 Having taken a shallow look at some of the various forces that manifest in the macroscopic world as a result of deeper things like the behavior of atoms, we're now going to take a slightly deeper dive into some of those forces themselves and explore new ones that are profound, common, and interesting in the world around us. The key ideas that we're going to explore in this section of the course are as follows. We're going to learn the empirical laws that govern the friction force, which we have briefly explored earlier, and we're going to learn something about the origins of those empirical laws who discovered them, the order in which they were discovered, and something a little bit deeper about the nature of where those laws emerge from. We're going to come to understand the mathematical description of friction, which despite the seeming complexities and perhaps even some of the counter-intuitive behaviors of the friction force, nonetheless has a fairly simple mathematical description. We're going to come to understand the much more complex origins of the drag force, which we have neglected in many considerations up until now. This also goes, for instance, by the term air resistance. It has a mathematical foundation, and we're going to explore briefly the details of its mathematical description. And finally, we're going to revisit uniform circular motion, and we're going to understand the force that is associated with its center-seeking acceleration that we have encountered previously in the course. Let's begin by looking a bit more closely at the friction force, which despite its seemingly approachable nature actually has some fairly profound behaviors whose origins really only have been illuminated thanks to revolutions in technology that began late in the 20th century. Let me begin with a brief and as is necessitated by a short format video like this, Incomplete History of Friction as a Study of the Natural World. In fact, perhaps the earliest recognition that there were empirical rules that governed the behavior of friction, that observation was made by Leonardo da Vinci, who's famous for many things. He's famous for his incredible works of art. He was famous for his engineering designs and his study of physiology, and for his work as a scientist into trying to understand the natural world. And in fact, it's widely recognized now that through his engineering work, which he came to understand deeply connected in its function to the way in which surfaces interact with one another, it's understood that he discovered the empirical laws of friction and even articulated them in his writings. But those particular writings were not commonly read, and it was a long time before they were rediscovered and understood to have contained what we would now think of as the modern empirical understanding of friction. When I say empirical laws, I mean laws that are gleaned by observation, but whose fundamental nature are not understood, at least not right away. They can be written down, they can be shown to hold observationally in the natural world, so they seem like reliable rules, laws, but their reason for being laws is not necessarily fully understood. We have a much better understanding now of these laws and we'll see them in a moment. Da Vinci is perhaps one of the first people who discovered them and wrote them down in a rigorous way. Most of the credit in modern writing about friction goes to this pair of individuals, Guillaume and Montan, who lived from 1663 to 1705, so almost a century after, a little more than a century after Da Vinci. He independently rediscovered the empirical laws of friction. Again, we'll meet these laws in a moment. The second of those laws was however enhanced by the work of another scientist, Charles Augustin de Coulomb, who lived from 1736 to 1806. Now, Coulomb is if not already a well-known name to you, a name you're going to come to know very quickly in the second semester of introductory physics when you study electricity and magnetism. One of the fundamental laws governing electricity and magnetism in nature bears his name. So he is renowned for his work in multiple fields and the study of surfaces interacting with other surfaces also happens to be one of those areas where he made some breakthroughs. Again, it's impossible to highlight the contributions of all of the people who have studied surface interactions and materials, but in some ways the reasons behind the empirical laws of friction to really understand why these rules hold in the natural world observationally required a 20th century explanation of the material world in order to make sense. In the 20th century physics, which could not have been accessed by Coulomb and Montaigne's da Vinci, they lived before these things were discovered, this was really needed in order to begin to make progress and understanding the origins and nature of the laws themselves for friction. So some of the insights were provided for instance by Frank Philip Bowden, pictured on the left, and David Tabor, pictured on the right, up here. Their work was done in 1950 and they discovered, for instance, that the actual amount of contact as opposed to the apparent amount of contact that you can see with your eyes between two surfaces, the real contact, which is very difficult to see without very precise instrumentation, is in fact likely very small and that is why, for instance, one of the laws appears to be independent of the apparent area of contact for two surfaces and we'll come and see that in a moment. The empirical laws of macroscopic friction, the friction that we experience in the material world at the scale of human technology, really only makes sense in light of the microscopic reality of nature which simply couldn't be probed until the end of the 1800s and the beginning of the 1900s with both the physics insights and the technological revolutions that occurred in parallel with those insights. So let's take a look at Emonton's and Coulomb's empirical laws of friction. These laws, as I have said, are so-called because they hold true observationally but their fundamental nature is connected to deeper and more fundamental laws of nature. For instance, quantum mechanics, the study of the behavior of the tiniest bits of matter in the universe. This is something that you encounter in third semester physics and beyond. You really need to have a foundation in that study of nature before you can really begin to appreciate the reasons why friction behaves the way that it does. But also electromagnetism, after all, as I've emphasized in the previous video lecture, is electromagnetism that governs the forces between and within the atom and until you can appreciate the course and specific details of electromagnetism and how those details interact with quantum mechanics, you really can't begin to make serious progress on understanding materials. The most fundamental behaviors of the bits of matter that make up materials ultimately govern the macroscopic properties that we observe in the world around us and if we want to make and control new materials, we have to understand these things. So the empirical laws are as