Classical Mechanics | Lecture 1

[Music] Stanford University classical mechanics is basically a set of rules about what those laws of motion look like it's a set of rules and in fact there are two varieties of questions the first question is I'm not sure about the order of them but let's take them first the first kind of question is what are the specific laws for particular kinds of system particular system could be a planet moving in the field of a heavy mass that has its own particular laws those laws are different than an electrically charged particle moving in the field of a magnet they're different then you know there are those specific specific laws of nature then there's a more general framework what are the rules for the allowable laws are there rules for the allowable laws what's the grand framework in which all of the various different specific laws are framed in and the second it's really the second question which we're really interested in the first question will provide us with illustrations illustrations of the principles that govern what the allowable laws of physics are okay so I like to start and I always do in these classes with a very simple set of illustrations who can guess what my illustrations are going to be coins right right of course that's because you've been here before going's let's take the very simplest system that I can think of a coin I don't have a coin I so when I say a coin I mean a abstract coin laid down on an abstract table the only thing about the coin which is relevant to us is whether it's heads or tails the coin has two states two states of being two configurations are two values however we want to call it heads or tails so there are two States this is our dynamical system it's one coin and it has two states heads or tails that's all there is to the world to this very simplified world question of course there's an even simpler world we can even go back to a simpler world it's a coin with only one side then it's even simpler all it can be is that one state right that's a little too simple nothing can happen nothing ever happens so this is our world what is an initial condition an initial condition is it's either heads or tails now what about laws of motion now we're going to make up some laws of motion the laws of motion of course for this coin that just sits here the law of motion is very very simple let's say let's put two circles here one stands for heads and one stands for tails what is the law of motion of this particular coin sitting on this particular table the answer is it stays the same so if we imagine breaking up time into little successive intervals the stroboscopic world of the squeak point like that then the only thing that happens is nothing happens heads goes - heads tails goes - tails or we can draw a picture heads goes - heads starting with heads we go back - heads tails goes - tails that's really a very boring law of physics but it's very powerful law of physics and that it tells you however you start you know where you will be arbic terally down the line and time alright if you start with heads the history of the world will be heads heads heads heads dot dot dot forever and ever so that's this law may be very boring but it's very powerful or if you start with tails it's going to be tails tails tails tails as far as the eye can see so that's an example of a dynamical system with a law of motion where the law of motion is just an updating but it's in this case very trivial there's only really one of the law of motion that you can imagine for this system and what is it it is that whatever it is at one instant at the next instant of the stroboscopic light it's the opposite in the next instance the opposite we can write it this way heads goes to tails tails goes to heads and we can make a picture of it we can make we could picture this law of motion by again arrows an arrow from head to tails which says that if you started hits the tail end of the arrow always means what you start with the head end of the arrow means where you'll end up and that would be this law of physics heads goes to tails tails because the heads heads goes to tails and the history now of the world would be if you start with heads you then go to tails and you go to heads and you go to tails not thought but if you start with tails you go to heads tails heads still pretty boring but a little more interesting than the original first law ah we could write we could write some mathematics for this in fact we can write an equation of motion let's invent a symbol a variable which takes two values let me call it Sigma why do I call it Sigma it's because Sigma is a traditional variable associated with two unison physics it has to do with the spin of the electron but about them or spin of particles but we don't need to know about that now Sigma is a variable it is either equal to one let's say or minus one one for heads - one for tails alright so we can instead of calling this heads and tails we can call it Sigma equals 1 and Sigma equals minus 1 alright we now have the idea of a configuration space which is labeled by the two possible values of a certain variable it's not as rich as the variables we will use for example positions of particles values of fields or all sorts of other things but nevertheless it's a mathematical symbol whose value tells you which of the configurations you're in alright let's take this first law of physics the first law of physics not the first law of physics the first law that I wrote down the previous law now as heads goes to heads and tails goes to tails let me write that in the mathematical form oh let's call time T and in our stroboscopic world T is an integer T is 0 1 2 3 it can also be minus 1 minus 2 minus 3 so in the scrubmites copic world time is discrete and it takes on integer values so let's see if we can write an equation for the board the most boring law in the world what it says is that whatever Sigma is at a given time let's put that over here Sigma at time T at the next instant T plus 1 it will be equal to whatever it was at the instant T this is just a law that says that the spin or that the coin doesn't flip it just stays the same whatever it is at one time it will be the same at the neighboring time at the next time ok so that's the simple law heads heads heads heads heads or tails tails tails tails tails what about the other law the one which says that the spin the coin does flip between each between each successive flash of the of the stroboscope that's easy that's just says Sigma of T plus 1 is equal to minus Sigma of T if it starts at 1 then at the next instant it's minus 1 if it starts at 1 the next min simmer it it's the opposite so there we've written down equations of motion for a very very simple system notice that it is completely predictive completely deterministic there's no ambiguity about what happens arbitrarily fire far down the future now of course I could come in and grab the coin and do something with it do something else with it that would disturb it in our language that we set up now that would be because the system was not closed I intervened and I was not part of the system so for a closed system laws of physics are completely deterministic in classical mechanics ok now let's go let's think about a more complicated system more complicated system not much more complicated instead of a coin let's take what's the next case friends a died about half of a half of a pair of dice you know I never knew until I started teaching this that as the singular of dice was die I really didn't somebody corrected me once I called it a I guess I called it a dice somebody said oh you mean a die and I guess I did ok so we have a die a die has six states 1 2 3 4 5 6 that's the only significance for our purposes today