 Consider the simple act of juggling a bunch of pins. This person is throwing objects into the air. Once he releases them, they're subject to the gravitational pull of the earth. Is this projectile motion? Is it the same projectile motion that we studied earlier in the class? And if so, what's the same about it? What's different? This is the motion of an extended object, no longer a simple point moving through space. But is there simplicity in this motion? And if there is, how can we apply these ideas to even more cosmic extended bodies? Consider a pair of black holes, the corpses of once very massive, bright-burning stars now orbiting each other in a death spiral. Can we understand this massive extended system using ideas we can develop in this class? Let's explore the physics of extended bodies. Let's begin to try to build an understanding of how one will handle extended physical bodies or systems of particles that together can be considered one thing in the context of the principles and laws of physics that we have explored so far in the course. The key ideas that we're going to explore in this section of the course are as follows. We're going to learn to transition, at least a little bit, from describing material objects in a very simplistic way to a bit more of a complex prescription. I would hesitate to say a far more complex prescription, but certainly we will begin to see how to go from single dots or particles representing complex systems to other things that still represent complex systems but allow for what will be evolving complexity to be considered. We will describe bodies by using a specific concept known as the center of mass concept. This will give us not only mathematical tools, but a real vocabulary for handling these kinds of systems. And we will consider other means to describe many component systems with an emphasis, of course, on Newton's laws and specifically the second law of motion to see how that law applied to the center of mass of a system of particles or an extended body will give us some insights into the kinds of collective if complicated motions that can occur. Let's begin by revisiting something we did at the very beginning of the course and that was to attempt to describe objects as simply as possible. So way back in the first lecture on motion in one dimension I noted that the world in reality is full of complicated large objects. The typical object like a human body or something like a planet for instance contains something at the level of Avogadro's number worth of atoms. I mean obviously planets are bigger than human bodies and trees probably have more atoms than say a rock, but nonetheless roughly speaking within a couple of orders of magnitude maybe even one order of magnitude so one power of 10. Things typically contain about Avogadro's number worth of atoms at the scale of human sized objects. And to describe things like that baseballs, basketballs, people jumping, all kinds of situations that are in reality quite complicated we have been making approximations. These are meant to be at least reasonable representations of the natural world intended to capture the essentials of the situation if imperfectly. So the example I started with way back in the motion on one dimension was the idea of a light rail train. A light rail train is obviously a very complicated things. It's got wheels that roll, it's got electrical systems, it's got transparent parts to it like windows, it's got opaque parts to it like walls and doors. This whole body can move along a rail system, it's quite massive, it's got a lot of atoms in it. And if I had to describe the motion of something this complicated by considering every single atom I would be unable to solve any problem. But the good news is is that we've seen, both by doing experiments and by doing calculations, that it's possible to approximate these large extended objects by considering as if all of their matter was compressed to a single point and then moved through space. So the reason we get away with this is because we've observed things like in a properly working train, one where all the pieces are actually coherently tied together by chemical bonds for instance, even though it's big and extended, when it moves along a rail all its atoms move in the same direction at once. So we don't need to describe every point. If we pick one point on the train and we call that point the representative point and we then concentrate all the mass of the train at that point and then consider its motions, we can then infer what everything else in the train is going to do. If an atom on the nose of the train moves one meter forward and the whole train is rigid, it's not coming apart at the seams, then an atom on the door in the second car will also move one meter forward. By picking one point that does what the gross body of the object does, I can then sort of say something about what all the atoms, all the points in the body do. So instead of drawing the entire train and trying to describe its motion, we picked a single point, for instance near the center of the train and the center will become better motivated today and we imagine that all of it is then described by that one point. So we reduced something large and complicated like this light rail train to something simple like this green dot with its mass M set to that of the whole train. Now this might work a lot and perhaps even quite well for things like linear motion, even motion in two dimensions, three dimensions, and motion along reasonable paths with a limited range of forces that don't tear the object apart or for instance cause the train to rotate wildly in a direction it wasn't moving, but obviously that's not going to work forever. There are plenty of examples of situations where the motion is complicated enough that you can't just get away with picking any random point on this thing and saying this one random point represents the gross motion of the whole body. Now to really emphasize