(b) Based on the lessons learned, best practices and any additional steps you came up with in part (a), what if project manager X then got a job at Bank of America. Would it be possible for him/her to implement lean in the banking industry based on experience from the previous positions held at the automotive plant and the pharmaceutical company? Please state yes or no and explain the logic clearly for the same. Also, explain the steps that project manager X could take to implement lean at Bank of America (in the service industry) [10 points] You can refer to your class notes and will also have to do research online for both parts (a) and (b). Please state all the references used for each question.

(b) Based on the lessons learned, best practices and any additional steps you came up with in part (a), what if project manager X then got a job at Bank of America. Would it be possible for him/her to implement lean in the banking industry based on experience from the previous positions held at the automotive plant and the pharmaceutical company? Please state yes or no and explain the logic clearly for the same. Also, explain the steps that project manager X could take to implement lean at Bank of America (in the service industry) [10 points] You can refer to your class notes and will also have to do research online for both parts (a) and (b). Please state all the references used for each question.

Yes, lean can be applied to the banking industry.   … Read More...
A cosmetic X marketing department has developed a linear trend equation that can be used to predict annual sales of its popular Hand & Foot cream. Yt = 80,000 + 15t; where Yt = Annual sales; t = 0 corresponds to year 2000 Predict the annual sales for the year 2005 using the equation.

A cosmetic X marketing department has developed a linear trend equation that can be used to predict annual sales of its popular Hand & Foot cream. Yt = 80,000 + 15t; where Yt = Annual sales; t = 0 corresponds to year 2000 Predict the annual sales for the year 2005 using the equation.

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2) _____ Coffee houses frequently serve coffee in a paper cup that has a corrugated paper jacket surrounding the cup. This corrugated jacket: a) Serves to keep the coffee hot. b) Increases the coffee-to-surroundings thermal resistance c) Lowers the temperature where the hand clasps the cup d) All of the above e) Only a and c

2) _____ Coffee houses frequently serve coffee in a paper cup that has a corrugated paper jacket surrounding the cup. This corrugated jacket: a) Serves to keep the coffee hot. b) Increases the coffee-to-surroundings thermal resistance c) Lowers the temperature where the hand clasps the cup d) All of the above e) Only a and c

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Problem 1xc) Stirling Expansion Stroke During the isothermal expansion stroke of the idealized Stirling cycle shown, the right hand piston remains fixed as the left hand piston travels downward. Problem 2xc The steam accumulator shown is used to maintain a ready source of new steam at 300°C. A piston of 63.6 metric tonnes maintains the required pressure, and the top of the cylinder is open to the atmosphere. 1 metric tonne = 1,000 kg The initial cold condition is shown at right. Heat is added to achieve the required end conditions The chamber contains no air. Find the temperature and steam quality when the piston begins to move Find the work done during the process and the final height of the piston at the desired condition of 300°C.

Problem 1xc) Stirling Expansion Stroke During the isothermal expansion stroke of the idealized Stirling cycle shown, the right hand piston remains fixed as the left hand piston travels downward. Problem 2xc The steam accumulator shown is used to maintain a ready source of new steam at 300°C. A piston of 63.6 metric tonnes maintains the required pressure, and the top of the cylinder is open to the atmosphere. 1 metric tonne = 1,000 kg The initial cold condition is shown at right. Heat is added to achieve the required end conditions The chamber contains no air. Find the temperature and steam quality when the piston begins to move Find the work done during the process and the final height of the piston at the desired condition of 300°C.

