In this circuit, V = 10 volts, R = 3,000 ohms, and C = 50 x 10-6 farads. The circuit has a time constant t, which depends on the resistance, R, and the capacitance, C, as t = R x C = 0.15 second. 1. Use a for loop. 2. Use the math library function exp(x) to compute ex. You will need to include the system header file math.h. 3. On Unix you will need –lm in your command line to tell the Linker to search the math library. 4. Use macro definition for all the constants. 5. Format the output so the output looks like the following. The time and voltage should display two digits after the decimal point.

In this circuit, V = 10 volts, R = 3,000 ohms, and C = 50 x 10-6 farads. The circuit has a time constant t, which depends on the resistance, R, and the capacitance, C, as t = R x C = 0.15 second. 1. Use a for loop. 2. Use the math library function exp(x) to compute ex. You will need to include the system header file math.h. 3. On Unix you will need –lm in your command line to tell the Linker to search the math library. 4. Use macro definition for all the constants. 5. Format the output so the output looks like the following. The time and voltage should display two digits after the decimal point.

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AUCS 340: Ethics in the Professions Homework Assignment: International and US Health Care Systems The following homework assignment will help you to discover some of the differences between the administration of health care in the United States and internationally. This is a research based assignment; remember the use of Wikipedia.com is not an acceptable reference site for this course. You must include a references cited page for this assignment; correctly formatted APA or MLA references are acceptable (simply stating s web address is NOT a complete reference). The answers should be presented in paragraph formation. Staple all pages together for presentation. The first question refers to a country other than the United States of America 1) Socialized Medicine – provide a definition of the term socialized medicine and discuss a country that currently has a socialized medicine system to cover all citizens; this discussion should include the types of services offered to the citizens of this country. When was this system first implemented in this country? What is the name of this country’s health insurance plan? Compare the ranking for the life expectancy for this country to that of the United States. Which is higher? Why? Compare the cost of financing healthcare in this country to the United States in comparison to the amount of annual funding in dollars and the percentage of gross domestic product spent on health care for each country. What rank does this country have in comparison to the United States for overall health of its citizens? (This portion of the assignment should be approximately one page in length and graphic data is acceptable to support some answers, however, graphic information should only be used to explain your written explanation not as the answer to the question.) Bonus: Is this country’s system currently financially stable? Why or why not? The following questions refer to the delivery of healthcare in the United States of America, as it was organized prior to the implementation of the Affordable Care Act (ACA). The ACA is currently being phased into coverage. It is estimated that the answers to the following questions will result in an additional two to three pages of written text in addition to the page for question number one. 2) Medicare – when was it enacted? Who does it cover? Who was President when Medicare was originally passed? What do the specific portions Part A, Part B and Part D cover? When was Part D enacted? Who was President when Part D was enacted? Is the Medicare system currently financially stable? Why or why not. Compare the average life expectancy for males and females when Medicare was originally passed and the average life expectancy of males and females as of 2010; more recent data is acceptable. Bonus: What does Part C cover and when was it enacted? 3) Health Maintenance Organization (HMO) – Define the term health maintenance organization. When did this type of health insurance plan become popular in the United States? How does this type of system provide medical care to the people enrolled? This answer should discuss in network versus out of network coverage. 4) Medicaid- when was it enacted? Who does it cover? Who was President when this insurance plan was enacted? Are the coverage benefits the same state to state? Why or why not? Is the system currently financially stable? Why or why not. What effect does passage of the ACA project to have on enrollment in the Medicaid system? Why? 5) Organ Transplants – What is the mechanism for placement of a patient’s name on the organ transplant list? What is the current length of time a patient must wait for a heart transplant? Explain at least one reason why transplants are considered an ethical issue. How are transplants financed? Give at least one example of how much any type of organ transplant would cost. 6) Health Insurance/Information Portability and Accountability Act (HIPAA) – When was it enacted? Who was President when this legislation as passed? What is the scope of this legislative for the medical community and the general community? (Hint: There are actually two reasons for HIPAA legislation; make sure to state both in your response) 7) Death with Dignity Act – what year was the Oregon Death with Dignity Act passed? What ethical issue is covered by the Death with Dignity Act? List the factors that must be met for a patient to use the Death with Dignity Act. List two additional states that have enacted Death with Dignity Acts and when was the legislation passed in these states? 8) Hospice – what is hospice care? When was it developed? What country was most instrumental in the development of hospice care? Do health insurance plans in the United States cover hospice care? What types of services are covered for hospice care? Grading: 1) Accuracy and completeness of responses = 60% of grade 2) Correct use of sentence structure, spelling and grammar = 30% of grade 3) Appropriate use of references and citations = 10% of grade Simply stating a web page is not an appropriate reference This assignment is due on the date published in the course syllabus.