this, we have been exploring forces and forces have points of application. So for instance, I can do a very simple example where I hold a meter stick balanced on my finger, for instance. In this case gravity is indeed attempting to pull the meter stick down, but it's pulling just as much material down on the left side of my finger as it is on the right and so the meter stick remains relatively well balanced there. It doesn't fall to the right, it doesn't fall to the left. No one side favors the other, but if I slide my finger someplace else so that the normal force exerted by my finger is now no longer at the center of the meter stick, but rather off-center of the meter stick and I let the meter stick go, gravity pulls unevenly on it. It's pulling on more atoms to the left of my finger now to the right and the normal force on my finger does not represent a location where the force of my finger is applied in such a way to grossly resist the pull of gravity. The meter stick will rotate and fall. So we can already see that because we can consider things like forces that can be applied in one place but not other places, we have to think about what it means to have an extended physical body or a system of particles bound together or not, as we'll see. And of course such applications have real physical consequence to the world around us. The world around us is in fact complicated and made of complicated and extended distributions of matter. So it's time we leave the comfort of picking any random point and start thinking a bit more carefully about what the right place to pick on say a train would be or on a meter stick or on a juggling pin or baton or something like that even on a ballet dancer or a basketball player leaping through the air. We need to think about what point we're going to choose that does in fact represent the sort of large-scale things that then happen to that body independent of all the details of the motion of its parts. So in order to do this we have to concentrate on a very specific point which is known as the center of mass. So this concept of representing a rigid body, a rigid body being one whose shape is not changed during motion or by the application of a force, you consider the following meter stick, even though it moved in a complicated way when it was no longer balanced half and half around the center by my finger, nonetheless the meter stick didn't rip itself apart it didn't have carbon atoms being pulled off as it fell. It maintains its shape even as it executes some complicated motion that's a rigid body. On the other hand we could also have something like a system of particles. A system of particles might be bound together by some force, glue for instance or some chemical bond of any kind or they may simply be a bunch of particles that we're describing that are all moving and they're all in the picture together. So consider a game of billiards or pool where a cue ball strikes a bunch of other balls and once the contact between the balls has ended they're all moving but they're not really touching each other anymore. Nonetheless we can still understand the gross behaviors of that larger system of pool balls or particles in this case. So it's not a wholly bad idea it turns out to use a single point to describe these things. We just have to pick the point wisely. We haven't been doing that up till now. Maybe you've been doing it with some intuition about where you think the right point on the body is but we're going to find that point today. So let's consider some extended system that's undergoing a complex motion. So the question of course that we're trying to answer here is can one point on the body be used to represent all of its gross motion. Is there a point on an object for instance that when we toss that object into the air even if it's whirling around spinning tumbling as it flies is there a point on that body whose motion can simply be described by very simple equations of motion without having to worry about the fact that this thing is now rotating in addition to moving in a so-called translational motion the whole of the body moving from one point in space to another to another to another even while the body executes some complicated motion of its parts. And it turns out that the answer is that yes there is such a point and this point is known as the center of mass. The center of mass is that point that can be used to track the motion of a rigid and extended body and its motion can be described as if all the body's mass were concentrated there at a single point. So for instance if you can find the center of mass of a basketball player and track the motion of that player's center of mass even while they jump into the air for instance to dunk a basketball in the hoop you will observe that the center of mass of their body while they stretch their limbs pull their legs up extend their neck in the in the arc of the jump as they prepare to put the ball into the hoop. Nonetheless despite the fact that all those internal parts are moving in complicated ways they are all bounded together by chemical bonds and the center of mass will simply move on a parabola just as if this basketball player were a little ball that had been thrown into the air the concentrating all the mass of a player at their center of mass and considering the motion of the player through space by picking just the center of mass will give you a large scale understanding of what that body is going to do even while all its parts are in motion. So let's pick a simple extended system a so-called two-body system with just two masses M1 and M2. Let's place them in this simplistic example along a single axis and I've drawn here M1 on the left and M2 on the right laying along a horizontal or x-axis. The question we would like to know the mathematical answer to is what is the center of mass where is this point located? I'm going to start with the example that the masses in question M1 and M2 are the same so for instance