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Chapter10 - CHAPTER FORCES I don't know what I may seem to...

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Unformatted text preview: CHAPTER FORCES I don't know what I may seem to the world, but, as to myself, I seem to have been only like a boy playing on the sea shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. Sir Isaac Newton 10.1 THE FUNDAMENTAL FORCES Using crude water clocks to time balls rolling down inclined planes, Galileo searched for and found a description of how bodies fall. His law of falling bodies, however, wasn’t a fundamental law of nature. Within half a century it was superseded by a deeper insight into nature — Newton’s universal law of gravity. Through the genius of Newton, the force of gravity, 193 194 FORCES (8.1) was revealed as a fundamental force of nature. Gravity acts on all matter, a fact reflected in its dependence on the masses of objects. Although its strength diminishes with distance, the effects of gravity are nevertheless felt across the far reaches of the universe. Gravity holds together planets and stars, organizes solar systems and galaxies; it orders the universe. Inspired by Newton, scientists in the eighteenth century sought to identify, classify, and mathematically describe the numerous forces observed in nature. Knowledge of these forces provided physics with a certain predictive power, because according to Newton’s second law, F = ma, forces shape the motion of all things. Through painstaking ex- periments, these scientists developed empirical descriptions of forces in the world about them: tensions, spring forces, viscosity, friction, electricity, magnetism, heat, light, chemical action. As the number of forces grew, so did the applications in an increasingly industrialized world. Yet there was a question confronting these physicists. Were all these forces fundamental, or could they be reduced to more basic forces? Not until late in the eighteenth century did another force emerge as fundamental — the electrical force. The French engineer Charles Augustin Coulomb assumed that, anal— ogous to the gravitational force between two masses, the electric force between two charges is proportional to the product of the charges. Experimentally he found that the electric force is similar to gravity in another way: the force between two charges decreases as the square of the distance between them. Summarized mathematically, the electric force F between two charges q] and q2 that are separated by a distance r is known as Coulomb’s law and written ‘1142 r2 F = K6 1". (10.1) Just as G is a universal constant for gravity, Ke is a universal constant for electricity. Magnetism was also identified as a fundamental force of nature. The attraction or repulsion between two magnets could be described by a force similar to Coulomb’s law. The progress of physics was a triumph of Newtonian mechanics: the forces of nature were successively reduced to attractions and repulsions between particles. The first 40 years of the nineteenth century, however, saw a growing reaction against such a division of phenomena, in favor of some kind of correlation of forces. The turn inward to unification of forces was spearheaded by James Clerk Maxwell. By the second half of the nineteenth century Maxwell succeeded in unifying two hitherto disparate forces, electricity and magnetism, into one — electromagnetism. Maxwell’s unification of elec- tricity and magnetism was expressed by a set of equations that interrelate electric and magnetic phenomena. Soon tensions, spring forces, friction, viscosity, chemical actions, and even light were recognized as arising fundamentally from the electromagnetic force; this force dominates the everyday world about us. Based on Maxwell’s success the search for a common mathematical description, or unification, of forces had begun. With the twentieth century came the discovery of radioactivity, the probing of atoms, and the subsequent realization that more than just gravity and electromagnetism would be needed to explain this new world. Experiments probing invisible atoms revealed that inside an atom there is a compact center — the nucleus — composed of positively charged 10.2 GRAVITATIONAL AND ELECTRIC FORCES ~ 195 protons and neutral neutrons. Negatively charged electrons orbit the nucleus, held by the electric force from the protons. This naturally led scientists to ask what held the nucleus together, since the protons in it would be expected to repel one another, and the neutrons in it should neither repel nor attract the protons or one another. Physicists realized that neither gravity nor electromagnetism held the compact nucleus together, but that a new force was at work. Aptly named the strong force, it overcomes the electric repulsion between protons and holds the nucleus together. Unlike gravity and electricity, the strong force does not extend to distant comers of the universe; it has a limited range — the size of a nucleus, 