about the fact that it's a by is that it has six states again we could label the six states one through six let's put them 1 2 3 4 5 six and these stand for the six different configurations of laying the dice down the died down on the table that's an initial condition well sorry an initial condition is a choice of one of these six configurations what about a law of physics what about a law of motion an equation not an equation I'm not going to write an equation is a little bit too complicated but what's a possible law of physics for this simple system well a very simple law would be or give you a very simple law a simple laws nothing happens nothing happens however you start you stay the same way to graph it we would just graph it like this whatever you start with it's what you get at the end that's about as boring as as the as the simple coin okay a more interesting law would be that you cycle around this collection of configurations for example a possible law for zexis would be 1 goes to 2 2 goes to 3 3 goes to 4 4 goes to 5 5 goes to 6 6 goes back to 1 I'll leave it to you to try to write an equation for this an equation of motion it's not hard to do I'll just leave it to you do ok but you can see what happens the history of the world would now be well give me the starting point for after that the history of the world is 4 5 6 1 2 3 4 5 6 1 2 3 endlessly cycling around and that would be the theory of this particular die with a particular equation or a particular law of evolution now we can write down other laws another law and the simplest way to write them down is to graph them Oh again notice that it's completely deterministic another law would be to cycle in the opposite direction 2 goes to 1 1 goes 6 6 goes to 5 or we could do more complicated things they're not really more complicated it's just more complicated to draw 1 goes to 2 2 goes to 5 5 goes to 3 3 goes to 4 4 goes to 6 and 6 goes to 1 now this law is logically equivalent to the one which just cycled around it's just a relabeling it's a relabeling of the states but it's again 1 cycle 1 to 2 2 to 5 5 to 3 3 to 4 4 to 6 6 back to 1 if I were to just rearrange them redraw them then it would just look exactly the same as the original cycle so this law would be what should we call logically equivalent to one of the others okay but let's think about some laws which would not be logically equivalent here's a law that's not logically equivalent again 6 6 points I won't bother labeling them with numbers 1 goes to 2 2 goes to 3 3 goes back to 1 4 goes to 5 5 goes sorry 6 goes to 5 5 goes to 4 4 goes back to 6 it's again completely deterministic wherever you begin the future is laid out for you completely if you start with 3 you go to 1 - 2 - 3 - 1 - 2 - 3 on the other hand if you start with 4 you go - whoops if you go to for you so you go to 6 6 2 5 5 back to 4 and so forth so it's again completely deterministic but it's not logically equivalent to any of the previous ones it now has two cycles and if you're on one of the cycles you will never get to I'm giving them a name now cycles okay if you're on one of the cycles you will never get to the other one so it's not that there are two systems is one diary only one die but with this particular law of physics you're trapped on one cycle or the other cycle there's a name for this kind of behavior it's called having a conserved quantity a conserved quantity is something non-trivial that you can label the system with which doesn't change with time for example we could label this cycle up here cycle number one and assign it a value one this cycle down here we could label with a value 2 and then we would say that if you start with 1 the quantity 1 or the kind of conserved quantity 1 you stay with 1 forever if you stay with to use if you start with - you stay with 2 forever this is called a conserved quantity in other words a quantitative quantity all quantities are quantitative but a quantitative quantity which just doesn't change it's conserved are let's call the conservation law so this cycle over here the single cycle system doesn't have a non-trivial conserved quantity you pass through every one of the states this has a conserved quantity to the 1 what did I say 1 or 0 I don't remember what I assign them but I assign them to different numbers this one in this one oh we can make up more complicated or not necessarily more complicated but other examples all right and this one one goes to warn if the die is it 1 the rule is it stays at 1 if it's at 2 it goes to 3 and if it's a 3 it goes back to 2 and if it's a 4 it goes too far five if it's a five it goes to six if it's six it goes back to one I've changed my color coding sorry about that all right again if you there are three cycles now there are three cycles and if you get on to any one of them if you start on any one of them you stay on it you could describe this by saying the first cycle over here corresponds to the value zero let's say corresponds to the conserved quantity having value 0 it could be 1 over here and it could be 2 over here if you start with a conserved quantity being 1 then it stays 1 if you start with one less sorry zero if I start with a conserved quantity being 0 it stays 0 if I start with it being one that doesn't tell me exactly where I start either here or here but it tells me I start with one of these two and I stay with that value I don't change I change the state but I don't change the value of the conserved quantity likewise over here so we can consider a variety of different logically different evolutions what is it that distinguishes them well really it's just a number of cycles the number of distinct cycles this is a 3 I'll know um it's a little more than that it's a little more than that but but you can see you can you can play around with this and investigate it and you'll very quickly get the point if you haven't already gotten it any questions up to now yes sir hhhh to go to write so those I mean I can imagine her the sequences of agency T yeah but if I were to try to construct one of those I think it would have to depend on more than just the preceding state in several state factors not for that stuff for those laws HT th that's a different right right right you can right that's right you could invent laws where to know what happens next you have to know not only the previous but the second previous state right that's correct yes yes it will become important it will become important but under those circumstances you would say that the state of the system is not characterized by just whether it's heads or tails it's characterized by the configuration and the previous configuration and so you will have in that case four configurations the four configurations would be heads preceded by tails heads preceded by heads tails proceed and then you were in and then you would do exactly the same thing except you would say there were four possible states right so and that of course is them I know exactly where you want me to go but I'm not going to go there yet but the right not yet so it's what I said before the state of a system or the configuration consists of all the things that you need to know to predict the future so in the case that you described what you need to do what you need to know the initial condition consists not only of what the coin is doing but what it was last doing good so that's a good point and we will come back to it all right so we have the idea of configurations we have the idea of isolated systems we have the idea of conservation laws let's just point out that there's nothing to prevent us from considering systems that have infinite number of states it doesn't have to have an infinite number of objects to have an infinite number of states it just needs to have one object which