2. In Graff and Birkenstein’s example from chapter one, what does the speaker at the academic conference do wrong? What could the speaker do to fix this problem?

2. In Graff and Birkenstein’s example from chapter one, what does the speaker at the academic conference do wrong? What could the speaker do to fix this problem?

2.    In Graff and Birkenstein’s example from chapter one, what … Read More...
Critical Essay Guidelines FORMAT: Prepare your paper as a Microsoft Word file. Single-space the body of your paper; you may double-space between the headings (Introduction, Background Explanation and Critical Evaluation) described below. Use 1” margins on all sides. Use a font that is no larger than Times New Roman at 12 pt. and no smaller than Times New Roman at 10 pt. Put your name, course name, section and the date in a header on top of all pages. Include page numbers. LENGTH, TOPIC, ETC.: Write a 2 – 3 page (single-spaced) (1500 words) critical response on your topic. Back up your discussion with direct quotation from the relevant text, preferably short quotes, such as single sentences and (even better) crucial phrases. Leave out words or phrases using…ellipses…, etc. Less than 1/4 page total of your paper should be direct quotation. Cite any direct quotes simply by giving text title and page number in parentheses; the page number will either be from the textbook or what’s posted on Blackboard. For example, such a citation might look like: (Schoedinger, 25). Include a “Works Cited” page at the end of your paper citing the primary philosophic text from Schoedinger’s textbook. No other sources should be used. Treat your intended audience as someone who has some familiarity with philosophy generally, but no familiarity with the details of what you are writing on. STRUCTURE: In this critical response, you will do all and only the following three things, putting each under its OWN SECTION HEADING: A. INTRODUCTION Begin with a one-sentence introductory paragraph where you very briefly say what you will be doing in the rest of the critical response, one which has the exact form: “In this critical response, I will consider <insert chosen topic>, and then I will argue that <insert statement of main thesis>.” For example: “In this critical response, I will consider Socrates’ views on a worthwhile life, and then I will argue that the worthwhile life is nothing more or less than the life of pleasure.” B. BACKGROUND EXPLANATION Explain (in one-half to 1 page), as clearly as you can, the background to your chosen topic, including any relevant discussion in the text, and also including any relevant theories, arguments, objections, crucial notions and distinctions, etc. C. CRITICAL EVALUATION Critically evaluate (in 1½ – 2 pages) your chosen topic. This involves explaining and defending your thesis on the topic. In doing this, address relevant material from your “ Background Explanation” section. Also, you are encouraged (but not required) to anticipate potential objections and reply to them. Throughout your critical evaluation, pay careful attention (even if just informally) to the criteria of a good argument. This applies both when you are considering others’ arguments and when you are giving your own. GRADING: Grading will be based partly on whether or not you have successfully followed the instructions above (including the format requirements). Each defect in terms of failure to satisfy the instructions will cost you points. Any paper which completely ignores all instructions, however, will receive a zero. Barring prior consent from me or documented and sufficiently excusing special contingency, late papers will be graded in accord with the late policy on the syllabus. Grading will also be based on the writing quality. Here I have in mind things like: is the paper clear, concise, grammatical and accurate? Does it provide necessary explanations and avoid irrelevant material?