10‘” cm. Outside this range, the strong force has no effect. If it did, the universe would be one very dense lump of subatomic particles. Natural radioactivity could not be explained by any of the known forces — strong, electromagnetic, or gravitational. Another force was responsible for the decays of nuclei — the weak force. This force is 105 times weaker than the strong force, but like the strong force it has a limited range, which is the size of a nucleus. Nevertheless, the weak force causes some stars ultimately to explode. Table 10.1 summarizes the four fundamental forces of nature, the strong, electro- magnetic, weak,.and gravitational; their relative strengths; and their respective ranges. Table 10.1 Characteristics of the Four Fundamental Forces Force Relative strength Range Importance Strong 1 10‘13 cm Holds nucleus together Electromagnetic 10‘2 Infinite Friction, tensions, etc. Weak 10‘5 10‘13 cm Nuclear decay Gravitational 10‘39 Infinite Organizes universe The behavior of each of the four forces is reasonably understood, but nobody knows why there should be four of them. Albert Einstein spent the last 20 years of his life unsuc- cessfully searching for a way to unify two of the forces, gravity and electromagnetism. In the twentieth century a search for unification of the fundamental forces has become an important part of physics. The water clocks and inclined planes of Galileo have been replaced by increasingly larger, more energetic particle accelerators. Emerging are unified and grand unified theories that present a coherent account of how these forces may have evolved from simpler laws in the infancy of the universe. The early universe ultimately may be the only experimental test for such theories. It may be the great ocean of truth that still lies undiscovered before us. 10.2 GRAVITATIONAL AND ELECTRIC FORCES , One of the great and deep mysteries of physics is that the laws describing gravitational force and electric force have the same mathematical form: M M ‘2 2 r (8.1) F=—G , 196 FORCES r. (10.1) It seems almost a minor point that each has an unknown universal constant in it. Yet for real-world applications, it is essential to know what those constants are. For gravity, we already know that G is related to the acceleration of a falling body near the surface of the earth, g, and the mass and radius of the earth: g = GME/Rg. The radius of the earth has been known for a long time, and we also know g, so measuring G amounts to finding the mass of the earth, ME. The determination of G was one of the classic experiments of physics. Henry Cavendish, a British physicist, performed the historic experiment to measure G in 1798. Cavendish was deeply inspired by Newton and regarded the Principia as the model for exact sciences, and the search for the forces between particles guided his scientific explorations. But Cavendish had fitful habits of publication; he left unpublished whatever did not fully satisfy him. Luckily the determination of G was an experiment of which he was proud. Figure 10.1, which is taken from Cavendish’s 1798 article, shows the apparatus he used for his delicate experiment to determine the value of G. In that experiment, he measured forces equal to one-billionth of the weights of the bodies involved. The two small lead balls are attached to a rigid rod, forming a dumbbell that is suspended by a thin fiber that allows the dumbbell to rotate freely. When the two larger lead balls are placed near the ends of the dumbbell, the smaller masses are attracted to the larger ones by the gravitational force. This force, although extremely small, nevertheless rotates the dumbbell and twists the fiber, which opposes the twisting. Using a telescope, Cavendish sighted the balls against scales illuminated by candles and thereby measured the amount of twisting. From that, he determined the force between the two balls, and then through Eq. (8.1) found the value of G. The accepted value is G = 6.67 X 10‘11 N mz/kgz. By weighing the world, Cavendish rendered the universal law of gravitation complete. The law was no longer a proportionality as Newton had stated it, but an exact law through which quantitative analyses could be made. It was the most important contribution to gravitation since Newton. Questions 1. Why are lead masses used instead of, say, rubber masses? 2. Newton’s universal law of gravity holds for particles, but the spheres used in the experiment are not particles but extended objects. Does this affect the result of the experiment? 