A Special Note to the Student

Transcription

1 Preface This book is an outgrowth of forty years of teaching introductory physics courses. The original manuscript has been used in the classroom for several of them. Because the book is written for students, it contains a great many of the intermediate steps that are often left out of the derivations and illustrative problem solutions in many traditional textbooks. Students new to physics often find it difficult to follow derivations when the intermediate steps are left out. In addition, the units of measurement are carried along, step by step, in the equations to make it easier for students to understand. This text gives a good, fairly rigorous, traditional physics coverage for students of science and engineering. There is more material here than most instructors can cover in a two or three semester course. Therefore instructors are expected to choose those topics they deem most important for their particular course. Students can read on their own the detailed descriptions found in those chapters, or parts of chapters, omitted from the course. Unfortunately, many interesting and important topics in modern physics are never covered in physics courses because of lack of time. These chapters, especially, are written in even more detail to enable students to read them on their own. The organization of the text follows the traditional sequence of mechanics, wave motion, heat, electricity and magnetism, optics, and modern physics. The emphasis throughout the book is on clarity. The book starts out at a very simple level, and advances as students understanding and ability with the calculus grows. There are a large number of diagrams and illustrative problems in the text to help students visualize physical ideas. Important equations are highlighted to help students find and recognize them. A summary of these important equations is given at the end of each chapter. Many students, taking physics for the first time, sometimes find the mathematics frightening. In order to help these students, I have computerized every illustrative example in the textbook using computer spreadsheets. These computerized Interactive Examples will allow the student to solve the example problem in the textbook, with all the in-between steps, many times over but with different numbers placed in the problem. The Interactive Examples are available on a computer disk for both the instructor and the student. More details on these Interactive Examples can be found in the section Computer Aided Instruction at the end of the Preface. These Interactive Examples make the book the first truly interactive physics textbook for students of science and engineering. Students sometimes have difficulty remembering the meanings of all the vocabulary associated with new physical ideas. Therefore, a section called The Language of Physics, found at the end of each chapter, contains the most important ideas and definitions discussed in that chapter. To comprehend the physical ideas expressed in the theory class, students need to be able to solve physics problems for themselves. Problem sets at the end of each chapter are grouped according to the section where the topic is covered. xvii

2 Problems that are a mix of different sections are found in the Additional Problems section. If you have difficulty with a problem, refer to that section of the chapter for help. The problems begin with simple, plug-in problems to develop students confidence and to give them a feel for the numerical magnitudes of some physical quantities. The problems then become progressively more difficult and end with some that are very challenging. The more difficult problems are indicated by a star (*). The starred problems are either conceptually more difficult or very long. Many problems at the end of the chapter are very similar to the illustrative problems worked out in the text. When solving these problems, students can use the illustrative problems as a guide, and use the Interactive Examples as a check on their work. To facilitate setting up a problem, the Hints for Problem Solving section, which is found before the problem set in chapter 2, should be studied carefully. A section called Interactive Tutorials, which also uses computer spreadsheets to solve physics problems, can be found at the end of the problems section in each chapter. These Interactive Tutorials are a series of physics problems, very much like the interactive examples, but are more detailed and more general. They are also available on a computer disk for both the instructor and the student. More details on these Interactive Tutorials can be found in the section Computer Aided Instruction at the end of the Preface. A series of questions relating to the topics discussed in the chapter is also included at the end of each chapter. Students should try to answer these questions to see if they fully understand the ramifications of the theory discussed in the chapter. Just as with the problem sets, some of these questions are either conceptually more difficult or will entail some outside reading. These more difficult questions are also indicated by a star (*). A word about units. Some forty years ago when I started teaching physics I said to my class, We use the British Engineering System of Units in our daily lives in the United States, but we are in the process of changing over to the metric system of units (Now called the International System of Units or SI units). In a few years the British Engineering System of Units will be obsolete and the entire world will use the metric system. Over forty years later I find myself giving that same lecture at the beginning of the course. Something is very wrong! Except for the United States and one or two small islands in the south pacific, the entire world uses the International System of Units. Even the British do not use the British Engineering System of Units. (I have even asked many of my colleagues individually how far away from the campus do they live. Each gave an answer in the British Engineering System of Units. If all the physics faculty still think in terms of the British Engineering System, what can we expect of our students and the rest of our society. Try asking your colleagues how far away from the campus do they live and see what they reply.) It is time for the United States to get in step with the rest of the world. To do that we must teach a new generation of physics students SI units almost exclusively so that they will think and work in SI units. Therefore in this book only SI units will be used in the description of physics. However, to help the student in this transition, the first chapter will show how to xviii