AUCS 340: Ethics in the Professions Homework Assignment: International and US Health Care Systems The following homework assignment will help you to discover some of the differences between the administration of health care in the United States and internationally. This is a research based assignment; remember the use of Wikipedia.com is not an acceptable reference site for this course. You must include a references cited page for this assignment; correctly formatted APA or MLA references are acceptable (simply stating s web address is NOT a complete reference). The answers should be presented in paragraph formation. Staple all pages together for presentation. The first question refers to a country other than the United States of America 1) Socialized Medicine – provide a definition of the term socialized medicine and discuss a country that currently has a socialized medicine system to cover all citizens; this discussion should include the types of services offered to the citizens of this country. When was this system first implemented in this country? What is the name of this country’s health insurance plan? Compare the ranking for the life expectancy for this country to that of the United States. Which is higher? Why? Compare the cost of financing healthcare in this country to the United States in comparison to the amount of annual funding in dollars and the percentage of gross domestic product spent on health care for each country. What rank does this country have in comparison to the United States for overall health of its citizens? (This portion of the assignment should be approximately one page in length and graphic data is acceptable to support some answers, however, graphic information should only be used to explain your written explanation not as the answer to the question.) Bonus: Is this country’s system currently financially stable? Why or why not? The following questions refer to the delivery of healthcare in the United States of America, as it was organized prior to the implementation of the Affordable Care Act (ACA). The ACA is currently being phased into coverage. It is estimated that the answers to the following questions will result in an additional two to three pages of written text in addition to the page for question number one. 2) Medicare – when was it enacted? Who does it cover? Who was President when Medicare was originally passed? What do the specific portions Part A, Part B and Part D cover? When was Part D enacted? Who was President when Part D was enacted? Is the Medicare system currently financially stable? Why or why not. Compare the average life expectancy for males and females when Medicare was originally passed and the average life expectancy of males and females as of 2010; more recent data is acceptable. Bonus: What does Part C cover and when was it enacted? 3) Health Maintenance Organization (HMO) – Define the term health maintenance organization. When did this type of health insurance plan become popular in the United States? How does this type of system provide medical care to the people enrolled? This answer should discuss in network versus out of network coverage. 4) Medicaid- when was it enacted? Who does it cover? Who was President when this insurance plan was enacted? Are the coverage benefits the same state to state? Why or why not? Is the system currently financially stable? Why or why not. What effect does passage of the ACA project to have on enrollment in the Medicaid system? Why? 5) Organ Transplants – What is the mechanism for placement of a patient’s name on the organ transplant list? What is the current length of time a patient must wait for a heart transplant? Explain at least one reason why transplants are considered an ethical issue. How are transplants financed? Give at least one example of how much any type of organ transplant would cost. 6) Health Insurance/Information Portability and Accountability Act (HIPAA) – When was it enacted? Who was President when this legislation as passed? What is the scope of this legislative for the medical community and the general community? (Hint: There are actually two reasons for HIPAA legislation; make sure to state both in your response) 7) Death with Dignity Act – what year was the Oregon Death with Dignity Act passed? What ethical issue is covered by the Death with Dignity Act? List the factors that must be met for a patient to use the Death with Dignity Act. List two additional states that have enacted Death with Dignity Acts and when was the legislation passed in these states? 8) Hospice – what is hospice care? When was it developed? What country was most instrumental in the development of hospice care? Do health insurance plans in the United States cover hospice care? What types of services are covered for hospice care? Grading: 1) Accuracy and completeness of responses = 60% of grade 2) Correct use of sentence structure, spelling and grammar = 30% of grade 3) Appropriate use of references and citations = 10% of grade Simply stating a web page is not an appropriate reference This assignment is due on the date published in the course syllabus.

Chapter 13 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, May 16, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Matter of Some Gravity Learning Goal: To understand Newton’s law of gravitation and the distinction between inertial and gravitational masses. In this problem, you will practice using Newton’s law of gravitation. According to that law, the magnitude of the gravitational force between two small particles of masses and , separated by a distance , is given by , where is the universal gravitational constant, whose numerical value (in SI units) is . This formula applies not only to small particles, but also to spherical objects. In fact, the gravitational force between two uniform spheres is the same as if we concentrated all the mass of each sphere at its center. Thus, by modeling the Earth and the Moon as uniform spheres, you can use the particle approximation when calculating the force of gravity between them. Be careful in using Newton’s law to choose the correct value for . To calculate the force of gravitational attraction between two uniform spheres, the distance in the equation for Newton’s law of gravitation is the distance between the centers of the spheres. For instance, if a small object such as an elephant is located on the surface of the Earth, the radius of the Earth would be used in the equation. Note that the force of gravity acting on an object located near the surface of a planet is often called weight. Also note that in situations involving satellites, you are often given the altitude of the satellite, that is, the distance from the satellite to the surface of the planet; this is not the distance to be used in the formula for the law of gravitation. There is a potentially confusing issue involving mass. Mass is defined as a measure of an object’s inertia, that is, its ability to resist acceleration. Newton’s second law demonstrates the relationship between mass, acceleration, and the net force acting on an object: . We can now refer to this measure of inertia more precisely as the inertial mass. On the other hand, the masses of the particles that appear in the expression for the law of gravity seem to have nothing to do with inertia: Rather, they serve as a measure of the strength of gravitational interactions. It would be reasonable to call such a property gravitational mass. Does this mean that every object has two different masses? Generally speaking, yes. However, the good news is that according to the latest, highly precise, measurements, the inertial and the gravitational mass of an object are, in fact, equal to each other; it is an established consensus among physicists that there is only one mass after all, which is a measure of both the object’s inertia and its ability to engage in gravitational interactions. Note that this consensus, like everything else in science, is open to possible