3. Using the values of G, g, and the radius of the earth, calculate the mass of the earth. From your value for the mass, determine the density of the earth by treating 102 GRAVITATIONAL AND ELECTRIC FORCES 197 Figure 10.1 Cavendish’s apparatus for measuring G. (From The Scientific Papers of the Honourable Henry Cavendish, Cambridge University Press.) the earth as a solid sphere. The average density of rocks on the earth’s surface is about 2.0 g/cm3. Comparing this value to your calculated value, what can you con— clude about the interior of the earth? Through a similar experiment in 1787, Coulomb showed that the electric force between two charges is similar to gravity: it decreases as the inverse square of the distance between the charges. In fact, Cavendish himself had already performed an even better experiment than Coulomb’s, but, characteristically, he never published it. In any case, the value of the electric constant Ke could be measured; its value is 9.0 X 109 N m2/ C2, where one coulomb (l C) is the unit of charge. But what is charge? The early Greeks had discovered that amber attracts bits of straw, and they identified that property of amber with charge. Charge is that which creates electric forces; even today that’s all we can say about it. We don’t know exactly what charge is any more than we know what mass is. Unlike mass, charge comes in two different varieties — positive and negative. There also is a smallest unit of charge — the charge of the proton (or electron). All charges come in multiples of this unit of electricity; the charge of a proton is 1.6 X 10‘19 C. Furthermore, like charges repel, whereas opposite charges 198 FORCES attract. Consequently, there can be attractive or repulsive forces between charges, and electricity can be neutralized. Because the electric force is comparatively strong, opposite charges attract and neutralize each other. Gravity, on the other hand, is always attractive. Atoms consist of positively charged nuclei and negatively charged electrons. A proton is about 2000 times heavier than an electron. The simplest atom, hydrogen, consists of one proton and one electron in a region of 10‘8 cm. This region is much bigger than the nucleus, so we imagine a low- density cloud of negative charge attached to the positive nucleus. The force that holds the electron cloud to the proton to make a hydrogen atom is the electric force. To construct a model of heavier atoms, we first construct nuclei with more protons and neutrons in them. Since the overall charge of atoms is zero, there are as many electrons in clouds about the nucleus as there are protons in the nucleus. Even though the electrons repel one another, they are held to the nucleus by the electric force. Atoms can in turn attract other atoms to make larger composites called molecules. The force that holds the atoms together to form molecules is again the electric force. Atoms and molecules can form larger agglomerations, which we see as liquids and solids. These too are held together by electric forces. Electricity is a fundamental force that governs the nature of the world around us. At distances small compared to the nucleus, the strong and weak forces dominate. Over distances large compared to the earth, gravity dominates. For the world of matter as we know it, electric forces are dominant. The gravitational and electric forces have the property that they act on distant objects through seemingly empty space. The moon, for example, feels a gravitational force from the earth 240,000 mi (390,000 km) away. Newton was bothered by the action-at—a-distance character of gravity and thought that there should be a physical mechanism for transmitting this force. Yet when pressed for such a mechanism, he declared, “I make no hypotheses.” Question 4. Compare the electric force between the proton and electron in a hydrogen atom with the gravitational force between them. Does your answer explain why grav1ty is not responsible for binding atoms together? (Use mp =1. 67 x 10 27 kg,m 9.11 X 10‘31 kg.) 10.3 CONTACT FORCES The fundamental forces of nature act at a distance: their effects can be experienced when the particles are not in contact. A second category is‘contact forces. These are forces that two objects exert on each other when they are physically in contact with each other, as, for example, when a book rests on a table. Contact forces are not fundamental forces; instead, they arise fundamentally from electric forces acting in complicated ways. The force a spring exerts on an object is an example of an electric force. Inside the spring are metal atoms that are bound together by electric forces. These electric forces keep the metal atoms a certain distance apart, called the equilibrium distance. When you stretch a spring, each atom is pulled a tiny bit out of the equilibrium distance. The electric forces try to pull the atoms back into the equilibrium position. The net result of all the electric forces acting on the atoms is what causes the end of the spring to pull on you, that is, to exert a macroscopic force. 