Conservation of Momentum

Conservation of Momentum
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Published Date:25-10-2017
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Pool balls exchange momentum. Chapter 5 Conservation of Momentum Physicist Murray Gell-Mann invented a wonderful phrase that has since entered into popular culture: “Everything not forbidden is compulsory.” Although people now use it as a sarcastic political statement, Gell-Mann was just employing politics as a metaphor for physics. What he meant was that the laws of physics forbid all the impossible things, and what’s left over is what really happens. Conservation of mass and energy prevent many things from happening. Objects can’t disappear into thin air, and you can’t run your car forever without putting gas in it. Some other processes are impossible, but not forbidden by these two conservation laws. In the martial arts movie Crouching Tiger, Hidden Dragon,thosewhohavereceivedmysticalenlightenmentareabletoviolate the laws of physics. Some of the violations are obvious, such as their ability to yfl , but others are a little more subtle. The rebellious young heroine/antiheroine Jen Yu gets into an argument while sitting at a table in a restaurant. A young tough, Iron Arm Lu, comes running toward her at full speed, and she puts up one arm and effortlessly makes him bounceback,withoutevengettingoutofherseatorbracingherselfagainst anything. She does all this between bites. It’s impossible, but how do we know it’s impossible? It doesn’t violate conservation of mass, because neither character’s mass changes. It conserves energy as well, since the rebounding Lu has the same energy he started with. Suppose you live in a country where the only laws are prohibitions against murder and robbery. One day someone covers your house with graffiti, and the authorities refuse to prosecute, because no crime was committed. You’reconvincedoftheneedforanewlawagainstvandalism. Similarly, the story of Jen Yu and Iron Arm Lu shows that we need a new conservation law. 895.1 Translation Symmetry Themostfundamentallawsofphysicsareconservationlaws,andNoether’s theorem tells us that conservation laws are the way they are because of symmetry. Time-translation symmetry is responsible for conservation of energy, but time is like a river with only two directions, past and future. What’simpossibleaboutLu’smotionistheabruptreversalinthedirection of his motion in space, but neither time-translation symmetry nor energy conservation tell us anything about directions in space. When you put gas in your car, you don’t have to decide whether you want to buy north gas or south gas, east, west, up or down gas. Energy has no direction. What we need is a new conserved quantity that has a direction in space, and such a conservation law can only come from a symmetry that relates to space. Sincewe’vealreadyhadsomeluckwithtime-translationsymmetry, it seems reasonable to turn now to space-translation symmetry, which I introduced on page 13 but haven’t mentioned since. Space-translation symmetry would seem reasonable to most people, but you’ll see that it ends up producing some very surprising results. To see how, it will be helpful to imagine the consequences of a violation of space-translation symmetry. What if, like the laws of nations, the laws of physics were different in different places? What would happen, and how would we detect it? We could try doing the same experiment in two different places and comparing the results, but it’s even easier than that. Tap your finger on this spot on the page × and then wait a second and do it again. Did both taps occur at the same point in space? You’re probably thinking that’s a silly question; am I just checking whether you followed my directions? Not at all. Consider the whole scene from the point of view of a Martian who is observing it through a powerful telescope from her home planet. (You didn’t draw the curtains, did you?) From her point of view, the earth is spinning on its axis and orbiting the sun, at speeds measured in thousands of kilometers per hour. According to her, your second finger tap happened at a point in space about 30 kilometers from the rs fi t. If you want to impress the Martians and win the Martian version of the Nobel Prize for detecting a violation of space-translation symmetry, all you have to do is perform a physics experiment twice in the same laboratory, and show that the result comes out different. But who’s to say that the Martian point of view is the right one? It gets a little thorny now. How do you know that what you detected was a violation of space-translation symmetry at all? Maybe it was just a violation of time-translation symmetry. The Martian Nobel committee isn’t going to give you the prize based on an experiment this ambiguous. A possible scheme for resolving the ambiguity would be to wait a year and do the same experiment a third time. After a year, the earth will have completed one full orbit around the sun, and your lab will be back in the same spot in space. If the third experiment comes out the same as the first one, then you can make a strong argument that what you’ve detected is an asymmetry of space, not time. There’s a problem, however. You and the Martians agree that the earth is back in the same place after a year, butwhataboutanobserverfromanothersolarsystem, whoseplanet orbits a different star? This observer says that our whole solar system is 90 Chapter 5 Conservation of Momentumin motion. To him, the earth’s motion around our sun looks like a spiral or a corkscrew, since the sun is itself moving. 