Knight Chapter 4 Forces and Newton's Laws of Motion PDF

LOOKING AHEAD • Forces Forces and Motion Reaction Forces A force is a push or a pull. ft is an interaction between rwo objects. the agent (here. rhe woman) and the object (the car). Forces cause objects ro accelerate. A forward acceleration of the sled requires a forward force. The hammer exert...

LOOKING AHEAD • Forces Forces and Motion Reaction Forces A force is a push or a pull. ft is an interaction between rwo objects. the agent (here. rhe woman) and the object (the car). Forces cause objects ro accelerate. A forward acceleration of the sled requires a forward force. The hammer exerts a downward force on the nail. Surprisingly, the nail exerts an equal force on the hammer, directed upward. In this chapter, you'll learn how to identify different forces, and you'll learn their properties. A larger acceleration requires a larger force. You'll learn this connection between force and motion, part of Newton's second law. You'll learn how to identify and reason with action/reaction pairs of forces according to ml Newton's third law. To establish the connection between force and motion. LOOKING BACK • Acceleration You learned in Chapters 2 and 3 that acceleration is a vecwr that points in the direction of the change in velocity. i4UJ.,IMMI A swan is landing on an icy lake. sliding across the ice and gradually coming to a stop. As the swan slides. the direction of the acceleration is A. To the left. If the velocity is changing, there is an acceleration. And so, as you'll learn in this chapter, there must be a net force. B. To the righr. C. Upward. D. Downward. 106 c HAP TE R 4 Forces and Newton's Laws of Motion 4.1 Motion and Forces Interstellar coasting A nearly perfect example of Newton's fu-st law is the pair of Voyager space probes launched in I 977. Both spacecraft long ago ran out of fuel and are now coasting through the frictionless vacuum of space. Although not entirely free of influence from the sun's gravity, they are now so far from the sun and other stars that gravitational influences are very nearly zero, and the probes will continue their motion for billions of years. The snake-necked tmtle in the photo at the beginning of the chapter can accelerate its head forward at 40 m/s 2 to capture prey. In Chapters 1 tlu·ough 3, we've learned how to describe this and other types of motion with pictures, graphs, and equations, and you know enough about the scale of things to see that th.isacceleration-which is about 4gis quite impressive. But until now we've said nothing to explain the motion, to say how the turtle is able to achieve tins feat. In this chapter, we will tum our attention to the cause of motion-forces. 1l1is topic is called dynamics, winch joins witl1kinematics to form mechanics, the general science of motion. We'll begin our study of dynamics qualitatively in this chapter and then add quantitative detail over the next four chapters. What Causes Motion? Let's stmt with a basic question: Do you need to keep pushing on sometl1ing-to keep applying a force-to keep it going? Your daily expe1ience n1ight suggest that the answer is Yes. If you slide your textbook across the desk and tl1enstop pushing, the book wiJl quickly come to rest. Other objects will continue to move for a longer time: When a hockey puck is slidi11gacross the ice, it keeps going for a long time, but it, too, comes to rest at some point. Now let's take a closer look to see whether this idea holds up to scrutiny. FIGURE 4.1 shows a series of motion experiments. Tyler slides down a hill on Ins sled and then out onto a hmizontal patch of smooth snow, as shown in Figure 4. la. Even if the snow is smooth, the f1iction between the sled and the snow will soon cause the sled to come to rest. What if Tyler slides down the hill onto some very slick ice, as in Figure 4.lb? 1l1e friction is much less, so tl1e sled could slide for quite a distance before stopping. Now, imagine the situation in Figw·e 4. lc, where the sled slides on idealized frictionless ice. In tl1iscase, the sled, once stmted in its motion, would continue in motion forever, moving in a straight lille with no loss of speed. FIGURE 4.1 A sled sliding on increasingly (a) Smooth snow .A7 ~ smooth surfaces. On smooth snow, the sled .A7 ~ (c) Frictionless surface ~ ............soon comes to rest. L t K > lf friction could be reduced to zero, the sled would never stop.•···.. :, LLLL In the absence of f1iction, if the sled is moving, it will stay in motion. It's also true that if the sled were sitting still, it wouldn't stmt moving on its own; if the sled is at rest, it will stay at rest. Careful experiments done over the past few centuries, notably by Galileo and then by Isaac Newton, verify that th.is is in fact the way the world works. What we have concluded for the sled is actually a general rule that applies to other sin1ilm·situations. We call the generalization Newton's first law of motion: Newton's first law Consider an object that has no forces acting on it. U it is at rest. it will remain at rest. If it is moving, it will continue to move in a straight line at a constant speed. As an important app]jcation of Newton's first law, consider the crash test of As the car contacts the wall, the wall exerts a force on the car and the car begins to slow. But the wall is a force on the ca,; not on the dununy. In accordance with FIGURE 4.2. 4.1 Motion and Forces 107 FIGURE 4.2 Newton's first law tells us: Wear your seatbelts! Al the instant of impact, the car and the dummy are moving m the same speed. The car slows as it hits, but the dummy continues at the same speed ... ... until it hits the now-stationary dashboard. Ouch! Newton's first law, the unbelted dummy continues to move straight ahead at its original speed. Sooner or later, a force will act to b1ing the dummy to rest. The only questions are when and how large the force will be. ln the case shown, the dummy comes to rest in a shott, violent collision with the dashboard of the stopped car. Seatbelts and air bags slow a dummy----or a dtiver-at a much lower rate and provide a much gentler stop. Forces We started with the question What causes motion? Our answer is Newton's first law, wl1ich tells us that an object in motion needs no cause-orjorce-to keep moving in a straight line forever. But this law does not explain in any detail exactly what a force is. The concept of force is best introduced by looking at examples of some common forces and considering the basic properties shared by all forces. Let's begin by examining the properties that all forces have in common, as presented in the table below. What is a force? A force is a push or a pull. Our commonsense idea of a force is that it is a push or a pull. We will refine this idea as we go along, but it is an adequate starting point. Notice our careful choice of words: We refer to "a force" rather than simply "force." We want to think of a force as a very specific action, so that we can talk about a single force or perhaps about two or tlU"eeindividual forces that we can clearly distinguish-hence the concrete idea of "a force" acting on an object. A force acts on an object. Implicit in our concept of force is that a force acts on an object. ln other words, pushes and pulls are applied to something-an object. From the object's perspective, ii has a force exerted