Critical Essay Guidelines FORMAT: Prepare your paper as a Microsoft Word file. Single-space the body of your paper; you may double-space between the headings (Introduction, Background Explanation and Critical Evaluation) described below. Use 1” margins on all sides. Use a font that is no larger than Times New Roman at 12 pt. and no smaller than Times New Roman at 10 pt. Put your name, course name, section and the date in a header on top of all pages. Include page numbers. LENGTH, TOPIC, ETC.: Write a 2 – 3 page (single-spaced) (1500 words) critical response on your topic. Back up your discussion with direct quotation from the relevant text, preferably short quotes, such as single sentences and (even better) crucial phrases. Leave out words or phrases using…ellipses…, etc. Less than 1/4 page total of your paper should be direct quotation. Cite any direct quotes simply by giving text title and page number in parentheses; the page number will either be from the textbook or what’s posted on Blackboard. For example, such a citation might look like: (Schoedinger, 25). Include a “Works Cited” page at the end of your paper citing the primary philosophic text from Schoedinger’s textbook. No other sources should be used. Treat your intended audience as someone who has some familiarity with philosophy generally, but no familiarity with the details of what you are writing on. STRUCTURE: In this critical response, you will do all and only the following three things, putting each under its OWN SECTION HEADING: A. INTRODUCTION Begin with a one-sentence introductory paragraph where you very briefly say what you will be doing in the rest of the critical response, one which has the exact form: “In this critical response, I will consider , and then I will argue that .” For example: “In this critical response, I will consider Socrates’ views on a worthwhile life, and then I will argue that the worthwhile life is nothing more or less than the life of pleasure.” B. BACKGROUND EXPLANATION Explain (in one-half to 1 page), as clearly as you can, the background to your chosen topic, including any relevant discussion in the text, and also including any relevant theories, arguments, objections, crucial notions and distinctions, etc. C. CRITICAL EVALUATION Critically evaluate (in 1½ – 2 pages) your chosen topic. This involves explaining and defending your thesis on the topic. In doing this, address relevant material from your “ Background Explanation” section. Also, you are encouraged (but not required) to anticipate potential objections and reply to them. Throughout your critical evaluation, pay careful attention (even if just informally) to the criteria of a good argument. This applies both when you are considering others’ arguments and when you are giving your own. GRADING: Grading will be based partly on whether or not you have successfully followed the instructions above (including the format requirements). Each defect in terms of failure to satisfy the instructions will cost you points. Any paper which completely ignores all instructions, however, will receive a zero. Barring prior consent from me or documented and sufficiently excusing special contingency, late papers will be graded in accord with the late policy on the syllabus. Grading will also be based on the writing quality. Here I have in mind things like: is the paper clear, concise, grammatical and accurate? Does it provide necessary explanations and avoid irrelevant material?

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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

