intro_physics_1

Preliminaries 23 to be learned, while using the material to be learned to accomplish specific goals, while teaching some of what you figure out to others who are sharing this whole experience while being taught by them in turn. The assignment is much more important than lecture, as it is entirely participatory, where real learning is far more likely to occur. You could, once you learn the trick of it, blow off lecture and do fine in a course in all other respects. If you fail to do the assignments with your entire spirit engaged, you are doomed. In other words, to learn you must do your homework, ideally at least partly in a group setting. The only question is: how should you do it to both finish learning all that stuff you sort-of-got in lecture and to re-attain the moment(s) of clarity that you then experienced, until eventually it becomes a permanent characteristic of your awareness and you know and fully understand it all on your own? There are two general steps that need to be iterated to finish learning anything at all. They are a lot of work. In fact, they are far more work than (passively) attending lecture, and are more important than attending lecture. You can learn the material with these steps without ever attending lecture, as long as you have access to what you need to learn in some media or human form. You in all probability will never learn it, lecture or not, without making a few passes through these steps. They are: a) Review the whole (typically textbooks and/or notes) b) Work on the parts (do homework, use it for something) (iterate until you thoroughly understand whatever it is you are trying to learn). Let’s examine these steps. The first is pretty obvious. You didn’t “get it” from one lecture. There was too much material. If you were lucky and well prepared and blessed with a good instructor, perhaps you grasped some of it for a moment (and if your instructor was poor or you were particu- larly poorly prepared you may not have managed even that) but what you did momentarily understand is fading, flitting further and further away with every moment that passes. You need to review the entire topic, as a whole, as well as all its parts. A set of good summary notes might contain all the relative factoids, but there are relations between those factoids – a temporal sequencing, mathematical derivations connecting them to other things you know, a topical association with other things that you know. They tell a story, or part of a story, and you need to know that story in broad terms, not try to memorize it word for word. Reviewing the material should be done in layers, skimming the textbook and your notes, creating a new set of notes out of the text in combination with your lecture notes, maybe reading in more detail to understand some particular point that puzzles you, reworking a few of the examples presented. Lots of increasingly deep passes through it (starting with the merest skim-reading or reading a summary of the whole thing) are much better than trying to work through the whole text one line at a time and not moving on until you understand it. Many things you might want to understand will only come clear from things you are exposed to later, as it is not the case that all knowledge is ordinal, hierarchical, and derivatory. You especially do not have to work on memorizing the content. In fact, it is not desire- able to try to memorize content at this point – you want the big picture first so that facts

24 Preliminaries have a place to live in your brain. If you build them a house, they’ll move right in with- out a fuss, where if you try to grasp them one at a time with no place to put them, they’ll (metaphorically) slip away again as fast as you try to take up the next one. Let’s understand this a bit. As we’ve seen, your brain is fabulously efficient at storing information in a compressed associative form. It also tends to remember things that are important – whatever that means – and forget things that aren’t important to make room for more important stuff, as your brain structures work together in understandable ways on the process. Building the cognitive map, the “house”, is what it’s all about. But as it turns out, building this house takes time. This is the goal of your iterated review process. At first you are memorizing things the hard way, trying to connect what you learn to very simple hierarchical concepts such as this step comes before that step. As you do this over and over again, though, you find that absorbing new information takes you less and less time, and you remember it much more easily and for a longer time without additional rehearsal. Sometimes your brain even outruns the learning process and “discovers” a missing part of the structure before you even read about it! By reviewing the whole, well-organized structure over and over again, you gradually build a greatly compressed representation of it in your brain and tremendously reduce the amount of work required to flesh out that structure with increasing levels of detail and remember them and be able to work with them for a long, long time. Now let’s understand the second part of doing homework – working problems. As you can probably guess on your own at this point, there are good ways and bad ways to do homework problems. The worst way to do homework (aside from not doing it at all, which is far too common a practice and a bad idea if you have any intention of learning the material) is to do it all in one sitting, right before it is due, and to never again look at it. Doing your homework in a single sitting, working on it just one time fails to repeat and rehearse the material (essential for turning short term memory into long term in nearly all cases). It exhausts the neurons in your brain (quite literally – there is metabolic energy consumed in thinking) as one often ends up working on a problem far too long in one sitting just to get done. It fails to incrementally build up in your brain’s long term memory the structures upon which the more complex solutions are based, so you have to constantly go back to the book to get them into short term memory long enough to get through a problem. Even this simple bit of repetition does initiate a learning process. Unfortunately, by not repeating them after this one sitting they soon fade, often without a discernable trace in long term memory. Just as was the case in our experiment with memorizing the number above, the prob- lems almost invariably are not going to be a matter of random noise. They have certain key facts and ideas that are the basis of their solution, and those ideas are used over and over again. There is plenty of pattern and meaning there for your brain to exploit in information compression, and it may well be very cool stuff to know and hence important to you once learned, but it takes time and repetition and a certain amount of meditation for the “gestalt” of it to spring into your awareness and burn itself into your conceptual memory as “high order understanding”.

Preliminaries 25 You have to give it this time, and perform the repetitions, while maintaining an optimistic, philosophical attitude towards the process. You have to do your best to have fun with it. You don’t get strong by lifting light weights a single time. You get strong lifting weights re- peatedly, starting with light weights to be sure, but then working up to the heaviest weights you can manage. When you do build up to where you’re lifting hundreds of pounds, the fifty pounds you started with seems light as a feather to you. As with the body, so with the brain. Repeat broad strokes for the big picture with increasingly deep and “heavy” excursions into the material to explore it in detail as the overall picture emerges. Intersperse this with sessions where you work on problems and try to use the material you’ve figured out so far. Be sure to discuss it and teach it to others as you go as much as possible, as articulating what you’ve figured out to others both uses a different part of your brain than taking it in (and hence solidifies the memory) and it helps you articulate the ideas to yourself! This process will help you learn more, better, faster than you ever have before, and to have fun doing it! Your brain is more complicated than you think. You are very likely used to working hard to try to make it figure things out, but you’ve probably observed that this doesn’t work very well. A lot of times you simply cannot “figure things out” because your brain doesn’t yet know the key things required to do this, or doesn’t “see” how those parts you do know fit together. Learning and discovery is not, alas, “intentional” – it is more like trying to get a bird to light on your hand that flits away the moment you try to grasp it. People who do really hard crossword puzzles (one form of great brain exercise) have learned the following. After making a pass through the puzzle and filling in all the words they can “get”, and maybe making a couple of extra passes through thinking hard about ones they can’t get right away, looking for patterns, trying partial guesses, they arrive at an impasse. If they continue working hard on it, they are unlikely to make further progress, no matter how long they stare at it. On the other hand, if they put the puzzle down and do something else for a while – especially if the something else is go to bed and sleep – when they come back to the puzzle they often can immediately see a dozen or more words that the day before were absolutely invisible to them. Sometimes one of the long theme answers (perhaps 25 characters long) where they have no more than two letters just “gives up” – they can simply “see” what the answer must be. Where do these answers come from? The person has not “figured them out”, they have “recognized” them. They come all at once, and they don’t come about as the result of a logical sequential process. Often they come from the person’s right brain22. The left brain tries to use logic and simple memory when it works on crosswork puzzles. This is usually good for some words, but for many of the words there are many possible answers and without any insight one can’t even recall one of the possibilities. The clues don’t suffice to connect you up to a word. Even as letters get filled in this continues to be the case, not because you don’t know the word (although in really hard puzzles this can sometimes be the case) but because you 22Note that this description is at least partly metaphor, for while there is some hemispherical specialization of some of these functions, it isn’t always sharp. I’m retaining them here (oh you brain specialists who might be reading this) because they are a valuable metaphor.

26 Preliminaries don’t know how to recognize the word “all at once” from a cleverly nonlinear clue and a few letters in this context. The right brain is (to some extent) responsible for insight and non-linear thinking. It sees patterns, and wholes, not sequential relations between the parts. It isn’t intentional – we can’t “make” our right brains figure something out, it is often the other way around! Working hard on a problem, then “sleeping on it” (to get that all important hippocampal involvement going) is actually a great way to develop “insight” that lets you solve it without really working terribly hard after a few tries. It also utilizes more of your brain – left and right brain, sequential reasoning and insight, and if you articulate it, or use it, or make some- thing with your hands, then it exercieses these parts of your brain as well, strengthening the memory and your understanding still more. The learning that is associated with this process, and the problem solving power of the method, is much greater than just working on a problem linearly the night before it is due until you hack your way through it using information assembled a part at a time from the book. The following “Method of Three Passes” is a specific strategy that implements many of the tricks discussed above. It is known to be effective for learning by means of do- ing homework (or in a generalized way, learning anything at all). It is ideal for “problem oriented homework”, and will pay off big in learning dividends should you adopt it, espe- cially when supported by a group oriented recitation with strong tutorial support and many opportunities for peer discussion and teaching. The Method of Three Passes Pass 1 Three or more nights before recitation (or when the homework is due), make a fast pass through all problems. Plan to spend 1-1.5 hours on this pass. With roughly 10- 12 problems, this gives you around 6-8 minutes per problem. Spend no more than this much time per problem and if you can solve them in this much time fine, otherwise move on to the next. Try to do this the last thing before bed at night (seriously) and then go to sleep. Pass 2 After at least one night’s sleep, make a medium speed pass through all problems. Plan to spend 1-1.5 hours on this pass as well. Some of the problems will already be solved from the first pass or nearly so. Quickly review their solution and then move on to concentrate on the still unsolved problems. If you solved 1/4 to 1/3 of the problems in the first pass, you should be able to spend 10 minutes or so per problem in the second pass. Again, do this right before bed if possible and then go immediately to sleep. Pass 3 After at least one night’s sleep, make a final pass through all the problems. Begin as before by quickly reviewing all the problems you solved in the previous two passes. Then spend fifteen minutes or more (as needed) to solve the remaining unsolved problems. Leave any “impossible” problems for recitation – there should be no more than three from any given assignment, as a general rule. Go immediately to bed. This is an extremely powerful prescription for deeply learning nearly anything. Here is the motivation. Memory is formed by repetition, and this obviously contains a lot of

Preliminaries 27 that. Permanent (long term) memory is actually formed in your sleep, and studies have shown that whatever you study right before sleep is most likely to be retained. Physics is actually a “whole brain” subject – it requires a synthesis of both right brain visualization and conceptualization and left brain verbal/analytical processing – both geometry and algebra, if you like, and you’ll often find that problems that stumped you the night before just solve themselves “like magic” on the second or third pass if you work hard on them for a short, intense, session and then sleep on it. This is your right (nonverbal) brain participating as it develops intuition to guide your left brain algebraic engine. Other suggestions to improve learning include working in a study group for that third pass (the first one or two are best done alone to “prepare” for the third pass). Teaching is one of the best ways to learn, and by working in a group you’ll have opportunities to both teach and learn more deeply than you would otherwise as you have to articulate your solutions. Make the learning fun – the right brain is the key to forming long term memory and it is the seat of your emotions. If you are happy studying and make it a positive experience, you will increase retention, it is that simple. Order pizza, play music, make it a “physics homework party night”. Use your whole brain on the problems – draw lots of pictures and figures (right brain) to go with the algebra (left brain). Listen to quiet music (right brain) while thinking through the sequences of events in the problem (left brain). Build little “demos” of problems where possible – even using your hands in this way helps strengthen memory. Avoid memorization. You will learn physics far better if you learn to solve problems and understand the concepts rather than attempt to memorize the umpty-zillion formulas, factoids, and specific problems or examples covered at one time or another in the class. That isn’t to say that you shouldn’t learn the important formulas, Laws of Nature, and all of that – it’s just that the learning should generally not consist of putting them on a big sheet of paper all jumbled together and then trying to memorize them as abstract collections of symbols out of context. Be sure to review the problems one last time when you get your graded homework back. Learn from your mistakes or you will, as they say, be doomed to repeat them. If you follow this prescription, you will have seen every assigned homework problem a minimum of five or six times – three original passes, recitation itself, a final write up pass after recitation, and a review pass when you get it back. At least three of these should occur after you have solved all of the problems correctly, since recitation is devoted to ensuring this. When the time comes to study for exams, it should really be (for once) a review process, not a cram. Every problem will be like an old friend, and a very brief review will form a seventh pass or eighth pass through the assigned homework. With this methodology (enhanced as required by the physics resource rooms, tutors, and help from your instructors) there is no reason for you do poorly in the course and every reason to expect that you will do well, perhaps very well indeed! And you’ll still be spending only the 3 to 6 hours per week on homework that is expected of you in any college course of this level of difficulty! This ends our discussion of course preliminaries (for nearly any serious course you