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1 15325 Pre-work assignment Preparing your conflict scenario (four copies of your scenario must be brought to the workshop) Dear Participant, This letter introduces some pre-course work that is essential for you to complete before arriving at the workshop for the subject Negotiations and Conflict Management: 15325 – in which you are enrolled. The workshop will combine theory and practice in a manner intended to use the wisdom in the room to bring together our thinking about enacting the practices you will learn about. You will bring with you a scenario to work through during the workshop. This letter explains how to write that. 1 The situation (you can give it a title if that helps to frame it for you) Your first task is to identify a situation that is (or in your opinion is) unresolved and has potential to escalate into a matter causing stress, tension, delay or confusion. This may be something at work or in a context where you have the power to take action. You will use fictional names and disguise other facts to ensure confidentiality, but it is essential that this is a real situation – not a hypothetical or fictional one. 2 The Details To enable others to understand the context you will need to describe the following – A The people. Describe each person using the following items – Name – Use a fictional name for each person and do not include more than four others apart from yourself. You can use your own name if you wish or also disguise that as well. General facts about each person – gender, age range, role title, marital status (if relevant) work/life location (if other than yours) Personal characteristics – select at least 5 key words/phrases chosen from the list at the end of this letter Relationship to others in the scenario – boss, subordinate, peer, family member, relative etc. B The context. Type of business or other relevant information to provide a general setting for the moment you will use to describe the unresolved issue. C The event (moment in time). This can be at least partly imagined in that you will need to summarise a lot of information and it might be easier to do so if you write it as conversation even if that has not happened. 2 A sample example written in this way follows. This is a real scenario written by a person who will not be attending the workshop. It took 40 minutes to write. That involved 10 minutes to collect thoughts, select words and frame the setting and then 30 minutes to put it into the words you are reading. The advice is to allow yourself at least this amount of time and also to find a quiet space and time to write your scenario. Example Case Study Title – Where is that space? Setting – a Sydney residential street, in a smallish inner city suburb. There is a main road at one end of the street and a large schoolyard at the other end. At the corner of the street and the main road is a temporary church site whose owners are seeking to extend and develop the site. On the opposite corner is a second hand car yard with the imaginative title of “Junk your Jalopy” (JyJ). Aside from a block of six flats next to the home Eva has lived in for 12 years, all the other residences are single storey homes most built in the first two decades of the 20th century. Most residents have at least one car – often two. Umberto works at JyJ and may be a part owner. He doesn’t live nearby. On a recent occasion Eva, who is reasonably laid back but can be forgetful, was moved to anger by the presence, in the street outside her front door, of a very old and battered panel van that she knew did not belong to any of the residents. It has been there for nearly two weeks and meant that she was parking her car out of sight in a side lane, on land owned by the church. This is not official parking for the street and is often blocked off by the church. Walking to the corner one morning she saw Umberto taking photos of a motorbike and went to raise the issue of the van with him. He is not particularly interested in others’ concerns about the lack of parking and merely wants to make a success of the business. If that means parking extra cars in the street and annoying a few residents he’s opportunistic enough to do so without compunction. Although she is usually fearful of conflict Eva was determined to do something to try and put a stop to JYJ’s habit of parking cars illegally in the residential area. She opened the conversation by asking if Umberto knew anything about the van. He denied all knowledge of it and became quite aggressive (or at least it seemed that way to Eva) about the matter of cars in the street, denying that any were from JyJ, suggesting she talk to the owners of the spare parts yard facing the main road. As Eva tried to ask him to consider the needs and rights of residents, Umberto became ever more inflexible disregarding her issue and suggesting she leave his premises. Although she is quite creative, and has worked for 30 years in a variety of roles Eva is not always able to speak her mind easily, and his denials were not helping. He even began whinging about having to ‘cop the s—t’ for the spare parts yard but resisted the idea of marking his cars so residents could see those parked illegally were not his. 3 As she walked away Eva heard herself say “well if you do nothing about it, then you’ll have to continue copping the s—t, and I hope it hurts”, realising as she did so that she would not be any better off for her efforts. When she got home that night the van was gone – but a different one had arrived within four days. The issue is unresolved. Words to describe the people in your scenario accurate inquisitive empire building adaptable knowledgeable erratic analytical logical fearful of conflict broad in outlook loyal forgetful calm & confident observant frightened of failure caring opportunistic fussy challenging original impatient clever outgoing impulsive competitive outspoken indecisive conscientious perfectionist inflexible conscious of priorities persistent insular consultative persuasive laid back 4 co-operative practical manipulative creative professionally dedicated not interested in others diplomatic Marking Criteria for the Case Study How to get the maximum marks for the case study! For 10 marks – the case study – Accurately uses more than the required number of suggested words to describe the people in the scenario. That is the words used to describe the people are descriptive and placed appropriately to ensure a reader is able to create an informative word picture of each person. The sequence of events is presented in a manner that ensures the current situation, and possible consequences of any future actions, are easily understood by a reader not familiar with the context. Includes enough information to ensure that a stranger does not need to ask additional questions to affirm understanding of the situation as described in the case study. For 8 – 9 marks – the case study – Uses the set minimum number of words. The words are used correctly. The sequence is reasonably ordered, but readers find they need to ask one or two questions about the actual context, order of events. There is less that a sufficient amount of information to ensure that a stranger will quickly understand the nature of issues that remain unresolved. For 5 – 7 – the case study – Uses the set minimum number of words. Not all words are used appropriately in the context, but a stranger is able to gain an impression of the people. The sequence of events – as presented in the case study text – needs some re-ordering in response to questions from other readers to enable them to understand the issues. Strangers will need to seek additional information before they feel able to understand the issue and/or the context. For F = less than 5 – the case study – Uses fewer than the set minimum number of words. They do not add to the information about the people. 5 The sequence of events is unclear and does not represent the issue/s in a manner that can be understood by a stranger. A good deal of additional information is required before a stranger can understand the nature of the issues and context.