HST 102: Paper 7 Formal essay, due in class on the day of the debate No late papers will be accepted. Answer the following inquiry in a typed (and stapled) 2 page essay in the five-paragraph format. Present and describe three of your arguments that you will use to defend your position concerning eugenics. Each argument must be unique (don’t describe the same argument twice from a different angle). Each argument must include at least one quotation from the texts to support your position (a minimum of 3 total). You may discuss your positions and arguments with other people on your side (but not your opponents); however, each student must write their own essay in their own words. Do not copy sentences or paragraphs from another student’s paper, this is plagiarism and will result in a failing grade for the assignment. HST 102: Debate 4 Eugenics For or Against? Basics of the debate: The term ‘Eugenics’ was derived from two Greek words and literally means ‘good genes’. Eugenics is the social philosophy or practice of engineering society based on genes, or promoting the reproduction of good genes while reducing (or prohibiting) the reproduction of bad genes. Your group will argue either for or against the adoption of eugenic policies in your society. Key Terms: Eugenics – The study of or belief in the possibility of improving the qualities of the human species or a human population, especially by such means as discouraging reproduction by persons having genetic defects or presumed to have inheritable undesirable traits (negative eugenics) or encouraging reproduction by persons presumed to have inheritable desirable traits (positive eugenics). Darwinism – The Darwinian theory that species originate by descent, with variation, from parent forms, through the natural selection of those individuals best adapted for the reproductive success of their kind. Social Darwinism – A 19th-century theory, inspired by Darwinism, by which the social order is accounted as the product of natural selection of those persons best suited to existing living conditions. Mendelian Inheritance – Theory proposed by Gregor Johann Mendal in 1865 that became the first theory of genetic inheritance derived from experiments with peas. Birth Control – Any means to artificially prevent biological conception. Euthanasia – A policy of ending the life of an individual for their betterment (for example, because of excessive pain, brain dead, etc.) or society’s benefit. Genocide – A policy of murdering all members of a specific group of people who share a common characteristic. Deductive Logic – Deriving a specific conclusion based on a set of general definitions. Inductive Logic – Deriving a general conclusion based on a number of specific examples. Brief Historical Background: Eugenics was first proposed by Francis Galton in his 1883 work, Inquiries into Human Faculty and its Development. Galton was a cousin of Charles Darwin and an early supporter of Darwin’s theories of natural selection and evolution. Galton defined eugenics as the study of all agencies under human control which can improve or impair the racial quality of future generations. Galton’s work utilized a number of other scientific pursuits at the time including the study of heredity, genes, chromosomes, evolution, social Darwinism, zoology, birth control, sociology, psychology, chemistry, atomic theory and electrodynamics. The number of significant scientific advances was accelerating throughout the 19th century altering what science was and what its role in society could and should be. Galton’s work had a significant influence throughout all areas of society, from scientific communities to politics, culture and literature. A number of organizations were created to explore the science of eugenics and its possible applications to society. Ultimately, eugenics became a means by which to improve society through policies based on scientific study. Most of these policies related to reproductive practices within a society, specifically who could or should not reproduce. Throughout the late 1800s and early 1900s a number of policies were enacted at various levels throughout Europe and the United States aimed at controlling procreation. Some specific policies included compulsory sterilization laws (usually concerning criminals and the mentally ill) as well as banning interracial marriages to prevent ‘cross-racial’ breeding. In the United States a number of individuals and foundations supported the exploration of eugenics as a means to positively influence society, including: the Rockefeller Foundation, the Carnegie Institution, the Race Betterment Foundation of Battle Creek, MI, the Eugenics Record Office, the American Breeders Association, the Euthanasia Society of America; and individuals such as Charles Davenport, Madison Grant, Alexander Graham Bell, Irving Fisher, John D. Rockefeller, Margaret Sanger, Marie Stopes, David Starr Jordan, Vernon Kellogg, H. G. Wells (though he later changed sides) Winston Churchill, George Bernard Shaw, John Maynard Keynes, Supreme Court Justice Oliver Wendell Holmes and Presidents Woodrow Wilson, Herbert Hoover and Theodore Roosevelt. Some early critics of eugenics included: Dr. John Haycroft, Halliday Sutherland, Lancelot Hogben, Franz Boaz, Lester Ward, G. K. Chesterton, J. B. S. Haldane, and R. A. Fisher. In 1911 the Carnegie Institute recommended constructing gas chambers around the country to euthanize certain elements of the American population (primarily the poor and criminals) considered to be harmful to the future of society as a possible eugenic solution. President Woodrow Wilson signed the first Sterilization Act in US history. In the 1920s and 30s, 30 states passed various eugenics laws, some of which were overturned by the Supreme Court. Eugenics of various forms was a founding principle of the Progressive Party, strongly supported by the first progressive president Theodore Roosevelt, and would continue to play an important part in influencing progressive policies into at least the 1940s. Many American individuals and societies supported German research on eugenics that would eventually be used to develop and justify the policies utilized by the NAZI party against minority groups including Jews, Africans, gypsies and others that ultimately led to programs of genocide and the holocaust. Following WWII and worldwide exposure of the holocaust eugenics generally fell out of favor among the public, though various lesser forms of eugenics are still advocated for today by such individuals as Dottie Lamm, Geoffrey Miller, Justice Ruth Bader Ginsberg, John Glad and Richard Dawson. Eugenics still influences many modern debates including: capital punishment, over-population, global warming, medicine (disease control and genetic disorders), birth control, abortion, artificial insemination, evolution, social engineering, and education. Key Points to discuss during the debate: • Individual rights vs. collective rights • The pros and cons of genetically engineering society • The practicality of genetically engineering society • Methods used to determine ‘good traits’ and ‘bad traits’ • Who determines which people are ‘fit’ or ‘unfit’ for future society • The role of science in society • Methods used to derive scientific conclusions • Ability of scientists to determine the future hereditary conditions of individuals • The value/accuracy of scientific conclusions • The role of the government to implement eugenic policies • Some possible eugenic political policies or laws • The ways these policies may be used effectively or abused • The relationship between eugenics and individual rights • The role of ethics in science and eugenics Strategies: 1. Use this guide to help you (particularly the key points). 2. Read all of the texts. 3. If needed, read secondary analysis concerning eugenics. 4. Identify key quotations as you read each text. Perhaps make a list of them to print out and/or group quotes by topic or point. 5. Develop multiple arguments to defend your position. 6. Prioritize your arguments from most persuasive to least persuasive and from most evidence to least evidence. 7. Anticipate the arguments of your opponents and develop counter-arguments for them. 8. Anticipate counter-arguments to your own arguments and develop responses to them.

