Problems Marking scheme 1. Let A be a nonzero square matrix. Is it possible that a positive integer k exists such that ?? = 0 ? For example, find ?3 for the matrix [ 0 1 2 0 0 1 0 0 0 ] A square matrix A is nilpotent of index k when ? ≠ 0 , ?2 ≠ 0 , … . . , ??−1 ≠ 0, ??? ?? = 0. In this task you will explore nilpotent matrices. 1. The matrix in the example given above is nilpotent. What is its index? ( 2 marks ) 2. Use a software program to determine which of the following matrices are nilpotent and find their indices ( 12 marks ) A. [ 0 1 0 0 ] B. [ 0 1 1 0 ] C. [ 0 0 1 0 ] D. [ 1 0 1 0 ] E. [ 0 0 1 0 0 0 0 0 0 ] F. [ 0 0 0 1 0 0 1 1 0 ] 3. Find 3×3 nilpotent matrices of indices 2 and 3 ( 2 marks ) 4. Find 4×4 nilpotent matrices of indices 2, 3, and 4 ( 2 marks ) 5. Find nilpotent matrix of index 5 ( 2 marks ) 6. Are nilpotent matrices invertible? prove your answer ( 3 marks ) 7. When A is nilpotent, what can you say about ?? ? prove your answer ( 3 marks ) 8. Show that if ? is nilpotent , then ? − ? is invertible ( 4 marks ) 30% 2. A radio transmitter circuit contains a resisitance of 2.0 Ω, a variable inductor of 100 − ? ℎ?????? and a voltage source of 4.0 ? . find the current ? in the circuit as a function of the time t for 0 ≤ ? ≤ 100? if the intial curent is zero. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 3. An object falling under the influence of gravity has a variable accelertaion given by 32 − ? , where ? represents the velocity. If the object starts from rest, find an expression for the velocity in terms of the time. Also, find the limiting value of the velocity. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 4. When the angular displacement ? of a pendulum is small ( less than 60), the pendulum moves with simple harmonic motion closely approximated by ?′′ + ? ? ? = 0 . Here , ?′ = ?? ?? and ? is the accelertaion due to gravity , and ? is the length of the pendulum. Find ? as a function of time ( in s ) if ? = 9.8 ?/?2, ? = 1.0 ? ? = 0.1 and ?? ?? = 0 when ? = 0 . sketch the cuve using any graphical tool. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 5. Find the equation relating the charge and the time in a electric circuit with the following elements: ? = 0.200 ? , ? = 8.00 Ω , ? = 1.00 ?? , ? = 0. In this circuit , ? = 0 and ? = 0.500 ? when ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 6. A spring is stretched 1 m by ? 20 − ? Weight. The spring is stretched 0.5 m below the equilibrium position with the weight attached and the then released. If it is a medium that resists the motion with a force equal to 12?, where v is the velocity, sketch and find the displacement y of the weight as a function of the time. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 7. A 20?? inductor, a 40.0 Ω resistor, a 50.0 ?? capacitor, and voltage source of 100 ?−100?are connected in series in an electric circuit. Find the charge on the capacitor as a function of time t , if ? = 0 and ? = 0 ?ℎ?? ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 10% quality and neatness and using Math equations in MS word. –

