newton new view

7/28/2019 Newton New View

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Intr oducti onNewton was born in

Woolsthorpe, Lincolnshire, inEngland. Newton, Sir Isaac (1642-1727), English physicist,mathematician, and naturalphilosopher, considered one ofthe most important scientists ofall time. Newton formulated lawsof universal gravitation and

motionlaws that explain howobjects move on Earth as well asthrough the heavens (Mechanics). He established themodern study of opticsor the behavior of lightandbuilt the first reflecting telescope. His mathematicalinsights led him to invent the area of mathematicscalled calculus.

Newton stated his ideas in several published

works, two of which, Philosophiae Naturalis PrincipiaMathematica (Mathematical Principles of NaturalPhilosophy, 1687) and Opticks (1704), are consideredamong the greatest scientific works ever produced.Newtons revolutionary contributions explained theworkings of a large part of the physical world inmathematical terms, and they suggested thatscience may provide explanations for otherphenomena as well.

Published in 1687, Newtons two-volumePhilosophiae Naturalis Principia Mathematica(Mathematical Principles of Natural Philosophy)contains his important three laws of motion, alsocalled Newtons laws, excerpted here.Newton carried


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Galileos equation forward by introducing the conceptof force. He began by restating Galileos law ofinertia as a situation applying to the absence offorce. Newtons first law states A body in motion

remains in motion (with constant velocity) unlessacted upon by an external force. Rest, in Newtonslaws, is merely an example of motion with zerovelocity. So force is defined as the agency thatchanges the state of motion, and thus the velocity, ofa body.

Newtons famous second law relates a to a forceacting on the object via the equation F = ma. The

quantity m is the stuff inside the object, whichNewton called inertial mass. The bigger the value ofm, the larger the force required to get the objectmovingthat is, accelerating. Applied to Galileosexperiments, F is the force of gravity tugging at theobject and aimed toward the center of Earth. F iscarefully defined as the sum of all forces.

Newtons second law accounted for the motion

of planets pulled by the Suns gravitational force; themotion of projectiles, influenced by air and the pull ofgravity; and the tides, which are caused by oceanwaters pulled by the Sun and the Moon. Newtonproved mathematically what Kepler had concludedfrom observationsthat planets move in ellipticalpaths. To make this proof, he had to know the preciseform of F. F must change, depending on the distance

of the planet from the Sun. So Newton had to guessthe way the force of the Sun on a planet growsweaker as the distance between these two objectsincreases. His guess was an inverse square law,which states that the force of gravity is inverselyproportional to the square of the distance between


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the two objects.Newtons equations also took into account the

fact that objects have two kinds of masses: inertialmass that resists motion and gravitational mass that

encourages motion. He wrote another equationillustrating that for any object, the two types ofmassesgravitational mass and inertial massareequal. Einstein would return to this idea in hisgeneral theory of relativity, wherein he made theequality of inertia and gravitational masses a keypoint.The curious behavior of objects in space capsules,

which we call weightlessness, works on the sameprinciple. The astronaut, his sandwich, and his drinkall float together, apparently without gravity. Butgravity is still pulling on the astronaut, and thecapsule, and the sandwich. They respond accordingto their inertia, and the two effects cancel each otherout.

Newtons work was vital to the evolution of

modern physics.Newton's third law-" To every action, there is equaland opposite reaction".

Wave Pheno me na in Gen era lWhen we speak of waves, we generally think of thosewe see in the water, but there are in fact two

different kinds. We have already discussed this in thecontext of the transmission of seismic waves throughthe Earths interior but it is so important that I repeatsome of that discussion here.

Longitudinal waves, which I demonstratedwith a `slinky toy' in class, consist of


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disturbances, which pass through a medium byvirtue of to-and-fro motions of and collisionsbetween atoms. Sound is an example. I speak,and my vibrating vocal cords cause the atoms

near me to `jiggle' in some way. As they moveaway from me and towards you, they bump intoother atoms and rebound, but the other atomscarry the motion forward until they in turn collidewith yet more atoms, and so on. The `jiggling'eventually reaches your ears (at the speed ofsound, about 300 metres per second in theatmosphere) and set your eardrums to vibrating

in a way that your brain can interpret as a soundwhich is comprehensible (I hope).

Transverse waves, which I demonstrated inclass by wiggling up and down one end of astretched rope, consist of disturbances which areacross (or transverse to) the direction in whichthe wave itself is going. For instance, if you were

a bug on the rope, you would be moving up anddown as the ripples pass by, but the disturbanceitself passes along the length of the rope. Thefamiliar wind-driven waves at the surface of abody of water are also transverse: a cork floatingin the water bobs up and down where it is, whilethe disturbance passes through the water alongthe surface.