HST 102: Paper 7 Formal essay, due in class on the day of the debate No late papers will be accepted. Answer the following inquiry in a typed (and stapled) 2 page essay in the five-paragraph format. Present and describe three of your arguments that you will use to defend your position concerning eugenics. Each argument must be unique (don’t describe the same argument twice from a different angle). Each argument must include at least one quotation from the texts to support your position (a minimum of 3 total). You may discuss your positions and arguments with other people on your side (but not your opponents); however, each student must write their own essay in their own words. Do not copy sentences or paragraphs from another student’s paper, this is plagiarism and will result in a failing grade for the assignment. HST 102: Debate 4 Eugenics For or Against? Basics of the debate: The term ‘Eugenics’ was derived from two Greek words and literally means ‘good genes’. Eugenics is the social philosophy or practice of engineering society based on genes, or promoting the reproduction of good genes while reducing (or prohibiting) the reproduction of bad genes. Your group will argue either for or against the adoption of eugenic policies in your society. Key Terms: Eugenics – The study of or belief in the possibility of improving the qualities of the human species or a human population, especially by such means as discouraging reproduction by persons having genetic defects or presumed to have inheritable undesirable traits (negative eugenics) or encouraging reproduction by persons presumed to have inheritable desirable traits (positive eugenics). Darwinism – The Darwinian theory that species originate by descent, with variation, from parent forms, through the natural selection of those individuals best adapted for the reproductive success of their kind. Social Darwinism – A 19th-century theory, inspired by Darwinism, by which the social order is accounted as the product of natural selection of those persons best suited to existing living conditions. Mendelian Inheritance – Theory proposed by Gregor Johann Mendal in 1865 that became the first theory of genetic inheritance derived from experiments with peas. Birth Control – Any means to artificially prevent biological conception. Euthanasia – A policy of ending the life of an individual for their betterment (for example, because of excessive pain, brain dead, etc.) or society’s benefit. Genocide – A policy of murdering all members of a specific group of people who share a common characteristic. Deductive Logic – Deriving a specific conclusion based on a set of general definitions. Inductive Logic – Deriving a general conclusion based on a number of specific examples. Brief Historical Background: Eugenics was first proposed by Francis Galton in his 1883 work, Inquiries into Human Faculty and its Development. Galton was a cousin of Charles Darwin and an early supporter of Darwin’s theories of natural selection and evolution. Galton defined eugenics as the study of all agencies under human control which can improve or impair the racial quality of future generations. Galton’s work utilized a number of other scientific pursuits at the time including the study of heredity, genes, chromosomes, evolution, social Darwinism, zoology, birth control, sociology, psychology, chemistry, atomic theory and electrodynamics. The number of significant scientific advances was accelerating throughout the 19th century altering what science was and what its role in society could and should be. Galton’s work had a significant influence throughout all areas of society, from scientific communities to politics, culture and literature. A number of organizations were created to explore the science of eugenics and its possible applications to society. Ultimately, eugenics became a means by which to improve society through policies based on scientific study. Most of these policies related to reproductive practices within a society, specifically who could or should not reproduce. Throughout the late 1800s and early 1900s a number of policies were enacted at various levels throughout Europe and the United States aimed at controlling procreation. Some specific policies included compulsory sterilization laws (usually concerning criminals and the mentally ill) as well as banning interracial marriages to prevent ‘cross-racial’ breeding. In the United States a number of individuals and foundations supported the exploration of eugenics as a means to positively influence society, including: the Rockefeller Foundation, the Carnegie Institution, the Race Betterment Foundation of Battle Creek, MI, the Eugenics Record Office, the American Breeders Association, the Euthanasia Society of America; and individuals such as Charles Davenport, Madison Grant, Alexander Graham Bell, Irving Fisher, John D. Rockefeller, Margaret Sanger, Marie Stopes, David Starr Jordan, Vernon Kellogg, H. G. Wells (though he later changed sides) Winston Churchill, George Bernard Shaw, John Maynard Keynes, Supreme Court Justice Oliver Wendell Holmes and Presidents Woodrow Wilson, Herbert Hoover and Theodore Roosevelt. Some early critics of eugenics included: Dr. John Haycroft, Halliday Sutherland, Lancelot Hogben, Franz Boaz, Lester Ward, G. K. Chesterton, J. B. S. Haldane, and R. A. Fisher. In 1911 the Carnegie Institute recommended constructing gas chambers around the country to euthanize certain elements of the American population (primarily the poor and criminals) considered to be harmful to the future of society as a possible eugenic solution. President Woodrow Wilson signed the first Sterilization Act in US history. In the 1920s and 30s, 30 states passed various eugenics laws, some of which were overturned by the Supreme Court. Eugenics of various forms was a founding principle of the Progressive Party, strongly supported by the first progressive president Theodore Roosevelt, and would continue to play an important part in influencing progressive policies into at least the 1940s. Many American individuals and societies supported German research on eugenics that would eventually be used to develop and justify the policies utilized by the NAZI party against minority groups including Jews, Africans, gypsies and others that ultimately led to programs of genocide and the holocaust. Following WWII and worldwide exposure of the holocaust eugenics generally fell out of favor among the public, though various lesser forms of eugenics are still advocated for today by such individuals as Dottie Lamm, Geoffrey Miller, Justice Ruth Bader Ginsberg, John Glad and Richard Dawson. Eugenics still influences many modern debates including: capital punishment, over-population, global warming, medicine (disease control and genetic disorders), birth control, abortion, artificial insemination, evolution, social engineering, and education. Key Points to discuss during the debate: • Individual rights vs. collective rights • The pros and cons of genetically engineering society • The practicality of genetically engineering society • Methods used to determine ‘good traits’ and ‘bad traits’ • Who determines which people are ‘fit’ or ‘unfit’ for future society • The role of science in society • Methods used to derive scientific conclusions • Ability of scientists to determine the future hereditary conditions of individuals • The value/accuracy of scientific conclusions • The role of the government to implement eugenic policies • Some possible eugenic political policies or laws • The ways these policies may be used effectively or abused • The relationship between eugenics and individual rights • The role of ethics in science and eugenics Strategies: 1. Use this guide to help you (particularly the key points). 2. Read all of the texts. 3. If needed, read secondary analysis concerning eugenics. 4. Identify key quotations as you read each text. Perhaps make a list of them to print out and/or group quotes by topic or point. 5. Develop multiple arguments to defend your position. 6. Prioritize your arguments from most persuasive to least persuasive and from most evidence to least evidence. 7. Anticipate the arguments of your opponents and develop counter-arguments for them. 8. Anticipate counter-arguments to your own arguments and develop responses to them.

