Book review The Shareholder Value Myth: How Putting Shareholders First Harms Investors, Corporations, and the Public by Lynn Stout Provide 1) a 900 word review of this book (word range 900-1,200) and 2) a 350 word reflection where you force yourself to relate the message of the book . As per the format of the review, I like the ones done by the folks of the WSJ. This is an example: http://forums.delphiforums.com/diversecity/messages?msg=17531.1264 or http://www.wsj.com/articles/book-review-how-adam-smith-can-change-your-life-by-russ-roberts-1413846808?KEYWORDS=book+reviews

Book review The Shareholder Value Myth: How Putting Shareholders First Harms Investors, Corporations, and the Public by Lynn Stout Provide 1) a 900 word review of this book (word range 900-1,200) and 2) a 350 word reflection where you force yourself to relate the message of the book . As per the format of the review, I like the ones done by the folks of the WSJ. This is an example: http://forums.delphiforums.com/diversecity/messages?msg=17531.1264 or http://www.wsj.com/articles/book-review-how-adam-smith-can-change-your-life-by-russ-roberts-1413846808?KEYWORDS=book+reviews

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According to the object relations theorist D. W. Winnicott, the purpose of a transitional object is to ________. Question 6 options: compensate for an inferiority complex defend the superego from unconscious urges protect the true self by keeping it invisible compensate for the fact that we cannot have our loved ones with us all of the time

According to the object relations theorist D. W. Winnicott, the purpose of a transitional object is to ________. Question 6 options: compensate for an inferiority complex defend the superego from unconscious urges protect the true self by keeping it invisible compensate for the fact that we cannot have our loved ones with us all of the time

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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...
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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Tornado Eddy Investigation Abstract The objective of this lab was to write a bunch of jibberish to provide students with a formatting template. Chemical engineering, bioengineering, and environmental engineering are “process engineering” disciplines. Good abstracts contains real content, such as 560 mL/min, 35 deg, and 67 percent yield. Ideal degreed graduates are technically strong, bring broad system perspectives to problem solving, and have the professional “soft skills” to make immediate contributions in the workplace. The senior lab sequence is the “capstone” opportunity to realize this ideal by integrating technical skills and developing professional soft skills to ensure workforce preparedness. The best conclusions are objective and numerical, such as operating conditions of 45 L/min at 32 deg C with expected costs of $4.55/lb. Background Insect exchange processes are often used in bug filtration, as they are effective at removing either positive or negative insects from water. An insect exchange column is a packed or fluidized bed filled with resin beads. Water flows through the column and most of the insects from the water enter the beads, but some of them pass in between the beads, which makes the exchange of insects non-ideal. Insectac 249 resin is a cation exchange resin, as it is being used to attract cationic Ca2+ from the toxic waste stream. This means the resin is negatively charged, and needs to be regenerated with a solution that produces positively charged insects, in this case, salt water which contains Na+ insects. The resin contains acidic styrene backbones which capture the cationic insects in a reversible process. A curve of Ca2+ concentration concentration vs. time was obtained after a standard curve was made to determine how many drops from the low cost barium test kit from Aquarium Pharmaceuticals (API)1 bottle #2 would correspond to a certain concentration in solution. A standard curve works by preparing solutions with known concentrations and testing these concentrations using the kit to create a curve of number of drops from bottle #2 (obtained result) vs. concentration of Ca2+ in solution (desired response). The standard curve can then be used for every test on the prototype and in the field, to quickly and accurately obtain a concentration from the test kit. The barium concentration vs. time curve can be used to calculate the exchange capacity of the resin and, in later tests, the regeneration efficiency. The curves must be used to get the total amount of barium removed from the water, m. Seen in Equation 2, the volumetric flow rate of water, , is multiplied by the integral from tinitial to tfinal of the total concentration of Ca2+ absorbed by the resin as a function of time, C. (2) 1 http://aquariumpharm.com/Products/Product.aspx?ProductID=72 , date accessed: 11/26/10 CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 9 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE A graphical trapezoid method was used to evaluate the integral and get the final solution in equivalents of Ca2+ per L, it must be noted that there are 2 equivalents per mole of barium, as the charge of the barium insect is +2. An initial exchange capacity was calculated for the virgin resin, and an adjusted exchange capacity was calculated once the resin was regenerated. The regenerated resin capacity was found by multiplying the virgin resin capacity by the regeneration efficiency, expressed in Equation 3. (3) See Appendix A for the calculation of the exchange capacities and the regeneration efficiency. Materials and Methods Rosalie and Peter Johnson of Corvallis established the Linus Pauling Chair in Chemical Engineering to honor Oregon State University’s most famous graduate. Peter Johnson, former President and owner of Tekmax, Inc., a company which revolutionized battery manufacturing equipment, is a 1955 graduate of the College of Engineering.2 The Chair, also known as the Linus Pauling Distinguished Engineer or Linus Pauling Engineer (LPE), was originally designed to focus on the traditional “capstone” senior lab sequence in the former Department of Chemical Engineering. The focus is now extended to all the process engineering disciplines. The LPE is charged with establishing strong ties with industry, ensuring current and relevant laboratory experiences, and helping upperclass students develop skills in communication, teamwork, project management, and leadership. Include details about lab procedures not sufficiently detailed in the SOP, problems you had, etc. The bulk solution prepared to create the standard curve was used in the second day of testing to obtain the exchange capacity of the insectac 249 resin. The solution was pumped through a bathroom scale into the prototype insect exchange column. 45 mL of resin was rinsed and added to the column. The bed was fluidized as the solution was pumped through the resin, but for the creation of the Ca2+ concentration vs. time curve, the solution was pumped down through the column, as illustrated in the process flow diagram seen in Figure 1. Figure 1. Process sketch of the insect exchange column used for the project. Ref: http://www.generon.co.uk/acatalog/Chromatography.html 2 Harding, P. Viscosity Measurement SOP, Spring, 2010. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 10 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE A bathroom scale calibration curve was created to ensure that the 150 mL/min, used to calculate the breakthrough time, would be delivered to the resin. The bathroom scale used was a Dwyer brand with flowrates between 0 and 300 cc/min of water. Originally, values between 120 and 180 mL/min were chosen for the calibration, with three runs for each flowrate, however the bathroom scale values were so far away from the measure values the range was extended to 100 to 200 mL/min. The regeneration experiment was performed using a method similar to that used in the water softening experiment, however instead of using a 640 ppm Ca2+ solution to fill the resin, a 6000 ppm Na+ solution was used to eject the Ca2+ from the resin. Twelve samples times were chosen and adjusted as the experiment progressed, with more than half of the samples taken at times less than 10 minutes, and the last sample taken at 45 minutes. The bulk exit solution was also tested to determine the regeneration efficiency. Results and Discussion The senior lab sequence has its roots in the former Department of Chemical Engineering. CHE 414 and 415 were taught in Winter and Spring and included 6 hours of lab time per week. The School has endeavored to incorporate the courses into the BIOE and ENVE curriculum, and this will be complete in 2008-2009. Recent development of the senior lab course sequence is shown chronologically in Fig. 1. In 2006-2007, CHE 414 and 415 were moved to Fall and Winter to enable CHE 416, an elective independent senior project course. Also that year, BIOE students took BIOE 414 in the Fall and BIOE 415 was developed and taught. No BIOE students enrolled in the optional CHE. In 2007-2008, the program transitioned in a new Linus Pauling Engineer and ENVE 414 was offered. Also, approximately 30 percent of BIOE students enrolled in the optional CHE 416. Accommodating the academic calendars of the three disciplines required a reduction in weekly student lab time from 6 to 3 hours. The expected relationship between coughing rate, y, and length of canine, x, is Bx z y Fe− (1) where F is a pre-exponential constant, B is vitamin B concentration and z is the height of an average trapeze artist. 3 The 2008-2009 brings the challenge of the dramatic enrollment increase shown in Fig. 1 and the first offering of ENVE 415. The result, shown on the right in Fig. 1, is the delivery of the senior lab sequence uniformly across the process engineering disciplines. CBEE 416 is expected to drawn approximately of the students that take the 415 courses. In 2007-2008, 414 and 415 were required for CHEs, 414 and 415 for BIOEs, and only 414 for ENVEs. CHE 416 is ostensibly an elective for all disciplines. In 2008-2009, 414 and 415 is required for all disciplines and CHE 416 will be an elective. The content of 414 is essentially 3 Fundamentals of Momentum, Heat, and Mass Transfer, Welty, J.R. et al., 4th edition, John Wiley & Sons, Inc. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 11 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE identical for all three disciplines, 415 has discipline-specific labs, and 416 consists of senior projects with potentially cross-discipline teams of 2 to 4 students. Tremendous labor and struggling with the lab equipment resulted in the data shown in y = –‐0.29x + 1.71 y = –‐0.25x + 2.03 y = –‐0.135x + 2.20 –‐1.5 –‐1.0 –‐0.5 0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 ln y (units) x (units) ln y_1 ln y_2 ln y_3 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Case 1 Case 2 Case 3 Slope (units) (a) (b) Figure 1. (a) Data for y and x plotted for various values of z and (b) a comparison of slopes for the 3 cases investigate. The log plot slope yields the vitamin B concentration. The slopes were shown to be significantly at the 90% confidence level, but the instructor ran out of time and did not include error bars. The slope changed as predicted by the Snirtenhoffer equation. Improvements to the lab might include advice on how to legally change my name to something less embarrassing. My whole life I have been forced to repeat and spell it. I really feel that this has affected my psychologically. This was perhaps the worst lab I have ever done in my academic career, primarily due to the fact that there was no lab time. I simply typed in this entire report and filled it with jibberish. Some might think nobody will notice, but I know that …… Harding reads every word. Acknowledgments The author acknowledges his elementary teacher for providing truly foundational instruction in addition and subtraction. Jenny Burninbalm was instrumental with guidance on use of the RT-345 dog scratching device. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 12

Tornado Eddy Investigation Abstract The objective of this lab was to write a bunch of jibberish to provide students with a formatting template. Chemical engineering, bioengineering, and environmental engineering are “process engineering” disciplines. Good abstracts contains real content, such as 560 mL/min, 35 deg, and 67 percent yield. Ideal degreed graduates are technically strong, bring broad system perspectives to problem solving, and have the professional “soft skills” to make immediate contributions in the workplace. The senior lab sequence is the “capstone” opportunity to realize this ideal by integrating technical skills and developing professional soft skills to ensure workforce preparedness. The best conclusions are objective and numerical, such as operating conditions of 45 L/min at 32 deg C with expected costs of $4.55/lb. Background Insect exchange processes are often used in bug filtration, as they are effective at removing either positive or negative insects from water. An insect exchange column is a packed or fluidized bed filled with resin beads. Water flows through the column and most of the insects from the water enter the beads, but some of them pass in between the beads, which makes the exchange of insects non-ideal. Insectac 249 resin is a cation exchange resin, as it is being used to attract cationic Ca2+ from the toxic waste stream. This means the resin is negatively charged, and needs to be regenerated with a solution that produces positively charged insects, in this case, salt water which contains Na+ insects. The resin contains acidic styrene backbones which capture the cationic insects in a reversible process. A curve of Ca2+ concentration concentration vs. time was obtained after a standard curve was made to determine how many drops from the low cost barium test kit from Aquarium Pharmaceuticals (API)1 bottle #2 would correspond to a certain concentration in solution. A standard curve works by preparing solutions with known concentrations and testing these concentrations using the kit to create a curve of number of drops from bottle #2 (obtained result) vs. concentration of Ca2+ in solution (desired response). The standard curve can then be used for every test on the prototype and in the field, to quickly and accurately obtain a concentration from the test kit. The barium concentration vs. time curve can be used to calculate the exchange capacity of the resin and, in later tests, the regeneration efficiency. The curves must be used to get the total amount of barium removed from the water, m. Seen in Equation 2, the volumetric flow rate of water, , is multiplied by the integral from tinitial to tfinal of the total concentration of Ca2+ absorbed by the resin as