4. The Fundamental Interactions

Transcription

1 4. The Fundamental Interactions So far in our study of motion, we have learned that forces occur in interactions and cause accelerations. By itself, this information is of limited use until we learn how to identify interactions and predict the important features of the resulting forces. All the forces we directly encounter in ordinary life result from only two kinds of interactions gravitational and electromagnetic. The other fundamental interactions are nuclear, which are important only when the interacting objects are as close together as inside the nucleus of an atom. These will be discussed in a later chapter. As we discuss the electromagnetic interaction, we will also begin to examine the question of what matter is made. We will find that all matter, living as well as nonliving, is composed of electrically charged particles. The electromagnetic interaction between these is responsible for most of the forces we encounter. We begin our discussion of interactions with a consideration of some of the manifestations of the gravitational interaction. Falling Objects Consider the motion of an object, such as a baseball, dropped from a great height. As it falls, its speed steadily increases. If we were to make careful measurements, we would find that, in the absence of air friction, its speed would increase at a constant rate. After 1 sec, its speed would be 35 kilometers/hour; after 2 seconds, 70 kilometers/hour; and after 3 seconds, 105 kilometers/hour. The speed would increase at 35 kilometers/hour every second as long as it falls. This rate of acceleration is sometimes designated by the symbol g. The falling ball is clearly accelerating. If its motion is in harmony with the Second Law of Motion, and it is, some downward force must be acting upon it. The force that causes this acceleration is called the weight of the ball (Fig. 4.1). The next step is to drop a different object (e.g., a large anchor) from the same height. Before actually doing the experiment, we might expect the anchor to drop more rapidly than the baseball, since it obviously has more weight. But nature does not always behave the way we expect. In this case, the acceleration of the anchor is exactly the same as that of the baseball! Figure 4.1. Why does a falling ball accelerate? Somehow the force causing the acceleration, the weight of the anchor, has increased in exactly the same ratio as its mass so that the resulting acceleration, determined by force divided by mass, does not change (Fig. 4.2). This surprising result is true for all objects near the surface of the earth. Even light objects such as feathers and sheets of paper have exactly the same acceleration when allowed to fall in the absence of air resistance. The free-fall acceleration, g, is the same for all objects. This must mean that weight (the force causing the acceleration) and mass are proportional. If the mass of one object is two times the mass of another, its weight is larger in exactly the same ratio. The acceleration (force divided by mass) is then the same for both. These results suggest the following conclusion: Every object near the surface of the earth experiences a downward force, called its weight, the strength of which is exactly proportional to its mass. 29

2 Clear evidence also shows that the moon pulls on the earth. The most apparent results of this force are the lunar tides in which the level surfaces of oceans rise and fall as the moon passes overhead. The earth s attraction to the moon also causes the earth to accelerate slightly. Such accelerations are measured routinely by sophisticated navigational instruments such as those used on submarines. An additional feature of the moon s acceleration worth noting is calculated by using a mathematical definition of acceleration and measurements of the moon s orbit. The moon s sideways acceleration is almost exactly 1/3,600 the acceleration of an object falling near the earth s surface, so the earth s pull on the moon must be only 1/3,600 as strong as it would be if the moon were moved to the earth s surface. The Law of Universal Gravitation Figure 4.2. Why do a falling anchor and a falling ball accelerate at the same rate when air resistance is not important? From these observations alone, we do not really know where this force comes from, but apparently every object is pulled toward the earth. We might suspect that some kind of interaction between the object and the earth is responsible. The Moon s Orbit The moon circles the earth in an almost perfect circle every 27.3 days. Since it is not moving in a straight line, we know that it is accelerating and that its acceleration is caused by some force. What can we say about the force? Your understanding of the last chapter lets you know immediately that the moon is experiencing a force sideways to its direction of motion that causes the continuous change in the direction of its motion. Further, you know that the force is directed toward the center of the circular path. The moon moves as if it, too, is being pulled toward the earth (Fig. 4.3). Isaac Newton was the