Argument essay Argue for or aganist the advisability of getting involved in trying to help others.Support your position with details/examples from your experience. + 500 words ————————————— 3 creative topics related to your major (my major is Mechanical engineering) 3-5 pages

Argument essay Argue for or aganist the advisability of getting involved in trying to help others.Support your position with details/examples from your experience. + 500 words ————————————— 3 creative topics related to your major (my major is Mechanical engineering) 3-5 pages

Homework #8  Consider the veracity or falsehood of each of the following statements. For bonus, argue for those that you believe are true while providing a counterexample for those that you believe are false.  If the first and third rows of A are equal, then det A 0.  If P is a projection, then uCP if and only if Pu  u.  If P is a projection, and detP  0, then P  I .  If A has determinant 10, then 1 A has determinant 1 10 .  If B is invertible, 1 1 det(A B ) det A (detB) .  If P is a projection, and R  2P I , then 2 R  I .  If P is a projection, and P  I , then detP  0 .  Short Computations. All of the following do not involve long computations:  Suppose 1 2 1 5 1 8 A                  and 1 9 2 4 3 1 A                   . Compute 7 13 19 A         .  Compute               0 8 7 1 0 2 3 4 5 3 0 9 2 0 0 0 3 0 0 0 1 9 3 2 0 det .  Use Cramer’s Rule to find 5 x (hint: you do not need your calculator). 1 2 3 4 5 5x 2x 8x x 3x 13 1 3 3x 5x 0 1 3 5 3x 3x 3x 9 1 2 3 5 3x 2x x 2x 7 1 3 x 4x 0 Let A 1 2 3 4 1 3 4 6 2 5 13 15 4 10 15 31 . Given is that det A  61. Do the following:  1 1 2 4 2 3 5 10 3 4 13 15 4 6 15 31 det  det2A  1 3 4 6 2 4 6 8 2 5 13 15 4 10 15 31 det  1 3 4 6 2 5 13 15 4 10 15 31 1 2 3 4 det  Consider the matrix A  0 1 0 0 0 0 1 0 0 0 0 1 1 2 2 1           . Use row (or column) expansion to compute det(xI A) .  The matrix 4 1 1 2 1 1 1 4 1 1 2 1 1 1 4 1 1 2 2 1 1 4 1 1 1 2 1 1 4 1 1 1 2 1 1 4 1 6 P is the projection matrix for the column space of matrix A. This matrix A is also known to be of full rank. Answer the following, giving reasons for your answers.  Find a transparent basis and the dimension for the column space of P.  Find a basis and the dimension for the column space of A .  What size is the matrix A ?  Find a transparent basis and the dimension for the null space of P.  Find a transparent basis and the dimension for the row space of P.  Find a basis and the dimension for the null space of A.  For which of the following b can you find a solution to the system Ax b ? This does not mean you should find a solution, only whether one could or not. 10 17 19 14 10 17 19 14 13 10 17 19 14 13 23 1 1 1 1 1 1 .  It is known that certain vector u is a solution to the system Ax c . Give all solutions to Ax c .  It is also known that 1 2 3 4 5 6 Ax does not have a solution. How would you change the constant vector so that there would be a solution? Extra Problems.  Fill in the blank with the best possible expression to complete the sentence truthfully. Only that one will be counted correct. 1. matrix with two equal columns will have zero determinant. 1 2 3 Some Every No 2. If A is invertible, then A commute with its inverse. 1 2 3 must always can will not 3. If A is 6  9 , then the columns of A be linearly independent. While in AT , the columns be linearly independent. 1 2 3 can have to cannot 4. Let A be square, and suppose Ax  0 has a nontrivial solution. Then detA equal 0. 1 2 3 may cannot must 5. Let A and B be 3 3. Then det (AB) equal det(A)det(B) . 1 2 3 could must couldn’t 6. Let A be square and suppose detA  0. Then have an inverse 1 2 3 will not may must always 7. Let A and B be 2  2 . Then det (A B) equal det(A)  det(B) . 1 2 3 could must could not 8. exist a 6  6 matrix all of whose entries are whole numbers and its determinant is 2 5 . 1 2 3 There does There does not There might Bonus: Consider the matrix 0 0 1 0 2 0 n 0 . Give its determinant as a function of n.