Problems Marking scheme 1. Let A be a nonzero square matrix. Is it possible that a positive integer k exists such that ?? = 0 ? For example, find ?3 for the matrix [ 0 1 2 0 0 1 0 0 0 ] A square matrix A is nilpotent of index k when ? ≠ 0 , ?2 ≠ 0 , … . . , ??−1 ≠ 0, ??? ?? = 0. In this task you will explore nilpotent matrices. 1. The matrix in the example given above is nilpotent. What is its index? ( 2 marks ) 2. Use a software program to determine which of the following matrices are nilpotent and find their indices ( 12 marks ) A. [ 0 1 0 0 ] B. [ 0 1 1 0 ] C. [ 0 0 1 0 ] D. [ 1 0 1 0 ] E. [ 0 0 1 0 0 0 0 0 0 ] F. [ 0 0 0 1 0 0 1 1 0 ] 3. Find 3×3 nilpotent matrices of indices 2 and 3 ( 2 marks ) 4. Find 4×4 nilpotent matrices of indices 2, 3, and 4 ( 2 marks ) 5. Find nilpotent matrix of index 5 ( 2 marks ) 6. Are nilpotent matrices invertible? prove your answer ( 3 marks ) 7. When A is nilpotent, what can you say about ?? ? prove your answer ( 3 marks ) 8. Show that if ? is nilpotent , then ? − ? is invertible ( 4 marks ) 30% 2. A radio transmitter circuit contains a resisitance of 2.0 Ω, a variable inductor of 100 − ? ℎ?????? and a voltage source of 4.0 ? . find the current ? in the circuit as a function of the time t for 0 ≤ ? ≤ 100? if the intial curent is zero. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 3. An object falling under the influence of gravity has a variable accelertaion given by 32 − ? , where ? represents the velocity. If the object starts from rest, find an expression for the velocity in terms of the time. Also, find the limiting value of the velocity. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 4. When the angular displacement ? of a pendulum is small ( less than 60), the pendulum moves with simple harmonic motion closely approximated by ?′′ + ? ? ? = 0 . Here , ?′ = ?? ?? and ? is the accelertaion due to gravity , and ? is the length of the pendulum. Find ? as a function of time ( in s ) if ? = 9.8 ?/?2, ? = 1.0 ? ? = 0.1 and ?? ?? = 0 when ? = 0 . sketch the cuve using any graphical tool. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 5. Find the equation relating the charge and the time in a electric circuit with the following elements: ? = 0.200 ? , ? = 8.00 Ω , ? = 1.00 ?? , ? = 0. In this circuit , ? = 0 and ? = 0.500 ? when ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 6. A spring is stretched 1 m by ? 20 − ? Weight. The spring is stretched 0.5 m below the equilibrium position with the weight attached and the then released. If it is a medium that resists the motion with a force equal to 12?, where v is the velocity, sketch and find the displacement y of the weight as a function of the time. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 7. A 20?? inductor, a 40.0 Ω resistor, a 50.0 ?? capacitor, and voltage source of 100 ?−100?are connected in series in an electric circuit. Find the charge on the capacitor as a function of time t , if ? = 0 and ? = 0 ?ℎ?? ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 10% quality and neatness and using Math equations in MS word. –

Problems Marking scheme 1. Let A be a nonzero square … 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

AERN 45350 Avionics Name: _______________________________ 1 | P a g e Homework Set One (40 Points) Due: 25 September 2015 General Instructions: Answer the following questions, submitting your answers on Blackboard. SHOW YOUR WORK on any math problems. Consider the following voltage signal: V t 12sin377t 1. What is the peak voltage of the signal [Volts]? 2. What is the angular frequency [rad/sec]? 3. What is the frequency of the signal [Hz]? 4. What is the period of the signal [sec/cycle]? In a heterodyne receiver, the intermediate frequency of the receiver is 21.4 MHz. 5. What is the local oscillator frequency (f1) if the tuned frequency (f2) is 120.9 MHz? 6. If the local oscillator frequency (f1) is 145.7 MHz, what is the tuned frequency (f2)? The gain of a power amplifier is 5. 7. If 30W are coming in, what is the power going out? 8. What is the gain in decibels (dB)? The attenuation of a voltage attenuator is 10. 9. If 120V are coming in, what is the voltage going out? 10. What is the loss in decibels (dB)? 11. What is the component of the ILS that provides the extended centerline of the runway? 12. What is the component of the ILS that provides vertical guidance to the runway? 13. If the aircraft is on the correct trajectory, the airplane will arrive at the outer marker on the ILS corresponding to intercepting what? 14. If the aircraft is on the correct trajectory, the airplane will arrive at the middle marker on the ILS corresponding to reaching what? 15. All marker beacons transmit at what frequency? 16. Why doesn’t this cause problems (all marker beacons transmitting on the same frequency)? 17. What are the four components to an ILS? 18. What is the most common ILS category? 19. Which ILS category requires aircraft with the “auto-land” feature? 20. An attenuator leads to a power ratio of 0.5. What is that in decibels (dB)?