5.2 The Strong Principle of Inertia Symmetry and inertia This story shows that space-translation symmetry is closely related to therelativenatureofmotion. Ridinginatrainonalong, straighttrackat constant speed, how can you even tell you’re in motion? You can look at the scenery outside, but that’s irrelevant, because we could argue that the trees and cows are moving while you stand still. (The Martians say both train and scenery are moving.) The real point is whether you can detect your motion without reference to any external object. You can hear the repetitive thunk-thunk-thunk as the train passes from one piece of track to the next, but again this is just a reference to an external object — all that proves is that you’re moving relative to the tracks, but is there any way to tell that you’re moving in some absolute sense? Assuming no interaction with the outside world, is there any experiment you can do that will come out different when the train is in motion than when it’s at rest? You could if space-translation symmetry was violated. If the laws of physics were different in different places, then as the train moved it would pass through them. “Riding over” these regions would be like riding over the pieces of track, but you would be able to detect the transition from one region to the next simply because experiments inside the train came out different, without referring to any external objects. Rather than the thunk-thunk-thunk of the rails, you would detect increases and decreases in some quantity such as the gravitational constant G, or the speed of light, or the mass of the electron. Wecanthereforeconcludethatthefollowingtwohypothesesareclosely related. The principle of inertia (strong version) Experiments don’t come out different due to the straight-line, constant- speed motion of the apparatus. Space-translation symmetry The laws of physics are the same at every point in space. Specifically, ex- periments don’tcome outdifferentjustbecause yousetupyourapparatus in a different place. A state of absolute rest example 1 Suppose that space-translation symmetry is violated. The laws of phys- ics are different in one region of space than in another. Cruising in our spaceship, we monitor the fluctuations in the laws of physics by watch- ing the needle on a meter that measures some fundamental quantity such as the gravitational constant. We make a short blast with the ship’s engines and turn them off again. Now we see that the needle is wavering more slowly, so evidently it’s taking us more time to move from one region to the next. We keep on blasting with the ship’s engines until the fluctuations stop entirely. Now we know that we’re in a state of absolute rest. The violation of translation symmetry logically resulted in a violation of the principle of inertia. Section 5.2 The Strong Principle of Inertia 91Self-check A Suppose you do an experiment to see how long it takes for a rock to drop one meter. This experiment comes out different if you do it on the moon. Does this violate space-translation symmetry? . Answer, p. 120 People have a strong intuitive belief that there is a state of absolute rest, and that the earth’s surface defines it. But Copernicus proposed as a mathematical assumption, and Galileo argued as a matter of physical reality, that the earth spins on its axis, and also circles the sun. Galileo’s opponents objected that this was impossible, because we would observe the effects of the motion. They said, for example, that if the earth was moving, then you would never be able to jump up in the air and land in the same place again — the earth would have moved out from under you. Galileorealizedthatthiswasn’treallyanargumentabouttheearth’s motion but about physics. In one of his books, which were written in the formofdialogues,hehasthethreecharactersdebatewhatwouldhappenif a ship was cruising smoothly across a calm harbor and a sailor climbed up tothetopofitsmastanddroppedarock. Wouldithitthedeckatthebase of the mast, or behind it because the ship had moved out from under it? Thisisthekindofexperimentreferredtointhestrongprincipleofinertia, and Galileo knew that it would come out the same regardless of the ship’s motion. His opponents’ reasoning, as represented by the dialog’s stupid character Simplicio, was based on the assumption that once the rock lost contact with the sailor’s hand, it would naturally start to lose its forward motion. In other words, they didn’t even believe in the weak principle of inertia (page 28), which states that motion doesn’t naturally slow down. Thestrongprincipleofinertiasaysmorethanthat. Itsaysthatmotion isn’t even real: to a sailor standing on the deck of the ship, the deck and the masts and the rigging are not even moving. People on the shore can tell him that the ship and his own body are moving in a straight line at constantspeed. Hecanreply,“No,that’sanillusion. I’matrest. Theonly reason you think I’m moving is because you and the sand and the water are moving in the opposite direction.” The strong principle of inertia says thatstraight-line, constant-speedmotionisamatterofopinion. Theweak principle of inertia is then a logical byproduct: things can’t “naturally” slow down and stop moving, because we can’t even agree on which things are moving and which are at rest. If observers in different frames of reference disagree on velocities, it’s natural to want to be able to convert back and forth. For motion in one dimension, this can be done by simple addition. A sailor running on the deck example 2 . A sailor is running toward the front of a ship, and the other sailors say that in their frame of reference, fixed to the deck, his velocity is 7.0 m/s. The ship is moving at 1.3 m/s relative to the shore. How fast does an observer on the beach say the sailor is moving? . They see the ship moving at 7.0 m/s, and the sailor moving even faster than that because he’s running from the stern to the bow. In one second, the ship moves 1.3 meters, but he moves 1.3 + 7.0 m, so his velocity relative to the beach is 8.3 m/s. The only way to make this rule come out consistent is if we dene fi velocitiesinonedirectionaspositiveandvelocitiesintheoppositedirection as negative. 