Homework #8  Consider the veracity or falsehood of each of the following statements. For bonus, argue for those that you believe are true while providing a counterexample for those that you believe are false.  If the first and third rows of A are equal, then det A 0.  If P is a projection, then uCP if and only if Pu  u.  If P is a projection, and detP  0, then P  I .  If A has determinant 10, then 1 A has determinant 1 10 .  If B is invertible, 1 1 det(A B ) det A (detB) .  If P is a projection, and R  2P I , then 2 R  I .  If P is a projection, and P  I , then detP  0 .  Short Computations. All of the following do not involve long computations:  Suppose 1 2 1 5 1 8 A                  and 1 9 2 4 3 1 A                   . Compute 7 13 19 A         .  Compute               0 8 7 1 0 2 3 4 5 3 0 9 2 0 0 0 3 0 0 0 1 9 3 2 0 det .  Use Cramer’s Rule to find 5 x (hint: you do not need your calculator). 1 2 3 4 5 5x 2x 8x x 3x 13 1 3 3x 5x 0 1 3 5 3x 3x 3x 9 1 2 3 5 3x 2x x 2x 7 1 3 x 4x 0 Let A 1 2 3 4 1 3 4 6 2 5 13 15 4 10 15 31 . Given is that det A  61. Do the following:  1 1 2 4 2 3 5 10 3 4 13 15 4 6 15 31 det  det2A  1 3 4 6 2 4 6 8 2 5 13 15 4 10 15 31 det  1 3 4 6 2 5 13 15 4 10 15 31 1 2 3 4 det  Consider the matrix A  0 1 0 0 0 0 1 0 0 0 0 1 1 2 2 1           . Use row (or column) expansion to compute det(xI A) .  The matrix 4 1 1 2 1 1 1 4 1 1 2 1 1 1 4 1 1 2 2 1 1 4 1 1 1 2 1 1 4 1 1 1 2 1 1 4 1 6 P is the projection matrix for the column space of matrix A. This matrix A is also known to be of full rank. Answer the following, giving reasons for your answers.  Find a transparent basis and the dimension for the column space of P.  Find a basis and the dimension for the column space of A .  What size is the matrix A ?  Find a transparent basis and the dimension for the null space of P.  Find a transparent basis and the dimension for the row space of P.  Find a basis and the dimension for the null space of A.  For which of the following b can you find a solution to the system Ax b ? This does not mean you should find a solution, only whether one could or not. 10 17 19 14 10 17 19 14 13 10 17 19 14 13 23 1 1 1 1 1 1 .  It is known that certain vector u is a solution to the system Ax c . Give all solutions to Ax c .  It is also known that 1 2 3 4 5 6 Ax does not have a solution. How would you change the constant vector so that there would be a solution? Extra Problems.  Fill in the blank with the best possible expression to complete the sentence truthfully. Only that one will be counted correct. 1. matrix with two equal columns will have zero determinant. 1 2 3 Some Every No 2. If A is invertible, then A commute with its inverse. 1 2 3 must always can will not 3. If A is 6  9 , then the columns of A be linearly independent. While in AT , the columns be linearly independent. 1 2 3 can have to cannot 4. Let A be square, and suppose Ax  0 has a nontrivial solution. Then detA equal 0. 1 2 3 may cannot must 5. Let A and B be 3 3. Then det (AB) equal det(A)det(B) . 1 2 3 could must couldn’t 6. Let A be square and suppose detA  0. Then have an inverse 1 2 3 will not may must always 7. Let A and B be 2  2 . Then det (A B) equal det(A)  det(B) . 1 2 3 could must could not 8. exist a 6  6 matrix all of whose entries are whole numbers and its determinant is 2 5 . 1 2 3 There does There does not There might Bonus: Consider the matrix 0 0 1 0 2 0 n 0 . Give its determinant as a function of n.