a function of time, C. (2) 1 http://aquariumpharm.com/Products/Product.aspx?ProductID=72 , date accessed: 11/26/10 CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 9 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE A graphical trapezoid method was used to evaluate the integral and get the final solution in equivalents of Ca2+ per L, it must be noted that there are 2 equivalents per mole of barium, as the charge of the barium insect is +2. An initial exchange capacity was calculated for the virgin resin, and an adjusted exchange capacity was calculated once the resin was regenerated. The regenerated resin capacity was found by multiplying the virgin resin capacity by the regeneration efficiency, expressed in Equation 3. (3) See Appendix A for the calculation of the exchange capacities and the regeneration efficiency. Materials and Methods Rosalie and Peter Johnson of Corvallis established the Linus Pauling Chair in Chemical Engineering to honor Oregon State University’s most famous graduate. Peter Johnson, former President and owner of Tekmax, Inc., a company which revolutionized battery manufacturing equipment, is a 1955 graduate of the College of Engineering.2 The Chair, also known as the Linus Pauling Distinguished Engineer or Linus Pauling Engineer (LPE), was originally designed to focus on the traditional “capstone” senior lab sequence in the former Department of Chemical Engineering. The focus is now extended to all the process engineering disciplines. The LPE is charged with establishing strong ties with industry, ensuring current and relevant laboratory experiences, and helping upperclass students develop skills in communication, teamwork, project management, and leadership. Include details about lab procedures not sufficiently detailed in the SOP, problems you had, etc. The bulk solution prepared to create the standard curve was used in the second day of testing to obtain the exchange capacity of the insectac 249 resin. The solution was pumped through a bathroom scale into the prototype insect exchange column. 45 mL of resin was rinsed and added to the column. The bed was fluidized as the solution was pumped through the resin, but for the creation of the Ca2+ concentration vs. time curve, the solution was pumped down through the column, as illustrated in the process flow diagram seen in Figure 1. Figure 1. Process sketch of the insect exchange column used for the project. Ref: http://www.generon.co.uk/acatalog/Chromatography.html 2 Harding, P. Viscosity Measurement SOP, Spring, 2010. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 10 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE A bathroom scale calibration curve was created to ensure that the 150 mL/min, used to calculate the breakthrough time, would be delivered to the resin. The bathroom scale used was a Dwyer brand with flowrates between 0 and 300 cc/min of water. Originally, values between 120 and 180 mL/min were chosen for the calibration, with three runs for each flowrate, however the bathroom scale values were so far away from the measure values the range was extended to 100 to 200 mL/min. The regeneration experiment was performed using a method similar to that used in the water softening experiment, however instead of using a 640 ppm Ca2+ solution to fill the resin, a 6000 ppm Na+ solution was used to eject the Ca2+ from the resin. Twelve samples times were chosen and adjusted as the experiment progressed, with more than half of the samples taken at times less than 10 minutes, and the last sample taken at 45 minutes. The bulk exit solution was also tested to determine the regeneration efficiency. Results and Discussion The senior lab sequence has its roots in the former Department of Chemical Engineering. CHE 414 and 415 were taught in Winter and Spring and included 6 hours of lab time per week. The School has endeavored to incorporate the courses into the BIOE and ENVE curriculum, and this will be complete in 2008-2009. Recent development of the senior lab course sequence is shown chronologically in Fig. 1. In 2006-2007, CHE 414 and 415 were moved to Fall and Winter to enable CHE 416, an elective independent senior project course. Also that year, BIOE students took BIOE 414 in the Fall and BIOE 415 was developed and taught. No BIOE students enrolled in the optional CHE. In 2007-2008, the program transitioned in a new Linus Pauling Engineer and ENVE 414 was offered. Also, approximately 30 percent of BIOE students enrolled in the optional CHE 416. Accommodating the academic calendars of the three disciplines required a reduction in weekly student lab time from 6 to 3 hours. The expected relationship between coughing rate, y, and length of canine, x, is Bx z y Fe− (1) where F is a pre-exponential constant, B is vitamin B concentration and z is the height of an average trapeze artist. 3 The 2008-2009 brings the challenge of the dramatic enrollment increase shown in Fig. 1 and the first offering of ENVE 415. The result, shown on the right in Fig. 1, is the delivery of the senior lab sequence uniformly across the process engineering disciplines. CBEE 416 is expected to drawn approximately of the students that take the 415 courses. In 2007-2008, 414 and 415 were required for CHEs, 414 and 415 for BIOEs, and only 414 for ENVEs. CHE 416 is ostensibly an elective for all disciplines. In 2008-2009, 414 and 415 is required for all disciplines and CHE 416 will be an elective. The content of 414 is essentially 3 Fundamentals of Momentum, Heat, and Mass Transfer, Welty, J.R. et al., 4th edition, John Wiley & Sons, Inc. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 11 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE identical for all three disciplines, 415 has discipline-specific labs, and 416 consists of senior projects with potentially cross-discipline teams of 2 to 4 students. Tremendous labor and struggling with the lab equipment resulted in the data shown in y = –‐0.29x + 1.71 y = –‐0.25x + 2.03 y = –‐0.135x + 2.20 –‐1.5 –‐1.0 –‐0.5 0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 ln y (units) x (units) ln y_1 ln y_2 ln y_3 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Case 1 Case 2 Case 3 Slope (units) (a) (b) Figure 1. (a) Data for y and x plotted for various values of z and (b) a comparison of slopes for the 3 cases investigate. The log plot slope yields the vitamin B concentration. The slopes were shown to be significantly at the 90% confidence level, but the instructor ran out of time and did not include error bars. The slope changed as predicted by the Snirtenhoffer equation. Improvements to the lab might include advice on how to legally change my name to something less embarrassing. My whole life I have been forced to repeat and spell it. I really feel that this has affected my psychologically. This was perhaps the worst lab I have ever done in my academic career, primarily due to the fact that there was no lab time. I simply typed in this entire report and filled it with jibberish. Some might think nobody will notice, but I know that …… Harding reads every word. Acknowledgments The author acknowledges his elementary teacher for providing truly foundational instruction in addition and subtraction. Jenny Burninbalm was instrumental with guidance on use of the RT-345 dog scratching device. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 12

For any additional help, please contact: info@checkyourstudy.com Call / Whatsapp … Read More...