follows. The first is what's known as Emonton's first law. The force of friction is directly proportional to the applied load. Now what do I mean by that? If you drag a material across a surface, you will notice a certain amount of resistance to motion. This is the friction force. It points in the direction opposite the motion of an object. So if you were to slide a material along a surface, you would measure a certain force of resistance due to the friction between the object and say the tabletop on which it's sliding. If you increase the load on that material, for instance by piling weight on top of it and then drag it across the surface again, if you double the weight, you double the friction force. That's the observational reality of friction. If you double the load between the surfaces, then you observe that the friction force between those surfaces also doubles. The force of friction is directly proportional to the applied load. That's the first law. The second law, which is also known as the Emonton's Coulomb law, states something that's a bit more counterintuitive. You could almost convince yourself that it makes sense that if you double the mass of an object and carefully study how the friction force behaves when you have the original mass and you drag it across the surface, when you double the mass and drag it across the surface, you could convince yourself hand-wavingly that that makes intuitive sense. The second law does not make intuitive sense. The second law states the force of friction is independent to the apparent area of contact. That is, if I keep the mass of an object the same, but double the amount of area that I, with my eye, appear to see in contact between the two surfaces, I do not double the force of friction. Now that seems counterintuitive. If I make more surface come in contact, shouldn't I expect friction to go up? And the empirical second law of friction says no. And the reason for that will come back to, it's quite counterintuitive at the macroscopic level, but it's because your lying eyes are deceiving you about what's really going on in the space where those two surfaces are meeting. And I touched on that a little bit in the previous lecture, but I'll return to that point in a bit. In order to see this second law of friction in action, let's do a simple experiment. I can take a brick made of iron, which has a mass of about 4 to 5 kilograms. This brick is 20 centimeters long, it is 10 centimeters wide, and it is 5 centimeters deep. It has a narrow long side that is 5 centimeters by 20 centimeters, or 100 centimeters square in area. And it has a broad side, a wider side, that is 10 centimeters by 20 centimeters, or 200 centimeters squared in area. What will I measure for the friction force if I lay it on its narrow long side or on its broad side? That's the question that the second law of friction attempts to answer, and it tells us that we should observe no significant difference, certainly not a doubling in the force of friction, due to doubling the contact area. Let's do the experiment. I have the brick shown here. It has a very low mass string tied around it, barely affects the overall mass of the brick. On the end of the string, I attach a force measuring device, essentially a scale, and it will tell us the force in Newton's that the, basically my hand is experiencing attempting to slide the brick along the surface. So let's do the first experiment. Let's take this long narrow side, which is 100 centimeters squared in area, and let's slide it along the surface and measure the force. We observe that the force is something like 13 to 14 Newtons. It's something in that range. This is the force of resistance that the brick feels when it's being moved along this surface. If I now flip the brick over on its wide side, which is 10 centimeters by 20 centimeters, or 200 centimeters squared, double the contact area that we had before, I can repeat the experiment, pulling the brick along the surface. Let's see what the scale observes. What we see is that there's really not much difference in the force that's measured this time. Certainly, the force has not doubled from the first experiment, which had half the area in contact, apparently, to the second experiment, which had the larger area in contact, twice the first experiment's area. We didn't double the force. It might have changed a little. That could be within the noise of this kind of very basic experiment. But indeed, what you can see from this is that even though we have doubled the apparent area of contact of this iron brick with the surface on which it's sliding, we have not doubled the friction force. It's barely moved, maybe by no more than half a Newton. Finally, there is the third law, Coulomb's law of friction. And this states that kinetic friction, the friction that is specifically associated with surfaces that are in relative motion, sliding over one another, for instance, is independent of the velocity at which the body slides. So if I take an object, and I slide it over a surface, and I double the speed at which I slide it over the surface, I do not double the force of friction. These are the observations of the behavior of friction as a force. It is directly proportional to the applied load. Double the weight, double the friction. It is independent to the apparent area of contact. Double the area of contact. You do not change the force of friction. And finally, doubling the speed of an object does not double the amount of friction force experienced by that object. It remains independent of the velocity at which the body slides. By material properties of the two surfaces in contact with each other, independent of the relative motion, or the amount of apparent, that is, area of contact I can see with my eyes, the apparent area of contact. Now, I've hinted at this in the previous slide, but there are two kinds of friction that you need to think about. A close look at the empirical law suggests that both of these exist. One of them is the friction that's associated with an unmoving body. If I have a body on a surface at rest and the surfaces are not in relative motion to each other, then they are experiencing what is known as static friction. And the way you manifest the static friction is you attempt to pull on one of the objects and slide it over the other. And as you know from your own experience, if you have a relatively difficult to slide material, like rubber, resting on a surface where it makes really good sticky contact, and you attempt to pull that object to the side, you have to apply a lot of force to get the rubber to slide. And that's because rubber has an extremely high, what's known as coefficient of static friction. We'll come back to that concept in a bit, but basically this quantifies in a single number the amount of resistance to sliding that this material possesses when in contact with the other surface. So, static friction is that friction force that holds a body at rest until you apply so much force to exceed some threshold, and then that body begins to slide over the other surface. Now once you have two materials in relative motion to each other, their