section measurements, you may have heard of the recent ex-

perimental evidence for the existence of a particle called the topquark. Of the twelve subatomic particles currently believed to be thesmallest constituents of matter, six form a family called the quarks,distinguished from the other six by the intense attractive forces thatmake the quarks stick to each other. (The other six consist of theelectron plus five other, more exotic particles.) The only two types of quarks found in naturally occurring matter are the “up quark” and“down quark,” which are what protons and neutrons are made of,but four other types were theoretically predicted to exist, for a totalof six. (The whimsical term “quark” comes from a line by JamesJoyce reading “Three quarks for master Mark.”) Until recently, onlyfive types of quarks had been proven to exist via experiments, andthe sixth, the top quark, was only theorized. There was no hopeof ever detecting a top quark directly, since it is radioactive, andonly exists for a zillionth of a second before evaporating. Instead,the researchers searching for it at the Fermi National AcceleratorLaboratory near Chicago measured cross-sections for scattering of nuclei off of other nuclei. The experiment was much like those of Rutherford and Chadwick, except that the incoming nuclei had to




be boosted to much higher speeds in a particle accelerator. Theresulting encounter with a target nucleus was so violent that bothnuclei were completely demolished, but, as Einstein proved, energycan be converted into matter, and the energy of the collision createsa spray of exotic, radioactive particles, like the deadly shower of wood fragments produced by a cannon ball in an old naval battle.

Among those particles were some top quarks. The cross-sectionsbeing measured were the cross-sections for the production of certaincombinations of these secondary particles. However different thedetails, the principle was the same as that employed at the turn of the century: you smash things together and look at the fragmentsthat fly off to see what was inside them. The approach has beencompared to shooting a clock with a rifle and then studying thepieces that fly off to figure out how the clock worked.


A The diagram, showing alpha particles being deflected by a goldnucleus, was drawn with the assumption that alpha particles came in on

lines at many different distances from the nucleus. Why wouldn’t they allcome in along the same line, since they all came out through the sametube?

B Why does it make sense that, as shown in the figure, the trajectoriesthat result in 19 ◦ and 20 ◦ scattering cross each other?

C Rutherford knew the velocity of the alpha particles emitted by radium,and guessed that the positively charged part of a gold atom had a chargeof about +100e (we now know it is +79e ). Considering the fact that somealpha particles were deflected by 180 ◦, how could he then use conserva-tion of energy to derive an upper limit on the size of a gold nucleus? (Forsimplicity, assume the size of the alpha particle is negligible compared tothat of the gold nucleus, and ignore the fact that the gold nucleus recoils

a little from the collision, picking up a little kinetic energy.)

2.4 The Structure of Nuclei

The proton

The fact that the nuclear charges were all integer multiples of esuggested to many physicists that rather than being a pointlike object, the nucleus might contain smaller particles having individualcharges of +e. Evidence in favor of this idea was not long in arriv-ing. Rutherford reasoned that if he bombarded the atoms of a verylight element with alpha particles, the small charge of the target

nuclei would give a very weak repulsion. Perhaps those few alphaparticles that happened to arrive on head-on collision courses wouldget so close that they would physically crash into some of the targetnuclei. An alpha particle is itself a nucleus, so this would be a col-lision between two nuclei, and a violent one due to the high speedsinvolved. Rutherford hit pay dirt in an experiment with alpha par-ticles striking a target containing nitrogen atoms. Charged particles

Section 2.4 The Structure of Nuclei



n / Examples of the constructionof atoms: hydrogen (top) andhelium (bottom). On this scale,the electrons’ orbits would be the

size of a college campus.

were detected flying out of the target like parts flying off of cars ina high-speed crash. Measurements of the deflection of these parti-cles in electric and magnetic fields showed that they had the samecharge-to-mass ratio as singly-ionized hydrogen atoms. Rutherfordconcluded that these were the conjectured singly-charged particlesthat held the charge of the nucleus, and they were later named

protons. The hydrogen nucleus consists of a single proton, and ingeneral, an element’s atomic number gives the number of protonscontained in each of its nuclei. The mass of the proton is about 1800times greater than the mass of the electron.

The neutron

It would have been nice and simple if all the nuclei could havebeen built only from protons, but that couldn’t be the case. If youspend a little time looking at a periodic table, you will soon noticethat although some of the atomic masses are very nearly integermultiples of hydrogen’s mass, many others are not. Even where themasses are close whole numbers, the masses of an element otherthan hydrogen is always greater than its atomic number, not equalto it. Helium, for instance, has two protons, but its mass is fourtimes greater than that of hydrogen.

Chadwick cleared up the confusion by proving the existence of a new subatomic particle. Unlike the electron and proton, whichare electrically charged, this particle is electrically neutral, and henamed it the neutron. Chadwick’s experiment has been describedin detail in chapter 4 of book 2 of this series, but briefly the methodwas to expose a sample of the light element beryllium to a stream of alpha particles from a lump of radium. Beryllium has only four pro-

tons, so an alpha that happens to be aimed directly at a berylliumnucleus can actually hit it rather than being stopped short of a col-lision by electrical repulsion. Neutrons were observed as a new formof radiation emerging from the collisions, and Chadwick correctlyinferred that they were previously unsuspected components of thenucleus that had been knocked out. As described in Conservation

Laws, Chadwick also determined the mass of the neutron; it is verynearly the same as that of the proton.