2. Career development process is complex and rapidly evolving and new theories are continually developing presenting challenges to traditional understandings. Discuss why an understanding of career development processes is critical to management, employee and organizational success.

2. Career development process is complex and rapidly evolving and new theories are continually developing presenting challenges to traditional understandings. Discuss why an understanding of career development processes is critical to management, employee and organizational success.

Studies are at the present extrapolative huge employment income in … Read More...
5 { GRAVITATION Last Updated: July 16, 2012 Problem List 5.1 Total mass of a shell 5.2 Tunnel through the moon 5.3 Gravitational eld above the center of a thin hoop 5.4 Gravitational force near a metal-cored planet surrounded by a gaseous cloud 5.5 Sphere with linearly increasing mass density 5.6 Jumping o Vesta 5.7 Gravitational force between two massive rods 5.8 Potential energy { Check your answer! 5.9 Ways of solving gravitational problems 5.10 Rod with linearly increasing mass density 5.11 Sphere with constant internal gravitational eld 5.12 Throwing a rock o the moon These problems are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Un- ported License. Please share and/or modify. Back to Problem List 1 5 { GRAVITATION Last Updated: July 16, 2012 5.1 Total mass of a shell Given: Marino { Fall 2011 Consider a spherical shell that extends from r = R to r = 2R with a non-uniform density (r) = 0r. What is the total mass of the shell? Back to Problem List 2 5 { GRAVITATION Last Updated: July 16, 2012 5.2 Tunnel through the moon Given: Marino { Fall 2011 Imagine that NASA digs a straight tunnel through the center of the moon (see gure) to access the Moon’s 3He deposits. An astronaut places a rock in the tunnel at the surface of the moon, and releases it (from rest). Show that the rock obeys the force law for a mass connected to a spring. What is the spring constant? Find the oscillation period for this motion if you assume that Moon has a mass of 7.351022 kg and a radius of 1.74106 m. Assume the moon’s density is uniform throughout its volume, and ignore the moon’s rotation. Given: Pollock { Spring 2011 Imagine (in a parallel universe of unlimited budgets) that NASA digs a straight tunnel through the center of the moon (see gure). A robot place a rock in the tunnel at position r = r0 from the center of the moon, and releases it (from rest). Use Newton’s second law to write the equation of motion of the rock and solve for r(t). Explain in words the rock’s motion. Does the rock return to its initial position at any later time? If so, how long does it takes to return to it? (Give a formula, and a number.) Assume the moon’s density is uniform throughout its volume, and ignore the moon’s rotation. Given: Pollock { Spring 2012 Now lets consider our (real) planet Earth, with total mass M and radius R which we will approximate as a uniform mass density, (r) = 0. (a) Neglecting rotational and frictional e ects, show that a particle dropped into a hole drilled straight through the center of the earth all the way to the far side will oscillate between the two endpoints. (Hint: you will need to set up, and solve, an ODE for the motion) (b) Find the period of the oscillation of this motion. Get a number (in minutes) as a nal result, using data for the earth’s size and mass. (How does that compare to ying to Perth and back?!) Extra Credit: OK, even with unlimited budgets, digging a tunnel through the center of the earth is preposterous. But, suppose instead that the tunnel is a straight-line \chord” through the earth, say directly from New York to Los Angeles. Show that your nal answer for the time taken does not depend on the location of that chord! This is rather remarkable – look again at the time for a free-fall trip (no energy required, except perhaps to compensate for friction) How long would that trip take? Could this work?! Back to Problem List 3 5 { GRAVITATION Last Updated: July 16, 2012 5.3 Gravitational eld above the center of a thin hoop Given: Pollock { Spring 2011, Spring 2012 Consider a very (in nitesimally!) thin but massive loop, radius R (total mass M), centered around the origin, sitting in the x-y plane. Assume it has a uniform linear mass density  (which has units of kg/m) all around it. (So, it’s like a skinny donut that is mostly hole, centered around the z-axis) (a) What is  in terms of M and R? What is the direction of the gravitational eld generated by this mass distribution at a point in space a distance z above the center of the donut, i.e. at (0; 0; z) Explain your reasoning for the direction carefully, try not to simply \wave your hands.” (The answer is extremely intuitive, but can you justify that it is correct?) (b) Compute the gravitational eld, ~g, at the point (0; 0; z) by directly integrating Newton’s law of gravity, summing over all in nitesimal \chunks” of mass along the loop. (c) Compute the gravitational potential at the point (0; 0; z) by directly integrating ?Gdm=r, sum- ming over all in nitesimal \chunks” dm along the loop. Then, take the z-component of the gradient of this potential to check that you agree with your result from the previous part. (d) In the two separate limits z << R and z >> R, Taylor expand your g- eld (in the z-direction)out only to the rst non-zero term, and convince us that both limits make good physical sense. (e) Can you use Gauss’ law to gure out the gravitational potential at the point (0; 0; z)? (If so, do it and check your previous answers. If not, why not?) Extra credit: If you place a small mass a small distance z away from the center, use your Taylor limit for z << R above to write a simple ODE for the equation of motion. Solve it, and discuss the motion Back to Problem List 4 5 { GRAVITATION Last Updated: July 16, 2012 5.4 Gravitational force near a metal-cored planet surrounded by a gaseous cloud Given: Pollock { Spring 2011 Jupiter is composed of a dense spherical core (of liquid metallic hydrogen!) of radius Rc. It is sur- rounded by a spherical cloud of gaseous hydrogen of radius Rg, where Rg > Rc. Let’s assume that the core is of uniform density c and the gaseous cloud is also of uniform