After reading the supplement article on Business Analytics linked to the week 1 schedule, write an essay on how business analytics impacts you today, or its potential role in your chosen career path. Do research for your paper, or interview someone who works in your area. The goals of this paper are two-fold: (1) focus on high quality writing, using the COBE Writing Styles Guide for writing help and citations. (2) consider the importance of BI from a personal/work/career perspective.

After reading the supplement article on Business Analytics linked to the week 1 schedule, write an essay on how business analytics impacts you today, or its potential role in your chosen career path. Do research for your paper, or interview someone who works in your area. The goals of this paper are two-fold: (1) focus on high quality writing, using the COBE Writing Styles Guide for writing help and citations. (2) consider the importance of BI from a personal/work/career perspective.

  Business analytics importance and its potential     Introduction … Read More...
4. Using your knowledge of the Stevenson’s career management model identify and briefly describe one activity that should be included in an organization’s career management program. Identify which element of the model the activity you identified fits within.

4. Using your knowledge of the Stevenson’s career management model identify and briefly describe one activity that should be included in an organization’s career management program. Identify which element of the model the activity you identified fits within.

Discipline Expertise- There is an apparent type of interdisciplinary in … Read More...
. Imagine yourself teaching a carefully planned lesson. Do the activities you imagine doing with your students correlate to your own strengths? Why or why not?

. Imagine yourself teaching a carefully planned lesson. Do the activities you imagine doing with your students correlate to your own strengths? Why or why not?

