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

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

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

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WEEKLY ASSIGNMENT #2 YOU 1. Verify for the Cobb-Douglas production function P(L;K) = 1:01L:75K:25 that the production will be doubled if both the amount of labor and the amount of capital are doubled. How much must you increase capital K to double production? How much must you increase labor by to double production? 1 2. Let F(x; y) = 1+ p 4 ? y2. Evaluate F(3; 1). Find and sketch the domain of F. Find the range of F. 2 3. Draw a contour map of the function showing several level curves. (a) g(x; y) = x2 ? y2 (b) s(x; y) = y=(x2 + y2) 3 4. Find the limit if it exists or show that the limit does not exist. You do not have to use the epsilon delta method so it will either be “obviously” continuous or you will have to show that it is not by finding two paths which give different results. (a) lim (x;y)!(2;?1) x2y + xy2 x2 ? y2 (b) lim (x;y)!(0;0) x4 ? 4y2 x2 + 2y2 (c) lim (x;y)!(0;0) xy p x2 + y2 4 5. The temperature T at a location in the Norther Hemisphere depends on the longitude x, the latitude y, and the time t. What are the meaning of the partial derivatives @T=@t; @T=@x; @T=@y? Moscow lies at 46:73N; 117W. Suppose that at 9 am on January 1st the wind is blowing hot air to the northeast so the air to the west and south is warm, and the air to the north and east is cooler. Would you expect fx(117; 4673; 9); fy(117; 4673; 9); ft(117; 4673; 9) to be positive negative or positive? 5 6. Find the first partial derivatives of the following functions. (a) f(x; y) = x4 + 5xy3 (b) g(x; y) = t2e?t (c) h(s; t) = ln(s + t2) (d) i(x; y) = x y (e) R(p; q) = arctan pq2 6 7. Find @z=@x and @z=@y for the following, assuming that f and g are differentiable single variable functions Hint: Your answer should use f0 and/or g0. z = f(x)g(y) ; z = f(x=y) 7

WEEKLY ASSIGNMENT #2 YOU 1. Verify for the Cobb-Douglas production function P(L;K) = 1:01L:75K:25 that the production will be doubled if both the amount of labor and the amount of capital are doubled. How much must you increase capital K to double production? How much must you increase labor by to double production? 1 2. Let F(x; y) = 1+ p 4 ? y2. Evaluate F(3; 1). Find and sketch the domain of F. Find the range of F. 2 3. Draw a contour map of the function showing several level curves. (a) g(x; y) = x2 ? y2 (b) s(x; y) = y=(x2 + y2) 3 4. Find the limit if it exists or show that the limit does not exist. You do not have to use the epsilon delta method so it will either be “obviously” continuous or you will have to show that it is not by finding two paths which give different results. (a) lim (x;y)!(2;?1) x2y + xy2 x2 ? y2 (b) lim (x;y)!(0;0) x4 ? 4y2 x2 + 2y2 (c) lim (x;y)!(0;0) xy p x2 + y2 4 5. The temperature T at a location in the Norther Hemisphere depends on the longitude x, the latitude y, and the time t. What are the meaning of the partial derivatives @T=@t; @T=@x; @T=@y? Moscow lies at 46:73N; 117W. Suppose that at 9 am on January 1st the wind is blowing hot air to the northeast so the air to the west and south is warm, and the air to the north and east is cooler. Would you expect fx(117; 4673; 9); fy(117; 4673; 9); ft(117; 4673; 9) to be positive negative or positive? 5 6. Find the first partial derivatives of the following functions. (a) f(x; y) = x4 + 5xy3 (b) g(x; y) = t2e?t (c) h(s; t) = ln(s + t2) (d) i(x; y) = x y (e) R(p; q) = arctan pq2 6 7. Find @z=@x and @z=@y for the following, assuming that f and g are differentiable single variable functions Hint: Your answer should use f0 and/or g0. z = f(x)g(y) ; z = f(x=y) 7