first to suggest that the attraction of the moon to the earth is due to the same kind of interaction that causes free objects near the earth to fall. These are called gravitational interactions. The observations we have described tell us much about the forces resulting from gravitational interaction. First, the two interacting objects always attract each other. Each force is proportional to the mass of the object on which it acts because g is the same for all objects. The two forces in each interaction obey the Third Law of Motion. (Remember the lunar tides and the acceleration of the earth due to the moon s attraction.) Finally, the force is weaker if the two objects are farther apart, since the moon s acceleration is only 1/3,600 as much as it would be if it were near the earth s surface. In fact, the strength of the force depends on the square of the distance between the centers of the interacting objects. The moon is 60 times farther from the center of the earth than is the earth s surface. Notice that 60 2 is 3600, the observed factor by which the gravitational force on the moon is diminished. With these insights, Newton suggested that every m M Figure 4.3. Something must be pulling or pushing the moon toward the earth. How do we know? What is it? Figure 4.4. Every object is attracted to every other object through gravitational interaction. The two forces have the same strength. 30

3 object in the universe interacts with every other object through gravitational interaction (Fig. 4.4). Since the strengths of the resulting forces depend on mass, they are ordinarily too small to be noticed for most objects. Only if one of the interacting objects has a large mass, like the earth, does the force become appreciable. This Universal Law of Gravitation (or the Law of Gravity) can be summarized as follows: Every object in the universe attracts every other object by a long-range gravitational interaction that obeys Newton s Third Law. The strength of the attractive force, F, varies with the masses, M and m, of the two objects and the distance, d, between their centers according to the relationship F GmM Newton s hypothesis is subject to experimental verification. It was confirmed in every detail over 70 years after Newton s death by Henry Cavendish ( ), who finally developed a method of measuring the gravitational attraction between such ordinary-sized objects as two large lead balls. Earlier support had come by studying the planets and their moons, whose motions through space can be explained in terms of Newton s Universal Law of Gravitation. The number G that appears in the equation for the strength of the gravitational force is called the gravitational constant. It must be measured experimentally and is so small that the mutual attractive gravitational force between two 100-kilogram balls placed 30 centimeters apart would be equivalent to the earth weight of only 0.01 gram of mass. (The actual value of G is in the metric system of units.) No wonder we ordinarily do not notice these forces, which were measured only in fairly recent times. Some Simple Experiments with Electricity It has been known, at least since early Greek times, that certain pairs of materials become electrified or charged when they are rubbed together. Suppose we rub one end of a hard rubber rod with a piece of fur and then hang the rod from a string without touching the rubbed end. Then we similarly rub one end of a second rod and hold it near the first. You will see from the motion of the hanging rod that the rods repel each other even when they are some distance apart. There is an interaction between the two rods. Further careful testing would show that the repulsion becomes greater as the two rods come closer together. Two glass rods that are rubbed with silk react similarly. The glass rods repel each other with a force that Figure 4.5. Two rubbed rubber rods or two rubbed glass rods repel each other. Yet a rubber rod and a glass rod are attracted. Why? increases as the rods get closer. A new feature is revealed, however, when we bring a charged glass rod near a charged rubber rod. The two dissimilar rods attract each other with a force that becomes larger as the rods come closer together. We are dealing with something more complicated than gravity, since these forces can be either attractive or repulsive, depending on the circumstances. Other kinds of materials can be electrified by rubbing. When they are, pairs of similar rods always repel each other. Some, however, are attracted to a charged rubber rod and some are repelled by it. Those that are attracted to the charged rubber rod are repelled by a charged glass rod and vice versa. Those attracted to the rubber rod are said to be positively charged; those attracted to the glass, negatively charged (Fig. 4.5). The explanation of these experiments requires two new broad insights. First, we need to know more about how materials are made and what it is that changes when they become charged. The second major part of the puzzle has to do with the law governing the interac- 31