can be rearranged in infinite number of ways for example if we had an infinite line and on that infinite line we marked off the integers and we said as a particle what does a particle mean a particle now just means a thing which occupies one of the places the places now consists only of the integers what's in-between doesn't matter doesn't count all right so a particle is simply a object which sits at one of the integers you could have an infinite array like this and then you would say the particle has an infinite number of states namely the infinite number of possibilities of where it can be we can label them by an integer in and then we could invent laws such as wherever you are move to the right one unit wherever you are move to the right one unit and that would be a picture that would look like this it doesn't cycle nevertheless we're going to call such a thing a cycle but but you can see it doesn't cycle it just goes on and on forever and ever we could have another law the other law could be wherever you are jump two units ahead wherever you are jump two units ahead you know I think I'll do it this way if you're over here jump two units ahead if you're over here jump two units ahead if you're over here if you're over here and so forth all right so we make a picture then of a kind of law of physics which tells us to jump two units ahead again make up an equation for that make up an equation for that it's easy to do in the first case there was no interesting conservation law there was only one cycle if you like wherever you are you will always either get to any other point or if you go back into the past if you imagine running it back into the past you will have come from that point so on so there cannot be any interesting conserved quantity because you'll pass through every single one of these integers and so they can't be distinguished by a conserved quantity in the second case there is a conserved quantity and you can call it the oddness or evenness of the position of the particle you could give all odd numbers are represent them by a value of a quantity which is zero and all even that's that's weird isn't it let's make the odd numbers have integer value one and the even numbers be labeled by integer values zero and then you would say there's a conserved quantity which can either be 0 or 1 if it starts at one of them it stays there it starts at the other it stays there all right so having an infinite number of configurations doesn't change the picture very much it does open the possibility that there's a kind of endless evolution which never repeats itself so in that sense it gets a little more interesting and you can think of all sorts of generalizations of this all right so I laid out some I'll call them allowable laws of physics allowable rules let me talk now about some rules which are not allowable no not allowable by whom by me but of course I'm simply reflecting the way classical real classical physics works when I say allowable and not allowable I give you an let me draw a picture of a non allowable or unallowable M in a stick it completely predicts the future but there's something different about it then the laws that I've drawn over here I'm going to do it by drawing the picture for it it's got something we can make up many many examples but for this particular example it's got three states so this is a three sided coin a three sided coin it's got heads tails and no sides heads till I was going to think something a little more risque but the tails tails us about as rescales we're going to get ya an edge sorry edge that's pretty edgy heads tails and edges okay all right here's a law of a kind which represents something that we will not allow in classical physics heads goes to tails tails goes to edges and edges goes back to tails all right now wherever you start wherever you start the history is completely predictive if I start with tails i go to i go to edges if a tails edges edge tail circles tails edge tails edge tails edge tails edge you just follow the lines if you start at heads you go to tails then edge then tails then edge then tail so here's very hit some histories first of all starting with heads it's heads tails edges tails edges tails popped on top if I start a tails I get tails edges tails edges pop I thought if I start on edges now what is it it's odd about this law what's on about this law is that it is completely predictive into the future but it is not predictive into the past so to speak um if you know that you're a tails you don't know where you came from did you come from edges well maybe if but you could have also come from heads so while you can predict the future you can't read icked what's the opposite of pre bit richer dick that's the word I'm looking for you can't retro dick the past from this law of motion one configuration or several configurations run into the same configuration and so you can't tell if you're over here whether you came from here or here the word for this is that it's not reversible here's the way to think about it reverse every arrow now you have an unpredicted situation if you're a tails you don't know whether to go to heads or whether to go to edges so it's a predictive situation one way but not retroactive the other way the word for this is not reversible I won't call it irreversible that's a little too definite it's not reversible it can't you can go one way but not the other way this is the kind of law that is not allowed in classical physics yes sir well ma'am I can't see well okay that of course depends on what you're trying to represent um [Music] classical physics doesn't allow probability so I think I could escape from the question by just saying we don't do that in classical physics but we could we can imagine because we can imagine anything we want in fact we can imagine this and we can study its properties the point is in one way or another it conflicts with the rules of classical mechanics probability or let's call non deterministic laws also conflict with the rules of classical mechanics so it's a very good question and it's something we want to come back to whether when when quantum mechanics for example which is not deterministic does quantum mechanics have a analog of this reversibility idea even though it's not deterministic and the answer is yes but not tonight well classical statistical mechanics also has keep in mind that the rules of classical statistical mechanics begin with the laws of mechanics okay we begin by assuming alright let's talk a little bit about the limits on predictability for a moment if we have a perfectly predictive system of equations it won't allow us to be completely predictive if we don't know the initial conditions exactly so we need to know two things to predict the future one is what the rules are laws and one is what the what the initial conditions are now in these very very simple systems it's easy to imagine that we could know exactly what the initial conditions are because we may not we may know that we again with either not this one but this one we may know that we begin either with edges or heads we don't know which and then there would be some ambiguity in what happens not ambiguity because the equations have ambiguity in them but because our knowledge of the initial state is imperfect ok this is easy to understand for these simple discrete systems it's easy to imagine that we can do enough experiment and very quickly just look at a coin and know the initial condition perfectly in the real world where we're faced with degrees of freedom which are continuous meaning to say they can be any number on the continuous real axis any number of numbers the positions and velocities of all the particles in the world you can never know them perfectly no matter how many decimal