on it. Forces do not exist in isolation from the object that experiences them. A force requires an agent. Every force has an agent, something that acts or pushes or pulls; that is, a force has a specific, identifiable cause. As you throw a ball, it is your hand, while in contact with the ball, that is the agent or the cause of the force exe1ted on the ball. (fa force is being exened on an object, you must be able to identify a specific cause (i.e., the agent) of that force. Conversely, a force is not exe,ted on an object unless you cim identify a specific cause or agent. Note that an agent can be an ine,t, inanimate object such as a tabletop or a wall. Such agents are the cause of many common forces. A force is a vector. If you push an object, you can push either gently or very hard. Similarly. you can push either left or right, up or down. To quantify a push, we need to specify both a magnitude a11da direction. It should thus come as no surprise that a force is a vector quantity. The general symbol for a force is the vector symbol F.The size or strength of such a force is its magnitude F. A force can be either a contact force ... There are two basic classes of forces, depending on whether the agent touches the object or not. Contact forces are forces that act on an object by touching it at a point of contact. The bat must touch the ball to hit it. A string must be tied to an object to pull it. The majority of forces that we will examine are contact forces . . . . or a long-range force. Long-range forces are forces that act on an object without physical contact. Magnetism is an example of a long-range force. You have undoubtedly held a magnet over a paper clip and seen the paper clip leap up to the magnet. A coffee cup released from your hand is pulled to the earth by the long-range force of gravity. 108 c HA PTER 4 Forces and Newton's Laws of Motion There's one more important aspect of forces. If you push against a door (the object) to close it, the door pushes back against your hand (the agent). If a tow rope pulls on a car (the object), the car pulls back on the rope (the agent). In general, if an agent exerts a force on an object, the object exerts a force on the agent. We really need to think of a force as an interaction between two objects. Although the interaction perspective is a more exact way to view forces, it adds complications that we would like to avoid for now. Our approach will be to start by focusing on how a single object responds to forces exerted on it. Later, in Section 4.7, we'll return to the larger issue of how two or more objects interact with each other. Force Vectors We can use a simple diagram to visualize how forces are exerted on objects. Because we are using the particle model, in which objects are treated as points, the process of drawing a force vector is straightforward: TACTICS 80 x 4 .1 . Drawing force vectors O Represent the object as a particle. •······• .• ~ Place the tail of the force vector ·············" on the particle. €} Draw the force vector as an airnw pointing•·····" in the direction that the force acts, and with a length proportional to the size of the force. ························~f O Give the vector an appropriate label.····· Exercise 1 ifll Step 2 may seem contrary to what a "push" should do (it may look as if the force arrow is pulling the object rather than pushing it), but recall that moving a vector does not change it as long as the length and angle do not change. The vector Fis the same regardless of whether the tail or the tip is placed on the particle. Our reason for using the tail will become clear when we consider how to combine several forces. FIGURE 4.3 shows three examples of force vectors. One is a pull, one a push, and one a long-range force, but in all three the tail of the force vector is placed on the particle that represents the object. Although the generic symbol for a force is F,as the figure shows we often use special symbols for certain forces that arise frequently, such as Tfor the tension force in a rope, F,p for the force of a spring, and wfor the force of gravity (an object's weight). FIGURE 4.3 Three force vectors. The box is the object. Pictorial representation of box Particle representation of box T'7 rope is the agent. ... The sprin~l the agent. □l ... Box Long-range force of gravity iv Box _______ _. Pulling force of rope f Box • Pushing force of spring f;P Earth is the agent. 4.2 Combining 109 A Short Catalog of Forces Forces Force is a vector quantity. We saw in • SECTION 3.1 how to find the vector sum of two vectors. FIGURE 4.4a shows a top view of a box being pulled by two ropes, each exerting a force on the box. How will the box respond? Experiments show that when several forces F1, F2, F1,. . . are exe1ted on an object, they combine to form a net force that is the vector sum of all the forces: FIGURE 4.4 Two forces applied to a box. (a) Top view of box (4.1) Two ropes exerting tension forces on a box That is, the single force F,,ei causes the exact same motion of the object as this combination of original forces Fi,F2, F3, . ... FIGURE 4.4b shows the net force on the box. lmIO ► It is important to realize that the net force in addition to the original forces original forces being replaced by i\ F2 , F3 , . ... F.,ei· F'nei is not a new force acting Instead, we should think of the ◄ (b) Pulling forces----------.... of the ropes \J T Two of the three forces exerted on an object are shown. The net force points directly to the left. Which is the missing third force? STOP TO THINK 4.1 '·.,, I I_,. ... /.,,Fnet ... =Fi+ F2 ,•, F , 'Tl;e f,G. .. Fi F3 Two of the three forces exerted on an object A. B. • 2 y ' ,,1 C. net force is the vector sum of all the forces acting on the box . D. 4.2 A Short Catalog of Forces There are many forces we will deal with over and over. This section will introduce you to some of them and to the symbols we use to represent them. Weight A falling rock is pulled toward the earth by the long-range force of gravity. Gravity is what keeps you in your chair, keeps the planets in their orbits around the sun, and shapes the large-scale structure of the universe. We'll have a thorough look at gravity in Chapter 6. For now we'll concentrate on objects on or near the surface of the earth (or other planet). The gravitational pull of the emth on an object on or near the surface of the earth is called weight. The symbol for weight is w.Weight is the only long-range force we will encounter in the next few chapters. The agent for the weight force is the entire earth pulling on an object. The weight force is in some ways the simplest force we'll study. As FIGURE 4.5 shows, an object's weight vector always points vertically downward, no matter how the object is moving. ~ ► We often refer to "the weight" of an object. This is an informal expression for w, the magnitude of the weight force exerted on the object. Note that weight is not the same thing as mass. We will briefly examine mass later in the chapter, and we'll explore the connection between weight and mass in Chapter 5. ◄ FIGURE 4.5 Weight always points vertically downward. Free fall, Free fall, moving moving up clown \ \ Pro1ect1le motion .......