amendments in the future. In this problem, you will answer several questions that require the use of Newton’s law of gravitation. Part A Two particles are separated by a certain distance. The force of gravitational interaction between them is . Now the separation between the particles is tripled. Find the new force of gravitational Fg m1 m2 r Fg = G m1m2 r2 G 6.67 × 10−11 N m2 kg2 r r rEarth F  = m net a F0 interaction . Express your answer in terms of . ANSWER: Part B A satellite revolves around a planet at an altitude equal to the radius of the planet. The force of gravitational interaction between the satellite and the planet is . Then the satellite moves to a different orbit, so that its altitude is tripled. Find the new force of gravitational interaction . Express your answer in terms of . You did not open hints for this part. ANSWER: Part C A satellite revolves around a planet at an altitude equal to the radius of the planet. The force of gravitational interaction between the satellite and the planet is . Then the satellite is brought back to the surface of the planet. Find the new force of gravitational interaction . Express your answer in terms of . ANSWER: F1 F0 F1 = F0 F2 F0 F2 = F0 F4 F0 Typesetting math: 81% Part D Two satellites revolve around the Earth. Satellite A has mass and has an orbit of radius . Satellite B has mass and an orbit of unknown radius . The forces of gravitational attraction between each satellite and the Earth is the same. Find . Express your answer in terms of . ANSWER: Part E An adult elephant has a mass of about 5.0 tons. An adult elephant shrew has a mass of about 50 grams. How far from the center of the Earth should an elephant be placed so that its weight equals that of the elephant shrew on the surface of the Earth? The radius of the Earth is 6400 . ( .) Express your answer in kilometers. ANSWER: The table below gives the masses of the Earth, the Moon, and the Sun. Name Mass (kg) Earth Moon Sun F4 = m r 6m rb rb r rb = r km 1 ton = 103 kg r = km 5.97 × 1024 7.35 × 1022 1.99 × 1030 Typesetting math: 81% The average distance between the Earth and the Moon is . The average distance between the Earth and the Sun is . Use this information to answer the following questions. Part F Find the net gravitational force acting on the Earth in the Sun-Earth-Moon system during the new moon (when the moon is located directly between the Earth and the Sun). Express your answer in newtons to three significant figures. You did not open hints for this part. ANSWER: Part G Find the net gravitational force acting on the Earth in the Sun-Earth-Moon system during the full moon (when the Earth is located directly between the moon and the sun). Express your answer in newtons to three significant figures. ANSWER: ± Understanding Newton’s Law of Universal Gravitation Learning Goal: To understand Newton’s law of universal gravitation and be able to apply it in two-object situations and (collinear) three-object situations; to distinguish between the use of and . 3.84 × 108 m 1.50 × 1011 m Fnet Fnet = N Fnet Fnet = N Typesetting math: 81% G g In the late 1600s, Isaac Newton proposed a rule to quantify the attractive force known as gravity between objects that have mass, such as those shown in the figure. Newton’s law of universal gravitation describes the magnitude of the attractive gravitational force between two objects with masses and as , where is the distance between the centers of the two objects and is the gravitational constant. The gravitational force is attractive, so in the figure it pulls to the right on (toward ) and toward the left on (toward ). The gravitational force acting on is equal in size to, but exactly opposite in direction from, the gravitational force acting on , as required by Newton’s third law. The magnitude of both forces is calculated with the equation given above. The gravitational constant has the value and should not be confused with the magnitude of the gravitational free-fall acceleration constant, denoted by , which equals 9.80 near the surface of the earth. The size of in SI units is tiny. This means that gravitational forces are sizeable only in the vicinity of very massive objects, such as the earth. You are in fact gravitationally attracted toward all the objects around you, such as the computer you are using, but the size of that force is too small to be noticed without extremely sensitive equipment. Consider the earth following its nearly circular orbit (dashed curve) about the sun. The earth has mass and the sun has mass . They are separated, center to center, by . Part A What is the size of the gravitational force acting on the earth due to the sun? Express your answer in newtons. F  g m1 m2 Fg = G( ) m1m2 r2 r G m1 m2 m2 m1 m1 m2 G G = 6.67 × 10−11 N m2/kg2 g m/s2 G mearth = 5.98 × 1024 kg msun = 1.99 × 1030 kg r = 93 million miles = 150 million km Typesetting math: 81% You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). Part F N Typesetting math: 81% This question will be shown after you complete previous question(s). Understanding Mass and Weight Learning Goal: To understand the distinction between mass and weight and to be able to calculate the weight of an object from its mass and Newton’s law of gravitation. The concepts of mass and weight are often confused. In fact, in everyday conversations, the word “weight” often replaces “mass,” as in “My weight is seventy-five kilograms” or “I need to lose some weight.” Of course, mass and weight are related; however, they are also very different. Mass, as you recall, is a measure of an object’s inertia (ability to resist acceleration). Newton’s 2nd law demonstrates the relationship among an object’s mass, its acceleration, and the net force acting on it: . Mass is an intrinsic property of an object and is independent of the object’s location. Weight, in contrast, is defined as the force due to gravity acting on the object. That force depends on the strength of the gravitational field of the planet: , where is the weight of an object, is the mass of that object, and is the local acceleration due to gravity (in other words, the strength of the gravitational field at the location of the object). Weight, unlike mass, is not an intrinsic property of the object; it is determined by both the object and its location. Part A Which of the following quantities represent mass? Check all that apply. ANSWER: Fnet = ma w = mg w m g 12.0 lbs 0.34 g 120 kg 1600 kN 0.34 m 411 cm 899 MN Typesetting math: 81% Part B This question will be shown after you complete previous question(s). Using the universal law of gravity, we can find the weight of an object feeling the gravitational pull of a nearby planet. We can write an expression , where is the weight of the object, is the gravitational constant, is the mass of that object, is mass of the planet, and is the distance from the center of the planet to the object. If the object is on the surface of the planet, is simply the radius of the planet. Part C The gravitational field on the surface of the earth is stronger than that on the surface of the moon. If a rock is transported from the moon to the earth, which properties of the rock change? ANSWER: Part D This question will be shown after you complete previous question(s). Part E If acceleration due to gravity on the earth is , which formula gives the acceleration due to gravity on Loput? You did not open hints for this part. ANSWER: w = GmM/r2 w G m M r r mass only weight only both mass and weight neither mass nor weight g Typesetting math: 81% Part F This question will be shown after you complete previous question(s). Part G This question will be shown after you complete previous question(s). Part H This question will be shown after you complete previous question(s). ± Weight on a Neutron Star Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun but a much smaller diameter. g 1.7 5.6 g 1.72 5.6 g 1.72 5.62 g 5.6 1.7 g 5.62 1.72 g 5.6 1.72 Typesetting math: 81% Part A If you weigh 655 on