3 convert from the British Engineering System to SI units and vice versa if it is ever necessary. If a student is told that a car is traveling at 55 mph, the student will convert that to 88.6 km/hr and then solve the entire problem in SI units. An interactive computer tutorial will be found at the end of chapter 1 that will help the student to make any of these conversions. After that, the rest of the book will be in SI units almost exclusively. Occasionally, a few problems throughout the book will still have some numbers in the British Engineering System of Units. When this occurs the student should convert these numbers into SI units, and proceed in solving the problem in the International System of Units. Scattered throughout the text, at the ends of chapters, are sections entitled Have you ever wondered? These are a series of essays on the application of physics to areas such as meteorology, astronomy, aviation, space travel, the health sciences, the environment, philosophy, traffic congestion, sports, and the like. Many students are unaware that physics has such far-reaching applications. These sections are intended to engage students varied interests but can be omitted, if desired, without loss of continuity in the physics course. The Have you ever wondered? section on the application of physics to traffic congestion is the only major item still expressed in British Engineering Systems of Units in this book. The relation between theory and experiment is carried throughout the book, emphasizing that our models of nature are good only if they can be verified by experiment. Concepts presented in the lecture and text can be well demonstrated in a laboratory setting. To this end, Experiments In Physics, second edition by Peter J. Nolan and Raymond E. Bigliani is available from Whittier Publishing Co. Experiments in Physics contains some 55 experiments covering all the traditional topics in a physics laboratory. The second edition of Experiments in Physics also contains a complete computer analysis of the data for every experiment. A Bibliography, given at the end of the book, lists some of the large number of books that are accessible to students taking physics. These books cover such topics in modern physics as relativity, quantum mechanics, and elementary particles. Although many of these books are of a popular nature, they do require some physics background. After finishing this book, students should be able to read any of them for pleasure without difficulty. Finally, we should note that we are living in a rapidly changing world. Many of the changes in our world are sparked by advances in physics, engineering, and the high-technology industries. Since engineering and technology are the application of physics to the solution of practical problems, it behooves every individual to get as much background in physics as possible. You can depend on the fact that there will be change in our society. You can be either the architect of that change or its victim, but there will be change. xix

4 A Special Note to the Student One thing I have learned in a long life: that all our science measured against reality, is primitive and childlike--and yet it is the most precious thing we have. Albert Einstein as quoted by Banesh Hoffmann in Albert Einstein, Creator and Rebel The language of physics is mathematics, so it is necessary to use mathematics in our study of nature. However, just as sometimes you cannot see the forest for the trees, you must be careful or you will not see the physics for the mathematics. Remember, mathematics is only a tool used to help describe the physical world. You must be careful to avoid getting lost in the mathematics and thereby losing sight of the physics. When solving problems, a sketch or diagram that represents the physics of the problem should be drawn first, then the mathematics should be added. Physics is such a logical subject that when a student sees an illustrative problem worked out, either in the textbook or on the blackboard, it usually seems very simple. Unfortunately, for most students, it is simple only until they sit down and try to do a problem on their own. Then they often find themselves confused and frustrated because they do not know how to get started. If this happens to you, do not feel discouraged. It is a normal phenomenon that happens to many students. The usual approach to overcoming this difficulty is going back to the illustrative problem in the text. When you do so, however, do not look at the solution of the problem first. Read the problem carefully, and then try to solve the problem on your own. At any point in the solution, when you cannot proceed to the next step on your own, peek at that step and only that step in the illustrative problem. The illustrative problem shows you what to do at that step. Then continue to solve the problem on your own. Every time you get stuck, look again at the appropriate solution step in the illustrative problem until you can finish the entire problem. The reason you had difficulty at a particular place in the problem is usually that you did not understand the physics at that point as well as you thought you did. It will help to reread the appropriate theory section. Getting stuck on a problem is not a bad thing, because each time you do, you have the opportunity to learn something. Getting stuck is the first step on the road to knowledge. I hope you will feel comforted to know that most of the students who have gone before you also had these difficulties. You are not alone. Just keep trying. Eventually, you will find that solving physics problems is not as difficult as you first thought; in fact, with time, you will find that they can even be fun to solve. The more problems that you solve, the easier they become, and the greater will be your enjoyment of the course. xx