AERN 45350 Avionics Name: _______________________________ 1 | P a g e Homework Set One (40 Points) Due: 25 September 2015 General Instructions: Answer the following questions, submitting your answers on Blackboard. SHOW YOUR WORK on any math problems. Consider the following voltage signal: V t 12sin377t 1. What is the peak voltage of the signal [Volts]? 2. What is the angular frequency [rad/sec]? 3. What is the frequency of the signal [Hz]? 4. What is the period of the signal [sec/cycle]? In a heterodyne receiver, the intermediate frequency of the receiver is 21.4 MHz. 5. What is the local oscillator frequency (f1) if the tuned frequency (f2) is 120.9 MHz? 6. If the local oscillator frequency (f1) is 145.7 MHz, what is the tuned frequency (f2)? The gain of a power amplifier is 5. 7. If 30W are coming in, what is the power going out? 8. What is the gain in decibels (dB)? The attenuation of a voltage attenuator is 10. 9. If 120V are coming in, what is the voltage going out? 10. What is the loss in decibels (dB)? 11. What is the component of the ILS that provides the extended centerline of the runway? 12. What is the component of the ILS that provides vertical guidance to the runway? 13. If the aircraft is on the correct trajectory, the airplane will arrive at the outer marker on the ILS corresponding to intercepting what? 14. If the aircraft is on the correct trajectory, the airplane will arrive at the middle marker on the ILS corresponding to reaching what? 15. All marker beacons transmit at what frequency? 16. Why doesn’t this cause problems (all marker beacons transmitting on the same frequency)? 17. What are the four components to an ILS? 18. What is the most common ILS category? 19. Which ILS category requires aircraft with the “auto-land” feature? 20. An attenuator leads to a power ratio of 0.5. What is that in decibels (dB)?

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

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

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Chapter 14 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 Harmonic Oscillator Equations Learning Goal: To derive the formulas for the major characteristics of motion as functions of time for a horizontal spring oscillator and to practice using the obtained formulas by answering some basic questions. A block of mass is attached to a spring whose spring constant is . The other end of the spring is fixed so that when the spring is unstretched, the mass is located at . . Assume that the +x direction is to the right. The mass is now pulled to the right a distance beyond the equilibrium position and released, at time , with zero initial velocity. Assume that the vertical forces acting on the block balance each other and that the tension of the spring is, in effect, the only force affecting the motion of the block. Therefore, the system will undergo simple harmonic motion. For such a system, the equation of motion is , and its solution, which provides the equation for , is . Part A At what time does the block come back to its original equilibrium position ( ) for the first time? Express your answer in terms of some or all of the variables: , , and . You did not open hints for this part. ANSWER: m k x = 0 A t = 0 a(t) = − x(t) km x(t) x(t) = Acos( t) km −−  t1 x = 0 A k m

Chapter 14 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 Harmonic Oscillator Equations Learning Goal: To derive the formulas for the major characteristics of motion as functions of time for a horizontal spring oscillator and to practice using the obtained formulas by answering some basic questions. A block of mass is attached to a spring whose spring constant is . The other end of the spring is fixed so that when the spring is unstretched, the mass is located at . . Assume that the +x direction is to the right. The mass is now pulled to the right a distance beyond the equilibrium position and released, at time , with zero initial velocity. Assume that the vertical forces acting on the block balance each other and that the tension of the spring is, in effect, the only force affecting the motion of the block. Therefore, the system will undergo simple harmonic motion. For such a system, the equation of motion is , and its solution, which provides the equation for , is . Part A At what time does the block come back to its original equilibrium position ( ) for the first time? Express your answer in terms of some or all of the variables: , , and . You did not open hints for this part. ANSWER: m k x = 0 A t = 0 a(t) = − x(t) km x(t) x(t) = Acos( t) km −−  t1 x = 0 A k m