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As a student of ECON1005, suppose you were asked to assist a co-worker in investigating whether there is an association between gender and annual salary of researchers in your country. Data was gathered from researchers in your country in your country. MINITAB was used to generate stem-and-leaf diagrams for the salaries of both the female and male researchers. See Exhibit 1 below. Exhibit 1 Stem-and-Leaf Display: Salary Female, Salary Male Stem-and-leaf of Salary Female N = 15 Leaf Unit = 1000.0 1 5 8 5 6 1345 (3) 7 148 7 8 389 4 9 2245 (a) Calculate the mean salary for both the female and the male researchers. All relevant working must be clearly shown. (b) Calculate the standard deviation of the salaries for both the female and the male researchers. All relevant working must be clearly shown. (c) Comment on your answers for parts (c) and (d), in relation to the purpose of your study, that is, the association between gender and annual salary of researchers in your country. MINITAB was used to generate box-and-whisker diagrams for the salaries of both the female and male researchers. See Exhibit 2 below. Exhibit 2 (d) With reference to the box-and-whisker diagrams, compare the salaries of the researchers selected in your sample, by gender. Ensure that you comment on the skewness, the median, the interquartile range, the minimum and the maximum values of both diagrams. MINITAB was used to generate the descriptive statistics for all the 35 researchers selected in the sample. See Exhibit 3 below. Exhibit 3 Descriptive Statistics: ResearcherSalary Variable N N* Mean SE Mean TrMean StDev Minimum Q1 Median Q3 Salary 35 0 82951 2266 83200 13404 58100 74800 83800 94300 Variable Maximum Salary 104500 (e) What does TrMean represent? Comment on the value of the TrMean and show how this value was calculated. (f) For further analysis, a table is drawn showing the number of females and the number of males whose salaries were below the median salary and equal to or above the median salary. Complete the table below: Salary < $83800 Salary ? $83800 Total Female Male Total 35 (g) Using your table in part (h), determine the probability that a randomly selected researcher from your sample, is a female OR has a salary < $83800. (h) Using your table in part (h), determine the probability that a randomly selected researcher from your sample, is a female AND has a salary < $83800. (i) Given that a randomly selected researcher from your sample is a female, what is the probability that her annual salary is < $83800? (j) Are the events “female researcher” and “salary < $83800” mutually independent events? Support your answer with relevant calculations or explanations.

As a student of ECON1005, suppose you were asked to assist a co-worker in investigating whether there is an association between gender and annual salary of researchers in your country. Data was gathered from researchers in your country in your country. MINITAB was used to generate stem-and-leaf diagrams for the salaries of both the female and male researchers. See Exhibit 1 below. Exhibit 1 Stem-and-Leaf Display: Salary Female, Salary Male Stem-and-leaf of Salary Female N = 15 Leaf Unit = 1000.0 1 5 8 5 6 1345 (3) 7 148 7 8 389 4 9 2245 (a) Calculate the mean salary for both the female and the male researchers. All relevant working must be clearly shown. (b) Calculate the standard deviation of the salaries for both the female and the male researchers. All relevant working must be clearly shown. (c) Comment on your answers for parts (c) and (d), in relation to the purpose of your study, that is, the association between gender and annual salary of researchers in your country. MINITAB was used to generate box-and-whisker diagrams for the salaries of both the female and male researchers. See Exhibit 2 below. Exhibit 2 (d) With reference to the box-and-whisker diagrams, compare the salaries of the researchers selected in your sample, by gender. Ensure that you comment on the skewness, the median, the interquartile range, the minimum and the maximum values of both diagrams. MINITAB was used to generate the descriptive statistics for all the 35 researchers selected in the sample. See Exhibit 3 below. Exhibit 3 Descriptive Statistics: ResearcherSalary Variable N N* Mean SE Mean TrMean StDev Minimum Q1 Median Q3 Salary 35 0 82951 2266 83200 13404 58100 74800 83800 94300 Variable Maximum Salary 104500 (e) What does TrMean represent? Comment on the value of the TrMean and show how this value was calculated. (f) For further analysis, a table is drawn showing the number of females and the number of males whose salaries were below the median salary and equal to or above the median salary. Complete the table below: Salary < $83800 Salary ? $83800 Total Female Male Total 35 (g) Using your table in part (h), determine the probability that a randomly selected researcher from your sample, is a female OR has a salary < $83800. (h) Using your table in part (h), determine the probability that a randomly selected researcher from your sample, is a female AND has a salary < $83800. (i) Given that a randomly selected researcher from your sample is a female, what is the probability that her annual salary is < $83800? (j) Are the events “female researcher” and “salary < $83800” mutually independent events? Support your answer with relevant calculations or explanations.

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Reading Guide 3 CHEM 101 Check here if you want your paper returned Chapter 3 – Section 3.1-3.4 Introduction to Chemistry Dr. Bragg Printed Last Name: First Name: WKUID: 1. Express in your own words the meaning of these terms: a. Hypothesis b. Law c. Theory d. Conservation e. Proportion f. Radioactive g. Atomic Number h. Mass Number i. Isotope j. Spectrum k. Ground State l. Excited State m. Quantum n. Valence o. Shell p. Subshell q. Orbital 2. Briefly describe the main points of Dalton’s Atomic Theory. On Time: Complete: Questions: Total Score: 3. Who experimentally verified the Law of Conservation of Matter? 4. Who experimentally verified the Law of Definite Proportions? 5. What are the three most important subatomic particles, and what is the charge on each? 6. Who discovered natural radioactivity? 7. What are the three main radioactive ‘particles,’ and what is the charge on each? 8. Who was the student that set up the experiments and made the observations that lead to the discovery of the nucleus of the atom? 9. Considering atomic numbers and mass numbers, which is the same among a set of isotopes and which is different? 10. What is the difference between a continuous spectrum and a line spectrum? 11. Who proposed the Shell Model of the hydrogen atom based on small energy steps between adjacent levels for electrons? 12. Which end of the electromagnetic spectrum is higher in ENERGY, ı-rays or radio waves? 13. Who proposed the mathematical wave theory that explained the existence of orbitals? 14. Give the general subshell filling order for electrons in ground state atoms. Reading Guide 3 CHEM 101 Dr. Bragg Chapter 3 – Sections 3.1 – 3.4 Introduction to Chemistry Page 2