points you may have that you have may have measured decimal places you may have measured about the location of a particle you still don't know it exactly so there is always a degree of ambiguity in your knowledge of the initial conditions that degree of ambiguity may or may not sort of blow up in your face that small changes in what the initial conditions are may or may not give rise to large changes in what happens in the future so the right way to say it is if you knew well we are we have to be quantitative but if you know the initial conditions perfectly then you could predict the future forever and ever in a true classical mechanical system and you could also predict the past perfectly if you have imperfect knowledge of the initial conditions you want to quantify that how imperfect is it and if you can quantify it it may allow you to answer the question how long into the future can you predict things so if you knew enough about the initial conditions of the atmosphere you might be able to predict the weather for 3 days but as your knowledge is limited and because the atmosphere is one of these systems where little errors build up they called chaotic because the atmosphere is chaotic no matter how well you know the initial conditions it is always true that if you wait long enough you won't be able to predict the future on the other hand if you say I want to be able to predict the future for X number of years it should be possible to say how precisely you have to know the initial conditions okay so when we come to the real world this idea of predictability becomes a little more complicated but I started on purpose with very simple discrete systems ok so laws that are allowed laws that are not allowed the laws of our classical mechanics are not only deterministic into the future but they deterministic into the past which makes them reversible now how do you look at one of these pictures and decide whether it's a legal law or not well it's very simple if every state has one outgoing and one incoming arrow that means that you know where it came from and where it's going so when you draw one of these pictures if you want to know whether it is an allowable legal law in the sense that I've defined I'm using legal now just as a term for for reversible systems if you want to know whether it's reversible look at the arrows and if each state has one incoming and only one incoming and one outgoing only one outgoing then it is both deterministic and reversible there are analogs of all the things I'm telling you now about more complex systems and more interesting systems such as particles moving around okay so that's a that's sort of warm up preliminary about what classical mechanics is about it's about predicting the future or using the predictability the fact that in principle you could predict the future in other ways such as mystical ways to to limit what will happen all right now we want to move on to a more realistic world and in fact the world of particles we're going to be interested in the world of particles moving and a particle for our purposes can be thought of as a point particle if we want to make a complicated system we'll make systems of particles points particles but we'll consider point particles moving around in space so what do we have to know what are the configurations of a point particle what do we need to know well ok before we do so I think we should do a little bit of mathematical preliminary I want to remind you for those who don't know or who know knew but don't remember or remember but just barely what vectors are and what coordinate frames are a coordinate system is just a way of describing space quantitatively and incidentally for our purposes today and largely in general we will of course assume that spaces three-dimensional and so a point of space will be labeled by three coordinates but we're perfectly free to think about systems which are higher dimensional or lower dimensional and we will do so since we're interested in formulating the basic principles we don't have to restrict ourselves to very very specific examples a particle could move in one direct or one dimension it could move in five dimensions and and we will be interested in all the possibilities but for the moment let's just think of particles as things which move in three dimensions so in order to be quantitative about the location of particles we introduce a coordinate system coordinate system Cartesian coordinates will be the usual things we'll introduce later on we'll introduce other ways of describing locations of particles but for the moment we take space we identify an origin the origin is up to us where we want to put it I can put it over here I can put it over here it should be that the important questions that were interested in should not depend on the convention of where we put the origin but it's useful to fix it once and for all and say the origin of coordinates is located in Palo Alto wherever wherever we want to put it here it is then introduced axes the axes are taken to be mutually perpendicular you can check that the perpendicular by by with a t-square or whatever it is that you use to align axes and you label them we can label them XY and Z or X 1 X 2 and X 3 we'll use both kinds of notations but there's also an ambiguity about the orientation of the axis given that they're mutually perpendicular we still have to decide you know I think you know what I mean which directions they go in and so that's like fixing the origin of coordinates we also have to fix an orientation for the X and y axis once we fix the orientation for the x and y axis the third one is fixed it's perpendicular to it incidentally there's a convention and the convention is called a right-handed coordinate system the convention is when you've picked x and y you still need to know one discrete piece of information is Z pointing out of the blackboard or is it pointing into the blackboard and we settle on that by a rule it's a convention it's arbitrary the right-hand rule if we take our thumb and our index finger thumb along X index finger along Y then Z is the middle finger the direction of the middle finger right so that's the right-hand rule and it selects out this coordinate system from the other one where Z goes in the other direction okay so that's the idea of a right-handed coordinate system we also have to mark off distances along here so distances are marked off equal distances with a roller and of course in saying that we mark off distances we're also assuming a set of units the units could be meters it could be inches it could be feet it could be lightyears so again another ambiguity but another convention is to choose our units but once we've chosen I chosen our units we can lay off distances along here and then every point in space can be labeled by the value of XY and Z let's see how to redraw this it has a height Y it has an X X and it has a Z Z if you like you can think about how do you get to this point from the origin you go a certain number of steps along X you go a certain number of steps along Y and then you scroll a certain number of steps along Z and those quantities XY and z are the coordinates of the point so a point is labeled by a set XY and z now I know you all know this but let's spell it out anyway okay so that's the way we describe a point that's the way we describe a particle and of course if we have a system of particles many particles then of course we just have to put in one such point for each particle okay vectors what is a vector a vector is an object which has both length and direction for example a very simple vector is the position of this point relative to the origin is the origin and the position of this point relative to the origin has a magnitude which is the distance