In some instances the educator plans and teaches mini-lessons using … Read More...
SUPPLY CHAIN MANAGEMENT AT BOSE CORPORATION Bose Corporation, headquartered in Framingham, Massachusetts, offers an excellent example of integrated supply chain management. Bose, a producer of audio premium speakers used in automobiles, high-fidelity systems, and consumer and commercial broadcasting systems, was founded in 1964 by Dr. Bose of MIT. Bose currently maintains plants in Massachusetts and Michigan as well as Canada, Mexico, and Ireland. Its purchasing organization, while decentralized, has some overlap that requires coordination between sites. It manages this coordination by using conference calls between managers, electronic communication, and joint problem solving. The company is moving toward single sourcing many of its 800 to 1,000 parts, which include corrugated paper, particle board and wood, plastic injected molded parts, fasteners, glues, woofers, and fabric. Some product components, such as woofers, are sourced overseas. For example, at the Hillsdale, Michigan, plant, foreign sourcing accounts for 20% of purchases, with the remainder of suppliers located immediately within the state of Michigan. About 35% of the parts purchased at this site are single sourced, with approximately half of the components arriving with no incoming inspection performed. In turn, Bose ships finished products directly to Delco, Honda, and Nissan and has a record of no missed deliveries. Normal lead time to customers is 60 working days, but Bose can expedite shipments in one week and airfreight them if necessary. The company has developed a detailed supplier performance system that measures on-time delivery, quality performance, technical improvements, and supplier suggestions. A report is generated twice a month from this system and sent to the supplier providing feedback about supplier performance. If there is a three-week trend of poor performance, Bose will usually establish a specific goal for improvement that the supplier must attain. Examples include 10% delivery improvement every month until 100% conformance is achieved, or 5% quality improvement until a 1% defect level is reached over a four-month period. In one case, a supplier sent a rejected shipment back to Bose without explanation and with no corrective action taken. When no significant improvement occurred, another supplier replaced the delinquent supplier. Bose has few written contracts with suppliers. After six months of deliveries without rejects, Bose encourages suppliers to apply for a certificate of achievement form, signifying that they are qualified suppliers. One of the primary criteria for gaining certification involves how well the supplier responds to corrective action requests. One of the biggest problems observed is that suppliers often correct problems on individual parts covered by a corrective action form without extending these corrective actions to other part families and applicable parts. Bose has adopted a unique system of marrying just-in-time (JIT) purchasing with global sourcing. Approximately half of the dollar value of Bose’s total purchases are made overseas, with the majority of the sourcing done in Asia. Because foreign sourcing does not support just-in-time deliveries, Bose “had to find a way to blend low inventory with buying from distant sources,” says the director of purchasing and logistics for Bose. Visualizing itself as a customer-driven organization, Bose now uses a sophisticated transportation system—what Bose’s manager of logistics calls “the best EDI system in the country.” Working closely with a national less-than-truckload carrier for the bulk of its domestic freight movements, including shipments arriving at a U.S. port from oversees, Bose implemented an electronic data interchange (EDI) system that does much more than simple tracking. The system operates close to real time and allows two-way communication between every one of the freight handler’s 230 terminals and Bose. Information is updated several times daily and is downloaded automatically, enabling Bose to perform shipping analysis and distribution channel modeling to achieve reliable lowest total cost scenarios. The company can also request removal from a terminal of any shipment that it must expedite with an air shipment. This state-of-the-art system provides a snapshot of what is happening on a daily basis and keeps Bose’s managers on top of everyday occurrences and decisions. Management proactively manages logistics time elements in pursuit of better customer service. The next step is to implement this system with all major suppliers rather than just with transportation suppliers. In the future, Bose plans to automate its entire materials system. Perhaps one of the most unique features of Bose’s procurement and logistics system is the development of JIT II. The basic premise of JIT II is simple: The person who can do the best job of ordering and managing inventory of a particular item is the supplier himself. Bose negotiated with each supplier to provide a full-time employee at the Bose plant who was responsible for ordering, shipping, and receiving materials from that plant, as well as managing on-site inventories of the items. This was done through an EDI connection between Bose’s plant and the supplier’s facility. Collocating suppliers and buyers was so successful that Bose is now implementing it at all plant locations. In fact, many other companies have also begun to implement collocation of suppliers. Assignment Questions The following assignment questions relate to ideas and concepts presented throughout this text. Answer some or all of the questions as directed by your instructor. 1. Discuss how the strategy development process might work at a company like Bose. 2. What should be the relationship between Bose’s supply management strategy and the development of its performance measurement system? 3. Why is purchased quality so important to Bose? 4. Can a just-in-time purchase system operate without total quality from suppliers? 5. Why can some components arrive at the Hillsdale, Michigan, plant with no incoming inspection required? 6. Discuss the reasons why Bose has a certificate of achievement program for identifying qualified suppliers. 7. Bose is moving toward single sourcing many of its purchased part requirements. Discuss why the company might want to do this. Are there any risks to that approach? 8. Discuss some of the difficulties a company like Bose might experience when trying to implement just-in-time purchasing with international suppliers. 9. Why does Bose have to source so much of its purchase requirements from offshore suppliers? 10. What makes the JIT II system at Bose unique? Why would a company pursue this type of system? 11. Why is it necessary to enter into a longer-term contractual arrangement when pursuing arrangements like the one Bose has with its domestic transportation carrier? 12. Why is it important to manage logistics time elements proactively when pursuing higher levels of customer service? 13. What role does information technology play at Bose? 14. What advantages do information technology systems provide to Bose that might not be available to a company that does not have these systems? 15. Why has Bose developed its supplier performance measurement system? 16. Do you think the performance measurement systems at Bose are computerized or manual? Why?