To summarize, atoms are made of three types of particles:

charge mass in units of

the proton’s mass

location in atom

proton +e 1 in nucleusneutron 0 1.001 in nucleus

electron −e 1/1836 orbiting nucleus

The existence of neutrons explained the mysterious masses of the elements. Helium, for instance, has a mass very close to fourtimes greater than that of hydrogen. This is because it containstwo neutrons in addition to its two protons. The mass of an atom is




o / A version of the Thomsapparatus modified for measurithe mass-to-charge ratios ions rather than electrons. small sample of the element question, copper in our exampis boiled in the oven to creaa thin vapor. (A vacuum pum

is continuously sucking on tmain chamber to keep it fro

accumulating enough gas to stthe beam of ions.) Some of tatoms of the vapor are ionized a spark or by ultraviolet light. Iothat wander out of the nozand into the region betwethe charged plates are thaccelerated toward the top of t

figure. As in the Thomson expement, mass-to-charge ratios ainferred from the deflection of tbeam.

essentially determined by the total number of neutrons and protons.The total number of neutrons plus protons is therefore referred toas the atom’s mass number .

Isotopes

We now have a clear interpretation of the fact that helium is

close to four times more massive than hydrogen, and similarly forall the atomic masses that are close to an integer multiple of themass of hydrogen. But what about copper, for instance, which hadan atomic mass 63.5 times that of hydrogen? It didn’t seem rea-sonable to think that it possessed an extra half of a neutron! Thesolution was found by measuring the mass-to-charge ratios of singly-ionized atoms (atoms with one electron removed). The techniqueis essentially that same as the one used by Thomson for cathoderays, except that whole atoms do not spontaneously leap out of thesurface of an object as electrons sometimes do. Figure o shows anexample of how the ions can be created and injected between thecharged plates for acceleration.

Injecting a stream of copper ions into the device, we find a sur-prise — the beam splits into two parts! Chemists had elevated todogma the assumption that all the atoms of a given element wereidentical, but we find that 69% of copper atoms have one mass, and31% have another. Not only that, but both masses are very nearlyinteger multiples of the mass of hydrogen (63 and 65, respectively).Copper gets its chemical identity from the number of protons in itsnucleus, 29, since chemical reactions work by electric forces. Butapparently some copper atoms have 63 − 29 = 34 neutrons whileothers have 65− 29 = 36. The atomic mass of copper, 63.5, reflects

the proportions of the mixture of the mass-63 and mass-65 varieties.The different mass varieties of a given element are called isotopesof that element.

Isotopes can be named by giving the mass number as a subscriptto the left of the chemical symbol, e.g., 65Cu. Examples:

protons neutrons mass number 1H 1 0 0+1 = 14He 2 2 2+2 = 412C 6 6 6+6 = 1214C 6 8 6+8 = 14262Ha 105 157 105+157 = 262

self-check AWhy are the positive and negative charges of the accelerating plates

reversed in the isotope-separating apparatus compared to the Thomson

apparatus? Answer, p. 205

Chemical reactions are all about the exchange and sharing of electrons: the nuclei have to sit out this dance because the forcesof electrical repulsion prevent them from ever getting close enough

Section 2.4 The Structure of Nuclei



q / The strong nuclear forcecuts off very sharply at a range ofabout 1 fm.

to make contact with each other. Although the protons do have avitally important effect on chemical processes because of their elec-trical forces, the neutrons can have no effect on the atom’s chemicalreactions. It is not possible, for instance, to separate 63Cu from 65Cuby chemical reactions. This is why chemists had never realized thatdifferent isotopes existed. (To be perfectly accurate, different iso-

topes do behave slightly differently because the more massive atomsmove more sluggishly and therefore react with a tiny bit less inten-sity. This tiny difference is used, for instance, to separate out theisotopes of uranium needed to build a nuclear bomb. The smallnessof this effect makes the separation process a slow and difficult one,which is what we have to thank for the fact that nuclear weaponshave not been built by every terrorist cabal on the planet.)

Sizes and shapes of nuclei

Matter is nearly all nuclei if you count by weight, but in termsof volume nuclei don’t amount to much. The radius of an individualneutron or proton is very close to 1 fm (1 fm=10−15 m), so even a biglead nucleus with a mass number of 208 still has a diameter of onlyabout 13 fm, which is ten thousand times smaller than the diameterof a typical atom. Contrary to the usual imagery of the nucleus as asmall sphere, it turns out that many nuclei are somewhat elongated,like an American football, and a few have exotic asymmetric shapeslike pears or kiwi fruits.


A Suppose the entire universe was in a (very large) cereal box, andthe nutritional labeling was supposed to tell a godlike consumer what per-centage of the contents was nuclei. Roughly what would the percentagebe like if the labeling was according to mass? What if it was by volume?

2.5 The Strong Nuclear Force, Alpha Decay

and Fission

Once physicists realized that nuclei consisted of positively chargedprotons and uncharged neutrons, they had a problem on their hands.The electrical forces among the protons are all repulsive, so thenucleus should simply fly apart! The reason all the nuclei in yourbody are not spontaneously exploding at this moment is that there

is another force acting. This force, called the strong nuclear force, isalways attractive, and acts between neutrons and neutrons, neutronsand protons, and protons and protons with roughly equal strength.The strong nuclear force does not have any effect on electrons, whichis why it does not influence chemical reactions.