4 tion itself. What determines the strength and direction of the resulting forces? The Electrical Model of Matter An important conclusion of these experiments is that matter is made of more basic pieces. Rubber, glass, silk, fur, and all other materials presumably have some important constituents in common. The experiments suggest at least two kinds. Many objects do not seem to be either attracted or repelled by charged objects, yet can be charged by rubbing. This suggests that the materials normally contain both kinds of constituents. When the constituents occur in equal amounts, they cancel each other s effects; the material is not charged, and is said to be electrically neutral. When this balance is disturbed by rubbing one of the constituents either off or onto the object, for example, the object becomes charged. If it has more of one constituent, it is said to have a positive charge; if more of the other, a negative charge (Fig. 4.6). b a Figure 4.6. All matter contains electric charge. The object in (a) is electrically neutral, in (b) it is positively charged, and in (c) it is negatively charged. More sophisticated research that is discussed later reveals that protons are an important constituent of all matter. These are tiny, dense particles in the center of all atoms. All protons are exactly alike, and each carries one unit of positive electric charge. The negative charge in matter is supplied by electrons, each of which can balance the positive charge of a proton exactly. Electrons have little mass only about 1/1,836 that of protons. The unit of charge used in calculations is the coulomb, equivalent to the charge of about protons. We could measure electric charge by simply counting the number of excess electrons or protons, but this is usually impractical because of the large numbers involved. Electric charge has one property that we have not encountered previously. It is discrete; that is, it occurs only in multiples of a fundamental unit, the charge of a single proton. Other physical quantities, such as mass, c force, and speed, can have any value and are said to be continuous. In many materials some of the electrons can be removed from the surface by rubbing. When rubber is rubbed with fur, some of the electrons in the fur are transferred to the rubber, which becomes negatively charged. (The fur and rod are attracted to each other, incidentally.) The electrons carry so little mass that the objects seem the same as before, except that they are now electrically charged. Protons, because of their larger mass, are held rigidly in place in all solid materials. This picture of matter might be termed the Electrical Model of Matter. It leaves many questions unanswered (e.g., how these charged particles are arranged in matter, how they combine to create the almost numberless kinds of materials found in living and nonliving matter, and what happens when materials change form). But it does provide an adequate model for explaining a wide range of experiences. We can summarize the model as follows: All matter contains two kinds of electrically charged particles: positive protons and negative electrons. Electrons have little mass and can be quite mobile and transferable from one object to another. Protons are held rigidly in place in solid materials. Objects that have equal numbers of protons and electrons are electrically neutral. Objects with more electrons than protons are negatively charged. Those with fewer electrons than protons are positively charged. The amount of extra charge of either kind is called the charge of an object. The Electrical Force Law By now you have probably guessed the main features of the electrical interaction. Objects with the same kind of charge repel each other. (Remember that in our experiments identical rods always repelled each other.) Objects with opposite charges one positive, the other negative attract each other. The forces, attractive or repulsive, become stronger when the charged objects are closer together. Careful measurements have shown that the strength of the force varies with separation in exactly the same way as for gravitational force inversely as the square of the distance between the interacting objects. The strength also depends on the amount of extra charge possessed by each object, increasing in exact proportion to the net charge on each. Finally, electrical forces obey Newton s Third Law. These important features of the electrical interaction are summarized in the following statement (Fig. 4.7): Pairs of objects with similar charges repel each other and pairs with dissimilar charges attract 32