SUPPLY CHAIN MANAGEMENT AT BOSE CORPORATION Bose Corporation, headquartered in Framingham, Massachusetts, offers an excellent example of integrated supply chain management. Bose, a producer of audio premium speakers used in automobiles, high-fidelity systems, and consumer and commercial broadcasting systems, was founded in 1964 by Dr. Bose of MIT. Bose currently maintains plants in Massachusetts and Michigan as well as Canada, Mexico, and Ireland. Its purchasing organization, while decentralized, has some overlap that requires coordination between sites. It manages this coordination by using conference calls between managers, electronic communication, and joint problem solving. The company is moving toward single sourcing many of its 800 to 1,000 parts, which include corrugated paper, particle board and wood, plastic injected molded parts, fasteners, glues, woofers, and fabric. Some product components, such as woofers, are sourced overseas. For example, at the Hillsdale, Michigan, plant, foreign sourcing accounts for 20% of purchases, with the remainder of suppliers located immediately within the state of Michigan. About 35% of the parts purchased at this site are single sourced, with approximately half of the components arriving with no incoming inspection performed. In turn, Bose ships finished products directly to Delco, Honda, and Nissan and has a record of no missed deliveries. Normal lead time to customers is 60 working days, but Bose can expedite shipments in one week and airfreight them if necessary. The company has developed a detailed supplier performance system that measures on-time delivery, quality performance, technical improvements, and supplier suggestions. A report is generated twice a month from this system and sent to the supplier providing feedback about supplier performance. If there is a three-week trend of poor performance, Bose will usually establish a specific goal for improvement that the supplier must attain. Examples include 10% delivery improvement every month until 100% conformance is achieved, or 5% quality improvement until a 1% defect level is reached over a four-month period. In one case, a supplier sent a rejected shipment back to Bose without explanation and with no corrective action taken. When no significant improvement occurred, another supplier replaced the delinquent supplier. Bose has few written contracts with suppliers. After six months of deliveries without rejects, Bose encourages suppliers to apply for a certificate of achievement form, signifying that they are qualified suppliers. One of the primary criteria for gaining certification involves how well the supplier responds to corrective action requests. One of the biggest problems observed is that suppliers often correct problems on individual parts covered by a corrective action form without extending these corrective actions to other part families and applicable parts. Bose has adopted a unique system of marrying just-in-time (JIT) purchasing with global sourcing. Approximately half of the dollar value of Bose’s total purchases are made overseas, with the majority of the sourcing done in Asia. Because foreign sourcing does not support just-in-time deliveries, Bose “had to find a way to blend low inventory with buying from distant sources,” says the director of purchasing and logistics for Bose. Visualizing itself as a customer-driven organization, Bose now uses a sophisticated transportation system—what Bose’s manager of logistics calls “the best EDI system in the country.” Working closely with a national less-than-truckload carrier for the bulk of its domestic freight movements, including shipments arriving at a U.S. port from oversees, Bose implemented an electronic data interchange (EDI) system that does much more than simple tracking. The system operates close to real time and allows two-way communication between every one of the freight handler’s 230 terminals and Bose. Information is updated several times daily and is downloaded automatically, enabling Bose to perform shipping analysis and distribution channel modeling to achieve reliable lowest total cost scenarios. The company can also request removal from a terminal of any shipment that it must expedite with an air shipment. This state-of-the-art system provides a snapshot of what is happening on a daily basis and keeps Bose’s managers on top of everyday occurrences and decisions. Management proactively manages logistics time elements in pursuit of better customer service. The next step is to implement this system with all major suppliers rather than just with transportation suppliers. In the future, Bose plans to automate its entire materials system. Perhaps one of the most unique features of Bose’s procurement and logistics system is the development of JIT II. The basic premise of JIT II is simple: The person who can do the best job of ordering and managing inventory of a particular item is the supplier himself. Bose negotiated with each supplier to provide a full-time employee at the Bose plant who was responsible for ordering, shipping, and receiving materials from that plant, as well as managing on-site inventories of the items. This was done through an EDI connection between Bose’s plant and the supplier’s facility. Collocating suppliers and buyers was so successful that Bose is now implementing it at all plant locations. In fact, many other companies have also begun to implement collocation of suppliers. Assignment Questions The following assignment questions relate to ideas and concepts presented throughout this text. Answer some or all of the questions as directed by your instructor. 1. Discuss how the strategy development process might work at a company like Bose. 2. What should be the relationship between Bose’s supply management strategy and the development of its performance measurement system? 3. Why is purchased quality so important to Bose? 4. Can a just-in-time purchase system operate without total quality from suppliers? 5. Why can some components arrive at the Hillsdale, Michigan, plant with no incoming inspection required? 6. Discuss the reasons why Bose has a certificate of achievement program for identifying qualified suppliers. 7. Bose is moving toward single sourcing many of its purchased part requirements. Discuss why the company might want to do this. Are there any risks to that approach? 8. Discuss some of the difficulties a company like Bose might experience when trying to implement just-in-time purchasing with international suppliers. 9. Why does Bose have to source so much of its purchase requirements from offshore suppliers? 10. What makes the JIT II system at Bose unique? Why would a company pursue this type of system? 11. Why is it necessary to enter into a longer-term contractual arrangement when pursuing arrangements like the one Bose has with its domestic transportation carrier? 12. Why is it important to manage logistics time elements proactively when pursuing higher levels of customer service? 13. What role does information technology play at Bose? 14. What advantages do information technology systems provide to Bose that might not be available to a company that does not have these systems? 15. Why has Bose developed its supplier performance measurement system? 16. Do you think the performance measurement systems at Bose are computerized or manual? Why?

