PHYS 200: Fundamentals of Physics I
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Fundamentals of Physics I
PHYS 200 - Lecture 1 - Course Introduction and Newtonian Mechanics
Chapter 1. Introduction and Course Organization [00:00:00]
Professor Ramamurti Shankar: This is a first part of the year-long course introducing you to all the major ideas in physics, starting from Galileo and Newton right up to the big revolutions of the last century, which was on relativity and quantum mechanics. The target audience for this course is really very broad. In fact, I’ve always been surprised at how broad the representation is. I don’t know what your major is; I don’t know what you are going to do later so I picked the topics that all of us in physics find fascinating. Some may or may not be useful, but you just don’t know. Some of you are probably going to be doctors and you don’t know why I’m going to do special relativity or quantum mechanics, but you don’t know when it will come in handy. If you’re a doctor and you’ve got a patient who’s running away from you at the speed of light, you’ll know what to do. Or, if you’re a pediatrician with a really small patient who will not sit still, it’s because the laws of quantum mechanics don’t allow an object to have a definite position and momentum. So these are all things you just don’t know when they will come in handy, and I teach them because these are the things that turn me on and got me going into physics and whether or not you go into physics, you should certainly learn about the biggest and most interesting revolutions right up to present day physics.
All right. So that’s what the subject matter’s going to be, and I’m going to tell you a little bit about how the course is organized. First thing is, this year it’s going to be taped. You can see some people in the back with cameras as part of an experimental pilot program funded by the Hewlett Foundation and at some point they will decide what they will do with these lectures. Most probably they’ll post them somewhere so people elsewhere can have the benefit of what you have sitting in the classroom. So I’ve been told that from now on we just ignore the camera and do business as usual. Nothing’s going to be changed. I tried to negotiate a laugh track so that if the jokes don’t work we can superimpose some laughter. I was told “no.” I just got to deal with it as it happens. So it’s going to be–it’s like one of the reality shows where things are going to be as they are and hopefully after a while we’ll learn to act and behave normally and not worry about its presence. Then, coming to the rest of the details of the course. By the way, there are more details on the website that I posted, that was given to me by the university, if you want to know more about what all this is about.
The course organization is fairly simple. We’re going to meet Monday and Wednesday in this room, 11:30-12:45. I will give you some problems to do on Wednesday and I’ll post them on the website. You guys should get used to going to the class’ website. I’m really, really dependent on that now. I finally learned how to use it. I will use that to post information, maybe once in a while send e-mail to the whole class. If you want to get those e-mails, you got to sign up for the course because I push a button and it goes to anybody who’s signed up there. The homework will be given on Wednesday and it’s due before class the following Wednesday.
Let me introduce you to our head TA, Mara Daniel, who’s recently Mara Baraban. So Mara’s going to be the person who will see you after class and she will take the problem sets that you have submitted before class and she’ll give you the graded ones after class. Okay? That will be sorted up, it’ll be up there. So you should drop the homework before you come into class, rather than furiously work on it during class, and the solutions will be posted the same afternoon. So there is not much point in giving homework that’s late. But once in a while, you know, you will come up with a reason that I just cannot argue with. You got married, you’re getting a transplant, whatever it is. That’s fine. You got a transplant, I want to see the old body part. You got married, I want to see your spouse. If something happened to a grandparent, I’m counting. Up to four I don’t get suspicious. Go five, six, seven, eight, I will have to look into the family tree. But, you know, any reasonable excuse will be entertained.
Relative importance given to these different things, there’s 20% for your homework, 30% for the Midterm, which will be sometime in October, and 50% for the Final. That’ll be the weighted average. But I have another plan called the “Amnesty Plan” in which I also compare just your final grade, what you did on the Final exam, and whichever is higher of the two is what I will take to determine your overall course grade. This is something I used to announce near the end but then some people felt that it’s not fair not to know this from the beginning. So, I’m telling you from the beginning, but don’t dream and think that somehow the Final’s going to be so much different from your regular day-to-day performance, but to give you some reason to live after the Midterm. So, you feel there is hope. I can change everything overnight; it does happen. I put that in for a reason because sometimes some of you have not taken a physics course and you don’t know how to do well in physics and slowly you catch on and by the time it’s Final exam you crack the code; you know how to do well. As far as I’m concerned, that’s just fine. If at the end of the semester you take a three-hour exam in a closed environment and you answer everything, I don’t care what you did in your homework or your Midterm. That’s not relevant.
So that’s how the grading will be done. We have Mara’s group of TAs. She is the head TA and she’s the one you should write to whenever you have a problem. Then we also have two faculty members. One is a Postdoctoral Fellow, Mark Caprio. So he will have a discussion section on Tuesdays between 1:00-2:00 in Sloane Lab. And Steve Furlanetto–I don’t know if Steve is here or not. There’s Steve, our new Assistant Professor. He will have his section on Tuesday night in Dunham Lab, in Room 220. Tuesday night is the night when you people realize homework is due on Wednesday. So we know that, so he will be there to comfort you and give you whatever help you need. All right. My own office hours I’ve not determined yet. I will have to find out when it is good for you. You know, I live and work out of Sloane Lab up on the hill and it was easy to have office hours before or after class but now you have to make a special trip. So, just give me a little bit of time to find out maybe by soliciting e-mail responses from you what would be a good time for my office hours. But for any procedural things, like, you know, this problem set was not graded properly, and so on, there’s no point e-mailing me because I’m going to send it to Mara anyway. So directly deal with the powers that be.
Okay, finally I want to give you some tips on how to do well in this course and what attitude you should have. First, I advise that you should come to the lectures. It’s not self-serving; it’s not so much for my benefit. I think there is something useful about hearing the subject presented once orally. Secondly, the book, you can see, one of you had a book here, it’s about 1,100 pages and when I learned physics it was, like 300 pages. Now, I look around this room, I don’t see anybody whose head is three times bigger than mine, so I know that you cannot digest everything the books have. So I have to take out what I think is the really essential part and cover them in the lecture. So, you come to class to find out what’s in and what’s not in. If you don’t do that, there’s a danger you will learn something you don’t have to, and we don’t want that. Okay, so that’s why you come to class.
