What is Law of Conservation of Mass and energy

how is the law of conservation of mass and energy how do the laws of conservation of mass and energy apply to living systems difference between conservation of mass and energy
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Dr.NaveenBansal,India,Teacher
Published Date:25-10-2017
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The pool skater trades two forms of energy back and forth: kinetic and gravitational. More photos of this insane pastime are at the web site www.sonic.net/∼ shawn. When I first came across it in 1998, I assumed these guys weren’t likely to stay alive for long, but they seem to have survived — or at least their web site has. Chapter 4 Conservation of Mass and Energy Inchapter1, Ipromisedthatasyoulearnedmoreandmoreaboutphysics, you would see it becoming more and more simple. The unifying principle that brings order and sanity to all of physics is Noether’s theorem, which so far you’ve only seen stated in a very rough form: the laws of physics havetobethewaytheyarebecauseofsymmetry. Thisbook’spresentation of physics so far has been suffused with symmetry arguments, but much of what you’ve learned has consisted of specific, practical applications, like the formation of images by lenses and mirrors. What have you learned so far that deserves to be called a fundamental law of physics? The only law of physics you’ve learned is the principle of inertia: a ray of light or a material object continues moving in the same direction and at the same speed if it is not interacting with anything else. That’s all very well, but the universe would be dull if it consisted only of individual atoms and rays of light crisscrossing space and never coming close enough to interact with each other — it would be like a game of pool played on an infinite table, with only one ball in sight. Your everyday life, to which we’d like to apply physics, involves vast numbers of particles. 30 Your own body, for instance, contains something like 10 atoms (that’s scientific notation for one followed by thirty zeroes). How can we make sense out of such incredible complexity? 614.1 Conservation of Mass Whatmakesourcomplexworldcomprehensibletothehumanmindisthat thefundamentallawsofphysicsareallconservationlaws: lawsstatingthat the total amount of something stays the same. You’ve already discovered some evidence in lab for such a law: the law of conservation of mass. Even when you carried out complex operations involving huge numbers of atoms, the total mass of the atoms never changed. The wonderful thing aboutconservationlawsisthattheyallowustomakesenseoutofcomplex processes. The law of conservation of mass probably didn’t surprise you very much, since you’ve known about atoms since an early age, and in every- day life we don’t encounter processes in which atoms change their masses noticeably, or in which atoms are created or destroyed. That argument wasn’t obvious to your ancestors, however. It’s not even hard to think of examples that would raise doubts in the minds of modern people. A log weighs more than its ashes. Did some mass simply disappear? It seems to be an exception to the rule. The French chemist Antoine-Laurent Lavoisier was the first scientist to realize that there were no such exceptions. Lavoisier hypothesized that when wood burns, for example, the supposed loss of mass is actually ac- counted for by the escaping hot gases that the flames are made of. Before Lavoisier, chemists had almost never weighed their chemicals to quantify 1 the amount of each substance that was undergoing reactions. They also didn’tcompletelyunderstandthatgaseswerejustanotherstateofmatter, and hadn’t tried performing reactions in sealed chambers to determine whether gases were being consumed from or released into the air. For this they had at least one practical excuse, which is that if you perform a gas-releasing reaction in a sealed chamber with no room for expansion, you get an explosion Lavoisier invented a balance that was capable of measuring milligram masses, and gured fi out how to do reactions in an upside-down bowl in a basin of water, so that the gases could expand by pushingoutsomeofthewater. Inonecrucialexperiment,Lavoisierheated a / Portrait of Monsieur Lavoisier a red mercury compound, which we would now describe as mercury oxide and His Wife, by Jacques-Louis (HgO), in such a sealed chamber. A gas was produced (Lavoisier later David, 1788. Lavoisier invented namedit“oxygen”), drivingoutsomeofthewater, andtheredcompound the concept of conservation of was transformed into silvery liquid mercury metal. The crucial point was mass. The husband is depicted that the total mass of the entire apparatus was exactly the same before with his scientific apparatus, and after the reaction. Based on many observations of this type, Lavoisier while in the background on the proposed a general law of nature, that mass is always conserved. left is the portfolio belonging Self-check A to Madame Lavoisier, who is In ordinary speech, we say that you should “conserve” something, be- thought to have been a student of cause if you don’t, pretty soon it will all be gone. How is this different David’s. from the meaning of the term “conservation” in physics? . Answer, p. 88 Although Lavoisier was an honest and energetic public official, he was caught up in the Terror and sentenced to death in 1794. He requested a fifteen-day delay of his execution so that he could complete some exper- iments that he thought might be of value to the Republic. The judge, Coffinhal, infamously replied that “the state has no need of scientists.” As a scientific experiment, Lavoisier decided to try to determine how long 1 Isaac Newton was a notable exception. 62 Chapter 4 Conservation of Mass and Energyhis consciousness would continue after he was guillotined, by blinking his eyes for as long as possible. He blinked twelve times after his head was chopped off. Ironically, Judge Coffinhal was himself executed only three months later, falling victim to the same chaos. A stream of water example 1 The stream of water is fatter near the mouth of the faucet, and skinnier lower down. This can be understood using conservation of mass. Since water is being neither created nor destroyed, the mass of the water that leaves the faucet in one second must be the same as the amount that flows past a lower point in the same time interval. The water speeds up as it falls, so the two quantities of water can only be equal if the stream is narrower at the bottom. 