Unlike electric forces, whose strengths are given by the simpleCoulomb force law, there is no simple formula for how the strongnuclear force depends on distance. Roughly speaking, it is effec-




p / A nuclear power plant at Ctenom, France. Unlike the cand oil plants that supply mof the U.S.’s electrical powernuclear power plant like this o

releases no pollution or greehouse gases into the Earth’s

mosphere, and therefore doescontribute to global warming. Twhite stuff puffing out of tplant is non-radioactive water vpor. Although nuclear powplants generate long-lived nuclewaste, this waste arguably pos

much less of a threat to the bsphere than greenhouse gaswould.

tive over ranges of ∼ 1 fm, but falls off extremely quickly at largerdistances (much faster than 1/r2). Since the radius of a neutron orproton is about 1 fm, that means that when a bunch of neutrons andprotons are packed together to form a nucleus, the strong nuclearforce is effective only between neighbors.

Figure r illustrates how the strong nuclear force acts to keepordinary nuclei together, but is not able to keep very heavy nucleifrom breaking apart. In r/1, a proton in the middle of a carbonnucleus feels an attractive strong nuclear force (arrows) from each

of its nearest neighbors. The forces are all in different directions,and tend to cancel out. The same is true for the repulsive electricalforces (not shown). In figure r/2, a proton at the edge of the nucleushas neighbors only on one side, and therefore all the strong nuclearforces acting on it are tending to pull it back in. Although all theelectrical forces from the other five protons (dark arrows) are allpushing it out of the nucleus, they are not sufficient to overcomethe strong nuclear forces.

In a very heavy nucleus, r/3, a proton that finds itself near theedge has only a few neighbors close enough to attract it significantlyvia the strong nuclear force, but every other proton in the nucleusexerts a repulsive electrical force on it. If the nucleus is large enough,the total electrical repulsion may be sufficient to overcome the at-traction of the strong force, and the nucleus may spit out a proton.Proton emission is fairly rare, however; a more common type of ra-dioactive decay1 in heavy nuclei is alpha decay, shown in r/4. The

1Alpha decay is more common because an alpha particle happens to be avery stable arrangement of protons and neutrons.

Section 2.5 The Strong Nuclear Force, Alpha Decay and Fission



r / 1. The forces cancel. 2. The forces don’t cancel. 3. In a heavy

nucleus, the large number of electrical repulsions can add up to a forcethat is comparable to the strong nuclear attraction. 4. Alpha emission. 5.Fission.

imbalance of the forces is similar, but the chunk that is ejected is an

alpha particle (two protons and two neutrons) rather than a singleproton.

It is also possible for the nucleus to split into two pieces of roughly equal size, r/5, a process known as fission. Note that inaddition to the two large fragments, there is a spray of individualneutrons. In a nuclear fission bomb or a nuclear fission reactor,some of these neutrons fly off and hit other nuclei, causing them toundergo fission as well. The result is a chain reaction.

When a nucleus is able to undergo one of these processes, it issaid to be radioactive, and to undergo radioactive decay. Some of the naturally occurring nuclei on earth are radioactive. The term

“radioactive” comes from Becquerel’s image of rays radiating outfrom something, not from radio waves, which are a whole differ-ent phenomenon. The term “decay” can also be a little misleading,since it implies that the nucleus turns to dust or simply disappears– actually it is splitting into two new nuclei with an the same totalnumber of neutrons and protons, so the term “radioactive transfor-mation” would have been more appropriate. Although the originalatom’s electrons are mere spectators in the process of weak radioac-




tive decay, we often speak loosely of “radioactive atoms” rather than“radioactive nuclei.”

Randomness in physics

How does an atom decide when to decay? We might imaginethat it is like a termite-infested house that gets weaker and weaker,

until finally it reaches the day on which it is destined to fall apart.Experiments, however, have not succeeded in detecting such “tick-ing clock” hidden below the surface; the evidence is that all atomsof a given isotope are absolutely identical. Why, then, would oneuranium atom decay today while another lives for another millionyears? The answer appears to be that it is entirely random. Wecan make general statements about the average time required for acertain isotope to decay, or how long it will take for half the atomsin a sample to decay (its half-life), but we can never predict thebehavior of a particular atom.

This is the first example we have encountered of an inescapable

randomness in the laws of physics. If this kind of randomness makesyou uneasy, you’re in good company. Einstein’s famous quote is“...I am convinced that He [God] does not play dice.“ Einstein’sdistaste for randomness, and his association of determinism withdivinity, goes back to the Enlightenment conception of the universeas a gigantic piece of clockwork that only had to be set in motioninitially by the Builder. Physics had to be entirely rebuilt in the20th century to incorporate the fundamental randomness of physics,and this modern revolution is the topic of book 6 in this series. Inparticular, we will delay the mathematical development of the half-life concept until then.

2.6 The Weak Nuclear Force; Beta Decay

All the nuclear processes we’ve discussed so far have involved re-arrangements of neutrons and protons, with no change in the totalnumber of neutrons or the total number of protons. Now considerthe proportions of neutrons and protons in your body and in theplanet earth: neutrons and protons are roughly equally numerousin your body’s carbon and oxygen nuclei, and also in the nickel andiron that make up most of the earth. The proportions are about50-50. But, as discussed in more detail in optional section 2.10, theonly chemical elements produced in any significant quantities by the

big bang2 were hydrogen (about 90%) and helium (about 10%). If the early universe was almost nothing but hydrogen atoms, whosenuclei are protons, where did all those neutrons come from?