5 Computer Assisted Instruction Interactive Examples Many students, taking physics for the first time, sometimes find the mathematics frightening. In order to help these students, I have computerized every illustrative example in the textbook. These computerized Interactive Examples will allow the student to solve the example problem in the textbook, with all the in-between steps, many times over but with different numbers placed in the problem. Figure 1 shows an example from Chapter 2 of the textbook for solving a problem in Kinematics. It is a problem in kinematics in which a car, initially traveling at 30.0 km/hr, accelerates at the constant rate of 1.50 m/s 2. The student is then asked how far will the car travel in 15.0 s? The example in the textbook shows all the steps and reasoning done in the solution of the problem. Example 2.6 Using the kinematic equation for the displacement as a function of time. A car, initially traveling at 30.0 km/hr, accelerates at the constant rate of 1.50 m/s 2. How far will the car travel in 15.0 s? Solution To express the result in the proper units, km/hr is converted to m/s as v 0 km 1 hr 1000 m m/s hr 3600 s 1 km The displacement of the car, found from equation 2.14, is x = v0t + 1 at 2 2 m 1 m s s 2 s 2 s = 125 m m = 294 m 2 The first term in the answer, 125 m, represents the distance that the car would travel if there were no acceleration and the car continued to move at the velocity 8.33 m/s for 15.0 s. But there is an acceleration, and the second term shows how much farther the car moves because of that acceleration, namely 169 m. The xxi

6 total displacement of 294 m is the total distance that the car travels because of the two effects. To go to this interactive example click on this sentence. Figure 1 Example 2.6 in the textbook. The last sentence in blue type in the example (To go to this interactive example click on this sentence.) allows the student to access the interactive example for this same problem. Clicking on the blue sentence, the spreadsheet shown in figure 2 opens. Notice that the problem is stated in Example 3.6 Using the kinematic equation for the displacement as a function of time. A car, initially traveling at 30.0 km/hr, accelerates at the constant rate of 1.50 m/s 2. How far will the car travel in 15.0 s? Initial Conditions Initial velocity v o = 30 km/hr acceleration a = 1.5 m/s 2 time t = 15 s Solution. To express the result in the proper units, km/hr is converted to m/s as v o = ( 30 km/hr) x [ 1 hr / ( 3600 s) ] x ( 1000 m/km) v o = ( m/s The displacement of the car, found from equation 3.14, is x = v o t + 1/2 a t 2 x = [( m/s) x ( 15 s)] + 1/2 [( 1.5 m/s 2 ) x ( 15 s) 2 ] x = [( 125 m)] + [ m)] x = m The first term in the answer, 125 m, represents the distance that the car would travel if there were no acceleration and the car continued to move at the velocity m/s for 15 s. But there is an acceleration, and the second term shows how much farther the car moves because of that acceleration, namely m. The total displacement of m is the total distance that the car travels because of the two effects. Figure 2 Interactive Example 2.6 in Microsoft Excel Spreadsheet. xxii

7 the identical manner as in the textbook. Directly below the stated problem is a group of yellow-colored cells labeled Initial Conditions. Into these yellow cells are placed the numerical values associated with the particular problem. For this problem the initial conditions consist of the initial velocity v0 of the car, the acceleration a of the car, and the time t that the car is moving, as shown in figure 2. The problem is now solved in the identical way it is solved in the textbook. Words are used to describe the physical principles and then the equations are written down. Then the in-between steps of the calculation are shown in light green-colored cells, and the final result of the calculation is shown in a light blue-colored cell. The entire problem is solved in this manner, as shown in figure 2. If the student wishes to change the problem by using a different initial velocity or a different time or acceleration, he or she then changes these values in the yellow-colored cells of the initial conditions. When the initial conditions are changed the computer spreadsheet recalculates all the new in-between steps in the problem and all the new final answers to the problem. In this way the problem is completely interactive. It changes for every new set of initial conditions. The Interactive Examples make the book a living book. The examples can be changed many times over to solve for all kinds of special cases. When the student is finished with the interactive example, and is accessing it from a CD, he or she just clicks on the X in the extreme upper right-hand corner of the spreadsheet screen, returning him or her to the original example in the textbook chapter. If the student is accessing the interactive example from a web page, then he or she presses the go Back button on the top of the browser page. When Excel closes, you will be returned to the first page of the present chapter. You can then go to wherever page you want in that chapter by sliding the Scroll Bar box on the right-hand side of the screen. These Interactive Examples are a very helpful tool to aid in the learning of physics if they are used properly. The student should try to solve the particular problem in the traditional way using paper and an electronic calculator. Then the student should open the spreadsheet, insert the appropriate data into the Initial Conditions cells and see how the computer solves the problem. Go through each step on the computer and compare it to the steps you made on paper. Does your answer agree? If not, check through all the in-between steps on the computer and your paper and find where your made a mistake. Do not feel bad if you make a mistake. There is nothing wrong in making a mistake, what is wrong is not learning from your mistake. Now that you understand your mistake, repeat the problem using different Initial Conditions on the computer and your paper. Again check your answers and all the in-between steps. Once you are sure that you know how to solve the problem, try some special cases. What would happen if you changed an angle?, a weight?, a force? etc. In this way you can get a great deal of insight into the physics of the problem and also learn a great deal of physics in the process. You must be very careful not to just plug numbers into the Initial Conditions and look at the answers without understanding the in-between steps and the actual xxiii