density g. What is the gravitational force on an object of mass m that is located at a radius r from the center of Jupiter? Note that you must consider the cases where the object is inside the core, within the gas layer, and outside of the planet. Back to Problem List 5 5 { GRAVITATION Last Updated: July 16, 2012 5.5 Sphere with linearly increasing mass density Given: Pollock { Spring 2011 A planet of mass M and radius R has a nonuniform density that varies with r, the distance from the center according to  = Ar for 0  r  R. (a) What is the constant A in terms of M and R? Does this density pro le strike you as physically plausible, or is just designed as a mathematical exercise? (Brie y, explain) (b) Determine the gravitational force on a satellite of mass m orbiting this planet. In words, please outline the method you plan to use for your solution. (Use the easiest method you can come up with!) In your calculation, you will need to argue that the magnitude of ~g(r; ; ) depends only on r. Be very explicit about this – how do you know that it doesn’t, in fact, depend on  or ? (c) Determine the gravitational force felt by a rock of mass m inside the planet, located at radius r < R. (If the method you use is di erent than in part b, explain why you switched. If not, just proceed!) Explicitly check your result for this part by considering the limits r ! 0 and r ! R. Back to Problem List 6 5 { GRAVITATION Last Updated: July 16, 2012 5.6 Jumping o Vesta Given: Pollock { Spring 2011 You are stranded on the surface of the asteroid Vesta. If the mass of the asteroid is M and its radius is R, how fast would you have to jump o its surface to be able to escape from its gravitational eld? (Your estimate should be based on parameters that characterize the asteroid, not parameters that describe your jumping ability.) Given your formula, look up the approximate mass and radius of the asteroid Vesta 3 and determine a numerical value of the escape velocity. Could you escape in this way? (Brie y, explain) If so, roughly how big in radius is the maximum the asteroid could be, for you to still escape this way? If not, estimate how much smaller an asteroid you would need, to escape from it in this way? Figure 1: Back to Problem List 7 5 { GRAVITATION Last Updated: July 16, 2012 5.7 Gravitational force between two massive rods Given: Pollock { Spring 2011 Consider two identical uniform rods of length L and mass m lying along the same line and having their closest points separated by a distance d as shown in the gure (a) Calculate the mutual force between these rods, both its direction and magnitude. (b) Now do several checks. First, make sure the units worked out (!) The, nd the magnitude of the force in the limit L ! 0. What do you expect? Brie y, discuss. Lastly, nd the magnitude of the force in the limit d ! 1 ? Again, is it what you expect? Brie y, discuss. Figure 2: Given: Pollock { Spring 2012 Determining the gravitational force between two rods: (a) Consider a thin, uniform rod of mass m and length L (and negligible other dimensions) lying on the x axis (from x=-L to 0), as shown in g 1a. Derive a formula for the gravitational eld \g" at any arbitrary point x to the right of the origin (but still on the x-axis!) due to this rod. (b) Now suppose a second rod of length L and mass m sits on the x axis as shown in g 1b, with the left edge a distance \d" away. Calculate the mutual gravitational force between these rods. (c) Let's do some checks! Show that the units work out in parts a and b. Find the magnitude of the force in part a, in the limit x >> L: What do you expect? Brie y, discuss! Finally, verify that your answer to part b gives what you expect in the limit d >> L. ( Hint: This is a bit harder! You need to consistently expand everything to second order, not just rst, because of some interesting cancellations) Fig 1a Fig 1b L m +x x=0 L x=0 x=d m Fig 1a Fig 1b L m +x x=0 L +x x=0 x=d L m m Back to Problem List 8 5 { GRAVITATION Last Updated: July 16, 2012 5.8 Potential energy { Check your answer! Given: Pollock { Spring 2011 On the last exam, we had a problem with a at ring, uniform mass per unit area of , inner radius of R, outer radius of 2R. A satellite (mass m) sat a distance z above the center of the ring. We asked for the gravitational potential energy, and the answer was U(z) = ?2Gm( p 4R2 + z2 ? p R2 + z2) (1) (a) If you are far from the disk (on the z axis), what do you expect for the formula for U(z)? (Don’t say \0″ – as usual, we want the functional form of U(z) as you move far away. Also, explicitly state what we mean by \far away”. (Please don’t compare something with units to something without units!) (b) Show explicitly that the formula above does indeed give precisely the functional dependence you expect. Back to Problem List 9 5 { GRAVITATION Last Updated: July 16, 2012 5.9 Ways of solving gravitational problems Given: Pollock { Spring 2011, Spring 2012 Infinite cylinder ρ=cr x z (a) Half-infinite line mass, uniform linear mass density, λ x (b) R z  P Figure 3: (a) An in nite cylinder of radius R centered on the z-axis, with non-uniform volume mass density  = cr, where r is the radius in cylindrical coordinates. (b) A half-in nite line of mass on the x-axis extending from x = 0 to x = +1, with uniform linear mass density . There are two general methods we use to solve gravitational problems (i.e. nd ~g given some distribution of mass). (a) Describe these two methods. We claim one of these methods is easiest to solve for ~g of mass distribution (a) above, and the other method is easiest to solve for ~g of the mass distribution (b) above. Which method goes with which mass distribution? Please justify your answer. (b) Find ~g of the mass distribution (a) above for any arbitrary point outside the cylinder. (c) Find the x component of the gravitational acceleration, gx, generated by the mass distribution labeled (b) above, at a point P a given distance z up the positive z-axis (as shown). Back to Problem List 10 5 { GRAVITATION Last Updated: July 16, 2012 5.10 Rod with linearly increasing mass density Given: Pollock { Spring 2012 Consider a very (in nitesimally!) thin but massive rod, length L (total mass M), centered around the origin, sitting along the x-axis. (So the left end is at (-L/2, 0,0) and