92 Chapter 5 Conservation of MomentumRunning back toward the stern example 3 . The sailor of example 2 turns around and runs back toward the stern at the same speed relative to the deck. How do the other sailors describe this velocity mathematically, and what do observers on the beach say? . Since the other sailors described his original velocity as positive, they have to call this negative. They say his velocity is now− 7.0 m/s. A person on the shore says his velocity is 1.3 + (− 7.0) =− 5.7 m/s. Inertial and noninertial frames Let’s not overstate this. Is all motion a matter of opinion? No — try telling that to the brave man in figure a He’s the one who feels the effects of the motion, not the observers standing by the track. Even if he can pull his face together enough to speak, he won’t have much luck convincing themthathismotionisanillusion,andthatthey’retheoneswhoarereally moving backward while his rocket sled is standing still. Only straight- line, constant-speed motion is a matter of opinion. His speed is changing, and the change in speed produces real effects. Experiments do come out different if your apparatus is changing its speed. A frame of reference whosemotionischangingiscalledanoninertialframeofreference,because the principle of inertia doesn’t apply to it. a / This Air Force doctor volun- teered to ride a rocket sled as a medical experiment. The obvious effects on his head and face are not because of the sled’s speed but because of its rapid changes in speed: increasing in 2 and 3, and decreasing in 5 and 6. In 4 his speed is greatest, but be- cause his speed is not increasing or decreasing very much at this moment, there is little effect on him. Experiments also come out different if your apparatus is changing its direction of motion. The landscape around you is moving in a circle right now due to the rotation of the Earth, and is therefore changing the direc- tion of its motion continuously on a 24-hour cycle. However, the curve of themotionissogentlethatunderordinaryconditionswedon’tnoticethat the local dirt’s frame of reference isn’t quite inertial. The first demonstra- tion of the noninertial nature of the earth-fixed frame of reference was by Foucault using a very massive pendulum whose oscillations would persist for many hours. Although Foucault did his demonstration in Paris, it’s easier to imagine what would happen at the north pole: the pendulum would keep swinging in the same plane, but the earth would spin under- Section 5.2 The Strong Principle of Inertia 93neath it once every 24 hours. To someone standing in the snow, it would appear that the pendulum’s plane of motion was twisting. The effect at latitudes less than 90 degrees turns out to be slower, but otherwise sim- ilar. The Foucault pendulum was the first denit fi ive experimental proof that the earth really did spin on its axis, although scientists had been con- vinced of its rotation for a century based on more indirect evidence about the structure of the solar system. Often when we adopt a noninertial frame of reference, there is a vivid illusion that the laws of physics are being violated. It might seem like the Foucault pendulum was being influenced by evil spirits, if you forgot that b / Foucault demonstrates his it was actually the ground that was twisting around, not the pendulum. pendulum to an audience at a A simpler example is shown in gure fi c. A bowling ball is in the back lecture in 1851. of a pickup truck, and the driver steps on the brakes. Because the truck is changing its speed, a frame of reference that moves with the truck is noninertial. For the driver, there is a strong psychological tendency to adopt this bad frame of reference, c/1, but then the bowling ball seems to be violating the laws of physics: according to the weak principle of inertia, the ball has no reason to start rolling toward the front of the truck. It’s not interacting with any other object that would cause it to do this. In figure c/2, we watch the motion in an (approximately) inertial frame of reference fixed to the sidewalk, and everything makes sense. The ball obeys the weak principle of inertia, and moves equal distances in equal time intervals. In this frame, it’s the truck that changes its speed, which makessense,becausethetruck’swheelsareinteractingwiththepavement. c / A bowling ball in the back of a pickup truck is viewed in a non- 1 2 inertial frame, 1, and an inertial one, 2. PopularbeliefhasGalileobeingprosecutedbytheCatholicChurchfor sayingtheearthrotatedonitsaxisandalsoorbitedthesun,butFoucault’s pendulum was still centuries in the future, so Galileo had no hard proof; hisinsightsintorelativeversusabsolutemotionsimplymadeitmoreplau- sible that the world could be spinning without producing dramatic effects, but didn’t disprove the contrary hypothesis that the sun, moon, and stars wentaroundtheearthevery24hours. Furthermore, theChurchwasmuch d / Galileo on trial before the more liberal and enlightened than most people believe. It didn’t (and still Inquisition. doesn’t)requirealiteralinterpretationoftheBible, andoneoftheChurch 94 Chapter 5 Conservation of Momentumofficials involved in the Galileo affair wrote that “the Bible tells us how to gotoheaven,nothowtheheavensgo.” Inotherwords,religionandscience shouldbeseparate. TheactualreasonGalileogotintroubleisshroudedin mystery, since Italy in the age of the Medicis was a secretive place where unscrupulous people might settle a score with poison or a false accusation of heresy. What is certain is that Galileo’s satirical style of scientific writ- ing made many enemies among the powerful Jesuit scholars who were his intellectualopponents—hecomparedonetoasnakethatdoesn’tknowits own back is broken. Galileo and the Pope were old friends, but someone started a rumor that the stupid character Simplicio in Galileo’s dialogs was really meant to represent the Pope. It’s also possible that the Church was far less upset by his astronomical work than by his