28 Preliminaries might take, not just physics courses) and it is time to get on with the actual material for this course. Mathematics Physics, as was noted in the preface, requires a solid knowledge of all mathematics through calculus. That’s right, the whole nine yards: number theory, algebra, geometry, trigonometry, vectors, differential calculus, integral calculus, even a smattering of differen- tial equations. You may well be reading this book intending to use it to learn physics (either on your own – good for you! – or in an actual class) never having taken a class in calculus, or perhaps you took a class in calculus but barely squeaked by with a C- or D+. I strongly advise against attempting this! At Duke University (where I teach this course) calculus is a strict prerequisite for taking calculus based physics. This is for two reasons: First, a calculus class may well be the only place where you are exposed to things like series, summation symbols, vectors (including vector products) that are often not taught in high school algebra and trigonometry classes. Second, Newton invented calculus just so that he could invent a consistent and successful theory of physics. Learning physics without calculus (as it is taught in most high school physics classes and sadly, in some University-level physics classes) is in my opinion a nearly complete waste of time. It becomes an exercise in the memorization of formulas because one can literally not understand where anything comes from or how it all fits together without cal- culus. It tends to concentrate on constant force/constant acceleration problems simply because they are pretty much the only ones one can solve without calculus, to such a fault that I usually have to help students unlearn the algebraic solutions they memorized in high school if this is their only exposure to physics before we can move forward and learn physics correctly. A final problem is that physics based only on algebraic solutions to constant accelera- tion kinematics is boring, and students understandably come out of such a course bored and frustrated with physics through no fault of the discipline (which is anything but boring). Learning physics is hard work, without doubt. It can be frustrating if it is poorly taught, taught as an exercise in memorization and graded primarily on how well you can do the simple arithmetic of substituting this or that set of numbers into a memorized formula in a single step. Substantial research in the teaching and learning of physics have demon- strated that there is a huge gap between achieving conceptual understanding of even the most elementary physics and this strictly algebra+arithmetic approach to studying physics. Of course, even students who have taken calculus successfully can have a bit of a gap there as well. Many calculus classes – perhaps understandably, perhaps not, I don’t want to judge too harshly – concentrate for better or for worse on skills, on the algebraic manipulations of calculus. Those courses are easy to identify – they had a student do- ing regular homework assignments consisting of page after page of taking derivatives of that, doing integrals of another thing, without one single exercise having the slightest bit of meaning! Again it is all too common for students to have treated the course as an exer- cise in memorization more than an invitation of mastery even of the limited tools required

Preliminaries 29 to do most of the problems, an understandable student response when presented with what appears to be an overwhelming mountain of meaningless symbols instead of a much smaller mountain of symbols that actually mean something and have some use when you are done. This creates two distinct problems when students start to learn physics, with calculus. First, well over 95% of the calculus problems they were drilled on turn out to be useless in any scientific discipline and are ‘valuable’ only as a hobby for people who like to solve for analytical derivatives or integrals of functions that don’t represent anything whatsoever in the real world. Second, because of the tremendous dilution of efforts wasted on these obscure and – to anyone but a future math major – useless problems, they end up inad- equately skilled in the 5% of the introductory calculus problems they might have studied that are really, really valuable in real world problems. Let me be very specific. One can succeed in – nay, thrive! – in introductory physics if you have truly mastered only five basic integration/differentiation rules, plus the product rule for differentiation (and the corresponding integration by parts rule for integration), plus the chain rule for differentiation (and the corresponding u-substitution rule for integration). If I were to add anything to that list it would be the hyperbolic differentials (and integrals). Let’s list them (as indefinite integrals, although some of them, such as the natural logs, are more useful as definite integrals when it comes to handling the physical dimensions of the arguments): • dun = nun−1 and undu = n 1 1 un+1 + C du + • d ln |u| = 1 and du = ln |u| + C du u u • deu = eu and eudu = eu + C du • d sin u = cos u and cos u du = sin u + C du • d cos u = − sin u and sin u du = − cos u + C du plus the two “extra” formulas (that can be easily enough derived from the exponential rule above, as can the trig integrals for that matter): • d sinh u = cosh u and cosh u du = sinh u + C du • d cosh u = − sinh u and sinh u du = − cosh u + C du where sinh u = 1 eu − e−u and cosh u = 1 eu + e−u. Tangent and cotangent are just a matter 2 2 of using the product rule, as are the hyperbolic equivalents. Trig substitution derivatives and integrals are similarly just a matter using the chain rule or u-substitution (but with some clever pictures that allow one to visualize them as e.g. triangles). It also helps to at least know about the Taylor series expansion of smooth functions and the slightly more specific

30 Preliminaries binomial expansion, and the idea of convergence. The actual use of all of this, extended right into the evaluation of multidimensional integrals and derivatives as needed, is taught in a self contained way in a typical course, as physics instructors have long since learned not to rely on the calculus supposedly learned in calculus classes by incoming students. To put it another way, the fundamental calculus formulas needed to completely master a one year (two semester) introductory physics sequence – mechanics, electricity, mag- netism, optics, and diverse applications of all of the above all entirely based on calculus – can easily fit on one single page. That isn’t to say that memorizing that page is sufficient preparation in math, though. Physics builds on skills in algebra, geometry, trigonometry, the idea of vector spaces and vector decomposition, some familiarity with at least three separable coordinate frames (much of which, again, is taught or retaught as need be in the physics courses them- selves), series, complex numbers – it really uses mathematics throughout, where every single mathematical tool and idea used has meaning and is not an empty exercise with meaningless symbols. The skill that is in some sense the least important is – perhaps surprisingly, given the public perception of the discipline – arithmetic. That isn’t to say that physicists don’t care about numbers or that you can get by in a physics class when you are unable to add, subtract, multiply or divide numbers with nothing but a pen and piece of paper, it’s just recognition of the fact that once one understands what is going on and can solve problems correctly algebraically, plugging in the numbers is a trivial final step, one that can easily be done with a calculator or computer if need be, where even an arithmetical savant, somebody capable of multiplying 8 digit numbers instantly in their head, is helpless in physics if they cannot actually perform the conceptual reasoning, visualization, algebra, and dimensional analysis required to take a word problem, convert it into a picture with coordinates attached, decorate it with given forces or interactions, express the whole thing in the algebraic forms associated with the relevant physical laws, and then use all of the mathematical and algebraic skills one needs to obtain an algebraic solution that can be checked with dimensional analysis for consistency and rechecked with some simple does- it-make-sense rules. Sure, once that formula is obtained, they might be better and faster than a computer, but given the formula, a sixth grader with the same numerical data and a calculator can get just as accurate a solution in only a little more time, provided only that they know what those trig functions and so on on their calculator are good for. In any event, if you are preparing to study calculus-based physics (from this book or any other), here is a list of a few of the kinds of things you’ll have to be able to do during the next two semesters of physics. Don’t worry just yet about what they mean – that is part of what you will learn along the way. The question is, can you (perhaps with a short review of things you’ve learned and knew at one time but have not forgotten) evaluate these mathematical expressions or solve for the algebraic unknowns? You don’t necessarily have to be able to do all of these things right this instant, but you should at the very least recognize most of them and be able to do them with just a very short review: • What are the two values of α that solve: α2 + R α + 1 = 0? L LC

Preliminaries 31 • What is: Q(r) = ρ0 4π r • What is: R r′3dr′? 0 d cos(ωt + δ) ? dt y A? θ ?x • What are the x and y components of a vector of length A that makes an angle of θ with the positive x axis (proceeding, as usual, counterclockwise for positive θ)? • What is the sum of the two vectors A = Axxˆ + Ayyˆ and B = Bxxˆ + Byyˆ? • What is the inner/dot product of the two vectors A = Axxˆ+Ayyˆ and B = Bxxˆ+Byyˆ? • What is the cross product of the two vectors r = rxxˆ and F = Fyyˆ (magnitude and direction)? If all of these items are unfamiliar – you don’t remember the quadratic formula (needed to solve the first one), can’t integrate xndx (needed to solve the second one), don’t recall how to differentiate a sine or cosine function, don’t recall your basic trigonometry so that you can’t find the components of a vector from its length and angle or vice versa, and don’t recall what the dot or cross product of two vectors are, then you are going to have to add to the burden of learning physics per se the burden of learning, or re-learning, all of the basic mathematics that would have permitted you to answer these questions easily. Here are the answers, see if this jogs your memory: • Here are the two roots, found with the quadratic formula: − R ± R 2 − 4 R R2 1 L L LC 2L 4L2 LC α± = = − ± − 2 • r r′4 r ρ0πr4 4 R Q(r) = ρ0 4π r′3dr′ = ρ0 4π = R R 0 0 • d cos(ωt + δ) dt = −ω sin(ωt + δ)

32 Preliminaries • Ax = A cos(θ) Ay = A sin(θ) • A + B = (Ax + Bx)xˆ + (Ay + By)yˆ • A · B = AxBx + AyBy • r × F = rxxˆ × Fyyˆ = rxFy(xˆ × yˆ) = rxFyzˆ My strong advice to you, if you are now feeling the cold icy grip of panic because in fact you are signed up for physics using this book and you couldn’t answer any of these questions and don’t even recognize the answers once you see them, is to seek out the course instructor and review your math skills with him or her to see if, in fact, it is advisable for you to take physics at this time or rather should wait and strengthen your math skills first. You can, and will, learn a lot of math while taking physics and that is actually part of the point of taking it! If you are too weak going into it, though, it will cost you some misery and hard work and some of the grade you might have gotten with better preparation ahead of time. So, what if you could do at least some of these short problems and can remember once learning/knowing the tools, like the Quadratic Formula, that you were supposed to use to solve them? Suppose you are pretty sure that – given a chance and resource to help you out – you can do some review and they’ll all be fresh once again in time to keep up with the physics and still do well in the course? What if you have no choice but to take physics now, and are just going to have to do your best and relearn the math as required along the way? What if you did in fact understand math pretty well once upon a time and are sure it won’t be much of an obstacle, but you really would like a review, a summary, a listing of the things you need to know someplace handy so you can instantly look them up as you struggle with the problems that uses the math it contains? What if you are (or were) really good at math, but want to be able to look at derivations or reread explanations to bring stuff you learned right back to your fingertips once again? Hmmm, that set of questions spans the set of student math abilities from the near- tyro to the near-expert. In my experience, everybody but the most mathematically gifted students can probably benefit from having a math review handy while taking this course. For all of you, then, I provide the following free book online: Mathematics for Introductory Physics It is located here: http://www.phy.duke.edu/∼rgb/Class/math for intro physics.php It is a work in progress, and is quite possibly still somewhat incomplete, but it should help you with a lot of what you are missing or need to review, and if you let me know what you are missing that you didn’t find there, I can work to add it!

Preliminaries 33 I would strongly advise all students of introductory physics (any semester) to visit this site right now and bookmark it or download the PDF, and to visit the site from time to time to see if I’ve posted an update. It is on my back burner, so to speak, until I finish the actual physics texts themselves that I’m working on, but I will still add things to them as motivated by my own needs teaching courses using this series of books. Summary That’s enough preliminary stuff. At this point, if you’ve read all of this “week”’s material and vowed to adopt the method of three passes in all of your homework efforts, if you’ve bookmarked the math help or downloaded it to your personal ebook viewer or computer, if you’ve realized that your brain is actually something that you can help and enhance in various ways as you try to learn things, then my purpose is well-served and you are as well-prepared as you can be to tackle physics.

34 Preliminaries Homework for Week 0 Problem 1. Skim read this entire section (Week 0: How to Learn Physics), then read it like a novel, front to back. Think about the connection between engagement and learning and how important it is to try to have fun in a physics course. Write a short essay (say, three paragraphs) describing at least one time in the past where you were extremely engaged in a course you were taking, had lots of fun in the class, and had a really great learning experience. Problem 2. Skim-read the entire content of Mathematics for Introductory Physics (linked above). Identify things that it covers that you don’t remember or don’t understand. Pick one and learn it. Problem 3. Apply the Method of Three Passes to this homework assignment. You can either write three short essays or revise your one essay three times, trying to improve it and enhance it each time for the first problem, and review both the original topic and any additional topics you don’t remember in the math review problem. On the last pass, write a short (two paragraph) essay on whether or not you found multiple passes to be effective in helping you remember the content. Note well: You may well have found the content boring on the third pass because it was so familiar to you, but that’s not a bad thing. If you learn physics so thoroughly that its laws become boring, not because they confuse you and you’d rather play World of Warcraft but because you know them so well that reviewing them isn’t adding anything to your understanding, well damn you’ll do well on the exams testing the concept, won’t you?