4.2 Conservation of Energy Noether’stheoremsaysthatconservationlawsresultfromsymmetries,but theconnectionbetweensymmetryandconservationofmasswon’tbeclear until the end of the chapter. As our rst fi full-efl dged example of Noether’s theorem in action, we’ll instead use conservation of energy. Energy means something specific and technical in physics, but let’s start by appealing to youreverydayknowledge. Energyiswhatyou’rebuyingatthegasstation, 2 and you also pay for it in your electric bill. Energy is why we need food. These forms of energy can be converted into others, such as the energy b / Example 1. your car has when it’s moving, the light from a lamp, or the body heat that we mammals must continuously produce. We’ll rst fi develop a real scientific definition of energy, and then relate it to symmetry in section 4.4. Kinetic energy Symmetry arguments led us to the conclusion that an isolated object or ray of light can never slow down, change direction or disappear entirely. But that falls short of being a conservation law. A full-efl dged conserva- tion law says that even when we have many objects interacting, the total amount of something stays constant. Is there any reason to believe that energy is conserved in general? The planet earth, c, is a large, complex c / The earth keeps spinning system consisting of a huge number of atoms. It keeps on spinning with- without slowing down. Energy is outslowingdown, whichisevidenceinfavorofenergyconservation. What conserved. about the spinning coin in figure d, however? Does its energy disappear gradually? Scientists would have thought so until the nineteenth century, when physicistJamesJoule(1818-1889)hadanimportantinsight. Joulewasthe wealthy heir to a Scottish brewery, and funded his own scientific research. As an industrialist, he had a practical interest in replacing steam engines with electric ones that would be more efficient, and cost less money to run. Scientists already knew that friction would cause a spinning coin to slow down, and that friction made engines less efficient. They also knew that friction heated things up, as when you rub your hands together on a cold day. Joule, however, realized that it went deeper than this: there d / The spinning coin slows was a conserved quantity, which ended up being called energy. When we down. It seems as though energy first start the coin spinning, its energy is in the form of motion, with its isn’t conserved, but it is. 2 Growing children also need to eat more than they excrete because conserva- tion of mass would otherwise make it impossible for them to grow. Section 4.2 Conservation of Energy 63atoms all going in circles. As it slows down, the energy isn’t disappearing, it’s being converted into another form: heat. We now know that heat is the random motion of atoms. As the coin rubs against the ground, the atoms in the two surfaces bump into each other, and the amount of random atomic motion increases. The organized motion of the atoms in the spinning coin is being converted into a disorganized form of motion, heat. Energy of motion is called kinetic energy. The simplest situations for calculating kinetic energy are those in which an object is moving through space without spinning or moving internally, e.g., a hockey puck sliding across the ice. All the atoms in the object are moving at the same speed, so the object’s kinetic energy just depends on two numbers, its mass and its speed. The actual equation can’t be proved based on logic; it can only be determined from experiments. Such experiments were rst fi done by English physicist Thomas Young, and in lab 4b you’re going to reproduce Young’s work and discover his equation for yourself. Whenenergyisbeingtransferredorchangedfromoneformtoanother, we use the term “power” to mean the amount of energy transferred per unit time. The metric unit of power is the watt (W), dened fi as one joule per second. Power of a lightbulb example 2 e / James Joule Every second, a 100 W lightbulb takes 100 J of energy from the wall socket. (Some of that energy is turned into light, and the rest just heats your house.) Gravitational energy If you toss a ball up in the air, it slows to a stop and then speeds up again on the way back down. As in the example of the spinning coin, it seems as though conservation of energy is being violated, but really we’re just seeing evidence that there is a new form of energy coming into play, gravitationalenergy. Thisformofenergydependsondistance,notmotion: the farther apart the earth and the ball are, the more gravitational energy there is. Self-check B We’ve discussed three kinds of energy so far: kinetic energy, heat en- ergy (which is really kinetic energy at the atomic level), and gravitational f / The water help up high behind energy. Energy can be converted from any of these forms into any other. Hoover Dam has gravitational Suppose a firefighter slides down the pole at the fire station, using her energy. grip to control her motion so that she neither speeds up nor slows down. How would you describe this in terms of energy? . Answer, p. 88 The metric unit of energy is the joule (J), and we’ll define it as the amount of energy needed to raise the temperature of 0.24 grams of water ◦ by 1 Celsius. (Don’t memorize that number) Gravity is a universal attraction between things that have mass. Here where we live on the earth’s surface, the atoms in the earth attract the atoms in all the objects around us, and measurements show that as a result of all that attraction, an energy of about 10 J is needed in order to lift a one-kilogram mass 3 by one meter. We say that the strength of the gravitational field, g, at the earth’s surface is 10 joules per kilogram per meter, or, in abbreviated 3 Amoreprecisevalueis9.8J,butthat’scloseto10,sowe’llusuallyroundoff to 10 to simplify numerical examples. In any case, don’t memorize the numbers. 