5 each other with forces, F, that obey Newton s Third Law and whose strength depends on the net charges, q and Q, on the objects and the distance, d, between them according to the relationship. F kqq. d 2 Q d q Imagine water flowing through a pipe loosely filled with gravel. Electric current in a metal wire is similar. The moving water represents the electrons; the stationary gravel represents the positive charges in the wire (together with the rest of the electrons, which are not free to move about). Notice that no part of the wire is charged, because there are always equal numbers of positive and negative charges in any part of the metal. Electric current flows in a circuit in which a battery plays the role that a pump plays in our water and pipe analogy (see Fig. 4.8). The circuit must be completed by closing the switch. Batteries produce a direct current of electrons that flows in only one direction through the circuit. The wall socket into which household appliances are connected is like the battery, except that it reverses the direction of current flow 60 times per second. Such a current flow is called alternating current. Q q Light Q q Figure 4.7. Every charged object is attracted or repelled by every other charged object through the electrical interaction. The two forces have the same strength in every case. (Only the excess charges are shown in these diagrams.) Switch + Conducting wire The constant, k, that appears in the strength equation is called the electrical force constant. As with the gravitational constant, it must be measured experimentally. The electrical force constant is large; its numerical value is about in the metric system. This means that the electrical force is easy to demonstrate, whereas the gravitational force between ordinary objects can be observed only in sensitive and careful experiments. In fact, the experiments described earlier involve the transfer of only a small fraction (about 1 out of every ) of the electrons actually present. If separating all the electrons from the protons in a single copper penny were possible, and the electrons and protons were placed 100 meters (about the length of a football field) from each other, the collection of particles would attract each other with a force of about tons. The electrical force can be strong indeed. Electric Currents Some of the electrons are free to move on the surface or through the interior of some materials which are known as conductors. Insulators are materials that do not permit this free interior motion of electrons. Semiconductors contain a few free electrons, but not as many as conductors. Moving charged particles form an electric current. Battery Figure 4.8. Electric current flows in a complete circuit. The arrows show the direction of motion of the electrons. What are the purposes of the switch and the battery? Electromagnetic Forces The electrical interaction described to this point is accurate for charges that are at rest. The total interaction between charged particles depends on the motion of the particles, as well as the factors already discussed. The changes that occur when charges are moving result in magnetic forces, some of which you have undoubtedly encountered. They are usually not important if the electrical interaction is operating, and they result only in small motion-dependent corrections. They can become important, however, when charges are moving inside electrically neutral objects, such as when current flows through a wire. The complete interaction due to electric charge is 33

6 known as the electromagnetic interaction. It includes the electrical interaction between charged particles, either moving or at rest, as well as the magnetic interactions between moving charged particles. Summary Four fundamental interactions gravitational, electromagnetic, weak, and strong (or nuclear) cause all the forces we know about. The gravitational interaction, together with the laws of motion, explains the motion of falling objects and is the source of the force called weight. The electromagnetic interaction is associated with all the other forces governing the motion of objects larger than atomic nuclei. The five important laws are the three laws of motion and the two macroscopic force laws. Together they make a tidy package that describes and predicts with amazing accuracy the motions of objects ranging in size from atoms to clusters of galaxies. All five are needed before we are ready to apply any of them. The gravitational acceleration of an object is the acceleration it would experience if the gravitational force were the only force acting on it. Gravitational acceleration depends on the location of an object, but not on its mass. That is, all objects have the same gravitational acceleration at a given point in space. The weight of an object is the gravitational force acting on it. It depends on the object s location. The weight of an object near the surface of the moon would be about 1/6 its weight near the earth. Weight on Jupiter is 2.7 times earth weight and weight near the sun s surface is 28 times earth weight. It is possible to measure the very small gravitational attraction between ordinary-sized objects by using a Cavendish balance. Such measurements provide direct experimental evidence supporting the Universal Law of Gravitation. The electric force can be a very strong force, even between ordinary objects. If it were possible to separate all the negative and positive charges in a penny from each other, for example, they would attract each other with a force of more than one trillion tons at a distance of 100 meters. Several common demonstrations illustrate the Electric Force Law and the motion of electric