A common feature in both of these types of waves isthat the medium itself does not suffer any netdisplacement: the atoms go back and forth or up anddown, but if you look again a few minutes later youwill see that they are still near where they were atthe start. Likewise, a cork floating in the water


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merely bobs up and down `in place'. There is roomfor confusion here, because you are used to seeingwaves breaking onto a beach. At the shore, individualatoms of water actually move quite a lot as the

waves spill up onto the sand; but this phenomenon iscaused by the complex interactions between thetransverse wave and the lake bottom, which itself issloped and irregular, causing friction and so on. Inthe open sea, the wave disturbance merely passesthrough the watery medium without carrying theconstituent atoms along with it.

In what follows, we will consider the evidence forlight being a transverse wave. (In preparation forthat, you should read carefully the definitions ofwavelength, frequency, and so on that you will findon page 164 of the text.) When we speak of light as awave phenomenon, we try to find ways in which itsbehaviour is not consistent with what you wouldexpect from a bunch of little energy-carrying`bullets'. Are there such ways?

Diffraction: Waves Spreading Out from a HoleConsider ocean waves arriving at a breakwater with a

hole in it, as shown inthe figure:

The breakwater could

consist of a cementwall, say, with anopening - the sort ofthing you find near ayacht harbour. Theboats enter through


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the gap in the wall, while the wall itself providessome protection against very strong waves crashingdirectly into the docks and the shore.

Now, if you were standing at point X on theshore, directly opposite the gap in the wall, youwould surely see waves breaking at your feet ontothe beach. But what if you were standing at point Y,perhaps fifty meters north along the shore? Wouldthe water there be as completely still as a millpond?No indeed! As you know from experience, thedisturbances in the water would indeed spread out asshown in the figure, and people along a wide stretchof the shoreline would see the water slosh up anddown, although less vigorously than would theperson directly opposite the gap.

This is quite distinct from the behaviour of lumpsthat move in straight lines - things like bullets, forinstance. If you are ever being shot at, just duckthrough the cover afforded by a doorway in a cement

wall and you are safe; there is no need to fear thatthe bullets will 'smear out' in all directions as theypass through the doorway.

But now, on a much smaller scale, think of asource of light to the left of the figure and shiningonto a little hole in an opaque card. If the light waslike bullets, you would expect to see a tiny distinctspot of light directly opposite the hole, but no light at

any other location. Instead, you will find that itspreads out into a fuzzy blob. (In fact, it does morethan produce just a fuzzy blob. We would also seesome interesting structure, including rings and soforth, which we will not concern ourselves with here,other than to note that this too is well understood in


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terms of the wave nature of light.

You may protest that you do not see this in day-to-day life! In general, an object held up in front of a

source of light seems to cast a very sharp shadow.(Imagine making hand-shadows to entertain yourkids at night, for instance.) That is true: mostshadows do look quite sharp. The effects ofdiffraction are really only important if the hole is verysmall (comparable to the wavelength of light itself).In other words, light which passes through anextremely tiny pinhole gets smeared out into arelatviely large fuzzy patch; but if the hole is muchlarger, the effect is not readily seen.

Diffra ctio n: Waves `Bending' Around CornersWe have considered a hole in a seawall (or, in

the context of the bullets, a doorway in a cementwall), but in fact the phenomenon is more general

than this. The effects of diffraction are seen whereverthere is a sharp discontinuity. Waves, for example,can `bend,' or diffract, around corners. Indeed, this ishow you can hear voices around corners, even ifthere are no walls or other buildings to provideechoes or reverberation. Sound waves diffract.

Light can be shown to do the same thing. Onceagain this occurs on very small scales, since thewavelengths of visible light are so short.Consequently, one doesn't notice such phenomena inday-to-day life. As evidence, however, I showed youa picture in class, one in which we could see thesharp edge of a razor blade and a so-called `shadow


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diffraction pattern,' a series of dark lines parallel tothe edges of the razor. (I do not want to go into thetechnical details of how these lines are formed.Suffice it to say that it is fundamentally due to this

`bending around corners.') This is further evidencethat light behaves like a wave.

By the way, you might be tempted to think thatthe phenomenon of twilight is caused by soemthinglike this. Does the light from the sun diffract aroundthe edge of the Earth so that we still see a bit ofsunlight even after it has set? The answer is no - or,at least, that this effect is utterly negligible. Twilightsare caused by the gas and dust in the Earth'satmosphere. If the Earth were completely airless, wewould be plunged into pitch darkness the instant thesun set. Instead, we enjoy lingering twilights becauseof the scattering (bouncing) of light off moleculesand atoms high in the Earth's atmosphere.

In terfere nce :Waves Interacting with OneAnother

Just above, we considered ocean waves arrivingat a breakwater with a single hole in it. A moreinteresting effect would be produced by a breakwater

with two holes in it, as shown in the following figure:


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Upon thearrival ofparallel

waves fromthe open sea,each holeacts as anindependentcentre of newsets of wavespropagating

out towardsthe beach. Inthe figure, I

have colour-coded the new waves, with the hole atthe top producing a set of expanding black ripples(one of which has been coloured green, for reasonsto become obvious in a moment) and the one at thebottom producing a set of expanding blue ripples

(one of which has been coloured red). At the right ofthe picture, the dotted parts of a few of the wavesshow how they would have continued had they notrun up against the beach, which is represented bythe thin black line.Now ask yourself what you would see if you werestanding on the beach in the various locationsindicated.