surfaces are in contact, one is sliding over the other, the friction associated with the motion is known as kinetic friction. Again, this is the friction force that acts to oppose the motion once the body has started moving. So static friction is a kind of threshold friction that you have to overcome and once it starts moving, it experiences kinetic friction. Let's take the case of static friction and think about it in light of the first two empirical laws of friction. This will allow us to come to a mathematical relationship that describes the behavior of static friction. If you have an object that is resting on a surface and you push on that object and it's making physical contact with the surface on which it rests and you observe that despite your pushing it remains at rest, then it is static friction that is opposing the push. By Newton's third law of motion, the force of your push is equal to the negative of the force of static friction. They oppose each other equally but in opposite directions and the object remains unaccelerated. Now, for any force that is below a certain threshold, one that is specific to the materials that are in contact, as you'll see in a bit, the friction will always remain zero. But, if you cross a threshold and the threshold of static friction is given by this equation here, this static friction threshold is equal to a number, mu, the Greek letter mu with a subscript s for static, times n vector, the normal force. If you exceed that threshold, which is specific to the two materials in contact with each other, and of course the mechanical load that is pressing one surface onto the other, if you can exceed that threshold with a push, then motion is possible. The object will transition from a static state, that is, acceleration is 0 meters per second squared, to a kinetic state, that is potentially non-zero acceleration, certainly one creating non-zero velocity. Up until that point, up until that threshold, contact between the surfaces prevents motion, but you can supply enough of a push to overcome that threshold and begin motion, to go from static friction to kinetic friction. Note that there is only a dependence on the normal force in this equation. This is an equation that describes observationally what we observe about static friction in the wild, in the world around us. It only depends on the mechanical load. The area of contact does not appear in this equation. Mu s is a constant, we'll come back to that in a bit, but it is a number that characterizes the summary of all interactions going on between these two surfaces. Essentially, from our perspective, appear at the macroscopic scale, the human scale, due to all these atomic interactions down at the atomic level between the surfaces. Friction, in general, appears as independent of the apparent area of contact, A. Now, if weight, for instance, is the source of the normal force, as opposed to something else, then all that matters is mass and or gravitational acceleration, not area. So, we know that if the normal force is supplied by weight, weight is equal to mass times gravitational acceleration, then all that matters is the mass and the gravitational acceleration. That's it. If I additionally pile on other forces that create an increased normal force between the surfaces, then there are other ways, of course, I can get the normal force to go up, but if it's just gravity that's doing all the work, then it's weight that is the source of the normal force and provides the mechanical load. The quantity mu with a subscript s, or mu sub s, or mu s, however you want to say it, is known as the coefficient of static friction, and it is a number associated with the contact between the two surfaces. So, for instance, the coefficient of static friction between aluminum and steel is different than the coefficient of static friction between aluminum and rubber. The two materials together define the coefficient, and so you will often need to think carefully when you are setting up problems involving friction, static, or kinetic. What are the materials? What materials are in contact? And for each point of contact, I'm going to have to figure out what mu s, or as you'll see in a moment, mu k, the kinetic coefficient of friction is for my problem. Speaking of kinetic friction, the first two empirical laws apply equally to the friction force experienced during motion. That is kinetic friction. Once the object starts moving, you've overcome static friction. It has entered a state of motion. Now kinetic friction is what matters, and once you've overcome that, then what kicks in is the coefficient of kinetic friction, which is written as a mu, again the Greek letter mu, with a subscript k. Now it is observed in the natural world, quite generally, that the coefficient of kinetic friction is less than or equal to the coefficient of static friction. That is, once you overcome static friction, the force required to keep the object in motion is generally less than the force that was required to overcome the static friction threshold in the first place. This is an interesting fact about the natural world. One that, again, requires a 20th or 21st century understanding of materials to really begin to understand at the deepest level, and it's those deep levels that have the big effects. The equation for kinetic friction force is very similar to that that gives us the threshold for overcoming the static friction force. F vector sub k is equal to the coefficient of kinetic friction times the normal force. That is the mechanical load that is compressing the two surfaces together. Now, of course, in addition to the first two empirical laws, because we're now dealing with kinetic friction, the third law is now in effect. And note that the force of kinetic friction between two bodies, two surfaces does not appear to depend on the relative speed of the two surfaces. The relative speed of one surface with respect to the other speed appears nowhere in this equation. It's just this coefficient, this number that describes the interaction between the two surfaces, times the normal force, the mechanical load. That's it. So this manifestly contains the third law as well. But how can all of this be? How can it be that the friction force doesn't depend on the apparent contact area, that I perceive with my eyes, you know, 1 meter squared, 1.5 meter squared, 10 centimeter squared, whatever it is, I perceive that if I double it, I don't see the friction force increase. How can that possibly be? And how can it be that the speed of motion doesn't have an impact on this either? And it turns out that if you really, really, really want to dig into this and understand the answers to these questions, you need to at least understand quantum mechanics. I warned you about that before, that's the theory of the very small. And thermodynamics, the study of heat energy or its more general form, which is statistical mechanics, statistical mechanics is the way in which the microscopic states of matter manifest into the macroscopic states of matter. So it really is about numbers of things and how you can arrange them, and really it's the foundation of