Second thing, most important thing for doing well in physics, is to do the homework. The 20% given to the homework is not a real measure of how important it is. Homework is when you really figure out how much you know and don’t know. If you watch me do the thing on the blackboard, it looks very reasonable. It looks like you can do it but the only way you’re going to find out is when you actually deal with the problem. That’s the only time you’re going to find out. So, I ask you to do the problems as and when they’re posted. So if I post it on Wednesday to cover the material for that week, then you should attempt it as quickly as possible because I’m going to assume you have done the problems when you come for the next few lectures. And in doing the homework, it is perfectly okay to work in groups. You don’t have to do it by yourself. That’s not how physics is done. I am now writing a paper with two other people. They are my experimental colleagues who write papers with 400 other people, maybe even 1, 000 other people. When they do the big collider experiments in Geneva or Fermilab, collaborations can run into hundreds. So, it’s perfectly okay to be part of a collaboration, but you’ve got to make sure that you’re pulling your weight. You’ve got to make sure that if you explain to others how to do this problem, then somebody else contributes to something else, but you know what everybody contributed in the end. So the game is not just to somehow or other get the solution to the problem set but to fully understand how it’s done, and the TAs will be there to help you.
Every day there’s going to be a TA in the undergraduate lounge. I would urge you to use that. That’s a beautiful new lounge that the Provost’s Office allowed us to build for physicists and chemists, or whoever happens to be in the building. If you go there on the third floor of Sloane, you may run into other people like you who are trying to work on problems. You may run into upper-class students, students who are more advanced, you will run into your TA. So that’s a good climate. There are coffee machines and there are lounge sofas and everything else. There are computers, there are printers, so it’s a good lounge, and I think if you go there one day a week to do your problem sets, more often that’s a good meeting place, I recommend that.
The final piece of advice, this is very important so please pay attention to this, which is, I ask you not to talk to your neighbors during lecture. Now, this looks like a very innocuous thing, but you will find out, it is the only thing that really gets my back up. Most of the time I don’t really care. I’m really liberal, but this disturbs me because I am looking at you, I’m trying to see from your reaction how much of my lecture you are following, and then it’s very distracting when people are talking. So please don’t do that. If you talk, I am going to assume you are talking about me. If you laugh, I’m going to assume you are laughing at me. That’s not really what I think, but that’s how disturbing it is when people talk, and very nice students who do not realize this often disrupt my line of thinking. So I ask you to keep that to a minimum. Once in a while you’ll have to talk to your neighbor and say, “Can you please pass me my pacemaker that fell down?” That’s fine. Then you go back to your business. But don’t do too much of that.
Finally, there is this ancient issue about sleeping in class. Now, my view is, it’s just fine, okay. I know you guys need the rest and interestingly, the best sleepers are in the first couple of rows. I haven’t met you guys. It’s not personal. I have found some people really have to come to the first and second row because they claim that if they don’t hear me they cannot really go to sleep. Now, that was true in Sloane but I think Luce has got very good acoustics so you can stretch out in the back. But my only criterion is if you talk in your sleep, now that’s not allowed because talking is not allowed. Next, if you’re going to sleep, I ask you to sit between two non-sleepers because sometimes what happens, the whole row will topple over. We don’t want the domino effect. Now, it’s going to be captured on tape and that’s going to be really bad for my reputation, so spread yourself around other people. All right. So that’s it in terms of class, you know, logistics and everything.
I’m going to start going into the physics proper. I will try to finish every lecture on time, but sometimes if I’m in the middle of a sentence or the middle of a derivation, I may have to go over by a couple of minutes; there’s no need to shuffle your feet and move stuff around. I know what time it is. I also want to get out like you guys, but let me finish something. Other days I may finish a few minutes before time. That’s because the ideas of physics don’t fall into 75-minute segments and sometimes they spill over a little bit. Also, I’m used to teaching this course three times a week and now it’s suddenly twice a week, and so things that fell into nice 50-minute units are now being snipped up different ways so it’s pretty difficult. So, even for me, some of it will be new and the timing may not be just right. I should tell you first of all that in this class, the taping is not going to affect you because the camera is going to be behind your head. I mentioned to you in the website that this is not the big opportunity you’ve been looking for to be a star. Only the back of your head will be seen. In some cases, the back of the head could be more expressive than the front, in which case this is your opportunity and I wish you luck. But otherwise, just don’t worry about it because you will be only heard. You may not even be heard. So, I’ve been asked that if a question is not very clear, I should repeat it so that people listening to it later will know what the question was.
Let me make one thing very clear. That is, I’m not in favor of your talking to each other because you’re distracting. Your stopping me at any time is just fine. I welcome that because I’ve seen this subject for God knows how many years. The only thing that makes it different for me is the questions that you people have. You can stop me any time and you should not feel somehow you are stopping the progress of the class. There is no fixed syllabus. We can move things around and it’s far more exciting for me to answer your questions than to have a monologue. So, don’t worry about that. So stop me anytime you don’t follow something, and don’t assume that you’re not following something because there’s something wrong with your level of comprehension. Quite often, you guys come up with questions that never cross my mind, so it’s very interesting. And things we’ve been repeating year after year after year, because they sound so reasonable, suddenly sound unreasonable when some of you point out some aspect of it that you didn’t follow. So, it could be very interesting for all of us to have issues to discuss in class, and quite often some questions are very common and your classmates will be grateful to you that you brought it up. Otherwise, you know, TAs get ten e-mails, all with the same question. Okay. So I’m going to start now. Anybody have any questions about class? The format? The Midterm? The exams? All right. Yes?
Student: You said there’s going to be two hours to be announced. How do we wait for [inaudible]
Professor Ramamurti Shankar: Oh, you mean my office hours?
Student: No. I thought there was an [inaudible]
Professor Ramamurti Shankar: No, the discussion sections are Tuesday afternoon from 1:00-2:00, and Tuesday night from 8:00-10:00, and the website has got all the details on when and where. Yes?