64 Chapter 4 Conservation of Mass and Energyform, g =10 J/kg/m. The pool skater example 3 On the way up the side of the pool, the skater on page 61 has con- verted all of his kinetic energy into gravitational energy. Figure g shows GE=3000 J KE=0 schematically how the two types of energy are traded off. (The numbers are just my estimates.) The birth of stars example 4 Orion is the easiest constellation to find. You can see it in the winter, even if you live under the light-polluted skies of a big city. Figure h shows GE=2000 J KE=1000 J an interesting feature of this part of the sky that you can easily pick out with an ordinary camera (that’s how I took the picture) or a pair of binoculars. The three stars at the top are Orion’s belt, and the stuff near the lower left corner of the picture is known as his sword — to the naked eye, it just looks like three more stars that aren’t as bright as the stars in the belt. The middle “star” of the sword, however, isn’t a star at all. GE=1000 J KE=2000 J It’s a cloud of gas, known as the Orion Nebula, that’s in the process of collapsing due to gravity. Like the pool skater on his way down, the gas is losing gravitational energy. The results are very different, however. The skateboard is designed to be a low-friction device, so nearly all of the lost gravitational energy is converted to kinetic energy, and very GE=0 KE=3000 J little to heat. The gases in the nebula flow and rub against each other, however, so most of the gravitational energy is converted to heat. This is the process by which stars are born: eventually the core of the gas cloud gets hot enough to ignite nuclear reactions. g / example 3 A lever example 5 Figure i shows two sisters on a seesaw. The one on the left has twice as much mass, but she’s at half the distance from the center. No energy input is needed in order to tip the seesaw. If the girl on the left goes up a certain distance, her gravitational energy will increase. At the same time, her sister on the right will drop twice the distance, which results in an equal decrease in energy, since her mass is half as much. Lifting a weight example 6 . At the gym, you lift a mass of 40 kg through a height of 0.5 m. How much gravitational energy is required? Where does this energy come from? . The strength of the gravitational field is 10 joules per kilogram per meter, so after you lift the weight, its gravitational energy will be greater by 10× 40× 0.5 = 200 joules. Energy is conserved, so if the weight gains gravitational energy, h / example 4 something else somewhere in the universe must have lost some. The energy that was used up was the energy in your body, which came from the food you’d eaten. This is what we refer to as “burning calories,” since calories are the units normally used to describe the energy in food, rather than metric units of joules. In fact, your body uses up even more than 200 J of food energy, be- cause it’s not very efficient. The rest of the energy goes into heat, which is why you’ll need a shower after you work out. We can summarize this as food energy→ gravitational energy + heat . i / example 5 Section 4.2 Conservation of Energy 65Lowering a weight example 7 . After lifting the weight, you need to lower it again. What’s happening in terms of energy? . Your body isn’t capable of accepting the energy and putting it back into storage. The gravitational energy all goes into heat. (There’s nothing fundamental in the laws of physics that forbids this. Electric cars can do it — when you stop at a stop sign, the car’s kinetic energy is absorbed back into the battery, through a generator.) Heavy objects don’t fall faster example 8 Stand up now, take off your shoe, and drop it alongside a much less massive object such as a coin or the cap from your pen. Did that surprise you? You found that they both hit the ground at the same time. The Greek philosopher Aristotle wrote that heavier objects fall faster than lighter ones. He was wrong, but Europeans believed him for thousands of years, partly because experiments weren’t an accepted way of learning the truth, and partly because the Catholic Church gave him its posthumous seal of approval as its official philosopher. Heavy objects and light objects have to fall the same way, because j / This photo was made with conservation laws are additive — we find the total energy of an object a special camera that records by adding up the energies of all its atoms. If a single atom falls through infrared light. The man’s warm a height of one meter, it loses a certain amount of gravitational energy skin emits quite a bit of infrared and gains a corresponding amount of kinetic energy. Kinetic energy light energy, while his hair, at a relates to speed, so that determines how fast it’s moving at the end of lower temperature, emits less. its one-meter drop. (The same reasoning could be applied to any point along the way between zero meters and one.) Now what if we stick two atoms together? The pair has double the mass, so the amount of gravitational energy transformed into kinetic energy is twice as much. But twice as much kinetic energy is exactly what we need if the pair of atoms is to have the same speed as the single atom did. Continuing this train of thought, it doesn’t matter how many atoms an object contains; it will have the same speed as any other object after dropping through the same height. Self-check C Part of the Aristotelian confusion was probably because of examples like dropping a feather. A feather won’t fall as quickly as a rock. Why is this? Our unspoken assumption was that the only energy transforma- tion going on was gravitational energy→ kinetic energy . Evidently this assumption fails — most of the feather’s gravitational en- ergy is being converted into something else besides kinetic energy. What other form of energy is there? . Answer, p. 88 k / An infrared camera distin- Emission and absorption of light guishes hot and cold areas. As the bike skids to a stop with its The example of the falling feather shows how tricky this can get. Of- brakes locked, the kinetic energy ten we miss something vital because it’s invisible. When a guitar string of the bike and rider is converted gradually stops vibrating, it may seem as though its energy was just dis- into heat in both the floor (top) appearing; sound has energy, but we may forget that because sound is and the tire (bottom). invisible. When the feather drops, the heating of the feather and the air are not only invisible but nearly undetectable without heroic measures. 