the earth, what would be your weight on the surface of a neutron star that has the same mass as our sun and a diameter of 19.0 ? Take the mass of the sun to be = 1.99×1030 , the gravitational constant to be = 6.67×10−11 , and the acceleration due to gravity at the earth’s surface to be = 9.810 . Express your weight in newtons. You did not open hints for this part. ANSWER: ± Escape Velocity Learning Goal: To introduce you to the concept of escape velocity for a rocket. The escape velocity is defined to be the minimum speed with which an object of mass must move to escape from the gravitational attraction of a much larger body, such as a planet of total mass . The escape velocity is a function of the distance of the object from the center of the planet , but unless otherwise specified this distance is taken to be the radius of the planet because it addresses the question “How fast does my rocket have to go to escape from the surface of the planet?” Part A The key to making a concise mathematical definition of escape velocity is to consider the energy. If an object is launched at its escape velocity, what is the total mechanical energy of the object at a very large (i.e., infinite) distance from the planet? Follow the usual convention and take the gravitational potential energy to be zero at very large distances. You did not open hints for this part. ANSWER: N km ms kg G N m2/kg2 g m/s2 wstar wstar = N m M R Etotal Typesetting math: 81% Consider the motion of an object between a point close to the planet and a point very very far from the planet. Indicate whether the following statements are true or false. Part B Angular momentum about the center of the planet is conserved. ANSWER: Part C Total mechanical energy is conserved. ANSWER: Part D Kinetic energy is conserved. ANSWER: Etotal = true false true false Typesetting math: 81% Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). A Satellite in a Circular Orbit Consider a satellite of mass that orbits a planet of mass in a circle a distance from the center of the planet. The satellite’s mass is negligible compared with that of the planet. Indicate whether each of the statements in this problem is true or false. Part A The information given is sufficient to uniquely specify the speed, potential energy, and angular momentum of the satellite. You did not open hints for this part. ANSWER: true false m1 m2 r true false Typesetting math: 81% Part B The total mechanical energy of the satellite is conserved. You did not open hints for this part. ANSWER: Part C The linear momentum vector of the satellite is conserved. You did not open hints for this part. ANSWER: Part D The angular momentum of the satellite about the center of the planet is conserved. You did not open hints for this part. ANSWER: true false true false Typesetting math: 81% Part E The equations that express the conservation laws of total mechanical energy and linear momentum are sufficient to solve for the speed necessary to maintain a circular orbit at without using . You did not open hints for this part. ANSWER: At the Galaxy’s Core Astronomers have observed a small, massive object at the center of our Milky Way galaxy. A ring of material orbits this massive object; the ring has a diameter of about 15 light years and an orbital speed of about 200 . Part A Determine the mass of the massive object at the center of the Milky Way galaxy. Take the distance of one light year to be . Express your answer in kilograms. You did not open hints for this part. true false R F = ma true false km/s M 9.461 × 1015 m Typesetting math: 81% ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). Properties of Circular Orbits Learning Goal: To teach you how to find the parameters characterizing an object in a circular orbit around a much heavier body like the earth. M = kg Typesetting math: 81% The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit–a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass . For all parts of this problem, where appropriate, use for the universal gravitational constant. Part A Find the orbital speed for a satellite in a circular orbit of radius . Express the orbital speed in terms of , , and . You did not open hints for this part. ANSWER: Part B Find the kinetic energy of a satellite with mass in a circular orbit with radius . Express your answer in terms of \texttip{m}{m}, \texttip{M}{M}, \texttip{G}{G}, and \texttip{R}{R}. ANSWER: Part C M G v R G M R v = K m R \texttip{K}{K} = Typesetting math: 81% This question will be shown after you complete previous question(s). Part D Find the orbital period \texttip{T}{T}. Express your answer in terms of \texttip{G}{G}, \texttip{M}{M}, \texttip{R}{R}, and \texttip{\pi }{pi}. You did not open hints for this part. ANSWER: Part E This question will be shown after you complete previous question(s). Part F Find \texttip{L}{L}, the magnitude of the angular momentum of the satellite with respect to the center of the planet. Express your answer in terms of \texttip{m}{m}, \texttip{M}{M}, \texttip{G}{G}, and \texttip{R}{R}. You did not open hints for this part. ANSWER: \texttip{T}{T} = Typesetting math: 81% Part G The quantities \texttip{v}{v}, \texttip{K}{K}, \texttip{U}{U}, and \texttip{L}{L} all represent physical quantities characterizing the orbit that depend on radius \texttip{R}{R}. Indicate the exponent (power) of the radial dependence of the absolute value of each. Express your answer as a comma-separated list of exponents corresponding to \texttip{v}{v}, \texttip{K}{K}, \texttip{U}{U}, and \texttip{L}{L}, in that order. For example, -1,-1/2,-0.5,-3/2 would mean v \propto R^{-1}, K \propto R^{-1/2}, and so forth. You did not open hints for this part. ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. \texttip{L}{L} = Typesetting math: 81%

Chapter 13 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, May 16, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Matter of Some Gravity Learning Goal: To understand Newton’s law of gravitation and the distinction between inertial and gravitational masses. In this problem, you will practice using Newton’s law of gravitation. According to that law, the magnitude of the gravitational force between two small particles of masses and , separated by a distance , is given by , where is the universal gravitational constant, whose numerical value (in SI units) is . This formula applies not only to small particles, but also to spherical objects. In fact, the gravitational force between two uniform spheres is the same as if we concentrated all the mass of each sphere at its center. Thus, by modeling the Earth and the Moon as uniform spheres, you can use the particle approximation when calculating the force of gravity between them. Be careful in using Newton’s law to choose the correct value for . To calculate the force of gravitational attraction between two uniform spheres, the distance in the equation for Newton’s law of gravitation is the distance between the centers of the spheres. For instance, if a small object such as an elephant is located on the surface of the Earth, the radius of the Earth would be used in the equation. Note that the force of gravity acting on an object located near the surface of a planet is often called weight. Also note that in situations involving satellites, you are often given the altitude of the satellite, that is, the distance from the satellite to the surface of the planet; this is not the distance to be used in the formula for the law of gravitation. There is a potentially confusing issue involving mass. Mass is defined as a measure of an object’s inertia, that is, its ability to resist acceleration. Newton’s second law demonstrates the relationship between mass, acceleration, and the net force acting on an object: . We can now refer to this measure of inertia more precisely as the inertial mass. On the other hand, the masses of the particles that appear in the expression for the law of gravity seem to have nothing to do with inertia: Rather, they serve as a measure of the strength of gravitational interactions. It would be reasonable to call such a property gravitational mass. Does this mean that every object has two different masses? Generally speaking, yes. However, the good news is that according to the latest, highly precise, measurements, the inertial and the gravitational mass of an object are, in fact, equal to each other; it is an established consensus among physicists that there is only one mass after all, which is a measure of both the object’s inertia and its ability to engage in gravitational interactions. Note that this consensus, like everything else in science, is open to possible amendments in the future. In this problem, you will answer several questions that require the use of Newton’s law of gravitation. Part A Two particles are separated by a certain distance. The force of gravitational interaction between them is . Now the separation between the particles is tripled. Find the new force of gravitational Fg m1 m2 r Fg = G m1m2 r2 G 6.67 × 10−11 N m2 kg2 r r rEarth F  = m net a F0 interaction . Express your answer in terms of . ANSWER: Part B A satellite revolves around a planet at an altitude equal to the radius of the planet. The force of gravitational interaction between the satellite and the planet is . Then the satellite moves to a different orbit, so that its altitude is tripled. Find the new force of gravitational interaction . Express your answer in terms of . You did not open hints for this part. ANSWER: Part C A satellite revolves around a planet at an altitude equal to the radius of the planet. The force of gravitational interaction between the satellite and the planet is . Then the satellite is brought back to the surface of the planet. Find the new force of gravitational interaction . Express your answer in terms of . ANSWER: F1 F0 F1 = F0 F2 F0 F2 = F0 F4 F0 Typesetting math: 81% Part D Two satellites revolve around the Earth. Satellite A has mass and has an orbit of radius . Satellite B has mass and an orbit of unknown radius . The forces of gravitational attraction between each satellite and the Earth is the same. Find . Express your answer in terms of . ANSWER: Part E An adult elephant has a mass of about 5.0 tons. An adult elephant shrew has a mass of about 50 grams. How far from the center of the Earth should an elephant be placed so that its weight equals that of the elephant shrew on the surface of the Earth? The radius of the Earth is 6400 . ( .) Express your answer in kilometers. ANSWER: The table below gives the masses of the Earth, the Moon, and the Sun. Name Mass (kg) Earth Moon Sun F4 = m r 6m rb rb r rb = r km 1 ton = 103 kg r = km 5.97 × 1024 7.35 × 1022 1.99 × 1030 Typesetting math: 81% The average distance between the Earth and the Moon is . The average distance between the Earth and the Sun is . Use this information to answer the following questions. Part F Find the net gravitational force acting on the Earth in the Sun-Earth-Moon system during the new moon (when the moon is located directly between the Earth and the Sun). Express your answer in newtons to three significant figures. You did not open hints for this part. ANSWER: Part G Find the net gravitational force acting on the Earth in the Sun-Earth-Moon system during the full moon (when the Earth is located directly between the moon and the sun). Express your answer in newtons to three significant figures. ANSWER: ± Understanding Newton’s Law of Universal Gravitation Learning Goal: To understand Newton’s law of universal gravitation and be able to apply it in two-object situations and (collinear) three-object situations; to distinguish between the use of and . 3.84 × 108 m 1.50 × 1011 m Fnet Fnet = N Fnet Fnet = N Typesetting math: 81% G g In the late 1600s, Isaac Newton proposed a rule to quantify the attractive force known as gravity between objects that have mass, such as those shown in the figure. Newton’s law of universal gravitation describes the magnitude of the attractive gravitational force between two objects with masses and as , where is the distance between the centers of the two objects and is the gravitational constant. The gravitational force is attractive, so in the figure it pulls to the right on (toward ) and toward the left on (toward ). The gravitational force acting on is equal in size to, but exactly opposite in direction from, the gravitational force acting on , as required by Newton’s third law. The magnitude of both forces is calculated with the equation given above. The gravitational constant has the value and should not be confused with the magnitude of the gravitational free-fall acceleration constant, denoted by , which equals 9.80 near the surface of the earth. The size of in SI units is tiny. This means that gravitational forces are sizeable only in the vicinity of very massive objects, such as the earth. You are in fact gravitationally attracted toward all the objects around you, such as the computer you are using, but the size of that force is too small to be noticed without extremely sensitive equipment. Consider the earth following its nearly circular orbit (dashed curve) about the sun. The earth has mass and the sun has mass . They are separated, center to center, by . Part A What is the size of the gravitational force acting on the earth due to the sun? Express your answer in newtons. F  g m1 m2 Fg = G( ) m1m2 r2 r G m1 m2 m2 m1 m1 m2 G G = 6.67 × 10−11 N m2/kg2 g m/s2 G mearth = 5.98 × 1024 kg msun = 1.99 × 1030 kg r = 93 million miles = 150 million km Typesetting math: 81% You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). Part F N Typesetting math: 81% This question will be shown after you complete previous question(s). Understanding Mass and Weight Learning Goal: To understand the distinction between mass and weight and to be able to calculate the weight of an object from its mass and Newton’s law of gravitation. The concepts of mass and weight are often confused. In fact, in everyday conversations, the word “weight” often replaces “mass,” as in “My weight is seventy-five kilograms” or “I need to lose some weight.” Of course, mass and weight are related; however, they are also very different. Mass, as you recall, is a measure of an object’s inertia (ability to resist acceleration). Newton’s 2nd law demonstrates the relationship among an object’s mass, its acceleration, and the net force acting on it: . Mass is an intrinsic property of an object and is independent of the object’s location. Weight, in contrast, is defined as the force due to gravity acting on the object. That force depends on the strength of the gravitational field of the planet: , where is the weight of an object, is the mass of that object, and is the local acceleration due to gravity (in other words, the strength of the gravitational field at the location of the object). Weight, unlike mass, is not an intrinsic property of the object; it is determined by both the object and its location. Part A Which of the following quantities represent mass? Check all that apply. ANSWER: Fnet = ma w = mg w m g 12.0 lbs 0.34 g 120 kg 1600 kN 0.34 m 411 cm 899 MN Typesetting math: 81% Part B This question will be shown after you complete previous question(s). Using the universal law of gravity, we can find the weight of an object feeling the gravitational pull of a nearby planet. We can write an expression , where is the weight of the object, is the gravitational constant, is the mass of that object, is mass of the planet, and is the distance from the center of the planet to the object. If the object is on the surface of the planet, is simply the radius of the planet. Part C The gravitational field on the surface of the earth is stronger than that on the surface of the moon. If a rock is transported from the moon to the earth, which properties of the rock change? ANSWER: Part D This question will be shown after you complete previous question(s). Part E If acceleration due to gravity on the earth is , which formula gives the acceleration due to gravity on Loput? You did not open hints for this part. ANSWER: w = GmM/r2 w G m M r r mass only weight only both mass and weight neither mass nor weight g Typesetting math: 81% Part F This question will be shown after you complete previous question(s). Part G This question will be shown after you complete previous question(s). Part H This question will be shown after you complete previous question(s). ± Weight on a Neutron Star Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun but a much smaller diameter. g 1.7 5.6 g 1.72 5.6 g 1.72 5.62 g 5.6 1.7 g 5.62 1.72 g 5.6 1.72 Typesetting math: 81% Part A If you weigh 655 on the earth, what would be your weight on the surface of a neutron star that has the same mass as our sun and a diameter of 19.0 ? Take the mass of the sun to be = 1.99×1030 , the gravitational constant to be = 6.67×10−11 , and the acceleration due to gravity at the earth’s surface to be = 9.810 . Express your weight in newtons. You did not open hints for this part. ANSWER: ± Escape Velocity Learning Goal: To introduce you to the concept of escape velocity for a rocket. The escape velocity is defined to be the minimum speed with which an object of mass must move to escape from the gravitational attraction of a much larger body, such as a planet of total mass . The escape velocity is a function of the distance of the object from the center of the planet , but unless otherwise specified this distance is taken to be the radius of the planet because it addresses the question “How fast does my rocket have to go to escape from the surface of the planet?” Part A The key to making a concise mathematical definition of escape velocity is to consider the energy. If an object is launched at its escape velocity, what is the total mechanical energy of the object at a very large (i.e., infinite) distance from the planet? Follow the usual convention and take the gravitational potential energy to be zero at very large distances. You did not open hints for this part. ANSWER: N km ms kg G N m2/kg2 g m/s2 wstar wstar = N m M R Etotal Typesetting math: 81% Consider the motion of an object between a point close to the planet and a point very very far from the planet. Indicate whether the following statements are true or false. Part B Angular momentum about the center of the planet is conserved. ANSWER: Part C Total mechanical energy is conserved. ANSWER: Part D Kinetic energy is conserved. ANSWER: Etotal = true false true false Typesetting math: 81% Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). A Satellite in a Circular Orbit Consider a satellite of mass that orbits a planet of mass in a circle a distance from the center of the planet. The satellite’s mass is negligible compared with that of the planet. Indicate whether each of the statements in this problem is true or false. Part A The information given is sufficient to uniquely specify the speed, potential energy, and angular momentum of the satellite. You did not open hints for this part. ANSWER: true false m1 m2 r true false Typesetting math: 81% Part B The total mechanical energy of the satellite is conserved. You did not open hints for this part. ANSWER: Part C The linear momentum vector of the satellite is conserved. You did not open hints for this part. ANSWER: Part D The angular momentum of the satellite about the center of the planet is conserved. You did not open hints for this part. ANSWER: true false true false Typesetting math: 81% Part E The equations that express the conservation laws of total mechanical energy and linear momentum are sufficient to solve for the speed necessary to maintain a circular orbit at without using . You did not open hints for this part. ANSWER: At the Galaxy’s Core Astronomers have observed a small, massive object at the center of our Milky Way galaxy. A ring of material orbits this massive object; the ring has a diameter of about 15 light years and an orbital speed of about 200 . Part A Determine the mass of the massive object at the center of the Milky Way galaxy. Take the distance of one light year to be . Express your answer in kilograms. You did not open hints for this part. true false R F = ma true false km/s M 9.461 × 1015 m Typesetting math: 81% ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). Properties of Circular Orbits Learning Goal: To teach you how to find the parameters characterizing an object in a circular orbit around a much heavier body like the earth. M = kg Typesetting math: 81% The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit–a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass . For all parts of this problem, where appropriate, use for the universal gravitational constant. Part A Find the orbital speed for a satellite in a circular orbit of radius . Express the orbital speed in terms of , , and . You did not open hints for this part. ANSWER: Part B Find the kinetic energy of a satellite with mass in a circular orbit with radius . Express your answer in terms of \texttip{m}{m}, \texttip{M}{M}, \texttip{G}{G}, and \texttip{R}{R}. ANSWER: Part C M G v R G M R v = K m R \texttip{K}{K} = Typesetting math: 81% This question will be shown after you complete previous question(s). Part D Find the orbital period \texttip{T}{T}. Express your answer in terms of \texttip{G}{G}, \texttip{M}{M}, \texttip{R}{R}, and \texttip{\pi }{pi}. You did not open hints for this part. ANSWER: Part E This question will be shown after you complete previous question(s). Part F Find \texttip{L}{L}, the magnitude of the angular momentum of the satellite with respect to the center of the planet. Express your answer in terms of \texttip{m}{m}, \texttip{M}{M}, \texttip{G}{G}, and \texttip{R}{R}. You did not open hints for this part. ANSWER: \texttip{T}{T} = Typesetting math: 81% Part G The quantities \texttip{v}{v}, \texttip{K}{K}, \texttip{U}{U}, and \texttip{L}{L} all represent physical quantities characterizing the orbit that depend on radius \texttip{R}{R}. Indicate the exponent (power) of the radial dependence of the absolute value of each. Express your answer as a comma-separated list of exponents corresponding to \texttip{v}{v}, \texttip{K}{K}, \texttip{U}{U}, and \texttip{L}{L}, in that order. For example, -1,-1/2,-0.5,-3/2 would mean v \propto R^{-1}, K \propto R^{-1/2}, and so forth. You did not open hints for this part. ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. \texttip{L}{L} = Typesetting math: 81%

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6. The primary operating goal of a publicly-owned firm trying to best serve its stockholders should be to a. Maximize managers’ own interests, which are by definition consistent with maximizing shareholders’ wealth. b. Maximize the firm’s expected EPS, which must also maximize the firm’s price per share. c. Minimize the firm’s risks because most stockholders dislike risk. In turn, this will maximize the firm’s stock price. d. Use a well-structured managerial compensation package to reduce conflicts that may exist between stockholders and managers. e. Since it is impossible to measure a stock’s intrinsic value, the text states that it is better for managers to attempt to maximize the current stock price than its intrinsic value.

6. The primary operating goal of a publicly-owned firm trying to best serve its stockholders should be to a. Maximize managers’ own interests, which are by definition consistent with maximizing shareholders’ wealth. b. Maximize the firm’s expected EPS, which must also maximize the firm’s price per share. c. Minimize the firm’s risks because most stockholders dislike risk. In turn, this will maximize the firm’s stock price. d. Use a well-structured managerial compensation package to reduce conflicts that may exist between stockholders and managers. e. Since it is impossible to measure a stock’s intrinsic value, the text states that it is better for managers to attempt to maximize the current stock price than its intrinsic value.