support for atom- ism, the idea that matter is made of atoms. Some theologians perceived atomism as contradicting transubstantiation, the Church’s doctrine that the holy bread and wine are literally transformed into the flesh and blood of Christ by the priest’s blessing. Section 5.2 The Strong Principle of Inertia 955.3 Momentum Conservation of momentum Let’s return to the impossible story of Jen Yu and Iron Arm Lu on page 89. For simplicity, we’ll model them as two identical, featureless pool balls, e. This may seem like a drastic simplification, but even a collision betweentwohumanbodiesisreallyjustaseriesofmanycollisionsbetween atoms. The film shows a series of instants in time, viewed from overhead. The light-colored ball comes in, hits the darker ball, and rebounds. It seems strange that the dark ball has such a big effect on the light ball without experiencing any consequences itself, but how can we show that this is really impossible? e / How can we prove that this col- lision is impossible? We can show it’s impossible by looking at it in a different frame of reference, f. This camera follows the light ball on its way in, so in this frame the incoming light ball appears motionless. (If you ever get hauled intocourtonanassaultchargeforhittingsomeone,trythisdefense: “Your honor, in my sfi t’s frame of reference, it was his face that assaulted my knuckles”) After the collision, the camera keeps moving in the same direction, because if it didn’t, it wouldn’t be showing us an inertial frame of reference. To help convince yourself that gur fi es e and f represent the same motion seen in two different frames, note that both films agree on the distances between the balls at each instant. After the collision, frame f shows the light ball moving twice as fast as the dark ball; an observer whoprefersframe eexplainsthisbysayingthatthecamerathatproduced film f was moving one way, while the ball was moving the opposite way. f / The collision of figure e is viewed in a different frame of ref- erence. Figureseandfrecordthesameevents,soifoneisimpossible,theother istoo. Butgur fi e fisdenitely fi impossible,becauseitviolatesconservation of energy. Before the collision, the only kinetic energy is the dark ball’s. Afterthecollision, lightballsuddenlyhassomeenergy,butwheredidthat energy come from? It can only have come from the dark ball. The dark ball should then have lost some energy, which it hasn’t, since it’s moving at the same speed as before. 96 Chapter 5 Conservation of MomentumFigure g shows what really does happen. This kind of behavior is familiar to anyone who plays pool. In a head-on collision, the incoming ball stops dead, and the target ball takes all its energy and flies away. In g/1,thelightballhitsthedarkball. Ing/2,thecameraisinitiallyfollowing the light ball; in this frame of reference, the dark ball hits the light one (“Judge, his face hit my knuckles”). The frame of reference shown in g/3 is particularly interesting. Here the camera always stays at the midpoint betweenthetwoballs. Thisiscalledthecenter-of-massframeofreference. g / This is what really happens. Three films represent the same collision viewed in three different frames of reference. Energy is conserved in all three frames. Self-check B In each picture in figure g/1, mark an x at the point half-way in between the two balls. This series of five x’s represents the motion of the camera that was used to make the bottom film. How fast is the camera moving? Does it represent an inertial frame of reference? . Answer, p. 120 What’sspecialaboutthecenter-of-massframeisitssymmetry. Inthis frame, both balls have the same initial speed. Since they start out with the same speed, and they have the same mass, there’s no reason for them to behave differently from each other after the collision. By symmetry, if the light ball feels a certain effect from the dark ball, the dark ball must feel the same effect from the light ball. This is exactly like the rules of accounting. Let’s say two big corpora- tionsaredoingbusinesswitheachother. IfGlutcorppaysamilliondollars to Slushco, two things happen: Glutcorp’s bank account goes down by a Section 5.3 Momentum 97million dollars, and Slushco’s rises by the same amount. The two compa- nies’ books have to show transactions on the same date that are equal in size, but one is positive (a payment) and one is negative. What if Glut- corp records− 1,000,000 dollars, but Slushco’s books say +920,000? This indicates that a law has been broken; the accountants are going to call the policeandstartlookingfortheemployeewho’sdrivinganew80,000-dollar Jaguar. Money is supposed to be conserved. Infigure g, let’sdene fi velocitiesaspositiveifthemotionistowardthe top of the page. In gur fi e g/1 let’s say the incoming light ball’s velocity is 1 m/s. velocity (meters per second) before the collision after the collision change 0 1 +1 1 0 − 1 The books balance. The light ball’s payment,− 1, matches the dark ball’s receipt, +1. Everything also works out ne fi in the center of mass frame, g/3: velocity (meters per second) before the collision after the collision change − 0.5 +0.5 +1 +0.5 − 0.5 − 1 Self-check C Make a similar table for figure g/2. What do you notice about the change in velocity when you compare the three tables? . Answer, p. 120 Accountingworksbecausemoneyisconserved. Apparently,something is also conserved when the balls collide. We call it momentum. Momen- tum is not the same as velocity, because conserved quantities have to be additive. Our pool balls are like identical atoms, but atoms can be stuck together to form molecules, people, and planets. Because conservation laws work by addition, two atoms stuck together and moving at a certain velocity must have double the momentum that a single atom would have had. We therefore define momentum as velocity multiplied by mass. Conservation of momentum The quantity defined by momentum=mv is conserved. This