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Select Case 1, 2, or 8 in the back of the textbook. After you have read the case, select at least one of the questions presented at the end.-If you select only one question, then you will need to elaborate with more examples and perspectives than if you select more than one, but the choice is yours. Fair warning: It is possible to fall into the trap of repeating oneself. To avoid that threat, think in advance of the different perspectives that you wish to explore. If you select more than one question, each answer will naturally be shorter. This may be a good approach if you discern that the questions lack strong potential to elicit in-depth answers. Remember to reply to the contributions of two other students in this exercise. This is a rule that we are only observing in the case analyses, given the relative complexity of the cases, compared to the chapter discussion questions. Always add value, from the textbook, news, personal experience, or all three. Indicate the case and question at the beginning, but avoid restating the question in your answer. In this respect, use the same method as in the chapter discussion questions, described in the Week 2 forum. Write at least 500 words (no minimum for replies, but do add value). Quoted passages do not contribute to the word count (so you will need to write more if you insert any quoted material). Post-edit your work carefully to catch errors. Avoid plagiarism at all cost. ——— Note on anomalous questions. Some questions will require you to work around selected details to fit the requisite discussion format. For example, Question 2 in Case 1 asks how your proposal will solve certain problems noted in answer to the previous question. If you have not actually answered Question 1, then you will have to assert one or more problems from the case, a proposed solution, and then an explanation of how your proposal may help. Question 3 is similar, in that you will need to identify a problem and a solution, followed by an argument about the budget. Although Alistair was expecting to hire a Project Engineer rather than a Quality Compliance Manager, the methods used to make the decision should be similar. The main difference in the Quality Compliance Manager position is that it is in a joint venture with a Hungarian government backed firm. International Joint Ventures (IJV) makes HRM practices more complicated because HRM practices and strategies are required for each IJV entity (Dowling, Festing, & Engle, 2013). HRM must address IJV in four stages, in which, each stage has an impact on the next. It is important for HRM to very thorough with each stage and communication through each stage is vital. To be successful, HRM must combine the IJV strategy along with the recruitment, selection, training, and development processes (Dowling et al., 2013). In light of the needs of the company and the new Quality Compliance Manager position, Alistair should choose the first candidate, Marie Erten-Loiseau. The fact that the job requires travel to France and Germany is a positive for Marie because she was born in France and was educated in France and Germany. The familiarity of these locations will help her as she meets with new business partners because she will have a good understanding of the policy and procedures required for companies in these two countries. Dowling et al., (2013), points out that the manager needs to be able to assess the desires of the stakeholders and be able to implement strategies based on their desires. Another reason for choosing Marie is that she has the most experience and has worked with Trianon for 13 years. The experience she has with the company is invaluable because she knows the goals of the company and strategies for implementing those goals. The last reason for choosing Marie is that she has been successful in her previous positions. She has lead two projects in two different countries and both were successful. This shows that she is able to adapt to the different practices of each country. There are many factors that Alistair should take into consideration to determine the correct choice for the Quality Compliance Manager position. The major factors that require consideration are the specificities of the entire situation, the reason for the assignment, and type of assignment. The four main specificities include context specificities, firm specific variables, local unit specificities, and IHRM practices (Dowling et al., 2013). The context specificities would include the differences in cultures between the assignment in Hungary and the base location for the Trianon, Marseilles. The firm specific variable includes any changes in the way operations in Hungary are conducted, whether it is strategy or HRM policies. The local unit specificities include the role of the joint venture in relation to Trianon and how this joint venture will fit into the long-term plan of the company. The company hopes that it will provide a good working relationship with the state supported airline, which will lead to more business in the future. The IHRM practices determine the employees that are hired and the training that is available to the employees. The reason for the assignment also is a major factor in determining the correct candidate. In the situation of Trianon, a joint venture with a Hungarian government back firm created a position that needed filling. The Quality Compliance Manager position allows Trianon to manage the joint venture operation, make sure it is