~ .. JJ J Rollin:1 IV Al res~! II' 110 c HAP TE R 4 Forces and Newton's Laws of Motion Spring Force Springs exert one of the most basic contact forces. A sp1ing can either push (when compressed) or pull (when stretched). FIGURE 4.6 shows the spring force. In both cases, pushing and pulling, the tail of the force vector is placed on the particle in the force diagram. There is no special symbol for a spring force, so we simply use a subscript label: Fsp· FIGURE 4.6 The spring force is parallel to the spring. (a) A compressedspring exerts a pushing force on an object. (b) A stretched spring exerts a pulling force on an object. When the flexible blade of this athlete's prosthesis hits the ground, it compresses just like an ordinary spring. • Box Fsr F,p Box Although you may think of a spring as a metal coil that can be stretched or compressed, this is only one type of spring. Hold a ruler, or any other thin piece of wood or metal, by the ends and bend it slightly. It flexes. When you let go, it "springs" back to its original shape. This is just as much a spring as is a metal coil. Tension Force FIGURE 4.7 Tension is parallel to the rope. The rope exerts a tension force on the sled. FIGURE 4.8 An atomic model of tension. ~rn Atoms Molecular bonds When a string or rope or wire pulls on an object, it exerts a contact force that we call the tension force, represented by f. The direction of the tension force is always in the direction of the string or rope, as you can see in FIGURE 4.7. When we speak of "the tension" in a string, this is an informal expression for T, the size or magnitude of the tension force. Note that the tension force can only pull in the direction of the string; if you try to push with a string, it will go slack and be unable to exert a force. We can think about the tension force using a microscopic picture. If you were to use a very powerful microscope to look inside a rope, you would "see" that it is made of atoms joined together by molecular bonds. Molecular bonds are not rigid connections between the atoms. They are more accurately thought of as tiny springs holding the atoms together, as in FIGURE 4.8. Pulling on the ends of a string or rope stretches the sp,ing-like molecular bonds ever so slightly. The tension within a rope and the tension force experienced by an object at the end of the rope are really the net spring force exerted by billions and billions of microscopic springs. This atomic-level view of tension introduces a new idea: a microscopic atomic model for understanding the behavior and properties of macroscopic (i.e., containing many atoms) objects. We will frequently use atomic models to obtain a deeper understanding of our observations. The atom.ic model of tension also helps to explain one of the basic properties of ropes and strings. When you pull on a rope tied to a heavy box, the rope in turn exerts a tension force on the box. If you pull harder, the tension force on the box becomes greater. How does the box "know" that you are pulling harder on the other end of the rope? According to our atomic model, when you pull harder on the rope, its microscopic springs stretch a bit more, increasing the spring force they exert on each other-and on the box they're attached to. Normal Force If you sit on a bed, the springs in the mattress compress and, as a consequence of the compression, exert an upward force on you. Stiffer springs would show less compression but would still exert an upward force. The compression of extremely stiff springs might be measurable only by sensitive instruments. Nonetheless, the springs would compress ever so slightly and exert an upward spring force on you. 4.2 FIGURE 4.9 shows a book resting on top of a sturdy table. The table may not visibly flex or sag, but-just as you do to the bed-the book compresses the molecular "springs" in the table. The compression is very small, but it is not zero. As a consequence, the compressed molecular springs push upward on the book. We say that "the table" exerts the upward force, but it is important to understand that the pushing is really done by molecular springs. Similarly, an object resting on the ground compresses the molecular springs holding the ground together and, as a consequence, the ground pushes up on the object. We can extend this idea. Suppose you place your hand on a wall and lean against it, as shown in FIGURE 4.10. Does the wall exert a force on your hand? As you lean, you compress the molecular springs in the wall and, as a consequence, they push outward against your hand. So the answer is Yes, the wall does exert a force on you. It's not hard to see this if you examine your hand as you lean: You can see that your hand is slightly deformed, and becomes more so the harder you lean. This deformation is direct evidence of the force that the wall exerts on your hand. Consider also what would happen if the wall suddenly vanished. Without the wall there to push against you, you would topple forward. The force the table surface exerts is vertical, while the force the wall exerts is horizontal. In all cases, the force exerted on an object that is pressing against a sutface is in a direction pe,pendicular to the su1face. Mathematicians refer to a line that is perpendicular to a surface as being nonna/ to the surface. In keeping with this terminology, we define the normal force as the force exerted by a surface (the agent) against an object that is pressing against the sutface. The symbol for the normal force is ii. We're not using the word "normal" to imply that the force is an "ordinary" force or to distinguish it from an "abnormal force." A surface exerts a force pe1pendicular (i.e., normal) to itself as the molecular springs press outward. FIGURE 4.11 shows an object on an inclined surface, a common situation. Notice how the normal force ii is perpendicular to the suiface. The normal force is a very real force arising from the very real compression of molecular bonds. It is in essence just a spring force, but one exerted by a vast number of microscopic springs acting at once. The normal force is responsible for the "solidness" of solids. It is what prevents you from passing right through the chair you are sitting in and what causes the pain and the lump if you bang your head into a door. Your head can then tell you that the force exerted on it by the door was very real! A Short Catalog of Forces FIGURE 4.9 An atomic model of the force exerted by a table. The compressed .•······molecular springs push •• upward on the object. Object (not to scale!) Atoms El A video to support a section's topic is embedded in the eText. Video Figure 4.9 Video What the Physics? Pushing Down to Move Upward FIGURE 4.10 The wall pushes outward against your hand. The compressed molecular springs·················• in the wall press outward against her hand. FIGURE 4.11 The normal force is perpen- dicular to the surface. Friction You've certainly observed that a rolling or sliding object, if not pushed or propelled, slows down and eventually stops. You've probably discovered that you can slide better across a sheet of ice than across asphalt. And you also know that most objects stay in place on a table without sliding off even if the table is tilted a bit. The force responsible for these sorts of behavior is friction. The symbol for friction is Friction, like the normal force, is exerted by a surface. Unlike the normal force, however, the frictional force is always parallel to the surface, not perpendicular to it. (In many cases, a surface will exert both a normal and a frictional force.) On a microscopic level, friction arises as atoms from the object and atoms on the surface run into each other. The rougher the su1face is, the more these atoms are forced into close proximity and, as a result, the larger the friction force. We will develop a simple model of friction in the next chapter that will be sufficient for our needs. For now, it is useful to distinguish between two kinds of friction: J. ■ Kinetic friction, denoted Jk, acts as an object slides across a surface. Kinetic ftiction is a force that always "opposes the motion," meaning that the friction force Jk on a sliding object points in the direction opposite to the direction of the object's motion. 