II: Elementary Mechanics 35



Preliminaries 37 OK, so now you are ready to learn physics. Your math skills are buffed and honed, you’ve practiced the method of three passes, you understand that success depends on your full engagement and a certain amount of hard work. In case you missed the previous section (or are unused to actually reading a math-y textbook instead of minimally skimming it to extract just enough “stuff” to be able to do the homework) I usually review its content on the first day of class at the same time I review the syllabus and set down the class rules and grading scheme that I will use. It’s time to embark upon the actual week by week, day by day progress through the course material. For maximal ease of use for you the student and (one hopes) your in- structor whether or not that instructor is me, the course is designed to cover one chapter per week-equivalent, whether or not the chapter is broken up into a day and a half of lecture (summer school), an hour a day (MWF), or an hour and a half a day (TTh) in a semester based scheme. To emphasize this preferred rhythm, each chapter will be referred to by the week it would normally be covered in my own semester-long course. A week’s work in all cases covers just about exactly one “topic” in the course. A very few are spread out over two weeks; one or two compress two related topics into one week, but in all cases the homework is assigned on a weekly rhythm to give you ample opportunity to use the method of three passes described in the first part of the book, culminating in an expected 2-3 hour recitation where you should go over the assigned homework in a group of three to six students, with a mentor handy to help you where you get stuck, with a goal of getting all of the homework perfectly correct by the end of recitation. That is, at the end of a week plus its recitation, you should be able to do all of the week’s homework, perfectly, and without looking or outside help. You will usually need all three passes, the last one working in a group, plus the mentored recitation to achieve this degree of competence! But without it, surely the entire process is a waste of time. Just finishing the homework is not enough, the whole point of the homework is to help you learn the material and it is the latter that is the real goal of the activity not the mere completion of a task. However, if you do this – attempt to really master the material – you are almost certain to do well on a quiz that terminates the recitation period, and you will be very likely to retain the material and not have to “cram” it in again for the hour exams and/or final exam later in the course. Once you achieve understanding and reinforce it with a fair bit of repetition and practice, most students will naturally transform this experience into remarkably deep and permanent learning. Note well that each week is organized for maximal ease of learning with the week/chapter review first. Try to always look at this review before lecture even if you skip reading the chapter itself until later, when you start your homework. Skimming the whole week/chapter guided by this summary before lecture is, of course, better still. It is a “first pass” that can often make lecture much easier to follow and help free you from the tyranny of note-taking as you only need to note differences in the presentation from this text and perhaps the answers to questions that helped you understand something during the discussion. Then read or skim it again right before each homework pass.

38 Week 1: Newton’s Laws

Week 1: Newton’s Laws Summary • Physics is a language – in particular the language of a certain kind of worldview. For philosophically inclined students who wish to read more deeply on this, I include links to terms that provide background for this point of view. – Wikipedia: http://www.wikipedia.org/wiki/Worldview – Wikipedia: http://www.wikipedia.org/wiki/Semantics – Wikipedia: http://www.wikipedia.org/wiki/Ontology Mathematics is the natural language and logical language of physics, not for any particularly deep reason but because it works. The components of the seman- tic language of physics are thus generally expressed as mathematical or logical coordinates, and the semantic expressions themselves are generally mathemati- cal/algebraic laws. • Coordinates are the fundamental adjectival modifiers corresponding to the differenti- ating properties of “things” (nouns) in the real Universe, where the term fundamental can also be thought of as meaning irreducible – adjectival properties that cannot be readily be expressed in terms of or derived from other adjectival properties of a given object/thing. See: – Wikipedia: http://www.wikipedia.org/wiki/Coordinate System • Units. Physical coordinates are basically mathematical numbers with units (or can be so considered even when they are discrete non-ordinal sets). In this class we will consistently and universally use Standard International (SI) units unless otherwise noted. Initially, the irreducible units we will need are: a) meters – the SI units of length b) seconds – the SI units of time c) kilograms – the SI units of mass All other units for at least a short while will be expressed in terms of these three, for example units of velocity will be meters per second, units of force will be kilogram- meters per second squared. We will often give names to some of these combinations, such as the SI units of force: 39

40 Week 1: Newton’s Laws 1 Newton = kg-m sec2 Later you will learn of other irreducible coordinates that describe elementary parti- cles or extended macroscopic objects in physics such as electrical charge, as well as additional derivative quantities such as energy, momentum, angular momentum, electrical current, and more. • Laws of Nature are essentially mathematical postulates that permit us to understand natural phenomena and make predictions concerning the time evolution or static re- lations between the coordinates associated with objects in nature that are consistent mathematical theorems of the postulates. These predictions can then be compared to experimental observation and, if they are consistent (uniformly successful) we in- crease our degree of belief in them. If they are inconsistent with observations, we decrease our degree of belief in them, and seek alternative or modified postulates that work better23. The entire body of human scientific knowledge is the more or less successful out- come of this process. This body is not fixed – indeed it is constantly changing be- cause it is an adaptive process, one that self-corrects whenever observation and prediction fail to correspond or when new evidence spurs insight (sometimes revolu- tionary insight!) Newton’s Laws built on top of the analytic geometry of Descartes (as the basis for at least the abstract spatial coordinates of things) are the dynamical principle that proved successful at predicting the outcome of many, many everyday experiences and experiments as well as cosmological observations in the late 1600’s and early 1700’s all the way up to the mid-19th century24. When combined with associated empirical force laws they form the basis of the physics you will learn in this course. • Newton’s Laws: a) Law of Inertia: Objects at rest or in uniform motion (at a constant velocity) in an inertial reference frame remain so unless acted upon by an unbalanced (net) force. We can write this algebraically25 as F i = 0 = ma = m dv ⇒ v = constant vector (1.1) dt i 23Students of philosophy or science who really want to read something cool and learn about the fundamental basis of our knowledge of reality are encouraged to read e.g. Richard Cox’s The Algebra of Probable Reason or E. T. Jaynes’ book Probability Theory: The Logic of Science. These two related works quantify how science is not (as some might think) absolute truth or certain knowledge, but rather the best set of things to believe based on our overall experience of the world, that is to say, “the evidence”. 24Although they failed in the late 19th and early 20th centuries, to be superceded by relativistic quantum mechanics. Basically, everything we learn in this course is wrong, but it nevertheless works damn well to describe the world of macroscopic, slowly moving objects of our everyday experience. 25For students who are still feeling very shaky about their algebra and notation, let me remind you that i F i stands for “The sum over i of all force F i”, or F 1 + F 2 + F 3 + .... We will often use as shorthand for summing over a list of similar objects or components or parts of a whole.

Week 1: Newton’s Laws 41 b) Law of Dynamics: The net force applied to an object is directly proportional to its acceleration in an inertial reference frame. The constant of proportionality is called the mass of the object. We write this algebraically as: F= Fi = ma = d(mv) = dp (1.2) dt dt i where we introduce the momentum of a particle, p = mv, in the last way of writing it. c) Law of Reaction: If object B exerts a force F AB on object A along a line con- necting the two objects, then object A exerts an equal and opposite reaction force of F BA = −F AB on object B. We write this algebraically as: F ij = −F ji (1.3) (1.4) (or) F ij = 0 i,j (the latter form means that the sum of all internal forces between particles in any closed system of particles cancel). • An inertial reference frame is a spatial coordinate system plus an indepen- dent time coordinate that is either at rest or moving at a constant speed, a non- accelerating set of coordinates that can be used to describe the locations of real objects as a function of time. However, this definition is inadequate, because ac- celeration itself is defined only relative to a frame. This leaves us with a problem: non-accelerating relative to what frame? We have to identify at least one inertial ref- erence frame before we can talk about other frames that do not accelerate relative to it. The way we can identify any inertial reference frame is as follows. As you will see, all actual physical forces in the laws of nature are interaction laws. For any observed force pushing an object, there exists another object somewhere that is doing the pushing, a “Newton’s Third Law partner”. No force exists in isolation, and no object can exert a force on itself. A signature of a non-inertial reference frame is that within it, there exists an observed “pseudoforce” that arises in Newton’s Second Law due to the acceleration of the frame. This pseudoforce has no Newton’s Third Law partner! A consistent definition of an inertial reference frame is therefore: Inertial Reference Frame: Any frame where all observed forces that occur in all statements of Newton’s Second Law for all particles are pairwise interactions between particles. In other words, there are no forces that act on any particle in complete isolation from and independent of the other particles in the system . In physics one has considerable leeway when it comes to choosing the (inertial) co- ordinate frame to be used to solve a problem – some lead to much simpler solutions than others!

42 Week 1: Newton’s Laws • Forces of Nature (weakest to strongest): a) Gravity b) Weak Nuclear c) Electromagnetic d) Strong Nuclear It is possible that there are more forces of nature waiting to be discovered. Because physics is not a dogma, this presents no real problem. If they are, we’ll simply give the discoverer a Nobel Prize and add their name to the “pantheon” of great physicists, add the force to the list above, and move on. Science, as noted, is self-correcting. • Force is a vector. For each force rule below we therefore need both a formula or rule for the magnitude of the force (which we may have to compute in the solution to a problem – in the case of forces of constraint such as the normal force (see below) we will usually have to do so) and a way of determining or specifying the direction of the force. Often this direction will be obvious and in corresponence with experience and mere common sense – strings pull, solid surfaces push, gravity points down and not up. Other times it will be more complicated or geometric and (along with the magnitude) may vary with position and time. • Force Rules The following set of force rules will be used both in this chapter and throughout this course. All of these rules can be derived or understood (with some effort) from the forces of nature, that is to say from “elementary” natural laws. a) Gravity (near the surface of the earth): Fg = mg (1.5) The direction of this force is down, so one could write this in vector form as F g = −mgyˆ in a coordinate system such that up is the +y direction. This rule follows from Newton’s Law of Gravitation, the elementary law of nature in the list above, evaluated “near” the surface of the earth where it is approximately constant. b) The Spring (Hooke’s Law) in one dimension: Fx = −k∆x (1.6) This force is directed back to the equilibrium point of unstretched spring, in the opposite direction to ∆x, the displacement of the mass on the spring from equilibrium. This rule arises from the primarily electrostatic forces holding the atoms or molecules of the spring material together, which tend to linearly oppose small forces that pull them apart or push them together (for reasons we will understand in some detail later). c) The Normal Force: F⊥ = N (1.7) This points perpendicular and away from solid surface, magnitude sufficient to oppose the force of contact whatever it might be! This is an example of a force

Week 1: Newton’s Laws 43 of constraint – a force whose magnitude is determined by the constraint that one solid object cannot generally interpenetrate another solid object, so that the solid surfaces exert whatever force is needed to prevent it (up to the point where the “solid” property itself fails). The physical basis is once again the electro- static molecular forces holding the solid object together, and microscopically the surface deforms, however slightly, more or less like a spring. d) Tension in an Acme (massless, unstretchable, unbreakable) string: Fs = T (1.8) This force simply transmits an attractive force between two objects on opposite ends of the string, in the directions of the taut string at the points of contact. It is another constraint force with no fixed value. Physically, the string is like a spring once again – it microscopically is made of bound atoms or molecules that pull ever so slightly apart when the string is stretched until the restoring force balances the applied force. e) Static Friction fs ≤ µsN (1.9) (directed opposite towards net force parallel to surface to contact). This is an- other force of constraint, as large as it needs to be to keep the object in question travelling at the same speed as the surface it is in contact with, up to the max- imum value static friction can exert before the object starts to slide. This force arises from mechanical interlocking at the microscopic level plus the electro- static molecular forces that hold the surfaces themselves together. f) Kinetic Friction fk = µkN (1.10) (opposite to direction of relative sliding motion of surfaces and parallel to surface of contact). This force does have a fixed value when the right conditions (sliding) hold. This force arises from the forming and breaking of microscopic adhesive bonds between atoms on the surfaces plus some mechanical linkage between the small irregularities on the surfaces. g) Fluid Forces, Pressure: A fluid in contact with a solid surface (or anything else) in general exerts a force on that surface that is related to the pressure of the fluid: FP = P A (1.11) which you should read as “the force exerted by the fluid on the surface is the pressure in the fluid times the area of the surface”. If the pressure varies or the surface is curved one may have to use calculus to add up a total force. In general the direction of the force exerted is perpendicular to the surface. An object at rest in a fluid often has balanced forces due to pressure. The force arises from the molecules in the fluid literally bouncing off of the surface of the object, transferring momentum (and exerting an average force) as they do so. We will study this in some detail and will even derive a kinetic model for a gas that is in good agreement with real gases.