successful, and build a strong relationship with Malev. The last major factor is the type of assignment. The Quality Compliance Manager assignment is long-term assignment because it is 3 years in duration. The joint venture is the first that the company has been involved in outside the UK so there is less familiarity on the administrative/compliance side. The candidate must act as an agent of direct control (Dowling et al., 2013) by assuring that compliance policies are followed and company strategy is implemented. Assessing whether a male or female would be the best fit for the position is also a factor that deserves consideration. The low number of female expatriates led Jessens, Cappellen, &Zanoni (2006) to research the following three myths: women have no desire to be in positions of authority in a foreign country, companies do not desire to place females in positions of authority while a foreign country, and women would be ineffective because of the views towards women in foreign countries. The research indicated that female expatriates do have conflict that arises related to their gender but the successful ones were able to turn the conflicts around based on the qualities that these women possess (Jessens et al., 2006). With all of these factors considered, I believe Marie Erten-Loiseau is the best candidate for the Quality Compliance Manager. References Dowling, P.J., Festing, M., & Engle, A.D. Sr. (2013). International Human Resource Management (6th ed.). Stamford, CT: Cengage Learning Janssens, M., Cappellen, T., &Zanoni, P. (2006). Successful female expatriates as agents: Positioning oneself through gender, hierarchy, and culture. Journal of World Business, 1-16. doi:10.1016/j.jwb.2006.01.001 2.) Case 8 – Questions 1 & 4 Multinational firms are often faced with recruiting and staffing decisions that could ultimately enhance or diminish the firm’s ability to be successful in a competitive global market. Perlmutter identified four staffing approaches for MNEs to consider based on the primary attitudes of international executives that would lay the foundation for MNEs during the recruitment and hiring process (Dowling, Festing, & Engle, 2013). At one point or another throughout the MacDougall family journey Lachlan and Lisa have served in one of the four capacities as an ethnocentric, polycentric, geocentric, and regiocentric employee. The ability to encompass all four attitudes that Perlmutter set forth is something that the MacDougall family has managed to do extremely well. The possibility for a multinational firm to recruit a family of this caliber that has been exposed and has an understanding of the positive and negative aspects of each attitude is phenomenal. This would be resourceful for any multinational firm. The MacDougal family’s exposure to cross-cultural management is also valuable. The diverse cultural background that the family has encountered on their international journey is a rarity. Cultural diversity and cross-cultural management play a critical role in MNEs because it produces a work environment that can transform the workplace into a place of learning and give the firm the availability to create new ideas for a more productive and competitive advantage over other firms (Sultana, Rashid, Mohiuddin, &Mazumder, 2013). This is something that is easy for the MacDougall family to bring to the table with the family’s given history. The expatriate lifestyle that has become second nature to the MacDougall family is beneficial for multinational firms for multifarious reasons Being raised around different cultures and then choosing to work internationally and learn different cultures has attributed to Lachlan’s successful career. The family’s ability to communicate and blend in socially among diverse cultures is an important aspect for international firms that want to stay competitive and be successful. The family has acclimated fairly easy to all of the places they have been and this is something that can be favorable when firms are recruiting employees. The MacDougall family has an upper-hand in the international marketplace naturally due to previous experiences with other countries and cultures. The exceptional way that the family has managed to conform to a multitude of other cultures and flourish is not an easy task. Marriage is not easy and many families experience a greater challenge avoiding divorcees when international mobility is involved. Lachlan and Lisa have been able to move together and this is an important aspect to the success of their marriage. Based on the case study they have a common desire to travel and both are successful in their careers. Lisa’s devotion to her husband’s successful career has put some strain on the marriage as she has had times where she felt she did not have her own identity. Military spouses experience this type of stress during long deployments and times that they have to hold the household together on their own. Another example is with employers who are transferred internationally for a short period of time or travel often. Separation of spouses can strain any marriage, but Lisa and Lachlan have been fortunate to avoid separation for any extended length of time. References Dowling, P.J., Festing, M., & Engle, A.D.Sr.(2013). International Human Resource Management. (6thed.). Stamford, CT: Cengage Sultana, M., Rashid, M., Mohiuddin, M. &Mazumder, M. (2013).Cross-cultural management and organizational performance.A Contnet analysis perspective.International Journal of Business and Management, 8(8), 133-146.