M1 might be one kilogram and M2 would then be one kilogram so they have the same mass and the physical location of the center of mass in this one-dimensional example is simply given by weighting each of these particles coordinates so x1 for M1 and x2 for M2 by those masses and adding these weighted coordinates together the sum of the weighted coordinates where the weights are determined by these masses gives you the location of the center of mass so you can already get a sense that if one of these masses were in fact bigger than the other it would have more weight in the sum and that weight would make its coordinate a bigger part of the center of mass at the end. We'll see this in a moment. The exact formula for the center of mass is given down here in this simple two body example the x coordinate of the center of mass is given by the weighted sum of x1 and x2 the x coordinates of the two masses. What are the weights? The weights are simply the mass of the object under consideration object one in this case so M1 divided by the total mass in the system in this case M1 plus M2 so this is the weight M1 divided by the sum of M1 and M2 we would then add to that weighted coordinate the weighted second coordinate M2 over the sum of M1 and M2 that weight times x2 so you can see here the weights now there are a few ways we can write this we note that these two equations these two bits of the equations have common denominators and so we can group the numerators together and then have a common denominator in the base of the fraction so for instance we could rewrite this as M1x1 plus M2x2 divided by the common denominator M where M is the sum of M1 and M2 the total mass of the system is denoted by this large capital M so in the case where M1 and M2 are the same number x1 and x2 get added together with equal weight and when you do this you find out that the center of mass lies exactly halfway between M1 and M2 go ahead and try this out on your own make up a coordinate location for M1 make up a coordinate location for M2 set M1 and M2 to be the same masses work out the x location of the center of mass no matter what you pick for x1 and x2 as long as M1 and M2 are equal the x coordinate of the center of mass will always lie halfway between the two masses and in fact the way I drew this graphic was by encoding this formula into the creation of my slides and setting the condition that M1 and M2 were the same the computer automatically calculated the location of this green dot that represents the center of mass and lies exactly halfway between M1 and M2 but let's consider a slightly different case where in fact M2 is bigger than M1 intuitively now we would expect that M2 being a bigger part of the weighted sum of coordinates M2 over the total mass will be a bigger fraction than M1 over the total mass x2 is going to play a bigger role in the location of the center of mass than x1 so this is intuitive if you think about this two particle representation here as standing in for something like a bowling pin a bowling pin has a wide end and it has a narrow end and so you would expect the center of mass to be somewhere closer to the wide end than to the narrow end as a result of this this kind of has a bowling pin shape to it so this simple two particle system could stand in for something that has an uneven distribution of mass in the physical shape something shaped like a for instance a bowling pin in this specific example I have made it so that M2 is double M1 that is to say M1 whatever its value is is half the value of M2 so if M2 is two kilograms and M1 is the original one kilogram mass that we gave it we see that this equation is true M1 is in fact one half of two so we have our intuition M2 contributes more so as a result of that you'd expect the center of mass to be pulled away from the center and closer to M2 and you can plug numbers into these equations and verify that and if you do all this you will find that now the center of mass which when the masses were equal lay half way between the location of M1 and M2 now lies much closer to M2 and in fact as I said before I generated these slides using a implementation of the center of mass calculation in the drawing program and so this in fact represents an accurate location in space given the condition that M2 is double M1 of where the center of mass should be now that's just two masses but this application being can be extended up to as many masses as you like so let's represent the total number of masses by some unknown number n where n is greater than 2 so this could be n equals 3 in which case we have M1 M2 and M3 or n could be Avogadro's number now that's a little insane because doing that by hand is impossible in one human lifetime but the point is that this prescription can be extended up to really any value of n particles in one dimension if we put all the particles on a single line and then we want to find the location of the center of mass on that line the prescription is just as it was for two particles we wait x1 by M1 over the total mass we wait x2 by M2 over the total mass etc all the way up to waiting xn by its mass divided by the total mass this long sum can be represented in a more compact notation using the summation notation of mathematics the sum from an index i whose value is ranged from 1 to n is given by mi over m the weight times xi now again m is itself a sum m is the sum of all the little masses at all their little points so m is the sum also for this index i equals 1 to n so the same number of terms in the sum but this time just of the masses mi now keep in mind that there is a special case if all these little mass pieces are exactly the same and if they are all uniformly distributed along say a line that is equally spaced from one another the distance from x1 to x2 is the same as the distance from x2 to x3 is the same as the distance from xn minus 1 to xn all those displacements between the masses are the same then in that very special case you