is our second example of Noether’s theorem: symmetry conserved quantity time translation ⇒ mass-energy space translation ⇒ momentum 98 Chapter 5 Conservation of MomentumConservation of momentum for pool balls example 4 . Is momentum conserved in figure g/1? . We have to check whether the total initial momentum is the same as the total final momentum. dark ball’s initial momentum + light ball’s initial momentum =? dark ball’s final momentum + light ball’s final momentum Yes, momentum was conserved: 0 + mv = mv + 0 Figure skaters push off from each other example 5 Let’s revisit the figure skaters from the example on page 11. I argued there that if they had equal masses, then mirror symmetry would imply that they moved off with equal speeds in opposite directions. Let’s check that this is consistent with conservation of momentum: left skater’s initial momentum + right skater’s initial momentum =? left skater’s final momentum + right skater’s final momentum Momentum was conserved: 0 + 0 = m× (− v) + mv h / example 5 This is an interesting example, because if these had been pool balls in- stead of people, we would have accused them of violating conservation of energy. Initially there was zero kinetic energy, and at the end there wasn’t zero. (Note that the energies at the end don’t cancel, because kinetic energy is always positive, regardless of direction.) The mystery is resolved because they’re people, not pool balls. They both ate food, and they therefore have chemical energy inside their bodies: food energy→ kinetic energy + kinetic energy + heat Unequal masses example 6 . Suppose the skaters have unequal masses: 50 kg for the one on the left, and 55 kg for the other. The more massive skater, on the right, moves off at 1.0 m/s. How fast does the less massive skater go? . Their momenta (plural of momentum) have to be the same amount, but with opposite signs. The less massive skater must have a greater velocity if her momentum is going to be as much as the more massive one’s. 0 + 0 = (50 kg)(− v) + (55 kg)(1.0 m/s) (50 kg)(v) = (55 kg)(1.0 m/s) (55 kg) v = (1.0 m/s) 50 kg = 1.1 m/s Section 5.3 Momentum 99Momentum compared to kinetic energy Momentumandkineticenergyarebothmeasuresoftheamountofmo- tion, and a sideshow in the Newton-Leibniz controversy over who invented calculus was an argument over which quantity was the “true” measure of motion. The modern student can certainly be excused for wondering why we need both quantities, when their complementary nature was not evi- dent to the greatest minds of the 1700’s. The following table highlights their differences. Kinetic energy... Momentum... has no direction in space. has a direction in space. isalwayspositive,andcannotcan- cancelswithmomentumintheop- cel out. posite direction. can be traded for forms of energy is always conserved. that do not involve motion. KE is not a conserved quantity by itself. is quadrupled if the velocity is is doubled if the velocity is dou- doubled (lab 4b). bled. Here are some examples that show the different behaviors of the two quantities. A spinning coin example 7 A spinning coin has zero total momentum, because for every moving point, there is another point on the opposite side that cancels its mo- mentum. It does, however, have kinetic energy. Momentum and kinetic energy in firing a rifle example 8 The rifle and bullet have zero momentum and zero kinetic energy to start with. When the trigger is pulled, the bullet gains some momen- tum in the forward direction, but this is canceled by the rifle’s backward momentum, so the total momentum is still zero. The kinetic energies of the gun and bullet are both positive numbers, however, and do not i / example 7 cancel. The total kinetic energy is allowed to increase, because both ob- jects’ kinetic energies are destined to be dissipated as heat — the gun’s “backward” kinetic energy does not refrigerate the shooter’s shoulder The wobbly earth example 9 As the moon completes half a circle around the earth, its motion re- verses direction. This does not involve any change in kinetic energy, because the moon doesn’t speed up or slow down, nor is there any change in gravitational energy, because the moon stays at the same 1 distance from the earth. The reversed velocity does, however, imply a reversed momentum, so conservation of momentum tells us that the j / example 10 earth must also change its momentum. In fact, the earth wobbles in a little “orbit” about a point below its surface on the line connecting it and the moon. The two bodies’ momenta always point in opposite directions and cancel each other out. The earth and moon get a divorce example 10 Why can’t the moon suddenly decide to fly off one way and the earth the other way? It is not forbidden by conservation of momentum, because the moon’s newly acquired momentum in one direction could be can- celed out by the change in the momentum of the earth, supposing the 1 Actually these statements are both only approximately true. The moon’s orbit isn’t exactly a circle. 100 Chapter 5 Conservation of Momentumearth headed the opposite direction at the appropriate, slower speed. The catastrophe is forbidden by conservation of energy, because both their kinetic energies would have increased greatly. Momentum and kinetic energy of a glacier example 11 12 A cubic-kilometer glacier would have a mass of about 10 kg — 1 fol- lowed by 12 zeroes. If it moves at a speed of 0.00001 m/s, then its 2 momentum is 10, 000, 000. This is the kind of heroic-scale result we expect, perhaps the equivalent of the space shuttle taking off, or all the cars in LA driving in the same direction at freeway speed. Its kinetic energy, however, is only 50 joules, the equivalent of the calories con- tained in a poppy seed or the energy in a drop of gasoline too small to be seen without a microscope. The surprisingly small kinetic energy is because kinetic energy is proportional to the square of the velocity, and the square of a small number is an even smaller number. Force Definition of force When momentum is being transferred, we refer to the rate of transfer 3 as the force. The metric unit of force is the newton (N). The relationship between force and momentum is like the relationship between power and energy, or the one between your cash flow and your bank balance: conserved quantity rate of transfer name units name units energy joules (J) power watts (W) momentum kg· m/s force newtons (N) A bullet example 12 4 . A bullet emerges from a gun with a momentum of 1.0 units, after having been acted on for 0.01 seconds by the force of the gases from the explosion of the gunpowder. What was the force on the bullet? 