Select Case 1, 2, or 8 in the back of the textbook. After you have read the case, select at least one of the questions presented at the end.-If you select only one question, then you will need to elaborate with more examples and perspectives than if you select more than one, but the choice is yours. Fair warning: It is possible to fall into the trap of repeating oneself. To avoid that threat, think in advance of the different perspectives that you wish to explore. If you select more than one question, each answer will naturally be shorter. This may be a good approach if you discern that the questions lack strong potential to elicit in-depth answers. Remember to reply to the contributions of two other students in this exercise. This is a rule that we are only observing in the case analyses, given the relative complexity of the cases, compared to the chapter discussion questions. Always add value, from the textbook, news, personal experience, or all three. Indicate the case and question at the beginning, but avoid restating the question in your answer. In this respect, use the same method as in the chapter discussion questions, described in the Week 2 forum. Write at least 500 words (no minimum for replies, but do add value). Quoted passages do not contribute to the word count (so you will need to write more if you insert any quoted material). Post-edit your work carefully to catch errors. Avoid plagiarism at all cost. ——— Note on anomalous questions. Some questions will require you to work around selected details to fit the requisite discussion format. For example, Question 2 in Case 1 asks how your proposal will solve certain problems noted in answer to the previous question. If you have not actually answered Question 1, then you will have to assert one or more problems from the case, a proposed solution, and then an explanation of how your proposal may help. Question 3 is similar, in that you will need to identify a problem and a solution, followed by an argument about the budget. Although Alistair was expecting to hire a Project Engineer rather than a Quality Compliance Manager, the methods used to make the decision should be similar. The main difference in the Quality Compliance Manager position is that it is in a joint venture with a Hungarian government backed firm. International Joint Ventures (IJV) makes HRM practices more complicated because HRM practices and strategies are required for each IJV entity (Dowling, Festing, & Engle, 2013). HRM must address IJV in four stages, in which, each stage has an impact on the next. It is important for HRM to very thorough with each stage and communication through each stage is vital. To be successful, HRM must combine the IJV strategy along with the recruitment, selection, training, and development processes (Dowling et al., 2013). In light of the needs of the company and the new Quality Compliance Manager position, Alistair should choose the first candidate, Marie Erten-Loiseau. The fact that the job requires travel to France and Germany is a positive for Marie because she was born in France and was educated in France and Germany. The familiarity of these locations will help her as she meets with new business partners because she will have a good understanding of the policy and procedures required for companies in these two countries. Dowling et al., (2013), points out that the manager needs to be able to assess the desires of the stakeholders and be able to implement strategies based on their desires. Another reason for choosing Marie is that she has the most experience and has worked with Trianon for 13 years. The experience she has with the company is invaluable because she knows the goals of the company and strategies for implementing those goals. The last reason for choosing Marie is that she has been successful in her previous positions. She has lead two projects in two different countries and both were successful. This shows that she is able to adapt to the different practices of each country. There are many factors that Alistair should take into consideration to determine the correct choice for the Quality Compliance Manager position. The major factors that require consideration are the specificities of the entire situation, the reason for the assignment, and type of assignment. The four main specificities include context specificities, firm specific variables, local unit specificities, and IHRM practices (Dowling et al., 2013). The context specificities would include the differences in cultures between the assignment in Hungary and the base location for the Trianon, Marseilles. The firm specific variable includes any changes in the way operations in Hungary are conducted, whether it is strategy or HRM policies. The local unit specificities include the role of the joint venture in relation to Trianon and how this joint venture will fit into the long-term plan of the company. The company hopes that it will provide a good working relationship with the state supported airline, which will lead to more business in the future. The IHRM practices determine the employees that are hired and the training that is available to the employees. The reason for the assignment also is a major factor in determining the correct candidate. In the situation of Trianon, a joint venture with a Hungarian government back firm created a position that needed filling. The Quality Compliance Manager position allows Trianon to manage the joint venture operation, make sure it is successful, and build a strong relationship with Malev. The last major factor is the type of assignment. The Quality Compliance Manager assignment is long-term assignment because it is 3 years in duration. The joint venture is the first that the company has been involved in outside the UK so there is less familiarity on the administrative/compliance side. The candidate must act as an agent of direct control (Dowling et al., 2013) by assuring that compliance policies are followed and company strategy is implemented. Assessing whether a male or female would be the best fit for the position is also a factor that deserves consideration. The low number of female expatriates led Jessens, Cappellen, &Zanoni (2006) to research the following three myths: women have no desire to be in positions of authority in a foreign country, companies do not desire to place females in positions of authority while a foreign country, and women would be ineffective because of the views towards women in foreign countries. The research indicated that female expatriates do have conflict that arises related to their gender but the successful ones were able to turn the conflicts around based on the qualities that these women possess (Jessens et al., 2006). With all of these factors considered, I believe Marie Erten-Loiseau is the best candidate for the Quality Compliance Manager. References Dowling, P.J., Festing, M., & Engle, A.D. Sr. (2013). International Human Resource Management (6th ed.). Stamford, CT: Cengage Learning Janssens, M., Cappellen, T., &Zanoni, P. (2006). Successful female expatriates as agents: Positioning oneself through gender, hierarchy, and culture. Journal of World Business, 1-16. doi:10.1016/j.jwb.2006.01.001 2.) Case 8 – Questions 1 & 4 Multinational firms are often faced with recruiting and staffing decisions that could ultimately enhance or diminish the firm’s ability to be successful in a competitive global market. Perlmutter identified four staffing approaches for MNEs to consider based on the primary attitudes of international executives that would lay the foundation for MNEs during the recruitment and hiring process (Dowling, Festing, & Engle, 2013). At one point or another throughout the MacDougall family journey Lachlan and Lisa have served in one of the four capacities as an ethnocentric, polycentric, geocentric, and regiocentric employee. The ability to encompass all four attitudes that Perlmutter set forth is something that the MacDougall family has managed to do extremely well. The possibility for a multinational firm to recruit a family of this caliber that has been exposed and has an understanding of the positive and negative aspects of each attitude is phenomenal. This would be resourceful for any multinational firm. The MacDougal family’s exposure to cross-cultural management is also valuable. The diverse cultural background that the family has encountered on their international journey is a rarity. Cultural diversity and cross-cultural management play a critical role in MNEs because it produces a work environment that can transform the workplace into a place of learning and give the firm the availability to create new ideas for a more productive and competitive advantage over other firms (Sultana, Rashid, Mohiuddin, &Mazumder, 2013). This is something that is easy for the MacDougall family to bring to the table with the family’s given history. The expatriate lifestyle that has become second nature to the MacDougall family is beneficial for multinational firms for multifarious reasons Being raised around different cultures and then choosing to work internationally and learn different cultures has attributed to Lachlan’s successful career. The family’s ability to communicate and blend in socially among diverse cultures is an important aspect for international firms that want to stay competitive and be successful. The family has acclimated fairly easy to all of the places they have been and this is something that can be favorable when firms are recruiting employees. The MacDougall family has an upper-hand in the international marketplace naturally due to previous experiences with other countries and cultures. The exceptional way that the family has managed to conform to a multitude of other cultures and flourish is not an easy task. Marriage is not easy and many families experience a greater challenge avoiding divorcees when international mobility is involved. Lachlan and Lisa have been able to move together and this is an important aspect to the success of their marriage. Based on the case study they have a common desire to travel and both are successful in their careers. Lisa’s devotion to her husband’s successful career has put some strain on the marriage as she has had times where she felt she did not have her own identity. Military spouses experience this type of stress during long deployments and times that they have to hold the household together on their own. Another example is with employers who are transferred internationally for a short period of time or travel often. Separation of spouses can strain any marriage, but Lisa and Lachlan have been fortunate to avoid separation for any extended length of time. References Dowling, P.J., Festing, M., & Engle, A.D.Sr.(2013). International Human Resource Management. (6thed.). Stamford, CT: Cengage Sultana, M., Rashid, M., Mohiuddin, M. &Mazumder, M. (2013).Cross-cultural management and organizational performance.A Contnet analysis perspective.International Journal of Business and Management, 8(8), 133-146.