111 The surface pushes outward against the bottom of the skis. The force is perpendicular to the surface. 112 CHAPTER 4 Forces and Newton's Laws of Motion ■ Static ji-iction, denoted .fs,is the force that keeps an object "stuck" on a surface and prevents its motion relative to the su1face. Finding the direction of fs is a little trickier than finding the direction of fk• Static f1iction points opposite the direction in which the object would move if there were no friction; that is, it points in the direction necessary to prevent motion. Examples of kinetic and static friction are shown in KEY CONCEPT The sled is moving .. Jk FIGURE 4.12. FIGURE 4.12 Kinetic and static friction 10 the right • ... becausea kinetic friction Sled force directed to the left ~...........•···oppose, this motion. are parallel to the surface. The woman is pulling to the left. but the crate doesn·t move ... ... becausea static friction force directed to the right Crate opposes this motion ..........................•··" J,, A frog is resting on a slope. Given what you learned in STOP TO THINK 4.2 Figure 4.12, what can you say about the friction force acting on the frog? A. B. C. D. E. There There There There There is no friction force. is a kinetic friction force directed up the slope. is a static friction force directed up the slope. is a kinetic friction force directed down the slope. is a static friction force directed down the slope. FIGURE 4.13 Air resistance is an example of drag. Air resistanceis a significant force on falling leaves.It points opposite the direction or motion. i t8 Leaf FIGURE 4.14 The thrust force on a rocket is opposite gases. the direction of the expelled Thrust force is exerted on a rocket by exhaust gases. Drag Friction at a su1face is one example of a resistive force, a force that opposes or resists motion. Resistive forces are also experienced by objects moving through fluidsgases (like air) and liquids (like water). This kind of resistive force-the force of a fluid on a moving object-is called drag and is symbolized as 8. Like kinetic fiiction, drag points opposite the direction of motion. FIGURE 4.13 shows an example of drag. Drag can be a large force for objects moving at high speeds or in dense fluids. Hold your arm out the window as you ride in a car and feel how hard the air pushes against your arm. Note also how the air resistance against your arm increases rapidly as the car's speed increases. For a small particle moving in water, such as a swimming Paramecium, drag can be the dominant force. On the other hand, for objects that are heavy and compact, moving in air, and with a speed that is not too great, the drag force of air resistance is fairly small. To keep things as simple as possible, you can neglect air resistance in all problems unless a problem explicitly asks you to include it. The error introduced into calculations by this approximation is generally pretty small. Thrust A jet airplane obviously has a force that propels it forward; likewise for the rocket in FIGURE 4.14. This force, called thrust, occurs when a jet or rocket engine expels gas molecules at high speed. Thrust is a contact force, with the exhaust gas being the agent that pushes on the engine. The process by which thrust is generated is rather subtle and requires an appreciation of Newton's third law, introduced later in this 4.3 Identifying Forces 113 chapter. For now, we need only consider that thrust is a force opposite the direction in which the exhaust gas is expelled. There's no special symbol for thrust, so we call it Fu,mst· Electric and Magnetic Forces Electricity and magnetism, like gravity, exert long-range forces. The forces of electricity and magnetism act on charged particles. We will study electric and magnetic forces in detail in Part VI of this text. These forces-and the forces inside the nucleus, which we will also see later in the text-won't be important for the dynamics problems we consider in the next several chapters. STOP TO THINK 4.3 A boy is using a rope to pull a sled to the right. What are the directions of the tension force and the friction force on the sled, respectively? B. Right, left D. Left, left A. Right, 1ight C. Left, right 4.3 Identifying It's not just rocket science @tu Rockets are propelledby thrust,but manyanimalsare as well.Scallopsare shellfishwith no feet and no fins,but theycan escapefrom predatorsor moveto newterritoryby using a form of jet propulsion.A scallopforciblyejects waterfromthe rearof its shell,resultingin a thrustforcethat movesit forward. Forces A typical physics problem describes an object that is being pushed and pulled in various directions. Some forces are given explicitly, while others are only implied. In order to proceed, it is necessary to determine all the forces that act on the object. It is also necessary to avoid including forces that do not really exist. Now that you have learned the properties of forces and seen a catalog of typical forces, we can develop a step-by-step method for identifying each force in a problem. A list of the most conunon forces we'll come across in the next few chapters is given in TABLE 4.1. II' 1111 Video IdentifyingForces lmJD ► Occasionally, you '11see labels for forces that aren't included in Table 4.1. For instance, if you push a book across a table, we might simply refer to the force you apply as firnnd· ◄ Tactics Box 4.2 will help you correctly identify the forces acting on an object. It's followed by two examples that explicitly illustrate the steps of the Tactics Box. TACTICS BOX 4.2 Identifying forces Common forces and their notation 0 Identify the object of interest. This is the object whose motion you wish TABLE 4.1 to study. Force f) Draw a picture of the situation. Show the object of interest and all other Notation General force l a closed curve around the object. Only the object of interest is inside the curve; everything else is outside. Weight \1, Spring force l,p 0 Locate every point on the boundary of this curve where other objects Tension f objects-such as ropes, springs, and surfaces-that touch it. @) Draw touch the object of interest. These are the points where contact forces are exerted on the object. 0 Name and label each contact force acting on the object. There is at least one force at each point of contact; there may be more than one. When necessary, use subscripts to distinguish forces of the same type. © Name and label each long-range force acting on the object. For now, the only long-range force we'll consider is weight. Exercises 4-8 g ormal force n Static friction ], Kinetic friction fk Drag jj Thrust f,hmst 114 CH APTER 4 Forces and Newton's CONCEPTUAL EXAMPLE 4.1 Laws of Motion Identifying forces on a bungee jumper A bungee jumper has leapt off a bridge and is nearing the bonorn of her fall. What forces are being exerted on the bungee jumper? REASON FIGURE4.15 Forces on a bungee jumper. Studentsketch Artist's version /Tension f TwionT" 1 •...