charges: walking across a rug, then touching a metal doorknob; lightning; the attraction and repulsion of rubber and glass rods rubbed with fur and silk; the operation of an electroscope when touched by a charged object. The electrical force is the glue that holds the particles of matter together. It is responsible for all the contact forces we ordinarily experience. Examples are friction; atmospheric pressure; the strength of bridges and buildings; the impact forces that occur, for example, when billiard balls or automobiles collide; and the forces exerted by gasoline and steam engines. Now that we have studied the fundamental motion and force laws one at a time, we are ready to consider some real applications. STUDY GUIDE Chapter 4: The Fundamental Interactions A. FUNDAMENTAL PRINCIPLES 1. The Universal Law of Gravitation: Every object in the universe attracts every other object by a longrange gravitational interaction that obeys Newton s Third Law. The strength of the attractive force, F, varies with the masses, M and m, of the two objects and the distance, d, between their centers according to the relationship F GmM 2. The Electric Force Law: Pairs of objects with similar charges repel each other and pairs with dissimilar charges attract each other with forces that obey Newton s Third Law and whose strength depends on the net charges, q and Q, on the objects and the distance, d, between them according to the relationship F kqq B. MODELS, IDEAS, QUESTIONS, OR APPLICATIONS 1. The Newtonian Model (sometimes, the Newtonian Synthesis): The model based on Newton s three laws of motion and the Universal Law of Gravitation which explains the motions of the heavens as well as the terrestrial motions of common experience. The Newtonian Model when applied to the motions of the planets replaces the medieval model which placed the earth at the center of the solar system and the universe. 2. Electrical Model of Matter: All matter contains two kinds of electrically charged particles: positive protons and negative electrons. Electrons have little mass and can be quite mobile and transferable from one object to another. Protons are held rigidly in place in solid materials. Objects that have equal numbers of protons and electrons are electrically neutral. Objects with more electrons than protons are negatively charged. Those with fewer electrons than protons are positively charged. The amount of extra charge of either kind is called the charge of the object. 34

7 3. Why do both heavy and light things accelerate at the same rate when only the gravitational force is acting on them? 4. How can the acceleration of the moon and the acceleration of a falling apple be accounted for by the same Universal Law of Gravitation? 5. What determines the strength of all gravitational forces? 6. What is the Electrical Model of Matter? 7. What determines the strength of electrical forces? 8. What interactions are responsible for all of the forces we observe in ordinary life experiences? C GLOSSARY 1. Circuit: A connected, continuous path along which electrical charge flows to produce an electrical current. 2. Conductor (specifically, of electricity): A substance which readily allows an electric current to flow through it. The opposite of an insulator (nonconductor). Copper wire is a conductor. 3. Continuous: Varying smoothly without distinct parts or discontinuous elements. Used here to mean the opposite of discrete. 4. Coulomb: The unit of charge used in calculations, equivalent to the charge of about protons. 5. Discrete: Separate or individually distinct, consisting of distinct parts or discontinuous elements. Used here to mean the opposite of continuous or smoothly varying. The electric charge of an electron is described as discrete since it cannot be smoothly subdivided into smaller parts. 6. Electrical Force Constant: The electrical force constant is usually represented by the symbol k. It is a constant of proportionality in the Electric Force Law which connects the strength of the electrical force to its dependence on the charges of objects and their separations. F kqq 7. Electric Current: A coherent motion of electrical charges constitutes an electrical current. The motion of electrons along or through a copper wire is an example of an electrical current. If the flow is only in one direction, the current is said to be direct. If the current periodically reverses its direction of flow, the current is said to be alternating. 8. Free-fall Acceleration, g: The acceleration of a falling object on which the only significant force is the gravitational force. Near the surface of the earth, the free-fall acceleration is about 35 kilometers per hour per second. 9. Gravitational Constant: The gravitational constant is usually represented by the symbol G. It is a constant of proportionality in Newton s Universal Law of Gravitation which connects the strength of the gravitational force to its dependence on the masses of the objects and their separations. F GmM 10. Insulator (specifically, of electricity): A substance which does not readily allow an electric current to flow through it. The opposite of a conductor. Glass is an insulator. 