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Research Paper Write a 10 to 15 page research paper double spaced on some aspect of logistics systems development of the student’s choosing. Note: line spacing format for research papers is double space RESEARCH REPORT FORMAT (MUST BE IN APA FORMAT) STATEMENT OF THE PROBLEM: What is the format that MGT 5061 students should use in their research papers? BACKGROUND: Here each student should comment on the background of their problem. The Background may be simply an academic issue or perhaps involve real problems at work. In any event, the importance of each student’s problem needs to be discussed so the reader understands why he or she should read more about it. DISCUSSION: Now comes the part where each student must write about the research done on his or her problem. What have other experts said about the problem? This information can come from other textbooks, magazine articles, newspapers, or personal interviews. If the student wants to use his or her own experience, that is fine, but the experience must be justified as to its validity. If a comparison between alternatives is being described, the choices should be discussed here in enough detail that a conclusion can be derived as to which one is better. Whenever an information source is cited, it must be referenced by footnotes so that the reader can find exactly where the information came from and what the author of the information exactly said. The style of footnoting used is not important; what is important is that each piece of information is referenced with a page number. Last, a bibliography must be included at the end of the paper. CONCLUSIONS: Once the background research is completed, the next step is to outline the conclusions that can be substantiated from the research DISCUSSION. The substantiation must be defended by saying why the research evidence supports the conclusions that are made. RECOMMENDATIONS: This section is to be used by those students who want to solve problems. By this time, the various choices have been examined in the DISCUSSION above and one was chosen in the CONCLUSION. So, the recommendation should be a simple statement that the preferred alternative be selected with the anticipated benefits that will be derived by the recommended selection.