1 15325 Pre-work assignment Preparing your conflict scenario (four copies of your scenario must be brought to the workshop) Dear Participant, This letter introduces some pre-course work that is essential for you to complete before arriving at the workshop for the subject Negotiations and Conflict Management: 15325 – in which you are enrolled. The workshop will combine theory and practice in a manner intended to use the wisdom in the room to bring together our thinking about enacting the practices you will learn about. You will bring with you a scenario to work through during the workshop. This letter explains how to write that. 1 The situation (you can give it a title if that helps to frame it for you) Your first task is to identify a situation that is (or in your opinion is) unresolved and has potential to escalate into a matter causing stress, tension, delay or confusion. This may be something at work or in a context where you have the power to take action. You will use fictional names and disguise other facts to ensure confidentiality, but it is essential that this is a real situation – not a hypothetical or fictional one. 2 The Details To enable others to understand the context you will need to describe the following – A The people. Describe each person using the following items – Name – Use a fictional name for each person and do not include more than four others apart from yourself. You can use your own name if you wish or also disguise that as well. General facts about each person – gender, age range, role title, marital status (if relevant) work/life location (if other than yours) Personal characteristics – select at least 5 key words/phrases chosen from the list at the end of this letter Relationship to others in the scenario – boss, subordinate, peer, family member, relative etc. B The context. Type of business or other relevant information to provide a general setting for the moment you will use to describe the unresolved issue. C The event (moment in time). This can be at least partly imagined in that you will need to summarise a lot of information and it might be easier to do so if you write it as conversation even if that has not happened. 2 A sample example written in this way follows. This is a real scenario written by a person who will not be attending the workshop. It took 40 minutes to write. That involved 10 minutes to collect thoughts, select words and frame the setting and then 30 minutes to put it into the words you are reading. The advice is to allow yourself at least this amount of time and also to find a quiet space and time to write your scenario. Example Case Study Title – Where is that space? Setting – a Sydney residential street, in a smallish inner city suburb. There is a main road at one end of the street and a large schoolyard at the other end. At the corner of the street and the main road is a temporary church site whose owners are seeking to extend and develop the site. On the opposite corner is a second hand car yard with the imaginative title of “Junk your Jalopy” (JyJ). Aside from a block of six flats next to the home Eva has lived in for 12 years, all the other residences are single storey homes most built in the first two decades of the 20th century. Most residents have at least one car – often two. Umberto works at JyJ and may be a part owner. He doesn’t live nearby. On a recent occasion Eva, who is reasonably laid back but can be forgetful, was moved to anger by the presence, in the street outside her front door, of a very old and battered panel van that she knew did not belong to any of the residents. It has been there for nearly two weeks and meant that she was parking her car out of sight in a side lane, on land owned by the church. This is not official parking for the street and is often blocked off by the church. Walking to the corner one morning she saw Umberto taking photos of a motorbike and went to raise the issue of the van with him. He is not particularly interested in others’ concerns about the lack of parking and merely wants to make a success of the business. If that means parking extra cars in the street and annoying a few residents he’s opportunistic enough to do so without compunction. Although she is usually fearful of conflict Eva was determined to do something to try and put a stop to JYJ’s habit of parking cars illegally in the residential area. She opened the conversation by asking if Umberto knew anything about the van. He denied all knowledge of it and became quite aggressive (or at least it seemed that way to Eva) about the matter of cars in the street, denying that any were from JyJ, suggesting she talk to the owners of the spare parts yard facing the main road. As Eva tried to ask him to consider the needs and rights of residents, Umberto became ever more inflexible disregarding her issue and suggesting she leave his premises. Although she is quite creative, and has worked for 30 years in a variety of roles Eva is not always able to speak her mind easily, and his denials were not helping. He even began whinging about having to ‘cop the s—t’ for the spare parts yard but resisted the idea of marking his cars so residents could see those parked illegally were not his. 3 As she walked away Eva heard herself say “well if you do nothing about it, then you’ll have to continue copping the s—t, and I hope it hurts”, realising as she did so that she would not be any better off for her efforts. When she got home that night the van was gone – but a different one had arrived within four days. The issue is unresolved. Words to describe the people in your scenario accurate inquisitive empire building adaptable knowledgeable erratic analytical logical fearful of conflict broad in outlook loyal forgetful calm & confident observant frightened of failure caring opportunistic fussy challenging original impatient clever outgoing impulsive competitive outspoken indecisive conscientious perfectionist inflexible conscious of priorities persistent insular consultative persuasive laid back 4 co-operative practical manipulative creative professionally dedicated not interested in others diplomatic Marking Criteria for the Case Study How to get the maximum marks for the case study! For 10 marks – the case study – Accurately uses more than the required number of suggested words to describe the people in the scenario. That is the words used to describe the people are descriptive and placed appropriately to ensure a reader is able to create an informative word picture of each person. The sequence of events is presented in a manner that ensures the current situation, and possible consequences of any future actions, are easily understood by a reader not familiar with the context. Includes enough information to ensure that a stranger does not need to ask additional questions to affirm understanding of the situation as described in the case study. For 8 – 9 marks – the case study – Uses the set minimum number of words. The words are used correctly. The sequence is reasonably ordered, but readers find they need to ask one or two questions about the actual context, order of events. There is less that a sufficient amount of information to ensure that a stranger will quickly understand the nature of issues that remain unresolved. For 5 – 7 – the case study – Uses the set minimum number of words. Not all words are used appropriately in the context, but a stranger is able to gain an impression of the people. The sequence of events – as presented in the case study text – needs some re-ordering in response to questions from other readers to enable them to understand the issues. Strangers will need to seek additional information before they feel able to understand the issue and/or the context. For F = less than 5 – the case study – Uses fewer than the set minimum number of words. They do not add to the information about the people. 5 The sequence of events is unclear and does not represent the issue/s in a manner that can be understood by a stranger. A good deal of additional information is required before a stranger can understand the nature of the issues and context.