5 . The force is 1.0 = 100 newtons . 0.01 There’s no new physics happening here, just a definition of the word “force.” Definitions are neither right nor wrong, and just because the Chinese call it instead, that doesn’t mean they’re incorrect. But when Isaac Newton rst fi started using the term “force” according to this techni- cal definition, people already had some definite ideas about what the word meant. Forces occur in equal-strength pairs In some cases Newton’s definition matches our intuition. In example 12, we divided by a small time, and the result was a big force; this is intuitively reasonable, since we expect the force on the bullet to be strong. In other situations, however, our intuition rebels against reality. 2 The units of this number are kilograms times meters per second, or kg· m/s. 3 This definition is known as Newton’s second law of motion. Don’t memorize that 4 metric units of kg· m/s 5 This is really only an estimate of the average force over the time it takes for the bullet to move down the barrel. The force probably starts out stronger than this, and then gets weaker because the gases expand and cool. Section 5.3 Momentum 101Extra protein example 13 . While riding my bike fast down a steep hill, I pass through a cloud of gnats, and one of them goes into my mouth. Compare my force on the gnat to the gnat’s force on me. . Momentum is conserved, so the momentum gained by the gnat equals the momentum lost by me. Momentum conservation holds true at every instant over the fraction of a second that it takes for the collision to happen. The rate of transfer of momentum out of me must equal the rate of transfer into the gnat. Our forces on each other have the same strength, but they’re in opposite directions. Mostpeoplewouldbewillingtobelievethatthemomentumgainedbythe gnat is the same as the momentum lost by me, but they would not believe that the forces are the same strength. Nevertheless, the second statement follows from the first merely as a matter of deni fi tion. Whenever two objects, A and B, interact, A’s force on B is the same strength as B’s force 6 on A, and the forces are in opposite directions. (A on B)=− (B on A) Excuse me, ma'am, but it appears that the money in your Using the metaphor of money, suppose Alice and Bob are adrift in a life purse would exactly cancel raft, and pass the time by playing poker. Money is conserved, so if they out my bar tab. count all the money in the boat every night, they should always come up with the same total. A completely equivalent statement is that their cash flowsareequalandopposite. IfAliceiswinningvfi edollarsperhour, then Bob must be losing at the same rate. This statement about equal forces in opposite directions implies to many students a kind of mystical principle of equilibrium that explains why things don’t move. That would be a useless principle, since it would 7 be violated every time something moved. The ice skaters of figure h on page 99 make forces on each other, and their forces are equal in strength k / It doesn’t make sense to and opposite in direction. That doesn’t mean they won’t move. They’ll add his debts to her assets. both move — in opposite directions. Thefallacycomesfromtryingtoaddthingsthatitdoesn’tmakesense toadd,assuggestedbythecartooningur fi e k. Weonlyaddforcesthatare acting on the same object. It doesn’t make sense to say that the skaters’ forces on each other add up to zero, because it doesn’t make sense to add them. One is a force on the left-hand skater, and the other is a force on the right-hand skater. In gure fi l, my ngers’ fi force and my thumbs’ force are both acting on the bathroom scale. It does make sense to add these forces, and they may possibly add up to zero, but that’s not guaranteed by the laws of physics. l / I squeeze the bathroom If I throw the scale at you, my thumbs’ force is stronger that my fingers’, scale. It does make sense to add and the forces no longer cancel: my fingers’ force to my thumbs’, because they both act on the (fingers on scale) =6 − (thumbs on scale) . same object — the scale. What’s guaranteed by conservation of momentum is a whole different re- 6 This is called Newton’s third law. Don’t memorize that name 7 DuringtheScopesmonkeytrial, WilliamJenningsBryanclaimedthatevery time he picked his foot up off the ground, he was violating the law of gravity. 102 Chapter 5 Conservation of Momentumlationship: (fingers on scale)= − (scale on fingers) (thumbs on scale)=− (scale on thumbs) The force of gravity How much force does gravity make on an object? From everyday 8 experience, we know that this force is proportional to the object’s mass. Let’s find the force on a one-kilogram object. If we release this object from rest, then after it has fallen one meter, its kinetic energy equals the strength of the gravitational field, 10 joules per kilogram per meter× 1 kilogram× 1 meter=10 joules . Using the equation for kinetic energy from lab 4b and doing a little simple algebra, we find that its final velocity is 4.4 m/s. It starts from 0 m/s, and endsat4.4m/s,soitsaveragevelocityis2.2m/s,andthetimetakestofall one meter is therefore (1 m)/(2.2 m/s)=0.44 seconds. Its final momentum is 4.4 units, so the force on it was evidently 4.4 =10 newtons . 