The answer is that there is another nuclear force, the weak nu-clear force, that is capable of transforming neutrons into protons

2The evidence for the big bang theory of the origin of the universe was dis-cussed in book 3 of this series.

Section 2.6 The Weak Nuclear Force; Beta Decay



and vice-versa. Two possible reactions are

n → p + e− + ν [electron decay]

and

p → n + e+ + ν . [positron decay]

(There is also a third type called electron capture, in which a protongrabs one of the atom’s electrons and they produce a neutron anda neutrino.)

Whereas alpha decay and fission are just a redivision of the pre-viously existing particles, these reactions involve the destruction of one particle and the creation of three new particles that did notexist before.

There are three new particles here that you have never previ-ously encountered. The symbol e+ stands for an antielectron, whichis a particle just like the electron in every way, except that its elec-

tric charge is positive rather than negative. Antielectrons are alsoknown as positrons. Nobody knows why electrons are so common inthe universe and antielectrons are scarce. When an antielectron en-counters an electron, they annihilate each other, producing gammarays, and this is the fate of all the antielectrons that are producedby natural radioactivity on earth. Antielectrons are an example of antimatter. A complete atom of antimatter would consist of antipro-tons, antielectrons, and antineutrons. Although individual particlesof antimatter occur commonly in nature due to natural radioactivityand cosmic rays, only a few complete atoms of antihydrogen haveever been produced artificially.

The notation ν stands for a particle called a neutrino, and ν means an antineutrino. Neutrinos and antineutrinos have no electriccharge (hence the name).

We can now list all four of the known fundamental forces of physics:

• gravity

• electromagnetism

• strong nuclear force

• weak nuclear force

The other forces we have learned about, such as friction and thenormal force, all arise from electromagnetic interactions betweenatoms, and therefore are not considered to be fundamental forces of physics.




Decay of 212Pb example 2

As an example, consider the radioactive isotope of lead 212Pb. It

contains 82 protons and 130 neutrons. It decays by the process

n → p + e − + ¯ ν . The newly created proton is held inside the

nucleus by the strong nuclear force, so the new nucleus contains

83 protons and 129 neutrons. Having 83 protons makes it the

element bismuth, so it will be an atom of212

Bi.In a reaction like this one, the electron flies off at high speed

(typically close to the speed of light), and the escaping electronsare the things that make large amounts of this type of radioactivitydangerous. The outgoing electron was the first thing that tippedoff scientists in the early 1900s to the existence of this type of ra-dioactivity. Since they didn’t know that the outgoing particles wereelectrons, they called them beta particles, and this type of radioac-tive decay was therefore known as beta decay. A clearer but lesscommon terminology is to call the two processes electron decay andpositron decay.

The neutrino or antineutrino emitted in such a reaction prettymuch ignores all matter, because its lack of charge makes it immuneto electrical forces, and it also remains aloof from strong nuclearinteractions. Even if it happens to fly off going straight down, itis almost certain to make it through the entire earth without in-teracting with any atoms in any way. It ends up flying throughouter space forever. The neutrino’s behavior makes it exceedinglydifficult to detect, and when beta decay was first discovered nobodyrealized that neutrinos even existed. We now know that the neu-trino carries off some of the energy produced in the reaction, but atthe time it seemed that the total energy afterwards (not counting

the unsuspected neutrino’s energy) was greater than the total en-ergy before the reaction, violating conservation of energy. Physicistswere getting ready to throw conservation of energy out the windowas a basic law of physics when indirect evidence led them to theconclusion that neutrinos existed.

The solar neutrino problem

What about these neutrinos? Why haven’t you heard of thembefore? It’s not because they’re rare — a billion neutrinos passthrough your body every microsecond, but until recently almostnothing was known about them. Produced as a side-effect of the

nuclear reactions that power our sun and other stars, these ghostlikebits of matter are believed to be the most numerous particles in theuniverse. But they interact so weakly with ordinary matter thatnearly all the neutrinos that enter the earth on one side will emergefrom the other side of our planet without even slowing down.

Our first real peek at the properties of the elusive neutrino hascome from a huge detector in a played-out Japanese zinc mine, s. Aninternational team of physicists outfitted the mineshaft with wall-

Section 2.6 The Weak Nuclear Force; Beta Decay



s / This neutrino detector is

in the process of being filled withultrapure water.

to-wall light sensors, and then filled the whole thing with water sopure that you can see through it for a hundred meters, compared toonly a few meters for typical tap water. Neutrinos stream throughthe 50 million liters of water continually, just as they flood every-thing else around us, and the vast majority never interact with awater molecule. A very small percentage, however, do annihilate

themselves in the water, and the tiny flashes of light they producecan be detected by the beachball-sized vacuum tubes that line thedarkened mineshaft. Most of the neutrinos around us come fromthe sun, but for technical reasons this type of water-based detectoris more sensitive to the less common but more energetic neutrinosproduced when cosmic ray particles strike the earth’s atmosphere.