(Conflict scenario) Title – Who steal the gold?   Setting: … Read More...
Question 1 In order to properly manage expenses, the company investigates the amount of money spent by its sales office. The below numbers are related to six randomly selected receipts provided by the staff. $147 $124 $93 $158 $164 $171 a) Calculate ̅ , s2 and s for the expense data. b) Assume that the distribution of expenses is approximately normally distributed. Calculate estimates of tolerance intervals containing 68.26 percent, 95.44 percent, and 99.73 percent of all expenses by the sales office. c) If a member of the sales office submits a receipt with the amount of $190, should this expense be considered unusually high? Explain your answer. d) Compute and interpret the z-score for each of the six expenses. Question 2 A survey presents the results of a concept study for the taste of new food. Three hundred consumers between 18 and 49 years old were randomly selected. After sampling the new cuisine, each was asked to rate the quality of food. The rating was made on a scale from 1 to 5, with 5 representing “extremely agree with the quality” and with 1 representing “not at all agree with the new food.” The results obtained are given in Table 1. Estimate the probability that a randomly selected 18- to 49-year-old consumer a) Would give the phrase a rating of 4. b) Would give the phrase a rating of 3 or higher. c) Is in the 18–26 age group; the 27–35 age group; the 36–49 age group. d) Is a male who gives the phrase a rating of 5. e) Is a 36- to 49-year-old who gives the phrase a rating of 2. f) Estimate the probability that a randomly selected 18- to 49-year-old consumer is a 27- to 49-year-old who gives the phrase a rating of 3. g) Estimate the probability that a randomly selected 18- to 49-year-old consumer would 1) give the phrase a rating of 2 or 4 given that the consumer is male; 2) give the phrase a rating of 4 or 5 given that the consumer is female. Based on the results of parts 1 and 2, is the appeal of the phrase among males much different from the appeal of the phrase among females? Explain. h) Give the phrase a rating of 4 or 5, 1) given that the consumer is in the 18–26 age group; 2) given that the consumer is in the 27–35 age group; 3) given that the consumer is in the 36–49 age group. Table 1. Gender Age Group Rating Total Male Female 18-26 27-35 36-49 Extremely Appealing (5) 151 68 83 48 66 37 (4) 91 51 40 36 36 19 (3) 36 21 15 9 12 15 (2) 13 7 6 4 6 3 Not at all appealing(1) 9 3 6 4 3 2 Question 3 Based on the reports provided by the brokers, it is concluded that the annual returns on common stocks are approximately normally distributed with a mean of 17.8 percent and a standard deviation of 29.3 percent. On the other hand, the company reports that the annual returns on tax-free municipal bonds are approximately normally distributed with a mean return of 4.7 percent and a standard deviation of 10.2 percent. Find the probability that a randomly selected a) Common stock will give a positive yearly return. b) Tax-free municipal bond will give a positive yearly return. c) Common stock will give more than a 13 percent return. d) Tax-free municipal bond will give more than a 11.5 percent return. e) Common stock will give a loss of at least 7 percent. f) Tax-free municipal bond will give a loss of at least 10 percent. Question 4 Based on a sample of 176 workers, it is estimated that the mean amount of paid time lost during a three-month period was 1.4 days per employee with a standard deviation of 1.3 days. It is also estimated that the mean amount of unpaid time lost during a three-month period was 1.0 day per employee with a standard deviation of 1.8 days. We randomly select a sample of 100 workers. a) What is the probability that the average amount of paid time lost during a three-month period for the 100 blue-collar workers will exceed 1.5 days? Assume σ equals 1.3 days. b) What is the probability that the average amount of unpaid time lost during a three-month period for the 100 workers will exceed 1.5 days? Assume σ equals 1.8 days. c) A sample of 100 workers is randomly selected. Suppose the sample mean amount of unpaid time lost during a three-month period actually exceeds 1.5 days. Would it be reasonable to conclude that the mean amount of unpaid time lost has increased above the previously estimated 1.0 day? Explain. Assume σ still equals 1.8 days.