our understanding of materials. However, you can just glean some insights into these laws from the following sort of hand-waving arguments about these things. I mentioned Bowden and Tabor before. It was in 1950 that they basically culminated their understanding of what they thought was going on with the second law, the second empirical law of friction. They came to understand that the reason for the Amonton's Coulomb law is because that while the apparent area of contact, the area you perceive can be large, if you look at the atomic scale, what's actually touching between the two surfaces at the atomic scale, it turns out that very little of those apparent areas are in physical contact with each other. Doubling the apparent area only doubles the amount of an actual very small amount of physical contact. Pressing harder on those two surfaces however, that is increasing the normal force, the mechanical load that compresses the two surfaces together, that much more quickly puts more of those two surfaces within physical contact with each other, and it winds up being a far larger and much more overwhelming effect. So if you link back to the picture of one surface being like a mountain range and another surface being like an inverted mountain range and they're kind of slipping and sticking over each other as the mountains fall into the valleys between the mountains on the other surface, you can begin to see how if it's only the mountain tops that touch the surface on the other side, pressing harder is going to slot more of those mountains and valleys into each other, but doubling the amount of areas is just going to put a bunch more tiny little mountain tops in contact with the other surface and it really isn't a noticeable effect, you don't really notice it. Doubling the area, doubling the apparent area of contact doesn't really double the physical contact as a big factor. It's a small factor in this effect. Now the independence of the kinetic friction force with velocity, doubling the velocity doesn't double the amount of kinetic friction. That has to do with the nature of friction as a force of impact and vibration and thus it's fundamentally about vibrating atoms around where they're sitting in the material and that is what we call heat at the macroscopic scale. When you heat something up you're making its atoms vibrate faster, when you cool something down you're making them vibrate slower. So fundamentally, heating is about vibrating atoms and again this all boils down to the behavior of the microscopic states of materials. Because of the way that new surface kind of appears at the front leading edge of the point of the area of contact, while at the back of the object you're kind of losing contact with the other surface, velocity it turns out has very little to no impact on this friction force, so it does result in increased heating and in fact if you do careful measurements of friction forces you'll find that the front of an object heats up much more than the back of an object. So there are effects that increase with velocity like heating, but the apparent kinetic friction force itself doesn't double when you double the speed of motion between two surfaces. So that's sort of the origins of these seemingly counter-intuitive features, doubling the apparent area doesn't double the friction force because very little of those two surfaces are actually making contact. It's the pressing together of those surfaces under mechanical load that really increases rapidly the amount of contact between the surfaces. Sliding one object over another doesn't double the friction force, but that's because of the nature of the way that energy is being transferred into the material and transferred out of the material and it doesn't double with speed. It double the speed does not double the kinetic friction. So you can see why these are empirical laws. They themselves don't have an obvious fundamental underpinning. Just observing them doesn't tell you what's going on. You have to dig deeper into nature to find out why friction behaves the way that it does, because it seems quite counter-intuitive at the macroscopic scale, but viewed from the perspective of studies like quantum mechanics, material science, thermodynamics, and statistical mechanics, which you would learn in much more advanced courses than this one, it makes much more sense. Now, how do you handle friction? Let's build a strategy. Let's build a toolkit for evaluating a situation where more than one surface is present and there's physical contact between at least one pair of surfaces. So the first thing you should always ask yourself when surfaces are available is, are any of them making contact? And if so, friction likely plays a role in the behavior of these surfaces as they move over each other, or as they don't move over each other, for instance, static friction keeps them from sliding. Are the surfaces in relative motion? Is there an acceleration? Or not? Is there a velocity? Or not? Is one surface moving against the other? If so, kinetic friction is in play. If not, static friction is in play. If no motion of the surfaces is present, but the surfaces still make that physical contact you look for in the first bullet, then, of course, as I said, static friction is what's happening. And if no motion is present, but an external force is then applied in a way to try to slide one surface over the other that is parallel to the contact area between the surfaces, then ask yourself, is that force less than F sub S, the threshold at which static friction can be overcome? If the force is less than F sub S, then there's no relative motion of the surfaces. Static friction wins. But if the force you're applying is equal to or exceeds the threshold for static friction to be overcome, motion can occur. And now kinetic friction takes over. And finally, you always want to consider what materials comprise those two surfaces. You need to think about what kinetic or static coefficients of friction you need to look up in order to apply to solve the problem. And again, you're just going to look this up to solve the problem. We've been talking about coefficients of friction, but we haven't really looked at what these numbers are. How big are they? How do they vary between the static friction case and the kinetic friction case? So here I provided just a small table of some fairly common materials that are often in contact with each other in the real world. So for instance, aluminum and steel may come into physical contact surface to surface in industrial applications. Robber in concrete. Rubber makes up bike tires. Concrete makes up bike paths, sidewalks, sometimes streets. And so rubber and concrete are often in contact in the real world. Wood in concrete and construction and home construction and things like that. Wood and metal. So there's another case in maybe a home construction or an industrial setting where you have both wood as part of the construction and metal as part of the construction. There's the opportunity for surfaces to be in contact. They can slide over each other. So what do these contacts look like from the perspective of