66 Chapter 4 Conservation of Mass and EnergyImagine how difficult it was for Joule to gure fi out all of this for the first time One challenge in his experiments is demonstrated in figure j. In general, light can heat matter (sunlight on your skin) and matter can also get rid of its heat energy by emitting light (a candle flame): heat↔light Light, however, includes more than just the spectrum of visible colors extending from red to violet on the rainbow. Hot objects, like the sun or a lightbulb filament, do emit visible light, but matter at lower temperatures gives off infrared light, a color of light that lies beyond the red end of the visible rainbow. Although the emission and absorption of infrared light was just a source of trouble and confusion for Joule, we can also use infrared pho- tography to gain insight into phenomena in which other types of energy are converted into heat. The heating of the tire and floor in figure k is something that the average person might have predicted in advance, but there are other situations where it’s not so obvious. When a ball slams into a wall, it doesn’t rebound with the same amount of kinetic energy. Was some energy destroyed? No. The ball and the wall heat up. Figure l shows a squash ball at room temperature (top), and after it has been playedwithforseveralminutes(bottom), causingittoheatupdetectably. l / A squash ball before and after several minutes of play. How many forms of energy? Howmanydifferenttypesofenergyarethere? Atthispoint,youmight worry that you were going to have to memorize a long list of them. The good news is that there aren’t really that many at all. m / At the atomic level, the energy in the bow is really electrical en- ergy In figure m, the bow evidently contains some stored energy, since we observe that the arrow gets kinetic energy from it. What kind of energy is this? Is it some new and mysterious “bow energy?” No. At the atomic level, thingsgetalotsimpler. Theenergyinthebowiselectricalenergyof the interacting atoms. Just as a rock can have more or less gravitational energy depending on its distance from the earth, an atom can have more or less electrical energy depending on its distance from another atom. Section 4.2 Conservation of Energy 67 expansion compressionMany other forms of energy turn out to be electrical energy in dis- guise, n. In particular, chemical reactions are based on electrical energy: in a reaction, atoms are rearranged like tinker toys, which changes their distances from one another. Food and gasoline are both fuels that store electrical energy. boiling Every type of energy you encounter in your day-to-day life is really just something from the following short list: kinetic energy (including heat) gravitational energy electrical and magnetic energy (including light, which is an bending electrical and magnetic wave) We’ll discuss electricity and magnetism in more detail in chapter 7. Two forms of nuclear energy can also be added to the list. One of the main goals of physics is to classify all the interactions: gravitational, electrical, breaking and so on. Physicists generally believe that there is an underlying simplicity to the laws of physics, and consider it a triumph when they can reveal part chemical of it. You might wonder, for instance, why electrical and magnetic energy reactions are shown as a single item on the list above. Well, just as we learned that “bow energy” and “food energy” are really both just types of electrical energy, we’llseeinchapter7thatelectricityandmagnetismarereallyjust n / All of these energy trans- two sides of the same coin. formations turn out at the atomic level to be changes in electrical Discussion Questions energy resulting from changes in the distances between atoms. A In figure o, a small amount of hot water is poured into the empty can, which rapidly fills up with hot steam. The can is then sealed tightly, and soon crumples. How can this be explained based on the idea that heat is a form of random motion of atoms? o / Discussion question A. 68 Chapter 4 Conservation of Mass and Energy4.3 Newton’s Law of Gravity Why does the gravitational efi ld on our planet have the particular valueitdoes? Forinsight, let’scomparewiththestrengthofgravity elsewhere in the universe: location g (joules per kg per m) asteroid Vesta (surface) 0.3 earth’s moon (surface) 1.6 Mars (surface) 3.7 earth (surface) 9.8 Jupiter (cloud-tops) 26 sun (visible surface) 270 12 typical neutron star (surface) 10 black hole (center) infinite according to some theories, on the order of 52 10 according to others A good comparison is Vesta versus a neutron star. They’re roughly the same size, but they have vastly different masses — a teaspoonful of neutron star matter would weigh a million tons The different mass must be the reason for the vastly different gravita- 12 tional fields. (The notation 10 means 1 followed by 12 zeroes.) This makes sense, because gravity is an attraction between things that have mass. The mass of an object, however, isn’t the only thing that deter- mines the strength of its gravitational efi ld, as demonstrated by the difference between the efi lds of the sun and a neutron star, despite their similar masses. The other variable that matters is distance. Because a neutron star’s mass is compressed into such a small space (comparable to the size of a city), a point on its surface is within a fairly short distance from every atom in the star. If you visited the surface of the sun, however, you’d be millions of miles away from most of its atoms. Asalessexoticexample,ifyoutravelfromtheseaportofGuaya- quil, Ecuador, to the top of nearby Mt. Cotopaxi, you’ll experience a slight reduction in gravity, from 9.7806 to 9.7624 J/kg/m. This is because you’ve gotten a little farther from the planet’s mass. Such differences in the strength of gravity between one location and an- other on the earth’s surface were first discovered because pendulum clocks that were correctly calibrated in one country were found to runtoofastortooslowwhentheywereshippedtoanotherlocation. Section 4.3 Newton’s Law of Gravity 69The general equation for an object’s gravitational field was dis- covered by Isaac Newton, by working backwards from the observed 4 motion of the planets: GM g = , 2 d where M is the mass of the object, d is the distance from the ob- ject,andGisaconstantthatisthesameeverywhereintheuniverse. 