•·············• ... ••0 Locate the points where other objects touch Lheobject of interest. Here the only point of contact is where the cord attaches to her ankles. 0 Identify the object of interest. Here the object is Lhebungee jumper. f} Draw a picrure of the siruation. 0 Name and label each contact force. The force exerted by the cord is a tension force. 0 Name and label long-range forces. Weight is the only one. ~ Draw a closed curve ······• around the object. 0 ....•··· We~trta,..........• Weight ii• CONCEPTUAL EXAMPLE 4.2 Identifying forces on a skier A skier is being towed up a snow-covered hill by a tow rope. What forces are being exerted on the skier? REASON FIGURE4.16 Forces on a skier. TwionT 0 l•···························· .0 Locate the points where other j objects touch the object of Identify the object of interest. Here the object is the skier. ! _../ f} Draw a picture of the situation. ···············/ 0 , ·······• Norml.ll',,,-u-n ~ Draw a closed ••• ..._____Normalforce ii Weight ii• Kinetic frictionJk curve around the object. IJJ6&1tW lvnuir, fricl,-.,, { ,••.............. interest. Here the rope and the ground Louch the skier. Name and label each contact force. The rope exerts a tension force, and the ground exerts both a normal and a kinetic friction force. ••••••••••••········ 0 Name and label long-range forces. Weight is the only one. lmIDI ► You might have expected two friction forces and two normal forces in Example 4.2, one on each ski. Keep in mind, however, that we're working within the particle model, which represents the skier by a single point. A particle has only one contact with the ground, so there is a single normal force and a single friction force. The particle model is valid if we want to analyze the motion of the skier as a whole, but we would have to go beyond the particle model to find out what happens to each ski. ◄ CONCEPTUAL EXAMPLE 4.3 Identifying forces on a rocket A rocket is flying upward through the air. high above the ground. Air resistance is not negligible. What forces are being exerted on the rocket? REASON FIGURE4.17 Forces on a rocket. 4.4 What Do Forces Do? You've just kicked a rock, and it is now sliding across the STOP TO THINK 4.4 ground about 2 meters in front of you. Which of these are forces acting on the rock? Include all that apply. A. B. C. D. E. Gravity, acting downward The normal force, acting upward The force of the kick, acting in the direction of motion Friction, acting opposite the direction of motion Air resistance, acting opposite the direction of motion 4.4 What Do Forces Do? The fundamental question is: How does an object move when a force is exerted on it? The only way to answer this question is to do experiments. To do experiments, however, we need a way to reproduce the same force again and again, and we need a standard object so that our experiments are repeatable. FIGURE 4.18 shows how you can use your fingers to stretch a rubber band to a certain length-say, 10 centimeters-that you can measure with a ruler. We'll call this the standard length. You know that a stretched rubber band exerts a force because your fingers feel the pull. Furthermore, this is a reproducible force. If you use the same rubber band stretched to the standard length, it will exert the same force. We' 11 call the magnitude of this force the standard.force F. Not surprisingly, two identical rubber bands, each stretched to the standard length, exert twice the force of one rubber band; three rubber bands exert three times the force; and so on. We'll also need several identical standard objects to which the force will be applied. As we learned in Chapter 1, the SI unit of mass is the kilogram (kg). For our standard objects, we will make ourselves several identical blocks, each with a mass of l kg. Now we're ready to start a virtual experiment. First, place one of the 1 kg blocks on a frictionless surface. (In a real experiment, we can nearly el.iminate friction by floating the block on a cushion of air.) Second, attach a rubber band to the block and stretch the band to the standard length. Then the block experiences the same force F as your finger did. As the block starts to move, in order to keep the pulling force constant you must move your hand in just the right way to keep the length of the rubber band-and thus the force-constant. FIGURE 4.19 shows the experiment being carried out. Once the motion is complete, you can use motion diagrams and kinematics to analyze the block's motion. FIGURE 4.19 Measuring the motion of a 1 kg block that is pulled with a constant force. Maintain constant stretch. I cBB Rubber band v~·••---• ►••~--~•►•~----•• z;--.. __.. __., ~ -Pull Frictionless surface Motion diagram The motion diagram in Figure 4.19 shows that the velocity vectors are getting longer, so the velocity is increasing: The block is accelerating. Furthermore, a close inspection of the motion diagram shows that the acceleration vectors are all the same length. Th.is is the first important finding of this experiment: An object pulled with a constant force moves with a constant acceleration. This finding could not have been anticipated in advance. It's conceivable that the object would speed up for a FIGURE 4.18 A reproducible force. / stretched the standard length exerts the standard force F. r------ Standard ~ength 2F / Two rubber bands stretched the standard length exert twice the standard t'orcc. 115 116 CHAPTER FIGURE 4.20 4 Forces and Newton's Graph of acceleration versus force. ,-:,, c.:501 Acceleration is directly proponiont.'.. to force. 0 "' " 4a 1 ~ E 3a1 ·•,••.=! 2 3 4 5 Force (number of rubber bands) FIGURE 4.21 Graph of acceleration versus number of blocks. With two blocks, the The acceleration of one block is a 1. acceleration is I/2 that with only one block. •• ••• J). f With ihree blocks, / / the acceleration is l/3 as much. k.' ai/3 a/4 0-+--~-~--~-~- o 2 Number of blocks 4 Laws of Motion while and then move with a steady speed. Or that it would continue to speed up, but the rate of increase, the acceleration, would steadily decline. But these descriptions do not match what happens. Instead, the object continues with a constant acceleration for as long as you pull it with a constant force. We'll call this constant acceleration of one block pulled by one band a 1. What happens if you increase the force by using several rubber bands? To find out, use two rubber bands. Stretch both to the standard length to double the force to 2F, then measure the acceleration. Measure the acceleration due to three rubber bands, then four, and so on. FIGURE 4.20 is a graph of the results. Force is the independent variable, the one you can control, so we've placed force on the horizontal axis to make an acceleration-versus-force graph. The graph reveals our second important finding: Acceleration is directly proportional to force. The final question for our virtual experiment is: How does the acceleration of an object depend on the mass of the object? To find out, glue two of the I kg blocks together, so that we have a block with twice as much matter as a l kg block-that is, a 2 kg block. Now apply the same force-a single rnbber band-as you applied to the single J kg block. FIGURE 4.21 shows that the acceleration is one-half as great as that of the single block. If we glue tlu·ee blocks together, making a 3 kg object, we find that the acceleration is only one-thi1d of the I kg block's acceleration. In general, we find that the acceleration is proportional to the in.verse of the mass of the object. So our third important result is: Acceleration is inversely proportional to an object's mass. L Inversely proportional relationships )' Two quantities are said to be inversely proportional to each other if one quantity is proportional to the inrerse of the other. Mathematically, this means that v=- , A X A ,,... ...... If xis halved, J doubles . ..... • A y=~ J is inverselyproportionalto x Here, A is a propmtionality constant. This relationship is sometimes written as y c,: l/x. SCALING RATIOS O+-,----+--+--~-,....