Lecture Notes: Understanding Forces and Motion in Physics - Prof. Michal R. Zochowski, Study notes of Physics

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Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Physics 135 Coursepack Physics for the Life Sciences I Author: Tim McKay Table of Contents Syllabus Important Dates to Remember Lecture Notes Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Physics 135: Physics for Life Sciences I Syllabus for Fall 2009 Lecture: MW: 11–12 Location: Room 170 Dennison Discussion: TuTh: 10-11, 11-12, or 12-1 Location: 1250 Undergraduate Science Building Primary Lecture: Professor Michal Zochowski • Office Hours: Wednesday 1:30-3:30 in the Physics Help Room • Email: michalz@umich.edu Discussion Instructor: Dr. Toby Eckhause • Office Hours: Mondays 10:00-12:00 and Tuesdays 2:00-4:00 in the Physics Help Room and by appointment • Email: eckhause@umich.edu Lecture # Date Lecture Title Pre-Lecture Reading Topics 1 Wednesday, Sept. 9 Physics and Life Lecture notes #1 Physics and life 2 Monday, Sept. 14 Tools for Mechanics: Forces and Vectors Lecture notes #2 Vectors: displacement, force, velocity 3 Wednesday, Sept. 16 Not Changing: Newton’s First and Third Laws Lecture notes #3 Newton’s Laws, forces, friction, Free Body Diagrams 4 Monday, Sept. 21 Standing Up and Staying Still Lecture notes #4 Tension, force transmission, and simple machines 5 Wednesday, Sept. 23 How Do Forces Happen I: Stretching and Squashing Lecture notes #5 Rotational equilibrium, Stress and Strain 6 Monday, Sept. 28 How Do Forces Happen II: Friction Lecture notes #6 Static friction, kinetic friction 7 Wednesday, Sept. 30 Describing Motion: Velocity and Acceleration Lecture notes #7 Position, speed, acceleration, PT, VT, AT graphs. Do this for both linear and rotational Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Important Dates to Remember Physics 135: Physics for Life Sciences I Exams: Exam 1 Thursday, Oct. 8: 8-10 pm Exam 2 Thursday, Nov. 5: 8-10 pm Exam 3 Thursday, Dec. 3: 8-10 pm Final Exam Monday, Dec. 21, 4:00-6:00 pm Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Physics for the Life Sciences: Fall 2008 Lecture #1 Physics 135: Physics for the Life Sciences This is a course for aspiring scientists; including biologists, chemists, medical doctors, bioengineers, biochemists, and physicists. The goal is to help you learn some very fundamental aspects of physics, especially those which most important for understanding life. In this course, we’ll teach you not only what the laws of physics are, but how to use them to analyze how life works. By covering many of the basics, we’ll also provide you with the physics framework you need to build a more detailed understanding of life later. This course is the first in a sequence of two courses especially designed for life scientists. During this first term, we will focus on mechanical and thermal aspects of life, including fluids. Here are some of the many questions we will address. • How does the inexorable pull of gravity affect the sizes and shapes of organisms? What must they do to move around? • How does inanimate matter apply forces? How do organisms use membranes, muscle, tendon, and bone to support themselves, get up and move about? • Why does it take so much effort to jog along at a constant speed? Why haven’t organisms evolved wheels to make this easier? • What is energy, and how do organisms take energy from or give energy to an object? What forms can energy take? • What is temperature and thermal energy? How does life harness purely random thermal motion to get things done? • How do organisms manage to survive winters and live in cold oceans? • How do living things get along within life’s two great fluids: air and water? In the second semester course, Physics 235, we’re going to learn about several new aspects of the physics important for life: • How do living things sense the world around them? Sound and light, imaging and detection. • How can we extend our senses? Instrumental imaging. • How does life send signals within an organism? Electric fields and potentials, electric currents and circuits, electricity and magnetism. • What is life built of? The elements, nuclei, radiation, and the origins of these. Each course will include a number of components: • Textbook: There will be no traditional textbook for this course. Instead we will provide detailed lecture notes for each class. These are available as a coursepack and also as a PDF file on the Ctools site. No text on the market approaches Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 physics in quite the way we will, so it doesn’t make sense to use one. That said, every introductory physics text provides a useful overview of many of the topics we will study. If you have one, feel free to use it. You may find it helpful in studying. • Supplementary Problems: We will ask you to purchase the Schaum’s Outline of College Physics. This is a very cheap, basic book which will give you another look at many of the topics we’re covering, and includes a lot of example problems which are likely to be useful for exam reviews. It should cost ~$15 purchased online, and you can also get it at the local bookstores. Here are some details: • Publisher: McGraw-Hill; 10 edition (November 15, 2005) • ISBN-10: 0071448144 • ISBN-13: 978-0071448147 • Readings: The time we spend in class will be focused on trying to understand the most difficult aspects of the material, rather than on providing a first-look introduction to each new topic. To make this work, you have to come to class prepared. This means you will have material to read and think about before every class. This will typically be 5-10 pages from lecture notes, occasionally accompanied by an additional reading. • Daily lecture preparation homework: To help you to prepare for class, you will have to answer a few simple questions and solve a few straightforward problems for each lecture meeting of the class. These are to be written out and turned in at front of the class before the start of lecture. Working through these will help make sure you’re ready for what we’ll do during the lecture period. These assignments will be available on the course Ctools site. Print each assignment, do the work in the space provided, and turn it in at the start of class. • Lecture/Demonstration: All students in this course will meet together twice a week (on Mondays and Wednesdays) in a lecture/demonstration setting in the Dennison building. During these sessions we will go over some details of the material, view and analyze demonstrations of the phenomena in question, and work through questions designed to challenge your understanding of the material. During lecture we will “Qwizdom” electronic response units to test your understanding of lectures in real time. You will need to purchase a Qwizdom unit from the Computer Showcase for this purpose. The details are available at: http://showcase.itcs.umich.edu/pages/remotes/ • Discussion: On Tuesdays and Thursdays you will be split out into smaller discussion sections, which will meet in the Undergraduate Science Building. During these sessions you will work in small groups solving problems, constructing models for biological problems, and occasionally working with auxiliary readings and the primary scientific literature. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 assumption; that life is shaped by and exists within the constraints of purely physical laws. We will ask whether living things ever evade the limits these laws place, as they might if life involved something beyond the physical, something metaphysical or supernatural. We will also ask how the extraordinary diversity we see in life could come about as a result of these physical laws. There is one extremely important tool needed to understand how the interplay between physical laws and life takes place: evolution. The idea that life evolves through natural selection of random variations provides our only tool for understanding the diversity of life. Evolution has allowed life to find incredibly various and seemingly ingenious ways to function. Evolution, working within the limits provided by physical laws, will allow us to understand why animals never evolved wheels, why cells are the size they are, why hummingbirds eat several times their weight in food each day, and why the largest animals which have ever lived all swim in the sea. What we’ll accomplish in this term and next will only scratch the surface of this profound and important topic. But I think you’ll find even a superficial look can teach you a lot about the inescapable unity between the physical and biological worlds. At a minimum you should learn how physical laws constrain organisms, and with luck this approach will change the way you think about life. Three examples to set the stage Let’s start with a few examples to illustrate how life has evolved to work around the limits placed on it by physical laws. Let’s start with one of the most obvious connections: size and shape. 1: The Spherical Cow and Modeling There is a famous joke about cows. A dairy farmer is having trouble making ends meet, and hopes to find a way to enhance the productivity of his farm. For reasons lost in the mists of time, he calls in a psychologist to help. The psychologist conducts a series of interviews with the farmer, his family, and the cows, and collates their reactions to a variety of visual stimuli. In the end, he tells the farmer “Your cows are suffering from Ruthvenian post-lactic stress disorder. To enhance your productivity you need to provide them with a more nurturing climate. Paint the walls of the barn a cool, neutral color, provide them with quietly energetic music, and be sure to reinforce their sense of self-worth daily.” This doesn’t work. Indeed the cows become very relaxed, but this only makes them harder to milk. Next he turns to a biologist, who sends a graduate student to take cell cultures from inside each cow’s mouth, extracts the DNA, and sequences it. Her report to the farmer suggests supporting a new research program to splice the DNA of a Minke Whale into the cow, allowing them to grow much larger, produce more and richer milk, and not wander about the farm so much. The Minke Whale, after all, has the richest milk of all mammals. The Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 farmer, imagining the bad press he would receive for this Frankenstein milk, politely thanks the biologist and sends her on her way. Finally, the farmer calls on a physicist. Unlike the others, the physicist doesn’t examine or consult the cows at all. Instead, she goes to the chalkboard, draws a large circle, and says: “Assume each cow is a sphere” and begins to write long equations on the board… The point of this joke, like so many, is to illustrate some partial truths. Psychologists do look for answers in the minds of their subjects. Biology has a strong focus on genetic research. And physicists achieve much of their success by building mathematical models. Let’s see where we can go with this “spherical cow approximation” (SCA)… If a cow were a sphere, it would be easy to calculate things about it. Tell me it’s radius, and I can quickly calculate both its volume (4/3πr3) and its surface area (4πr2). For a real cow, with its complicated shape, this is hard to do. What’s the formula for the volume of a cow given it’s height at the shoulder h? There isn’t one. But although these spherical answers are easy to calculate, they’re also obviously wrong. I don’t expect to precisely predict the mass of a cow from the SCA. But there are things I can accurately predict. What would happen if I changed the size of a cow? How would its volume and surface area change? Without a model, we could only do the experiment: grow a big cow and measure it. But given the SCA, we can predict what will happen. For our sphere, volume is related to radius through the equation V = 4/3πr3. If we double the size of r, the volume increases by a factor of 23, or 8. We can also predict how the surface area (S = 4πr2) changes: it should increase by a factor of 22, or 4. Why might the farmer care? Suppose he is raising beef cattle. Doubling the size of the cow would yield 8 times as much meat. What if he’s out to make leather? Doubling the size of the cow would yield 4 times as much hide. But of course there are other implications of size. The amount of food and oxygen an organism needs is largely governed by mass. Each cell must be kept alive, and more mass means more cells. So doubling the size of a cow increases its food needs by a factor of eight! A farmer out to make leather would, as a result, generally prefer a lot of smaller cows to a few big ones! We will make use of the SCA, along with many other simple models, quite often in this class. They will allow us to extract, from all the specificity of real biological circumstances, a few important facts. Some examples follow. 2: Convergent Evolution I have said that evolution, the random change and selection of subsequent generations of life, allows life to find strikingly effective ways to work within the limits placed by physical laws. A beautiful example of this is “convergent evolution”. Here’s the idea: if physical laws present a difficult problem for life, there may be very few workable solutions. When this is so, the same solutions will often be found, completely Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 independently, over and over during the course of evolution. While there are many examples of this, perhaps the most visible, well documented, and delightful, is flight. We all know what will happen when we release an object in the air: gravity pulls it to the ground. Sometimes the friction an object feels when moving through the air slows its fall, but in the end “what goes up must come down”. To fly, to remain in the air at will, climbing and diving when you want, you need to be able to generate forces large enough to overcome the gravity which pulls you down. As we will see in a bit, if you want something to push up on you (to lift you up more strongly than gravity pulls you down) you have to push down on it. Up there in the air, the only thing to push on is the air. If you want to push hard on the air, you need something big; you need a wing. Now it would also help to minimize how hard gravity pulls you down by being as light as possible. Life, through evolution, has found this solution at least four quite independent times, among the insects, reptiles, birds, and mammals. In each case the solutions are strikingly similar: large, thin, flexible wings are attached to bodies with many adaptations designed to reduce weight. Insects might seem the exception to this focus on slenderness. After all, a big fat June Bug flies along just fine. Why is that? Recall the SCA. The volume of a creature increases like size3, while the surface area increases like size2. So if shapes remain the same, the ratio of mass to wing area (volume to surface area) changes like size3 / size2, or like size. The bigger a flier is, the more important it will be for it to limit its weight and increase the relative size of its wings. The largest fliers, birds like condors and cranes, have quite enormous wingspans and surprisingly tiny masses. Insects live at the low end of this tradeoff, where the benefits of reducing weight are really not important. So there are lots of chubby shaped insects which can still fly. Why are insects always pretty small? It turns out their sizes are not limited by flight. We’ll see what does limit them in a bit too. 3: Diffusion Most of the motion we associate with life seems willful: I throw a ball, a bird flaps its wing, a snail crawls across the floor. But one kind of motion, incredibly important for life, clearly doesn’t require will: it just happens. This is transport on the molecular scale: what we would generally call “diffusion”. Imagine a rectangular box divided in two. The left hand side is filled with Nitrogen gas. In it huge numbers of N2 molecules fly freely through space, colliding occasionally with one another or the walls. The right hand side is completely empty; a vacuum. Now suppose we remove the dividing wall. What will happen? Some of the molecules which would have hit the divider and bounced back will now just continue into the empty side, eventually reaching the far wall and bouncing back. After a while (a very short while indeed in this case) there will be essentially equal numbers of molecules on both sides of the box. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 For other shapes, like a wombat or an automobile, we don’t have simple formulas for surface area and volume. There are some simple scaling rules though. If you change the size of an object, keeping the shape exactly the same, the surface area will increase like the size2 and the volume will increase like the size3. You can see this is true in the above trivial examples, and you can extend it to any shape you like by building that shape up out of tiny cubes. Density and Mass The average density of an object can be found by dividing its total mass by its total volume: ρav = M / V The density of an object is determined by what it’s made of. Most living things (like you) are made of a mix of things (muscle, bone, brain, etc.). In most animals the main ingredient is water, so often you’re not too far wrong if you use the density of water to estimate the density of an animal. Conveniently, water has a memorable density of about 1000 kg/m3. How could you use this to estimate your mass? There’s no formula to get your volume from your height. So let’s estimate it by imagining you’re a cylinder, say 1.8 m high and with a radius of 10 cm (about 4 inches). This would give us a volume of 0.23 m3 and a mass of 56 kilograms. Trigonometry and the Pythagorean Theorem As we discuss various geometric properties such as size, shape, motion, etc., you will have to use trigonometry in many basic ways. Here are some basics which would be good to recall. Given a right triangle with sides that have length A, B, and C, we can write the following: Here’s another useful thing to remember: angles can be measured either in radians (which run from 0 to 2π) or in degrees (which run from 0 to 360). You’ll need to be careful about which you’re using with your calculator. Some basic calculus Calculus is a branch of mathematics dedicated to describing change. Physics is all about change; not about how things are but about how they change. Calculus was invented, in large part by Newton, as the central tool of physics. As a result, any serious understanding of physics requires reference to calculus. It does not, fortunately, require a A B C θ A2 + B2 = C2 cosθ = A/C sinθ = B/C tanθ = B/A Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 very elaborate application of calculus. So while we will very often incorporate the ideas of calculus in what we discuss, you won’t have to deploy the many methods of calculus particularly often in this class. You will need to understand that derivatives of functions describe their slopes, their rates of change, and that integrals of functions describe areas under them. We will do some simple calculus derivations occasionally, and you should be comfortable with this. Estimation This provides a nice introduction to the topic of estimation. We will often estimate things in this course. Why not just be precise, use equations, and calculate exact answers? There are at least two different reasons. First, most of what happens in the world is incredibly complicated. This makes precise description, in the form of a perfect, tidy equation, impossible. Fortunately, this complexity doesn’t leave us helpless to describe or predict what will happen. It simply means we will have to approximate; to construct models which capture some of the most important features of the situation, while glossing over less significant details. Our spherical cow is a great example of this. It doesn’t tell us the volume of a cow. But it does give us an idea of how that volume changes as a cow grows. There is another important reason to estimate. Even if we had a perfectly precise, tidy analytic theory, we still may not perfectly know the parameters involved. For example, if we want to know the mass of a cow, we may not know its precise height or length. We may not know its density or detailed shape. What’s the density of a cow? How could you estimate this? Well, like most land animals, cows can swim, a little at least. This means they can nearly float. This means their density must be close to the density of water, which is one of those nice, useful, numbers you should just know… It is very useful, and important, to be able to make quantitative estimates in situations where perfect knowledge is absent. This appears in all arenas of life. If, for example, someone told you there were 5,000 piano tuners in Ann Arbor, should you believe them? Most of you have probably heard the five second rule: “if you drop a piece of food on the ground and pick it up in less than five seconds, it’s OK to eat it”. Is this nonsense or true? Why five seconds and not 2.5 or 10? What happens in five? Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Physics for the Life Sciences: Fall 2008 Lecture #2 Before we embark on any really serious analysis we need to make sure a few basic tools are in place, things we’ll need to use. Units of Time, Space, and Mass, and Questions of Scale Quantitative description of anything implies measurement: comparison to some frame of reference. All measurements in this class will involve comparisons to just three fundamental standards, measuring time, distance, and mass: Time: seconds (defined as time required for a Cesium atom to vibrate 9,192,631,170 times) Distance: meters (defined as distance traveled by light in 1/299,752,458 second, about 3.28 feet) Mass: Kilograms (defined as mass of a little cylinder kept in Paris, about 2.2 pounds in more familiar units) Now each thing you might measure, like distance, or time, might be measured in a variety of different units. Time, for example, might be measured in seconds, or hours, or days. Now some particular period of time, 45 seconds say, actually has only one duration. We might measure it in many different units, but it’s always really the same thing. To convert this one period from one set of units to another we can take advantage of conversion factors: 45 seconds * (1 minute / 60 seconds) = (45/60) minutes = 0.75 minutes 45 seconds * (1 day / 86400 seconds) = (45/86400) days = 5.2x10-4 days Notice what we do in each case. Start with what you are given (45 seconds), then multiply by a “conversion factor”; a ratio of two periods that are equal to one another, but measured in different units. Since the two are equal, the ratio is actually equal to one, and when you multiply by it, you leave the original period unchanged. What the conversion factor does, then, is to change the units without changing the value of the measured quantity. Here are a few more examples: 1.8 meters * (100 centimeters / 1 meter) = 180 centimeters 56 kilogram * (1 pound / 0.454 kilogram) = 123.4 pounds Since we will work with a variety of different units, you will need to develop some facility with doing these conversions. Sometimes they will be more complicated. Let’s convert speed in meters per second to miles per hour: Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 seem rather simple; they can be represented by a single number. A baseball has a mass. That mass is just a number (5 1/4 “ounces avoirdupois”, or about 149 gm, according to the rulebook). Everything there is to know about the baseball’s mass is represented in that number. It also has a circumference (officially “not less than nine nor more than 9 1/4 inches”, or about 23 cm). That number also tells you everything there is to know about its circumference. Physical properties which can be represented by just a single number are known as scalars. They are quite common. In addition to mass and diameter, they might include temperature, density, pressure, metabolic rate, pH, age, or even cost. Scalars are properties which can be fully described by just one number. We sometimes say that scalars are properties of things which have only a magnitude. Some things we want to measure, especially in physics, are more complex. If we want to describe the wind for a sailor, it’s not enough to simply list its speed. To usefully describe the wind, we need also to give its direction. Another example of a quantity like this is a force. To fully describe a force, to tell you everything you need to know about it, we have to give both its magnitude and its direction. There are other examples in physics, including displacement, velocity, acceleration, electric, magnetic, and gravitational fields. These properties are called vector quantities. These vectors are things which require us to specify both a magnitude and a direction to give a complete description. The point of this discussion is to draw your attention to this important difference. If you’re analyzing something and the answer you seek is a scalar, you only have to figure out and report how big it is. But if the answer to your question is a vector, you will have to determine and report both its magnitude and its direction. Displacement as an example vector: Scalars are pretty familiar things, so they don’t need much further introduction. Vectors are considerably less so, as they were invented for and are largely used in physics. So we will take some time to talk about what vectors are and how we do things like add, subtract, and multiply them. Let’s take as our example a displacement; a kind of instruction for a trip. To describe this trip, we have to say how far to travel, and also what direction you should go. One way to do this is to specify the magnitude of the vector, and describe its direction by measuring an angle relative to some reference direction. Here’s an example; you receive an instruction telling you to travel 30m in the direction North-East. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 We could represent that trip graphically as a little arrow that looks like this. Think a little about what this graph represents. Each of the two axes represent positions. The horizontal axis measures how far East or West you are from the (arbitrarily selected) origin, while the vertical axis measures how far North or South you are from the origin. The solid arrow shows the displacement “30m in the NE direction”. The dashed arrow also shows a displacement “30m in the NE direction”. Since these two displacement vectors have exactly the same magnitude and direction, they are precisely equal to one another. This is an important point. Vectors are not tied to particular points in a space. They don’t say go from this particular spot to that; they just tell you how far to go and in what direction. The reason for this is actually rather deep. The way things move can’t be affected by how we choose to draw our coordinate system. If they were we would never know what was going to happen until we defined a coordinate system. We’re talking about these displacement vectors because they’ll be useful in describing physics, so they’ll have to be independent of particular starting and ending points. Q: If I tell you a vector displacement like this is 10m North, what would it look like on a NSEW plot? Q: What if I tell you the displacement is -10m North? What does this mean? Q: Is the displacement "10m North" tied to particular points? Vectors: Displacement is the archetype of a "vector". We denote this in typed text by making the symbol boldface, so while s might be a scalar a distance, the boldface s is a displacement vector. On a chalkboard or in your written homework we would usually note this by putting an arrow over the symbol, sG , which is actually easier to remember. There are several ways we might want to manipulate vectors. The first is addition. Why we might to add vectors? If we take two trips, we undergo two displacements, and the sum of the two is the same as taking a single equivalent trip. Likewise, the sum of two forces is the same as a single equivalent force. N E Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Rather like scalars, the sum of two vectors a and b is equal to a single third vector, c, which is equivalent of doing a then doing b. So we can plausibly write: a + b = c The order doesn't matter, which if you remember your fifth grade math is called commutativity: c = a + b = b + a We see drawn here the first way to discover the sum of two vectors. This is called graphical addition, or the tip to tail method. It relies on the fact that vectors are NOT tied to particular points; they are only have magnitudes and directions. Because of this, you’re free to move them around and line up the tip of the first with the tail of the second. Note that we could go a step further, and consider what happens if we add three vectors together: a + b + c = d Notice from this example that the same resultant vector d is produced whether I take: (a + b) + c or a + (b + c) This different kind of independence from order is called associativity. a b c b a c d OR: d b c a a a a b b b c Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Add three vectors: 5m East = 5E 8m North = 8N 6m 30° East of North = (6*sin30)E + (6*cos30)N = 3E + 5.2N so the sum is: (5 + 3)E + (8 + 5.2)N = 8E + 13.2N We can work out the magnitude and direction of this final vector in the way we did for adding perpendicular vectors above: Magnitude m2 = 82 + 13.22 or m = 15.4m Direction tanθ = opp/adj = 13.2 / 8 = 1.65 Or θ = arctan(1.65) = 1.02 radians = 58.8° Notice that this would have been just as easy if there were 30 vectors instead of three. So whenever you have to add vectors, it is usually easiest to do it by components. Picking the right coordinate system: Now often you can greatly simplify a problem by using some feature of the arrangement of elements in a problem to simplify its solution. When we talk about "picking the right coordinate system" for a problem, this is usually what we mean. A couple of examples will give the general idea: First a simple one: Add two displacement vectors 4m NE and 3m SW. We could break this into N and E components, buts it easier to add them along the direction NE/SW. Then we immediately find: Sum = 1m NE Now a second, slightly more complicated example: We could resolve this into components along horizontal and vertical x and y axes, but that would be hard. Easier to think about a coordinate system rotated 30° counterclockwise: Writing these vectors in components along these x'-y' axes is simple: c a Resultant 4m b 60° 4m 30° 30° 1m What is the magnitude of the resultant? x 30° x’ y’ x y Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 a = 4x' b = -1x' c = 4y' So the sum of the three in this coordinate system is just: r = 3x' + 4y' Which is a vector with magnitude 5, and direction θ = tan-1(opp/adj) = tan-11.33 = 53.1°. Does it matter physically that we’ve defined this vector in an unusual coordinate system? Not at all. It’s always the same vector no matter how we choose to measure it. Components and vector equality We say that two vectors are equal when both their magnitude and their directions are the same. It’s also true that if any two vectors are equal, each of their individual components (along the x, y, and z, axes for example) must be equal. So if we have two vectors A and B and they’re equal, we can write: A = B or Ax = Bx and Ay = By and Az = Bz This alternate way of writing things will often be simpler to keep track of than the more general definition of vector equality. So a lot of times when we know two vectors are equal we’ll go ahead and write out three independent equations, one for each component. Since each equation is just a scalar equation, it’s simpler to work with. Velocity vectors: So now we have displacement vectors and we have some ideas about how to manipulate them. Apparently velocity also must be described with vectors, because we usually need to know both how fast we're going and in what direction. We can define an average velocity vector in a straightforward way from the displacement vector: vav = (Δs/Δt) if we determine this velocity over an infinitely short period of time, we speak of the instantaneous velocity: vinst = limΔt->0 (Δs/Δt) = ds/dt Notice carefully what this is. The velocity vector is really just a scaled version of the displacement vector. In other words it is just the displacement vector multiplied by a scalar number; the inverse of the time it took to make this displacement (1/Δt). What this means is that the velocity vector always points in the same direction as the displacement vector. Because motion takes place in three spatial dimensions, many things we will use to discuss motion this semester will be vectors; including forces, accelerations, stresses, flow rates, etc. It is important that you understand vectors very clearly, and that's why we're expending so much effort on them now. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 2. The vector product: a x b The vector product takes two vectors and makes a third, new vector out of them. a x b = c where the magnitude of c is given by: |c| = |a| * |b| * sinφ and the direction of c is perpendicular to the plane defined by a and b in a direction given by the right hand rule. The right hand rule says you should: • Take your right hand • Point your fingers in the direction of the first vector (a in this case) • Turn your hand until you can "curl" your fingers in the direction of the second vector (b) • Now your thumb defines the direction of the vector c. From this definition you can see that the vector c is always perpendicular to both a and b. The vector product is a kind of measure of the amount of perpendicularity of two vectors. Because the symbol ‘x’ is used to denote this operation, it is often called ‘the cross product’. Note that this vector product has the special property that it does not commute. That is: a x b ≠ b x a in fact it "anticommutes" (a x b) = - (b x a) Where will we use the vector product in physics? One good example has to do with rotation. If you want to get something to start rotating, you have to apply a force to it. The ability of the force you apply to make the object rotate depends on both where you apply the force and in what direction you push. First we define a radius vector r which goes from the center of rotation (the hinge of a door for example) to the point where the force is applied. Given this vector r and the force vector F, we will quantify this ‘ability to create rotation’ by defining the torque τ with the vector product: τ = r x F Don’t worry if this is confusing now. It’s just an example which you ought to recognize when we return to it later. Vector multiplication by components Remember that there are two kinds of vector multiplication: the scalar product and the vector product. In both cases, there’s a basic definition in terms of the magnitudes of the vectors and the angle θ between them: Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Relative velocity: The "relative" we're talking about here is the constant velocity motion of some object observed by two different observers, who are themselves moving relative to one another. Start with a simple example: Joe is on a train moving past the platform at 4m/s. He walks forward in the train with a speed of 1m/s relative to the train. What is Joe's speed relative to the platform? In vector form this problem is: Vtp = speed of train relative to platform = 4m/s Vjt = speed of Joe relative to the train = 1 m/s Vjp = Vjt + Vtp = 4 m/s + 1 m/s = 5 m/s So it's fairly obvious how this works for colinear motion. What about in two dimensions? A boat can travel at 3 m/s through the water. It steers straight across a river which flows past the shore at 5 m/s. What is the velocity of the boat relative to the shore? So the magnitude of the boat's velocity is: m2 = 32 + 52 or m=5.8 m/s And its direction is θ = tan-1(5/3) = 59° West of North Vtp Vjt Vjp vws vbw vbs E N Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Physics for the Life Sciences: Fall 2008 Lecture #3 Newton’s Laws and Statics Life on Earth faces many challenges. One of the most basic is dealing with the constant pull of the Earth’s gravity. Every living thing near the surface of the Earth (and every nonliving thing too) is constantly pulled downward. This downward pull is so steady and omnipresent we usually forget it’s there. But just one misstep on the staircase, one moment’s loss of balance on your bike, one slip of the cup off the edge of the table, and you’re reminded of the power of gravity with shocking suddenness. It’s not a stretch to claim that gravity is America’s number one killer, and it is certainly life’s number one mechanical challenge. In addition to standing up to gravity, many living things have to move around, pick things up and do things with them (carry them, chew them, throw them, etc.). To accomplish any of this, living things have to not only resists forces but apply them. So our first big task will be to understand forces and how they affect motion. This topic in physics is called “mechanics”, one of those otherwise everyday words which means something quite special in physics. The core of our understanding of mechanics is contained in three terse laws first gathered together by Newton in the 17th century. Newton's laws provide the rules we need to understand how objects react to forces, and to describe motions as various as the orbits of the planets and the swimming of a bacterium. Their ability to analyze almost every mechanical situation observed at the human scale makes them a remarkable part of the collective human intellectual legacy. They’re also incredibly useful for understanding what’s going on around you, and hopefully by the time you finish studying them you will see the world in a new and richer way. We will begin trying to understand in detail how to analyze cases in which objects aren’t moving, a subset of mechanics sometimes called “statics”. On Earth objects sitting still always experience a number of forces, but these are balanced, so that the total force on them is zero. Once we have that in hand, we’ll look at cases where the forces are not balanced, and learn to understand how these unbalanced forces make motions change. Newton’s first law Newton’s first law is this somewhat surprising assertion: Any body continues in a state of rest, or of uniform motion in a straight line, unless it is compelled to change its motion by unbalanced forces imposed on it. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 This is usually called the law of "inertia", or Newton's first law. It's not really Newton's, as even he would have freely acknowledged; everyone working seriously on motion at the time knew about it. Still, it is hardly obvious. The first bit is no surprise. Objects at rest stay at rest unless you do something to move them about. But the second part isn’t familiar at all. Daily experience does not suggest that an object in motion tends to stay in motion. What do you have to do to keep an object in motion? You have to push all the time. Aristotle, noting this ubiquitous experience, assumed that the “natural state” of an object was to be at rest, and that to have an object in motion required a motive force. He believed that "motion implies a mover". But even he allowed a tantalizing exception; objects in free- fall seemed to fall only because it was in their nature to do so. No “mover” was required to create free-fall motion. Aristotle gets a bad rap in physics, and we tend to dismiss Aristotelian beliefs too quickly. They in fact well describe a lot of ordinary experience. Seeing behind these "obvious" facts requires great care. It took a world of smart people thousands of years to see what you’re learning now. It was Galileo Galilei, one of the delightful Italians of the 1600s, who first clearly exploded the Aristotelian idea of motion. His argument, which elegantly encapsulates the idealization which has proven so powerful in physics, went like this: 1. Imagine a wedge shaped track. Roll a ball down one side and it rises up the other to almost the same height 2. Carefully clean and polish it all and the ball rolls still more closely to the starting height, so we might ascribe any "loss of height" to friction between the ball and the track. 3. Now decrease the angle of the second side. The ball still rises to the same height from which is was launched, but now travels much farther along the ramp. 4. Carry this to its logical conclusion: if we lower the second side to be horizontal, the ball will travel forever, always attempting to rise again to its original height. Notice the details here: the ball will roll forever with no help from anything, its “natural state” is to be in motion, and friction is the only thing which prevents that motion. Without friction everything that is moving would continue to move forever. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Fest ≅ Δp / Δt We know the force at each instant is given by the instantaneous rate of change of the momentum, so we can estimate it during the period while the momentum is changing by dividing the total change in momentum by the total amount of time this change took. Let’s look at this with vectors. It’s useful to think about it in several ways just to get it clear. • If you see the momentum changing very suddenly, dp/dt will be large, and there must be a large force acting. • If you want to create a certain change in momentum Δp, you can rearrange the second equation to see Δp = F Δt. This emphasizes that to achieve a particular change in momentum you can either apply a large force for a short time, or use a small force for a long time. In fact, this quantity F Δt (which is equal to Δp) has a special name; it is called “impulse”. So it is sometimes said that you “apply an impulse” to achieve a certain change in momentum. For the moment, we’ll set aside the second law, because we’re going to talk about static cases where the momentum doesn’t change. We’ll return to this in a week or two. Units for force What are the units of a force? From the second law, we see that a force is a change in momentum (which has the same units as momentum) divided by a time. So using the typical units of meters, seconds, and kilograms, we have: kilograms * (meters / second) / seconds = kilogram*meters / seconds2 Because forces are so important, this particular combination of the basic units is especially significant, and is given a name of its own. Because Newton was so important in understanding the importance of forces, this unit of force is named for him. 1 kilogram meter / second2 = 1 Newton 1 kgm/s2 = 1 N Though many other units of force exist, in this course we will measure forces almost always in these units. pi pf -pi pf Δp = pf - pi F = Δp / Δt Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Newton’s third law The most important and perhaps least obvious fact about forces is that they are never isolated actions; they happen only through interactions. Pick apart that word a little. “Inter-actions” are actions that take place between things. There is never a force that comes from nowhere and pushes on something. Every force comes from one thing and pushes (or pulls, or whatever) on another. Newton, realizing this, was the first to recognize a fundamental fact about interactions: they are always perfectly balanced. In absolutely every case, when one object applies a force to a second, the second applies a perfectly equal and opposite force on the first: F12 = -F21 That is, every interaction has exactly two forces, equal in magnitude and opposite in direction, one acting on each of the two interacting bodies. This 'third law' of Newton is often stated in words as: For every action there is an equal and opposite reaction. But it is perhaps more useful to rewrite this in more modern terms as: If object A exerts a force on object B, then object B exerts and equal and opposite force on object A Let’s be careful about the notation here. I have written F12. By this notation I mean the force applied by object 1 on object 2. Likewise the notation F21 stands for the force applied by object 2 on object 1. The minus sign in the way we have written the 3rd law just reflects the fact that while these two vectors are equal in magnitude, they have exactly opposite directions. How can the third law be true? The 3rd law is simple to state, but is quite surprising. No one before Newton ever recognized it. It almost seems it can’t be true. If every time I push on you, you push back equally on me, how can anything get anywhere? Don’t those two forces always cancel out? Consider the way you throw a ball. The 3rd law says that Fpb = -Fbp. That is, the force the ball exerts on the person is equal and opposite to the force the person exerts on the ball. But the ball goes flying off, and the person does not. What's going on? The reason for this asymmetry of outcome is that the force of the person on the ball is the only force acting on the ball, but the force of the ball on the person is not the only force acting on the person. The two sets of forces are drawn below: Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 The forces on the person are the force of the ball on the person (Fbp) and the force of friction between the person’s feet and the floor. So the total force on the person is zero, and her motion doesn’t change, but the total force on the ball is non-zero, and its momentum is suddenly increased. Note that I have not drawn this picture at a time after the ball was thrown, I have just drawn separate pictures of the ball and the person while it is being thrown. After it is released there is no interaction at all between the ball and the person. Also note that to simplify this picture I’ve left out the force of gravity which pulls down on both the person and the ball, along with the counteracting upward forces that balance the pull of gravity. We’ll have much more to say about this a bit later. Third Law Thought Experiments: Since the third law was somewhat surprising, it’s useful to consider a few thought experiments. These are offered up as examples which might help to ease you into acceptance of this very important idea. Consider what happens when a horse pulls a stone with a rope. In the process the rope is stretched. It will do whatever it can to "ease" the stretch, so it both pulls the rock forward, and pulls backward on the horse. The effect is that the force with which the horse pulls the rock