Student: So the lab times will still be [inaudible]
Professor Ramamurti Shankar: Yeah. There are many, many lab times and you have to go to the website for the lab. And, by the way, that reminds me. I’ve got here lots of flyers given to me by the director of the laboratories which will tell you which lab is the right lab for you, and they’re offered many times a week. Yes?
Student: As far as knowing the material, just from your class, how important is taking a lab concurrent with this class?
Professor Ramamurti Shankar: I think it’s a good idea to take the lab, particularly in this particular class because I don’t have any demonstrations. They’re all in the other building. So, this will remind you that physics is, after all, an experimental science and you will be able to see where all the laws of physics come from. So, if you’re going to take it, you should take it at the same time. Yes?
Student: Could you please talk about when you expect [inaudible]
Professor Ramamurti Shankar: Ah, very good. This is a calculus-based class and I expect everyone to know at least the rudiments of differential calculus. What’s a function, what’s a derivative, what’s a second derivative, how to take derivatives of elementary functions, how to do elementary integrals. Sometime later, I will deal with functions of more than one variable, which I will briefly introduce to you, because that may not be a prerequisite but certainly something you will learn and you may use on and off. But there are different ways of doing physics. Mine is to demonstrate over and over how little mathematics you need to get the job done. There are others who like to show you how much mathematics you could somehow insinuate into the process, okay.
There are different ways of playing the game, and some of us find great pride in finding the most simple way to understand something. That’s certainly my trademark; that’s how I do my research also. So, if you feel there’s not enough math used, I guarantee you that I certainly know enough eventually to snow the whole class, but that’s not the point. I will use it in moderation and use it to the best effect possible rather than use it because it is there. Okay. So I don’t know your mathematical background, but the textbook has an appendix, which is a reasonable measure of how much math you should know. You’ve got to know your trigonometry, you’ve got to know what’s a sine and what’s a cosine. You cannot say, “I will look it up.” Your birthday and social security number is what you look up. Trigonometry functions you know all the time. Okay. I will ask you, and you do. All right. And of course, there’s trigonometric identities you know from high school. Pages and pages of them, so no one expects you to know all those identities, but there are a few popular ones we will use. All right. Anything else? Yes?
Student: This may be a bit early, but when will we be having our Midterm?
Professor Ramamurti Shankar: Yeah. Midterm will be sometime around 20th of October. I have to find out exactly the right time. We have 24 lectures for this class and the first 12 roughly will be part of the Midterm, but after the 12th lecture I may wait a week so that you have time to do the problems and get the solutions. Then I will give you the Midterm. Yes?
Student: If wanting one of the two lab courses, which one do you recommend?
Professor Ramamurti Shankar: Yeah, this tells you in detail. This flyer answers exactly that. Okay, there was one more question from somebody? Yes?
Student: A few people I’ve talked to have recommended that we start taking the lab second semester instead of first. Would that be advisable or should we take both concurrently?
Professor Ramamurti Shankar: I don’t have a strong view. I think you should take the lab sometime but I don’t know how many semesters that you have to take. But I would say the advice of your predecessors is very important. If they tell you this is what works, that’s better than what somebody like me can tell you. Also, you should talk to Stephen Irons, who is the director of the labs. He has seen every possible situation. He will give you good advice.
Chapter 2. Newtonian Mechanics: Dynamics and Kinematics [00:21:25]
Let’s start now. Okay. So we are going to be studying in the beginning what’s called Newtonian mechanics. It’s pretty remarkable that the whole edifice is set up by just one person - Newton – and he sent us on the road to understanding all the natural phenomena until the year 18-hundred-and-something when Maxwell invented the laws of electromagnetism and wrote down the famous Maxwell equations. Except for electromagnetism, the basics of mechanics, which is the motion of billiard balls and trucks and marbles and whatnot, was set up by Newton. So that’s what we are going to focus on, and you will find out that the laws of physics for this entire semester certainly can be written on one of those blackboards or even half of those blackboards.
And the purpose of this course is to show you over and over and over again that starting with those one or two laws, you can deduce everything, and I would encourage you to think the same way. In fact, I would encourage you to think the way physicists do, even if you don’t plan to be a physicist, because that’s the easiest way to do this subject, and that is to follow the reasoning behind everything I give you. And my purpose will be not to say something as a postulate, but to show you where everything comes from, and it’s best for you if you try to follow the logic. That way, you don’t have to store too many things in your head. In the early days when there are four or five formulas, you could memorize all of them and you can try each one of them until something works, but after a couple of weeks you will have a hundred formulas and you cannot memorize all of them. You cannot resort to trial and error. So you have to know the logic. So the logical way is not just the way the physicists do it, it’s the easier way to do it. If there is another way that it will work for non-physicists, I won’t hesitate to teach it to you that way if that turns out to be the best way. So try to follow the logic of everything. Okay.
So, Newtonian mechanics is our first topic. So, Newtonian mechanics has two parts. All of physics is a two-part program. The plan, every time, is to predict the future given the present. That’s what we always do. When we do that right, we are satisfied. So the question is, “What do you mean by ‘predict the future?’” What do you mean by the future? What do you mean by the present? By “present,” we mean–we will pick some part of the universe we want to study and we will ask, “What information do I need to know for that system at the initial time, like, right now, in order to be able to predict the future?” So, for example, if you were trying to study the motion of some object, here is one example.
[throws a piece of candy for someone to catch]
Professor Ramamurti Shankar: See, that’s an example of Newtonian mechanics. I’ll give you one more demonstration. Let’s see who can catch this one.
[throws another piece]
Professor Ramamurti Shankar: That’s a good example. So, that was Newtonian mechanics at work, because what did I do? I released a piece of candy, threw it from my hand, and the initial conditions have to do with where did I release it and with what velocity. That’s what he sees with his eyes. Then that’s all you really need to know. Then he knows it’s going to go up, it’s going to curve, follow some kind of parabola, then his hands go there to receive it. That is verification of a prediction. His prediction was, the candy’s going to land here, then he put his hand there. He also knew where the candy was going to land, but he couldn’t get his hand there in time. But we can always make predictions. But this is a good example of what you need to know. What is it you have to know about this object that was thrown, I claim, is the initial location of the object and the initial velocity. The fact that it was blue or red is not relevant, and if I threw a gorilla at him it doesn’t matter what the color of the gorilla is, what mood it is in. These are things we don’t deal with in physics.