5 This is known as Newton’s law of gravity. It’s an inverse square law, which is reasonable since an object’s gravitational field is an effect that spreads outward from it in all directions. Newton’s law of gravity really gives the field of an individual atom, and the field of a many-atom object is the sum of the efi lds of the atoms. New- ton was able to prove mathematically that this scary sum has an unexpectedly simple result in the case of a spherical object such as a planet: the result is the same as if all the object’s mass had been concentrated at its center. p / Isaac Newton (1642-1727) Newton showed that his theory of gravity could explain the or- bits of the planets, and also finished the project begun by Galileo of driving a stake through the heart of Aristotelian physics. His book on the motion of material objects, the Mathematical Princi- ples of Natural Philosophy, was uncontradicted by experiment for 200 years, but his other main work, Optics, was on the wrong track due to his conviction that light was composed of particles rather than waves. He was an avid alchemist, an embarrassing fact that modern scientists would like to forget. Newton was on the winning sideoftherevolutionthatreplacedKingJamesIIwithWilliamand Mary of Orange, which led to a lucrative post running the English royal mint; he worked hard at what could have been a sinecure, and took great satisfaction from catching and executing counterfeitors. Newton’s personal life was less happy. Rejected by his mother at an early age, he never married or formed any close attachments, except for one intense emotional relationship with a younger man; around the time when this liaison ended, Newton experienced what 6 we would today probably describe as a nervous breakdown. q / example 9 60 1 4 Example 14 on page 104 shows the type of reasoning that Newton had to go through. 5 This is not the form in which Newton originally wrote the equation. 6 The historical record can’t be decoded with certainty. Seventeenth-century Englanddidn’tconceiveofmentalillnessinthesamewaywedonow. Homosexu- alitywasacapitaloffense,notapersonalpreference. IfNewtonwashomosexual, he had a strong motivation not to record the fact. 70 Chapter 4 Conservation of Mass and EnergyNewton’s apple example 9 A charming legend attested to by Newton’s niece is that he first con- ceived of gravity as a universal attraction after seeing an apple fall from a tree. He wondered whether the force that made the apple fall was the same one that made the moon circle the earth rather than flying off straight. Newton had astronomical data that allowed him to calcu- late that the gravitational field the moon experienced from the earth was 7 1/3600 as strong as the field on the surface of the earth. (The moon has its own gravitational field, but that’s not what we’re talking about.) The moon’s distance from the earth is 60 times greater than the earth’s radius, so this fit perfectly with an inverse-square law: 60× 60 = 3600. 7 See example 14 on page 104. Section 4.3 Newton’s Law of Gravity 714.4 Noether’s Theorem for Energy Now we’re ready for our rs fi t full-efl dged example of Noether’s theorem. Conservationofenergyisalawofphysics,andNoether’stheoremsaysthat the laws of physics come from symmetry. Specifically, Noether’s theorem says that every symmetry implies a conservation law. Conservation of energy comes from a symmetry that we haven’t even discussed yet, but one that is simple and intuitively appealing: as time goes by, the universe doesn’t change the way it works. This is a kind of translation symmetry, but in time, not space. We have strong evidence for time translation symmetry, because when we see a distant galaxy through a telescope, we’re seeing light that has taken billions of years to get here. A telescope, then, is like a time ma- chine. For all we know, alien astronomers with advanced technology may 8 be observing our planet right now, but if so, they’re seeing it not as it is now but as it was in the distant past, perhaps in the age of the dinosaurs, or before life even evolved here. As we observe a particularly distant, and therefore ancient, supernova, we see that its explosion plays out in exactly the same way as those that are closer, and therefore more recent. Nowsupposephysicsreallydoeschangefromyeartoyear,likepolitics, pop music, and hemlines. Imagine, for example, that the “constant” G in Newton’s law of gravity isn’t quite so constant. One day you might wake up and find that you’ve lost a lot of weight without dieting or exercise, simply because gravity has gotten weaker since the day before. IfyouknowaboutsuchchangesinGovertime,it’stheultimateinsider information. You can use it to get as rich as Croesus, or even Bill Gates. Ona day whenG is low, you payfor the energy needed to lift a large mass uphigh. Then, onadaywhengravityisstronger, youlowerthemassback down, extracting its gravitational energy. The key is that the energy you get back out is greater than what you originally had to put in. You can run the cycle over and over again, always raising the weight when gravity is weak, and lowering it when gravity is strong. Each time, you make a profit in energy. Everyone else thinks energy is conserved, but your secret technique allows you to keep on increasing and increasing the amount of energy in the universe (and the amount of money in your bank account). The scheme can be made to work if anything about physics changes over time, not just gravity. For instance, suppose that the mass of an elec- tron had one value today, and a slightly different value tomorrow. Elec- trons are one of the basic particles from which atoms are built, so on a day when the mass of electrons is low, every physical object has a slightly lowermass. Inproblem7onpage77,you’llworkoutawaythatthiscould be used to manufacture energy out of nowhere. Sorry, but it won’t work. Experiments show that G doesn’t change measurably over time, nor does there seem to be any time variation in any 9 of the other rules by which the universe works. The rules of the game 8 Our present technology isn’t good enough to let us pick the planets of other solar systems out from the glare of their suns, except in a few exceptional cases. 