If you were exactly half-way between the twoholes in the breakwater, at the point markedwith a large red letter 'A', you would be seeing apeak (an upward surge of water, represented bythe solid green line) in the wave coming from thetop hole, and a peak in the wave coming from


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the bottom hole (the solid red line). At thatlocation, therefore, the upward surge of thewater would be redoubled as the independentwaves arrive and their effects add. A moment

later, you would see a trough (a downwarddisplacement) in the waves arriving from the tophole, and another trough in the wave patternarriving from the bottom hole, so the the totaldownward effect would also be doubled. Theseeffects would repeat again and again at thefrequency of the original waves.

In other words, at point A you would see waveswhich are enhanced in effect because they arearriving in phase from the two holes in thebreakwater. This is called constructiveinterference. In short, if you were at position A,you would see the water going up and down withgreat vigour!

You can identify other positions where the

behaviour would be qualitatively similar. At thepoints labelled 'B1' and 'B2,' for instance, wewould expeience the coincident arrival of a wavefrom the top hole and another from the bottomhole. This situation differs from position A in thatthe arriving waves did not set off at the sametime from the two holes. (At point B1, we areseeing the arrival of the black wave which is just

ahead of the green one, whereas we see thearrival of the blue wave which is fourwaves infront of the red one.) But the effect is the same:we experience a vigourous up-and-down of thewater as these disturbances arrive in phase.


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Now look at the position marked C1. Here we seean upward peak from the top hole (the greenwave), but a downward trough from the bottomhole (we are exactly halfway between the

successive crests represented by the bluewaves). The net displacement of the water iszero, and that will always be the case. Whateverdisturbance is arriving from the top hole, adisturbance of equal size but in the oppositedirection will be arriving from the bottom hole.This, then, is a region ofdestructiveinterference, and the water should be as smooth

as a millpond all the time at that particularlocation -- in principle!

In real life, of course, things are not this simple.Water at the seaside is running up against theshore, which may be of irregular depth androughness, and the wave disturbances reboundoff the shore. Still, if you were sitting on theshore you would notice that there are regionswhere the water seems quite placid, and otherregions where the water is sloshing up and downwith great vigour.

The remarkable thing is that lightcan be shown toproduce the same kind of interference phenomena.There is a classical physics experiment known as theYoung double-slit experiment. It duplicates, on a tinyscale, the two-holed ocean breakwater situation justdescribed. Young cut two tiny slits into an opaquescreen onto which he shone a light of a well-definedcolour (which means, as we will see, that it containslight of just one wavelength). This did not producetwo bright images, one of each slit, on the wall


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A century or more ago, it was believed that therewas a medium through which light propagated, amedium called the luminiferous (`light-bearing')ether. Since light passes between the stars and

planets, it was obvious that the ether had to fill allspace. Yet the Earth has clearly been orbiting the sunin a more-or-less-unchanging orbit for many millionsof years, so it must experience a negligible amountof `wind resistance' from the ether. Thisconsideration, and the extremely high speed withwhich light travels, implied some very unusualproperties for the ether. A great goal of late 19th-

century physics, then, was to find absolute proof ofthe existence of the ether, and to learn more aboutits properties.

The death-knell came with a famous experiment,carried out by Michelson and Morley, which seemedto show that there was no ether. An almost directconsequence of this was Einstein's development ofhis special theory of relativity, in 1905, a theorywhich completely changed the way in which we thinkabout space and time. I will not describe thosedevelopments now since they fit in more naturallynearer the end of the course. But the important pointis that I want you to realize that light is notanalogous to sound and other mechanicaldisturbances which pass through a substance or amedium. It can, and does, travel through the true

vacuum of space.

Ele ctromag neti c Wave sThe modern physics interpretation is that light is

a wave which consists of changing electric and


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magnetic fields which propagate through space (seethe figure below). Before going further, let us remindourselves of what we mean by an electric field.

We say that an electric field exists in a location ifa charged particle, like a proton, feels an electricalforce there (perhaps because of the presence ofother charged particles in its vicinity, for instance).That is what makes a spark leap from your fingertipto a doorknob when you build up what we call `staticelectricity' -- the electrical forces cause thenegatively-charged electrons to leap across the gap.

Light can be thought of as a transverse wave movingat high speed (300,000 kilometers per second)through intervening space, and consisting of rapidlychanging electric and magnetic fields.

Can youpredict the

effects such awave mighthave as itpasses by?

Well, one answer is that a charged particle (like anelectron) sitting by itself in empty space should 'bobup and down' as the wave goes by, just as a corkbobs up and down in the water when a wave passes

by. (As you can see from the figure, there is also achanging magnetic field, which is at right angles tothe electric field. You can imagine a small compassturning back and forth in quick response to thischanging magnetic field as the wave passes by.)