Answer: d
Three forces FA, FB and FC are applied to a beam as shown in the figure. (a) Determine the moment of force FC about point O using the cross product definition of moment (M = r x F) (10 points) (b) Determine the moment of force FB about point A using the rectangular components of the force (that is, do not use the cross product) (5 points) (c) Resolve force FA into a force and a couple at point C

Three forces FA, FB and FC are applied to a beam as shown in the figure. (a) Determine the moment of force FC about point O using the cross product definition of moment (M = r x F) (10 points) (b) Determine the moment of force FB about point A using the rectangular components of the force (that is, do not use the cross product) (5 points) (c) Resolve force FA into a force and a couple at point C

Elastic Collision Write up for TA Jessica Andersen The following pages include what is expected for the PHY 112 Elastic Collision lab. Below each section heading are general tips for lab writing that can be applied to any lab in the future. Point values associated with each section are stated, as well are the points associated for topics within that section. Read through completely before beginning. Introduction ( 20 pts total ) Tips for a good Introduction section: Be thorough but do not write a five paragraph essay! Concisely present the purpose and background material. You don’t need to number equations unless you will be referring back to them. Simply explain what they apply to as you introduce them. A 2pt bullet should not correspond to more than two lines of writing in your report. – Include a statement of purpose for the lab. (5pts) – Define the necessary conditions of an Elastic Collision (5pts) – Introduce the concept of conservation of linear momentum and derive the equation for calculating linear momentum in the x-direction and the y direction. (5pts) – Introduce the concept of conservation of energy and derive the equation for calculating kinetic energy of the system before and after the collision. (5pts) Methods (10 pts total) Tips for a good Methods section: Don’t spend too much time on this section! Be very quick and to the point. Write as if you are giving instructions to someone else. This will sound much more professional and you won’t have to worry about the use of “I” or “we”, which can tend to make a lab report sound very informal. – Briefly describe the setup of the lab and what precautions were taken to ensure something close to an elastic collision (5pts) – What frequency was the “zapper” set to? (5pts) Results (25 pts total) Tips for a good Results section: This is an important section. It should be organized and formatted in a way that makes it very easy to read. Your tables should have borders and bolded headings where you see appropriate. Always include a brief description of each table at the opening of the section. REMEMBER, the Results section is about conveying your data in a readable and easy to understand way. • do not divide tables across pages • do not include more than 3 decimal places unless they are legitimately important – Include a table that summarizes all of the values recorded from the collision path. (5pts) – Include a table that displays the Kinetic Energy before and after the collision (5pts) – Include a table that displays the Linear Momentum in both directions before and after the collision (10pts) – Include a summary table that calculates the percent error between before collision values and after collision values. Use the before collision values as your theoretical value. (5pts) Discussion (40 pts total) Tips for a good Discussion section: This section is worth almost half of your report! I want to see that you put legitimate thought into your data and how it relates to what you learn in lecture. Show me that you understand the things we talked about in class. Be thorough, but remember that long and drawn out does not necessary achieve this. • do not present data as one large paragraph, make them smaller and easier to read • do not refer back to tables, actually state the values when asked for • you may refer back to graphs when necessary • do not use math vocabulary wrong, if you are unsure of a definition, look it up!!! – Present the percent error values for both momentum and energy calculations. (10pts) – Why was the energy and momentum BEFORE collision used as the theoretical value? (hint: It has to do with us assuming we have an Elastic Collision) (10pts) – Present the frequency of the “zapper”. What does this mean about the time that passes between each dot on the collision path? (10pts) – Discuss sources of error in this lab and how they may have affected our final result. (10pts) Appendix (5pts total) – Just staple on whatever notes you took in class.

Elastic Collision Write up for TA Jessica Andersen The following pages include what is expected for the PHY 112 Elastic Collision lab. Below each section heading are general tips for lab writing that can be applied to any lab in the future. Point values associated with each section are stated, as well are the points associated for topics within that section. Read through completely before beginning. Introduction ( 20 pts total ) Tips for a good Introduction section: Be thorough but do not write a five paragraph essay! Concisely present the purpose and background material. You don’t need to number equations unless you will be referring back to them. Simply explain what they apply to as you introduce them. A 2pt bullet should not correspond to more than two lines of writing in your report. – Include a statement of purpose for the lab. (5pts) – Define the necessary conditions of an Elastic Collision (5pts) – Introduce the concept of conservation of linear momentum and derive the equation for calculating linear momentum in the x-direction and the y direction. (5pts) – Introduce the concept of conservation of energy and derive the equation for calculating kinetic energy of the system before and after the collision. (5pts) Methods (10 pts total) Tips for a good Methods section: Don’t spend too much time on this section! Be very quick and to the point. Write as if you are giving instructions to someone else. This will sound much more professional and you won’t have to worry about the use of “I” or “we”, which can tend to make a lab report sound very informal. – Briefly describe the setup of the lab and what precautions were taken to ensure something close to an elastic collision (5pts) – What frequency was the “zapper” set to? (5pts) Results (25 pts total) Tips for a good Results section: This is an important section. It should be organized and formatted in a way that makes it very easy to read. Your tables should have borders and bolded headings where you see appropriate. Always include a brief description of each table at the opening of the section. REMEMBER, the Results section is about conveying your data in a readable and easy to understand way. • do not divide tables across pages • do not include more than 3 decimal places unless they are legitimately important – Include a table that summarizes all of the values recorded from the collision path. (5pts) – Include a table that displays the Kinetic Energy before and after the collision (5pts) – Include a table that displays the Linear Momentum in both directions before and after the collision (10pts) – Include a summary table that calculates the percent error between before collision values and after collision values. Use the before collision values as your theoretical value. (5pts) Discussion (40 pts total) Tips for a good Discussion section: This section is worth almost half of your report! I want to see that you put legitimate thought into your data and how it relates to what you learn in lecture. Show me that you understand the things we talked about in class. Be thorough, but remember that long and drawn out does not necessary achieve this. • do not present data as one large paragraph, make them smaller and easier to read • do not refer back to tables, actually state the values when asked for • you may refer back to graphs when necessary • do not use math vocabulary wrong, if you are unsure of a definition, look it up!!! – Present the percent error values for both momentum and energy calculations. (10pts) – Why was the energy and momentum BEFORE collision used as the theoretical value? (hint: It has to do with us assuming we have an Elastic Collision) (10pts) – Present the frequency of the “zapper”. What does this mean about the time that passes between each dot on the collision path? (10pts) – Discuss sources of error in this lab and how they may have affected our final result. (10pts) Appendix (5pts total) – Just staple on whatever notes you took in class.