--,--~x 0 2 }.,/ 4 If x is doubled ...... y is halved. ■ If you double x, you halve y. • If you triplex, y is reduced by a factor of 3. ■ If you halve x, y doubles. ■ If you reduce x by a factor of 3, y becomes 3 times as large. For any two values of x-say, x 1 and x 2 -we have A Y2=-x and - 2 Dividing the y 1 equation by the Y2equation, we find Y2 Alx 2 Xi A x1 That is, the ratio of y-values is the inverse of the ratio of the co1Tesponding values of x. • As x gets very large, y approaches zero. LIMITS ■ As x approaches zero, y gets very large. Exercises 10, 11 If.I 4.5 Newton's Second Law 117 You're familiar with this idea: It's much harder to get your car rolling by pushing it than to get your bicycle rolling, and it's harder to stop a heavily loaded grocery cart than to stop a skateboard. This tendency to resist a change in velocity (i.e., to resist speeding up or slowing down) is called inertia. Thus we can say that more massive objects have more inertia. Finding the mass of an unknown block EXAMPLE 4.4 When a rubber acceleration of mass is puJled 5.0 m/s 2 . What STRATEGIZE reasoning PREPARE to band is stretched to pull on a 1.0 kg block with a constant force. the the block is measured to be 3.0 m/s 2 . When a block with an unknown with the same rubber band, using the same force, its acceleration is is the mass of the unknown block? Because acceleration is inversely proportionaJ to mass, we will use ratio solve this problem. We denote the mass of the unknown block by m. SOLVE We can use the result of the Inversely proportional relationships box to write 3.0 m/s 2 111 5.0 m/s 2 1.0 kg Feel the difference Because of its high sugar content, a can of regular soda has a mass about 4% greater than that of a can of diet soda. If you try to judge which can is more massive by simply holding one in each hand, this smaU difference is almost impossible to detect. If you move the cans up and down, however, the difference becomes subtly but noticeably apparent: People evidently are more sensitive to how the mass of each can resists acceleration than they are lo the cans' weights alone. or Ill 3.0 m/s 2 , X ( 1.0 kg) :,.0 mis- =_ = 0.60 kg With the same force applied. the unknown block had a larger acceleration than the 1.0 kg block. It makes sense, then. that its mass-its resistance to acceleration-is less than LO kg. ASSESS 'II Uil Video Newton's Second Law Two rubber bands stretched to the standard length cause an object to accelerate at 2 m/s 2 . Suppose another object with twice the mass is pulled by four rubber bands stretched to the standard length. What is the acceleration of this second object? STOP TO THINK 4.5 A. 1 rn/s 2 B. 2 m/s 2 C. 4 m/s 2 D. 8 m/s2 E. 16 m/s2 4.5 Newton's Second Law We can now summarize the results of our experiments. We've seen that a force causes an object to accelerate. The acceleration a is directly proportional to the force F and inversely proportional to the mass m. We can express both these relationships in equation form as a=- F m (4.2) Note that if we double the size of the force F, the acceleration a will double, as we found experimentally. And if we triple the mass m, the acceleration will be only onethird as great, again agreeing with our experiments. Equation 4.2 tells us the magnitude of an object's acceleration in terms of its mass and the force applied. But our experiments also had another important finding: The direction of the acceleration was the same as the direction of the force. We can express this fact by writing Equation 4.2 in vector form as F ci= - m (4.3) fl Video Newton's Second Law Application 118 CHA PTEA 4 Forces and Newton's Laws of Motion Finally, our experiment was limited to looking at an object's response to a single applied force acting in a single direction. Realistically, an object is likely to be subjected to several distinct forces F1, F2, F3 , . .. that may point in different directions. What happens then? Experiments show that the acceleration of the object is determined by the net force acting on it. Recall from Figure 4.4 and Equation 4.1 that the net force is the vector sum of all forces acting on the object. So if several forces are acting, we use the net force in Equation 4.4. Newton was the first to recognize these connections between force and motion. This relationship is known today as Newton's second law. Newton's second law An object of mass 111subjected to forces wilJ undergo an acceleration a given by F1, F2 , F3 , ... a (4.4) ~ p.116m ~ IINERSE where the net force Fact = F1 + F 2 + F3 + · · · is the vector sum of all forces acting on the object. The acceleration vector ii points in the same direction as the net force vector Fnet· While some relationships are found to apply only in special circumstances, others seem to have universal applicability. Those equations that appear to apply at all times and under all conditions have come to be called "laws of nature." Newton's first and second laws are laws of nature; you will meet others as we go through this text. We can rewrite Newton's second law in the form (4.5) Size matters? Race car driver Danica Patrick was the subject of controversial comments by drivers who thought her relatively small mass of 45 kg gave her an unfair advantage. Because every driver's car must have the same mass, Patrick's overall racing mass was lower than any other driver's, so her car could be expected to have a slightly greater acceleration. CONCEPTUAL EXAMPLE4.5 which is how you'll see it presented in many textbooks and how, in practice, we'll often use the second law. Equations 4.4 and 4.5 are mathematically equivalent, but Equation 4.4 better describes the central idea of Newtonian mechanics: A force applied to an object causes the object to accelerate and the acceleration is in the direction of the net force. lmIDI ► When several forces act on an object, be careful not to think that the strongest force "overcomes" the others to determine the motion on its own. It is F'net, the sum of all the forces, that determines the acceleration a.