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The reality of this interpretation can be tested.Take a strip of metal which is a good conductor (thatis, one in which the electrons are fairly free to move)and send light of some wavelength (and associated

frequency) towards it. Then design some simpleelectronics to detect whether or not the electrons areindeed bobbing up and down, an effect which wouldbe tanatamount to producing small electric currentsof varying size inside the conductor.

This is exactly what happens in your radioantenna or TV antenna!. The signal which isbroadcast from the radio or TV station is not visiblelight, of course, so our eyes are not sensitive to it,but it is light (electromagnetic radiation) none theless. It makes the electrons in the radio antenna``bob up and down'', and the small electric currentsso generated are detected, amplified, and used todetermine how to make your speakers vibrate. Thisin turn creates the sound waves which you hear.

Please note an important distinction. Radiowaves are light, not sound. They are used by thecircuitry in your radio to determine how to make thespeakers vibrate, and that is where the sound comesfrom. Radio astronomers are collectingelectromagnetic radiation from the stars andgalaxies, not sound (which could never pass throughthe vacuum of space anyway).


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Gra vitati onNewton's Law of Gravity is not precise in

extreme circumstances, such as very high velocitiesor very strong gravity. For cases such as these,Einstein's General and Special Relativity theories areneeded. However, in most other cases, and

especially those that we are familiar with on Earth,Newton's Law works extremely well.

It is based upon his laws ofmotion, and it shows how two objectsexhibit a force upon the other. It isthe equation to the right. It says

that the gravitational force experienced is equal to agravitational constant times both masses divided by

the distance between them squared. The value "G" isan extremely small number, and therefore thegravitational force is extremely weak - the weakest ofthe four fundamental forces. This law also shows thatthe force of gravity dies off with the square of thedistance. This means that if you are twice as faraway from something, then the gravitational forceyou experience is 1/4 as much. if distance is trebled,

the force becomes one-ninth as much.He had also discovered the law stating the

centrifugal force (or force away from the center) of abody moving uniformly in a circular path. However,he still believed that the earth's gravity and themotions of the planets might be caused by the action


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of whirlpools, or vortices, of small corpuscles. Hethought of circular motion as the result of a balancebetween two forces--one centrifugal, the othercentripetal (toward the center)--rather than as the

result of one force, a centripetal force, whichconstantly deflects the body away from its inertialpath in a straight line. Earth's gravity extended tothe Moon, counterbalancing its centrifugal force.From his law of centrifugal force and Kepler's thirdlaw of planetary motion, Newton deduced that thecentrifugal (and hence centripetal) force of the Moonor of any planet must decrease as the inverse square

of its distance from the center of its motion. Newtonapplied his mathematical talents & proved that if abody obeys Kepler's second law (which states thatthe line joining a planet to the sun sweeps out equalareas in equal times), then the body is being actedupon by a centripetal force. This discovery revealedfor the first time the physical significance of Kepler'ssecond law.

Newton succeeded in showing that a bodymoving in an elliptical path and attracted to onefocus must indeed be drawn by a force that varies asthe inverse square of the distance. By commonconsent the Principia is the greatest scientific bookever written. Within the framework of an infinite,homogeneous, three-dimensional, empty space anda uniformly and eternally flowing "absolute" time,

Newton fully analyzed the motion of bodies inresisting and no resisting media under the action ofcentripetal forces. The results were applied toorbiting bodies, projectiles, pendulum, and free-fallnear the Earth.


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Law s O f M oti onNewton's Laws of Motion are still used by physicistsall over the world. . Everything in that genre ofphysics is based upon these three laws:

1. Every object has uniform motion unless actedupon by a force.

2. The force applied to an object is equal to theobject's mass times the resulting acceleration:

3. For every action, there is an equal and oppositereaction.

These laws are used to describe everything fromthrowing a ball to the merging of galaxies.


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The First Law: Inertia Formalized

As we noted before, there seem coincidentally to

be many examples of physical laws `in threes.' Hereis another example: Newton's famous three laws ofmotion. While these warrant careful consideration,and while they can be expressed in technical andmathematical terms, my earnest wish is that you willdevelop a complete intuitive understanding of whatthey mean qualitatively. To clarify yourunderstanding, therefore, let us consider them insimple conversational terms.

The first law is merely a restatement, intechnical terms, of the notion ofinertia, a conceptintroduced by Galileo. Newton now makes explicit theunderstanding that an object in any state of motion(including rest) will remain unchanged in that state(which means that those at rest will remain at rest)unless some unbalanced force is acting.

The word `unbalanced' merely acknowledgesthat we don't expect any motion to result frombalanced forces. If you and your friend both push ona car, one at the front and one at the back, the forceswill balance each other and nothing will happen. Thatis, all kinds of forces can be present, but unless thereis an excess force in some direction, there will be nochange in the state of motion of the body being

pushed or pulled.