Neutrinos were already known to come in three “flavors,” whichcan be distinguished from each other by the particles created whenthey collide with matter. An “electron-flavored neutrino” creates anordinary electron when it is annihilated, while the two other typescreate more exotic particles called mu and tau particles. Think of the three types of neutrinos as chocolate, vanilla, and strawberry.When you buy a chocolate ice cream cone, you expect that it willkeep being chocolate as you eat it. The unexpected finding fromthe Japanese experiment is that some of the neutrinos are changingflavor between the time when they are produced by a cosmic ray andthe moment when they wink out of existence in the water. It’s asthough your chocolate ice cream cone transformed itself magicallyinto strawberry while your back was turned.

How did the physicists figure out the change in flavor? Theexperiment detects some neutrinos originating in the atmosphere

above Japan, and also many neutrinos coming from distant parts of the earth. A neutrino created above the Atlantic Ocean arrives inJapan from underneath, and the experiment can distinguish theseupward-traveling neutrinos from the downward-moving local vari-ety. They found that the mixture of neutrinos coming from belowwas different from the mixture arriving from above, with some of the electron-flavored and tau-flavored neutrinos having apparentlychanged into mu-flavored neutrinos during their voyage through theearth. The ones coming from above didn’t have time to changeflavors on their much shorter journey.

This is interpreted as evidence that the neutrinos are constantly

changing back and forth among the three flavors. On theoreticalgrounds, it is believed that such a vibration can only occur if neu-trinos have mass. Only a rough estimate of the mass is possible atthis point: it appears that neutrinos have a mass somewhere in theneighborhood of one billionth of the mass of an electron, or about10−39 kg.

If the neutrino’s mass is so tiny, does it even matter? It mattersto astronomers. Neutrinos are the only particles that can be used




t / A detector being lowerdown a shaft at the IceCuneutrino telescope in Antarctic

to probe certain phenomena. For example, they are the only directprobes we have for testing our models of the core of our own sun,which is the source of energy for all life on earth. Once astronomershave a good handle on the basic properties of the neutrino, theycan start thinking seriously about using them for astronomy. As of 2006, the mass of the neutrino has been confirmed by an accelerator-

based experiment, and neutrino observatories have been operatingfor a few years in Antarctica, using huge volumes of natural ice inthe same way that the water was used in the Japanese experiment.

A In the reactions n → p + e− + ¯ ν and p → n + e+ + ν, verify thatcharge is conserved. In beta decay, when one of these reactions happensto a neutron or proton within a nucleus, one or more gamma rays mayalso be emitted. Does this affect conservation of charge? Would it bepossible for some extra electrons to be released without violating chargeconservation?

B When an antielectron and an electron annihilate each other, theyproduce two gamma rays. Is charge conserved in this reaction?

2.7 Fusion

As we have seen, heavy nuclei tend to fly apart because each protonis being repelled by every other proton in the nucleus, but is onlyattracted by its nearest neighbors. The nucleus splits up into twoparts, and as soon as those two parts are more than about 1 fmapart, the strong nuclear force no longer causes the two fragmentsto attract each other. The electrical repulsion then accelerates them,causing them to gain a large amount of kinetic energy. This release

of kinetic energy is what powers nuclear reactors and fission bombs.

It might seem, then, that the lightest nuclei would be the moststable, but that is not the case. Let’s compare an extremely lightnucleus like 4He with a somewhat heavier one, 16O. A neutron orproton in 4He can be attracted by the three others, but in 16O, itmight have five or six neighbors attracting it. The 16O nucleus istherefore more stable.

It turns out that the most stable nuclei of all are those aroundnickel and iron, having about 30 protons and 30 neutrons. Just as anucleus that is too heavy to be stable can release energy by splitting

apart into pieces that are closer to the most stable size, light nucleican release energy if you stick them together to make bigger nucleithat are closer to the most stable size. Fusing one nucleus withanother is called nuclear fusion. Nuclear fusion is what powers oursun and other stars.

Section 2.7 Fusion



u / 1. Our sun’s source of energy is nuclear fusion, so nuclear fusion is also the source of power for all

life on earth, including, 2, this rain forest in Fatu-Hiva. 3. The first release of energy by nuclear fusion throughhuman technology was the 1952 Ivy Mike test at the Enewetak Atoll. 4. This array of gamma-ray detectors iscalled GAMMASPHERE. During operation, the array is closed up, and a beam of ions produced by a particleaccelerator strikes a target at its center, producing nuclear fusion reactions. The gamma rays can be studied forinformation about the structure of the fused nuclei, which are typically varieties not found in nature. 5. Nuclearfusion promises to be a clean, inexhaustible source of energy. However, the goal of commercially viable nuclearfusion power has remained elusive, due to the engineering difficulties involved in magnetically containing aplasma (ionized gas) at a sufficiently high temperature and density. This photo shows the experimental JETreactor, with the device opened up on the left, and in action on the right.

2.8 Nuclear Energy and Binding Energies

In the same way that chemical reactions can be classified as exother-mic (releasing energy) or endothermic (requiring energy to react), sonuclear reactions may either release or use up energy. The energiesinvolved in nuclear reactions are greater by a huge factor. Thou-sands of tons of coal would have to be burned to produce as much




energy as would be produced in a nuclear power plant by one kg of fuel.

Although nuclear reactions that use up energy (endothermicreactions) can be initiated in accelerators, where one nucleus isrammed into another at high speed, they do not occur in nature, noteven in the sun. The amount of kinetic energy required is simply

not available.