44 Week 1: Newton’s Laws h) Drag Forces: Fd = −bvn (1.12) (directed opposite to relative velocity of motion through fluid, n usually between 1 (low velocity) and 2 (high velocity). This force also has a determined value, although one that depends in detail on the motion of the object. It arises first because the surface of an object moving through a fluid is literally bouncing fluid particles off in the leading direction while moving away from particles in the trailing direction so that there is a differential pressure on the two surfaces, second from “friction” between the fluid particles and the surface.

Week 1: Newton’s Laws 45 The first week summary would not be complete without some sort of reference to methodologies of problem solving using Newton’s Laws and the force laws or rules above. The following rubric should be of great use to you as you go about solving any of the prob- lems you will encounter in this course, although we will modify it slightly when we treat e.g. energy instead of force, torque instead of force, and so on. Dynamical Recipe for Newton’s Second Law a) Draw a good picture of what is going on. In general you should probably do this even if one has been provided for you – visualization is key to success in physics. b) On your drawing (or on a second one) decorate the objects with all of the forces that act on them, creating a free body diagram for the forces on each object. It is sometimes useful to draw pictures of each object in isolation with just the forces acting on that one object connected to it, although for simple problems this is not always necessary. Either way your diagram should be clearly drawn and labelled. c) Choose a suitable coordinate system for the problem. This coordinate system need not be cartesian (x, y, z). We sometimes need separate coordinates for each mass (with a relation between them) or will even find it useful to “wrap around a corner” (following a string around a pulley, for example) in some problems d) Decompose the forces acting on each object into their components in the (orthogo- nal) coordinate frame(s) you chose, using trigonometry and geometry. e) Write Newton’s Second Law for each object (summing the forces and setting the result to miai for each – ith – object for each dimension) and algebraically rearrange it into (vector) differential equations of motion (practically speaking, this means solving for or isolating the acceleration ai = d2xi of the particles in the equations of motion). dt2 f) Solve the independent 1 dimensional systems for each of the independent orthog- onal coordinates chosen, plus any coordinate system constraints or relations. In many problems the constraints will eliminate one or more degrees of freedom from consideration, especially if you have chosen your cooordinates wisely (for example, ensuring that one coordinate points in the direction of a known component of the acceleration such as 0 or Ω2r). Note that in most nontrivial cases, these solutions will have to be simultaneous solu- tions, obtained by e.g. algebraic substitution or elimination. g) Reconstruct the multidimensional trajectory by adding the vector components thus obtained back up (for a common independent variable, time). In some cases you may skip straight ahead to other known kinematic relations useful in solving the problem. h) Answer algebraically any questions requested concerning the resultant trajectory, using kinematic relations as needed. Some parts of this rubric will require experience and common sense to implement correctly for any given kind of problem. That is why homework is so critically important! We

46 Week 1: Newton’s Laws want to make solving the problems (and the conceptual understanding of the underlying physics) easy, and they will only get to be easy with practice followed by a certain amount of meditation and reflection, practice followed by reflection, iterate until the light dawns...

Week 1: Newton’s Laws 47 1.1: Introduction: A Bit of History and Philosophy It has been remarked by at least one of my colleagues that one reason we have such a hard time teaching Newtonian physics to college students is that we have to first unteach them their already prevailing “natural” worldview of physics, which dates all the way back to Aristotle. In a nutshell and in very general terms (skipping all the “nature is a source or cause of being moved and of being at rest” as primary attributes, see Aristotle’s Physica, book II) Aristotle considered motion or the lack thereof of an object to be innate properties of materials, according to their proportion of the big four basic elements: Earth, Air, Fire and Water. He extended the idea of the moving and the immovable to cosmology and to his Metaphysics as well. In this primitive view of things, the observation that (most) physical objects (being “Earth”) set in motion slow down is translated into the notion that their natural state is to be at rest, and that one has to add something from one of the other essences to pro- duce a state of uniform motion. This was not an unreasonable hypothesis; a great deal of a person’s everyday experience (then and now) is consistent with this. When we pick something up and throw it, it moves for a time and bounces, rolls, slides to a halt. We need to press down on the accelerator of a car to keep it going, adding something from the “fire” burning in the motor to the “earth” of the body of the car. Even our selves seem to run on “something” that goes away when we die. Unfortunately, it is completely and totally wrong. Indeed, it is almost precisely Newton’s first law stated backwards. It is very likely that the reason Newton wrote down his first law (which is otherwise a trivial consequence of his second law) is to directly confront the error of Aristotle, to force the philosophers of his day to confront the fact that his (Newton’s) the- ory of physics was irreconcilable with that of Aristotle, and that (since his actually worked to make precise predictions of nearly any kind of classical motion that were in good agree- ment with observation and experiments designed to test it) Aristotle’s physics was almost certainly wrong. Or at any rate, wronger than Newton’s. Newton’s discoveries were a core component of the Enlightment, a period of a few hundred years in which Europe went from a state of almost slavish, church-endorsed belief in the infallibility and correctness of the Aristotelian worldview to a state where humans, for the first time in history, let nature speak for itself by using a consistent framework to listen to what nature had to say26. Aristotle lost, but his ideas are slow to die because they closely parallel everyday experience. The first chore for any serious student of physics is thus to unlearn this Aristotelian view of things27. 26Students who like to read historical fiction will doubtless enjoy Neal Stephenson’s Baroque Cycle, a set of novels – filled with sex and violence and humor, a good read – that spans the Enlightenment and in which Newton, Liebnitz, Hooke and other luminaries to whom we owe our modern conceptualization of physics all play active parts. 27This is not the last chore, by the way. Physicists have long since turned time into a coordinate just like space so that how long things take depends on one’s point of view, eliminated the assumption that one can measure any set of measureable quantities to arbitrary precision in an arbitrary order, replaced the determin- ism of mathematically precise trajectories with a funny kind of stochastic quasi-determinism, made (some) forces into just an aspect of geometry, and demonstrated a degree of mathematical structure (still incomplete,

48 Week 1: Newton’s Laws This is not, unfortunately, an abstract problem. It is very concrete and very current. Because I have an online physics textbook, and because physics is in some very fun- damental sense the “magic” that makes the world work, I not infrequently am contacted by individuals who do not understand the material covered in this textbook, who do not want to do the very hard work required to master it, but who still want to be “magicians”. So they invent their own version of the magic, usually altering the mathematically precise meanings of things like “force”, “work”, “energy” to be something else altogether that they think that they understand but that, when examined closely, usually are dimensionally or conceptually inconsistent and mean nothing at all. Usually their “new” physics is in fact a reversion to the physics of Aristotle. They recre- ate the magic of earth and air, fire and water, a magic where things slow down unless fire (energy) is added to sustain their motion or where it can be extracted from invisible an inex- haustible resources, a world where the mathematical relations between work and energy and force and acceleration do not hold. A world, in short, that violates a huge, vast, truly stupdendous body of accumulated experimental evidence including the very evidence that you yourselves will very likely observe in person in the physics labs associated with this course. A world in which things like perpetual motion machines are possible, where free lunches abound, and where humble dilettantes can be crowned “the next Einstein” without having a solid understanding of algebra, geometry, advanced calculus, or the physics that everybody else seems to understand perfectly. This world does not exist. Seriously, it is a fantasy, and a very dangerous one, one that threatens modern civilization itself. One of the most important reasons you are taking this course, whatever your long term dreams and aspirations are professionally, is to come to fully and deeply understand this. You will come to understand the magic of science, at the same time you learn to reject the notion that science is magic or vice versa. There is nothing wrong with this. I personally find it very comforting that the individuals that take care of my body (physicians) and who design things like jet airplanes and auto- mobiles (engineers) share a common and consistent Newtonian28 view of just how things work, and would find it very disturbing if any of them believed in magic, in gods, in fairies, in earth, air, fire and water as constituent elements, in “crystal energies”, in the power of a drawn pentagram or ritually chanted words in any context whatsoever. These all repre- sent a sort of willful wishful thinking on the part of the believer, a desire for things to not follow the rigorous mathematical rules that they appear to follow as they evolve in time, for there to be a “man behind the curtain” making things work out as they appear to do. Or sometimes an entire pantheon. Let me be therefore be precise. In the physics we will study week by week below, the natural state of “things” (e.g. objects made of matter) will be to move uniformly. We will learn non-Aristotelian physics, Newtonian physics. It is only when things are acted on from outside by unbalanced forces that the motion becomes non-uniform; they will speed up or slow down. By the time we are done, you will understand how this can still lead to the we’re working on it) beyond the wildest dreams of Aristotle or his mathematical-mystic buddies, the Pythagore- ans. 28Newtonian or better, that is. Of course actual modern physics is non-Newtonian quantum mechanics, but this is just as non-magical and concrete and it reduces to Newtonian physics in the macroscopic world of our quotidian experience.

Week 1: Newton’s Laws 49 damping of motion observed in everyday life, why things do generally slow down. In the meantime, be aware of the problem and resist applying the Aristotelian view to real physics problems, and consider, based on the evidence and your experiences taking this course rejecting “magic” as an acceptable component of your personal worldview unless and until it too has some sort of objective empirical support. Which would, of course, just make it part of physics! 1.2: Dynamics Physics is the study of dynamics. Dynamics is the description of the actual forces of nature that, we believe, underlie the causal structure of the Universe and are responsible for its evolution in time. We are about to embark upon the intensive study of a simple description of nature that introduces the concept of a force, due to Isaac Newton. A force is considered to be the causal agent that produces the effect of acceleration in any massive object, altering its dynamic state of motion. Newton was not the first person to attempt to describe the underlying nature of causal- ity. Many, many others, including my favorite ‘dumb philosopher’, Aristotle, had attempted this. The major difference between Newton’s attempt and previous ones is that Newton did not frame his as a philosophical postulate per se. Instead he formulated it as a mathemat- ical theory and proposed a set of laws that (he hoped) precisely described the regularities of motion in nature. In physics a law is the equivalent of a postulated axiom in mathematics. That is, a physical law is, like an axiom, an assumption about how nature operates that not formally provable by any means, including experience, within the theory. A physical law is thus not considered “correct” – rather we ascribe to it a “degree of belief” based on how well and consistently it describes nature in experiments designed to verify and falsify its correspon- dence. It is important to do both. Again, interested students are are encouraged to look up Karl Popper’s “Falsifiability”29 and the older Postivism30 . A hypothesis must successfully withstand the test of repeated, reproducible experiments that both seek to disprove it and to verify that it has predictive value in order to survive and become plausible. And even then, it could be wrong! If a set of laws survive all the experimental tests we can think up and subject it to, 29Wikipedia: http://www.wikipedia.org/wiki/Falsifiability. Popper considered the ability to in principle disprove a hypothesis as an essential criterion for it to have objective meaning. Students might want to purchase and read Nassim Nicholas Taleb’s book The Black Swan to learn of the dangers and seductions of worldview- building gone awry due to insufficient skepticism or a failure to allow for the disproportionate impact of the unexpected but true anyway – such as an experiment that falsifies a conclusion that was formerly accepted as being verified. 30Wikipedia: http://www.wikipedia.org/wiki/Positivism. This is the correct name for “verifiability”, or the ability to verify a theory as the essential criterion for it to have objective meaning. The correct modern approach in physics is to do both, following the procedure laid out by Richard Cox and E. T. Jaynes wherein propositions are never proven or disproven per se, but rather are shown to be more or less “plausible”. A hypothesis in this approach can have meaning as a very implausible notion quite independent of whether or not it can be verified or falsified – yet.