There is a tall building, a standard physics problem. An object falls off a tall building. Object could be a person. So we don’t ask why is this guy ending it all today? We don’t know, and we cannot deal with that. So we don’t answer everything. We just want to know when he’s going to hit the pavement, and with what speed. So we ask very limited questions, which is why we brag about how accurately we can predict the future. So, we only ask limited goals and we are really successful in satisfying them. So, we are basically dealing with inanimate objects.
So the product of Newtonian mechanics of predicting the future given the present, has got two parts, and one is called kinematics and the other is called dynamics. So, kinematics is a complete description of the present. It’s a list of what you have to know about a system right now. For example, if you’re talking about the chalk–if I throw the chalk, you will have to know where it is and how fast it’s moving. Dynamics then tells you why the object goes up, why the object goes down and why is it pulled down and so on. That’s dynamics. The reason it comes down is gravity is pulling it. In kinematics, you don’t ask the reason behind anything. You simply want to describe things the way they are and then dynamics tells you how they changed and why they changed.
So, I’m going to illustrate the idea of kinematics by taking the simplest possible example. That’s going to be the way I’m going to do everything in this course. I’m going to start with the simplest example and slowly add on bells and whistles and make it more and more complicated. So, some of you might say, “Well, I’ve seen this before,” so maybe there is nothing new here. That may well be. I don’t know how much you’ve seen, but quite often the way you learned physics earlier on in high school is probably different from the way professional physicists think about it. The sense of values we have, the things that we get excited about are different, and the problems may be more difficult. But I want to start in every example, in every situation that I explain to you, with the simplest example, and slowly add on things.
Chapter 3. Average and Instantaneous Rate of Motion [00:28:20]
So, what we are going to study now is a non-living object and we’re going to pick it to be a mathematical point. So the object is a mathematical point. It has no size. If you rotate it, you won’t know. It’s not like a potato. You take a potato, you turn it around, it looks different. So, it’s not enough to say the potato is here. You’ve got to say which way the nose is pointing and so on. So, we don’t want to deal with that now. That comes later when we study what we call “rigid bodies”. Right now, we want to study an entity which has no spatial extent. So just a dot, and the dot can move around all over space. So we’re going to simplify that too. We’re going to take an entity that lives along the x axis.
[draws a line with integrals]
It moves along a line. So you can imagine a bead with a wire going through it and the bead can only slide back and forth. So, this is about the simplest thing. I cannot reduce the number of dimensions. One is the lowest dimension. I cannot make the object simpler than being just a mathematical point. Then, you’ve got to say, “What do I have to know about this object at the initial time? What constitutes the present, or what constitutes maximal information about the present?” So what we do is we pick an origin, call it zero, we put some markers there to measure distance, and we say this guy is sitting at 1, 2, 3, 4, 5. He is sitting at x = 5. Now, of course, we’ve got to have units and the units for lengths are going to be meters. The unit for time will be a second, and time will be measured in seconds. Then we’ll come to other units.
Right now, in kinematics, this is all you need. Now, there are some tricky problems in the book. Sometimes they give you the speed in miles per hour, kilometers per year, pounds per square foot, whatever it is. You’ve got to learn to transform them, but I won’t do them. I think that’s pretty elementary stuff. But sometimes I might not write the units but I’ve earned the right to do that and you guys haven’t so you’ll have to keep track of your units. Everything’s got to be in the right units. If you don’t have the units, then if you say the answer is 19, then we don’t know what it means. Okay.
So here’s an object. At a given instant, it’s got a location. So what we would like to do is to describe what the object does by drawing a graph of time versus space and the graph would be something like this. You’ve got to learn how to read this graph. I’m assuming everyone knows how to read it.
[draws a graph of x versus t]
This doesn’t mean the object is bobbing up and down. I hope you realize that. Even though the graph is going up and down, the object is moving from left to right. So, for example, when it does this, it’s crossed the origin and is going to the left of the origin. Now, at the left of the origin, it turns around and starts coming to the origin and going to the right. That is x versus t. So, in the language of calculus, x is a function of time and this is a particular function. This function doesn’t have a name. There are other functions which have a name. For example, this is x = t, x = t2, you’re going to have x = sin t and cos t and log t. So some functions have a name, some functions don’t have a name. What a particle tries to do generally is some crazy thing which doesn’t have a name, but it’s a function x (t). So you should know when you look at a graph like this what it’s doing.
So, the two most elementary ideas you learn are what is the average velocity of an object, as then ordered by the symbol v-bar. So, the average is found by taking two instants in time, say t1 and later t2, and you find out where it was at t2 minus where it was at t1 and divide by the time. So, the average velocity may not tell you the whole story. For example, if you started here and you did all this and you came back here, the average velocity would be zero, because you start and end at the same value of x, you get something; 0 over time will still be 0. So you cannot tell from the average everything that happened because another way to get the same 0 is to just not move at all. So the average is what it is. It’s an average, it doesn’t give you enough detail. So it’s useful to have the average velocity. It’s useful to have the average acceleration, which you can find by taking similar differences of velocities. But before you even do that, I want to define for you an important concept, which is the velocity at a given time, v (t). So this is the central idea of calculus, right? I am hoping that if you learned your calculus, you learned about derivatives and so on by looking at x versus t.