8 physics of the problem. You will only be deceiving yourself. Be careful, these spreadsheets can be extremely helpful if they are used properly. We should point out two differences in a text example and in a spreadsheet example. Powers of ten that are used in scientific notation in the text are written with the capital letter E in the spreadsheet. Hence, the number 5280, written in scientific notation as , will be written on the spreadsheet as 5.280E+3. Also, the square root symbol,, in the textbook is written as sqrt[ ] in a spreadsheet. Finally, we should note that the spreadsheets are protected by allowing you to enter data only in the designated light yellow-colored cells of the Initial Conditions area. Therefore, the student cannot damage the spreadsheets in any way, and they can be used over and over again. To access these Interactive Examples the student must have Microsoft s Excel computer spreadsheet installed on his computer. Computer Assisted Instruction Interactive Tutorials Besides the Interactive Examples in this text, I have also introduced a section called Interactive Tutorials at the end of the problem section in each chapter. These Interactive Tutorials are a series of physics problems, very much like the interactive examples, but are more detailed and more general. The Interactive Tutorials are available on the Internet, but the student must have Microsoft s Excel computer spreadsheet on his or her computer. To access the Interactive Tutorial on the Internet, the student will click on the sentence in blue type at the end of the Interactive Tutorials section (To go to this interactive tutorial click on this sentence.) Clicking on the blue sentence, opens the appropriate spreadsheet. Figure 3 show a typical Interactive Tutorial for a problem in chapter 4 on Kinematics In Two Dimensions. When the student opens this particular spreadsheet, he or she sees the problem stated in the usual manor. That is, this problem is an example of a projectile fired at a initial velocity vo = 53.0 m/s at an angle = , and it is desired to find the maximum height of the projectile, the total time the projectile is in the air, and the range of the projectile. Directly below the stated problem is a group of yellow-colored cells labeled Initial Conditions. Into these yellow cells are placed the numerical values associated with the particular problem. For this problem the initial conditions consist of the initial velocity vo, the initial angle, and the acceleration due to gravity g as shown in figure 3. The problem is now solved in the traditional way of a worked out example in the book. Words are used to describe the physical principles and then the equations are written down. Then the in-between steps of the calculation are shown in light green-colored cells, and the final result of the calculation is shown in a light blue-green-colored cell. The entire problem is solved in this manor as shown in figure 3. If the student wishes to change the problem by using a different initial xxiv