11. Semiconductors: Materials whose electrical conducting properties place them somewhere midway between conductors and insulators. Silicon is a semiconductor. 12. Weight: The gravitational force of attraction of a very massive object, usually a planet or moon, for a less massive object on or near its surface. D. FOCUS QUESTIONS 1. In each of the following situations: a. Describe what would be observed. b. Name and state in your own words the fundamental principle(s) that could explain what would happen. c. Explain what would happen in terms of the fundamental principle(s). (1) A penny and a feather are caused to fall toward the earth in a vacuum tube. They start to fall at the same time. (2) Suppose an elephant and a feather were to fall from a high cliff at exactly the same time. If air friction could be ignored, what would happen? (3) A rubber rod is rubbed with fur and placed on a wire rack suspended by a string. A second rubber rod that has been rubbed with fur is brought nearby. The second rod is then taken away and a glass rod that has been rubbed with a vinyl sheet is brought nearby. (Note: the rubber rod acquires extra electrons. The glass rod loses electrons.) E. EXERCISES 4.1. The earth pulls on you with a gravitational force of attraction, your weight. Describe the reaction to this force. Show that your answer is consistent with the Third Law of Motion If you are pulling on the earth with a gravitational force, why doesn t the earth move in the same way you do in response to that force? Show that your answer is consistent with the Second and Third Laws of Motion. 35

8 4.3. Why does an object weigh less near the surface of the moon than near the surface of the earth? 4.4. The sun has much more mass than the earth (about 330,000 times as much). Why aren t we pulled toward the sun with 330,000 times as much force as we are toward the earth? 4.5. Compare the weights of an object in three locations: (a) near the surface of the earth, (b) near the surface of the moon, and (c) in a place outside the solar system where there are almost no gravitational forces How does the mass of the object in the previous exercise change as it is taken to the same three locations? 4.7. Compare the definitions of weight and mass. Can you see why the weight of an object can change from place to place while its mass does not? Explain how this can be so A cannonball originally at rest and a marble originally at rest are dropped in a vacuum from the same height at the same time. (a) What happens when they are dropped? Compare the speed and acceleration of the cannonball with that of the marble. (b) Is the gravitational force of attraction larger on the cannonball than it is on the marble? Justify your answer using a fundamental law. (c) Does the cannonball require a larger force to provide the same acceleration as the marble? Justify your answer using the Second Law of Motion. (d) Show that your answers to (a), (b), and (c) are consistent with each other State the Universal Law of Gravitation and explain its meaning in your own words A small ball is dropped from the edge of a cliff. One-tenth of a second later a much heavier ball is dropped from the same position. Ignoring the effects of air friction, can the second ball overtake the first? Justify your answer using fundamental laws or principles Describe an experiment that demonstrates that there are two kinds of electric charge Describe the important properties of a proton Describe the important properties of an electron Describe the Electrical Model of Matter What is meant when we say that electric charge is discrete? Describe how the electrons and protons in an atom could be held together by the electrical force Explain why you experience a repulsive force when you slap a table with your hand Precisely what is electric current? A rubber rod rubbed with fur and then brought near a second, similarly prepared, rubber rod can illustrate the Electric Force Law. (a) Describe what happens when the two rods are brought near each other. (b) Explain how the observed results illustrate the Electric Force Law. (c) What happened to the rubber rods when they were rubbed with fur? (d) What would happen if the rubber rods were brought near a glass rod which had been rubbed with silk? (e) What additional feature of the Electric Force Law is illustrated by this second experiment? (f) What happened to the glass rod when it was rubbed with silk? State the Electric Force Law and explain its meaning How do we know that the Electrical Model of Matter and the Electric Force Law are valid descriptions of nature? When a glass rod is rubbed with rubber, it becomes positively charged. This is because (a) protons are transferred from rubber to glass. (b) protons are transferred from glass to rubber. (c) electrons are transferred from glass to rubber. (d) electrons are transferred from rubber to glass. (e) electrons and protons annihilate each other What is meant when we say that an object is charged? Describe what happens when a glass rod becomes positively charged by being rubbed with silk. 36