9 In 2002, there have been some reports that the properties of atoms as ob- served in distant galaxies are slightly different than those of atoms here and now. If so, then time translation symmetry is weakly violated, and so is con- servation of energy. However, this is a revolutionary claim, and it needs to be examined carefully. The change being claimed is large enough that, if it’s real, it 72 Chapter 4 Conservation of Mass and Energyare symmetric under time translation. If archaeologists nd fi a copy of this book thousands of years from now, they’ll be able to reproduce all the experiments you’re doing in this course. I’ve probably convinced you that if time-translation symmetry was violated, then conservation of energy wouldn’t hold. But does it work the other way around? If time-translation symmetry is valid, must there be a law of conservation of energy? Logically, that’s a different question. We may be able to prove that if A is false, then B must be false, but that doesn’t mean that if A is true, B must be true as well. For instance, if you’re not a criminal, then you’re presumably not in jail, but just because someone is a criminal, that doesn’t mean he is in jail — some criminals never get caught. Noether’s theorem does work the other way around as well: if physics has a certain symmetry, then there must be a certain corresponding con- servation law. This is a stronger statement. The full-strength version of Noether’s theorem can’t be proved without a model of light and matter more detailed than the one currently at our disposal. shouldbedetectablefromoneyeartothenextinultra-high-precisionlaboratory experiments here on earth. Section 4.4 Noether’s Theorem for Energy 734.5 Equivalence of Mass and Energy Mass-energy You’veencounteredtwoconservationlawssofar: conservationofmass and conservation of energy. If conservation of energy is a consequence of symmetry, is there a deeper reason for conservation of mass? Actually they’re not even separate conservation laws. Albert Einstein found, as a consequence of his theory of relativity, that mass and energy are equivalent, and are not separately conserved — one can be converted into the other. Imagine that a magician waves his wand, and changes a bowl of dirt into a bowl of lettuce. You’d be impressed, because you were expecting that both dirt and lettuce would be conserved quantities. Neither one can be made to vanish, or to appear out of thin air. However, there are processes that can change one into the other. A farmer changes dirt into lettuce, and a compost heap changes lettuce into dirt. At the most fundamental level, lettuce and dirt aren’t really different things at all;they’rejustcollectionsofthesamekindsofatoms—carbon,hydrogen, and so on. We won’t examine relativity in detail until chapter 6, but mass-energy equivalence is an inevitable implication of the theory, and it’s the only partofthetheorythatmostpeoplehaveheardof,viathefamousequation 2 E = mc . This equation tells us how much energy is equivalent to how much mass: the conversion factor is the square of the speed of light, c. Sincecabignumber,yougetareallyreallybignumberwhenyoumultiply 2 it by itself to get c . This means that even a small amount of mass is equivalent to a very large amount of energy. Gravity bending light example 10 Gravity is a universal attraction between things that have mass, and since the energy in a beam of light is equivalent to a some very small amount of mass, we expect that light will be affected by gravity, although the effect should be very small. The first experimental confirmation of relativity came in 1919 when stars next to the sun during a solar eclipse were observed to have shifted a little from their ordinary position. (If there was no eclipse, the glare of the sun would prevent the stars from being observed.) Starlight had been deflected by the sun’s gravity. Fig- ure r is a photographic negative, so the circle that appears bright is actually the dark face of the moon, and the dark area is really the bright corona of the sun. The stars, marked by lines above and below then, appeared at positions slightly different than their normal ones. Black holes example 11 A star with sufficiently strong gravity can prevent light from leaving. Quite a few black holes have been detected via their gravitational forces on neighboring stars or clouds of gas and dust. Because mass and energy are like two different sides of the same coin, wemayspeakofmass-energy,asingleconservedquantity,foundbyadding up all the mass and energy, with the appropriate conversion factor: E + 2 mc . A rusting nail example 12 . An iron nail is left in a cup of water until it turns entirely to rust. The energy released is about 500,000 joules. In theory, would a sufficiently 74 Chapter 4 Conservation of Mass and Energyr / example 10 precise scale register a change in mass? If so, how much? . The energy will appear as heat, which will be lost to the environment. The total mass-energy of the cup, water, and iron will indeed be less- ened by 500,000 joules. (If it had been perfectly insulated, there would have been no change, since the heat energy would have been trapped 8 in the cup.) The speed of light in metric units is c = 3× 10 meters per second (scientific notation for 3 followed by 8 zeroes), so converting to mass units, we have E m = 2 c 500, 000 =  2 8 3× 10 = 0.000000000006 kilograms . (The design of the metric system is based on the meter, the kilogram, and the second. The joule is designed to fit into this system, so the result comes out in units of kilograms.) The change in mass is too small to measure with any practical technique. This is because the square of the speed of light is such a large number in metric units. The correspondence principle The realization that mass and energy are not separately conserved is our rst fi example of a general idea called the correspondence principle. When Einstein came up with relativity, conservation of energy had been accepted by physicists for decades, and conservation of mass for over a hundred years. Doesanexamplelikethismeanthatphysicistsdon’tknowwhatthey’re talking about? There is a recent tendency among social scientists to deny Section 4.5 Equivalence of Mass and Energy 75thatthescientificmethodevenexists,claimingthatscienceisnomorethan asocialsystemthatdetermineswhatideastoacceptbasedonanin-group’s criteria. If science is an arbitrary social ritual, it would seem difficult to explain its effectiveness in building such useful items as airplanes, CD players and sewers. If voodoo and astrology were no less scientific in their methods than chemistry and physics, what was it that kept them from producing anything useful? This silly attitude was effectively skewered by a famous hoax carried out in 1996 by New York University physicist Alan Sokal. Sokal