coefficients of friction? We saw from the equations of friction that if the coefficient of friction is a big number, the friction force gets bigger. And if the coefficient of friction is a small number, the friction force gets smaller. The friction force is proportional to the normal force and the coefficient of friction. If you keep the normal force the same, but you reduce the coefficient of friction, you reduce the friction force. So let's take a look at a couple of these examples. I won't go through all of them, but just as kind of a fun example let's take a look at concrete and rubber. Car tires are made of rubber. Roads are made of concrete. Bike tires are made of rubber. Bike pads are made of concrete. So if the concrete and the rubber are totally dry, the coefficient of friction is about one. Now on the other hand, if the surface of the concrete or if the tire surface is wet from an oil slick or rain or a combination of those two things, then the coefficient of static friction drops from one to 0.3. It drops by 70%. You can begin to see here why it is very dangerous to drive the same way on a wet surface as you would otherwise drive, perhaps recklessly, on a dry surface. On a dry surface static friction, which would keep you from sliding, is big. But on a wet surface for the same materials, the static friction coefficient drops by 70%. Suddenly the risk of sliding goes way up. Suddenly you can begin to understand motion and the risks of motion in the natural world very quickly from the perspective of friction. It is the friction between rubber tires and the road surface that keeps the car from sliding if you lock the wheels while braking very hard. If you have an excellent coefficient of friction, then you will just stop. However, if your coefficient of static friction is low, it is easily overcome and you will begin to slide and you can slide into the car in front of you if you break too suddenly on a wet surface. Notice what I said before about the kinetic coefficient of friction versus the static coefficient of friction. Notice for rubber and concrete that the kinetic friction for either dry or wet is typically slightly less than the dry. There are ranges here because it depends on what the wetting of the surface is and what happens. Generally you can see here that this pattern that the kinetic coefficient of friction is lower than the static coefficient of friction holds. The study of two surfaces passing over each other is a very broad subject. You can find lots of engineering tables and books on the internet that will provide you the coefficients of static and kinetic friction for pairs of materials. The study of two surfaces passing over each other while in physical contact is known as tribology and you will learn about a kind of friction induced effect when you study electricity. Later in introductory physics known as tribal electricity or the tribal electric effect you've probably experienced this before if you've ever been to a party like a birthday party and there are balloons, rubber balloons. You can take those balloons you can rub the balloon against your hair if it's a very dry day, if your hair is dry and the balloon surface is dry, you'll notice that it starts to make your hair stand on end. The rubbing of two materials transferring electric charge from one to the other is known as the tribal electric effect and in fact it's using experiments by the tribal electric effect that the early properties of electricity were first gleaned. So this is a really important and fundamental study of the natural world. Now how do you visualize the friction force? Let me give you a kinetic example. I have here what is known as a free body diagram. You've seen these in your textbook and you've exercised already some of these in class for instance and these are diagrams that can help you to understand the interplay of forces along a surface or perpendicular to a surface, especially when two surfaces are in contact. So I have here an object shown as a gray box it's sitting on another surface which I've just labeled material and it's free to move horizontally over the surface but not vertically. So it's just resting on the surface. Now of course gravity is pulling the object down, so this thing has a weight associated with it, weight is a force and because the weight points straight down into the surface perpendicular to the surface we get the full component of weight on this object and the weight force is equal to the mass of the object times g, the acceleration due to gravity at the surface of the earth. The normal force that the surface of the material exerts back on the object according to Newton's third law of motion is labeled here as n vector it is equal in magnitude to the weight but opposite in direction and as we know from the study of the mathematical description of friction forces it is the normal force, the mechanical load on the surface between the two materials drives the friction force. So for instance if an object is moving at some velocity to the right and I've drawn its velocity vector here so it's not sitting still it is in motion this is kinetic friction that is at play here based on our little guidelines from before. The kinetic friction force that's acting to oppose the motion to the right points to the left it has a coefficient of kinetic friction mu k depending on the material that makes up the surface and the material that makes up the object and that is multiplied by the normal force which is just the weight, the mass times the vector g, the acceleration due to gravity. So drawing little diagrams like this will help you to begin to visualize the forces, what direction they point in with respect to surfaces and so forth to help you understand more how to set up and solve problems involving Newton's laws where you need to add forces, subtract forces in various directions. Let's move on to a more complicated force kind of a relative friction but very different in its character than friction and that is the drag force also known as air resistance. Now what is the origin of the drag force? Well in principle the drag force occurs any time a body moves through a liquid or a gas, a so-called fluid. Air is a fluid air can flow it can move from one place to another quite easily water is a fluid water can flow, water can move quite easily from one place to another it doesn't, it fills the container that it's in and it doesn't keep a specific form, if you change the shape of the container the water will arrange itself to fill the shape of the container. So liquids, gases, these are fluids and any time a body moves through a fluid, a liquid, or a gas you can in principle have a drag force a resistance to the motion of the object through that material. So for instance if you're a swimmer or a cyclist you are very familiar with the drag force swimmers moving through water encounter a tremendous drag force from the water that they're moving through resisting their motion through that fluid. Cyclists love a tailwind but hate a headwind. When you are cycling into a wind that's aimed in the opposite direction you want to go it feels like brutal torture to have