9 Chapter 3 Kinematics Computer Assisted Instruction Interactive Tutorial 72. A golf ball is hit with an initial velocity v o = 53.0 m/s at an angle = above the horizontal. (a) How high will the ball go? (b) What is the total time the ball is in the air? (c) How far will the ball travel horizontally before it hits the ground? Initial Conditions The magnitude of the initial velocity v o = 53 m/s The angle = 50 degrees The acceleration of gravity g = 9.8 m/s 2 The x-component of the initial velocity is found as v ox = v o cos v ox = ( 53 m/s) x cos( 50 ) v ox = m/s The y-component of the initial velocity is found as v oy = v o sin v oy = ( 53 m/s) x sin( 50 ) v oy = 40.6 m/s (a) The maximum height y max of the golf ball above the launch point is found from the kinematic equation v 2 y = v 2 oy - 2 g y When y = y max, v y = 0. Therefore 0 = v 2 oy - 2 g y max Solving for the maximum height y max above the launch point, we get y max = (v 2 oy ) / 2 g y max = ( 40.6 m/s) 2 / 2 ( 9.8 m/s 2 ) y max = 84.1 m (b) The total time t t the ball is in the air is found from the kinematic equation y = v oy t - (1/2) g t 2 When t is equal to the total time t t, the projectile is on the ground and y = 0, therefore 0 = v oy t t - (1/2) g t t 2 or dividing each term by t t we get 0 = v oy - (1/2) g t t upon solving for the total time t t we get t t = 2 v oy / g t t = 2 x ( 40.6 m/s) / ( 9.8 m/s 2 ) t t = 8.29 s (c) The maximum distance the ball travels in the x-direction before it hits the ground, is found from the kinematic equation for the displacement of the ball in the x-direction x = v ox t The maximum distance x max occurs for t = t t. Therefore x max = v ox t t x max = ( m/s) x ( 8.29 s) x max = m Figure 3. A typical Interactive Tutorial. velocity or a different launch angle, he or she then changes these values in the yellowed-colored cells of the initial conditions. When the initial conditions are xxv

10 changed the computer spreadsheet recalculates all the new in-between steps in the problem and all the new final answers to the problem. In this way the problem is completely interactive. It changes for every new set of initial conditions. The tutorials can be changed many times over to solve for all kinds of special cases. Another example of an Interactive Tutorial is shown in figure 4. It is an interactive tutorial for a problem on an RLC series AC circuit from chapter 28. The initial conditions in the yellow-colored cells set the values of the resistance R, the inductance L, the capacitance C, the frequency f, and the applied voltage V. The spreadsheet then completely solves the problem for these initial values. The student can follow through all the in-between steps of the problem in the light green-colored cells of the spreadsheet, and all the answers in the light blue-green-colored cells. The spreadsheet then draws a graph showing the phase relations for the voltage across the resistor, VR, the voltage across the inductance VL, the voltage across the capacitance, VC, and the resultant voltage V in the circuit depending upon the initial values in the yellow cells. As special cases the student can then let L = 0 in the yellow cell area, and observe the characteristics of an RC series circuit; then replace a value of L in the initial conditions and then let C = 0 and observe the characteristics of an RL series circuit. Finally inserting values for L and C and letting R = 0 the student can observe the characteristics of an LC series circuit. In all the cases the spreadsheet calculates all the quantities with all the in-between steps, and shows the phase relations of the voltages. Chapter 24 Alternating Current Circuits Computer Assisted Instruction Interactive Tutorial 56. RLC series circuit. An RLC series circuit has a resistance R = 800, Inductance L = 2.00 H, capacitance C = 5.55 x 10-6 F, and they are connected to an applied voltage V= 110 V, operating at a frequency f = 60 Hz. Find (a) the inductive reactance X L, (b) the capacitive reactance, X C, (c) the impedance Z of the circuit, (d) the current i in the circuit, (e) the voltage drop V R across the resistor, (f) the voltage drop V L across the inductor, (g) the voltage drop V C across the capacitor, (h) the total voltage drop across RLC, (i) the phase angle, and (j) the resonant frequency f o. (k) Plot the curves of the voltage across R, L, and C individually and the voltage across the combination. Show the phase relations for all these voltages. Initial Conditions R = 800 f = 60 Hz L = 2.00E+00 H V = 110 V C = 5.55E-06 F (a) The inductive reactance X L is found from equation as X L = 2 f L X L = ( 2 ) x ( 3.14 ) x ( 60 Hz) x ( 2 X L = 7.54E+02 (b) The capacitive reactance X C is found from equation as X C = 1 / [2 f C] X C = ( 1 ) / ( 6.28 ) x ( 60 Hz) x ( 5.55E-06 X C = 4.78E+02 (c) The impedance Z of the circuit is found from equation as Z = Sqrt[R 2 + (X L - X C ) 2 ] Z = Sqrt[( 800 ) 2 + (( ) - ( )) 2 Z = 8.46E+02 xxvi