Let us consider an immediate implication whichfollows from the First Law. Think of the space shuttleand its astronauts orbiting the Earth. The shuttledoes not move in a straight line, but rather follows a


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curved path around the Earth. This must mean thatsome force is acting on it!The force is gravity, as wewill see: if the space shuttle did not feel thegravitational force of the Earth, it would simply move

in a straight line, and gradually leave the Earthbehind. In other words, the shuttle is mostemphatically not beyond the Earth's gravity, as iscommonly believed, despite the fact that theastronauts experience weightlessness. (I will returnto this point later.) This consideration also makesclear the incorrectness of Galileo's thinking: hebelieved that the circular motion of the moon around

the Earth was a natural 'coasting' which was relatedto inertia withoutthe requirement of any forces atplay.

The Second Law: When Forces AreUnbalanced

The second law, perhaps the most famous in allof physics, merely makes quantitative somethingwhich we know already by everyday experience. Ifyou kick a tennis ball, it moves away quickly. If youkick a cannon ball, it does not, even if you use thesame amount of force. The difference is that massiveobjects (those containing many atoms and lots ofmaterial) are not easily set it motion. This is becausethey have lots ofinertia (resistance to beingaccelerated from one state of motion, such as rest,

into another state of motion).

The law is usually given in the form of a verysimple equation: F = ma. Let us think in words aboutwhat this means!


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Suppose first of all that you have an object of agiven mass (m) -- that is, an object containingsome total number of atoms, some total amountof matter. If a tiny force Facts on it, it will

accelerate at a given rate a, which will be rathersmall (since Fis small). If you increase the forceFto some larger value, the acceleration a will belarger. In other words, bigger forces makeobjects accelerate more rapidly than smallforces do. A powerful engine can get your car upto highway speed more quickly than a weakengine would.

Please note, though, that even a small force canproduce a large final speed if it is allowed to actfor long enough! The small force generates asmall acceleration, so the object gains speedonly very slowly; but if the force is applied formany minutes (or hours, or years) the body maywind up moving quite quickly after all. This willbecome important later on, when we considerinterstellar travel. We will have to consider thealternate merits of accelerating to high speedvery quickly, using powerful rockets for a brieftime, or accelerating rather slowly, using feeblerockets which are allowed to burn for a very longtime.

Now consider a situation in which you have a

force of fixed size (F) at your disposal - say, allthe strength you can muster with your two armsin trying to clear stones and boulders from yourgarden. If you apply the full force to a stone ofsmall mass, (m) you can really send it flying (andof course in practice you would not bother


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applying all your force to it, but rather conserveyour energy for the more taxing jobs). But a verymassive stone, with large m, would beaccelerated only a little, and you would really

have to struggle to make it move perceptibly.

By the way, this is an opportune moment to warnyou about a very common misuse of a word. Inphysics, a massive body is one which contains a lotof matter in total -- many atoms, or atoms whichthemselves are particularly massive because theycontain many protons and neutrons. This may havenothing to do with the size of the object. A small leadblock may be much more massive -- much moreresistant to being set in motion by a push, forinstance -- than a much larger beach ball.

My experience is that students often use the word"massive" to mean nothing more than "big" (often inthe sense of "awesome"). Later in the course, we willlearn, for example, that as the sun uses up its

nuclear fuel, it will expand enormously, becoming ared giant star of such large size that the Earth maywind up inside its outermost parts. But the sun willbe no more massive at that time than it is now -- itwill contain as many atoms as it ever did, and will nothave 'put on weight.' Of course, it will be much lessdensely packed on average: the atoms will be morewidely spread, but the total mass will not have

changed. Be very careful about how you use thisword, which has a very precise physical meaning!

The Third Law: Action and Reaction

The third law, also known as the "action-reaction" law, is one that causes many people a lot of


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confusion. Partly this is because it is used in somesituations as a kind of vague metaphor for humanbehaviour. (If you get mad at me, I'll react by gettingmad at you.) But in physics it has a very clearcut

meaning. When one body acts on another (as when Iuse my finger to push a book across the table), thenthere is a reaction of equal size acting the other way(so my finger feels a force which we register as theresistance of the book to being moved).

One reason for confusion is that a lot of peoplethink that if two forces are "equal and opposite" thennothing will happen -- they must cancel out. Whythen does anything ever move at all? The answer, ofcourse, is that the forces do notcancel! They act ondifferent bodies, and can have an effect. When youdo a pushup, for instance, you are pressing down onthe ground, pushing it away from you. The equal andopposite force (the "reaction" of the Earth acting onyou) pushes you up and away from the ground. Infact, under the influence of these two forces of equalsize, both you and the Earth move, but the Earth isso huge and massive that it budges an immeasurablysmall amount (remember Newton's second law!),while your body moves perceptibly.

By the way, it is worth thinking for a momentabout how these forces are transmitted. When youdo a pushup, you are flexing your arms in such a way

as to push your constituent atoms into the ground, orat least try to. If the ground had no structural rigidity-- imagine doing a pushup on water! -- your handswould merely slide seamlessly into it. As it is, though,the material is held in a rigid configuration by theelectric forces between all the constituent atoms,


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molecules, and crystals. You are trying to force yourconstituent atoms (those in your hands and fingers)into these already crowded regions. As your atomsare pushed ever closer to those of the ground, the

electric repulsion between the various particlesresists the motion and stops your progress. Thisforce, applied to the Earth, pushes it away from you;and the reaction force pushes you away from theEarth. Again, the reason that you are lifted bodily isbecause you have some rigidity of your own. If youhad arms like cooked spaghetti, you would merelyflop onto the floor. (Have you ever tried topush a car

with a rope?)