to cycle in that environment. So cyclists are also very familiar in the air environment with the drag force, race car drivers this is something that engineers who build fast cars and fast trains and fast planes and fast rocket ships that have to contend with getting through atmospheres they worry endlessly about the shape as you'll see the shape of the object to try to improve the way that it moves through air or water, whatever the fluid is. Basically you have to overcome this resistance from the fluid in order to maintain your velocity in the medium. Now as I said the drag force just like friction always appears to act to oppose the direction of motion. So if I'm running forward very quickly I will feel a resistance from the air pushing back trying to slow me down and so that again in character makes it very similar to the friction force but unlike the friction force the drag force depends extremely deeply not only on the properties of the liquid or the gas the fluid but also the geometric properties of the body moving through that fluid. Friction is independent of the apparent area of contact but drag force depends very much on the area that you present to the fluid in the direction you move through it and this is because the origin of the drag force is not in the way in which two surfaces passing over each other interact and make contact with each other it's about collisions. You are quite literally when you move forward through air running into the air molecules in front of you and creating a gap for the air molecules to fill behind you. You're making space behind you and you're compressing the air in front of you and basically you're running through a crowd. If you try to run through a crowd really fast you're going to run into one person and deflect off of them and they deflect off of you and then you're going to hit another person and it's brutal and moving through air is like moving through Avogadro's number worth of tiny little people and because of that you experience a resistive force. Those particles bounce off of you, they smash into each other, they bounce into you again and the net effect of that microscopic physics is to create a macroscopic resistance to motion that we call the drag force or air resistance in air fluid resistance in something like water. To fully understand and appreciate the drag force you really need a detailed study of fluid mechanics or aerodynamics. For those of you that may have delved a little into fluid mechanics you will live in fear of something called the Navier-Stokes equation. If you've never heard of it go look it up. If you can solve it then you have mastered a problem. The problem is solving the Navier-Stokes equation is extremely difficult and requires often numerical approaches and computational approaches you can't do it on pen and paper. Aerodynamics has the same considerations. Aerospace engineers are very concerned with how their vehicles will interact with the fluids they're moving through like air for instance and how those motions will result in energy being dissipated into the structure out of the structure and so forth. Engineers obviously and engineering of these kinds of vehicles requires a deep understanding of the drag force what impact it has and how to mitigate that impact. That's the art of aerospace engineering in a sense. So how do you describe the drag force mathematically? It is an extremely complex force in comparison to the equations that describe the force due to friction. This one carries a lot of baggage around. This owes to its complex nature of its origin in particle collisions. So Avogadro's number worth of things slamming tiny things but still slamming into a body as it moves through the fluid. So for a body that's moving at a speed v through a fluid the opposing drag force is given by this complex equation. The drag force including its magnitude and direction is one half times rho where rho is a Greek letter that represents the density of the fluid times the speed of the object squared times a number that is called the drag coefficient times the area that the object presents to the liquid or gas perpendicular to its direction of motion. This is its cross-sectional area, that's what this is referred to and all you have to do is picture the object and how it looks. Is it a square? Is it a circle? Is it something more complicated than that when viewed along the direction that it's moving? And then finally the direction of the drag force is given by the negative of the unit vector of its velocity. So if velocity points along the positive direction the drag force points along the let's say along x. So again, there are many pieces in this. The density of the fluid tells you something about kind of the number of things that can scatter per unit volume off of the object. The area tells you the target size off of which those objects can scatter. And the drag coefficient summarizes the complex interactions of the three-dimensional shape of the object with its area of density of the fluid. We'll come back to the drag coefficient in a moment but again keep in mind that whatever direction the object is moving through the fluid the drag force points in the opposite direction. It's a contrary force to the direction of motion. Let's talk about the drag coefficient because a lot of physics is summarized in this number and if you really wanted to understand where the drag coefficient comes from you need to study statistical mechanics probably also related to that thermodynamics and certainly fluid mechanics. The drag coefficient summarizes a number of complex things and you can think of them broadly as the interactions between the geometry of the body and the things that make up the fluid the liquid of the gas. Now the drag coefficient can be determined empirically in an experiment. So if you know all the other things, if you know the density of the fluid if you know the cross-sectional area of the object if you know the speed of the object you can then make measurements of the force that the fluid exerts on that object as you pull it through the fluid and then you can work back to the drag coefficient. Now shown at the right in this sort of graphical table are some values of the drag coefficient for common shapes and forms in various orientations relative to the motion of the liquid or gas around the object. So for instance if we have a sphere that is moving say to the right the drag coefficient for that sphere through any material would be 0.47. If it was a half sphere so now this looks more like a flat circle that would be pushed through the fluid that has a drag coefficient that's slightly less 0.42. What about a cone? Well a cone has a different shape on the back end and it turns out that these shapes create very complex situations where particles can move around the object, collide with other particles, collide with the object, even the backside of the sides of it. So this actually has a higher drag coefficient. This shape is perhaps you can think of it as not so easy to move through a fluid as a half sphere. In the language of aerodynamics we might say that the cone is a less aerodynamic object