forward is just equal to the force with which the rock pulls the horse backwards. Fbp Ff Fpb Frope-horse Fhorse-rope Frock-rope Frope-rock Wrock Fground-rock Ffriction-rock Fground-horse Ffriction-horse Whorse Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 The first of these has to do with the third law. It says that if the Earth is pulling you down with a force FEarth-You, you must be pulling the Earth upward with a force FYou-Earth which is equal and opposite to the first. That is: FEarth-You = -FYou-Earth While the Earth pulls you down, you pull it up. That action-reaction pair is always there. Why doesn’t the Earth come rushing up to meet you when you jump off a cliff? After all, you’re pulling it upward just as hard as it’s pulling you down. The reason for this disparity, which emerges always in unequal matches like this, is the relatively enormous mass of the Earth. Your weight (FEarth-You) is a big enough force to quite easily change your motion. But a force of the same size (FYou-Earth) is much too small to make any appreciable change in the motion of the Earth. So although these two forces act to pull you and the Earth together, it’s you who does all of the moving. The second thing worth discussing in some detail is the sensation of weight. What is it we feel when we feel our own weight? Can you feel the force of gravity upon you? Imagine what happens when you jump off a chair. For a moment you are floating freely in the air. Do you "feel" a force tugging on you? While you’re in the air, there is no sensation of force at all. Try this if you dare, but please be careful! So what is the sensation of weight? What is this feeling you get? When you are standing on the floor, the sensation that you feel is of pressure on your feet. This sensation is exactly what you would feel if, while you were lying down, someone pressed with a board on your feet. Now think about what you feel when you sit in your chair. Is there any pressure on your feet? Now the pressure seems to be on your backside. And if you stand on your head you "feel your weight" on your hands and head. So could what we feel really be the “weight”? Is what you feel really the force exerted on you by the gravity of the Earth? In fact it is not. The sensation you feel when you talk about weight is actually the force which something else applies to you to resist the downward pull of gravity, to prevent you from falling downward. When you are standing still your weight (the force of gravity on you) pulls you down. In order to remain stationary, some other force must balance this. That force is provided by the object which you are in contact with. The sensation you feel as weight is just the force of the floor (or your chair, or whatever) pushing up on you to resist your weight. Take away the floor, or the chair, and your sensation of weight would vanish, but the weight itself (the downward pull of gravity) would not. Contact forces and a second force: the normal force Most of the forces we encounter in our lives fit are contact forces. They arise from the direct "touching" interaction between two bodies. When two more or less solid bodies are in such direct, atom-to-atom contact, it will usually be useful to talk about the force between them as being composed of two parts. We will break the total contact force Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 between objects into a component perpendicular to the plane of contact between the surfaces (the normal force) and a component along the plane of contact between the surfaces (the friction force). The first of these is the part which prevents one object from moving through the other. Let’s look at an example. When I put a book on a table, the book's weight tries to pull it downward. To move downward it would have to pass through the table. The table prevents the book from doing this by pressing back up on it with a force which keeps it in place. This force (which acts to prevent the objects from passing through one another) always acts perpendicular to the plane of contact of the objects. Because of this it is called the “normal” force. It is NOT normal in the sense of "usual", but normal in the mathematical sense of “perpendicular” to the surface between the two objects. How does this normal force occur? Ultimately all of the forces (save gravity) which we will talk about in this course are consequences of electromagnetism. Electromagnetic interactions determine whether two atoms placed close together will resist being pushed closer together or attract one another (perhaps bonding together). In the second semester of this course, Physics 235, you will learn quite a lot about the nature of these electromagnetic forces. For now, we will encode a lot of complicated atomic interactions in a few simple phenomenological rules. There’s a real mystery here. How can an inanimate object like a table exert a force? More important, how can it “decide” how much force to apply; knowing to push upward very lightly on a pencil while providing a much larger force for lamp? The answer lies in the passive nature of the normal force. It exists only in response to some other force. Begin by thinking about a cushioned chair. When you sit on it, the chair compresses to the point where to compress it further would require a force larger than your weight. In other words, as it is "squashed", it pushes back up on you, harder and harder, until it is pushing up on you with a force equal to your weight. This squashing, this distortion, is what allows the chair to push back up on you. When you push atoms closer together, they push back. Now imagine something "harder" than this cushioned chair. If you stand on a plastic chair, it too is distorted until it pushes back on you just enough to balance your weight. Take this to its conclusion; when you stand on the floor, the floor actually distorts until it pushes back on you with a force just large enough to prevent you from falling through it. This distortion, which always accompanies forces applied to solid objects, is perfectly real, even when it is not apparent. BookBook Table FN FW Table Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 So this "normal force" prevents objects from passing through one another. How big is this force? What is its magnitude? The basic answer is "whatever it has to be". For this reason, the normal force is our first good example of a passive force. Passive forces have magnitudes which are not determined in advance. They arise, and adjust themselves, in response to active forces. Their magnitudes are determined based on the restrictions which give rise to them. They can usually be any size up to some limit at which the object which is creating them breaks. So, if I push down on a table, it pushes back up on my hand with a force just equal and opposite to my own. If I push harder the resisting normal force increases. If I stop pushing it goes away. The force adjusts itself to be just as large as it needs to be to prevent my hand from moving through the table. Free body diagrams Notice what we did there. In order to understand what happens to two bodies while they interact, we have drawn each of the bodies separate from the others, so that we can understand fully the forces on each one. Let's look at a couple of other examples of this. What happens when a book sits on the table? What are the forces on it: First we might draw the circumstance: Now, in order to understand it, I draw a free body diagram for each part of the problem. First consider the book. What are the forces acting on it? It experiences a weight, the gravitational force of the Earth pulling it downward. Since it is sitting still, it must also experience some other force which balances this. This is the force with which the table pushes back up on the book. This is called the normal force. Now in order for it not to move we know that: FN = FW = mbg. These forces must be equal in magnitude (and opposite in direction). So now we know that FN = mbgy. What does Newton's third law say about this? It says that for every force there must be an equal and opposite reaction force. What are they here? The table pushes on the book with a force: FN = mbgy, so the book must push on the table with a force FBT= -FN =-mbgy. FTable-Book = FN FEarth-Book = FW Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 The other part of the contact force: friction The normal force is what prevents objects from passing through one another. It is the part of the total force between two surfaces which is perpendicular to their plane of contact. The rest of the force between two bodies is that part of the force which is parallel to the plane of contact. This force acts to prevent one object from slipping over the other. It resists their relative motion. We call this part of the interaction between two objects the force of friction. We will have a lot more to say about friction and how it works next time. For now let’s just concentrate on the manner in which friction acts. Friction acts in an attempt to prevent relative motion along the plane of contact between two objects. Let's look at a simple example; an object sitting at rest on a slope: What are the forces on this? There is a weight acting straight down. Then there is some interaction between the book and the surface of the wedge it sits on. This total contact interaction has two parts; a normal force perpendicular to the surface, and a frictional force along the surface. I know (because its motion doesn’t change), that the total force Ftotal = FN + Ff + FW = 0. So now I can write the forces: Very often in a problem like this it is useful to work in a coordinate system which defines directions along and perpendicular to the surface between two bodies. Such a coordinate system is shown above. Now we know that since it doesn't move the sum of the forces must be zero, and that in turn means that the sum of the forces in each direction must be zero. Now let's add up these forces in each direction: ∑Falong = Fwsinθ - Ff = 0 ∑F⊥ = FN - FWcosθ = 0 So in this case we know that: Ff = FWsinθ = mgsinθ and FN = FWcosθ = mgcosθ FN Ff FN FW Ff θ perpendicular along FW θ Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Once again we see that the normal force is not equal to the weight (it rarely is), and in addition we see how we can use the motion (or lack of it) to figure out how large this "frictional force" must be. Now this picture, a block on an “inclined plane”, is the very icon of traditional introductory physics courses. Poor students have been learning to analyze these for literally hundreds of years, and they’ve always seemed completely unconnected from everyday experience. After all, most of you stopped playing with blocks some time ago. But in fact this example, while drawn in an abstract (and easy to draw) way, is very much an everyday experience. Here are two examples. The first is standing on a slope: Every time you stand on a slope a situation very like what we just described happens. If there is not enough frictional force preventing you from sliding down the slope you will slip downward. No doubt this is something you’re quite aware of, and once snow and ice arrive you’ll be careful about standing on slopes. The second example is the slope itself! Each layer of a hill would slip downward if not held in place by a frictional force. Sand dunes, like Sleeping Bear dune in Northwest Michigan, provide a vivid example of this. When they become too steep, the force required to hold the sand in place becomes larger than the frictional force available, and top layers of sand begin to slide downward. The maximum angle for a pile of sand is often given the poetic name “the angle of repose”. More impressive and dangerous versions of this are seen all the time in avalanches and landslides. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 A way to transmit force; ropes and tension We have seen how simple contact forces can occur due to the compression of bodies, as when a book sits on a table. It is also possible for bodies to exert forces when they are “stretched”. This process of attempting to stretch a body is called putting it “in tension”. A simple example of how an object in tension behaves is given by a mass hanging on a rope. Imagine a mass hanging from a rope attached to the ceiling. We can draw three free body diagrams: What do we know about these? We know that for the mass at the bottom: ∑Fy = Frm - mmg = 0 so Frm = mmg and for the rope: ∑Fy = Fcr - Fmr - mrg = 0 so Fcr = Fmr + mrg We also know that some of these are third law partners, so their magnitudes must be equal. In particular: Fcr = Frc Frm = Fmr So now we can say: Fcr = Fmr + mrg = Frm + mrg = mmg + mrg = (mm + mr)g So the force which the ceiling must exert to support the rope and the weight is just equal to the weight of the mass plus the weight of the rope. Not too surprising. What about the forces within the rope? Now let’s think about what's really happening in the rope….Picture the little piece of the rope at the bottom. The mass is pulling down on it with a force mmg. This little piece has some mass Δmr, so we can write the same kind of free body diagram for it…. Frm FEm = mmg Fcr Fmr Frc FEr = mrg Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Why would physicists talk about massless ropes: Let's go back a step to the simple hanging mass problem. We found that the force exerted by the ceiling on the rope was: Fcr = (mm + mr)g In other words the rope isn't a perfect force transmitter; it isn’t just transmitting a force from the ceiling to the hanging mass. Some extra force is needed to support the weight of the rope. How important is this? The answer depends on the details of the problem. If the mass of the rope is very much less than the mass of the hanging object mr<<mm, then we can say with some accuracy: Frc ≈ mmg In this case the rope is a very good force transmitter. It allows the ceiling to exert a force on the object below without adding anything to it. What if mr>=mm? Now the force the rope exerts on the ceiling is at least twice would it would be if we directly attached the mass. So in this case the rope is a really poor force transmitter. For some of the problems we will do we will assume that the mass of a rope is "small enough" to "not matter". If the rope is assumed to be "massless" in this way, then it becomes a perfect force transmitter. Any force applied to one end is directly transferred to the other end no matter what the circumstances. In technological cases, people always try to use ropes where this is the case, choosing a rope strong enough to support the load, but light enough to be able to transmit almost all of the force to the load. Very often it is a reasonable approximation. The one case where this becomes very difficult is with very long ropes, like those used to sample material at the sea floor, or the cables used in the constructions cranes which have become so common in Ann Arbor. When we assume such a massless rope, the force exerted on one end is directly transmitted to the other. This force which is transmitted is what we call the "tension" in the rope. So, if the tension in a rope stretched between two objects A and B is 50N, this means that object A is pulling on object B with a 50N force, and object B is pulling on object A with a 50N force. This force is perfectly "transmitted" by the rope, with no loss. Tension and force transmission: Consider the following situation. A block (mb) hangs on an essentially massless string. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Considering the block as we did above we find: The fact that the block is at rest implies that T = mg. With what force does the string pull on the wall? Now consider a slight variation. Instead of attaching the string to the wall, I hang a second, identical weight off a second pulley. What is the tension in the string now? It is still T=mg. Remember, all the string does is transmit a force from one end to another. It doesn't matter if that force comes from the wall or another block, it is still just transmitted. Reducing forces required with simple machines There is one last general topic to raise before moving on. We have been discussing the forces required to cause objects to move in various ways. We found, for example, that to lift an object at a constant rate straight up we have to apply a force just equal to its weight. I want you to think a little about what limits the actions people can take. When a person tries to do something, tries to make something happen, they are generally limited by the size of the maximum force they can apply. This is true whether you’re lifting a large stone or trying to open a jar; people are generally force limited. So how can we do things when the force required is too large for us to create? We have to get some help, and there is a general class of ‘tools’ which have long been known to help evade these limits. These tools are generally called ‘simple machines’, and they are probably best thought of as force magnifiers. Fsb = T FEb = W = mg Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 A first machine: the inclined plane Let’s start with the simplest simple machine: the inclined plane. If I want to lift an object straight upward at a constant rate without help, I have to apply a force equal to its weight: If I’m not strong enough to supply this large a force, I can take advantage of the simplest simple machine: the inclined plane. To slide a box up such a plane at a constant speed, I will have to push with a force Fpush such that: ΣFalong = Fpush - mgsinθ - Ffriction = 0 Fpush = mgsinθ + Ffriction ΣFperp = FN - mgcosθ = 0 FN = mgcosθ So long as I can keep the friction low, I can ‘lift’ an object of arbitrary mass with my small maximum force by making the angle θ small and sliding the block up the plane. This kind of simple machine was obviously known to the early Egyptian pyramid builders who combined large numbers of laborers (increasing Fmax) and shallow inclined planes to lift quite enormous blocks of stone. If you think about this a little more generally, this trick has allowed you to ‘magnify’ your force by a factor of 1/sinθ. You apply a force Fpush in a direction along the slope and you are able to generate a force approximately perpendicular to this which has a magnitude Fpush/sinθ. The force magnification of the inclined plane is at work in many other systems as well. An excellent example is a knife. What are you doing when you cut? You’re trying to tear a material apart, to pull the two sides of the material apart with so much force that the bonds holding it together come apart. Doing this generally requires a large force. By forcing in a very thin knife blade you can generate a large sideways force. In this case, if you push in with a force Fpush, you generate a sideways force with a magnitude on the order of Fpush/sinθ. This is what allows you to smoothly cut apart an object which would otherwise be almost impossible to rip apart. ΣFy = Flift – mg = 0 Flift = mg Fpush θ θ Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Physics for the Life Sciences: Fall 2006 Lecture #4 Torque and rotational statics We're going to extend our discussion of objects at equilibrium; things which aren't accelerating. We know from Newton's second law for translational motion that: ∑F = dp/dt so if dp/dt = 0, then ∑F = 0. This is the first condition for equilibrium: if the momentum of an object is not changing, the vector sum of the forces on the object must be zero. For point objects this is all we need to know. Now it is time to go beyond this, and begin thinking about how extended bodies will behave under the influence of forces. What happens if I have the following arrangement; a bar with a rope holding it up on one end? This is a body for which ∑Fx = ∑Fy = 0. Do you think it will remain at rest? No, it will begin to rotate. Whenever a body is extended, larger than a "point" object, it is necessary to know both what forces act on it, and also where the forces are applied. You all know from experience what we would have to do to prevent rotation in the system described above; we'd just hang it from two ropes: Why is this case stable when the other wasn't? Because here we have one force which "tends" to make the object rotate clockwise, and one which "tends" equally to make the object rotate counterclockwise. This is the basic idea of our second condition of equilibrium; the forces applied to an object at equilibrium must be applied in such a way that their tendency to make the object rotate cancels out. mg T = mg T = 1/2mg mg T=1/2mg Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Why haven't we talked about this before? A lot of the time we have considered point objects, and any object which does have extension really can't rotate. They're just points, and a point can't rotate. So any time I apply a force to a point, it can only translate. When we have considered problems involving extended objects, we have just ignored rotation. Not because it wasn't possible, just because we hadn't gotten to talking about it yet. We will look at some examples of how rotation comes into problems like those we have done later today. Quantifying the ability of a force to make something rotate: It should be clear from this little discussion that the ability of a force to make an object rotate depends not only on how large the force is, but also on where it is applied. If you want to open a door, to make it rotate around its hinges, you can either apply a large force close to the hinge, or a small force very far from the hinge. The direction you apply the force also affects the result. If you push on a door along a line which passes through the hinge, the door will never begin to rotate. The number which quantifies this "ability to cause an object to begin to rotate" is the called the "torque". A first definition of the torque which will generate rotation around a center c is: τc = r⊥F where r⊥ is called the "moment-arm" of the applied force and F is the magnitude of the force. This "moment-arm" is illustrated in the following figures: There are three lines drawn in each picture. In each the solid line is a vector representing the force F. The dot-dashed line is a vector drawn from the center of rotation to the point at which the force is applied. We call this position vector r. The dotted line perpendicular to the force F in each case represents the "moment-arm" associated with this force, r⊥. One thing to note is that I can move the force forward or back along its direction and produce exactly the same r⊥. A second thing to note is that the rotation which is produced by a force has a particular direction, it can cause rotation one way or the other. We record direction of rotation by using the "right hand rule". One way of stating this rule says that if you curl the fingers of your right hand in the direction of motion of the rotation, your thumb points in what we define as the direction of rotation. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Experience, and a second way of looking at torque Does this agree with experience? Let's look at a few limiting cases. 1. F is perpendicular to r. Now |r| = r⊥ and the torque is just τ = rF. This is the familiar case of opening a door by pushing perpendicular to its surface. I can create the same torque by either pushing with a large force close to the door hinge, or pushing with a small force far from the door hinge. The torque is the product of these two. 2. Force F is parallel to (or anti-parallel to) r. In this case the force points towards or away from the center of rotation. In this case r⊥ is zero, and the torque associated with this force is zero. Applying a force like this can never cause rotation about this center of rotation. 3. Force F is applied at the center of rotation. This is really a subset of the previous example, and it also generates no torque and causes no rotation. These facts suggest a second, equivalent way of looking at torque. We can take each force which acts on a body and break it up into two components, one directed along the line to the center of rotation and one perpendicular to it. The component along the line through the center of rotation will generate no rotation. Only that component of the force which is perpendicular to the line through the center will cause rotation. This suggests another way to determine the torque generated by a force: τ = rF⊥ So now we have two alternate ways of looking at it: τ = r⊥F = rF⊥ Compare these on the drawing: Torque and the cross-product There is a general way to see what the magnitude of the torque will be: τ = r⊥F = rF⊥ = rFsinθ where θ is the angle between the vectors r and F, as shown in the drawing: You can see this by noticing that Fsinθ = F⊥, or by seeing that rsinθ = r⊥ as shown below: r⊥ F⊥ θ θ F r Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 3. Neutral equilibrium: a small displacement leaves the object in a new equilibrium position These three are illustrated by thinking about a cone. When the cone is standing on its bottom, it is in stable equilibrium. Tip it a bit and it falls back into place. If it’s standing on its tip, it is in unstable equilibrium. Tip it a bit and it falls over completely. If you lay the cone down on its side it is in neutral equilibrium. Roll it over a bit and it just lies there. For simple cases of balance, like those which apply to many organisms the following rules apply: 1. If a tilt away from equilibrium raises the center of gravity, the object is in stable equilibrium 2. If a tilt away from equilibrium lowers the center of gravity, the object is in unstable equilibrium 3. If a tilt away from equilibrium leaves the height of the center of gravity unchanged, the object is in neutral equilibrium This is because of the torque exerted on the object by the force of gravity. If you’re raising the center of gravity when you tip it, the force of gravity will tend to pull it back down. If you’re lowering the center of gravity, the force of gravity will tend to pull it away from where it started. When will an object "tip" over? Any time the center of gravity of an object is not above the supporting surface of the object it will tip. This is because once the CG moves past the support point, gravity exerts a torque which tends to tip the object over. Consider these pictures to get an idea of how this works. As we tip this stiff little person over he is, at first, stable. We’re still raising the CG. Eventually, the CG moves outside the support point, and is now moving downward. This is unstable. Stable Unstable Neutral Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 A quantitative example of rotational equilibrium: a truck on a bridge A first useful example comes from analyzing a truck driving over a bridge which is supported on its two ends. What might we like to know about this? Imagine we know the weight of the truck, and we want to know how much force must be applied by the right and left hand supports to as the truck moves across the bridge. This is just the sort of thing an engineer or an architect might need to know to make sure the bridge is safely constructed. Consider the forces on the bridge slab. There is an upward force on each end, a downward force of the weight of the bridge, and a downward force due to the mass of the truck. We can find the the magnitudes of FL and FR by using the equations of equilibrium. ∑F = 0 = FL + FR - mtg - mbg and ∑τ = 0 = -mtgx - mbg(L/2) + FRL where for this second equation I have calculated the torques with a center of rotation at the left hand end of the bridge. Why do I choose this center? After all, I could calculate the torques around any center and they must always be equal. This choice is made purely for convenience. By picking a center through which one of the two unknown forces passes (FL goes through this spot) I know that only one of the unknown forces will appear in the torque equation I obtain. This just makes the algebra a little simpler than it would otherwise be. Solving the torque equation yields: FR = mtgx/L + mbg/2 And plugging this back into the first equation gives us the other unknown force FL = mtg(1 – x/L) + mbg What does this mean? It means that the upward force exerted by the right hand bridge support is half the weight of the bridge plus a fraction of the weight of the truck that varies as it drives across the bridge. When it is over the left hand support, all of the truck’s weight is supported on the left, when it reaches the middle, half of it is supported by each, and when x=L, all the weight is supported on the right. The upward force exerted by the left hand support makes up for the rest of the weight of the bridge and truck. Added together, FL + FR = mtg + mbg, all the time. This is a nice answer, which we might have anticipated, worked out using our equilibrium conditions. x L FL mtg mbg FR Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Example 2: Weight lifting Your bodies, and the structures of other living things, obey these same principles of statics. These simple ideas, that in a static situation the sum of forces and sum of torques must be zero, is all it takes to understand a lot about how your body works. Let's start with a simple example: lifting a one dumbbell: Imagine that the mass of your forearm is mfa, and the mass of the dumbbell is md. How do you hold this up? You have a bicep which attaches to your forearm just a few centimeters from the joint in your elbow. If you poke around the inside of your elbow now, you can feel the tendons which connect this muscle to your forearm. So the picture looks something like the situation on the right. Remember, the only way to analyze forces is to consider only a single object in a free body diagram. So let's look just at the bone in the forearm: If you just had the three forces we mentioned (weight of forearm, weight of barbell, and upward bicep force Fb), what would happen? Your forearm would rotate. So, there must be another force. What is it? It's the force of the end of your upper arm bone pushing down on the end of your forearm. This sort of thing is always happening when your body supports weight; you do this by pulling with muscles and pushing with bones. The combination of the two is what allows your full range of movement, and you need them both: muscles must have something to pull against. To determine the size of these various forces, we have two facts to work with: ∑F = 0 and ∑τ = 0 Here there are only y forces, so: ∑Fy = Fb - Fua - mfag – mdg = 0 Fua = Fb - mfag – mdg If we sum the torques around the end, we find: ∑τ = xFb - (L/2)mfag – Lmdg = 0 or Fb = (L/x)[1/2mfag + mdg] Fu mdg mfag Fb mdg mfa Fb Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Second, the force applied by the Achilles tendon is larger than half your weight, usually by quite a large amount. Just holding a ruler up to my foot I find Lach ≅ 5 cm and Ltoe ≅ 20 cm. So the upward force applied by each of your Achilles tendons is about four times as large as half your weight, or twice your total weight! This is why your calf muscles are probably substantially larger than your biceps. You rarely lift twice your weight with your biceps, but you do it all the time with your calves, with every step you take. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Physics for the Life Sciences: Fall 2008 Lecture #5 Last time we learned about the two conditions for equilibrium. At least in some cases, these rules allow us to understand how organisms stand up to the pull of gravity. Today we will see that these two rules are, almost always, not enough to really figure out what’s happening. To really understand organisms and other structures, we need to understand how the materials of which they are made apply forces. How can a bone, or a chair, or a string, apply a force? Answer that and you have the crucial piece for understanding equilibrium. The problem with our two equilibrium conditions: What’s the problem with our equilibrium conditions? The condition for equilibrium requires: ΣF = 0 and Στ = 0 (around every center) We have noted these conditions before, when we first started to consider rotational motion. Note that since each of these is a vector equation, there are really six different equations here. These include the forces in each of the three directions, and torques around each of three perpendicular axes. Remember the equilibrium problems we did: In each of these cases, we looked at the sum of the forces in each direction, and at the sum of the torques around any center, and used the constraints of equilibrium to determine the required sizes of unknown forces. Let’s try a new example to remind you how this works. Consider a ladder resting on a floor which exerts a frictional force on it but which leans on a frictionless wall: mbg mfag F x L Ff FNW mLg FNF θ Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 How large is the frictional force applied by the floor? Use the constraints: ΣFx = FN – Ff = 0 ΣFy = FNF – mLg = 0 or FNF = mlg What about the torques? Let’s calculate the sum of the torques around the point at the top. Defining counterclockwise rotation as positive, I have: Στ = FNFLcosθ - mLg(L/2)cosθ - FfLsinθ = 0 or mLg(1/2)cosθ - Ffsinθ = 0 or Ff = mLg/(2tanθ) As the angle gets smaller, you’re going to need a larger and larger amount of friction to prevent the ladder from slipping. Your intuition will tell you that there’s a limit here. If you try to stand the ladder up at too steep an angle it will slip. No big surprise in this. Now let’s consider a just slightly more complicated case. The same situation, but now imagine that there is some friction with the vertical wall. Of course there would be in any real case. How are the equations changed? ΣFx = FN – Ff = 0 ΣFy = FNF + FfW – mLg = 0 and still Στtop = FNFLcosθ - mLg(L/2)cosθ - FfLsinθ = 0 But now I can no longer identify FNF with mLg, because part of the weight may be supported by friction with the wall! What’s going on here? I have three equations to work with, and four unknowns: FNF, Ff, FNW and FfW Ff FNW mLg FNF θ FfW Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 tissues are made of many tiny “cells”. In fact Hooke invented the name. In many ways, this book was a starting point for the life sciences. You can look at some of this exceptionally beautiful book online at: http://archive.nlm.nih.gov/proj/ttp/flash/hooke/hooke.html For our purposes today though, Hooke’s most important discoveries had to do with how objects support loads. His main realizations are part of what is now called Hooke’s law. 1. Objects can support loads only by yielding to them. If you push on an object, it will push back by distorting. 2. Most solids are elastic as opposed to plastic. This means that if you squash them and let them go, they spring completely back to their original shapes. Elastic objects (like rubber) do this. Plastic objects (like clay) change shape permanently when you distort them. 3. Hooke quantified all of this by noting that the amount of deformation is, in many solids, proportional to the load These observations are encoded in a general form as a very simple law: F = -kΔx If you deform a solid (change its length by an amount Δx), it will push back with a force which resists your deformation. The solid is trying to return to its original shape. The constant “k” tells you how hard it will push for a given deformation Δx. Each object has a different constant, which you can determine by apply a force F and seeing how far it distorts (Δx). If the constant k is small, the object is easy to deform; you would call it flexible. If the constant k is large, the object is hard to deform, you would call it stiff. The minus sign in this equation just means that if you stretch an object out, so that Δx is positive, the force the object exerts on you will be negative, in the opposite direction. Limitations to Hooke’s picture: it only works for individual objects The problem with Hooke’s law for practical purposes is that it makes no predictions about what k will be for a particular object. If I’m interested in one particular object I can measure k for that object, and then predict exactly how it will deform under a load. But this is not too practical in making a new building for example. You sort of have to build it first before you can see whether it will work. This was, in fact, how cathedrals were built in Europe. A fair number were constructed, fell down, and then were rebuilt with more extensive supports. The basic problem is that Hooke focused on individual objects, rather than the materials of which they were made. He could tell you about an individual spring or beam, but not usually about a new one. To give a specific example, Hooke could not answer the question: “I know the constant k for this particular beam. What will the new constant k’ be if I make a beam of the same material which is twice as long?” Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Focusing on materials instead of objects What’s the key to answering this? To begin, let’s think about a toy model of what a solid object is like. Solids are built of atoms more or less locked in place by bonding with their neighbors. You can think of this as an array of atoms held together by rather stiff springs. If I apply a force which is spread out over the top of this solid, it will have to squash all of these springs. If I apply the force to just a little spot, it will have to squash only a few springs. This is easier to do, so that same force applied to a small spot will squash things more; it will create a larger distortion. To quantify this, consider a 1 and 2 spring model, where each individual spring has the same spring constant k. This would be the case, for example, for each of the little springs that connect the atoms in a solid. If I apply a force F to one spring, it is compressed a distance Δx = F/k. If I put two such springs in parallel, the combined spring constant will be 2k, and the same force F will compress them only half as much Δx2 springs = F/2k You can see from this example that what matters is the force per spring. If I measure the force per spring, all objects which are made of this material (whether big or small) will behave the same way. Now we can’t actually measure the “force per spring” unless we know exactly how far apart the atoms are. So instead, we just account for this by asking whether the force is spread out over more springs or fewer. We can do this by just measuring the area over which the force is applied. If we double the area, there will be twice as many springs, etc. So instead of just measuring the force applied to an object, what we will care about is the force per unit area. In this application the quantity force per unit area is called the stress, and it’s defined as: stress = Force / Area = F / A = σ Here F is the total force, and A encodes the area (proportional to the number of atoms) this force is applied over. The symbol σ (“sigma”) is usually used for stress. Notice that stress is really the same as the more familiar “pressure”. There’s no clear distinction between the two, though the term pressure is used more often in cases of fluids, as for air pressure or hydrostatic pressure. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Now think about the displacements in a real material. One spring again displaces by a distance Δx = F/k. Now imagine I have two springs in series, stacked on top of one another. In this case each spring will compress by a distance Δx, because the same force will be transmitted through both. The combination of two springs will compress twice as much as one. This is somewhat tricky, so think about it carefully. This suggests that what matters for displacement is the compression per spring. Again, we can’t really do this per spring without knowing exactly how far apart the atoms are, so instead we use the thickness of the material as a way of tracking whether there are more or fewer atoms. If the material is twice as thick, there will be twice as many atoms and the material will compress twice as far. To keep track of this, we measure not just the change in length of the material (Δx) but the fractional change in length, which is called the strain. It is defined as ε = ΔL / L = strain Here ΔL is the total displacement, and L keeps track of the number of atoms which are compressed in series. Stress and Strain: avoiding the obvious confusion The words “stress” and “strain” are great examples of the problem with using ordinary words to describe physics concepts. In everyday language, both stress and strain mean something similar. But in physics they mean very particular, quite different things. The “stress” is a measure of how much force per unit area is applied to an object. It has units of N/m2. The “strain” is our measure of how much an object is distorted by the stress which is applied to it. Strain, measured as the ratio of the distortion ΔL to the total size L, is dimensionless. You’ll have to find a way to remember, unfailingly, the difference between these two. One way is to practice the mantra: “stress causes strain…stress causes strain…stress causes strain…” until you can’t think of it any other way. If you invent some other good mnemonic for this please share it with your peers and your instructor. It’s been a problem for students throughout the ages. Maybe you can solve it. Stress causes strain: a material dependent version of Hooke’s Law For every different kind of material, there will be some relation between stress and strain. If you apply a stress σ, you will measure some amount of strain ε. Very often it is the case, at least for small stresses, that the stress and strain will follow a simple, linear relation. Now, although you would probably do the experiment by applying a stress and measuring a strain, people usually plot it by showing the strain as a function of the stress. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 legs will vary like size2. This implies that the stress in the rhino’s legs, which is force per unit area, will vary like F/A ∝ size3/size2 = size. If you increase the size of a rhino by ten times, the stress in its leg bones will increase by a factor of 10. Remembering that the material making up the bones stays the same, you can see how just making an organism larger becomes risky very fast. For this reason, large animals have different shapes from small ones. They have evolved proportionately thicker legs. They have also adopted postures which tend to keep the legs straight and under the body. On the other end of the size spectrum, tiny organisms have no trouble at all keeping the stress in their limbs low. As a result, they can adopt a much wider array of shapes, with long spindly legs that allow them to walk uninterrupted over the extremely varied terrain they see at their size. So you can see that an understanding of how materials support loads, of the importance of stress, helps us to understand a lot about the diversity of form we see in living things. Organisms don’t simply choose the shapes they take. These forms are, in a very real way, imposed on them by physical constraints. There is much more we could say on this topic, and you’ll have an outside reading2 to give you a broader perspective on it. Other kinds of stress and strain There are several other kinds of stress and strain. Since they affect the “springs between the atoms” differently, they have to be accounted for differently. The first new kind of stress is called “shear”. Shear is what happens when you try to shove the top of something sideways relative to the bottom. As an example, consider laying a textbook on the table, then sliding the front cover of a textbook to the left while pushing the bottom to the right. 2 Michael Labarbara, “The Strange Laboratory of Dr. Lababara”, University of Chicago Magazine, Oct.- Dec., 1996 F Δx L Can you tell which is smaller? Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Stress = F / Area of top Strain = Δx / L This is called ‘shear stress’ and ‘shear strain’. Usually this would be written: σshear = Sεshear or F/A = S(Δx/L) For every material there is a constant associated with the response to this stress. For shear stresses this is called the “shear modulus” and usually denoted S. Shear stress tries to make one layer of a material slide over another. Note that the shear modulus and the Young’s modulus can be very different. So to know how a material will respond, you need to know what kind of stress is applied to it. A third kind of stress and strain is called bulk stress and strain, or “hydrostatic” stress and strain. Stress = F/A Strain = ΔV/V This is the kind of stress and strain encountered when the object is under pressure which squeezes in from every direction, like when it is deep under water. For this we write: σbulk = Bεbulk or F/A = B(ΔV/V) For every material there is a constant associated with the response to this stress. For shear stresses this is called the “bulk modulus” and usually denoted B. You can see how all three of these stress/strain relations are really the same. They’re all just expressions of the basic model of a solid as a collection of springs connecting atoms. Limitations to the “Hooke’s law” model of linear stress/strain relations Hooke’s law, the linear model of how stress creates strain, is an empirical, phenomenological, law. Many materials behave like this under modest stresses. We certainly expect it to break down eventually. It can’t be right when the stress becomes large enough to break the object. Likewise, we know that there are materials with more exotic behaviors. Let’s talk about these limitations in turn. 1. Strength: what happens if the stress becomes too large, and you stretch the object too much? F Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 a. First, it stops behaving elastically, and no longer returns to its original shape when you remove the stress. For each material, this happens at some stress which is called the ‘elastic limit’ of the material. If the stress rises above this, it will become ‘plastic’ and permanently deforms. This behavior might seem familiar to you if you think about bending something made of metal, like a paper clip. Bend it a little, like when you use it normally, and it springs back to its original shape. Bend it a lot, and it stays permanently deformed. b. Eventually the material breaks. We sometimes talk about this happening at the ‘breaking stress’ of the material. Actually, what happens when things break is much more complicated. We will look at some features of tearing and shattering after we discuss energy. c. Note that since stress is related to strain in materials, we could talk about either the breaking stress or the breaking strain of a material. Which one we use may depend on the application. We can divide materials up in several interesting ways based on how they break: • “Strong” material: high breaking stress (supports big loads) • “Weak” material: low breaking stress (can’t support big loads) • “Stiff” material: low breaking strain (can’t be stretched much before breaking, no matter how large that stress is…) • “Flexible” material: high breaking strain (can be stretched a lot before breaking) Here are some illustrative examples of various kinds of materials: • Steel: stiff, and strong • Biscuit: stiff, but weak • Nylon: flexible and strong • Jello: flexible and weak 2. Nonlinear, non-Hookian flexible materials: Most man-made objects are made of relatively stiff materials that have linear stress/strain relations extending to a large fraction of their breaking stress. They are mostly dry, hard, solids which behave more or less according to Hooke’s law. By contrast, most biological structures are often made of wet, squishy stuff which doesn’t obey a linear Hooke’s law stress/strain relation. These materials are flexible, able to stretch a lot before breaking, but still very much elastic. They easily deform and then spring back to their original shape. Nonetheless, they are often quite strong, with high breaking strains. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Physics for the Life Sciences: Fall 2008 Lecture #6 Now that we have learned a bit about statics, about the balances of forces and torques needed to keep objects in place, it’s time to delve a little deeper into how some of these individual forces work. This time we will concentrate on friction, a very important player in the lives of organisms, and in all of our technology. Without it there is no way we could get around at all. Place of friction in motion Friction is the tendency of objects in contact to resist relative motion, to stick together. It is the dominant factor in most motion which we observe in our world. Its tendency to bring any moving body to rest is what so reinforced the Aristotalean view that "motion implies a mover". Because of the resistance which friction provides, experience suggests that a continuous force is required to keep something moving. To uncover the real principles of dynamics, Galileo had to imagine a world without friction. In such a world motion could be perpetual, and no force at all would be required to maintain it. Constant motion is as natural a state as rest. Because of this, Newton focused our attention on forces as the cause of changes in motion. He also showed that to predict the motion of an object, all you need to do is to understand the forces which act on it. So if we wish to understand the influence of friction on the motion of bodies, we need to understand how to predict the magnitude and direction of the frictional force. We will start by examining one particular kind of friction, the "sliding" friction which occurs when two dry, solid surfaces slip across one another. This is the force which causes a book to slow to a halt when you slide it across the table. To understand forces like friction, we often will often seek a "force law". Such a force law will tell us how large a force on a body would be if it had a particular set of properties (such as mass, composition, surface condition etc). Establishing force laws is a basic task in physics, and the force laws which we derive tend to fall into two categories: Fundamental forces: (like gravity). These laws seem to reflect an underlying reality at a level which suggests that they are "True" with a capital T. Their behavior tends to become simpler and simpler as we look at them more closely. These laws approach the underlying basics. It turns out there are only four of these basic kinds of forces in nature, and even some of these are closely coupled. The four fundamental forces are: • Gravity: Every object with mass attracts every other. This attraction holds planets, stars and galaxies together, and keeps you on the surface of the Earth. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 • Electricity & Magnetism: This combination is responsible for chemistry, and all the bonding between atoms that makes matter interesting. Every force you’ll see in this class, except gravity, ultimately arises just from E&M. • Strong Nuclear Force: This very short range force holds atomic nuclei together, allowing for all the various elements of the periodic table. • Weak Nuclear Force: This short range force is responsible for the radioactive decay of some atomic nuclei. While both the strong and weak force are crucial for the existence and nature of chemical elements, they are also remote, in the sense that about all they do is help create the periodic table. You won’t see them acting more directly in your lives. Phenomenological forces: (like the force of friction). While all forces ultimately arise due to the four fundamental forces, this is often far from clear. When forces are more complex, we describe them with “laws” that would more appropriately be called models. In them we attempt to quantify an often very complicated set of phenomena by a series of approximations. The distinguishing feature of phenomenological models is that the more closely you study the phenomena, the more complicated the law you are using to describe it becomes. This is considered evidence that the understanding you have is ad-hoc, approximate, and not fundamental. That doesn’t mean these models are not accurate reflections of reality or that they aren’t “true”. It’s just that by acknowledging that they gloss over details, we confine them to being “true” with a lower-case t. We know for sure that there are other details hiding beneath these general principles. To find a force law, we first try to describe the basic behavior. We try to predict correctly the approximate size and direction of forces. We try to understand approximately how these forces will change when we change the properties of the objects in question. This is called understanding the problem in the "first approximation". Then, once we have a handle on the basic behavior, we look at things at the next level of detail (in the second approximation), and so on. Phenomenological laws are never perfect, but they can be enormously useful, and in complicated cases like friction they are absolutely necessary. It’s worth noting here that the structure and behavior of living systems is often extraordinarily complex. As a result, such systems almost always require description with these kinds of phenomenological approaches. Quantitative models of biological systems almost always begin simple and gradually add complexity, and accuracy, in this way. This is called “mathematical modeling”, and these days it’s a very important part of life science research. Research in these areas is very hot, and is reflected at Michigan in programs like Mathematical Biology in the Math Department, the Biostatistics program in the School of Public Health, and the Complex Systems Program in Physics. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 How does friction act? Friction always acts so as to resist relative motion between two surfaces which are in contact. Let's consider two examples to see what this means. First the simplest: imagine I slide a block over the table. I push it for a bit and then let it slide to a halt. While it is sliding to a halt the friction between the surfaces will generate the force which acts to prevent this motion, and which decelerates the object to rest. In this case the frictional force acts in a direction opposite to the direction of motion. This is kind of frictional effect that brings objects to rest, and which led Aristotle to conclude that the natural state of objects was to be at rest. Now let's consider another case. A heavy block sits at rest on a surface. I touch it on the side with my finger and apply a force; but the block doesn’t move. It remains at rest. Why? Since we see the block remain at rest, some force must balance the push I applied, keeping the net force on it zero. This balancing force, which appears when it’s needed to oppose my force, is also a frictional force. For reasons which we will describe in a minute, we think of these two examples as involving two different kinds of friction. The first case, in which the two objects are already in motion relative to one another, is called "kinetic" friction, because it refers to objects which are in motion. Once they are in motion, this kind of friction acts to oppose their relative motion. The second kind of friction, acting before the objects begin to slip, is called "static" friction. Static friction acts to prevent objects from beginning to slide over one another. Direction of Motion FN W Ff W FN Ff Fpush Fpush Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 better considered an active force than a passive one. Its magnitude is determined all the time by the equation: Ffk = μkFN In this equation μk is the "coefficient of kinetic friction" and FN is again the normal force between the two objects. So once an book is moving, the frictional force becomes independent of the size of the force I push with and independent of the rate of motion. Let's think about this for a moment. I apply a force to get an object moving. If I start out with a small force and gradually increase it, the object will first remain at rest, as the static frictional force gradually increases to match my push. Then, when my push exceeds μsFN, the object will begin to move. Once it is moving, regardless of the force with which I push, the friction force which resists motion will always be Ffk = μkFN Since this frictional force is now constant, different things can happen depending on how large the other applied forces are. If I apply a force less than Ffk, perhaps by no longer pushing, the unbalanced frictional force will decelerate the object and slow it down. If I apply a force larger than Ffk, my unbalanced force will accelerate the object forward. Only if I apply a force exactly to Ffk, no more and no less, will the object move along the surface at constant speed, because only then will the net force along the surface be zero. This is just what happens when you try to push something heavy, let’s say a cabinet, across the floor. When you first push, it goes nowhere. Then you shove harder, and eventually it breaks loose and starts to slip. Once this happens you adjust your force (not too hard, not too weak) so that it slips along at a constant rate. The rate of motion now is not set by how hard you push (you’re always just balancing the friction, which is independent of how fast the couch moves) but rather by how quickly you’re prepared to move along with it. Coefficients of friction, static and kinetic The constants in these equations, the μ’s, determine the size of frictional forces. Their values depend primarily on the composition of the two surfaces which are placed in contact. In detail, they depend on many other things, such as how the surfaces are prepared (are they rough or smooth), or the temperature of the surfaces. But for starters we will stick with the most important factor, what the surfaces are made of. Some examples: Materials μs μk Steel on Steel 0.6 0.4 Rope on wood 0.5 0.3 Tires on dry concrete 1.0 0.75 Tires on icy concrete 0.3 0.02 Teflon on teflon 0.04 0.04 Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 A larger table can be found here: http://www.roymech.co.uk/Useful_Tables/Tribology/co_of_frict.htm#coef and in many other places online. There are several things to note. First, the static coefficient μs it larger than the kinetic coefficient μk for almost every set of materials. It is harder to start something sliding than to keep it moving. If you have ever tried to move furniture you will have experienced this. You push hard trying to get something sliding, then once it "breaks free", it slips along more easily. The one exception to this general rule shown here is teflon, for which μs ≈ μk. Because of this, it is as easy to start sliding motion with teflon as it is to maintain it, there is no sudden jerk as the two surfaces break free. This property makes teflon on telfon contacts very useful for artificial joints, like knee repair. Try moving some of your joints, bending your elbow for example. There is no apparent need to apply an extra large force to get the motion started. This is because your joints, unlike the dry interfaces discussed here, are nicely lubricated. Second, there is a relatively large range of coefficients of friction, another hint that what's really going on here is very complicated. And remember, all we're talking about here is the contact between clean, dry surfaces. Imagine how many more different coefficients we would have to know if we wanted to predict the frictional forces for surfaces with varying degrees of contamination, or surfaces which are wet or otherwise lubricated. This is what I mean when I say the laws of friction are "first approximation" and "phenomenological". They are not a wild guess, real objects do behave in roughly this way. But such phenomenological laws are only capable of giving us a ballpark idea of what's going on, and we must use them with caution. With these basics in hand, let’s consider some examples of how friction acts. Examples of the role of friction: Let's consider two typical examples of friction. The first involves the slippage of a block down a slope. If I have a block sitting on a board like this and I gradually increase the angle which the board makes with the vertical, what happens? The force with which gravity pulls the object down the slope gradually increases. For a while, the static frictional force increases in step with this, causing the body to remain in place. But at some point, the pull of gravity down the slope overcomes the maximum possible static frictional force, and the block slips. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Now let’s analyze this situation. At a particular angle θ, we sum the forces along and perpendicular to the slope and find: Before the block begins to slide we have: ∑Falong = mgsinθ - Ff = maalong = 0 or Ff = mgsinθ ∑Fperpendicular = FN - mgcosθ = maperpendicular = 0 or FN = mgcosθ But remember, there is a maximum static frictional force, it cannot be larger than μsN. From this information we can calculate the angle at which it will start to slip. This will happen when: Ff = μsFN = mgsinθ Or μs(mgcosθ) = mgsinθ or μs = tanθ Notice what's going on here. There are two effects. First the frictional force required to hold it on the slope is increasing as we increase the angle θ. Second, the maximum available static friction is decreasing as we increase θ, because the normal force between the surfaces is decreasing. Both these facts tend to make objects slip down slopes more easily. This application of the laws of friction tells us that for a particular coefficient of friction, there exists some maximum slope beyond which the object will slip. Surely this is familiar to you from standing on slopes. If the slope is too steep, you slip downward. You probably have also experienced the important dependence of this critical slope on the nature of the two materials. Stand on a slope in sneakers and you can avoid slipping to quite steep angles. Do it in dress shoes (or on ice!) and the slope you can manage is much less steep. This result also suggests that the critical angle at which things will slip is independent of the nature of the object on the slope. If you stand on a slippery slope with a two year old, you’ll both start to slip at the same angle. This might be somewhat surprising, but it’s a clear prediction of this result. Going back to the original example, once static friction is overcome and the block starts slipping, then the friction becomes purely kinetic, and you know exactly how large it is: ∑Falong = mgsinθ - Ff = maalong = dpalong/dt ≠ 0 ∑Fperpendicular = FN - mgcosθ = maperpendicular = 0 or FN = mgcosθ θ N W = mg Ff Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 But when we look at objects on an atomic scale, they are very rough, with peaks and valleys that are typically many 100s of atoms tall. We can do this now (look at surfaces on the atomic scale), especially using instruments like Scanning- Tunneling Microscopes and Atomic Force Microscopes. What we find when we image typical surfaces is something that looks like the Alps. So when we put two surfaces in contact, it's like taking the Alps in Switzerland and turning them over on top of the Alps of Bavaria. In such a case, the actual microscopic area of contact is a tiny fraction of the total area of the two objects. At these points where the objects do meet, the atoms actually bond, and the materials stick to each other (or "adhere"). This happens through the same kind of chemical bonds which hold the object themselves together. This "sticking" of these points, is what we perceive as friction. With this picture in mind, we can start to understand da Vinci’s two rules and some other features of friction. Stick-and-slip friction, μs, and μk The first thing we want explore is why μs and μk are different, and why μs is typically larger. When I try to move a stationary object I gradually stretch these “merged mountaintops”, causing them to distort and resist the force which I apply. Once the object breaks free and starts to actually move, what happens goes something like this: • The distorted object breaks free at the surface, releasing the stretched points of contact which spring forward until the "catch up" with the bulk of the object • Then the two surfaces are again essentially at rest. New points of contact bond, generating a new source of frictional force, grabbing ahold of the surface below. • As we continue to pull, these new contacts are stretched, until finally they break loose, allowing the contact points to "jump" again. This cycle, which is known as "stick-and-slip" friction, is the reason why static and kinetic friction are related in the way they are. Kinetic friction is really a bunch of repeated applications of static friction. In each step the static frictional force builds up from zero to it’s maximum value, then breaks free and starts again. Each time you break free the bonds between these two surfaces, the top material jumps forward. After this, you have to build up the force to overcome static friction in this new spot. This makes the average value of kinetic friction somewhat less than the maximum for static friction. If you want to feel this stick-and-slip friction in action, try putting the eraser of your pencil down on the desk, while you push down fairly hard. As you drag it across the table, it will perform just this kind of stick-slip motion. The horrible shriek of chalk on a blackboard is also just this stick-slip happening, now at a high frequency, so it generates a high pitched noise. Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 How large can μs be? There is, in principle, no limit to the size of μs. It often seems that since Ff = μFN, that the frictional force can never be larger than FN. But this is not the case. Friction depends on adhesion, so it is possible for the friction between objects to be much larger than FN. This is just what we use glue to do, to make the adhesion large enough to dramatically increase the maximum static friction. Then objects will not slip over one another. The laws of friction and independence of area Ultimately all friction is caused by this bonding between atoms, and all such bonds are ultimately due to electromagnetic interactions between the atoms. The same interactions that hold matter together create the stickiness which underlies friction. How large this effect is depends on both the materials you use and the nature of the surfaces (polished so that many atoms come into contact, or Alps-on-Alps so that very few come into contact). But since most surfaces are actually quite rough, the simple laws da Vinci first discovered give a pretty accurate estimate of what will happen. Probably the most surprising thing about sliding friction is that it is independent of contact area. How can this be, particularly in light of the fact that friction is really due to adhesion? The trick to this, which was not understood until at least the 1950s, is that friction does depend on the area of contact, but not on the apparent area of contact. It depends instead on the tiny bit of contact at the tops of those mountains, what we might write Acontactmicroscopic. With this knowledge we can write Ff ∝ Acontactmicroscopic And for most solids, which are very rough this microscopic area of contact depends on how hard you squeeze the two surfaces together: Acontactmicroscopic ∝ FN Hence Ff ∝ FN This is a great example of how going deeper, looking beneath the first level of phenomenological laws, can be revealing Understanding the reason why friction is independent of contact area makes it possible to better appreciate the limitations of this general rule, as we will see in a moment. Breaking the rules of friction As I have several times said, these rules are basic, and generally work quite well for dry, solid surfaces. They don't work for a lot of practically important cases, especially for ones involving biological materials. A nice example is your fingertips (Need to do a little experiment on friction with fingertips!) Another area in which substantial violations of these rules occurs is with materials that are neither dry, nor rigid. Many biological cases are like this, in your joints for example. In these lubricated cases, frictional forces can be very different from what is described Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 here, depending on factors like temperature and velocity, instead of just on the nature of the surfaces and the normal force. A good technological example, drawn from biology, is rubber shoes and tires. Rubber is a substance which distorts pretty easily, following the typical biological J-curve stress vs. strain relation. The fact that it can distort so easily means that I can make the microscopic area of contact between the rubber and a floor very large without pushing down on it too hard. This large area of microscopic contact means a lot of adhesion, which in turn means large friction. You can see the efficacy of this as you jump across the floor. Your rubber shoes can provide the relatively enormous frictional force required to stop you. If instead you place a sheet of paper on the floor, and jump onto it, your feet will slide and you’ll land on your can. You may have experienced this with dress shoes. We can see how this works with a simple model. If we approximate this effect by saying that: Acontactmicroscopic ∝ FN2 We would expect to find: Ff ∝ FN2 Which is approximately what we see. Other violations of these general rules come from surfaces which are unlike the "typical" surfaces considered here in other ways. It is possible to make extremely smooth and clean surfaces. This is often done in machining to make something called "gauge blocks" out of metal. If I clean to of these gauge blocks carefully and put them together, they will form such a large area of contact that they will essentially merge into one block in a process known as "cold welding". All the atoms from one surface bond with all the atoms from the other, and the surfaces essentially disappear. Rolling friction Another practically important example is rolling friction. Why is it that a wheel can roll along for so long, apparently unaffected by friction, while a box that you slide so rapidly skids to a halt? The very small friction associated with a wheel comes about because, perhaps surprisingly, the point of contact between the wheel and the ground does not move. Since there is no relative motion between the wheel and the ground, there is no sliding friction. How can this be? Consider the picture below: Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 Physics for the Life Sciences: Fall 2008 Lecture #7 Today’s class will be a bit of a pause. For a while now we have been examining forces and how they play against one another in cases of equilibrium (either rest or uniform motion). Today we’ll begin to discuss how motions change, and what happens when forces acting on objects are not balanced. To do this, we must first focus a bit on refining our description of motion. We’ll spend a good bit of today on what is called “kinematics”; the pure description of motion. We will see that if we know the full path of an object, its position at each instant of time, we know everything about the motion. To understand that motion, we will need to speak of the velocity of the object (how rapidly its position is changing), and its acceleration (how rapidly the velocity is changing). For starters, we’ll talk just about motion along a straight line. Later we will see that motion in two and three dimensions is a rather straightforward extension of one dimensional motion. Let’s discuss a simple example first, just to get a sense of where we’re headed. Picture in your mind a sprinter prepared to run the 100m dash. Before the start she is still on the starting blocks. During this time her position remains the same from instant to instant, her speed is zero, and since her speed is not changing her acceleration is zero. Then the gun fires, she bursts forward from the blocks, heading quickly for her top speed. During this period, her position changes from instant to instant. In addition, her speed changes from instant to instant, becoming larger and larger. This changing speed means that she is accelerating as well. After just a few seconds, our sprinter is going full out, running at absolutely her top speed. During this period, her position continues to change from instant to instant, but her speed does not. Since her speed is not changing her acceleration is now zero. After bursting through the finish tape, our sprinter cruises to a stop. During this period, her position continues to increase, always moving farther from the starting blocks. Now her speed is gradually decreasing, and this changing speed implies an acceleration. Finally, she stops, hugging her coach in victory. Now her position remains the same from instant to instant, her speed is zero, and since her speed does not change, her acceleration is zero as well. In today’s lecture we will develop in more detail the tools we need to describe this motion fully. We will speak of position, velocity, and acceleration, each determined instant-by-instant along the path. Position, and intervals of distance: The motion of real, extended objects is complex. You can see this by considering what we mean by a cloud traveling at 10 mph, or a horse racing at 35 mph. When we talk informally about this our meaning is clear (the cloud is ‘on average’ moving along at 10 Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 mph), and to begin we will simplify complicated motions in this way. We will begin by ignoring any internal structure in an object and treat it as a "point object". If we do this we can talk about a complicated motion (like a horse racing down the track with its legs churning along and its rider bobbing up and down) in a simple way. To describe a motion we beginning by setting up a reference frame, a standard scale against which to measure. -3 -2 -1 0 1 2 3 To describe motion we simply record the position of the object at every instant. We’ll use this kind of labeling scheme. s = position at a particular instant s1 = position at the instant t1 Δs21 = s2 - s1 = interval between the two instants t1 and t2 = displacement Notice the notation, the symbol Δ (the Greek letter "delta") is used to denote a change in a quantity. Q: Is s1 a distance? No, it's just a location. We merely label it with its distance from the origin. Q: What do the signs of Δs mean? What if Δs is positive? The object is moving to the right. What if Δs is negative? The object is moving to the left. Can you have s2<0, s1<0, and Δs>0? Yes. You should think of an example. Does Δs=0 imply that no distance has been traveled at all during this period? Instants and Intervals of Time: Time too must be measured. How do we do this? We measure time by comparison to something which happens “regularly”. What do we compare to? Over the years, many steady timekeepers have been used. The oldest are astronomical, including the rotation of the Earth and its orbit around the Sun. These allow us to mark off days and keep track of the years. To measure shorter periods of time requires something which repeats more often. For this purpose, many different tools have been used, including the pulse, water clocks, pendula, and more recently the very regular, rapid oscillations of quartz crystals and atoms. In a manner very similar to the way we described positions and intervals of distance, we also talk about instants and intervals of time: t1 = time of a particular instant when something happens (an “event”) Physics 135 Fall 2009: Lecture Notes Copyright: Regents of the University of Michigan, 2009 t2 = time of a second instant when something happens Δt = t2 - t1 = interval of time between two events Like a position, an instant is a "location" in time. It's really just a marker, without units. Only an interval, the time between two instants, has units of seconds. Motion: s vs. t "histories" Since each position si corresponds to a particular instant ti, we can represent the series of events which makes up the motion of an object graphically: If the motion is "smooth" we can reliably fill in the history between these points with a curve "describing" the motion. With no extra information, other histories are possible. Particularly when motions change very fast the sampling of positions must be very dense in time to have an adequately accurate history. What does the above picture describe? This object starts out at a position which is negative (to the left of the zero point), moves farther to the left, stops briefly, moves towards the right, passes the zero point, travels a little farther, and then stops and stays at the same place for a while. Interpreting position time graphs properly can be very helpful. In class we will do quite a bit of practicing. Now we want to extend our discussion of motion to include the idea of speed. Rate of Motion: What do we mean by speed? Note that we’ll distinguish, somewhat loosely, between speed and velocity. By speed, we mean the magnitude of the velocity vector, and by velocity, we mean the full vector, including both magnitude and direction. Speed is a good example of something we talk about all the time without care. It clearly has something to do with how fast you go, how much distance you travel in some short period of time, but what's a "short period of time"? For a cross country trip, it might make sense to consider the entire length of the trip. If we use this in determining speed we will learn an average speed. If, instead, we wish to s t