◄ Acceleration of a wind-blown basketball You drop a basketball while a stiff breeze is blowing In what direction does the ball accelerate? to the right. REASON Wind is just air in motion. If the air is moving to the righr with respect to the ball. then the ball is moving to the left with respect to the air. There will be a drag force opposite the velocity of the ball relative to the air. 10 the right. So. as FIGURE 4.22a shows, two forces are acting on the ball: its weight,,, directed downward and the drag force fj directed to the right. Newton's second law tells us that the direct.ion of the acceleration is the same as the direction of the net force Fne,-ln FIGURE 4.22b we find Fne, by graphical vector addition of 111 and D.We see that Fne,and therefore a poin1 downward and to the right. FIGURE 4.22 (a) A basketball falling in a strong breeze. The drag force is to the right. ~ ·., jj '<fJJ _ II' r----The weight force poin1sdown. (b) The acceleration is in the direction of Fnct· ...... . \ ;; ASSESS This makes sense on the basis of your experience. Weight pulls the ball down. and the wind pushes the ball to the right. The net result is an acceleration down and to the right. 4.5 The Unit of Force Because F.,ei = ma, the units Newton's Approximate some typical forces magnitude of TABLE 4.2 of force must be the unit of mass (kg) multiplied by the unit of acceleration (m/s 2); thus the units of force are kg· m/s 2 . This unit of force is called the newton: k0 m I newton = I N = l ~ • s- The abbreviation for the newton is N. TABLE 4.2 lists some typical forces in newtons. The newton is a secondary unit, meaning that it is defined in terms of the prima,y unils of kilograms, meters, and seconds. The unit of force in the English system is the pound (abbreviated lb). Although the definition of the pound has varied, it is now defined in terms of the newton: 119 Second Law Approximate magnitude (newtons) Force Weight of a U.S. nickel 0.05 Weight of 1/4 cup of sugar 0.5 Weight of a I pound object s Weight of a typical house cat 50 Weight of a I IO pound person 500 l pound = l lb = 4.45 N Propulsion force of a car 5000 You very likely associate pounds with kilograms rather than with newtons. Everyday language often confuses the ideas of mass and weight, but we're going to need to make a elem·distinction between them. We'll have more to say about this in the next chapter. Thrust force of a small jet engine 50,000 Pulling force of a locomotive 500,000 Racing down the runway EXAMPLE 4.6 A Boeing 737-a small, short-rangejet with a mass of 51.000 kgsits at rest at the start of a runway. The pilot turns the pair of jet engines to full thronle, and the thrust accelerates the plane down the runway. After traveling 940 m. the plane reaches its takeoff speed of 70 m/s and leaves the ground. What is the thrust of each engine? We don't know how much time it took the plane to reach its takeoff speed. but we do know that it traveled a distance of 940 m. We can solve for the acceleration by using Equation 2.13 of Synthesis 2.1 : If we assume that 1he plane undergoes a constant acceleration (a reasonable assumption), we can use kinematics to find the magnitude of that acceleration. Then we will use Newton·s second law to find the force-the thrust-that produced this acceleration. The displacement is .lx = xr - xi = 940 m. and the initial velocity is 0. We can rearrange the equation to solve for the acceleration: STRATEGIZE SOLVE (vx)l = (v.,)? + 2ax.h (v.r)r2 (70m/s)2 ar=--=----=2.61 • 2 .l.x 2(940 m) mis2 FIGURE 4.23 is a visual overview of the airplane·s PREPARE motion. . FIGURE4.23 Visual overview of the accelerating airplane. ii ........ ,;• ... •.,• .,• ...... We"ve kept an extra significant figure because this isn"t our final result-we are asked to solve for the thrust. We complete the solution by using Newton's second law: F =ma_,= (51,000 kg)(2.6I m/s 2) ... = 133.000 N The thrust of eacb engine is half of this total force: Thrust of one engine= 67.000 N = 67 k.N ~x 0 An acceleration of about ¼g seems reasonable for an airplane: It's zippy. but it"s nor a thrill ride. And the final value we find for the thrust of each engine is close to the value given in Table 4.2. This gives us confidence that our final result makes good physical sense. ASSESS Known x; = 0 m. (1:r);= 0 m/s Find a., and Fne, xr = 9-lO m, (1:r)r= 70 m/s 44..;s.,.rnm,,, Three forces act on an object. In which diJection does the object accelerate? A. B. C. D. E. 120 CHAPTER 4 ForcesandNewton'sLawsofMotion 4.6 Free-Body Diagrams When we solve a dynamics problem, it is useful to assemble all of the information about the forces that act on an object (or "body") into a single diagram called a free-body diagram. A free-body diagram represents the object as a particle and shows all of the forces that act on the object. Learning how to draw a correct free-body diagram is a very important skill, one that in the next chapter will become a critical part of our approach to solving motion problems. Drawing a free-body diagram TACTICS BOX 4.3 0 Identify all forces acting on the object. This step was described in Tactics Box4.2. @ Draw a coordinate system. Use the axes defined in your pictorial representation (see Tactics Box 2.2). lf those axes are tilted, for motion along an incline, then the axes of the free-body diagram should be similarly tilted. @ Represent the object as a dot at the origin of the coordinate axes. This is the particle model. 0 Draw vectors representing each of the identified forces. This was described in Tactics Box 4.1. Be sure to label each force vector. 0 Draw and label the net force vector Fnet· Draw this vector beside the diagram, not on the particle. Then check that l 0 • 1 points in the same direction as the acceleration vector a on your motion diagram. Or, if appropriate, write F'nel = 0. Exercises 17-22 EXAMPLE 4.7 Forces on an elevator An elevator, suspended by a cable, speeds up as it moves upward from the ground floor. Draw a free-body diagram of the elevator. (downward) weight force iv. We have sbown I.hisin the free-body diagram by drawing the tension vector longer than the weight vector. We'll follow the steps of Tactics Box 4.3. We note that the elevator is moving upward and its speed is increasing. That means the acceleration is directed upward, so that F001 must be directed upward as well. STRATEGIZE Let"s take a look at our picture and see if it makes sense. The coordinate axes, with a vertical y-axis, are the ones we would use in a pictorial representation of the motion. so we've chosen the correct axes. And, as noted, the tension force is drawn longer than the weight, which indicates an upward net force and hence an upward acceleration. ASSESS PREPARE 1n FIGURE 4.24 we illustrate the steps listed in Tactics Box 4.3. Because the net force is directed upward, the magnitude of the (upward) tension force T must be greater than that of the FIGURE 4.24 Free-body diagram of an elevator accelerating upward. Force identification Free-body diagram .!:J ...@ Draw a coordinate system. ~ ................. •· T ./ /J.Jt.i,j,f w 0 Identify all forces acting on the object. ............... @ Represent 1heobjecl .·• ..••••• .,: X as a dol at the oriein . t ",t~'. 