Question 1 In order to properly manage expenses, the company investigates the amount of money spent by its sales office. The below numbers are related to six randomly selected receipts provided by the staff. $147 $124 $93 $158 $164 $171 a) Calculate ̅ , s2 and s for the expense data. b) Assume that the distribution of expenses is approximately normally distributed. Calculate estimates of tolerance intervals containing 68.26 percent, 95.44 percent, and 99.73 percent of all expenses by the sales office. c) If a member of the sales office submits a receipt with the amount of $190, should this expense be considered unusually high? Explain your answer. d) Compute and interpret the z-score for each of the six expenses. Question 2 A survey presents the results of a concept study for the taste of new food. Three hundred consumers between 18 and 49 years old were randomly selected. After sampling the new cuisine, each was asked to rate the quality of food. The rating was made on a scale from 1 to 5, with 5 representing “extremely agree with the quality” and with 1 representing “not at all agree with the new food.” The results obtained are given in Table 1. Estimate the probability that a randomly selected 18- to 49-year-old consumer a) Would give the phrase a rating of 4. b) Would give the phrase a rating of 3 or higher. c) Is in the 18–26 age group; the 27–35 age group; the 36–49 age group. d) Is a male who gives the phrase a rating of 5. e) Is a 36- to 49-year-old who gives the phrase a rating of 2. f) Estimate the probability that a randomly selected 18- to 49-year-old consumer is a 27- to 49-year-old who gives the phrase a rating of 3. g) Estimate the probability that a randomly selected 18- to 49-year-old consumer would 1) give the phrase a rating of 2 or 4 given that the consumer is male; 2) give the phrase a rating of 4 or 5 given that the consumer is female. Based on the results of parts 1 and 2, is the appeal of the phrase among males much different from the appeal of the phrase among females? Explain. h) Give the phrase a rating of 4 or 5, 1) given that the consumer is in the 18–26 age group; 2) given that the consumer is in the 27–35 age group; 3) given that the consumer is in the 36–49 age group. Table 1. Gender Age Group Rating Total Male Female 18-26 27-35 36-49 Extremely Appealing (5) 151 68 83 48 66 37 (4) 91 51 40 36 36 19 (3) 36 21 15 9 12 15 (2) 13 7 6 4 6 3 Not at all appealing(1) 9 3 6 4 3 2 Question 3 Based on the reports provided by the brokers, it is concluded that the annual returns on common stocks are approximately normally distributed with a mean of 17.8 percent and a standard deviation of 29.3 percent. On the other hand, the company reports that the annual returns on tax-free municipal bonds are approximately normally distributed with a mean return of 4.7 percent and a standard deviation of 10.2 percent. Find the probability that a randomly selected a) Common stock will give a positive yearly return. b) Tax-free municipal bond will give a positive yearly return. c) Common stock will give more than a 13 percent return. d) Tax-free municipal bond will give more than a 11.5 percent return. e) Common stock will give a loss of at least 7 percent. f) Tax-free municipal bond will give a loss of at least 10 percent. Question 4 Based on a sample of 176 workers, it is estimated that the mean amount of paid time lost during a three-month period was 1.4 days per employee with a standard deviation of 1.3 days. It is also estimated that the mean amount of unpaid time lost during a three-month period was 1.0 day per employee with a standard deviation of 1.8 days. We randomly select a sample of 100 workers. a) What is the probability that the average amount of paid time lost during a three-month period for the 100 blue-collar workers will exceed 1.5 days? Assume σ equals 1.3 days. b) What is the probability that the average amount of unpaid time lost during a three-month period for the 100 workers will exceed 1.5 days? Assume σ equals 1.8 days. c) A sample of 100 workers is randomly selected. Suppose the sample mean amount of unpaid time lost during a three-month period actually exceeds 1.5 days. Would it be reasonable to conclude that the mean amount of unpaid time lost has increased above the previously estimated 1.0 day? Explain. Assume σ still equals 1.8 days.