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Capital Punishment For this paper, please read both the Ernest Van den Haag article and the Larry Tifft article. PICK A SIDE, either pro-capital punishment (death penalty) or abolitionist (anti-death penalty). This IS an ARGUMENTATIVE PAPER!!!!!!!!!!!! Using both the articles, your text book, (and a general source if you wish such as the Bible or dictionary), argue for your stance. You may also use the Marx/Durkheim handouts if you so wish, so long as there are SOCIOLOGY TERMS in your paper! You must: 1. Provide three points to back up your argument and include them in your thesis paragraph and explain them in the body of the paper; 2. Give voice to the opposing side. If you’re abolitionist, bring up any validity to Van den Haag, if you’re pro-capital punishment, bring up any validity to Tifft’s argument. 3. Use both a thesis opening paragraph and a solid conclusion paragraph. 4. Use SOCIOLOGY TERMS! 5. NO PERSONAL PRONOUNS! TECH requirements: • 12 point, Arial or Times Roman Numeral font. • 3 page long NOT including cover sheet and references page • Double spaced

Capital Punishment For this paper, please read both the Ernest Van den Haag article and the Larry Tifft article. PICK A SIDE, either pro-capital punishment (death penalty) or abolitionist (anti-death penalty). This IS an ARGUMENTATIVE PAPER!!!!!!!!!!!! Using both the articles, your text book, (and a general source if you wish such as the Bible or dictionary), argue for your stance. You may also use the Marx/Durkheim handouts if you so wish, so long as there are SOCIOLOGY TERMS in your paper! You must: 1. Provide three points to back up your argument and include them in your thesis paragraph and explain them in the body of the paper; 2. Give voice to the opposing side. If you’re abolitionist, bring up any validity to Van den Haag, if you’re pro-capital punishment, bring up any validity to Tifft’s argument. 3. Use both a thesis opening paragraph and a solid conclusion paragraph. 4. Use SOCIOLOGY TERMS! 5. NO PERSONAL PRONOUNS! TECH requirements: • 12 point, Arial or Times Roman Numeral font. • 3 page long NOT including cover sheet and references page • Double spaced

http://www.youtube.com/watch?v=UnKEFSVAiNQ Watch the video, and then answer the questions below. According to realism, which of the following represents something that states would NOT seek? A. security B. prestige C. autonomy D. wealth E. permanent cooperation Schweller suggests that realists are wary of interdependence. If that is true, which of the following might be the most acceptable to a realist? A. creating a permanent pact of nonviolence with all English-speaking countries B. establishing an alliance to defend the U.S. against an invading country C. turning North America into something similar to the European Union, with a unified currency D. permitting the United Nations to run a global military so that the U.S. can reduce its military spending E. entering into a global production agreement in which the U.S. only manufactures computers Based on the video, which of the following statements about realists would seem to be false? A. Realists see the world as perpetually violent and full of war. B. Realists see humans as basically self-interested. C. Realists believe that the absence of a threat means a country should retrench. D. Realists believe that intervening in other countries to spread democracy is dangerous. E. Realists believe that autonomy is better than interdependence. What does Schweller mean by his statement that “there is no 911”? A. There is no global authority that is guaranteed to help any state in trouble. B. The world needs a central government to provide a universal social safety net. C. States need to cooperate more with each other in order to provide greater security for all. D. The United Nations is terrible at dealing with international emergencies. E. Islamic terrorists were not responsible for the attacks of September 11, 2001. Which of the following would be the best way to convince a realist to go to war? A. argue that we signed a treaty to protect that country B. argue that the country we are helping to defend was an ally in a prior war C. argue that it will provide the world with a chance at long-term peace and stability D. argue that the other country is a direct threat to our interests E. argue that if we do not intervene, the United Nations will