0.44 This is like one of those card tricks where the magician makes you go through a bunch of steps so that you end up revealing the card you had chosen — the result is just equal to the gravitational efi ld, 10, but in units of newtons If algebra makes you feel warm and fuzzy, you may want to replay the derivation using symbols and convince yourself that it had to come out that way. If not, then I hope the numerical result is enough to convince you of the general fact that the force of gravity on a one- kilogram mass equals g. For masses other than one kilogram, we have the handy-dandy result that (force of gravity on a mass m)=mg . In other words, g can be interpreted not just as the gravitational energy per kilogram per meter of height, but also as the gravitational force per kilogram. Motion in two dimensions Projectile motion Galileo was an innovator in more than one way. He was arguably the inventorofopen-source software: he inventeda mechanicalcalculating device for certain engineering applications, and rather than keeping the device’sdesignsecretashiscompetitorsdid,hemadeitpublic,butcharged students for lessons in how to use it. Not only that, but he was the first physicist to make money as a military consultant. Galileo understood projectiles better than anyone else, because he understood the principle of inertia. Even if you’re not planning on a career involving artillery, projectile motion is a good thing to learn about because it’s an example of how to handle motion in two or three dimensions. 8 This follows from the additivity of forces. Section 5.3 Momentum 103Figure m shows a ball in the process of falling — or rising, it really doesn’t matter which. Let’s say the ball has a mass of one kilogram, each square in the grid is 10 meters on a side, and the positions of the ball are shown at time intervals of one second. The earth’s gravitational force on theballis10newtons,sowitheachsecond,theball’smomentumincreases by 10 units, and its speed also increases by 10 m/s. The ball falls 10 m in the first second, 20 m in the next second, and so on. Self-check D What would happen if the ball’s mass was 2 kilograms? . Answer, p. 120 Now let’s look at the ball’s motion in a new frame of reference, n, which is moving at 10 meters per second to the left compared to the frame of reference used in gur fi e m. An observer in this frame of reference sees the ball as moving to the right by 10 meters every second. The ball traces m / A ball is falling (or rising). an arc of a specific mathematical type called a parabola: 1 step over and 1 step down 1 step over and 2 steps down 1 step over and 3 steps down 1 step over and 4 steps down ... It doesn’t matter which frame of reference is the “real” one. Both diagrams show the possible motion of a projectile. The interesting point here is that the vertical force of gravity has no effect on the horizontal motion, and the horizontal motion also has no effect on what happens in the vertical motion. The two are completely independent. If the sun is directly overhead, the motion of the ball’s shadow on the ground seems perfectly natural: there are no horizontal forces, so it either sits still or moves at constant velocity. (Zero force means zero rate of transfer of momentum.) The same is true if we shine a light from one side and cast n / The same ball is viewed the ball’s shadow on the wall. Both shadows obey the laws of physics. in a frame of reference that is moving horizontally. The moon example 14 In example 9 on page 71, I promised an explanation of how Newton knew that the gravitational field experienced by the moon due to the earth was 1/3600 of the one we feel here on the earth’s surface. The radius of the moon’s orbit had been known since ancient times (see page 27), so Newton knew its speed to be 1,100 m/s (expressed in modern units). If the earth’s gravity wasn’t acting on the moon, the moon would fly off straight, along the straight line shown in figure p, and it would cover 1,100 meters in one second. We observe instead that it travels the arc of a circle centered on the earth. Straightforward geom- etry shows that the amount by which the arc drops below the straight line is 1.6 millimeters. Near the surface of the earth, an object falls 5 9 meters in one second, which is indeed about 3600 times greater than o / The drops of water travel 1.6 millimeters. in parabolic arcs. 9 Its initial speed is 0, and its final speed is 10 m/s, so its average speed is 5 m/s over the first second of falling. 104 Chapter 5 Conservation of MomentumThe tricky part about this argument is that although I said the path of 1100 m 1.6 mm a projectile was a parabola, in this example it’s a circle. What’s going on here? What’s different here is that as the moon moves 1,100 meters, it changes its position relative to the earth, so down is now in a new to earth to earth direction. We’ll discuss circular motion more carefully soon, but in this example, it really doesn’t matter. The curvature of the arc is so gentle p / example 14 that a parabola and a circle would appear almost identical. (Actually the curvature is so gentle — 1.6 millimeters over a distance of 1,100 meters — that if I had drawn the figure to scale, you wouldn’t have even been able to tell that it wasn’t straight.) As an interesting historical note, Newton claimed that he first did this calculation while confined to his family’s farm during the plague of 1666, and found the results to “answer pretty nearly.” His notebooks, however, show that although he did the calculation on that date, the result didn’t quite come out quite right, and he became uncertain about whether his theory of gravity was correct as it stood or needed to be modified. Not until 1675 did he learn of more accurate astronomical data, which convinced him that his theory didn’t need to be tinkered with. It appears that he rewrote his own