It does not take volition or conscious intent tomake a force act on a body, or to generate a reactionforce. Consider a brick sitting on the floor, forinstance. I will anticipate Newton's introduction ofgravitation to point out what you already know: thegravity of the Earth is pulling down on the brick, andif there were no floor there it would merelyaccelerate downwards, or fall, in accordance withNewton's second law. But, just as with you and thepushup, the gravitational tug downwards has theeffect of trying to intermingle the atoms of the brickwith those of the floor. The repulsive force betweenthe electrically-charged constituents of the atomsresists that action (Newton's third law) and is strongenough to hold the brick up.


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Opti csNewton's optical research began during his

undergraduate years at Cambridge. In 1665-1666,Newton performed a number of experiments on thecomposition of light. Guided initially by the writingsof Kepler and Descartes, Newton's main discoverywas that visible (white) light is heterogeneous--thatis, white light is composed of colors that can beconsidered primary. Through a brilliant series of

experiments, Newton demonstrated that prismsseparate rather than modify white light.Newton also demonstrated that the colors of the

spectrum, once thought to be qualities, correspondto an observed and quantifiable 'degree ofRefrangibility. 'Newton's most famous experiment,the experimentum crucis, demonstrated his theory ofthe composition of light. Briefly, in a dark room

Newton allowed a narrow beam of sunlight to passfrom a small hole in a window shutter through aprism, thus breaking the white light into an oblongspectrum on a board. Then, through a small aperturein the board, Newton selected a given color (forexample, red) to pass through yet another apertureto a second prism, through which it was refractedonto a second board. What began as ordinary whitelight was thus dispersed through two prisms?

Newton's 'crucial experiment' demonstrated thata selected color leaving the first prism could not beseparated further by the second prism. The selectedbeam remained the same color, and its angle ofrefraction was constant throughout. Newton


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concluded that white light is a 'Heterogeneousmixture of differently refrangible Rays' and thatcolors of the spectrum cannot themselves beindividually modified, but are 'Original and connate

properties. 'The Opticks of 1704, which first appearedin English, is Newton's most comprehensive andreadily accessible work on light and color. InNewton's words, the purpose of the Opticks was 'notto explain the Properties of Light by Hypotheses, butto propose and prove them by Reason andExperiments.' Divided into three books, the Opticksmoves from definitions, axioms, propositions, and

theorems to proof by experiment. A subtle blend ofmathematical reasoning and careful observation, theOpticks became the model for experimental physicsin the 18th century.

Newton showed that the Spectrum was too longto be explained by the accepted theory of thebending (or Refraction) of light by dense media. Theold theory said that all rays of white light striking the

prism at the same angle would be equally refracted.These discoveries led Newton to the logical, buterroneous, conclusion that telescopes usingrefracting lenses could never overcome thedistortions of chromatic dispersion. He thereforeproposed and constructed a reflecting telescope (seeTelescope, Optical), the first of its kind, and theprototype of the largest modern optical telescopes. In

1671 he donated an improved version to the RoyalSociety of London, the foremost scientific society ofthe day. Newton's Opticks appeared the followingyear. It dealt with the theory of light and color andwith Newton's investigations of the colors of thinsheets, of "Newton's rings," and of the phenomenon


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of diffraction of light.

Proj ecti les & Pl anetsLet us now turn to the central topic of the Principia,the universality of the gravitational force. The legendis that Newton saw an apple fall in his garden inLincolnshire, thought of it in terms of an attractive

gravitational force towards the earth, and realizedthe same force might extend as far as the moon. Hewas familiar with Galileo's work on projectiles, andsuggested that the moon's motion in orbit could beunderstood as a natural extension of that theory. Tosee what is meant by this, consider a gun shooting a


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projectile horizontally from a very high mountain,and imagine using more and more powder insuccessive shots to drive the projectile faster andfaster.

The parabolic paths would become flatter and flatter,and, if we imagine that the mountain is so high thatair resistance can be ignored, and the gun issufficiently powerful, eventually the point of landingis so far away that we must consider the curvature ofthe earth in finding where it lands.