11 (d) The effective current in the circuit is found from Ohm s law, equation 24.19, as i = V / Z i = ( 110 V) / ( 8.46E+02 ) i = 1.30E-01 A (e) The voltage drop V R across the resistor is found from equation 24.13, as V R = i R V R = ( 1.30E-01 A) x ( 8.00E+02 ) V R = 1.04E+02 V (f) The voltage drop V L across the inductor is found from equation 24.14, as V L = i X L V L = ( 1.30E-01 A) x ( 7.54E+02 ) V L = 9.80E+01 V (g) The voltage drop V C across the capacitor is found from equation 24.16, as V C = i X C V C = ( 1.30E-01 A) x ( 4.78E+02 ) V C = 6.21E+01 V (h) The total voltage across R, L, C in series is found from equation as V = Sqrt[V 2 R + (V L - V C ) 2 ] V = Sqrt[( ) 2 + (( 98 ) - ( )) 2 V = 1.10E+02 V which is equal to the applied voltage as expected. (i) The phase angle is found from equation as = tan -1 [(V L - V C ) / V R ] = tan -1 [(( 98 V) - ( V)) / ( V) ] = degrees (j) The resonant frequency f o is found from equation as f o = 1 / [2 sqrt(l C)] f o = ( 1 ) / ( 6.28 ) x sqrt( 2 H) x ( 5.55E-06 f o = 4.78E+01 Hz The angular frequency of the alternating current is related to the frequency of the current by = 2 f = ( 2 ) x ( 3.14 ) x ( 160 Hz) = 3.77E+02 rad/s The maximum value of the current, i max, is found from rearranging equation 24.5 as i max = i eff / i max = ( 0.13 A) / ( 0.71 ) i max = 0.18 A The equation for the alternating current is given by i = i max sin[ t] i = ( 0.18 A) x sin[( 3.77E+02 rad/s) t] The data to plot the current as a function of time is calculated from this equation and the results are shown in the graph below. The data to plot the voltage drop across the resistor as a function of time is calculated from the equation V R = i R V R = i max R sin[ t] and the results are shown in the graph below. The data to plot the voltage drop across the inductor as a function of time is calculated from the equation V L = i X L V L = i max X L sin[ t + / 2] The term + / 2 in the sine term is to show that V L leads V R by The results are shown in the graph below. The data to plot the voltage drop across the capacitor as a function of time is calculated from the equation V C = i X C V C = i max X C sin[ t - / 2] xxvii

12 Potential volts Preface The term - / 2 in the sine term is to show that V C lags V R by The results are shown in the graph below. V is the sum of the voltages across R, L, and C in series and is given by V = V R + V L + V C and is shown in the graph below. Phase relations in an RLC series circuit V VR VL VC degrees Angle Since V R is in phase with the current i in the circuit, we can compare the phase relationship between the voltage V and the current i, by comparing the voltage V and the voltage V R. The phase angle can best be determined from this diagram by observing where the V curve intersects the horizontal axis, and where the V R curve intersects the horizontal axis. The difference between these two values represents the phase angle which was already calculated in part (i). Special Cases 1) An RC series circuit. Let L = 0. 2) An RL series circuit. Let C = 0. 3) An LC series circuit. Let R = 0. Figure 4. An Interactive Tutorial for Chapter 28 Alternating Current Circuits These Interactive Tutorials are a very helpful tool to aid in the learning of physics if they are used properly. The student should try to solve the particular problem in the traditional way using paper and an electronic calculator. Then the student should open the spreadsheet, insert the appropriate data into the Initial Conditions cells and see how the computer solves the problem. Go through each step on the computer and compare it to the steps you made on paper. Does your answer agree? If not, check through all the in-between steps on the computer and your paper and find where you made a mistake. Repeat the problem using different Initial Conditions on the computer and your paper. Again check your answers and all the in-between steps. Once you are sure that you know how to solve the problem, try some special cases. What would happen if you changed an angle?, a weight?, a xxviii

13 force? etc. In this way you can get a great deal of insight into the physics of the problem and also learn a great deal of physics in the process. You must be very careful not to just plug numbers into the Initial Conditions and look at the answers without understanding the in-between steps and the actual physics of the problem. You will only be deceiving yourself. Be careful, these spreadsheets can be extremely helpful if they are used properly. When the student is finished with the interactive tutorial, and is accessing it from a CD, he or she just clicks on the X in the extreme upper right-hand corner of the spreadsheet screen, returning him or her to the original tutorials in the textbook chapter. If the student is accessing the interactive tutorial from a web page, then he or she presses the go Back button on the top of the browser page. When Excel closes, you will be returned to the first page of the present chapter. You can then go to wherever page you want in that chapter by sliding the Scroll Bar box on the right-hand side of the screen. Click on this sentence to go to the Brief Table of Contents which will allow you to go to any chapter in this book. xxix