50 Week 1: Newton’s Laws we consider it likely that it is a good approximation to the true laws of nature; if it passes many tests but then fails others (often failing consistently at some length or time scale) then we may continue to call the postulates laws (applicable within the appropriate milieu) but recognize that they are only approximately true and that they are superceded by some more fundamental laws that are closer (at least) to being the “true laws of nature”. Newton’s Laws, as it happens, are in this latter category – early postulates of physics that worked remarkably well up to a point (in a certain “classical” regime) and then failed. They are “exact” (for all practical purposes) for massive, large objects moving slowly com- pared to the speed of light31 for long times such as those we encounter in the everyday world of human experience (as described by SI scale units). They fail badly (as a basis for prediction) for microscopic phenomena involving short distances, small times and masses, for very strong forces, and for the laboratory description of phenomena occurring at rela- tivistic velocities. Nevertheless, even here they survive in a distorted but still recognizable form, and the constructs they introduce to help us study dynamics still survive. Interestingly, Newton’s laws lead us to second order differential equations, and even quantum mechanics appears to be based on differential equations of second order or less. Third order and higher systems of differential equations seem to have potential problems with temporal causality (where effects always follow, or are at worst simultaneous with, their causes in time); it is part of the genius of Newton’s description that it precisely and sufficiently allows for a full description of causal phenomena, even where the details of that causality turn out to be incorrect. Incidentally, one of the other interesting features of Newton’s Laws is that Newton in- vented calculus to enable him to solve the problems they described. Now you know why calculus is so essential to physics: physics was the original motivation behind the invention of calculus itself. Calculus was also (more or less simultaneously) invented in the more useful and recognizable form that we still use today by other mathematical-philosophers such as Leibnitz, and further developed by many, many people such as Gauss, Poincare, Poisson, Laplace and others. In the overwhelming majority of cases, especially in the early days, solving one or more problems in the physics that was still being invented was the motivation behind the most significant developments in calculus and differential equa- tion theory. This trend continues today, with physics providing an underlying structure and motivation for the development of much of the most advanced mathematics. 1.3: Coordinates Think about any thing, any entity that objectively exists in the real, visible, Universe. What defines the object and differentiates it from all of the other things that make up the Uni- verse? Before we can talk about how the Universe and its contents change in time, we have to talk about how to describe its contents (and time itself) at all. As I type this I’m looking over at just such a thing across the room from me, an object that I truly believe exists in the real Universe. To help you understand this object, I have to use language. I might tell you how large it is, what its weight is, what it looks like, where 31c = 3 × 108 meters/second

Week 1: Newton’s Laws 51 it is, how long it has been there, what it is for, and – of course – I have to use words to do this, not just nouns but a few adjectival modifiers, and speak of an “empty beer glass sitting on a table in my den just to my side”, where now I have only to tell you just where my den is, where the table is in the den, and perhaps differentiate this particular beer glass from other beer glasses you might have seen. Eventually, if I use enough words, construct a detailed map, make careful measurements, perhaps include a picture, I can convey to you a very precise mental picture of the beer glass, one sufficiently accurate that you can visualize just where, when and what it is (or was). Of course this prose description is not the glass itself! If you like, the map is not the territory32! That is, it is an informational representation of the glass, a collection of symbols with an agreed upon meaning (where meaning in this context is a correspondence between the symbols and the presumed general sensory experience of the glass that one would have if one looked at the glass from my current point of view). Constructing just such a map is the whole point of physics, only the map is not just of mundane objects such as a glass; it is the map of the whole world, the whole Universe. To the extent that this worldview is successful, especially in a predictive sense and not just hindsight, the physical map in your mind works well to predict the Universe you perceive through your sensory apparatus. A perfect understanding of physics (and a knowledge of certain data) is equivalent to a possessing a perfect map, one that precisely locates every thing within the Universe for all possible times. Maps require several things. It is convenient, but not necessary, to have a set of single term descriptors, symbols for various “things” in the world the map is supposed to describe. So this symbol might stand for a house, that one for a bridge, still another one for a street or railroad crossing. Another absolutely essential part of a map is the actual coordinates of things that it is describing. The coordinate representation of objects drawn in on the map is supposed to exist in an accurate one-to-one but abstract correspondence with the concrete territory in the real world where the things represented by the symbols actually exist and move about33. Of course the symbols such as the term “beer glass” can themselves be abstractly modeled as points in some sort of space; Complex or composite objects with “simple” coordinates can be represented as a collection of far more coordinates for the smaller objects that make up the composite object. Ultimately one arrives at elementary objects, objects that are not (as far as we know or can tell so far ) made up of other objects at all. The various kinds of elementary objects, the list of their variable properties, and their spatial and temporal coordinates are in some deep sense all coordinates, and every object in the universe can be thought of as a point in or volume of this enormous and highly complex coordinate space! In this sense “coordinates” are the fundamental adjectival modifiers corresponding to the differentiating properties of “named things” (nouns) in the real Universe, where the 32This is an adage of a field called General Semantics and is something to keep carefully in mind while studying physics. Not even my direct perception of the glass is the glass itself; it is just a more complex and dynamical kind of map. 33Of course in the old days most actual maps were stationary, and one had to work hard to see “time” on them, but nowadays nearly everybody has or at least has seen GPS maps and video games, where things or the map coordinates themselves move.

52 Week 1: Newton’s Laws term fundamental can also be thought of as meaning elementary or irreducible – adjectival properties that cannot be readily be expressed in terms of or derived from other adjectival properties of a given object/thing. Physical coordinates are, then, basically mathematical numbers with units (and can be so considered even when they are discrete non-ordinal sets). At first we will omit most of the details from the objects we study to keep things simple. In fact, we will spend most of the first part of the course interested in only three quantities that set the scale for the coordinate system we need to describe the classical physics of a rather generic “particle”: space (where it is), time (when it is there), and mass (an intrinsic property). This is our first idealization – the treatment of an extended (composite) object as if it were itself an elementary object. This is called the particle approximation, and later we will justify this approximation a posteriori (after the fact) by showing that there really is a special point in a collective system of particles that behaves like a particle as far as Newton’s Laws are concerned. For the time being, then, objects such as porpoises, puppies, and ponies are all idealized and will be treated as particles34. We’ll talk more about particles in a page or two. We need units to describe intervals or values in all three coordinates so that we can talk or think about those particles (idealized objects) in ways that don’t depend on the listener. In this class we will consistently and universally use Standard International (SI) units unless otherwise noted. Initially, the irreducible units we will need are: a) meters – the SI units of length b) seconds – the SI units of time c) kilograms – the SI units of mass All other units for at least a short while will be expressed in terms of these three. For example units of velocity will be meters per second, units of force will be kilogram- meters per second squared. We will often give names to some of these combinations, such as the SI units of force: 1 Newton = kg-m sec2 Later you may learn of other irreducible coordinates that describe elementary particles or extended macroscopic objects in physics such as electrical charge, as well as additional derivative quantities such as energy, momentum, angular momentum, electrical current, and more. As for what the quantities that these units represent are – well, that’s a tough question to answer. I know how to measure distances between points in space and times between events that occur in space, using things like meter sticks and stopwatches, but as to just 34I teach physics in the summers at the Duke Marine Lab, where there are porpoises and wild ponies visible from the windows of our classroom. Puppies I threw in for free because they are cute and also begin with “p”. However, you can think of a particle as a baseball or bullet or ball bearing if you prefer less cute things that begin with the letter “b”, which is a symmetry transformed “p”, sort of.

Week 1: Newton’s Laws 53 what the space and time in which these events are embedded really is I’m as clueless as a cave-man. It’s probably best to just define distance as that which we might measure with a meter stick or other “standard” of length, time as that which we might measure with a clock or other “standard” of time, and mass that which we might measure compared to some “standard” of mass using methods we’ll have to figure out below. Existential properties cannot really be defined, only observed, quantified, and understood in the context of a complete, consistent system, the physical worldview, the map we construct that works to establish a useful semantic representation of that which we observe. Sorry if that’s difficult to grasp, but there it is. It’s just as difficult for me (after most of a lifetime studying physics) as it is for you right now. Dictionaries are, after all, writ- ten in words that are in the dictionaries and hence are self-referential and in some deep sense should be abstract, arbitrary, meaningless – yet somehow we learn to speak and understand them. So it is, so it will be for you, with physics, and the process takes some time. y y(t) m ∆x x(t) x(t+ ∆t) x(t) x Figure 2: Coordinatized visualization of the motion of a particle of mass m along a trajec- tory x(t). Note that in a short time ∆t the particle’s position changes from x(t) to x(t+∆t). Coordinates are enormously powerful ideas, the very essence of mapmaking and knowl- edge itself. To assist us in working with them, we introduce the notion of coordinate frame – a system of all of the relevant coordinates that describe at least the position of a particle (in one, two or three dimensions, usually). In figure 2 is a picture of a simple single particle with mass m (that might represent my car) on a set of coordinates that describes at least part of the actual space where my car is sitting. The solid line on this figure represents the trajectory of my car as it moves about. Armed with a watch, an apparatus for measuring mass, meter sticks and some imagination, one can imagine a virtual car rolling up or down along its virtual trajectory and compare its motion in our conceptual map35 with the corre- spondent happenings in the world outside of our minds where the real car moves along a real track. 35This map need not be paper, in other words – I can sit here and visualize the entire drive from my house to the grocery store, over time. Pictures of trajectories on paper are just ways we help our brains manage this sort of understanding.

54 Week 1: Newton’s Laws Note well that we idealize my car by treating the whole thing as a single object with a single position (located somewhere “in the middle”) when we know that it is really made up steering wheels and bucket seats that are “objects” in their own right that are further assembled into a “car” All of these wheels and panels, nuts and bolts are made up of still smaller objects – molecules – and molecules are made up of atoms, and atoms are made of protons and neutrons and electrons, and protons and neutrons are made up of quarks, and we don’t really know for certain if electrons and quarks are truly elementary particles or are themselves composite objects36. Later in this semester we will formally justify our ability to do this, and we will improve on our description of things like cars and wheels and solid composite objects by learning how they can move, rotate, and even explode into little bits of car and still have some parts of their collective coordinate motion that behaves as though the ex-car is still a “single point-like object”. In the meantime, we will simply begin with this idealization and treat discrete solid ob- jects as particles – masses that are at a single point in space at a single time. So we will treat objects such as planets, porpoises, puppies, people, baseballs and blocks, cars and cannonballs and much more as if they have a single mass and a single spatial location at any single instant in time – as a particle. One advantage of this is that the mathemat- ical expressions for all of these quantities become functions of time37 and possibly other coordinates. In physical dynamics, we will be concerned with finding the trajectory of particles or systems – the position of each particle in the system expressed as a function of time. We can write the trajectory as a vector function on a spatial coordinate system (e.g. cartesian coordinates in 2 dimensions): x(t) = x(t)xˆ + y(t)yˆ (1.13) Note that x(t) stands for a vector from the origin to the particle, where x(t) by itself (no boldface or vector sign) stands for the x-component of this vector. An example trajectory is visualized in figure 2 (where as noted, it might stand for the trajectory of my car, treated as a particle). In all of the problems we work on throughout the semester, visualization will be a key component of the solution. The human brain doesn’t, actually, excel at being able to keep all of these details on- board in your “mind’s eye”, the virtual visual field of your imagination. Consequently, you must always draw figures, usually with coordinates, forces, and other “decorations”, when you solve a physics problem. The paper (or blackboard or whiteboard) then becomes an extension of your brain – a kind of “scratch space” that augments your visualization ca- pabilities and your sequential symbolic reasoning capabilities. To put it bluntly, you are more intelligent when you reason with a piece of paper and pen than you are when you are forced to rely on your brain alone. To succeed in physics, you need all of the intelligence you can get, and you need to synthesize solutions using both halves of your brain, visual- ization and sequential reason. Paper and pen facilitate this process and one of the most important lessons of this course is how to use them to attain the full benefit of the added intelligence they enable not just in physics problems, but everywhere else as well. 36Although the currently accepted belief is that they are. However, it would take only one good, reproducible experiment to make this belief less plausible, more probably false. Evidence talks, belief walks. 37Recall that a function is a quantity that depends on a set of argument(s) that is single-valued, that is, has a single value for each unique value of its argument(s).