In fact, the real situation is more dramatic---theearth's curvature may mean the projectile neverlands at all. This was envisioned by Newton in thePrincipia. The following diagram is from his laterpopularization,A Treatise of the System of the World,written in the 1680's:

The mountaintop at V is supposed to be above theearth's atmosphere, and for a suitable initial speed,the projectile orbits the earth in a circular path. Infact, the earth's curvature is such that the surfacefalls away below a truly flat horizontal line by aboutfive meters in 8,000 meters (five miles). Recall thatfive meters is just the vertical distance an initiallyhorizontally moving projectile will fall in the first

second of motion. But this implies that if the(horizontal) muzzle velocity were 8,000 meters persecond, the downward fall of the cannonball wouldbe just matched by the earth's surface falling away,and it would never hit the ground! This is just themotion, familiar to us now, of a satellite in a low


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orbit, which travels at about 8,000 meters (fivemiles) a second, or 18,000 miles per hour. (Actually,Newton drew this mountain impossibly high, nodoubt for clarity of illustration. A satellite launched

horizontally from the top would be far above theusual shuttle orbit, and go considerably more slowlythan 18,000 miles per hour.)

Newton realized that the moon's circular patharound the earth could be caused in this way by thesame gravitational force that would hold such acannonball in low orbit, in other words, the sameforce that causes bodies to fall.

Moon's motion, beginning at some particularinstant, as deviating downwards from some initial"horizontal" line, just as for the cannonball shothorizontally from a high mountain.

The first question is: does the moon fall fivemeters below the horizontal line, that is, towards theearth, in the first second? This was not difficult forNewton to check, because the path of the moon wasprecisely known by this time. The moon's orbit isapproximately a circle of radius about 384,000kilometers (240,000 miles), which it goes around in amonth (to be precise, in 27.3 days), so the distancecovered in one second is, conveniently, very close toone kilometer. It is then a matter of geometry tofigure out how far the curved path falls below a

"horizontal" line in one second of flight, and theanswer turns out to be not five meters, but only alittle over one millimeter! (Actually around 1.37millimeters.) Thus the "natural acceleration" of themoon towards the earth, measured by how far it fallsbelow straight line motion in one second, is less than


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that of an apple here on earth by the ratio of fivemeters to 1.37 millimeters, which works out to beabout 3,600.

What can be the significance of this muchsmaller rate of fall? Newton's answer was that thenatural acceleration of the moon was much smallerthan that of the cannonball because they were bothcaused by a force---a gravitational attraction towardsthe earth, and that the gravitational force becameweaker on going away from the earth.

The Gr eat Conserv atio nLawsA lot of very profound physics is encapsulated in

the so-called conservation laws, which arestatements that certain quantities are `conserved'(unchanging in total) in isolated systems. We haveencountered this twice before, once in my discussion

of how the conservation of energy explains the factthat stars are hot, and again when I explained howthe conservation of angular momentum explainedthe stability of the spin of the Earth. The time is nowright, however, to explore the issue a little moredeeply.


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In Newton's time, the concept of the conservationlaws was not as developed as nowadays, so thisperspective is not one that Newton had fullyavailable to him. In modern terminology, we believe

that the following quantities are conserved in aclosed system (that is, one in which no externalinfluences or forces intrude):

The total linear momentum (to be definedbelow)

The total angular momentum

The total electrical charge The total energy

The total mass

I demonstrated some of this in class. Consider, forinstance, the conservation of energy. If I lift a piece ofchalk above the table, I have done some workagainst gravity (my muscles have expended somestored chemical energy by burning up sugars andother fuels). My virtue of its new position, the chalknow possesses some "gravitational potential energy."When I drop it, the potential energy vanishes, orrather is converted to kinetic energy, the energy ofmotion. When the chalk hits the table, its directedmotion stops but the total energy is still conserved; itgoes into the heating of the table (the impact makes

the atoms jiggle around more vigorously) and thenoise, which you hear (the impact jiggles the atomsin the air, and this disturbance spreads out as asound and rattles your eardrums).

Let us turn now to linear momentum, which is, as


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the name implies, a measure of the momentum (aword which may have some intuitive meaning foryou) carried by an object moving along someparticular direction, or line, of motion. In fact, the

amount of linear momentum an object carries isgiven by its total mass times its speed of motion.Again, this equation is sterile on its own; so let usthink of some applications. In a football game,stopping a fullback from crossing the goal line ismore difficult if he is moving at speed, with a gooddeal of momentum, and there is an obviousadvantage if the fullback is a large (massive) player

rather than someone of very slight build. Likewise,we all know that a baseball thrown at ninety miles anhour carries more ``punch'' than a ping-pong ballmoving at the same speed. In a sense, it is as thoughyou were to ask how much damage a moving objectcould do if it should be involved in a collision.

Interestingly, there are two ways to quantify this``punch'' (or, if you prefer, this "ability to dosignificant damage"). One is to consider the kineticenergy(the energy of motion) of the body; the other,which is not the same, is to consider the linearmomentum. Why two different ways? It turns out thatthe conservation of linear momentum is intimatelyrelated to Newton's Second Law, while theconservation of energy has very broad-reachingimplications in a host of physical situations. Those of

you who have taken physics courses know that sometypes of problems are more easily solved byconsidering the energetic; in other problems,considering the momentum may be the key to aquick solution. (No matter what your approach, ofcourse, you should get the same answer.) In general,


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given starting situation.