from the origin and it has a direction namely just the direction of the of the arrow connecting the point with here think of that vector as an object which has a length in a direction but don't think of it as being located anywheres think of it as being the same no matter where I draw it in space in fact I don't even have to think about drawing it in space it is what it is it's a magnitude and a direction in space that's called a vector and from now on we will label vectors by putting a little bar on top of them if I'm really conscientious I'll put a little arrow on top of that if I forget the top of the arrow or I get bored writing arrows I'll just put a little bar on top and here and there I may even forget to put anything on top but you'll remember you'll remind me okay so a vector has a magnitude the magnitude is its length it does not have to necessarily be a relative position it could be a velocity it could be an acceleration something there are other things it could be an electric field with all sorts of things the criteria for it to be a vector is that it has a length and a direction okay so that's that's the notion of vector every vector can be described or has associated with it a length let's call the length of it in fact for the length of it I don't have to put a vector sign two bars on either side of it the absolute value of it are called its length so this is a way of writing its length okay so that's the length of the vector and the vector it's always positive or zero it could be zero if the vector has no length at all in other words at this point or right at the origin would have no length at all it's always are the positive or zero so every vector has a length and it has a direction which is not so easy to write down okay it can also be described by components a components the components of the vector are exactly what I said before if you wanted to go from the origin to this point over here you would go a certain number of units of X a certain number of units of Y and a certain number of units of Z the other way to define it is to drop a perpendicular from the point to each one of the axes I don't draw this very well let's see I think how are we going to do this I want to get this vector to be out of the blackboard how can I get it to be out of the blackboard but draw it on the blackboard anytime I draw it in the blackboard always looks like it's on the blackboard okay from here to here okay we drop a perpendicular from the tip of the arrow to each one of the axes and that lays off for us a distance along those axes which are the three components of a vector we'll call them V X V Y and VZ when I'm writing the components I have no need to put the arrow on top of them the components themselves are numbers what is the length of this vector whose components are VX v y and VZ ya the square of the length is VX square plus V Y square plus VC squared Pythagoras is theorem except in three dimensions I'll assume it we won't prove it that the length of V the magnitude of V is the square root of the sums of the components that's the notion of the magnitude of vector and the the direction of the vector is encoded in the ratios of the different components for example if the X component is much bigger than the Y component then the vector is pointed more or less along the x axis and so forth the components can be positive or negative if they're negative they're pointing along the negative axis and so forth all right but as I I want to emphasize that now again that the notion of a vector is not necessarily tied to the location of anything in space it is what it is it's a length or magnitude and a direction and if you move it around it doesn't change the vector that's just a mathematical definition of the way you think about vectors you don't think of them as being tied to a point in space you can I mean there were circumstances where you may want to tie a vector to a point of space for example you might want to ask what is the electric field at a particular point of space then that vector is tied to that particular point in space but the notion of the vector transcends that doesn't matter where you put it all right now we have to talk about the algebra of vectors I really feel sorry for all of you people who've sat through this endless number of times before but it's funny there are some things like a book which no matter how good the book is pretty much except for some extremely special cases you really only want to read it once you know you may say you want to read that book you could you love that book so much you could read it endlessly but it's not really true you read it once there are other things like a good piece of music which you want to hear over and over and it doesn't matter how many times you've heard it it just is always good to hear I assume my lectures are like that okay alright so let's talk about the algebra of vectors the algebra of vectors are first of all you can add them you can subtract them you can multiply them by numbers so let's talk about that a little bit given a vector let's call the vector a again it's a length and it's a direction you can multiply it by a number an ordinary number not an integer but a real number let's multiply it by the number see what's the definition of that it depends on whether C is positive or negative if C is positive it's a vector along the same direction except its length is multiplied by C so twice a vector means a vector of exactly the same direction but twice as long and so forth and so on we can also define the negative of a vector in other words we could let C be minus one I can either put here minus 1 or just minus the vector and of course the negative of the vector is exactly the same vector except with a direction in the opposite direction that's the definition of multiplying a vector by a number by an ordinary number now you can also add vectors let's just remind ourselves about the rules for adding vectors there are three ways of thinking about adding vectors in the first way let's say a plus B we take a and we lay it out is there is one end of it tail end of it here's the arrow end of it then we take B and we put the tail of B at the head of a we draw a triangle the triangle has its third leg C which is the sum of a plus B it might be a degenerate triangle it might be that B is along the same direction is a in which case C will also be along the same direction it'll look something like this in which case it's a kind of a degenerated triangle but still we just think of it as a triangle we can add vectors by this role we can multiply by vectors by numbers oh before I do that let's just talk about the two other ways of adding vectors they're trivial in according to this rule it looks like it might depend on which one I laid down first but of course it doesn't you can choose a rule which is symmetric between the two of them if this is a and this is B you put the tails of a and B down together you draw a parallelogram and you draw the diagonal of the parallelogram and that's called C and in this form it's completely manifestly clear that it doesn't matter whether you put a down first or B down first so a plus B is the same as B plus a ok so that's that's vector addition you can multiply vectors by numbers either positive or negative let's call this a we know exactly what that means we can multiply B by a number B B and we can add them we know how to multiply by numbers we know how to add and that means we know how to construct the vector a times vector a plus B times vector B we know exactly what that means okay the third way of adding vectors is to use the components so if we have two vectors with components a X a Y and a Z and B X B Y and B Z then CX is just equal to ax plus BX and likewise C Y and cz we could