will find that the center of masses location along this one dimensional axis x is simply one half times the sum of the first coordinate plus the last coordinate the earliest coordinate to the latest coordinate so if we organize this sum from smallest value of x to largest value of x then this is the smallest coordinate and this is the largest coordinate in other words it's halfway along the length of the object if you want to think about just I have an object of a certain length call it capital L I know that the center of mass is at one half L one half the length wherever that is in space that's the point that represents the center of mass that's why it would make it in this case a literal center but this is a special case if the mass bits are all the same but they're unevenly distributed in space more of them clump over on the right side of the object than the left again kind of like a bowling pin there's more mass clumping over on one end than on the other then the center of mass won't actually be in the physical center along the length of the object it'll be located closer to where more of the mass is clumped now we don't have to limit ourselves to one dimension we can do this in three dimensions and the good news is is that all this does is multiply your problem by a factor of three now you have to locate the coordinate in the horizontal axis using the technique I just outlined but you can repeat this for the y coordinates and the z coordinates of all those little bits of matter m1 m2 etc it just grows by a factor of three not a big deal so if you have three particles and you have to locate the the the center of mass in in two dimensions you have to do one sum for x and one sum for y and you're done you have the x and y coordinate of the center of mass in three dimensions you also need to repeat that for z so for a body made for many pieces I mean even like a hundred pieces actually even ten pieces these sums are possible to calculate by hand but I wouldn't recommend it of course everything is more efficiently calculated by using repetitive so-called looping techniques within a computational system so for instance using a language like Python or C++ or Java heck even JavaScript in a web browser you could write up a very quick calculation for the center of mass for n particles where maybe n is only a few hundred thousands or a few millions or so and just do this using a computer much better use of your time right a short program that'll maybe take you an hour but it will save you human years of actual on paper calculation work so that's totally worth it computers are intended to do repetitive tasks like this to help you save time why waste that resource now even computers have their limitations so for instance if we have a truly continuous distribution of matter and we're talking about matter matter made from atoms where there's avagadro's number of atoms you really don't have enough computer time to do avagadro's numbers worth of terms and a sum in a computer so it's much better at that point to bust out calculus and instead model the system of being made from a large value and going to infinity avagadro's numbers pretty big so we can approximate that by infinity but where each of those pieces of matter is nearly infinitesimal in mass itself so for instance and we'll go to infinity m will be replaced by dm which means a tiny bit of mass and that is being sent down to zero in size and of course in that case this is beginning to sound a lot like a sum and the limit of the pieces going to zero size while the number of pieces goes to infinity that sounds a lot like an integral and in fact it is an integral and the way we relate position and mass is through the density of the material density tells us the mass per unit something length area or volume so we have the linear density the area density and the volume density regardless of which one you have you can use that to relate position and mass to each other that's what the job of the density function does is it relates position or location in space or volume or area to mass so for instance let's consider a simple one-dimensional example of this where we're given what's known as the linear mass density this is often denoted by the symbol lambda it's a Greek letter lowercase lambda and that's shown here lambda could be a function of x it could be that there's more mass at small values of position than at large values of position so the linear mass density sort of mass per unit length is changing as a function of length the average linear mass density would be the whole mass divided by the whole length but lambda itself could be a function of x now l is the length of the object and m is its total mass so in this case our sum goes from being a sum to being an integral the location for instance in x of the center of mass is the integral over all the little bits of mass dm of this function x over m now it's typical in a class like this and in general fairly early in physics and engineering to tend to deal with objects of so-called uniform density so for instance in one dimension what that means is that lambda is a constant the amount of matter per unit length is not changing regardless of how much length you look at if I look at the mass in one atom in that material and I divide by the length of that atom I'll get a certain value for lambda if I instead take the whole length of the material like a meter of it and I take the mass of that material and I divide by that length I'll get the same value of lambda that is lambda is a constant independent of the size scale that you're looking at and in that case the density of any piece is the same as the density of the whole and I'm writing this out explicitly now giving you an example lambda would be defined as mass over length so the whole mass divided by the whole length but that in this case is exactly equal to any little piece of the mass divided by any little piece of the length that it occupies dm over dx so this