Reading Guide 3 CHEM 101 Check here if you want your paper returned Chapter 3 – Section 3.1-3.4 Introduction to Chemistry Dr. Bragg Printed Last Name: First Name: WKUID: 1. Express in your own words the meaning of these terms: a. Hypothesis b. Law c. Theory d. Conservation e. Proportion f. Radioactive g. Atomic Number h. Mass Number i. Isotope j. Spectrum k. Ground State l. Excited State m. Quantum n. Valence o. Shell p. Subshell q. Orbital 2. Briefly describe the main points of Dalton’s Atomic Theory. On Time: Complete: Questions: Total Score: 3. Who experimentally verified the Law of Conservation of Matter? 4. Who experimentally verified the Law of Definite Proportions? 5. What are the three most important subatomic particles, and what is the charge on each? 6. Who discovered natural radioactivity? 7. What are the three main radioactive ‘particles,’ and what is the charge on each? 8. Who was the student that set up the experiments and made the observations that lead to the discovery of the nucleus of the atom? 9. Considering atomic numbers and mass numbers, which is the same among a set of isotopes and which is different? 10. What is the difference between a continuous spectrum and a line spectrum? 11. Who proposed the Shell Model of the hydrogen atom based on small energy steps between adjacent levels for electrons? 12. Which end of the electromagnetic spectrum is higher in ENERGY, ı-rays or radio waves? 13. Who proposed the mathematical wave theory that explained the existence of orbitals? 14. Give the general subshell filling order for electrons in ground state atoms. Reading Guide 3 CHEM 101 Dr. Bragg Chapter 3 – Sections 3.1 – 3.4 Introduction to Chemistry Page 2