wrote an article titled “Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity,” and got it 10 accepted by a cultural studies journal called Social Text. The scientific content of the paper is a carefully constructed soup of mumbo jumbo, using technical terms to create maximum confusion; I can’t make heads or tails of it, and I assume the editors and peer reviewers at Social Text understood even less. The physics, however, is mixed in with cultural relativist statements designed to appeal to them — “...the truth claims ofscienceareinherentlytheory-ladenandself-referential”—andfootnoted references to academic articles such as “Irigaray’s and Hayles’ exegeses of gender encoding in uid fl mechanics ...and ...Harding’s comprehensive critique of the gender ideology underlying the natural sciences in general and physics in particular...” On the day the article came out, Sokal published a letter explaining that the whole thing had been a parody — one that apparently went over the heads of the editors of Social Text. What keeps physics from being merely a matter of fashion is that it has to agree with experiments and observations. If a theory such as conservationofmassorconservationofenergybecameacceptedinphysics, it was because it was supported by a vast number of experiments. It’s just that experiments never have perfect accuracy, so a discrepancy such as the tiny change in the mass of the rusting nail in example 12 was undetectable. The old experiments weren’t all wrong. They were right, within their limitations. If someone comes along with a new theory he claims is better, it must still be consistent with all the same experiments. In computer jargon, it must be backward-compatible. This is called the correspondence principle: new theories must be compatible with old ones insituationswheretheyarebothapplicable. Thecorrespondenceprinciple tellsusthatwecanstilluseanoldtheorywithintherealmwhereitworks, so for instance I’ll typically refer to conservation of mass and conservation of energy in this book rather than conservation of mass-energy, except in cases where the new theory is actually necessary. Ironically, the extreme cultural relativists want to attack what they see as physical scientists’ arrogant claims to absolute truth, but what they fail to understand is that science only claims to be able to nd fi partial, provisional truth. The correspondence principle tells us that each of to- day’s scientific truth can be superseded tomorrow by another truth that is more accurate and more broadly applicable. It also tells us that today’s truth will not lose any value when that happens. 10 The paper appeared in Social Text 46/47 (1996) pp. 217- 252. The full text is available on professor Sokal’s web page at www.physics.nyu.edu/faculty/sokal/. 76 Chapter 4 Conservation of Mass and EnergyProblems 1 You jump up straight up in the air. When do you have the greatest gravitational energy? The greatest kinetic energy? (Based on a problem by Serway and Faughn.) 2 AnyaandIvanleanoverabalconysidebyside. Anyathrowsapenny downward with an initial speed of 5 m/s. Ivan throws a penny upward withthesamespeed. Bothpenniesenduponthegroundbelow. Compare their kinetic energies and velocities on impact. 3 (a) If weight B moves down by a certain amount, how much does weight A move up or down? (b) What should the ratio of the two weights be if they are to balance? Explain in terms of conservation of energy. 4 How high above the surface of the earth should a rocket be in order to have 1/100 of its normal weight? Express your answer in units of earth radii. 5 (a) You release a magnet on a tabletop near a big piece of iron, and the magnet leaps across the table to the iron. Does the magnetic energy increase or decrease? Explain. B (b) Suppose instead that you have two repelling magnets. You give them an initial push towards each other, so they decelerate while approaching A each other. Does the magnetic energy increase or decrease? Explain. Problem 3. 6 A closed system can be a bad thing — for an astronaut sealed inside a space suit, getting rid of body heat can be difficult. Suppose a 60-kg astronaut is performing vigorous physical activity, expending 200 watts of power. If none of the heat can escape from her space suit, how long ◦ ◦ will it take before her body temperature rises by 6 C (11 F), an amount sufficient to kill her? Assume that the amount of heat required to raise ◦ herbodytemperatureby1 Cisthesameasitwouldbeforanequalmass of water. Express your answer in units of minutes. 7 As suggested on page 72, imagine that the mass of the electron rises and falls over time. (Since all electrons are identical, physicists generally talk about “the electron” collectively, as in “the modern man wants more than just beer and sports.”) The idea is that all electrons are increasing and decreasing their masses in unison, and at any given time, they’re all identical. They’re like a litter of puppies whose weights are all identical on any given day, but who all change their weights in unison from one month to the next. Suppose you were the only person who knew about these small day-to-day changes in the mass of the electron. Find a plan for violating conservation of energy and getting rich. 8 A typical balance like the ones used in school classes can be read to − 4 an accuracy of about plus or minus 0.1 grams, or 10 kg. What if the laws of physics had been designed around a different value of the speed of light? To make mass-energy equivalence detectable in example 12 on page 74 using an ordinary balance, would c have to be smaller than it is in our universe, or bigger? Find the value of c for which the effect would be just barely detectable. 