the right end is at (+L/2,0,0) Assume the mass density  (which has units of kg/m)is not uniform, but instead varies linearly with distance from the origin, (x) = cjxj. (a) What is that constant \c” in terms of M and L? What is the direction of the gravitational eld generated by this mass distribution at a point in space a distance z above the center of the rod, i.e. at (0; 0; z) Explain your reasoning for the direction carefully, try not to simply \wave your hands.” (The answer is extremely intuitive, but can you justify that it is correct?) (b) Compute the gravitational eld, ~g, at the point (0; 0; z) by directly integrating Newton’s law of gravity, summing over all in nitesimal \chunks” of mass along the rod. (c) Compute the gravitational potential at the point (0; 0; z) by directly integrating ?Gdm=r, sum- ming over all in nitesimal \chunks” dm along the rod. Then, take the z-component of the gradient of this potential to check that you agree with your result from the previous part. (d) In the limit of large z what do you expect for the functional form for gravitational potential? (Hint: Don’t just say it goes to zero! It’s a rod of mass M, when you’re far away what does it look like? How does it go to zero?) What does \large z” mean here? Use the binomial (or Taylor) expansion to verify that your formula does indeed give exactly what you expect. (Hint: you cannot Taylor expand in something BIG, you have to Taylor expand in something small.) (e) Can you use Gauss’ law to gure out the gravitational potential at the point (0; 0; z)? (If so, do it and check your previous answers. If not, why not?) Back to Problem List 11 5 { GRAVITATION Last Updated: July 16, 2012 5.11 Sphere with constant internal gravitational eld Given: Pollock { Spring 2012 (a) Imagine a planet of total mass M and radius R which has a nonuniform mass density that varies just with r, the distance from the center. For this (admittedly very unusual!) planet, suppose the gravitational eld strength inside the planet turns out to be independent of the radial distance within the sphere. Find the function describing the mass density  = (r) of this planet. (Your nal answer should be written in terms of the given constants.) (b) Now, determine the gravitational force on a satellite of mass m orbiting this planet at distance r > R. (Use the easiest method you can come up with!) Explain your work in words as well as formulas. For instance, in your calculation, you will need to argue that the magnitude of ~g(r; ; ) depends only on r. Be explicit about this – how do you know that it doesn’t, in fact, depend on  or ? (c) As a nal check, explicitly show that your solutions inside and outside the planet (parts a and b) are consistent when r = R. Please also comment on whether this density pro le strikes you as physically plausible, or is it just designed as a mathematical exercise? Defend your reasoning. Back to Problem List 12 5 { GRAVITATION Last Updated: July 16, 2012 5.12 Throwing a rock o the moon Given: Pollock { Spring 2012 Assuming that asteroids have roughly the same mass density as the moon, make an estimate of the largest asteroid that an astronaut could be standing on, and still have a chance of throwing a small object (with their arms, no machinery!) so that it completely escapes the asteroid’s gravitational eld. (This minimum speed is called \escape velocity”) Is the size you computed typical for asteroids in our solar system? Back to Problem List 13

5 { GRAVITATION Last Updated: July 16, 2012 Problem List 5.1 Total mass of a shell 5.2 Tunnel through the moon 5.3 Gravitational eld above the center of a thin hoop 5.4 Gravitational force near a metal-cored planet surrounded by a gaseous cloud 5.5 Sphere with linearly increasing mass density 5.6 Jumping o Vesta 5.7 Gravitational force between two massive rods 5.8 Potential energy { Check your answer! 5.9 Ways of solving gravitational problems 5.10 Rod with linearly increasing mass density 5.11 Sphere with constant internal gravitational eld 5.12 Throwing a rock o the moon These problems are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Un- ported License. Please share and/or modify. Back to Problem List 1 5 { GRAVITATION Last Updated: July 16, 2012 5.1 Total mass of a shell Given: Marino { Fall 2011 Consider a spherical shell that extends from r = R to r = 2R with a non-uniform density (r) = 0r. What is the total mass of the shell? Back to Problem List 2 5 { GRAVITATION Last Updated: July 16, 2012 5.2 Tunnel through the moon Given: Marino { Fall 2011 Imagine that NASA digs a straight tunnel through the center of the moon (see gure) to access the Moon’s 3He deposits. An astronaut places a rock in the tunnel at the surface of the moon, and releases it (from rest). Show that the rock obeys the force law for a mass connected to a spring. What is the spring constant? Find the oscillation period for this motion if you assume that Moon has a mass of 7.351022 kg and a radius of 1.74106 m. Assume the moon’s density is uniform throughout its volume, and ignore the moon’s rotation. Given: Pollock { Spring 2011 Imagine (in a parallel universe of unlimited budgets) that NASA digs a straight tunnel through the center of the moon (see gure). A robot place a rock in the tunnel at position r = r0 from the center of the moon, and releases it (from rest). Use Newton’s second law to write the equation of motion of the rock and solve for r(t). Explain in words the rock’s motion. Does the rock return to its initial position at any later time? If so, how long does it takes to return to it? (Give a formula, and a number.) Assume the moon’s density is uniform throughout its volume, and ignore the moon’s rotation. Given: Pollock { Spring 2012 Now lets consider our (real) planet Earth, with total mass M and radius R which we will approximate as a uniform mass density, (r) = 0. (a) Neglecting rotational and frictional e ects, show that a particle dropped into a hole drilled straight through the center of the earth all the way to the far side will oscillate between the two endpoints. (Hint: you will need to set up, and solve, an ODE for the motion) (b) Find the period of the oscillation of this motion. Get a number (in minutes) as a nal result, using data for the earth’s size and mass. (How does that