Week 1: Newton’s Laws 55 If we know the trajectory function of a particle, we know a lot of other things too. Since we know where it is at any given time, we can easily tell how fast it is going in what direction. This combination of the speed of the particle and its direction forms a vector called the velocity of the particle. Speed, we all hopefully know from our experience in real life doing things like driving cars, is a measure of how far you go in a certain amount of time, expressed in units of distance (length) divided by time. Miles per hour. Furlongs per fortnight. Or, in a physics course, meters per second38. The average velocity of the particle is by definition the vector change in its position ∆x in some time ∆t divided by that time: vav = ∆x (1.14) ∆t Sometimes average velocity is useful, but often, even usually, it is not. It can be a rather poor measure for how fast a particle is actually moving at any given time, especially if averaged over times that are long enough for interesting changes in the motion to occur. For example, I might get in my car and drive around a racetrack at speed of 50 meters per second – really booking it, tires squealing on the turns, smoke coming out of my engine (at least if I tried this in my car, as it would likely explode if I tried to go 112 mph for any extended time), and screech to a halt right back where I began. My average velocity is then zero – I’m back where I started! That zero is a poor explanation for the heat waves pulsing out from under the hood of the car and the wear on its tires. More often we will be interested in the instantaneous velocity of the particle. This is basically the average velocity, averaged over as small a time interval as possible – one so short that it is just long enough for the car to move at all. Calculus permits us to take this limit, and indeed uses just this limit as the definition of the derivative. We thus define the instantaneous velocity vector as the time-derivative of the position vector: v(t) = lim x(t + ∆t) − x(t) = lim ∆x = dx (1.15) ∆t ∆t dt ∆t→0 ∆t→0 Sometimes we will care about “how fast” a car is going but not so much about the direction. Speed is defined to be the magnitude of the velocity vector: v(t) = |v(t)| (1.16) We could say more about it, but I’m guessing that you already have a pretty good intu- itive feel for speed if you drive a car and pay attention to how your speedometer reading corresponds to the way things zip by or crawl by outside of your window. The reason that average velocity is a poor measure is that (of course) our cars speed up and slow down and change direction, often. Otherwise they tend to run into things, because it is usually not possible to travel in perfectly straight lines at only one speed and drive to the grocery store. To see how the velocity changes in time, we will need to consider 38A good rule of thumb for people who have a practical experience of speeds in miles per hour trying to visualize meters per second is that 1 meter per second is approximately equal to 2 1 miles per hour, hence 4 four meters per second is nine miles per hour. A cruder but still quite useful approximation is (meters per second) equals (miles per hour/2).

56 Week 1: Newton’s Laws the acceleration of a particle, or the rate at which the velocity changes. As before, we can easily define an average acceleration over a possibly long time interval ∆t as: aav = v(t + ∆t) − v(t) = ∆v (1.17) ∆t ∆t Also as before, this average is usually a poor measure of the actual acceleration a particle (or car) experiences. If I am on a straight track at rest and stamp on the accelerator, burning rubber until I reach 50 meters per second (112 miles per hour) and then stamp on the brakes to quickly skid to a halt, tires smoking and leaving black streaks on the pavement, my average acceleration is once again zero, but there is only one brief interval (between taking my foot off of the accelerator and before I pushed it down on the brake pedal) during the trip where my actual acceleration was anything close to zero. Yet, my average acceleration is zero. Things are just as bad if I go around a circular track at a constant speed! As we will shortly see, in that case I am always accelerating towards the center of the circle, but my average acceleration is systematically approaching zero as I go around the track more and more times. From this we conclude that the acceleration that really matters is (again) the limit of the average over very short times; the time derivative of the velocity. This limit is thus defined to be the instantaneous accleration: a(t) = lim ∆v = dv = d2x , (1.18) ∆t dt dt2 ∆t→0 the acceleration of the particle right now. Obviously we could continue this process and define the time derivative of the accel- eration39 and still higher order derivatives of the trajectory. However, we generally do not have to in physics – we will not need to continue this process. As it turns out, the dynamic principle that appears sufficient to describe pretty much all classical motion will involve force and acceleration, and pretty much all of the math we do (at least at first) will involve solving backwards from a knowledge of the acceleration to a knowledge of the velocity and position vectors via integration or more generally (later) solving the differential equation of motion. We are now prepared to formulate this dynamical principle – Newton’s Second Law. While we’re at it, let’s study his First and Third Laws too – might as well collect the complete set... 39A quantity that actually does have a name – it is called the jerk – but we won’t need it.

Week 1: Newton’s Laws 57 1.4: Newton’s Laws The following are Newton’s Laws as you will need to know them to both solve problems and answer conceptual questions in this course. Note well that they are framed in terms of the spatial coordinates defined in the previous section plus mass and time. a) Law of Inertia: Objects at rest or in uniform motion (at a constant velocity) in an inertial reference frame remain so unless acted upon by an unbalanced (net, total) force. We can write this algebraically as: F= F i = 0 = ma = m dv ⇒ v = constant vector (1.19) dt i b) Law of Dynamics: The total force applied to an object is directly proportional to its acceleration in an inertial reference frame. The constant of proportionality is called the mass of the object. We write this algebraically as: F= Fi = ma = d(mv) = dp (1.20) dt dt i where we introduce the momentum of a particle, p = mv, in the last way of writing it. c) Law of Reaction: If object B exerts a (named) force F AB on object B along a line connecting the two objects, then object A exerts an equal and opposite reaction force of F BA = −F AB on object B. Because we can have a lot more than just two particles A and B, we write this algebraically as: F ij = −F ji (1.21) (1.22) (or) F ij = 0 i,j where i and j are arbitrary particle labels. The latter form will be useful to us later; it means that the sum of all internal forces between particles in any closed system of particles cancels!. Note that these laws are not all independent as mathematics goes. The first law is a clear and direct consequence of the second. The third is not – it is an independent statement. The first law historically, however, had an important purpose. It rejected the dynamics of Aristotle, introducing the new idea of intertia where an object in motion contin- ues in that motion unless acted upon by some external agency. This is directly opposed to the Aristotelian view that things only moved when acted on by an external agency and that they naturally came to rest when that agency was removed. A second important purpose of the first law is that – together with the third law – it helps us define an inertial reference frame as a frame where the first law is true. The second law is our basic dynamical principle. It tells one how to go from a problem description (in words) plus a knowledge of the force laws of nature to an “equation of motion” (typically a statement of Newton’s second law). The equation of motion, generally solved for the acceleration, becomes a kinematical equation from which we can develop

58 Week 1: Newton’s Laws a full knowledge of the solution using mathematics guided by our experience and physical intuition. The third law leads (as we shall see) to the surprising result that systems of particles behave collectively like a particle! This is indeed fortunate! We know that something like a baseball is really made up of a lot of teensy particles itself, and yet it obeys Newton’s Second law as if it is a particle. We will use the third law to derive this and the closely related Law of Conservation of Momentum in a later week of the course. An inertial reference frame is a coordinate system (or frame) that is either at rest or mov- ing at a constant speed, a non-accelerating frame of reference. For example, the ground, or lab frame, is a coordinate system at rest relative to the approximately non-accelerating ground or lab and is considered to be an inertial frame to a good approximation. A (coordi- nate system inside a) car travelling at a constant speed relative to the ground, a spaceship coasting in a region free from fields, a railroad car rolling on straight tracks at constant speed are also inertial frames. A (coordinate system inside a) car that is accelerating (say by going around a curve), a spaceship that is accelerating, a freight car that is speeding up or slowing down – these are all examples of non-inertial frames. All of Newton’s laws suppose an inertial reference frame (yes, the third law too) and are generally false for accelerations evaluated in an accelerating frame as we will prove and discuss next week. In the meantime, please be sure to learn the statements of the laws including the con- dition “in an inertial reference frame”, in spite of the fact that you don’t yet really understand what this means and why we include it. Eventually, it will be the other important meaning and use of Newton’s First Law – it is the law that defines an inertial reference frame as any frame where an object remains in a state of uniform motion if no forces act on it! You’ll be glad that you did. 1.5: Forces Classical dynamics at this level, in a nutshell, is very simple. Find the total force on an object. Use Newton’s second law to obtain its acceleration (as a differential equation of motion). Solve the equation of motion by direct integration or otherwise for the position and velocity. That’s it! Well, except for answering those pesky questions that we typically ask in a physics problem, but we’ll get to that later. For the moment, the next most important problem is: how do we evaluate the total force? To answer it, we need a knowledge of the forces at our disposal, the force laws and rules that we are likely to encounter in our everyday experience of the world. Some of these forces are fundamental forces – elementary forces that we call “laws of nature” be- cause the forces themselves aren’t caused by some other force, they are themselves the actual causes of dynamical action in the visible Universe. Other force laws aren’t quite so fundamental – they are more like “approximate rules” and aren’t exactly correct. They are also usually derivable from (or at least understandable from) the elementary natural laws,

Week 1: Newton’s Laws 59 Particle Location Size Up or Down Quark Nucleon (Proton or Neutron) Proton pointlike Neutron Nucleus 10−15 meters Nucleus Nucleus 10−15 meters Electron 10−15 meters Atom Atom Molecule Atom pointlike Molecules or Objects ∼ 10−10 meters Objects > 10−10 meters Table 1: Basic building blocks of normal matter as of 2011, subject to change as we dis- cover and understand more about the Universe, ignoring pesky things like neutrinos, pho- tons, gluons, heavy vector bosons, heavier leptons that physics majors (at least) will have to learn about later... although it may be quite a lot of work to do so. We quickly review the forces we will be working with in the first part of the course, both the forces of nature and the force rules that apply to our everyday existence in approximate form. 1.5.1: The Forces of Nature At this point in your life, you almost certainly know that all normal matter of your everyday experience is made up of atoms. Most of you also know that an atom itself is made up of a positively charged atomic nucleus that is very tiny indeed surrounded by a cloud of negatively charged electrons that are much lighter. Atoms in turn bond together to make molecules, atoms or molecules in turn bind together (or not) to form gases, liquids, solids – “things”, including those macroscopic things that we are so far treating as particles. The actual elementary particles from which they are made are much tinier than atoms. It is worth providing a greatly simplified table of the “stuff” from which normal atoms (and hence molecules, and hence we ourselves) are made: In this table, up and down quarks and electrons are so-called elementary particles – things that are not made up of something else but are fundamental components of nature. Quarks bond together three at a time to form nucleons – a proton is made up of “up-up- down” quarks and has a charge of +e, where e is the elementary electric charge. A neutron is made up of “up-down-down” and has no charge. Neutrons and protons, in turn, bond together to make an atomic nucleus. The simplest atomic nucleus is the hydrogen nucleus, which can be as small as a single proton, or can be a proton bound to one neutron (deuterium) or two neutrons (tritium). No matter how many protons and neutrons are bound together, the atomic nucleus is small – order of 10−15 meters in diameter40. The quarks, protons and neutrons are bound together by means of a force of nature called the strong nuclear interaction, which is the strongest force we know of relative to the mass of the interacting particles. 40...with the possible exception of neutrons bound together by gravity to form neutron stars. Those can be thought of, very crudely, as very large nuclei.

60 Week 1: Newton’s Laws The positive nucleus combines with electrons (which are negatively charged and around 2000 times lighter than a proton) to form an atom. The force responsible for this binding is the electromagnetic force, another force of nature (although in truth nearly all of the inter- action is electrostatic in nature, just one part of the overall electromagnetic interaction). The light electrons repel one another electrostatically almost as strongly as they are attracted to the nucleus that anchors them. They also obey the Pauli exclusion principle which causes them to avoid one another’s company. These things together cause atoms to be much larger than a nucleus, and to have interesting “structure” that gives rise to chemistry and molecular bonding and (eventually) life. Inside the nucleus (and its nucleons) there is another force that acts at very short range. This force can cause e.g. neutrons to give off an electron and turn into a proton or other strange things like that. This kind of event changes the atomic number of the atom in question and is usually accompanied by nuclear radiation. We call this force the weak nuclear force. The two nuclear forces thus both exist only at very short length scales, basically in the quantum regime inside an atomic nucleus, where we cannot easily see them using the kinds of things we’ll talk about this semester. For our purposes it is enough that they exist and bind stable nuclei together so that those nuclei in turn can form atoms, molecules, objects, us. Our picture of normal matter, then, is that it is made up of atoms that may or may not be bonded together into molecules, with three forces all significantly contributing to what goes on inside a nucleus and only one that is predominantly relevant to the electronic structure of the atoms themselves. There is, however, a fourth force (that we know of – there may be more still, but four is all that we’ve been able to positively identify and understand). That force is gravity. Gravity is a bit “odd”. It is a very long range, but very weak force – by far the weakest force of the four forces of nature. It only is signficant when one builds planet or star sized objects, where it can do anything from simply bind an atmosphere to a planet and cause moons and satellites to go around it in nice orbits to bring about the catastrophic collapse of a dying star. The physical law for gravitation will be studied over an entire week of work – later in the course. I put it down now just for completeness, but initially we’ll focus on the force rules in the following section. F 21 = − Gm1m2 rˆ12 (1.23) r122 Don’t worry too much about what all of these symbols mean and what the value of G is – we’ll get to all of that but not now. Since we live on the surface of a planet, to us gravity will be an important force, but the forces we experience every day and we ourselves are primarily electromagnetic phenom- ena, with a bit of help from quantum mechanics to give all that electromagnetic stuff just the right structure. Let’s summarize this in a short table of forces of nature, strongest to weakest: a) Strong Nuclear b) Electromagnetic