To find the amount of energy consumed or released in a nuclearreaction, you need to know how much nuclear interaction energy,U nuc, was stored or released. Experimentalists have determined theamount of nuclear energy stored in the nucleus of every stable el-ement, as well as many unstable elements. This is the amount of mechanical work that would be required to pull the nucleus apartinto its individual neutrons and protons, and is known as the nuclearbinding energy.

A reaction occurring in the sun example 3

The sun produces its energy through a series of nuclear fusionreactions. One of the reactions is

1H +2 H →3 He + γ

The excess energy is almost all carried off by the gamma ray (not

by the kinetic energy of the helium-3 atom). The binding energies

in units of pJ (picojoules) are:

1H 0 J2H 0.35593 pJ3He 1.23489 pJ

The total initial

nuclear energy is 0 pJ+0.35593 pJ, and the final nuclear energy

is 1.23489 pJ, so by conservation of energy, the gamma ray must

carry off 0.87896 pJ of energy. The gamma ray is then absorbed

by the sun and converted to heat.

self-check B

Why is the binding energy of 1H exactly equal to zero? Answer, p.

205

Conversion of mass to energy and energy to mass

If you add up the masses of the three particles produced in thereaction n → p + e− + ¯ ν, you will find that they do not equal the mass ofthe neutron, so mass is not conserved. An even more blatant example is

the annihilation of an electron with a positron, e−

+e+

→ 2 γ, in which theoriginal mass is completely destroyed, since gamma rays have no mass.Nonconservation of mass is not just a property of nuclear reactions. Italso occurs in chemical reactions, but the change in mass is too smallto detect with ordinary laboratory balances.

The reason why mass is not being conserved is that mass is be-ing converted to energy, according to Einstein’s celebrated equationE = mc 2, in which c stands for the speed of light. In the reaction

e− + e+ → 2 γ, for instance, imagine for simplicity that the electron and

Section 2.8 Nuclear Energy and Binding Energies



positron are moving very slowly when they collide, so there is no signif-icant amount of energy to start with. We are starting with mass and noenergy, and ending up with two gamma rays that possess energy butno mass. Einstein’s E = mc 2 tells us that the conversion factor betweenmass and energy is equal to the square of the speed of light. Sincec is a big number, the amount of energy consumed or released by achemical reaction only shows up as a tiny change in mass. But in nu-

clear reactions, which involve large amounts of energy, the change inmass may amount to as much as one part per thousand. Note that inthis context, c is not necessarily the speed of any of the particles. Weare just using its numerical value as a conversion factor. Note also thatE = mc 2 does not mean that an object of mass m has a kinetic energyequal to mc 2; the energy being described by E = mc 2 is the energyyou could release if you destroyed the particle and converted its massentirely into energy, and that energy would be in addition to any kineticor potential energy the particle had.

Have we now been cheated out of two perfectly good conservationlaws, the laws of conservation of mass and of energy? No, it’s justthat according to Einstein, the conserved quantity is E + mc 2, not E or

m individually. The quantity E + mc

2

is referred to as the mass-energy,and no violation of the law of conservation of mass-energy has yet beenobserved. In most practical situations, it is a perfectly reasonable totreat mass and energy as separately conserved quantities.

It is now easy to explain why isolated protons (hydrogen nuclei) arefound in nature, but neutrons are only encountered in the interior ofa nucleus, not by themselves. In the process n → p + e− + ¯ ν, thetotal final mass is less than the mass of the neutron, so mass is beingconverted into energy. In the beta decay of a proton, p → n+e+ + ν, , thefinal mass is greater than the initial mass, so some energy needs to besupplied for conversion into mass. A proton sitting by itself in a hydrogenatom cannot decay, since it has no source of energy. Only protonssitting inside nuclei can decay, and only then if the difference in potential

energy between the original nucleus and the new nucleus would resultin a release of energy. But any isolated neutron that is created in naturalor artificial reactions will decay within a matter of seconds, releasingsome energy.

The equation E = mc 2 occurs naturally as part of Einstein’s theoryof special relativity, which is not what we are studying right now. Thisbrief treatment is only meant to clear up the issue of where the masswas going in some of the nuclear reactions we were discussing.

Figure v is a compact way of showing the vast variety of thenuclei. Each box represents a particular number of neutrons andprotons. The black boxes are nuclei that are stable, i.e., that wouldrequire an input of energy in order to change into another. Thegray boxes show all the unstable nuclei that have been studied ex-perimentally. Some of these last for billions of years on the aver-age before decaying and are found in nature, but most have muchshorter average lifetimes, and can only be created and studied inthe laboratory.




v / The known nuclei, represented on a chart of proton number versus neutron number. Note the twonuclei in the bottom row with zero protons. One is simply a single neutron. The other is a cluster of founeutrons. This “tetraneutron” was reported, unexpectedly, to be a bound system in results from a 2002experiment. The result is controversial. If correct, it implies the existence of a heretofore unsuspected type o

matter, the neutron droplet, which we can think of as an atom with no protons or electrons.

The curve along which the stable nuclei lie is called the line of stability. Nuclei along this line have the most stable proportionof neutrons to protons. For light nuclei the most stable mixtureis about 50-50, but we can see that stable heavy nuclei have twoor three times more neutrons than protons. This is because theelectrical repulsions of all the protons in a heavy nucleus add upto a powerful force that would tend to tear it apart. The presenceof a large number of neutrons increases the distances among theprotons, and also increases the number of attractions due to the

strong nuclear force.