Things do not always work out that neatly, by theway! Some physical situations are horrendously more

complicated. Think, for instance, of a row of stoppedcars at an intersection. If an inattentive motorist runsinto the back of the row at high speed, you will notgenerally see a single car pop off the front of the rowwith the other cars left sitting there unscathed! Thedifference is that the cars are designed to absorbsome of the energy of the collision by crumpling andbreaking apart. (This absorption of energy, seen in itsextreme example in racing car accidents, actuallyprotects the occupants by soaking up much of theenergy of the collision.) Any calculation of thebehaviour would have to take into account all theseeffects. The billiard-ball executive toy is especiallysimple in that the resilient balls collide elastically,which means that essentially all of the kinetic energystays in that form. The balls escape the collisionunscathed.

A fine example of the conservation of linearmomentum is to imagine yourself standing in anunmoving canoe on a placid lake, with a heavy stoneheld against your chest. Since everything ismotionless, there is no linear momentum associatedwith this system of people and objects. Now fling thestone away from you as fast as you can! Since it is

now moving to the right(let us say) with somespeed, something else must be moving to the leftifthe total linear momentum is to be conserved (i.e. ifit is still to add up to zero, as it did before). This, ofcourse, is accomplished by the sudden backwardmotion of you and the canoe -- with the likely result


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that you lose your balance and fall into the water.

This can equally well be considered from the pointof view of Newton's Laws. The First Law reminds us

that when you are standing still, you and the stonedon't change your state of motion since nounbalanced forces are at play. Suddenly yourmuscles twitch and apply an unbalanced force to thestone, which is accelerated to the right, inaccordance with Newton's Second Law. (The stone isaccelerated only as long as you keep pushing on it,applying the unbalanced force; as soon as it leavesyour grip, the force vanishes and the stone fliesfreely through the air.) Meanwhile, Newton's ThirdLaw reminds us that the unbalanced force youapplied to the stone is matched by one of equal sizeacting onyourselfbut pointing in the oppositedirection. This will set you in motion, according toNewton's Second Law. Of course, if you are verymassive (or if the canoe is filled with other peopleand stones), you will not accelerate very much -- butmove you will!

How Rocke ts WorkAs we will see in the next section, Newton's Laws

plus the Law of Universal Gravitation explain howand why the planets orbit the sun as they do. Inconsidering these matters, Newton imagined the way


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in which various objects would move if they were tobe launched horizontally at high speed from amountaintop. When we consider such objectsorbiting the Earth, one tends inevitably to visualise

rockets, such as those used in the space program.This visualisation can be a little misleading, so I wantto comment on it in a couple of different ways. Thefirst of these is to consider how rockets actually work.

Earlier, we considered an implication of Newton'sThird Law: when you do a pushup you actually movethe Earth! You push on it, and the reaction forcepushes on you, so that you lift up from the surface.The forces are of the same size (``equal andopposite''), so you move more than the Earth itselfdoes (since you are so much less massive: seeNewton's Second Law), but in principle both move,even if the Earth's motion is immeasurably small.

Now consider a rocket. When most people think ofsuch devices, they visualize:

1. a huge flame and hot gas pouring out the back;and

2. the rocket pushing against the ground or the air,rather in the way that you push the blade of yourpaddle against the water when you propel acanoe.

The first of these aspects is misleading; the second is

just plain wrong. The rocket actually works by virtue of Newton's

Third Law (or alternatively and equivalentlythrough the Conservation of Linear Momentum).Within the rocket engine, the burning of the fuelheats the gases; this raises the pressure so that


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the gases try to expand in all directions. Sincethere is a nozzle at the back, the gases rush outthat way at high speed. The equal and oppositereaction force pushes the rocket the other way.

(Alternatively, we can just recognize that thetotal linear momentum has to be conserved.)

The important point is that this would workregardless of the nature of the stuff thrown outthe back.You could, for instance, build a littletreadmill device to throw bricks out the back andthereby accelerate the rocket the otherdirection! The reason we use a hot flame issimply that the rocket is accelerated mostefficiently if the ejected material moves at highspeed, and the burning of liquid fuel heats it somuch that the gases come out very fast indeed.So it is merely a matter of efficiency.

The rocket needs nothing to `push against' andwill function in the vacuum of space perfectly

well. (In the 1920s, by the way, the New YorkTimes published a strident editorial in which theycriticised a physics professor who, they said, hadcompletely forgotten his basic physics in evendiscussing the prospects of future space travel.According to the Times, rockets would neverfunction in the vacuum of space! Events haveproven them wrong, of course.) Indeed, rockets

benefitfrom the lack of air resistance, whichmerely retards their acceleration as they climbaway from the Earth's surface. Nor do rocketsneed to be streamlined, except insofar as ithelps get them up out of the Earth's atmospherewith minimal drag. The Space Shuttle looks like


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an airplane because that is what it turns into onreturn to the Earth: it has to use itsaerodynamics to glide to a safe landing. But ahundred years from now, it is possible that we

may see a manned interstellar spacecraft, builtin and launched from the vacuum of near-Earthorbit, which could be shaped like a cauliflower,for all that it matters.

newton new view

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