summarize this all by saying C sub I where I could be X Y or Z or 1 2 or 3 is equal to AI plus bi and that's an equivalent way useful way to add vectors since often we specify the vectors by the by their components all right if we can add vectors and arm subtract vectors and multiply them by numbers the natural question is can we multiply vectors can we divide vectors well now we're not going to divide vectors the idea of vector division is not a well-defined idea but there are vector products there are two kinds of vector products two distinct concepts of multiplying vectors in one of them when we multiply two vectors we don't get another vector we get a number the thing that's called a scalar sometimes the number is just called a scalar so one definition of the product of two vectors is called the dot product let's talk about the dot product of two vectors a dot B a dot B is defined in the following way you take a you take B and now you think of the component of a along the axis of B what does that mean that means you drop a perpendicular from the end of a to the axis not to the x-axis not to the y-axis not to the z-axis but to the axis defined by B so you drop a perpendicular and now you take the length of let's call this here a sub B the component of a along the B axis you take the component of a along the B axis and you multiply it by the component of B along the B axis what's the component of B along the B axis it's the magnitude of B all right so you take the component of a along the B axis you multiply it by the component of B along the B axis alright and that's called the dot product well I'll write it out in a minute more definitely that's called a dot product but the way they defined it it looks like a dot B might not be the same as B dot a II after all the rule for B dot a would beat the drop of perpendicular from B onto the axis defined by a which is a different thing and then multiply them together is it obvious that a dot B and B dot a are the same thing well if we can write them in a manner which is symmetric between the two of them then we'll know they're the same thing okay so let's call this angle here theta the magnitude of a is the length of a the magnitude of B is the length of B what is the component of a along the B axis a sub B is equal to the magnitude of a time's the cosine of the angle between them right let's call it theta a B theta is an angle now and it's the angle between a and B that's the component of a along the B axis we now multiply that by the magnitude of B to get the dot product all right so dot product is a dot B and it's equal to a B cosine theta I won't bother writing theta a B it's a along the component of a along the B axis that's this times the component to be along the B axis which is just a magnitude of B it's a B magnitude of a B cosine of theta now in this form it's completely clear that it's completely symmetric between a and B and it doesn't matter in which order you multiply them together the cosine of the angle is just the cosine of the angle is a dot B always positive and under what circumstances if not under what circumstances would it be negative if the cosine is negative when is the cosine negative the cosine is negative if the angle is bigger than 90 degrees so that would be the situation for example where a was pointing like that then the component of a in the B direction would be negative cosine of theta would be negative for an angle bigger than 90 degrees the cosine is negative what about two perpendicular vectors zero cosine of theta is zero for 90 degrees so a diagnostic for deciding whether two vectors are perpendicular or not is the calculate the dot product now that can be useful and the reason it can be useful is because we can express the dot product in component form if we express the dot product and confirm the component form I won't prove this this is not hard to prove that in terms of the components of the vectors this is equal to ax BX plus a y be y plus AZ BZ in other words you multiply the components in your ad that can be easily proved a little bit of a little bit of trigonometry not much that that this is equal to this over here now that's nice let me show you how you can use that well let's see do we really need that well if I just what to give you the components of a three numbers and the components of B three numbers and asked you to compute the angle between a and B your first reaction might be to throw up your hands and say well I don't know get me a protractor and I'll try to I'll try to measure it but here's a tool now you calculate the dot product between them that gives you a dot B you can also calculate what is the magnitude of a it is the dot product of a with itself let's write eight oh sorry that's not quite right a dot a what is that that's just a square of the magnitude of a so that's a squared likewise for B so you could calculate the magnitude of a or the square of the square of it is ax squared plus say Y square plus AZ squared likewise for the magnitude of B and then you would calculate the angle between them by calculating the dot product of the two of the vectors so from the magnitude of each one of them and the dot product of two of them you would compute the cosine of the angle between them all right so that's a good trick if you want to know the angle between two vectors let's let's prove the law of cosines let's prove the law of cosines there's a simple elementary thing supposing we have two vectors a and B and we want think of it as two sides of a triangle and we want to compute the length of the other side of the triangle so let's call that C C is equal to a plus B how do we compute the length of C that's just C is equal to a plus B squared or a plus B dotted with a plus B the square of the length of a vector is the dot product of a vector with itself so the vector happens to be C this is C dot C D that's the square of the length of C let's get but I'm a chemist a my sorry a minus B a minus B thank you a minus B C is a minus B it's not a plus B a plus B would be over here we could do it that way couldn't we but let's do it that's a minus B and you can work out very quickly that this is a minus B do it yourself okay so it's a minus B squared what is this equal to this is equal to a dot a plus B dot B minus two a dot B right just multiplying them together and this is just the square of a this is a magnitude of a squared plus the magnitude of B squared and this is minus twice the magnitude of a time's the magnitude of B times cosine of the angle between them that's called the law of cosines that the size of the third leg of a triangle is given by the sums of the squares minus twice the product of the lengths times the cosine of the angle between them okay so that's a that's a vector addition volta vector multiplication vector subtraction even vector multiplication of that product will we'll worry about the cross product another time not today all right now one not more important but more interestingly we have not only a algebra of vectors which means adding subtracting and so forth Oh as I said there's no notion of division of two vectors and notice that for the dot product the product of two vectors is not a vector it's a number the cross product is another matter that we'll come to another time okay now let's talk first of all about a particular vector which characterizes the position of a particle here's the position of a particle now think of it as a particle in fact we're going to allow it to move around in a little while and the origin is a particular special point that we've picked out we do have to worry about what happens if we change the origin but we're not going to do that today and therefore the position of the particle defines a vector that vector is usually called