9 Physics in the modern sense of the word began in the seventeenth century, with Galileo and Newton, but conservation of energy wasn’t dis- covered until the nineteenth century. In the intervening period, there was no scientific reason to think that it was impossible to make a perpetual Problems 77motion machine, which today we would describe as a machine that creates more energy than it takes in. For instance, people tried to make cars that would run forever without requiring fuel. We now know this is impossible because of conservation of energy; as a car rolls, a great deal of frictional heating occurs, and the amount of heat created must be the same as the amount of energy consumed by burning the fuel. Even so, people still try to make perpetual motion machines. The U.S. patent office long ago elim- inated its general requirement that a working model accompany a patent application, but the requirement still applies to attempts to patent a per- petual motion machine; since a working model is forbidden by the laws of physics, this has the effect of making it impossible to patent a perpetual motion machine. Nowadays, enthusiasts tend to talk about “free energy” or “vacuum energy” rather than “perpetual motion.” (Vacuum energy is legitimate physics, but these people are trying to say it can be used to violate conservation of energy, which is wrong.) Websurf, and try to nd fi some examples of people promoting or selling perpetual motion machines or designs for them. Is it clear where the border lies between science and pseudoscience? If you form opinions about which people’s web pages are scams, would you be able to convince someone who hadn’t taken a physics course? Can you find any free-energy nuts within otherwise respectable organizations such as NASA? — in Google (google.com), for instance, you can do an advanced search in which you ask only for results from a spe- cific domain like nasa.gov. What about category-based guides to the Web, such as Open Directory (dmoz.org) or Yahoo (yahoo.com)? How do their editors seem to treat pseudoscience sites? Do you agree with their deci- sions? Back up all your statements with specific descriptions of the data you collected by websurfing. Problem 10 is to be done after you’ve completed lab 4b, and know the equation for an object’s kinetic energy in terms of its mass and speed. Problem 10. 10 The multiflash photograph below shows a collision between two pool balls. The ball that was initially at rest shows up as a dark image in its initial position, because its image was exposed several times before it was struck and began moving. By making measurements on the gur fi e, determine whether or not energy appears to have been conserved in the collision. What systematic effects would limit the accuracy of your test? 78 Chapter 4 Conservation of Mass and Energy(From an example in PSSC Physics.) Problems 79Lab 4a: Conservation Laws Apparatus two gallons of gasoline is twice as much as the amount of energy contained in one gal- Part A: lon; energy is additive. An example of a vacuum pump .................................1 non-additive quantity is temperature. Two electronic balance (large capacity) .............1 cups of coffee do not have twice as high a plastic-coated as fl k ......................1/group temperature as one cup. Part B: propyl alcohol ....................200 mL/group Conservation laws always refer to the total canola oil .........................200 mL/group amount of the quantity when you add it all funnels ................................. 2/group up. If you add it all up at one point in time, 100-mL volumetric as fl k .................1/group and then come back at a later point in time rubber stopper, ttin fi g in and add it all up, it will be the same. volumetric flask .........................1/group 1-ml pipette and bulb ...................1/group Howcanwepindownmoreaccuratelytheconcept magnetic stirrer .........................1/group of the “amount of a substance”? Should a gallon triple-beam balance .....................1/group ofshavingcreambeconsidered“moresubstantial” Introduction thanabrick? Atleasttwopossiblequantitiescome to mind: mass and volume. Is either conserved? Styles in physics come and go, and once-hallowed Both? Neither? To nd fi out, we will have to make principlesgetmodifiedasmoreaccuratedatacome measurements. along,butsomeofthemostdurablefeaturesofthe science are its conservation laws. A conservation We can measure mass by the “see-saw method” law is a statement that something always remains — when two children are sitting on the opposite constantwhenyouadditallup. Mostpeoplehave sides of a see-saw, the more massive one has to a general intuitive idea that the amount of a sub- move closer in to the fulcrum to make it balance. stance is conserved. That objects do not simply If we enslave some particular child as our perma- appearordisappearisaconceptualachievementof nent mass standard, then any other child’s mass babies around the age of 9-12 months. Beginning can simply be measured by balancing them on the at this age, they will for instance try to retrieve a other side and measuring their distance from the toy that they have seen being placed under a blan- fulcrum. A more practical version of the same ba- ket, rather than just assuming that it no longer sic principle that does not involve human rights violations is the familiar pan balance with sliding exists. Conservation laws in physics have the fol- weights. lowing general features: Volume is not necessarily so easy to measure. For Physicists trying to nd fi new conservation instance, shavingcreamismostlyair, soshouldwe lawswilltrytofindameasurable, numerical nd fi a way to measure just the volume of the bub- quantity, so that they can check quantita- bly film itself? Precise measurements of volume tively whether it is conserved. One needs an can most easily be done with liquids and gases, operationaldefinitionofthequantity, mean- whichconformtoavesselinwhichtheyareplaced. ingadenit fi ionthatspellsouttheoperations Shouldagas,suchasair,becountedashavingany required to measure it. substanceatall? EmpedoclesofAcragas(bornca. 492 BC) was the originator of the doctrine that all Conservation laws are only true for closed material substances are composed of mixtures of systems. For instance, the amount of water four elements: earth, fire, water and air. The idea inabottlewillremainconstantaslongasno seems amusingly naive now that we know about water is poured in or out. But if water can the chemical elements and the periodic table, but get in or out, we say that the bottle is not it was accepted in Europe for two thousand years, a closed system, and conservation of matter and the inclusion of air as a material substance cannot be applied to it. was actually a nontrivial concept. Air, after all, The quantity should be additive. For in- was invisible, seemed weightless, and had no defi- stance, the amount of energy contained in nite shape. Empedocles decided air was a form of 80 Chapter 4 Conservation of Mass and Energy

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