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ME4575/5575 Renewable and Alternative Energy Fall 2015 Project 1 In this project, you will design a two piston Stirling engine (as in the example given in the class) of 1 kW power. The engine will be operated at TH=650 0C. The waste energy will be rejected at the temperature of TC=40 0C. The objective of the design is to maximize the efficiency and minimize the system weight. For the given temperature ratio, you have to select and optimize the piston diameter and piston stroke. The weight is proximately equal to the hot and cold piston volume multiplying by steel density. You can use Excel spreadsheet (or other engineering software) to create a Stirling modeling file to iterate on the piston diameter and stroke until the best combination of efficiency and weight (cost) is achieved. The project report has to contain a short introduction, technical description of the problem, details of analyses, and final conclusion of the design (size, weight, and efficiency).

ME4575/5575 Renewable and Alternative Energy Fall 2015 Project 1 In this project, you will design a two piston Stirling engine (as in the example given in the class) of 1 kW power. The engine will be operated at TH=650 0C. The waste energy will be rejected at the temperature of TC=40 0C. The objective of the design is to maximize the efficiency and minimize the system weight. For the given temperature ratio, you have to select and optimize the piston diameter and piston stroke. The weight is proximately equal to the hot and cold piston volume multiplying by steel density. You can use Excel spreadsheet (or other engineering software) to create a Stirling modeling file to iterate on the piston diameter and stroke until the best combination of efficiency and weight (cost) is achieved. The project report has to contain a short introduction, technical description of the problem, details of analyses, and final conclusion of the design (size, weight, and efficiency).

Page 1 of 2 Name ________________________ ENGR350-01 Learning Exercise 7: Problem 1 [3 points]: For the circuit below, we want to solve for Vc(t). Assume that for t < 0, switch S1 has been closed long enough for Vc(t) to reach a constant value. The switch S1 opens at t=0. Note that the steady state model for a capacitor is an open circuit (since ?????=?). 1a) Find Vc just before t=0 and also for t. 1b) Find τ for t>0 (after the switch opens). 1c) Find Vc(t) mathematically and graph it for the first 50 milliseconds after the switch opens. Make the graph big enough to clearly show the natural response and steady state response. Page 2 of 2 Problem 2 [7 points]: For the circuit below, we want to calculate iL(t). For t<0, you can assume the voltage source has been at +5V for a long time prior to t=0. At t=0, the voltage source drops to -5V and stays. Note that the steady state model for an inductor is a wire (since ?????=?). 2a) Find the value of iL(t) just prior to t=0. 2b) Find the value of iL(t) for t. 2c) Find the time constant τ. 2d) Write the mathematical expression describing iL(t) for t>0. 2e) Based on 2d, find VL(t) for t>0. 2f) Use nodal analysis to find the differential equation governing iL(t) for this circuit, with circuit values (such as R1, R2, L, V1) in addition to iL(t) and ?????. 2g) In this circuit, R2 is actually modeling the resistive loss within a non-ideal inductor. Calculate the point in time when the power dissipated in R2 is minimum. Hint: first think about the point in time that (iL)2 is minimum, since P=i2R for a resistor. +5 Volts -5 Volts V1