compare to ying to Perth and back?!) Extra Credit: OK, even with unlimited budgets, digging a tunnel through the center of the earth is preposterous. But, suppose instead that the tunnel is a straight-line \chord” through the earth, say directly from New York to Los Angeles. Show that your nal answer for the time taken does not depend on the location of that chord! This is rather remarkable – look again at the time for a free-fall trip (no energy required, except perhaps to compensate for friction) How long would that trip take? Could this work?! Back to Problem List 3 5 { GRAVITATION Last Updated: July 16, 2012 5.3 Gravitational eld above the center of a thin hoop Given: Pollock { Spring 2011, Spring 2012 Consider a very (in nitesimally!) thin but massive loop, radius R (total mass M), centered around the origin, sitting in the x-y plane. Assume it has a uniform linear mass density  (which has units of kg/m) all around it. (So, it’s like a skinny donut that is mostly hole, centered around the z-axis) (a) What is  in terms of M and R? What is the direction of the gravitational eld generated by this mass distribution at a point in space a distance z above the center of the donut, i.e. at (0; 0; z) Explain your reasoning for the direction carefully, try not to simply \wave your hands.” (The answer is extremely intuitive, but can you justify that it is correct?) (b) Compute the gravitational eld, ~g, at the point (0; 0; z) by directly integrating Newton’s law of gravity, summing over all in nitesimal \chunks” of mass along the loop. (c) Compute the gravitational potential at the point (0; 0; z) by directly integrating ?Gdm=r, sum- ming over all in nitesimal \chunks” dm along the loop. Then, take the z-component of the gradient of this potential to check that you agree with your result from the previous part. (d) In the two separate limits z << R and z >> R, Taylor expand your g- eld (in the z-direction)out only to the rst non-zero term, and convince us that both limits make good physical sense. (e) Can you use Gauss’ law to gure out the gravitational potential at the point (0; 0; z)? (If so, do it and check your previous answers. If not, why not?) Extra credit: If you place a small mass a small distance z away from the center, use your Taylor limit for z << R above to write a simple ODE for the equation of motion. Solve it, and discuss the motion Back to Problem List 4 5 { GRAVITATION Last Updated: July 16, 2012 5.4 Gravitational force near a metal-cored planet surrounded by a gaseous cloud Given: Pollock { Spring 2011 Jupiter is composed of a dense spherical core (of liquid metallic hydrogen!) of radius Rc. It is sur- rounded by a spherical cloud of gaseous hydrogen of radius Rg, where Rg > Rc. Let’s assume that the core is of uniform density c and the gaseous cloud is also of uniform density g. What is the gravitational force on an object of mass m that is located at a radius r from the center of Jupiter? Note that you must consider the cases where the object is inside the core, within the gas layer, and outside of the planet. Back to Problem List 5 5 { GRAVITATION Last Updated: July 16, 2012 5.5 Sphere with linearly increasing mass density Given: Pollock { Spring 2011 A planet of mass M and radius R has a nonuniform density that varies with r, the distance from the center according to  = Ar for 0  r  R. (a) What is the constant A in terms of M and R? Does this density pro le strike you as physically plausible, or is just designed as a mathematical exercise? (Brie y, explain) (b) Determine the gravitational force on a satellite of mass m orbiting this planet. In words, please outline the method you plan to use for your solution. (Use the easiest method you can come up with!) In your calculation, you will need to argue that the magnitude of ~g(r; ; ) depends only on r. Be very explicit about this – how do you know that it doesn’t, in fact, depend on  or ? (c) Determine the gravitational force felt by a rock of mass m inside the planet, located at radius r < R. (If the method you use is di erent than in part b, explain why you switched. If not, just proceed!) Explicitly check your result for this part by considering the limits r ! 0 and r ! R. Back to Problem List 6 5 { GRAVITATION Last Updated: July 16, 2012 5.6 Jumping o Vesta Given: Pollock { Spring 2011 You are stranded on the surface of the asteroid Vesta. If the mass of the asteroid is M and its radius is R, how fast would you have to jump o its surface to be able to escape from its gravitational eld? (Your estimate should be based on parameters that characterize the asteroid, not parameters that describe your jumping ability.) Given your formula, look up the approximate mass and radius of the asteroid Vesta 3 and determine a numerical value of the escape velocity. Could you escape in this way? (Brie y, explain) If so, roughly how big in radius is the maximum the asteroid could be, for you to still escape this way? If not, estimate how much smaller an asteroid you would need, to escape from it in this way? Figure 1: Back to Problem List 7 5 { GRAVITATION Last Updated: July 16, 2012 5.7 Gravitational force between two massive rods Given: Pollock { Spring 2011 Consider two identical uniform rods of length L and mass m lying along the same line and having their closest points separated by a distance d as shown in the gure (a) Calculate the mutual force between these rods, both its direction and magnitude. (b) Now do several checks. First, make sure the units worked out (!) The, nd the magnitude of the force in the limit L ! 0. What do you expect? Brie y, discuss. Lastly, nd the magnitude of the force in the limit d ! 