than a half sphere. Cubes are terrible. They have really high coefficients of drag. This would make a drag force much bigger for the same fluid, the same area and so forth. Same speed. An angled cube is better than a cube with its face pushed into the fluid. If you tilt the cube so that its corner is pushed into the fluid it becomes more aerodynamic, more fluid dynamic. It moves more easily through the fluid with less drag force. Long cylinder, short cylinder, etc. A streamlined body, a highly engineered body for instance that's moving through a fluid, something that's been highly engineered to allow the fluid to move efficiently and effectively around the object even if it still makes collisions can have extremely small drag coefficients, as small as 0.04 for instance for imagine the wing of a plane, which is a highly engineered structure designed to achieve certain goals while moving through air, a fluid. The drag force is driven by a relative motion between a body and the fluid in which it's immersed. If the fluid and the body are co-moving, that is moving together in the same inertial reference frame, no drag force exists. So if I take a whole volume of air and I can somehow on average make all the molecules of that body of air move at the same speed as the object inside the air, then the drag force should essentially go down to nearly 0. This is to say that they're co-moving. They're moving together in the same way. But when relative motion begins the object and the air on average start to move at different speeds relative to one another, then the drag force increases and it always opposes the direction of relative motion. This picture will help you to visualize the drag force a little bit better. So in this case the sort of blueish-grayish background represents some kind of fluid, a liquid or a gas, and that is at rest with respect to us, the observers. But this object here in dark gray is moving through the fluid and it's moving to the right. So here's the velocity of this object inside the fluid. This is its speed with respect to the fluid. So it's a relative speed. Now we're at rest, the fluid's at rest, so that speed is also with respect to us. The drag force points in the opposite direction. It points in the sense opposite the direction of the velocity of this object through the fluid. And so here's the full glory of that equation written out one more time one-half times the density, times the speed of the object squared, times the drag coefficient, times the cross-sectional area, and then the directionality comes from the direction opposite the velocity vector. So if you are introducing the drag force into a graphical representation of an object moving through a fluid this will help you to think about how to write the arrows in that picture. Finally let's discuss the centripetal force. This is the force that's associated with the center seeking acceleration we studied earlier, centripetal acceleration. Let's think back to the topic of centripetal acceleration. It's depicted here on the left. The blue circle represents a circular path that some object is following. It does so at a constant speed, but with an ever-changing direction. And at a given moment, let's say the object is this red dot. We have represented the velocity vector of the object. It is a line tangent to the circle, and it makes a right angle with respect to the radius that points from the center of the circular motion out to the edge where the object is currently located. The centripetal acceleration is what is keeping this in a circular path. Objects in motion would like to remain in their current state of motion. And that means constant velocity, but an outside force acting on the object will change the velocity. That can be the direction of the velocity or the magnitude of the velocity or both. And in uniform circular motion, the speed remains unchanged. So the magnitude of the velocity is constant, but its direction is ever-changing as the velocity vector points in different ways as you go around this circular path. So, again, remember when an object executes a uniform circular motion, its velocity vector changes direction, but not magnitude. Any change in velocity, however, either in magnitude or direction, is necessarily an acceleration. Acceleration is change in velocity divided by change in time. Direction changing for a speed means the velocity is changing. Even if the magnitude of the velocity, the speed doesn't change. This acceleration, as we learned earlier, points from the object direction toward the center of the motion. It's always at a right angle to the velocity vector as drawn here, and it always always points to the center of the motion. This is what is meant by centripetal or center-seeking acceleration. But if we think of this in the context of Newton's laws, we know that there must also be some associated force. For every unbalanced force that causes an acceleration, we can relate force and acceleration through Newton's laws. So if there is an acceleration, which there is in this case, it is a direction changing velocity, there is a net unbalanced force. We can very quickly derive the centripetal force from Newton's second law. We already know the equation that gives the acceleration in uniform circular motion. If that object can be located relative to the center of motion using a vector r that points from the center of the motion to the object, then we can write the centripetal acceleration as follows. The full centripetal acceleration vector is equal to the velocity squared divided by the r, the radius of the circular path, and it points in a direction that is opposite the line that goes from the center to the object. It points from the object to the center. Now, Newton's second law tells us exactly the associated force. It's just equal mA. And if that's true, if we plug in the acceleration A, then we find that the centripetal force is equal to mV squared over r times negative r hat. That is, it points into the center just like the acceleration does. So we need only know the mass of the object m that is moving in uniform circular motion, and given its speed and the radius of the circular path, the force is exactly determined. It points in the same direction as the centripetal acceleration. The force in a sense is also center seeking. Its magnitude is given by m times V squared over r. That's it. So this is kind of friendly compared certainly to the drag force that we looked at. It's almost as friendly as the friction force in the sense of the simplicity of the equation that describes it. And yet it's extremely powerful. Once we start thinking about these circular motions in terms of forces, we begin to learn a lot about the motion of objects that are going in circular paths. In order to better understand the centripetal force, let's use an example. Let's employ it to understand the situation that we have visited before in the course. So consider the moment in time depicted over here on the right. A car has just come off a road, entered a circular on-ramp, and is about to enter onto a highway. At the current moment in time, the car is rounding the last bend of the circular