allows us to write a relationship between little bits of mass and little bits of coordinate space dm is equal to lambda dx and you get that just by rearranging the left side of this equation and the right side of this equation and in that case we can do some substituting so for instance now our integral for the center of mass of x over m dm becomes x over m lambda dx we've substituted for dm here lambda is a constant m is a constant and so we can pull that ratio lambda over m out in front of the whole integral and then we just have to worry about integrating x dx and then you'll notice that lambda divided by m which you can get from over here lambda divided by m is just equal to 1 over l so we can rewrite this one more time is 1 over the length times the integral of x dx and so this is how you can exercise the integral calculus concept when handling extremely large numbers of particles all comprising an extended body of say some length l with some mass big m in the context of the center of mass we can revisit Newton's second law but now thinking about multi particle systems systems where there really are more than one piece that are all moving they may be all moving together as one they might be a rigid body or they may just be a loosely affiliated system of particles that we wish to describe sort of grossly so we now have a language and a methodology for pinpointing the place in a multi body system and or an extended rigid body that represents the the bulk positional information about the system where is the mass concentrating as if it were a single particle and so with that in mind we can revisit Newton's second law which relates the forces acting on a body to the acceleration of the body and specifically we can think about how a force acting on such a body or many forces added together for instance gravity acting on a falling juggling club can be converted into overall effects like the acceleration of specifically the center of mass and so what will be true is that the sum of all the external forces acting on this body of mass m will yield an acceleration of the center of mass that's completely consistent with Newton's second law note that I emphasize the word external here all these forces that I'm adding together to get f vector net they are only forces external to the body so for instance internal forces anything that holds the particles together or was present in the system at the beginning but which can be converted into say other forms of motion and I'll come to an example of that in a moment are not included here their internal to the system they may cause the parts of the system to change their orientation but it doesn't affect what the center of mass is going to do under the influence of external forces the m here is defined as before it's the sum of all the little bits of mass that make up this extended body that's the total mass of the body and this a vector co m is the acceleration of specifically the center of mass point of the system so we can think about this a bit more deeply by considering some very interesting examples of the motion of the center of mass of a complex system so one example that comes up immediately and I I've shown a movie of this before and I mentioned it earlier in this lecture is a game of pool or billiards so depending on kind of where you're from you might have seen a variation in this game before you have a special ball the cue ball you're allowed to strike that ball to hit other balls on the table which is sort of the game board for this whole game you can never directly strike any other ball on the table but then you can use the cue ball to try to achieve your aim which is usually to get the other balls into pockets in the table so consider a game of pool where we initially get the cue ball moving by applying an external force to it we strike it with a cue so this is a stick that we use to hit the cue ball once we're done striking the cue ball it's now free to move and in principle if we ignore friction and drag then it's got no external forces now acting on it but it can strike other balls on the table and cause them to move by giving up some of its kinetic energy to provide kinetic energy for the other balls on the table so what we're going to see now though from this f net equals m a center of mass equation so Newton's second law but now for net forces acting on a system of objects the pool balls in this case is that once once the cue ball is moving and it has no more external forces acting on it it means that for this whole system of pool balls all of them including the the cue ball there are no more external forces acting no one's striking physically any of those pool balls on the table so it must be true that the net force on all of these is zero and so from that from Newton's second law we would conclude that there are no accelerations of the center of mass of this system of pool balls that is to say that the center of mass the system will now simply continue to move at constant speed with unchanging direction so despite the fact that the the cue ball strikes a whole bunch of other balls and transfers kinetic energy to them and then they all fly off in various directions the center of mass of this system will always at all points during the motion no matter how complex internally of the system move at constant velocity with unchanging direction now of course in reality friction and drag will slow the pool balls down and some of the pool balls will strike the walls of the pool table and that will begin to alter their direction of motion the pool table itself will offer external forces that can change this but after that initial contact between the cue ball and all the other balls on the table the center of mass of all of the pool balls will continue to move along a predictable trajectory even though the balls themselves have seemingly very complicated motions related to this is the concept of an explosion so for instance it's possible to create objects that