little R or I suppose stands for radius and in fact the magnitude of the vector R is the radial distance from the origin to the particle that is true or presumably I don't know where the notation came from I think it was for radius but for now it stands for the position of the particle what are the components of our that just the x y&z of the particle they're just the coordinates of the particle x y&z so R has coordinates XY and Z or let me write it that way R sub X is equal to X the position X and so forth and so on that's the notion the simple notion of the location of a particle described as a vector now in general we are interested in the motion of particles the motion of particles is what classical mechanics is about how the motion goes from one instant to the next in other words how it's updated from one instant to the next and so and should think about R as a function of time in general it moves around however it moves around we'll assume it moves around continuously differentiable smoothly at the same time its components if R is a function of time then R sub X is also a function of time likewise for y sub y of T and Z of T and so the motion of the particle is summarized in this case by three functions of time x y and z what about the velocity of a particle the velocity of a particle is also a vector it has a direction it's not necessarily the direction of the position of the particle from the origin for example the particle might be over here but moving this way moving out of the blackboard in that case its velocity would be this way but its position would be this way and so they're two separate distinct vectors the position could be any vector and the velocity any other vector but how do we describe the velocity the velocity is the time derivative of the position alright I'm not going to spend a lot of time explaining that fact I think you all probably know if not you're probably in a little over your heads but I will assume you know this that the velocity of a particle is the time derivative of the position of the particle so we can write that in a number of ways we can first of all say that the components of the velocity the velocity along the x axis is the derivative with respect to time of the position of the X component of the position or we can write it either as R sub X or we can just write it as DX DT unlike Y's for the other two components of velocity so the velocity is also a three thing with three components it's also a thing with a length the magnitude of the velocity which is called the speed the magnitude of the velocity and it has a direction and the components of velocity are just the derivatives of the components of the position so there's a velocity vector and it's easiest to specify by specifying its components the X well I won't write equals it it has components XY and Z which are the X DT dy DT and the comer and DZ DT that's the notion of the velocity vector and I will frequently in almost always use a notation which many of you know but I will introduce it here anyway differentiating anything with respect to time calculating its rate of change as time proceeds the a traditional notation for it some of you know it some of you don't I'll tell you right now the derivative of anything let's call it any function of time with respect to time is just labeled this is just 1/2 this is just in order to keep from having to write D by DT over and over is just to put a little dot on top of it dot definition of a dot on top of a function is its derivative with respect to time it doesn't mean the general you wouldn't use it for derivatives with respect to space you wouldn't use it for derivatives with respect to anything else dot means derivative with respect to time okay so the we could rewrite this saying that the components of velocity let's write it in one formula V sub I where I could be X Y or Z is equal first of all to the X sub I by DT which is the same as X sub I dot that's the notation for velocity velocities are important let's say I work out an example well know before we work out an example let's talk about acceleration what is acceleration it's the rate at which the velocity is changing acceleration is zero if velocity is not changing whenever the velocity changes this acceleration acceleration is also a vector it's not just 30 miles an hour per hour or 30 miles 3 feet per second per second it's also got a direction thing can accelerate that way it can accelerate that way if an object is moving slowly along the plus so that's your plus x-axis if it's moving that way slowly and then it speeds up you would say the X component of acceleration is positive if it's moving fast and it slows down we would ordinarily call that deceleration but mathematically it's negative acceleration which means that the X component of the acceleration is negative it could be exactly the same acceleration as if you started from rest and accelerated along the neck of x axis in both cases a change of the velocity along the x axis is called acceleration and it's simply the time derivative of the velocity all right so we write acceleration the components of acceleration are the derivatives of the velocity with respect to time or we could write it as V of course VDOT re but that makes it the second derivative second derivative with respect to time of the position the second derivative is usually labeled with two dots on top of it two dots means second derivative one that means first derivative no dots mean don't differentiate it at all three dots means a third dot time derivative and so forth so acceleration is the second time derivative of position with respect left second time derivative position period we can also write this in vector form we could write that the velocity as a vector is equal to the time derivative of the possessor time derivative of the position vector or just this gets a little annoying r dot r with an arrow to indicate it's a vector dot to indicate the time derivative likewise acceleration acceleration is the second derivative R double dot okay now that we have these concepts let's work out an example or to two examples two specific examples of position velocity and acceleration are we going first example motion along the line particle has a position X this is the x-axis a particle is labeled as having a position X of T along one axis it's hardly worth thinking of it as a vector we just think of it as X of T it is a one-dimensional vector but one dimensional vectors are too trivial to even call vectors just X of T and let's as an example let's write down a particular formula for X of T this is X of T which we're going to assume is a constant some number plus B times T plus C times T squared now there's nothing to prevent us from going on but let's suppose that is the formula which tells us where the particle is in any given time in particular at the start let's take the start to be at T equals 0 at T equals 0 the particle is at a let's calculate the velocity and the acceleration you all know how to do this to calculate the velocity we write X dot and that's the first time derivative the derivative of a is 0 this is the constant derivative b times t b and the derivative of c t squared plus 2 CT right alright so this tells us now that the velocity at time 0 is B to start with but then the velocity starts increasing of course it depends on whether C is positive or negative or whether increases one way or the other but whatever it is it starts at B and then as time goes on it either increases or decreases depending on C linearly with time what about the acceleration we just differentiate again what's the derivative of B and the derivative of 2 of 2 C T - C okay so this is the acceleration it's twice C it's constant with time it's constant with time it doesn't