So, I will remind you, again, this is not a course in calculus. I don’t have to do it in any detail. I will draw the famous picture of some particle moving and it’s here at t of some value of x. A little later, which is t + Δt. So Δt is going to stand always for a small finite integral of time; infinitesimal interval of time not yet 0. So, during that time, the particle has gone from here to there, that is x + Δx, and the average velocity in that interval is Δx/ Δt. Graphically, this guy is Δ x and this guy is Δt, and Δx over Δt is a ratio. So in calculus, what you want to do is to get the notion of the velocity right now. We all have an intuitive notion of velocity right now. When you’re driving in your car, there’s a needle and the needle says 60; that’s your velocity at this instant. It’s very interesting because velocity seems to require two different times to define it – the initial time and the final time. And yet, you want to talk about the velocity right now. That is the whole triumph of calculus is to know that by looking at the position now, the position slightly later and taking the ratio and bringing later as close as possible to right now, we define a quantity that we can say is the velocity at this instant.
So v of t, v(t) is the limit, Δt goes to 0 of Δx over Δt and we use the symbol dx/dt for velocity. So technically, if you ask what does the velocity stand for–Let me draw a general situation. If a particle goes from here to here, Δx over Δt, I don’t know how well you can see it in this figure here, is the slope of a straight line connecting these two points, and as the points come closer and closer, the straight line would become tangent to the curve. So the velocity at any part of the curve is tangent to the curve at that point. The tangent of, this angle, this θ, is then Δx over Δt.
Okay, once you can take one derivative, you can take any number of derivatives and the derivative of the velocity is called the acceleration, and we write it as the second derivative of position. So I’m hoping you guys are comfortable with the notion of taking one or two or any number of derivatives. Interestingly, the first two derivatives have a name. The first one is velocity, the second one is acceleration. The third derivative, unfortunately, was never given a name, and I don’t know why. I think the main reason is that there are no equations that involve the third derivative explicitly. F = ma. The a is this fellow here, and nothing else is given an independent name. Of course, you can take a function and take derivatives any number of times. So you are supposed to know, for example, if x(t) is tn, you’re supposed to know dx/dt is ntn-1. Then you’re supposed to know derivatives of simple functions like sines and cosines. So if you don’t know that then, of course, you have to work harder than other people. If you know that, that may be enough for quite some time.
Chapter 4. Motion at Constant Acceleration [00:37:56]
Okay, so what I’ve said so far is, a particle moving in time from point to point can be represented by a graph, x versus t. At any point on the graph you can take the derivative, which will be tangent to the curve at each point, and its numerical value will be what you can call the instantaneous velocity of that point and you can take the derivative over the derivative and call it the acceleration. So, we are going to specialize to a very limited class of problems in the rest of this class. A limited class of problems is one in which the acceleration is just a constant. Now, that is not the most general thing, but I’m sure you guys have some idea of why we are interested in that. Does anybody know why so much time is spent on that? Yes?
Professor Ramamurti Shankar: Pardon me?
Professor Ramamurti Shankar: Right. The most famous example is that when things fall near the surface of the Earth, they all have the same acceleration, and the acceleration that’s constant is called g, and that’s 9.8 meters/second2. So that’s a very typical problem. When you’re falling to the surface of the Earth, you are describing a problem of constant acceleration. That’s why there’s a lot of emphasis on sharpening your teeth by doing this class of problems. So, the question we are going to ask is the following, “If I tell you that a particle has a constant acceleration a, can you tell me what the position x is?”
Normally, I will give you a function and tell you to take any number of derivatives. That’s very easy. This is the backwards problem. You’re only given the particle has acceleration a, and you are asked to find out what is x? In other words, your job is to guess a function whose second derivative is a, and this is called integration, which is the opposite of differentiation, and integration is just guessing. Integration is not an algorithmic process like differentiation. If I give you a function, you know how to take the derivative. Change the independent variable, find the change in the function, take the ratio and that’s the derivative. The opposite is being asked here. I tell you something about the second derivative of a function and ask you what is the function. The way we do that is we guess, and the guessing has been going on for 300 years, so we sort of know how to guess.
So, let me think aloud and ask how I will guess in this problem. I would say, okay, this guy wants me to find a function which reduces to the number a when I take two derivatives, and I know somewhere here, this result, which says that when I take a derivative, I lose a power of t. In the end, I don’t want any powers of t. It’s very clear I’ve got to start with a function that looks like t2. This way when I take two derivatives, there will be no t left. Well, unfortunately, we know this is not the right answer, because if you take the first derivative, I get 2t. If I take the second derivative I get 2, but I want to get a and not 2. Then it’s very clear the way you patch it up is you multiply it by this constant and now we’re all set. This function will have the right second derivative. So, this certainly describes a particle whose acceleration is a. The a is not dependent on time. But the question is, is this the most general answer, or is it just one answer, and I think you all know that this is not the most general answer. It is one answer. But I can add to this some number, like 96, that’ll still have the property that if you take two derivatives, you’re going to get the same acceleration. So 96 now is a typical constant, so I’m going to give the name c to that constant.
Everyone knows from calculus that if you’re trying to find a function about which you know only the derivative, you can always add a constant to one person’s answer without changing anything. But I think here, you know you can do more, right? You can add something else to the answer without invalidating it, and that is anything with one power of t in it, because if you take one derivative it’ll survive, but if you take two derivatives, it’ll get wiped out. Now, it’s not obvious but it is true that you cannot add to this anymore. The basic idea in solving these equations and integrating is you find one answer, so then when you take enough derivatives, the function does what it’s supposed to do. But then having found one answer, you can add to it anything that gets killed by the act of taking derivatives. If you’re taking only one derivative you can add a constant. If you’re taking two derivatives you can add a constant and something linear in t.. If you knew only the third derivative of the function, you can have something quadratic in t without changing the outcome.
So, this is the most general position for a particle of constant acceleration, a. Now, you must remember that this describes a particle going side to side. I can also describe a particle going up and down. If I do that, I would like to call the coordinate y, then I will write the same thing. You’ve got to realize that in calculus, the symbols that you call x and y are completely arbitrary. If you know the second derivative of y to be a, then the answer looks like this. If you knew the second derivative of x, the answer looks like that. Now, we have to ask what are these numbers, b and c.