10.3 CONTACT FORCES 199 Trying to describe how a spring works by examining the electric forces acting between the atoms is impossible, because of the sheer numbers of atoms to consider; there might be 1024 atoms in a mousetrap spring. Rather than attempting to describe all these com- plicated interactions in terms of a fundamental force, we describe them by empirical rules, which are experimental summaries of the net result of all the complications. Most of these empirical descriptions were deduced by eighteenth-century scientists. The empirical law for a spring is simple: the force exerted by a spring (on an object) is proportional to the change in length of the spring. This is known as Hooke’s law (after Robert Hooke, a contemporary of Newton) and may be expressed as (10.2) Hooke’s law: where x is the change in length of the spring. The constant k is called the spring constant and is a measure of the stiffness of a spring; the stiffer the spring, the larger the value of k. The direction of the force is always opposite to the displacement of the end of the spring from its unstretched position. Whenx > 0, the spring is stretched and F is negative; when x < O, the spring is compressed and F is positive. The spring force always acts to restore the spring to its unstretched length, as Fig. 10.2 illustrates. WW equilibrium length I I Figure 10.2 Force exerted by a spring described by Hooke’s law. Example 1 A 1.5—kg block on a frictionless table is attached to a spring with spring constant k = 0.5 N/m. If the spring is stretched 3.0 cm and released, what is the acceleration of the block at the instant it is released? From Hooke’s law, we know that the force exerted on the block by the spring is F = — kx, where x = 0.03 m. By Newton’s second law, this force accelerates the block according to F = ma. Therefore the acceleration of the block at the instant it is released is 200 . FORCES a : F /m = — kx/m, a = —(0.5 N/m)(0.03 m)/(1.5 kg) = —0.01 m/sz. Since the force F = — kx changes as the spring contracts, the acceleration is not constant. The tension in a rope or string is another example of an electric force. As in a spring, the atoms in the rope have equilibrium positions at which electric forces tend to keep them. When you pull on one end each atom electrically tugs on its neighbors, but unlike a spring, the rope doesn’t stretch because it can’t uncoil. The pull is transmitted to the other end of the rope, usually undiminished in force, much like a chain link, as Fig. 10.3 illustrates. Figure 10.3 Tension in a rope arises from electric interactions. Whenever any object is pressing against another there is a contact force between the two objects, known as the normal force. (Here normal means perpendicular, not the opposite of abnormal.) This force is a result of repulsion between the atoms of the two objects. The magnitude of the normal force depends on how hard the two objects press against each other. The direction of the normal force acting on an object, however, is always perpendicular to the surface. Figure 10.4 indicates normal forces between different objects. Figure 10.4 Illustration of the normal force between different objects. Friction is an inescapable example of an electric force. At times we wish that we could do away with it, so as, for example, to improve engine performance, yet without it we couldn’t walk. Even though a highly polished object may appear smooth, when 10.3 CONTACT FORCES 201 examined through a microscope it appears very rough, having many tiny surface irreg- ularities, as shown in Fig. 10.5. When two objects are placed in contact, the many contact points resulting from the (microscopic) rough edges actually become welded together by electric forces. When one object moves across another, these tiny welds rupture and continually reform. The net result is friction — a force parallel to the surface which opposes the motion of the object. Since the number of welds is proportional to pressure from the object on the surface, the force of friction is proportional to the normal force on the object. Figure 10.5 Microscopic examination of a highly polished surface reveals irregularities. Example 2 A mover pulls on a 20.0—kg crate resting on a floor with a force of 80.0 N at an angle of 37°, but the crate does not move. What is the normal force of the floor on the crate? Let’s first list the forces that act on the crate: (a) tension T = 80.0 N from the rope, (b) weight W = mg = 196 N, (c) normal force N from the floor, (d) friction f from the floor. In the diagram we indicate these forces as well as a set of coordinate axes. Since the block has zero acceleration (it doesn’t move), Newton’s second law implies E F = 0, which gives us two scalar equations, 2F,=0, 213:0. The normal force is in the y direction, so let’s add up the components of all the forces in that direction and, by the second law above, set them equal to zero. The result is 2Fy=N—mg+Tsin37°=0. 202 FORCES Solving for N, we get N = mg — Tsin 37° = 196 — (80) sin 37° = 148 N. Does this make sense? If the crate were simply sitting on the floor with no rope pulling on it, the floor would support the entire weight of the crate and then N = mg. But the rope has a vertical component, which helps support the weight of the crate and decreases the normal force from the floor, just as we found. In the Principia Newton considered the motion of objects in resi...
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