store chemical energy that's internal energy I can make for instance fireworks these are explosive shells that can be launched into the air and then they blow up okay so they can have a fuse that times the explosion or they can be triggered externally but the idea is that they have this stored chemical energy and then they can release it and when they do this they blow apart the shell that the chemical was stored in and of course fireworks are quite pretty explosives are less pretty because they're usually intended to hurt other people but they're both examples of objects that store chemical energy with the intention of turning it into kinetic energy of all the parts of the thing later on now if the container itself is moving again launching fire a firework rocket up into the air we will learn from this that although that once rigid object may be blown into many pieces that chemical energy is internal energy and so thinking about fireworks for a second when you fire a fireworks shell up into the sky it's essentially once launched in projectile motion gravity is trying to accelerate it down toward the surface of the earth and it will follow a parabolic trajectory and when it blows up although it will scatter its pieces into many directions in the sky because there are no external net forces acting on this besides gravity the center of mass will merely continue to accelerate as if under the influence of gravity at a single point so the center of mass of a firework for instance even once exploded will continue to accelerate along a parabolic trajectory the same trajectory that the original firework shell was traveling along before it exploded this information is extremely crucial to understanding complex systems of motion and and figuring out grossly what's going to happen so this is why for instance it's possible to do forensic analysis of the breakup of say a plane or a rocket that has an accident in the sky if we know the original trajectory of that object before it broke up and we know that once it loses control that the only force acting on it is something like gravity drag because air is a real force in reality we can use a computer simulation knowing that the object was rigid at the beginning but then blew apart and that was in free you know free to fall basically under the influence of drag and gravity but no other forces we can track down to the ground along x along the the ground where the parts of that now exploded rocket or aircraft should be found so this is an essential ingredient in artillery calculations if you work in the military and you're tasked with directing arms at an enemy arms that explode in midair nonetheless this gives you the ability to predict the trajectory even of the pieces on the ground now that's a bit scary that's not something that I like to think about rather I like to think about the good that one can do in the case of say an airline tragedy or the re-entry of a spacecraft from outer space that's carrying astronauts but that suffers some catastrophic accident and breaks up in the atmosphere this gives us as humans the ability to do some good to find the wreckage to figure out where the largest pieces should go because the largest pieces should track closer to the center of mass of the object using the reasoning from before so you see that this concept although we still haven't really dealt with the full complexity of a system the center of mass gives us the ability to say okay I know at least where the center of mass of the system should have gone and I know for instance that the largest pieces will be closer to the center of mass usually than the smallest pieces so this allows me to find maybe specific parts of a craft after it breaks up or certainly at least to find the center of the collision on the ground where the all the pieces should have landed in a scatter pattern around that location that gives search team search crews the ability to focus where they're going to search for debris in order to analyze the wreckage and figure out what happened to do some detective work on the accident itself so you can see that this concept is all kinds of applications in the real world and although we still are dealing with a complicated system we have an anchor point for the system that can be used to then extend our knowledge of the overall behavior of the system thinking from the center of mass out to the pieces let's review the key ideas that we have seen in this section of the course we've seen how to begin to transition from describing material objects in an overly simplistic way to a more complex prescription but nonetheless we're doing so by collapsing these bodies to a well-defined point the center of mass and then we have been looking at how the center of mass perhaps under a revisited Newton's second law gives us the ability to describe grossly the behavior of a many component system at least allowing us to figure out where the center of mass of the system would be going under Newton's second law so that we can then expand outward and figure out where all the pieces of the system may have landed in relation to the center of mass so we have begun a small step into more complex systems I've given you a taste of where computation can play a role in all of this taking large sums of masses and coordinates to find center of masses and I've given you a sense of how calculus can be used to handle the most complicated situations where it's just not possible to physically do the addition of all the weighted coordinates to get the center of mass you've got to use density information the relationship between mass and space in order to figure out something about the center of mass of a very large number of pieces complex object so this is just a taste of complexity you can see how this can scale more now from this point but we will begin to use these ideas to explore another fundamental concept the last major concept that we're going to encounter in this course and that is momentum