So let me go back now to this expression, x(t) = at2/ 2 + c + bt. It is true mathematically, you can add two numbers, but you’ve got to ask yourself, “What am I doing as a physicist when I add these two numbers?” What am I supposed to do with a and b? I mean, with the b and c? What value should I pick? The answer is that simply knowing the particle has an acceleration is not enough to tell you where the particle will be. For example, let’s take the case where the particle is falling under gravity. Then you guys know, you just told me, acceleration is -9.8, my g is -9.8. We call it “minus” because it’s accelerating down and up was taken to be the positive direction. In that case, y(t) will be -1/2gt2 + c + bt.
So, the point is, every object falling under gravity is given by the same formula, but there are many, many objects that can have many histories, all falling under gravity, and what’s different from one object and the other object is, when was it dropped, from what height, and with what initial speed. That’s what these numbers are going to tell us and we can verify that as follows. If you want to know what the number c is, you say, let’s put time t = 0. In fact, let me go back to this equation here. You’ll put time t = 0, x(0) doesn’t have this term, doesn’t have this term, and it is c. So I realize that the constant, c, is the initial location of the object, and it’s very common to denote that by x0.
So the meaning of the constant c is where was the object at the initial time? It could’ve been anywhere. Simply knowing the acceleration is not enough to tell you where it was at the initial time. You get to pick where it was at the initial time. Then, to find the meaning of b, we take one derivative of this, dx/dt, that’s velocity as a function of time, and if you took the derivative of this guy, you will find as at + b. That’s the velocity of the object. Then, you can then understand that v(0) is what b is, which we write as v0. Okay, so the final answer is that x(t) looks like x0 + v0 t + 1/2 at2. Okay. So what I’m saying here is we are specializing to a limited class of motion where the particle has a definite acceleration, a. Then, in every situation where the body has an acceleration a, the location has to have this form, where this number (x0) is where it was initially, this (v0 ) was the initial velocity of the object. So, when I threw that thing up and you caught it, what you are doing mentally was immediately figuring out where it started and at what speed. That was your initial data. Then in your mind, without realizing it, you found the trajectory at all future times.
Now, there is one other celebrated formula that goes with this. I’m going to find that, then I’ll give you an example. Now, I’m fully aware that this is not the flashiest example in physics, but I’m not worried about that right now. You’ll see enough things that will confound you, but right now I want to demonstrate a simple paradigm of what it means to know the present and what it means to say this is what the future behavior will be. We want to do that in the simplest context, then we can make the example more and more complicated, but the phenomenon will be the same. So, what we have found out so far, I’m purposely going from x to y because I want you to know that the unknown variable can be called an x or can be called a y. It doesn’t matter, as long as the second derivative is a; that’s the answer.
Now there’s a second formula one derives from this. You guys probably know that too from your days at the daycare, but I want to derive the formula and put it up, then we’ll see how to use it. Second formula tries to relate the final velocity of some time, t, to the initial velocity and the distance traveled with no reference to time. So the trick is to eliminate time from this equation.
So let’s see how we can eliminate time. You know that if you took a derivative of this, you will find v(t) is v0 + at. What that means is, if you know the velocity of the given time and you know the initial velocity, you know what time it is. The time, in fact, is v - v0 over a. If I don’t show you any argument for v, it means v at time t and the subscript of 0 means t is zero. So what this says is, you can measure time by having your own clock. A clock tells you what time it is, but you can also say what time it is by seeing how fast the particle is moving because you know it started with some speed. It’s gaining speed at some rate a. So, if the speed was so and so now, then the time had to be this. So time can be indirectly inferred from these quantities. Then you take that formula here (t) and you put it here, (y(t)) to see a times t, you put this expression. So what will you get? We’ll get an expression in which there is no t; t has been banished in favor of v. So, I’m not going to waste your time by asking what happens if you put it in. I will just tell you want happens. What happens is, you will find that v2 = vo2 + 2a times (y- y0). [Note: The Professor said x when he meant y] How many people have seen this thing before? Okay. That’s a lot. Look, I know you’ve seen this.
At the moment, I have to go through some of the more standard material before we go to the more non-standard material. If this part’s very easy for you, there’s not much I can do right now. So let me draw a box. Drawing a box to you guys means important. These are the two important things. Remember, I want you to understand one thing. How much of this should you memorize? Suppose you’ve never seen this in high school. How much are you supposed to memorize? I would say, keep that to a minimum, because what the first formula tells you should be so intuitive that you don’t have to cram this. We are talking about particles of constant acceleration. That means, when I take two derivatives, I want to get a, then you should know enough calculus to know it has to be something like at2, and half comes from taking two derivatives. The other two you know are stuff you can add, and you know where you’re adding those things, because the particle has a head start. It’s got an initial position. Even at = 0, and it has an initial velocity, so even without any acceleration, it will be moving from y0 to y0 + vt. The acceleration gives you an extra stuff, quadratic in time. Once you’ve got that, one derivative will give you the velocity, then in a crunch you can eliminate t and put it into this formula. But most people end up memorizing these two because you use it so many times. It eventually sticks in you but you shouldn’t try to memorize everything.
Chapter 5. Example Problem: Physical Meaning of Equations [00:52:37]
So, we are now going to do one standard problem where we will convince ourselves we can apply this formulae and predict the future given the present. So the problem I want to do–there are many things you could do but I just picked one, and this is the one with round numbers so I can do it without a calculator. Here’s the problem. There is this building and it’s going to be 15 meters high, and I’m going to throw something and it’s going to go up and come down. It’s something I throw up has an initial speed of 10 meters per second. So we have to ask now, now that my claim is, you can ask me any question you want about this particle and I can answer you. You can ask me where it will be nine seconds from now, eight seconds from now, how fast will it be moving. I can answer anything at all. But what I needed to do this problem was to find these two unknowns. So, you’ve got to get used to the notion of what will be given in general and what is tailor-made to the occasion. So, we know in this example the initial height should be 15 meters and the initial velocity should be 10, and for acceleration, I’m going to use -g and to keep life simple, I’m going to call it -10. As you know, the correct answer is 9.8, but we don’t want to use the calculator now so we’ll call it -10. Consequently, for this object the position y, at any time t is known to be 15 + 10t - 5t2. That is the full story of this object. Of course, you’ve got to be a little careful when you use it. For example, let’s put t equal to 10,000 years. What are you going to get? When t is equal to 10,000 years or 10000 seconds, you’re going to find y is some huge negative number. You know, that’s not right, what’s wrong with that reasoning?