1 ? Again, is it what you expect? Brie y, discuss. Figure 2: Given: Pollock { Spring 2012 Determining the gravitational force between two rods: (a) Consider a thin, uniform rod of mass m and length L (and negligible other dimensions) lying on the x axis (from x=-L to 0), as shown in g 1a. Derive a formula for the gravitational eld \g" at any arbitrary point x to the right of the origin (but still on the x-axis!) due to this rod. (b) Now suppose a second rod of length L and mass m sits on the x axis as shown in g 1b, with the left edge a distance \d" away. Calculate the mutual gravitational force between these rods. (c) Let's do some checks! Show that the units work out in parts a and b. Find the magnitude of the force in part a, in the limit x >> L: What do you expect? Brie y, discuss! Finally, verify that your answer to part b gives what you expect in the limit d >> L. ( Hint: This is a bit harder! You need to consistently expand everything to second order, not just rst, because of some interesting cancellations) Fig 1a Fig 1b L m +x x=0 L x=0 x=d m Fig 1a Fig 1b L m +x x=0 L +x x=0 x=d L m m Back to Problem List 8 5 { GRAVITATION Last Updated: July 16, 2012 5.8 Potential energy { Check your answer! Given: Pollock { Spring 2011 On the last exam, we had a problem with a at ring, uniform mass per unit area of , inner radius of R, outer radius of 2R. A satellite (mass m) sat a distance z above the center of the ring. We asked for the gravitational potential energy, and the answer was U(z) = ?2Gm( p 4R2 + z2 ? p R2 + z2) (1) (a) If you are far from the disk (on the z axis), what do you expect for the formula for U(z)? (Don’t say \0″ – as usual, we want the functional form of U(z) as you move far away. Also, explicitly state what we mean by \far away”. (Please don’t compare something with units to something without units!) (b) Show explicitly that the formula above does indeed give precisely the functional dependence you expect. Back to Problem List 9 5 { GRAVITATION Last Updated: July 16, 2012 5.9 Ways of solving gravitational problems Given: Pollock { Spring 2011, Spring 2012 Infinite cylinder ρ=cr x z (a) Half-infinite line mass, uniform linear mass density, λ x (b) R z  P Figure 3: (a) An in nite cylinder of radius R centered on the z-axis, with non-uniform volume mass density  = cr, where r is the radius in cylindrical coordinates. (b) A half-in nite line of mass on the x-axis extending from x = 0 to x = +1, with uniform linear mass density . There are two general methods we use to solve gravitational problems (i.e. nd ~g given some distribution of mass). (a) Describe these two methods. We claim one of these methods is easiest to solve for ~g of mass distribution (a) above, and the other method is easiest to solve for ~g of the mass distribution (b) above. Which method goes with which mass distribution? Please justify your answer. (b) Find ~g of the mass distribution (a) above for any arbitrary point outside the cylinder. (c) Find the x component of the gravitational acceleration, gx, generated by the mass distribution labeled (b) above, at a point P a given distance z up the positive z-axis (as shown). Back to Problem List 10 5 { GRAVITATION Last Updated: July 16, 2012 5.10 Rod with linearly increasing mass density Given: Pollock { Spring 2012 Consider a very (in nitesimally!) thin but massive rod, length L (total mass M), centered around the origin, sitting along the x-axis. (So the left end is at (-L/2, 0,0) and the right end is at (+L/2,0,0) Assume the mass density  (which has units of kg/m)is not uniform, but instead varies linearly with distance from the origin, (x) = cjxj. (a) What is that constant \c” in terms of M and L? What is the direction of the gravitational eld generated by this mass distribution at a point in space a distance z above the center of the rod, i.e. at (0; 0; z) Explain your reasoning for the direction carefully, try not to simply \wave your hands.” (The answer is extremely intuitive, but can you justify that it is correct?) (b) Compute the gravitational eld, ~g, at the point (0; 0; z) by directly integrating Newton’s law of gravity, summing over all in nitesimal \chunks” of mass along the rod. (c) Compute the gravitational potential at the point (0; 0; z) by directly integrating ?Gdm=r, sum- ming over all in nitesimal \chunks” dm along the rod. Then, take the z-component of the gradient of this potential to check that you agree with your result from the previous part. (d) In the limit of large z what do you expect for the functional form for gravitational potential? (Hint: Don’t just say it goes to zero! It’s a rod of mass M, when you’re far away what does it look like? How does it go to zero?) What does \large z” mean here? Use the binomial (or Taylor) expansion to verify that your formula does indeed give exactly what you expect. (Hint: you cannot Taylor expand in something BIG, you have to Taylor expand in something small.) (e) Can you use Gauss’ law to gure out the gravitational potential at the point (0; 0; z)? (If so, do it and check your previous answers. If not, why not?) Back to Problem List 11 5 { GRAVITATION Last Updated: July 16, 2012 5.11 Sphere with constant internal gravitational eld Given: Pollock { Spring 2012 (a) Imagine a planet of total mass M and radius R which has a nonuniform mass density that varies just with r, the distance from the center. For this (admittedly very unusual!) planet, suppose the gravitational eld strength inside the planet turns out to be independent of the radial distance within the sphere. Find the function describing the mass density  = (r) of this planet. (Your nal answer should be written in terms of the given constants.) (b) Now, determine the gravitational force on a satellite of mass m orbiting this planet at distance r > R. (Use the easiest method you can come up with!) Explain your work in words as well as formulas. For instance, in your calculation, you will need to argue that the magnitude of ~g(r; ; ) depends only on r. Be explicit about this – how do you know that it doesn’t, in fact, depend on  or ? (c) As a nal check, explicitly show that your solutions inside and outside the planet (parts a and b) are consistent when r = R. Please also comment on whether this density pro le strikes you as physically plausible, or is it just designed as a mathematical exercise? Defend your reasoning. Back to Problem List 12 5 { GRAVITATION Last Updated: July 16, 2012 5.12 Throwing a rock o the moon Given: Pollock { Spring 2012 Assuming that asteroids have roughly the same mass density as the moon, make an estimate of the largest asteroid that an astronaut could be standing on, and still have a chance of throwing a small object (with their arms, no machinery!) so that it completely escapes the asteroid’s gravitational eld. (This minimum speed is called \escape velocity”) Is the size you computed typical for asteroids in our solar system? Back to Problem List 13