Week 1: Newton’s Laws 61 c) Weak Nuclear d) Gravity Note well: It is possible that there are more forces of nature waiting to be discovered. Because physics is not a dogma, this presents no real problem. If a new force of nature (or radically different way to view the ones we’ve got) emerges as being consistent with observation and predictive, and hence possibly/plausibly true and correct, we’ll simply give the discoverer a Nobel Prize, add their name to the “pantheon of great physicists”, add the force itself to the list above, and move on. Science, as noted above, is a self-correcting system of reasoning, at least when it is done right. 1.5.2: Force Rules The following set of force rules will be used both in this chapter and throughout this course. All of these rules can be derived or understood (with some effort) from the forces of nature, that is to say from “elementary” natural laws, but are not quite laws themselves. a) Gravity (near the surface of the earth): Fg = mg (1.24) The direction of this force is down, so one could write this in vector form as F g = −mgyˆ in a coordinate system such that up is the +y direction. This rule follows from Newton’s Law of Gravitation, the elementary law of nature in the list above, evaluated “near” the surface of the earth where it is varies only very slowly with height above the surface (and hence is “constant”) as long as that height is small compared to the radius of the Earth. The measured value of g (the gravitational “constant” or gravitational field close to the Earth’s surface) thus isn’t really constant – it actually varies weakly with latitude and height and the local density of the earth immediately under your feet and is pretty complicated41 . Some “constant”, eh? Most physics books (and the wikipedia page I just linked) give g’s value as something like: meters second2 g ≈ 9.81 (1.25) (which is sort of an average of the variation) but in this class to the extent that we do arithmetic with it we’ll just use g ≈ 10 meters (1.26) second2 because hey, so it makes a 2% error. That’s not very big, really – you will be lucky to measure g in your labs to within 2%, and it is so much easier to multiply or divide by 10 than 9.80665. 41Wikipedia: http://www.wikipedia.org/wiki/Gravity of Earth. There is a very cool “rotating earth” graphic on this page that shows the field variation in a color map. This page goes into much more detail than I will about the causes of variation of “apparent gravity”.

62 Week 1: Newton’s Laws b) The Spring (Hooke’s Law) in one dimension: Fx = −k∆x (1.27) This force is directed back to the equilibrium point (the end of the unstretched spring where the mass is attached) in the opposite direction to ∆x, the displacement of the mass on the spring away from this equilibrium position. This rule arises from the primarily electrostatic forces holding the atoms or molecules of the spring material together, which tend to linearly oppose small forces that pull them apart or push them together (for reasons we will understand in some detail later). c) The Normal Force: F⊥ = N (1.28) This points perpendicular and away from solid surface, magnitude sufficient to op- pose the force of contact whatever it might be! This is an example of a force of constraint – a force whose magnitude is determined by the constraint that one solid object cannot generally interpenetrate another solid object, so that the solid surfaces exert whatever force is needed to prevent it (up to the point where the “solid” prop- erty itself fails). The physical basis is once again the electrostatic molecular forces holding the solid object together, and microscopically the surface deforms, however slightly, more or less like a spring to create the force. d) Tension in an Acme (massless, unstretchable, unbreakable) string: Fs = T (1.29) This force simply transmits an attractive force between two objects on opposite ends of the string, in the directions of the taut string at the points of contact. It is another constraint force with no fixed value. Physically, the string is like a spring once again – it microscopically is made of bound atoms or molecules that pull ever so slightly apart when the string is stretched until the restoring force balances the applied force. e) Static Friction fs ≤ µsN (1.30) (directed opposite towards net force parallel to surface to contact). This is another force of constraint, as large as it needs to be to keep the object in question travelling at the same speed as the surface it is in contact with, up to the maximum value static friction can exert before the object starts to slide. This force arises from mechanical interlocking at the microscopic level plus the electrostatic molecular forces that hold the surfaces themselves together. f) Kinetic Friction fk = µkN (1.31) (opposite to direction of relative sliding motion of surfaces and parallel to surface of contact). This force does have a fixed value when the right conditions (sliding) hold. This force arises from the forming and breaking of microscopic adhesive bonds between atoms on the surfaces plus some mechanical linkage between the small irregularities on the surfaces.

Week 1: Newton’s Laws 63 g) Fluid Forces, Pressure: A fluid in contact with a solid surface (or anything else) in general exerts a force on that surface that is related to the pressure of the fluid: FP = P A (1.32) which you should read as “the force exerted by the fluid on the surface is the pressure in the fluid times the area of the surface”. If the pressure varies or the surface is curved one may have to use calculus to add up a total force. In general the direction of the force exerted is perpendicular to the surface. An object at rest in a fluid often has balanced forces due to pressure. The force arises from the molecules in the fluid literally bouncing off of the surface of the object, transferring momentum (and exerting an average force) as they do so. We will study this in some detail and will even derive a kinetic model for a gas that is in good agreement with real gases. h) Drag Forces: Fd = −bvn (1.33) (directed opposite to relative velocity of motion through fluid, n usually between 1 (low velocity) and 2 (high velocity). It arises in part because the surface of an object moving through a fluid is literally bouncing fluid particles off in the leading direction while moving away from particles in the trailing direction, so that there is a differential pressure on the two surfaces, in part from “kinetic friction” that exerts a force compo- nent parallel to a surface in relative motion to the fluid. It is really pretty complicated – so complicated that we can only write down a specific, computable expression for it for very simple geometries and situations. Still, it is a very important and ubiquitous force and we’ll try to gain some appreciation for it along the way. 1.6: Force Balance – Static Equilibrium Before we start using dynamics at all, let us consider what happens when all of the forces acting on an object balance. That is, there are several non-zero (vector) forces acting on an object, but those forces sum up to zero force. In this case, Newton’s First Law becomes very useful. It tells us that the object in question will remain at rest if it is initially at rest. We call this situation where the forces are all balanced static force equilibrium: F tot = F i = ma = 0 (1.34) i This works both ways; if an object is at rest and stays that way, we can be certain that the forces acting on it balance! We will spend some time later studying static equilibrium in general once we have learned about both forces and torques, but for the moment we will just consider a single example of what is after all a pretty simple idea. This will also serve as a short introduction to one of the forces listed above, Hooke’s Law for the force exerted by a spring on an attached mass.

64 Week 1: Newton’s Laws m ∆x Figure 3: A mass m hangs on a spring with spring constant k. We would like to compute the amount ∆x by which the string is stretched when the mass is at rest in static force equilibrium. Example 1.6.1: Spring and Mass in Static Force Equilibrium Suppose we have a mass m hanging on a spring with spring constant k such that the spring is stretched out some distance ∆x from its unstretched length. This situation is pictured in figure 3. We will learn how to really solve this as a dynamics problem later – indeed, we’ll spend an entire week on it! Right now we will just write down Newton’s laws for this problem so we can find a. Let the x direction be up. Then (using Hooke’s Law from the list above): Fx = −k(x − x0) − mg = max (1.35) or (with ∆x = x − x0, so that ∆x is negative as shown) ax = − k ∆x − g (1.36) m Note that this result doesn’t depend on where the origin of the x-axis is, because x and x0 both change by the same amount as we move it around. In most cases, we will find the equilibrium position of a mass on a spring to be the most convenient place to put the origin, because then x and ∆x are the same! In static equilibrium, ax = 0 (and hence, Fx = 0) and we can solve for ∆x: ax = − k ∆x − g = 0 m k ∆x = g m mg ∆x = k (1.37) You will see this result appear in several problems and examples later on, so bear it in mind.

Week 1: Newton’s Laws 65 1.7: Simple Motion in One Dimension Finally! All of that preliminary stuff is done with. If you actually read and studied the chapter up to this point (many of you will not have done so, and you’ll be SORRReeee...) you should: a) Know Newton’s Laws well enough to recite them on a quiz – yes, I usually just put a question like “What are Newton’s Laws” on quizzes just to see who can recite them perfectly, a really useful thing to be able to do given that we’re going to use them hundreds of times in the next 12 weeks of class, next semester, and beyond; and b) Have at least started to commit the various force rules we’ll use this semester to memory. I don’t generally encourage rote memorization in this class, but for a few things, usually very fundamental things, it can help. So if you haven’t done this, go spend a few minutes working on this before starting the next section. All done? Well all rightie then, let’s see if we can actually use Newton’s Laws (usually Newton’s Second Law, our dynamical principle) and force rules to solve problems. We will start out very gently, trying to understand motion in one dimension (where we will not at first need multiple coordinate dimensions or systems or trig or much of the other stuff that will complicate life later) and then, well, we’ll complicate life later and try to understand what happens in 2+ dimensions. Here’s the basic structure of a physics problem. You are given a physical description of the problem. A mass m at rest is dropped from a height H above the ground at time t = 0; what happens to the mass as a function of time? From this description you must visualize what’s going on (sometimes but not always aided by a figure that has been drawn for you representing it in some way). You must select a coordinate system to use to describe what happens. You must write Newton’s Second Law in the coordinate system for all masses, being sure to include all forces or force rules that contribute to its motion. You must solve Newton’s Second Law to find the accelerations of all the masses (equations called the equations of motion of the system). You must solve the equations of motion to find the trajectories of the masses, their positions as a function of time, as well as their velocities as a function of time if desired. Finally, armed with these trajectories, you must answer all the questions the problem poses using algebra and reason and – rarely in this class – arithmetic! Simple enough. Let’s put this simple solution methodology to the test by solving the following one di- mensional, single mass example problem, and then see what we’ve learned.

66 Week 1: Newton’s Laws Example 1.7.1: A Mass Falling from Height H Let’s solve the problem we posed above, and as we do so develop a solution rubric – a recipe for solving all problems involving dynamics42! The problem, recall, was to drop a mass m from rest from a height H, algebraically find the trajectory (the position function that solves the equations of motion) and velocity (the time derivative of the trajectory), and then answer any questions that might be asked using a mix of algebra, intuition, experience and common sense. For this first problem we’ll postpone actually asking any question until we have these solutions so that we can see what kinds of questions one might reasonably ask and be able to answer. The first step in solving this or any physics problem is to visualize what’s going on! Mass m? Height H? Drop? Start at rest? Fall? All of these things are input data that mean something when translated into algebraic ”physicsese”, the language of physics, but in the end we have to coordinatize the problem (choose a coordinate system in which to do the algebra and solve our equations for an answer) and to choose a good one we need to draw a representation of the problem. Figure 4: A picture of a ball being dropped from a height H, with a suitable one-dimensional coordinate system added. Note that the figure clearly indicates that it is the force of gravity that makes it fall. The pictures of Satchmo (my border collie) and the tree and sun and birds aren’t strictly necessary and might even be distracting, but my right brain was bored when I drew this picture and they do orient the drawing and make it more fun! Physics problems that you work and hand in that have no figure, no picture, not even additional hand-drawn decorations on a provided figure will rather soon lose points in the grading scheme! At first we (the course faculty) might just remind you and not take points off, but by your second assignment you’d better be adding some relevant artwork to every solution43. Figure 4 is what an actual figure you might draw to accompany a problem might 42At least for the next couple of weeks... but seriously, this rubric is useful all the way up to graduate physics. 43This has two benefits – one is that it actually is a critical step in solving the problem, the other is that

Week 1: Newton’s Laws 67 look like. Note a couple of things about this figure. First of all, it is large – it took up 1/4+ of the unlined/white page I drew it on. This is actually good practice – do not draw postage-stamp sized figures! Draw them large enough that you can decorate them, not with Satchmo but with things like coordinates, forces, components of forces, initial data reminders. This is your brain we’re talking about here, because the paper is functioning as an extension of your brain when you use it to help solve the problem. Is your brain postage-stamp sized? Don’t worry about wasting paper – paper is cheap, physics educations are expensive. Use a whole page (or more) per problem solution at this point, not three problems per page with figures that require a magnifying glass to make out. When I (or your instructor) solve problems with you, this is the kind of thing you’ll see us draw, over and over again, on the board, on paper at a table, wherever. In time, physicists become pretty good schematic artists and so should you. However, in a textbook we want things to be clearer and prettier, so I’ll redraw this in figure 5, this time with a computer drawing tool (xfig) that I’ll use for drawing most of the figures included in the textbook. Alas, it won’t have Satchmo, but it does have all of the important stuff that should be on your hand-drawn figures. +y v = 0 @ t = 0 +y’ v’ = 0 @ t = 0 Hm m mg x’ mg H x Figure 5: The same figure and coordinate system, drawn “perfectly” with xfig, plus a second (alternative) coordinatization. Note that I drew two alternative ways of adding coordinates to the problem. The x-y coordinate system on the left is appropriate if you visualize the problem from the ground, looking up like Satchmo, where the ground is at zero height. This might be e.g. dropping a drawing engages the right hemisphere of your brain (the left hemisphere is the one that does the algebra). The right hemisphere is the one that controls formation of long term memory, and it can literally get bored, independently of the left hemisphere and interrupt your ability to work. If you’ve ever worked for a very long time on writing something very dry (left hemisphere) or doing lots of algebraic problems (left hemisphere) and found your eyes being almost irresistably drawn up to look out the window at the grass and trees and ponies and bright sun, then know that it is your “right brain” that has taken over your body because it is dying in there, bored out of its (your!) gourd. To keep the right brain happy while you do left brained stuff, give it something to do – listen to music, draw pictures or visualize a lot, take five minute right-brain-breaks and deliberately look at something visually pleasing. If your right brain is happy, you can work longer and better. If your right brain is engaged in solving the problem you will remember what you are working on much better, it will make more sense, and your attention won’t wander as much. Physics is a whole brain subject, and the more pathways you use while working on it, the easier it is to understand and remember!