Professor Ramamurti Shankar: So you cannot use the formula once it hits the ground because once it hits the ground, the fundamental premise that a was a constant of -9.8 or -10 is wrong. So that’s another thing to remember. Once you get a formula, you’ve got to always remember the terms under which the formula was derived. If you blindly use it beyond its validity, you will get results which don’t make any sense. Conversely, if you get an answer and it doesn’t seem to make sense, then you’ve got to go back and ask, am I violating some of the assumptions, and here you will find the assumption that the particle had that acceleration a is true as long it’s freely falling under gravity but not when you hit the ground. Now, if you dug a hole here until there, and of course it may work until that happens, okay. But you’ve got them every time. This is so obvious in this problem, but when you see more complicated formula, you may not know all the assumptions that went into the derivation and quite often you will be using it when you shouldn’t. All right.
See, this you agree, is a complete solution to this miniature, tiny, Mickey-Mouse problem. You give me the time and I’ll tell you where it is. If you want to know how fast it’s moving at a given time, if you want to know the velocity, I just take the derivative of this answer, which is 10 - 10t. So let me pick a couple of trivial questions one can ask. One can ask the following question. How high does it go? How high will it rise? To what height will it rise? So, we know it’s going to go up and turn around and come down. We’re trying to see how high that is. So, that is a tricky problem to begin with because if you take this formula here, it tells you y if you know t, but no, we’re not saying that. We don’t know the time and we don’t know how high it’s rising so you can ask, “How am I supposed to deal with this problem?” Then you put something else that you know in your mind, which is that the highest point is the point when it’s neither going up nor coming down. If it’s going up, that’s not the highest point. If it’s coming down, that’s not the highest point. So at the highest point it cannot go up and it cannot go down. That’s the point where velocity is 0. If you do that, let’s call the particular time t*, then 10t* - 10 = 0, or t* is 1 second. So we know that it’ll go up for one second then it will turn around and come back.
Now, we are done because now we can ask how high does it go, and you go back to your, and y (1) is 15 + 10 - 5, which is what? Twenty meters. By the way, you will find that I make quite a lot of mistakes on the blackboard. You’re going to find out, you know, one of these years when you start teaching that when you get really close to a blackboard, you just cannot think. There’s definitely some inverse correlation between your level of thinking and the proximity to the blackboard. So if you find me making a mistake, you’ve got to stop me. Why do you stop me? For two reasons. First of all, I’m very pleased when this happens, because I’m pretty confident that I can do this under duress, but I may not do it right every time. But if my students can catch me making a mistake, it means they are following it and they are not hesitating to tell me. Secondly, as we go to the more advanced part of the course, we’ll take a result from this part of the blackboard, stick it into the second part and keep manipulating, so if I screwed up in the beginning and you guys keep quiet, we’ll have to do the whole thing again.
I would ask you when you follow this thing to do it actively. Try to be one step ahead of me. For example, if I’m struck by lightning, can you do anything? Can you guess what I’m going to say next? Do you have any idea where this is going? You should have a clue. If I die and you stop, that’s not a good sign, okay. You’ve got to keep going a little further because you should follow the logic. So, for example, you know, I’m going to calculate next when it hits the ground. You should have some idea of how I’ll do it. But this is not a spectator sport. If you just watch me, you’re going to learn nothing. It’s like watching the U.S. Open and thinking you’re some kind of a player. You will have to shed the tears and you’ve got to bang your head on the wall and go through your own private struggle. I cannot do that for you. I cannot even make it look hard because I have memorized this problem from childhood, so there is no way I can make this look difficult. That’s your job.
All right. So, we know this point at one second is 20 meters, so let’s just ask one other question and we’ll stop. One other question may be, “When does it hit the ground and at what speed?” – a typical physics question. So when does it hit the ground? Well, I think you must know now how to formulate that question. “When does it hit the ground” is “When is y = 0”? By the way, I didn’t tell you this but I think you know that I picked my origin to be here and measured y positively to be upwards and I called that 15 meters. You can call that your origin. If you call that your origin, your y0 will be 0, but ground will be called -15. So, in the end, the physics is the same but the numbers describing it can be different. We have to interpret the data differently. But the standard origin for everybody is the foot of the building. You can pick your origin here, some crazy spot. It doesn’t matter. But some origins are more equal than others because there is some natural landmark there. Here, the foot of the building is what I call the origin. So, in that notation, I want to ask, when is y = 0? I ask when y = 0, then I say 0 = 15 + 10t - 5t2. Or I’m canceling the 5 everywhere and changing the sign here I get t2 - 2t - 3 = 0. That’s when it hits the ground. So let’s find out what the time is. So t is then 2 + or - or + 12 over 2, which is 2 + or - 4 over 2, which is -1 or 3. Okay, so you get two answers when it hits the ground. So it’s clear that we should pick 3. But you can ask, “Why is it giving me a second solution?” Anybody have an idea why?
Student: Because there was an entire parabola [inaudible]
Professor Ramamurti Shankar: That’s correct. So her answer was, if it was a full parabola, then we know it would’ve been at the ground before I set my clock to 0. First of all, negative time should not bother anybody; t = 0 is when I set the clock, I measured time forward, but yesterday would be t = -1 day, right? So we don’t have any trouble with negative times. So the point is, this equation, it does not know about the building. Doesn’t know the whole song and dance that you went to a building and you threw up a rock or anything. What does the mathematics know? It knows that this particle happened to have a height of 15, a time 0, and a velocity of 10, a time 0, and it is falling under gravity with an acceleration of -10. That’s all it knows. If that’s all it knows, then in that scenario there is no building or anything else; it continues a trajectory both forward in time and backward in time, and it says that whatever seconds, one second before you set your clock to 0, it would’ve been on the ground. What it means is if you’d release a rock at that location one second before with a certain speed that we can calculate, it would’ve ended up here with precisely the position and velocity it had at the beginning of our experiment.