Page 1 of 2 Name ________________________ ENGR350-01 Learning Exercise 7: Problem 1 [3 points]: For the circuit below, we want to solve for Vc(t). Assume that for t < 0, switch S1 has been closed long enough for Vc(t) to reach a constant value. The switch S1 opens at t=0. Note that the steady state model for a capacitor is an open circuit (since ?????=?). 1a) Find Vc just before t=0 and also for t. 1b) Find τ for t>0 (after the switch opens). 1c) Find Vc(t) mathematically and graph it for the first 50 milliseconds after the switch opens. Make the graph big enough to clearly show the natural response and steady state response. Page 2 of 2 Problem 2 [7 points]: For the circuit below, we want to calculate iL(t). For t<0, you can assume the voltage source has been at +5V for a long time prior to t=0. At t=0, the voltage source drops to -5V and stays. Note that the steady state model for an inductor is a wire (since ?????=?). 2a) Find the value of iL(t) just prior to t=0. 2b) Find the value of iL(t) for t. 2c) Find the time constant τ. 2d) Write the mathematical expression describing iL(t) for t>0. 2e) Based on 2d, find VL(t) for t>0. 2f) Use nodal analysis to find the differential equation governing iL(t) for this circuit, with circuit values (such as R1, R2, L, V1) in addition to iL(t) and ?????. 2g) In this circuit, R2 is actually modeling the resistive loss within a non-ideal inductor. Calculate the point in time when the power dissipated in R2 is minimum. Hint: first think about the point in time that (iL)2 is minimum, since P=i2R for a resistor. +5 Volts -5 Volts V1

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modeling and simulation APA style Please post an abstract for your white paper. with peer reviewed paper related to this subject The da Vinci® Surgical System I want to develop this robotic to make it as a doctor to send it to Infested places, or places where natural disasters or serious illnesses that arise to serve the work as real doctor and to control it remotely. In this way we protect the medical staff from any risks related to their lives, such as murder, kidnapping, or incidence of these diseases

modeling and simulation APA style Please post an abstract for your white paper. with peer reviewed paper related to this subject The da Vinci® Surgical System I want to develop this robotic to make it as a doctor to send it to Infested places, or places where natural disasters or serious illnesses that arise to serve the work as real doctor and to control it remotely. In this way we protect the medical staff from any risks related to their lives, such as murder, kidnapping, or incidence of these diseases

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– 1 – Laboratory 1 Introduction: In this lab you will look at two problems that are at the heart of calculus. Each of these experiments illustrates a core calculus concept. You should perform each experiment taking notes and pictures. You will use these to write up your results. You are expected to use a word processor to produce the laboratory. Graphing software should be used to draw your graphs and illustrations. You can also include pictures you have taken. Equations should be written using “equation editor” software. In short, the laboratory should have a professional look and feel to it. It should be of publishable quality. You report should be printed on 8.5 x 11 inch paper and include a title page (format will be discussed in class). Each page should be numbered. You can work in groups of 3 on this laboratory. If you do this, you must include a page right after the title page and before the report that includes a list of the contributions of each member of the group has made. Question 1 Suppose you start 10 feet away from a wall and walk 5 feet toward the wall and stop. Now walk 2.5 feet toward the wall and stop. Keep going each time walking half the distance of your previous walk toward the wall. 1. Where are you after three walks? 2. Where are you after 2, 3, 4, 5, 10 walks? 3. Create a function where n is the number of the walk and f(n) is the distance from the wall. 4. Graph this function. 5. Using your modeling skills find a model for this function. 6. If you walk forever, were will you end up? For this one write a paragraph defending your location. 7. If instead of walking one half as far as the previous walk, walk one third. That is start 9 feet away from the wall and walk 3 feet, then 1 foot, then 1/3 of a foot, etc. Where do you end up this time? Again write a paragraph. 8. Discuss you experiment in relation Zeno’s Paradox called Achilles and Tortoise. – 2 – Question 2 Here you are going to find the circumference and area of a circle by approximating it with polygons. 1. Start by drawing a circle with radius 3” on a sheet of paper. (You should include your drawings in laboratory report. You should be able to get two per page.) 2. Divide the circle into 3 equal parts. 3. Now connect adjacent points on the circumference to form 3 triangles as shown below. You need to find the area of these isosceles triangles and the length of the bases (red lines). 4. In a table keep track of the following: a. The number of triangles. b. The sum of the lengths of the bases. This is your approximation for the circumference. Label this column, C. c. The sum of the areas of the triangles. This is your approximation for the area of the circle. Label this column , A. d. In a column divide your approximation for the circumference by 2*r. This value should be 6 since r is the radius of your circle is 3. Label this column P1 e. In a column divide your approximation for the area by r2 or 9. Label this column P2. – 3 – 5. Repeat this process for n = 4 … 15 recording your results in the correct columns. 6. Create the two functions described below. You should the graph for each of these functions separately. a. C(n) which associates n to the corresponding approximation of the circumference. b. A(n) which associates n to the corresponding approximation of the area. 7. For the two functions created in step 6 find a model for each function. 8. If we were to continue this experiment — let n grow larger without bound then what values do C and A will approach. Write a paragraph for each variable explaining your reasoning. 9. Then examine the P1 and P2 columns of your table. Write a paragraph on what you if n is allowed to grow larger without bound.