0 Draw veclors for the identified forces. ~ 0DraudSbel f~ beside the diagram. jg 121 4.6 Free-Body Diagrams Forces on the world's fastest car EXAMPLE 4.8 The speed record for the world's fastest car is held by the jetpowered Thrust SSC. whose top speed exceeds the speed of sound. At high speeds. air drag is significant. The Thrust SSC is the world's fastest car. but when the car is just starting to accelerate from rest_friction and drag are negligiblecompared to the jet thrust Draw a visual overview-motion diagram, force identification picture. and free-body diagram-for the car as it starts to move forward from rest. STRATEGIZE To draw the visual overview. we'll use the steps listed in Tactics Box 4.3. PREPARE We treat the car as a particle. The visuaJ overview is shown in FIGURE 4.25. The motion diagram tells us that the acceleration is in the positive x-direction. According to the rules of vector addition, this can be true only if the upward-pointing ii and the downwardpointing ii'•are equaJ in magnitude and thus cancel each other. The vectors have been drawn accordingly, and this leaves the net force vector pointing toward the right, in agreement with ii from the motion diagram. ASSESS FIGURE 4 25 Visual overview for a carbon dioxide racer. Free-body diagram Force identification Motion diagram Thratf firce- F.,,..,,.,- . ii 7 ii -------x. ~1rt;;;\ F............... EXAMPLE 4.9 \t'---~ ... ... ... iJ ~ _ Norr11/force.ii ....-.Fnrl ••·························· Check thatFnctpoints in the same directionas ii............................••••••••••••• Forces on a towed skier A tow rope pulls a skier up a snow-covered hilJ at a constant speed. Draw a full visual overview of the skier. This is example 4.2 again, with the additional information that the skier is moving at a constant speed. Because his acceleration is zero, the net force acting on him must be zero as well. STRATEGIZE lf we were doing a kinematics problem, the pictorial representation would use a tiJtedcoordinate system with the x-axis parallel to the slope, so we use these same tilted coordinate axes for the free-body diagram. The full visual overview is shown in FIGURE 4.26. PREPARE We have shown f pulling parallel to the slope and Jk, which opposes the direction of motion. pointing down the slope. The normaJ force ii is perpendicular to the surface and thus along the y-axis. Finally. and this is important, the weight 111is vertically downward, 1101 along the negative y-axis. T_!leskier moves in a straight line with constant speed. so ii = 0. Newton ·s second law then tells us that Fnet = = 0. Thus we have drawn the vectors such that the forces add to zero. we·11 learn more about how to do this in Chapter 5. ASSESS ma FIGURE 4.26 Visual overview for a skier being towed at a constant speed. )' X v? a=o "··. Normal force ii ' ••••.. Weight ii• ••••• .............. Check that fu~oox Fnet = 0 F,.,, matches ii. .................................... /' w ~--. The weight vector always points venicallydownward. 122 CHAPTER 4 Forces and Newton's Laws of Motion Free-body diagrams will be our major tool for the next several chapters. Careful practice with the workbook exercises and homework in this chapter will pay immediate benefits in the next chapter. Indeed, it is fair to say that a problem is more than half solved when you correctly complete the free-body diagram. STOP TO THINK 4.7 An elevator suspended by a cable is moving upward and slowing to a stop. Whjch free-body diagram is c01Tect? y y t t t X ,v A. y .Y t 1\/ B. F'ele\'alor .r X ,V C. X l\c, = 6 tt, D. E. 4.7 Newton's Third Law FIGURE 4.27 The hammer and nail each exert a force on the other. The hammer exerts a force on the nai I . ·.°":-• ..... □ ·:u a ) .. •••••... but the nail also exerts a force on 1hehammer. FIGURE 4.28 An action/reaction pair of forces. B A f'"AnO 0 Thus far, we've focused on the motion of a single particle responding to well-defined forces exerted by other objects or to long-range forces. A skier sliding downhill, for instance, is subject to frictional and normal forces from the slope and the pull of gravity on hjs body. Once we have identified these forces, we can use Newton's second law to calculate the acceleration, and hence the overall motion, of the skier. But motion in the real world often involves two or more objects interacting with each other. Consider the hammer and nail in FIGURE 4.27. As the hammer hjts the nail, the nail pushes back on the hammer. A bat and a ball, your foot and a soccer ball, and the earth-moon system are other examples of interacting objects. We need to consider how the forces on these interacting objects are related to each other. Interacting Objects Think about the hanm1er and nail in Figure 4.27 again. The hammer certainly exerts a force on the nail as it drives the nail forward. At the same time, the nail exerts a force on the hammer. If you are not sure that it does, imagine hitting the nail with a glass hammer. It's the force of the nail on the hammer that would cause the glass to shatter. Indeed, any time that object A pushes or pulls on object B, object B pushes or pulls back on object A. As you push on a filing cabinet to move it, the cabinet pushes back on you. (If you pushed forward without the cabinet pushing back, you would fall forward in the same way you do if someone suddenly opens a door you're leaning against.) Your chair pushes upward on you (the normal force that keeps you from falling) while, at the same time, you push down on the chair, compressing the cushion. These are examples of what we call an interaction. An interaction is the mutual influence of two objects on each other. These examples illustrate a key aspect of interactions: The forces involved in an interaction between two objects always occur as a pair: To be more specific, if object A exerts a force F'Aon 8 on object B, then object B exerts a force F8011A on object A. Thjs prur of forces, shown in FIGURE 4.28, is called an action/reaction pair. Two objects interact by exerting an action/reaction pair of forces on each other. Notice the very explicit subscripts on the force vectors. The first letter is the agent-the source of the force-and the second letter is the object on which the force acts. F'AonB is thus the force exerted by A on B. 4.7 Newton's 123 Third Law (mIB ► The name "action/reaction pair" is somewhat rrusleading. The forces occur simultaneously, and we cannot say which is the "action" and which the "reaction." Neither is there any implication about cause and effect: The action does not cause the reaction. An action/reaction pair of forces exists as a pair, or not at all. For action/reaction pairs, the labels are the key: Force FAonB is paired with force FsonA· ◄ Reasoning with Newton's Third Law Two objects always interact via an action/reaction pair of forces. Newton was the first to recognize how the two members of an action/reaction pair of forces are related to each other. Today we know this as Newton's third law: Newton's third law Every force occurs as one member of an action/reaction pair of forces. ■ The two members of an action/reaction pair act on two different objects. ■ The two members of an action/reaction pair point in opposite directions and are equal in magnitude. Newton's third law is often stated: "For every action there is an equal but opposite reaction." While this is a catchy plu·ase, it lacks the preciseness of our prefeJTed version. In particular, it fails to capture an essential feature of the two members of an action/reaction pair-that each acts on a different object. This is shown in FIGURE4.29. where a hanu11erhitting a nail exerts a force F'hammeronnail on the nail, and by the third law, the nail must exert a force Fnailonhammer to complete the action/reaction pair. Figure 4.29 also illustrates