life story a little bit in order to make it appear that his work was more advanced at an earlier date, which would have helped him in his dispute with Leibniz over priority in the invention of calculus. The memory of motion There’s another useful way of thinking about motion along a curve. The weak principle of inertia tells us that in the absence of a force, an object will continue moving in the same speed and in the same direction. One of my students invented a wonderful phrase for this: the memory of motion. Over the first second of its motion, the ball in figure q moved 1 square over and 1 square down, which is 10 meters and 10 meters. The default for the next one-second interval would be to repeat this, ending up at the location marked with the first dashed circle. The earth’s 10-newton q / The memory of motion: gravitational force on the ball, however, changes the vertical part of the the default would be for the ball ball’s momentum by 10 units. The ball actually ends up 10 meters (1 to continue doing what it was square) below the default. already doing. The force of grav- ity makes it deviate downward, ending up one square below the default. Section 5.3 Momentum 105Circular motion Figure r shows how to apply the memory-of-motion idea to circular motion. It should convince you that only an inward force is needed to produce circular motion. One of the reasons Newton was the first to make any progress in analyzing the motion of the planets around the sun was thathiscontemporarieswereconfusedonthispoint. Mostofthemthought thatinadditiontoanattractionfromthesun,asecond,forwardforcemust exist on the planets, to keep them from slowing down. This is incorrect Aristotelian thinking; objects don’t naturally slow down. Car 1 in figure s only needs a forward force in order to cancel out the backward force of friction; the total force on it is zero. Similarly, the forward and backward forces on car 2 are canceling out, and the only force left over is the inward one. There’s no friction in the vacuum of outer space, so if car 2 was a planet, thebackwardforcewouldn’texist; theforwardforcewouldn’texist either, because the only force would be the force of the sun’s gravity. r / A large number of gentle taps gives a good approximation to cir- cular motion. A steady inward force would give exactly circular motion. On page 94 we saw that when we tried to visualize motion in a non- inertial frame of reference, we experienced the vivid illusion of a violation of the laws of physics. In circular motion, this temptation is especially 1 strong. Frame t/1, attached to the turning truck, is noninertial, because 2 it changes the direction of its motion. The ball violates the weak principle of inertia by accelerating from rest for no apparent reason. Is there some mysterious outward force that is slamming the ball into the side of the truck’s bed? No. By analyzing everything in a proper inertial frame of reference, t/2, we see that it’s the truck that swerves and hits the ball. That makes sense, because the truck is interacting with the asphalt. s / The forces on car 1 can- cel, and the total force on it is zero. The forward and backward forces on car 2 also cancel. Only the inward force remains. 106 Chapter 5 Conservation of Momentum1 2 t / A bowling ball is in the back of a pickup truck turning left. The motion is viewed first in a frame that turns along with the truck, 1, and then in an inertial frame, 2. Section 5.3 Momentum 107Problems 1 The beer bottle shown in the figure is resting on a table in the dining car of a train. The tracks are straight and level. What can you tell about themotionofthetrain? Canyoutellwhetherthetrainiscurrentlymoving forward, movingbackward, orstandingstill? Canyoutellwhatthetrain’s speed is? 2 You’reapassengerintheopenbaskethangingunderahot-airballoon. The balloon is being carried along by the wind at a constant velocity. If you’re holding a ag fl in your hand, will the flag wave? If so, which way? (Based on a question from PSSC Physics.) 3 Driving along in your car, you take your foot off the gas, and your speedometer shows a reduction in speed. Describe an inertial frame in which your car was speeding up during that same period of time. Problem 1. 4 If all the air molecules in the room settled down in a thin lm fi on the floor,wouldthatviolateconservationofmomentumaswellasconservation of energy? 5 A bullet flies through the air, passes through a paperback book, and then continues to fly through the air beyond the book. When is there a force? When is there energy? 6 (a) Continue figure n farther to the left, and do the same for the numerical table in the text. (b)Sketchasmoothcurve(aparabola)throughallthepointsonthegur fi e, including all the ones from the original figure and all the ones you added. Identify the very top of its arc. (c) Now consider figure m. Is the highest point shown in the figure the top Problem 2 of the ball’s up-down path? Explain by comparing with your results from parts a and b. 7 Criticize the following statement about the top panel of figure g on page 97: In the rst fi few pictures, the light ball is moving up and to the right, while the dark ball moves directly to the right. 8 The gure fi on page 109 shows a ball dropping to the surface of the earth. Energy is conserved: over the whole course of the lm, fi the gravi- tational energy between the ball and the earth decreases by 1 joule, while the ball’s kinetic energy increases by 1 joule. (a) How can you tell directly from the figure that the ball’s speed isn’t staying the same? (b) Draw what the lm fi would look like if the camera was following the ball. (c) Explain how you can tell that in this new frame of reference, energy is not conserved. (d) Does this violate the strong principle of inertia? Isn’t every frame of reference supposed to be equally valid? 108 Chapter 5 Conservation of Momentum

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