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MA 3351 – Fall 2015 Homework #3 Due Friday 18 September 1. Find eigenvalues and eigenvectors of the following matrices  1 2 2 4   3 1 1 2   3 0 0 4   1 2 1 3   0 −1 1 0   −2 1 0 1 −2 0 0 0 1   0 1 0 −1 0 0 0 0 1  . Do calculations by hand, though you can use Mathematica to check your results. 2. Find eigenvectors and eigenvalues of A =  2 0 1 1 2 −1 0 0 3  . Show that one of the eigenvalues is defective. Do calculations by hand, though you can use Mathematica to check your results. 3. Solve the initial value problem y′ = Ay, y (0) = y0 for the following cases (a) A =  −4 1 1 −4  y0 =  1 2  (b) A =  −1 1 0 −2  y0 =  −1 3  (c) A =  1 0 0 0 −2 1 0 1 −2  y0 =  1 0 2  Do all calculations by hand. 4. Repeat problem 3 using Mathematica to do all calculations. MORE PROBLEMS ON BACK OF PAGE 1 5. Use Mathematica’s Eigensystem function to find eigenvalues and eigenvectors of A =  −2 1 0 0 0 1 −2 1 0 0 0 1 −2 1 0 0 0 1 −2 1 0 0 0 1 −2  . Suppose you are interested in solutions to y′ = Ay. Without constructing the full solution, answer the following questions: (a) Does the solution grow or decay in time (or a mix of both)? (b) What is the smallest (in magnitude) rate constant? (c) What is the largest (in magnitude) rate constant? (d) As t → ¥, the solution will be dominated by one eigenvector times an exponen- tial. Which eigenvector, and what is the rate constant of the exponential? 6. Use diagonalization to compute (Is − A)−1, where A =  −2 1 0 1 −2 1 0 1 −2  . You may use Mathematica. I suggest running FullSimplify on your result. 2