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Homework Assignment 7. Due March 19 1. Consider the differential equation: ?? ?? = − 1 2 ? sin? ? with initial condition given by ?(0) = 1 Solve this equation from t = 0 to t = 8π using the following methods: (a) Solve analytically by separating variables and integrating. (b) Solve using the 4th-order Runge-Kutta method (write your own code for this, do not use the MATLAB provided ODE solvers) for the following two step sizes: I. Maximum step size for stability (don’t try and do this analytically – try out your code for different step sizes to find the stability limit). II. Maximum step size for a time-accurate solution. “Good” accuracy can be defined in several ways, but use the definition that the numerical solution remains within 2% of the true solution a t = nπ. (c) Solve using the MATLAB function ode45. 2. A car and its suspension system traveling over a bumpy road can be modeled as a mass/spring/damper system. In this model, ?? is the vertical motion of the wheel center of mass, ?? is the vertical motion of the car chassis, and ?? represents the displacement of the bottom of the tire due to the variation in the road surface. Applying Newton’s law to the two masses yields a system of second-order equations: ???̈? + ??(?̇? − ?̇?) + ??(?? − ??) + ???? = ???? ???̈? − ??(?̇? − ?̇?) − ??(?? − ??) + ???? = 0 (a) Convert the two second-order ODE’s into a system of 4 first-order ODE’s. Write them in standard “state-space” form. (b) Assume the car hits a large pothole at t = 0 so that ??(?) = ?−0.2 m 0 ≤ ? < 0.2 s 0 ? > 0.2 s Create a MATLAB function that returns the right hand sides of the state-space equations for an input t and an input state vector. (c) Solve the system on the time interval [0 60] seconds using the MATLAB function ode45. Find the displacement and velocity of the chassis and the wheel as a function of time. Use the following data: ?? = 100 kg, ?? = 1900 kg, ?? = 145 N/m, ?? = 25 N/m, ?? = 150 N-s/m 3. Write a MATLAB program to simulate the dynamics of a helicopter lifting a survivor. When lifting the survivor into the helicopter with a constant speed winch, the resulting dynamics are non-linear, and stability is dependent upon the winch speed. Using polar coordinates, we can find the equations of motion to be: −?? sin ? = ????̈ + 2?̇?̇? ?̇ = constant (negative) Notice that the mass of the survivor factors out and thus the solution is independent of the mass of the person being lifted. In these equations, r is the instantaneous length of the winch cable, g, is the gravitational constant, and θ is the angle of the swing. You may choose to use either your Runge-Kutta solver from problem 1 or ode45 to integrate the equations of motion. This problem is of particular interest to the survivor since an unstable condition can cause the angle of the swing to exceed 90⁰, essentially placing him/her in danger of being beheaded by the rotor blades of the rescue helicopter. Also, it is desirable to retrieve the survivor as fast as possible to get away from the danger. Use your program to determine the maximum winch speed for which the survivor will not swing above the helicopter attach point for a lift from the initial conditions: ?? = 0.1 ??? ?? ̇ = 0 ?? = 34 ? And ending when ? = 0.5 ?. The maximum lifting speed of the winch is 5 m/s. Present your results for the above problems in an appropriate fashion. For problem 1, be sure to include a comparison of the numerical methods with each other and with the true solution. Be sure to discuss your findings with respect to the notions of stability and accuracy of the numerical methods. For problem 2, ensure that your results are easily interpreted by a reader. Students receiving a score of 70% or above on these two problems will receive credit for outcome #5. For problem 3, if you receive at least 70% of the points, you will receive credit for outcome #4.

Homework Assignment 7. Due March 19 1. Consider the differential equation: ?? ?? = − 1 2 ? sin? ? with initial condition given by ?(0) = 1 Solve this equation from t = 0 to t = 8π using the following methods: (a) Solve analytically by separating variables and integrating. (b) Solve using the 4th-order Runge-Kutta method (write your own code for this, do not use the MATLAB provided ODE solvers) for the following two step sizes: I. Maximum step size for stability (don’t try and do this analytically – try out your code for different step sizes to find the stability limit). II. Maximum step size for a time-accurate solution. “Good” accuracy can be defined in several ways, but use the definition that the numerical solution remains within 2% of the true solution a t = nπ. (c) Solve using the MATLAB function ode45. 2. A car and its suspension system traveling over a bumpy road can be modeled as a mass/spring/damper system. In this model, ?? is the vertical motion of the wheel center of mass, ?? is the vertical motion of the car chassis, and ?? represents the displacement of the bottom of the tire due to the variation in the road surface. Applying Newton’s law to the two masses yields a system of second-order equations: ???̈? + ??(?̇? − ?̇?) + ??(?? − ??) + ???? = ???? ???̈? − ??(?̇? − ?̇?) − ??(?? − ??) + ???? = 0 (a) Convert the two second-order ODE’s into a system of 4 first-order ODE’s. Write them in standard “state-space” form. (b) Assume the car hits a large pothole at t = 0 so that ??(?) = ?−0.2 m 0 ≤ ? < 0.2 s 0 ? > 0.2 s Create a MATLAB function that returns the right hand sides of the state-space equations for an input t and an input state vector. (c) Solve the system on the time interval [0 60] seconds using the MATLAB function ode45. Find the displacement and velocity of the chassis and the wheel as a function of time. Use the following data: ?? = 100 kg, ?? = 1900 kg, ?? = 145 N/m, ?? = 25 N/m, ?? = 150 N-s/m 3. Write a MATLAB program to simulate the dynamics of a helicopter lifting a survivor. When lifting the survivor into the helicopter with a constant speed winch, the resulting dynamics are non-linear, and stability is dependent upon the winch speed. Using polar coordinates, we can find the equations of motion to be: −?? sin ? = ????̈ + 2?̇?̇? ?̇ = constant (negative) Notice that the mass of the survivor factors out and thus the solution is independent of the mass of the person being lifted. In these equations, r is the instantaneous length of the winch cable, g, is the gravitational constant, and θ is the angle of the swing. You may choose to use either your Runge-Kutta solver from problem 1 or ode45 to integrate the equations of motion. This problem is of particular interest to the survivor since an unstable condition can cause the angle of the swing to exceed 90⁰, essentially placing him/her in danger of being beheaded by the rotor blades of the rescue helicopter. Also, it is desirable to retrieve the survivor as fast as possible to get away from the danger. Use your program to determine the maximum winch speed for which the survivor will not swing above the helicopter attach point for a lift from the initial conditions: ?? = 0.1 ??? ?? ̇ = 0 ?? = 34 ? And ending when ? = 0.5 ?. The maximum lifting speed of the winch is 5 m/s. Present your results for the above problems in an appropriate fashion. For problem 1, be sure to include a comparison of the numerical methods with each other and with the true solution. Be sure to discuss your findings with respect to the notions of stability and accuracy of the numerical methods. For problem 2, ensure that your results are easily interpreted by a reader. Students receiving a score of 70% or above on these two problems will receive credit for outcome #5. For problem 3, if you receive at least 70% of the points, you will receive credit for outcome #4.

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