2.9 Biological Effects of Ionizing Radiation

As a science educator, I find it frustrating that nowhere in the mas-sive amount of journalism devoted to the Chernobyl disaster doesone ever find any numerical statements about the amount of radia-tion to which people have been exposed. Anyone mentally capable of

Section 2.9 Biological Effects of Ionizing Radiation



w / A map showing levels of

radiation near the site of theChernobyl disaster. At theboundary of the most highlycontaminated (bright red) areas,people would be exposed to

about 13,000 µ Sv per year, orabout four times the natural back-ground level. In the pink areas,which are still densely populated,the exposure is comparable

to the natural level found in ahigh-altitude city such as Denver.

understanding sports statistics or weather reports ought to be ableto understand such measurements, as long as something like thefollowing explanatory text was inserted somewhere in the article:

Radiation exposure is measured in units of Sieverts (Sv). Theaverage person is exposed to about 2000 µSv (microSieverts) eachyear from natural background sources.

With this context, people would be able to come to informedconclusions based on statements such as, “Children in Finland re-ceived an average dose of µSv above natural backgroundlevels because of the Chernobyl disaster.”

What is a Sievert? It measures the amount of energy per kilo-gram deposited in the body by ionizing radiation, multiplied by a“quality factor” to account for the different health hazards posedby alphas, betas, gammas, neutrons, and other types of radiation.Only ionizing radiation is counted, since nonionizing radiation sim-ply heats one’s body rather than killing cells or altering DNA. Forinstance, alpha particles are typically moving so fast that their ki-netic energy is sufficient to ionize thousands of atoms, but it ispossible for an alpha particle to be moving so slowly that it wouldnot have enough kinetic energy to ionize even one atom.

Notwithstanding the pop culture images of the Incredible Hulkand Godzilla, it is not possible for a multicellular animal to become“mutated” as a whole. In most cases, a particle of ionizing radiationwill not even hit the DNA, and even if it does, it will only affectthe DNA of a single cell, not every cell in the animal’s body. Typ-ically, that cell is simply killed, because the DNA becomes unableto function properly. Once in a while, however, the DNA may be

altered so as to make that cell cancerous. For instance, skin cancercan be caused by UV light hitting a single skin cell in the body of a sunbather. If that cell becomes cancerous and begins reproducinguncontrollably, she will end up with a tumor twenty years later.

Other than cancer, the only other dramatic effect that can resultfrom altering a single cell’s DNA is if that cell happens to be asperm or ovum, which can result in nonviable or mutated offspring.Men are relatively immune to reproductive harm from radiation,because their sperm cells are replaced frequently. Women are morevulnerable because they keep the same set of ova as long as theylive.

A whole-body exposure of 5,000,000 µSv will kill a person withina week or so. Luckily, only a small number of humans have ever beenexposed to such levels: one scientist working on the ManhattanProject, some victims of the Nagasaki and Hiroshima explosions,and 31 workers at Chernobyl. Death occurs by massive killing of cells, especially in the blood-producing cells of the bone marrow.

Lower levels, on the order of 1,000,000 µSv, were inflicted on




x / A typical example of ra

ation hormesis: the health mice is improved by low levof radiation. In this study, youmice were exposed to fairly hilevels of x-rays, while a contgroup of mice was not exposeThe mice were weighed, atheir rate of growth was takas a measure of their health. levels below about 50,000 µ S

the radiation had a beneficeffect on the health of the micpresumably by activating celludamage control mechanismThe two highest data poiare statistically significant at t99% level. The curve is a fita theoretical model. Redrafrom T.D. Luckey, Hormesis wIonizing Radiation , CRC Pre

1980.

some people at Nagasaki and Hiroshima. No acute symptoms resultfrom this level of exposure, but certain types of cancer are signifi-cantly more common among these people. It was originally expectedthat the radiation would cause many mutations resulting in birthdefects, but very few such inherited effects have been observed.

A great deal of time has been spent debating the effects of very

low levels of ionizing radiation. The following table gives some sam-ple figures.

maximum beneficial dose per day ∼ 10,000 µSvCT scan ∼ 10,000 µSvnatural background per year 2,000-7,000 µSvhealth guidelines for exposure to a fetus 1,000 µSvflying from New York to Tokyo 150 µSvchest x-ray 50 µSv

Note that the largest number, on the first line of the table, is themaximum beneficial dose. The most useful evidence comes from

experiments in animals, which can intentionally be exposed to sig-nificant and well measured doses of radiation under controlled con-ditions. Experiments show that low levels of radiation activate cel-lular damage control mechanisms, increasing the health of the or-ganism. For example, exposure to radiation up to a certain levelmakes mice grow faster; makes guinea pigs’ immune systems func-tion better against diptheria; increases fertility in trout and mice;improves fetal mice’s resistance to disease; increases the life-spans of flour beetles and mice; and reduces mortality from cancer in mice.3

This type of effect is called radiation hormesis.