– 1 – Laboratory 1 Introduction: In this lab you will look at two problems that are at the heart of calculus. Each of these experiments illustrates a core calculus concept. You should perform each experiment taking notes and pictures. You will use these to write up your results. You are expected to use a word processor to produce the laboratory. Graphing software should be used to draw your graphs and illustrations. You can also include pictures you have taken. Equations should be written using “equation editor” software. In short, the laboratory should have a professional look and feel to it. It should be of publishable quality. You report should be printed on 8.5 x 11 inch paper and include a title page (format will be discussed in class). Each page should be numbered. You can work in groups of 3 on this laboratory. If you do this, you must include a page right after the title page and before the report that includes a list of the contributions of each member of the group has made. Question 1 Suppose you start 10 feet away from a wall and walk 5 feet toward the wall and stop. Now walk 2.5 feet toward the wall and stop. Keep going each time walking half the distance of your previous walk toward the wall. 1. Where are you after three walks? 2. Where are you after 2, 3, 4, 5, 10 walks? 3. Create a function where n is the number of the walk and f(n) is the distance from the wall. 4. Graph this function. 5. Using your modeling skills find a model for this function. 6. If you walk forever, were will you end up? For this one write a paragraph defending your location. 7. If instead of walking one half as far as the previous walk, walk one third. That is start 9 feet away from the wall and walk 3 feet, then 1 foot, then 1/3 of a foot, etc. Where do you end up this time? Again write a paragraph. 8. Discuss you experiment in relation Zeno’s Paradox called Achilles and Tortoise. – 2 – Question 2 Here you are going to find the circumference and area of a circle by approximating it with polygons. 1. Start by drawing a circle with radius 3” on a sheet of paper. (You should include your drawings in laboratory report. You should be able to get two per page.) 2. Divide the circle into 3 equal parts. 3. Now connect adjacent points on the circumference to form 3 triangles as shown below. You need to find the area of these isosceles triangles and the length of the bases (red lines). 4. In a table keep track of the following: a. The number of triangles. b. The sum of the lengths of the bases. This is your approximation for the circumference. Label this column, C. c. The sum of the areas of the triangles. This is your approximation for the area of the circle. Label this column , A. d. In a column divide your approximation for the circumference by 2*r. This value should be 6 since r is the radius of your circle is 3. Label this column P1 e. In a column divide your approximation for the area by r2 or 9. Label this column P2. – 3 – 5. Repeat this process for n = 4 … 15 recording your results in the correct columns. 6. Create the two functions described below. You should the graph for each of these functions separately. a. C(n) which associates n to the corresponding approximation of the circumference. b. A(n) which associates n to the corresponding approximation of the area. 7. For the two functions created in step 6 find a model for each function. 8. If we were to continue this experiment — let n grow larger without bound then what values do C and A will approach. Write a paragraph for each variable explaining your reasoning. 9. Then examine the P1 and P2 columns of your table. Write a paragraph on what you if n is allowed to grow larger without bound.

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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

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

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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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