on-ramp. It has a fixed distance from the center of this uniform circular motion. It's moving at a constant speed forward along the circumference of the circle. The radius of the circle is not changing. It's neither sliding further outward from the center nor moving closer inward to the center. In many ways this car is like the tennis ball tied to the string spun in my hand. Something tethers it to the center of motion. What is it that keeps the car from moving further out along the circle's radius, or moving closer in toward the center of the circle? What is it that keeps changing the direction of the velocity of the car, but not its magnitude? That, of course, is the centripetal acceleration it experiences, and so there must be, from Newton's laws, an associated centripetal force. This car is in uniform circular motion. What is it that provides the role of the centripetal force? In the tennis ball example, it's the string that connects from my hand where I grip the string to the ball where the string is then tied through the ball in a knot so that it can't move further out along the string or move closer in toward my hand along the string. When I whip the string in my hand, the ball goes in a circle. That's what keeps the whole thing together. But a car is not tied to a rope to the center of the on-ramp. What is it that, in effect, tethers the car in this path of circular motion? Well, the force is something that we have looked at earlier in this lecture, and that is friction. The tires are made of rubber. The road is made of something like concrete. Do concrete have a coefficient both of static friction and of kinetic friction? What is it that is actually playing a role here? Well, the tires are not sliding. That is the point of contact that the rubber tire makes with the road is fixed directly beneath the car right where it makes its contact, and that contact is not sliding. The car is not sliding further out along the radius of the circle, nor moving further in along the radius of the circle. There is no slipping of the point of contact. And that means static friction between the rubber and the road. It must be what is acting. It is this force of friction that provides the centripetal force that yields the centripetal acceleration that is associated with the changing direction of motion of the car. The car maintains a uniform speed of say, V, whatever that is, maybe that is 50 miles an hour, 40 miles an hour, something like that. Its direction constantly changes and it is friction that provides the centripetal force. It is the tether that binds the car to the road and keeps it makes it possible basically for this car to continue moving in a circle. So as we can see here, since the car does not slide further out in a wider circle, nor move closer in on a tighter circle, the friction force must be the origin of the centripetal force in this problem. And I've illustrated that here in this picture. Here's the car. Its velocity vector points perpendicular to the radius of the circle. Here is the radial line going in toward the center of the motion. Here is the centripetal force that must be keeping the car's speed direction changing and it is friction that provides the source of this centripetal force. But what if something changed? What if, for instance, the car accelerated? Attempting to maintain a circular path while increasing its speed, the driver accelerates from 40 miles an hour to 45 and then to 50 miles an hour and then to 55 and then to 60 all while attempting to keep the wheel turned and the car moving in a circle. At some point the centripetal force exceeds the static friction force and at that point it must be true that the car would begin to slide after all as we saw earlier in this video if one exceeds the threshold at which static friction can act against any other force then the object will begin to slide and kinetic friction will begin. And as we've seen before kinetic friction does not require as much force to overcome as static friction does. So we would expect the car to begin to slide outward moving further from the center of the circle and this could be quite dangerous. The car could smash into a roadway boundary or something like that. And it's why when you drive on a circular on-ramp onto a highway there are posted guidelines about the maximum speed that you should drive. There are of course tolerances built in by the engineers who set these limits for these roadways but nonetheless while there may still be a margin of safety above the posted speed limit it's wise not to exceed it if you can in order to avoid questionable road conditions from suddenly causing your car to slide and lose control. So this is an example of employing the centripetal force and then thinking about the other forces that we've studied and asking the question what is it that provides the centripetal force that keeps the car from moving closer into the center of the circle or further out into a wider circle. And in this specific example there is no rope, there is no steel cable that keeps the car moving this way. It's friction that does the job, in this case keeping the car tethered into its uniform circular motion. Let's review the key ideas that we have explored in this section of the course. We have seen the empirical laws that govern the friction force. We've learned something about the origins and history of those force but also in the deeper laws of nature that one would require much more time to not only encounter but to grapple with and to come to some understanding of in order to look back on friction and go ah that's why those counterintuitive things are happening with friction. We've seen the mathematical description of friction it's really not that bad if you can determine the normal force in a circumstance and you can use your information about the materials that are present make up the surfaces that are in contact you can quite simply describe the force due to friction even in spite of the fact that it has particularly complex origins at the atomic level and certainly involving physics that goes beyond the introductory physics sequence. We have seen the more complex origins of a much more complicated force and that is the drag force. We've explored a little bit its mathematical details of course exercising these on your own will be the way that you really begin to learn more about this force or at least how to use it but we can see that the drag force unlike friction forces is quite deeply shaped by not only the orientation of a material as it moves through a fluid but the cross sectional area that it presents to the fluid along the line of motion, the degree of its motion, its speed and the density of the fluid through which it moves. And finally we have revisited uniform circular motion in lieu of Newton's laws of motion and we've seen that there is of course an associated force because there is an acceleration and that is the centripetal force and we've begun to think a little bit about when there are situations where an object moves in uniform circular motion what might it be that provides the tether that keeps an object moving in that circular path.