68 Week 1: Newton’s Laws ball off of the top of Duke Chapel, for example, with you on the ground watching it fall. The x′-y′ coordinate system on the right works if you visualize the problem as some- thing like dropping the same ball into a well, where the ground is still at “zero height” but now it falls down to a negative height H from zero instead of starting at H and falling to height zero. Or, you dropping the ball from the top of the Duke Chapel and counting “y′ = 0 as the height where you are up there (and the initial position in y′ of the ball), with the ground at y′ = −H below the final position of the ball after it falls. Now pay attention, because this is important: Physics doesn’t care which coordinate system you use! Both of these coordinatizations of the problem are inertial reference frames. If you think about it, you will be able to see how to transform the answers obtained in one coordinate system into the corresponding answers in the other (basically subtract a constant H from the values of y in the left hand figure and you get y′ in the right hand figure, right?). Newton’s Laws will work perfectly in either inertial reference frame44, and truthfully there are an infinite number of coordinate frames you could choose that would all describe the same problem in the end. You can therefore choose the frame that makes the problem easiest to solve! Personally, from experience I prefer the left hand frame – it makes the algebra a tiny bit prettier – but the one on the right is really almost as good. I reject without thinking about it all of the frames where the mass m e.g. starts at the initial position yi = H/2 and falls down to the final position yf = −H/2. I do sometimes consider a frame like the one on the right with y positive pointing down, but it often bothers students to have “down” be positive (even though it is very natural to orient our coordinates so that F points in the positive direction of one of them) so we’ll work into that gently. Finally, I did draw the x (horizontal) coordinate and ignored altogether for now the z coordinate that in principle is pointing out of the page in a right-handed coordinate frame. We don’t really need either of these because no aspect of the motion will change x or z (there are no forces acting in those directions) so that the problem is effectively one-dimensional. Next, we have to put in the physics, which at this point means: Draw in all of the forces that act on the mass as proportionate vector arrows in the direction of the force. The “proportionate” part will be difficult at first until you get a feeling for how large the forces are likely to be relative to one another but in this case there is only one force, gravity that acts, so we can write on our page (and on our diagram) the vector relation: F = −mgyˆ (1.38) or if you prefer, you can write the dimension-labelled scalar equation for the magnitude of the force in the y-direction: Fy = −mg (1.39) Note well! Either of these is acceptable vector notation because the force is a vector (magnitude and direction). So is the decoration on the figure – an arrow for direction labelled mg. What is not quite right (to the tune of minus a point or two at the discretion of the grader) is to just write F = mg on your paper without indicating its direction somehow. 44For the moment you can take my word for this, but we will prove it in the next week/chapter when we learn how to systematically change between coordinate frames!

Week 1: Newton’s Laws 69 Yes, this is the magnitude of the force, but in what direction does it point in the particular coordinate system you drew into your figure? After all, you could have made +x point down as easily as −y! Practice connecting your visualization of the problem in the coordinates you selected to a correct algebraic/symbolic description of the vectors involved. In context, we don’t really need to write Fx = Fz = 0 because they are so clearly irrelevant. However, in many other problems we will need to include either or both of these. You’ll quickly get a feel for when you do or don’t need to worry about them – a reasonable “rule” for this is represented in the figure above – the particle has no x velocity, there are no forces at all in the x-direction, and we could even make the initial x coordinate of the particle zero. Nothing happens that is at all interesting in the x direction, so we more or less ignore it. Now comes the key step – setting up all of the algebra that leads to the solution. We write Newton’s Second Law for the mass m, and algebraically solve for the acceleration! Since there is only one relevant component of the force in this one-dimensional problem, we only need to do this one time for the scalar equation for that component.: Fy = −mg = may may = −mg ay = −g d2y = dvy = −g (1.40) dt2 dt where g = 10 m/second2 is the constant (within 2%, close to the Earth’s surface, remem- ber). We are all but done at this point. The last line (the algebraic expression for the accel- eration) is called the equation of motion for the system, and one of our chores will be to learn how to solve several common kinds of equation of motion. This one is a constant acceleration problem. Let’s do it. Here is the algebra involved. Learn it. Practice doing this until it is second nature when solving simple problems like this. I do not recommend memorizing the solution you obtain at the end, even though when you have solved the problem enough times you will probably remember it anyway for the rest of your share of eternity. Start with the equation of motion for a constant acceleration: dvy = −g Next, multiply both sides by dt to get: dt dvy = −g dt Then integrate both sides: dvy = − g dt doing the indefinite integrals to get: vy(t) = −gt + C (1.41) The final C is the constant of integration of the indefinite integrals. We have to evaluate it using the given (usually initial) conditions. In this case we know that: vy(0) = −g · 0 + C = C = 0 (1.42) (Recall that we even drew this into our figure to help remind us – it is the bit about being

70 Week 1: Newton’s Laws “dropped from rest” at time t = 0.) Thus: (1.43) vy(t) = −gt We now know the velocity of the dropped ball as a function of time! This is good, we are likely to need it. However, the solution to the dynamical problem is the trajectory function, y(t). To find it, we repeat the same process, but now use the definition for vy in terms of y: dy = vy(t) = −gt Multiply both sides by dt to get: dt dy = −gt dt Next, integrate both sides: dy = − gt dt to get: y(t) = − 1 gt2 + D (1.44) 2 The final D is again the constant of integration of the indefinite integrals. We again have to evaluate it using the given (initial) conditions in the problem. In this case we know that: y(0) = − 1 g · 02 + D = D = H (1.45) 2 because we dropped it from an initial height y(0) = H. Thus: y(t) = − 1 gt2 + H (1.46) 2 and we know everything there is to know about the motion! We know in particular exactly where it is at all times (until it hits the ground) as well as how fast it is going and in what direction. Sure, later we’ll learn how to evaluate other quantities that depend on these two, but with the solutions in hand evaluating those quantities will be (we hope) trivial. Finally, we have to answer any questions that the problem might ask! Note well that the problem may not have told you to evaluate y(t) and vy(t), but in many cases you’ll need them anyway to answer the questions they do ask. Here are a couple of common questions you can now answer using the solutions you just obtained: a) How long will it take for the ball to reach the ground? b) How fast is it going when it reaches the ground? To answer the first one, we use a bit of algebra. “The ground” is (recall) y = 0 and it will reach there at some specific time (the time we want to solve for) tg. We write the condition that it is at the ground at time tg: y(tg ) = − 1 gtg2 + H = 0 (1.47) 2 If we rearrange this and solve for tg we get: tg = ± 2H (1.48) g

Week 1: Newton’s Laws 71 Hmmm, there seem to be two times at which y(tg) equals zero, one in the past and one in the future. The right answer, of course, must be the one in the future: tg = + 2H/g, but you should think about what the one in the past means, and how the algebraic solution we’ve just developed is ignorant of things like your hand holding the ball before t = 0 and just what value of y corresponds to “the ground”... That was pretty easy. To find the speed at which it hits the ground, one can just take our correct (future) time and plug it into vy! That is: vg = vy(tg) = −gtg = −g 2H =− 2gH (1.49) g Note well that it is going down (in the negative y direction) when it hits the ground. This is a good hint for the previous puzzle. What direction would it have been going at the negative time? What kind of motion does the overall solution describe, on the interval from t = (−∞, ∞)? Do we need to use a certain amount of common sense to avoid using the algebraic solution for times or values of y for which they make no sense, such as y < 0 or t < 0 (in the ground or before we let go of the ball, respectively)? The last thing we might look at I’m going to let you do on your own (don’t worry, it’s easy enough to do in your head). Assuming that this algebraic solution is valid for any reasonable H, how fast does the ball hit the ground after falling (say) 5 meters? How about 20 = 4 ∗ 5 meters? How about 80 = 16 ∗ 5 meters? How long does it take for the ball to fall 5 meters, 20 meters, 80 meters, etc? In this course we won’t do a lot of arithmetic, but whenever we learn a new idea with parameters like g in it, it is useful to do a little arithmetical exploration to see what a “reasonable” answer looks like. Especially note how the answers scale with the height – if one drops it from 4x the height, how much does that increase the time it falls and speed with which it hits? One of these heights causes it to hit the ground in one second, and all of the other answers scale with it like the square root. If you happen to remember this height, you can actually estimate how long it takes for a ball to fall almost any height in your head with a division and a square root, and if you multiply the time answer by ten, well, there is the speed with which it hits! We’ll do some conceptual problems that help you understand this scaling idea for homework. This (a falling object) is nearly a perfect problem archetype or example for one dimen- sional motion. Sure, we can make it more complicated, but usually we’ll do that by having more than one thing move in one dimension and then have to figure out how to solve the two problems simultaneously and answer questions given the results. Let’s take a short break to formally solve the equation of motion we get for a constant force in one dimension, as the general solution exhibits two constants of integration that we need to be able to identify and evaluate from initial conditions. Note well that the next problem is almost identical to the former one. It just differs in that you are given the force F itself, not a knowledge that the force is e.g. “gravity”.

72 Week 1: Newton’s Laws Example 1.7.2: A Constant Force in One Dimension This time we’ll imagine a different problem. A car of mass m is travelling at a constant speed v0 as it enters a long, nearly straight merge lane. A distance d from the entrance, the driver presses the accelerator and the engine exerts a constant force of magnitude F on the car. a) How long does it take the car to reach a final velocity vf > v0? b) How far (from the entrance) does it travel in that time? As before, we need to start with a good picture of what is going on. Hence a car: y t=0 t=tf vf m v0 v0 F d x D Figure 6: One possible way to portray the motion of the car and coordinatize it. In figure 6 we see what we can imagine are three “slices” of the car’s position as a function of time at the moments described in the problem. On the far left we see it “entering a long, nearly straight merge lane”. The second position corresponds to the time the car is a distance d from the entrance, which is also the time the car starts to accelerate because of the force F . I chose to start the clock then, so that I can integrate to find the position as a function of time while the force is being applied. The final position corresponds to when the car has had the force applied for a time tf and has acquired a velocity vf . I labelled the distance of the car from the entrance D at that time. The mass of the car is indicated as well. This figure completely captures the important features of the problem! Well, almost. There are two forces I ignored altogether. One of them is gravity, which is pulling the car down. The other is the so-called normal force exerted by the road on the car – this force pushes the car up. I ignored them because my experience and common sense tell me that under ordinary circumstances the road doesn’t push on the car so that it jumps into the air, nor does gravity pull the car down into the road – the two forces will balance and the car will not move or accelerate in the vertical direction. Next week we’ll take these forces into explicit account too, but here I’m just going to use my intuition that they will cancel and hence that the y-direction can be ignored, all of the motion is going to be in the x-direction as I’ve defined it with my coordinate axes. It’s time to follow our ritual. We will write Newton’s Second Law and solve for the accel- eration (obtaining an equation of motion). Then we will integrate twice to find first vx(t) and then x(t). We will have to be extra careful with the constants of integration this time, and