Research Paper Write a 10 to 15 page research paper double spaced on some aspect of logistics systems development of the student’s choosing. Note: line spacing format for research papers is double space RESEARCH REPORT FORMAT (MUST BE IN APA FORMAT) STATEMENT OF THE PROBLEM: What is the format that MGT 5061 students should use in their research papers? BACKGROUND: Here each student should comment on the background of their problem. The Background may be simply an academic issue or perhaps involve real problems at work. In any event, the importance of each student’s problem needs to be discussed so the reader understands why he or she should read more about it. DISCUSSION: Now comes the part where each student must write about the research done on his or her problem. What have other experts said about the problem? This information can come from other textbooks, magazine articles, newspapers, or personal interviews. If the student wants to use his or her own experience, that is fine, but the experience must be justified as to its validity. If a comparison between alternatives is being described, the choices should be discussed here in enough detail that a conclusion can be derived as to which one is better. Whenever an information source is cited, it must be referenced by footnotes so that the reader can find exactly where the information came from and what the author of the information exactly said. The style of footnoting used is not important; what is important is that each piece of information is referenced with a page number. Last, a bibliography must be included at the end of the paper. CONCLUSIONS: Once the background research is completed, the next step is to outline the conclusions that can be substantiated from the research DISCUSSION. The substantiation must be defended by saying why the research evidence supports the conclusions that are made. RECOMMENDATIONS: This section is to be used by those students who want to solve problems. By this time, the various choices have been examined in the DISCUSSION above and one was chosen in the CONCLUSION. So, the recommendation should be a simple statement that the preferred alternative be selected with the anticipated benefits that will be derived by the recommended selection.

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Which of the following IS NOT a limitation of groups? a) Group think mentality b) Group members can come to realize that they are not alone, unique, or abnormal in their problems or concerns c) The problems of individuals may not be dealt with in enough depth d) Individuals may use the group to escape or for selfish purposes and disrupt the group process.

Which of the following IS NOT a limitation of groups? a) Group think mentality b) Group members can come to realize that they are not alone, unique, or abnormal in their problems or concerns c) The problems of individuals may not be dealt with in enough depth d) Individuals may use the group to escape or for selfish purposes and disrupt the group process.

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