MA 3351 – Fall 2015 Homework #3 Due Friday 18 September 1. Find eigenvalues and eigenvectors of the following matrices  1 2 2 4   3 1 1 2   3 0 0 4   1 2 1 3   0 −1 1 0   −2 1 0 1 −2 0 0 0 1   0 1 0 −1 0 0 0 0 1  . Do calculations by hand, though you can use Mathematica to check your results. 2. Find eigenvectors and eigenvalues of A =  2 0 1 1 2 −1 0 0 3  . Show that one of the eigenvalues is defective. Do calculations by hand, though you can use Mathematica to check your results. 3. Solve the initial value problem y′ = Ay, y (0) = y0 for the following cases (a) A =  −4 1 1 −4  y0 =  1 2  (b) A =  −1 1 0 −2  y0 =  −1 3  (c) A =  1 0 0 0 −2 1 0 1 −2  y0 =  1 0 2  Do all calculations by hand. 4. Repeat problem 3 using Mathematica to do all calculations. MORE PROBLEMS ON BACK OF PAGE 1 5. Use Mathematica’s Eigensystem function to find eigenvalues and eigenvectors of A =  −2 1 0 0 0 1 −2 1 0 0 0 1 −2 1 0 0 0 1 −2 1 0 0 0 1 −2  . Suppose you are interested in solutions to y′ = Ay. Without constructing the full solution, answer the following questions: (a) Does the solution grow or decay in time (or a mix of both)? (b) What is the smallest (in magnitude) rate constant? (c) What is the largest (in magnitude) rate constant? (d) As t → ¥, the solution will be dominated by one eigenvector times an exponen- tial. Which eigenvector, and what is the rate constant of the exponential? 6. Use diagonalization to compute (Is − A)−1, where A =  −2 1 0 1 −2 1 0 1 −2  . You may use Mathematica. I suggest running FullSimplify on your result. 2

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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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2. According to the video: Semmelweis and Pasteur: Germ Theory (5 points) a) What is childbed fever? infection of womb and cases septicimea. b) What was Semmelweis’s hypothesis as to why more women were dying of childbed fever under the doctor’s care rather than the midwives? chlorine hand wash. c) Why were Semmelweis’s ideas not accepted in 1846? no scientific reasoning. d) Why was Pasteur so concerned with finding out about disease transmission? transmission is to be studied to identify the cure or prevention. e) What industry was he working for? microbiology. f) Pasteur’s experiment led to what theory? Describe this theory. transmission principle. dna is responsible for transmission not the protein. g) Your thoughts: Can all diseases be explained by the germ theory? no, non infectious diseases cannot come under this theory.

2. According to the video: Semmelweis and Pasteur: Germ Theory (5 points) a) What is childbed fever? infection of womb and cases septicimea. b) What was Semmelweis’s hypothesis as to why more women were dying of childbed fever under the doctor’s care rather than the midwives? chlorine hand wash. c) Why were Semmelweis’s ideas not accepted in 1846? no scientific reasoning. d) Why was Pasteur so concerned with finding out about disease transmission? transmission is to be studied to identify the cure or prevention. e) What industry was he working for? microbiology. f) Pasteur’s experiment led to what theory? Describe this theory. transmission principle. dna is responsible for transmission not the protein. g) Your thoughts: Can all diseases be explained by the germ theory? no, non infectious diseases cannot come under this theory.

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