So sometimes the extra solution is very interesting and you should always listen to the mathematics when you get extra solutions. In fact, when a very famous physicist, Paul Dirac, was looking for the energy of a particle in relativistic quantum mechanics, he found the energy of a particle is connected to its momentum, this p is what we call momentum, and its mass by this relation. It’s a particle of mass m and momentum p has this energy so you solve for the energy, you get two answers. Now, your temptation is to keep the first answer because you know energy is not going to be negative. Particle’s moving, it’s got some energy and that’s it. But the mathematicians told Dirac, “You cannot ignore the negative energy solution because it tells you there’s a second solution and you cannot throw them out,” and it turns out the second solution, with negative energy, was when the theory is telling you, hey, there are particles and there are anti-particles, and the negative energy when properly interpreted will describe anti-particles. So the equations are very smart.
The way the physics works is you will find some laws of motion in mathematical form, you put in the initial conditions of whatever, you solve the equations, and the answer that comes, you have no choice. You have to accept the answer, but there are new answers besides the one you were looking for. You’ve got to think about what they mean, and that’s one of the best things about physics because here’s a person who is not looking for anti-particles. He was trying to describe electrons, but the theory said there are two roots in the quadratic equation and the second root is mathematically as interesting as the first one. It has to be part of a theory, and then trying to adjust it so it can be incorporated, you discover anti-particles. So always amazing to us how we go into the problem, our eye or mind can see one class of solutions, but the math will tell you sometimes there are new solutions and you’ve got to respect it and understand and interpret the unwanted solutions, and this is a simple example where you can follow what the meaning of the second solution is. It means that to the problem you pose, there’s more than the answers that you could imagine. Here it meant particle that was released from the ground earlier. There it meant something much more interesting, mainly anti-particles accompanying particles. They are going to accompany particles surely as every quadratic equation has two solutions.
All right, so now in this problem, we can do something slightly different, and let’s use this expression here, and I will do that, then I’ll stop for today. If you were asking questions, like, how high does it go, but you don’t ask when does it go to the highest point, then you don’t have to go through the whole process of finding the time at which it turned around. I don’t know where that is, that disappeared on the blackboard, then putting the time equal to 1 second into this formula. If the question of time is not explicitly brought up, then you should know that you have to use this formula. So how do we get it here? Well, we say at the top of the loop, when it goes up and comes down the velocity is 0. Therefore, you say 02 = initial velocity2 + 2 times -g, that’s my acceleration, times y - y0. If you solve for that, you find y - y0 = v02 over 2g, and if you put in the v0 I gave you, which was what, 10? That’s 100 over 20, which is 5 meters. So y = y0 + 5 meters, and that was the height to which it rises. I think we got it somewhere else. We found the maximum height to be 20 meters. Another thing you can do is you can find the speed here. If you want to find the speed there, you put the equation v2 = v02 + 2 times -g (y - y0). What is y - y0? The final y is 0, the initial y is 15. You solve for that equation and you will find the final velocity. So, if time is not involved, you can do it that way.
Chapter 6. Derive New Relations Using Calculus Laws of Limits [01:08:42]
I want to derive the last result in another way, then I will stop, and that’s pretty interesting because it tells you the use and abuse of calculus. So I’m going to find for you this result using calculus in a different way. So, from the calculus we know dv/dt = a. Now, multiply both sides by v. Now you have to know from elementary calculus that v times dv/dt is really d by dt of v2 over 2. Now, I hope you guys know that much calculus, that when you take a derivative of a function of a function, namely v2 over 2 is a function of v, and v itself is a function of t, then the rule for taking the derivative is first take the v derivative of this object, then take the d by dt of t, which is this one. On the right-hand side, I’m going to write as a dx/dt. This much is standard.
I’m going to do something which somehow we are told never, ever to do, which is to just cancel the dts. You all know that when you do dy/dx, you’re not supposed to cancel that d. That’s actually correct. You don’t want to cancel the d in the derivative. But this happens to be completely legitimate, so I’m going to assume it’s true and I’ll maybe take a second and explain why it’s legitimate. What this really means is in a given time, Δt, the change in this quantity is a times the change in this quantity. Therefore, you can multiply both sides by the Δt, but the only thing you should understand is Δt, as long as it’s small and finite, will lead to some small infinite errors in the formula, because the formula is really the limit in which Δx and Δt both go to 0. So what you have to do is multiply both sides by Δt, but remember it’s got to be in the end made to be vanishingly small. As long as we understand that, we can do this cancellation and this says on the left-hand side the change in the quantity v2 over 2 is a times the change in the quantity x. So add up all the changes or what you mean by integral. Same thing. Add up all the changes. The change in v2 over 2 will be the final v2 over 2 - the initial v2 over 2 and the other side will be a times change in x; x - x0 and that’s the formula I wrote for you: v2 is v02 + 2a (x - x0).
So, the point is whenever you have derivatives with something over dt, do not hesitate to cancel the dts and think of them as Δv2 over 2 is equal to a times Δ of x. This will be actually true as long as both quantities are vanishingly small. They will become more and more true as Δx and Δv2 become vanishingly small, in the limit in which they are approaching 0, the two will be, in fact, equal. If x is a finite amount, like 1 second, this will not be true because in the starting equation, Δx and Δt and Δv2 were all assumed to be infinitesimal. So don’t hesitate to do manipulations of this type, and I will do them quite often. So you’ve got to understand when it’s okay and when it’s not okay. What this means is, in a time Δt, this quantity changes by some amount, and in the same time, Δt, that quantity changes by some amount, then keeping the Δt equal to some number we may equate the changes of the two quantities, provided it is understood that Δv2 over 2 is a change in v2 over 2 in the same time in which the particle moved a distance, Δx. Adding the differences, we eliminate time and we get this final result.
All right. So if you go to your website today, you will find I’ve assigned some problems and you should try to do them. They apply to this chapter. Then next week we’ll do more complicated problems that involve motion in higher dimensions, how to go to two dimensions or three dimensions.
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