change so this is a uniformly accelerated particle okay that's a uniformly accelerated particle that has acceleration to see all right that's the kind of motion you have of an object falling in a gravitational field with a constant acceleration but here it is and here's the the reason why what I want to do before we finish is one more example which is more interesting and it's circular motion motion in a circle this will teach us something interesting and new let's think of a particle moving around in a circle it starts at time T equals 0 along the x axis so here's X and here is y x and y are the components of its position for us now we're going to ignore the third Direction Z plays no role it's a particle moving on a plane it has to coma coordinates x and y and at any given time it's on the circle the angle increases linearly the angle just continues to increase and increase of course when it goes all the ways around the angle goes from 2pi back to zero but we don't have to say zero we can we can keep letting it go wind up higher and higher the angle is that so here's the angle theta and the angle increases with time according to the rule fada is equal to some constant called Omega times T it's not a w it's an Omega a Greek letter Omega how long does it take to go all the ways around the circle well to answer that we just say how long does it take for it to go from theta equals 0 to theta equals 2 pi 2 pi is 360 degrees in radians or working in radians of course so we simply solve the equation 2 pi is equal to Omega times T or 2 pi Omega is the amount of time that it takes to go all the ways around and that's called the period the period of the motion is 2 pi over over Omega and then of course determines what Omega is if we know the period we solve this equation for Omega if we want a particle it moves around in a circle and 1/10 of a second we put in for the period a tenth of a second then we calculate Omega all right so that's what Omega is it's called the angular frequency but let's now consider what are the X components and what are what if the X and the y components of the position of the particle and that's easy oh let's have it moving around on the unit circle for simplicity let's having it move around on the unit circle which means radius equal to 1 okay what is the X component of a position cosine theta right cosine theta where is the y-component sine theta okay so now we know the components of position as functions of time X of T is equal to cosine of Omega T y of T is equal to sine of Omega T and now we can start calculating the velocity and the acceleration and so let's calculate the velocity and the acceleration I assume that you know how to differentiate a function like cosine Omega T so we just have to compute the first time derivative to calculate V sub X and the time derivative of cosine Omega T is what minus Omega times the sine of Omega T okay everybody know that anybody not know that when you're in the notes I suspect this there's a whole firm primer on calculus and one of the things that I think goes discussed is derivatives of trigonometric functions so to differentiate cosine the derivative of cosine is minus the sine but because of the omega here is an Omega here so this much I expect that you will be able to either either you know you recognize it or you'll be able to go home and figure out why this is true what about V sub y Omega times cosine Omega T okay now I have an interesting question for you what's the angle between the velocity and the position the position is over here what's the angle between the position and the and the velocity it's pretty obvious that it's 90 degrees you can see that just by saying well we know the velocity is going to be moving along there but how can we check it we can check it by checking the dot product here's x and y are the components of a unit vector V X V and Y are the components of the velocity vector what's the dot product between the position vector let's call R and the velocity vector V well it's the product of the X component of position times the X component of the velocity plus the product of the Y component of position and Y components of velocity so it's cosine Omega T times minus sine Omega T plus sine Omega T times cosine Omega T they cancel this product here has a minus sign in front of it this product here has no sign both of them have cosine times sine in them so there's a direct calculation of the dot product of position and velocity and it's zero what does it tell you that the dot product of two vectors is zero it tells you that they're perpendicular so without you know intuitively it's obvious that the that the velocity and position are perpendicular to each other but here's the calculation that proves it okay let's go another step and calculate the acceleration the acceleration is another derivative all right so what's the Dare what's the derivative of minus Omega sine Omega T well it's minus Omega times the derivative of the sine which is another factor of Omega and a cosine so this would give us minus Omega squared cosine Omega T and what about a sub y derivative of Omega we first of all have a factor of 2 factors of Omega and the derivative of cosine equal minus sine well let's compare a with the position itself the position is the vector cosine sine the acceleration is minus Omega squared times the same vector okay so apart from let's the Omega squared is important it tells us how fast the velocity is but what's important here apart from the Omega squared is the fact that a lies in the same direction is R itself except not quite it up it lies in the opposite direction there's a minus sign here so that tells us that when the particle is over here its acceleration R is pointing outward the acceleration is pointing inward the acceleration is pointing inward toward the origin we all know that that the acceleration of a particle moving in a circle is toward the origin it's a centrifugal acceleration but here's the mathematics that demonstrates it that the velocity is perpendicular to the variant of them to the position of the particle and the acceleration is parallel but in the opposite direction how about the magnitude the magnitude of the position is one I put it on the unit circle what about the magnitude of the velocity velocity this is Omega right just this the x squared plus V y squared is equal to the square of the magnitude sine squared plus cosine squared equal at the one so we're just left with Omega squared for the for the square the velocity the velocity of this particle is just Omega then that makes sense the larger omega is the faster this thing circles around here and in fact the velocity is just proportional to omega how about the acceleration the magnitude of the acceleration to make a squared okay so the bigger omega is the bigger the velocity but even more so for the acceleration but acceleration opposite to the direction of the position of the particle alright i think i think that's all we wanted to do for today we've covered some material i wanted to go through the simple elements of acceleration velocity circular motion motion on the line and and the next time we'll start to talk about what is what are the initial conditions for a particle what is the configure what is the space of initial conditions and what are the laws of motion what are the things which tell us how the particle moves from one instant of time to the next okay

Channel: Stanford

Views: 1,037,434

Rating: 4.9190006 out of 5

Keywords: science, modern physics, law, motion, time, system, cycle, configuration, conservation, particle, vector, coordinate, component, axis, dot product, magnitude, velocity, position, acceleration, circular

Id: ApUFtLCrU90

Length: 89min 10sec (5350 seconds)

Published: Thu Dec 15 2011