There is also some evidence that in humans, small doses of ra-diation increase fertility, reduces genetic abnormalities, and reducesmortality from cancer. The human data, however, tend to be verypoor compared to the animal data. Due to ethical issues, one cannotdo controlled experiments in humans. For example, one of the bestsources of information has been from the survivors of the Hiroshimaand Nagasaki bomb blasts, but these people were also exposed tohigh levels of carcinogenic chemicals in the smoke from their burningcities; for comparison, firefighters have a heightened risk of cancer,and there are also significant concerns about cancer from the 9/11attacks in New York. The direct empirical evidence about radiationhormesis in humans is therefore not good enough to tell us anythingunambiguous,4 and the most scientifically reasonable approach is to

assume that the results in animals also hold for humans: small dosesof radiation in humans are beneficial, rather than harmful. However,

3Radiation Hormesis Overview, T.D. Luckey, www.radpro.com/641luckey.pdf 4For two opposing viewpoints, see Tubiana et al., “The Linear No-Threshold

Relationship Is Inconsistent with Radiation Biologic and Experimental Data,”Radiology, 251 (2009) 13 and Little et al., “ Risks Associated with Low Dosesand Low Dose Rates of Ionizing Radiation: Why Linearity May Be (Almost) theBest We Can Do,” Radiology, 251 (2009) 6.

Section 2.9 Biological Effects of Ionizing Radiation



aa / The Crab Nebula is remnant of a supernova expsion. Almost all the elements oplanet is made of originated such explosions.

ab / Construction of the ULAC accelerator in Germany, oof whose uses is for experimento create very heavy artificelements. In such an experimefusion products recoil throughdevice called SHIP (not showthat separates them based

their charge-to-mass ratios it is essentially just a scaled-version of Thomson’s appatus. A typical experiment rufor several months, and out

the billions of fusion reactioinduced during this time, oone or two may result in tproduction of superheavy atomIn all the rest, the fused nucle

breaks up immediately. SHis used to identify the smnumber of “good” reactions aseparate them from this intenbackground.

the Hanford, Washington weapons plant.


A Should the quality factor for neutrinos be very small, because theymostly don’t interact with your body?

B Would an alpha source be likely to cause different types of cancerdepending on whether the source was external to the body or swallowedin contaminated food? What about a gamma source?

2.10 The Creation of the Elements

Creation of hydrogen and helium in the Big bang

Did all the chemical elements we’re made of come into being inthe big bang?6 Temperatures in the first microseconds after the bigbang were so high that atoms and nuclei could not hold togetherat all. After things had cooled down enough for nuclei and atomsto exist, there was a period of about three minutes during which

the temperature and density were high enough for fusion to occur,but not so high that atoms could hold together. We have a good,detailed understanding of the laws of physics that apply under theseconditions, so theorists are able to say with confidence that theonly element heavier than hydrogen that was created in significantquantities was helium.

We are stardust

In that case, where did all the other elements come from? As-tronomers came up with the answer. By studying the combinationsof wavelengths of light, called spectra, emitted by various stars, theyhad been able to determine what kinds of atoms they contained.

(We will have more to say about spectra at the end of this book.)They found that the stars fell into two groups. One type was nearly100% hydrogen and helium, while the other contained 99% hydrogenand helium and 1% other elements. They interpreted these as twogenerations of stars. The first generation had formed out of cloudsof gas that came fresh from the big bang, and their compositionreflected that of the early universe. The nuclear fusion reactionsby which they shine have mainly just increased the proportion of helium relative to hydrogen, without making any heavier elements.The members of the first generation that we see today, however, areonly those that lived a long time. Small stars are more miserly with

their fuel than large stars, which have short lives. The large stars of the first generation have already finished their lives. Near the end of its lifetime, a star runs out of hydrogen fuel and undergoes a seriesof violent and spectacular reorganizations as it fuses heavier andheavier elements. Very large stars finish this sequence of events byundergoing supernova explosions, in which some of their material is

6The evidence for the big bang theory of the origin of the universe was dis-cussed in book 3 of this series.

Section 2.10 The Creation of the Elements



flung off into the void while the rest collapses into an exotic objectsuch as a black hole or neutron star.

The second generation of stars, of which our own sun is an exam-ple, condensed out of clouds of gas that had been enriched in heavyelements due to supernova explosions. It is those heavy elementsthat make up our planet and our bodies.

Artificial synthesis of heavy elements

Elements up to uranium, atomic number 92, were created bythese astronomical processes. Beyond that, the increasing electricalrepulsion of the protons leads to shorter and shorter half-lives. Evenif a supernova a billion years ago did create some quantity of anelement such as Berkelium, number 97, there would be none left inthe Earth’s crust today. The heaviest elements have all been createdby artificial fusion reactions in accelerators. As of 2006, the heaviestelement that has been created is 116.7

Although the creation of a new element, i.e., an atom with a

novel number of protons, has historically been considered a glam-orous accomplishment, to the nuclear physicist the creation of anatom with a hitherto unobserved number of neutrons is equally im-portant. The greatest neutron number reached so far is 179. Onetantalizing goal of this type of research is the theoretical predictionthat there might be an island of stability beyond the previously ex-plored tip of the chart of the nuclei shown in section 2.8. Just ascertain numbers of electrons lead to the chemical stability of the no-ble gases (helium, neon, argon, ...), certain numbers of neutrons andprotons lead to a particularly stable packing of orbits. Calculationsdating back to the 1960’s have hinted that there might be relatively

crowell physics selection

Documents