http://www.youtube.com/watch?v=UnKEFSVAiNQ Watch the video, and then answer the questions below. According to realism, which of the following represents something that states would NOT seek? A. security B. prestige C. autonomy D. wealth E. permanent cooperation Schweller suggests that realists are wary of interdependence. If that is true, which of the following might be the most acceptable to a realist? A. creating a permanent pact of nonviolence with all English-speaking countries B. establishing an alliance to defend the U.S. against an invading country C. turning North America into something similar to the European Union, with a unified currency D. permitting the United Nations to run a global military so that the U.S. can reduce its military spending E. entering into a global production agreement in which the U.S. only manufactures computers Based on the video, which of the following statements about realists would seem to be false? A. Realists see the world as perpetually violent and full of war. B. Realists see humans as basically self-interested. C. Realists believe that the absence of a threat means a country should retrench. D. Realists believe that intervening in other countries to spread democracy is dangerous. E. Realists believe that autonomy is better than interdependence. What does Schweller mean by his statement that “there is no 911”? A. There is no global authority that is guaranteed to help any state in trouble. B. The world needs a central government to provide a universal social safety net. C. States need to cooperate more with each other in order to provide greater security for all. D. The United Nations is terrible at dealing with international emergencies. E. Islamic terrorists were not responsible for the attacks of September 11, 2001. Which of the following would be the best way to convince a realist to go to war? A. argue that we signed a treaty to protect that country B. argue that the country we are helping to defend was an ally in a prior war C. argue that it will provide the world with a chance at long-term peace and stability D. argue that the other country is a direct threat to our interests E. argue that if we do not intervene, the United Nations will

http://www.youtube.com/watch?v=UnKEFSVAiNQ   Watch the video, and then answer the questions … Read More...
How Soccer Explains the World Essay Prompt Foer argues that studying soccer different parts of the soccer world helps us understand globalization. He examines teams, fans, criminal groups, ownership, and the spread of soccer as ways to look at the international system. Each chapter is a different part of the soccer world and a different explanation of globalization. Chose 1 team or team rivalry or aspect of the soccer world he covers, and briefly summarize it, whether it be the team club, the fan base, or whatever part of the soccer world he is trying to explain. What does the team or fan base he is studying explain? What is the history involved and how are people reacting to it? What do the songs fans sing, or the owners of teams, or the way a community relates with the team tell us about the world, according to Foer? What are the fans or the team doing, and why? And what does he argue it tells us about international politics? You should explain the history of the team, the community, or the fan base, if provided. You should address the nature of a rivalry and on what basis this rivalry exists. Be sure to identify what feature of globalization Foer is trying to explain by looking at that particular part of the soccer world. Is it corruption? Cultural influence? The rise and fall of nationalism? Is it ancient ethnic, racial, or religious hatreds? Economic disparity? Summarize the story Foer is trying to tell by looking at the part of the soccer world you have decided to write about. You can choose any aspect of the soccer world Foer writes about- any team, any fan group, any ownership, or any soccer trend on which he writes. Just chose one, and recount Foer’s argument about it. 5 pages, double spaced. In text citation of Foer’s book required No other outside sources are required, but you may use them if you desire Due 30 OCT in class or in my office by 5 pm (NOTE THE SCHEDULE CHANGE) Hard copy only. Typed. Double spaced. Stapled. Spelling and Grammar count. See syllabus. No electronic submission. You must bring it to class on or before 30 OCT OR turn it into my office (JT 621) by 5pm 30 OCT.

How Soccer Explains the World Essay Prompt Foer argues that studying soccer different parts of the soccer world helps us understand globalization. He examines teams, fans, criminal groups, ownership, and the spread of soccer as ways to look at the international system. Each chapter is a different part of the soccer world and a different explanation of globalization. Chose 1 team or team rivalry or aspect of the soccer world he covers, and briefly summarize it, whether it be the team club, the fan base, or whatever part of the soccer world he is trying to explain. What does the team or fan base he is studying explain? What is the history involved and how are people reacting to it? What do the songs fans sing, or the owners of teams, or the way a community relates with the team tell us about the world, according to Foer? What are the fans or the team doing, and why? And what does he argue it tells us about international politics? You should explain the history of the team, the community, or the fan base, if provided. You should address the nature of a rivalry and on what basis this rivalry exists. Be sure to identify what feature of globalization Foer is trying to explain by looking at that particular part of the soccer world. Is it corruption? Cultural influence? The rise and fall of nationalism? Is it ancient ethnic, racial, or religious hatreds? Economic disparity? Summarize the story Foer is trying to tell by looking at the part of the soccer world you have decided to write about. You can choose any aspect of the soccer world Foer writes about- any team, any fan group, any ownership, or any soccer trend on which he writes. Just chose one, and recount Foer’s argument about it. 5 pages, double spaced. In text citation of Foer’s book required No other outside sources are required, but you may use them if you desire Due 30 OCT in class or in my office by 5 pm (NOTE THE SCHEDULE CHANGE) Hard copy only. Typed. Double spaced. Stapled. Spelling and Grammar count. See syllabus. No electronic submission. You must bring it to class on or before 30 OCT OR turn it into my office (JT 621) by 5pm 30 OCT.

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