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Chapter 07 Reading Questions Due: 11:59pm on Friday, May 23, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Chapter 7 Reading Quiz Question 17 Part A A lake is currently at high pool, with the same amount of water flowing into the lake as is flowing over the spillway. Which of the following temporary changes would increase the resident time of water in this lake? ANSWER: Chapter 7 Reading Quiz Question 16 Part A A large reservoir behind a dam is rapidly rising, as rain and melting snow add more water than is being released out of the dam’s spillway. In this situation, _____. ANSWER: Chapter 7 Reading Quiz Question 1 Part A Which one of the following statements is correct? ANSWER: Double the rate of water flow into the lake and double the rate of water flow out of the lake, while keeping the lake at the same level. Keep the inflow into the lake the same, but release twice as much water from the lake, resulting in a lowering of the lake level. Decrease the inflow into the lake by half, and decrease the outflow of the lake by half. None of the choices would increase the resident time in the lake. the net flux is positive and the capital of water within the reservoir is decreasing the net flux is positive and the capital of water within the reservoir is increasing the net flux is negative and the capital of water within the reservoir is increasing the net flux is negative and the capital of water within the reservoir is decreasing Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 1 of 8 5/21/2014 8:01 PM Chapter 7 Reading Quiz Question 18 Part A A raging river cascades down a granite mountain and eventually reaches the ocean. At the mouth of the river is a beautiful sandy beach composed of fine grains of granite particles from the river. The entire process of producing this sand is a result of _____. ANSWER: Chapter 7 Reading Quiz Question 4 Part A The physical and chemical properties of soils are primarily determined by _____. ANSWER: Chapter 7 Reading Quiz Question 19 Part A Several inches of rain fall over a field of tall corn, soaking into the soil and draining into ditches. Within an hour, there is no standing water and the humidity over the field rises quickly. At a nearby shopping mall, the rainwater fell onto blacktop and drained to sewer pipes, which carried the water directly into a stream. Which of the following occurred in The cycling time of an element or molecule in an ecosystem is equal to the sum of all the flux times. The cycling time is how long it takes an element or molecule to pass through a biogeochemical cycle. The cycling time of water moving through an ecosystem is typically shorter than the resident time in any pool in this system. The amount of time that water spends in an ocean is the cycling time. mineral evaporation erosion, weathering, transport, and then deposition erosion, dissolution, and precipitation organisms consuming and eroding granite the properties of rock from which the soils develop the amount of precipitation that the soil experiences the range of temperatures that the soil experiences the types of animals that live and move through the soils Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 2 of 8 5/21/2014 8:01 PM the cornfield but not in the parking lot? ANSWER: Chapter 7 Reading Quiz Question 6 Part A Most of the water on Earth is found in _____. ANSWER: Chapter 7 Reading Quiz Question 5 Part A Which one of the following primarily results from the effects of solar energy? ANSWER: Chapter 7 Reading Quiz Question 20 Part A A rural Minnesota farmer grows a variety of vegetables to feed her family. In addition, she cuts down some of her dead trees for firewood to heat her home in the winter. This farmer is adding to the flux of the carbon cycle in her region by _____. precipitation evaporation runoff transpiration the polar ice caps lakes and streams aquifers the oceans evaporation of water from a lake the formation of ice on the top of a pond movement of ocean tides the movement of water over a waterfall Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 3 of 8 5/21/2014 8:01 PM ANSWER: Chapter 7 Reading Quiz Question 8 Part A In a terrestrial ecosystem, most carbon is stored in the biomass of _____. ANSWER: Chapter 7 Reading Quiz Question 7 Part A In which of the following countries would we expect that the terrestrial ecosystems have the highest net primary production and biomass? ANSWER: Chapter 7 Reading Quiz Question 22 Part A Some farmers in the Midwest of the United States rotate their crops from year to year, switching from soybeans to corn on the same fields. What is one of the advantages of doing this? encouraging photosynthesis as she raises crops burning carbon-based fuels by consuming vegetables grown on her farm All of the choices are correct. the animals living there air the top layers of soil containing dead organisms living plants China Australia Brazil United States Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 4 of 8 5/21/2014 8:01 PM ANSWER: Chapter 7 Reading Quiz Question 10 Part A Most nitrogen enters the biosphere through the process of _____ ANSWER: Chapter 7 Reading Quiz Question 9 Part A Where do we expect to find the least amount of nitrogen? ANSWER: Chapter 7 Reading Quiz Question 12 Part A Along the west coast of the United States, upwellings bring deep ocean waters to the surface, carrying with them _____, which greatly increases NPP. ANSWER: The corn crop benefits from reactive nitrogen added to the soil by the soybean crop. Both crops require the same fertilizing supplies, so farmers save by buying fertilizer in bulk. Soybeans add large amounts of carbon dioxide to the soil, which helps the corn crop. Corn adds large amounts of phosphorus to the soil, which helps the soybean crop. nitrogen fixation in which bacteria convert N2 to NH3 cellular respiration, in which animals convert N2 to NH4 fermentation in which bacteria convert N2 to HNO3 photosynthesis, in which plants convert N2 to NO2 in Earth’s crust in plants in animals in the atmosphere Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 5 of 8 5/21/2014 8:01 PM Chapter 7 Reading Quiz Question 11 Part A Which one of the following statements about the carbon, phosphorus, and nitrogen cycles is true? ANSWER: Chapter 7 Reading Quiz Question 24 Part A A large coal-burning power plant is about 50 miles upwind from a lake that used to be popular for fishing. But now, just five years after the plant was constructed, the fish populations are decreasing dramatically. Which one of the following impacts of this coal-burning power plant is most likely hurting the fish populations in this downwind lake? ANSWER: Chapter 7 Reading Quiz Question 14 Part A Which one of the following statements about sulfur is correct? ANSWER: oxygen phosphate carbon sulfur Phosphorus is virtually absent in the atmosphere. The major source of carbon used by plants is the soil. Bacteria drive the phosphorus cycle. The major source of nitrogen used by plants is the air. insufficient sunlight reaching the lake low oxygen levels from burning fossil fuels eutrophication of the lake acidification of the lake Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 6 of 8 5/21/2014 8:01 PM Chapter 7 Reading Quiz Question 13 Part A Nitrogen and sulfur are important to all organisms because they are important constituents of _____. ANSWER: Chapter 7 Reading Quiz Question 25 Part A In Iowa, a small, deep lake in the summer becomes stratified with warmer, less-dense water at the surface and colder, denser water near the bottom. As fall air temperatures decrease, the surface water cools and then drops toward the bottom, mixing the lake levels together. As a result of this mixing, _____. ANSWER: Chapter 7 Reading Quiz Question 15 Part A A fire spreads across hundreds of acres of prairie, burning most of the plant parts above the ground. Compared to before the fire, right after this fire the pool of nutrients in the prairie plants _____. The main pool of sulfur is in the atmosphere where the flux is high and the residence time is long. The main pool of sulfur is in rocks. The flux of sulfur through the atmosphere is high and the residence is short. The main pool of sulfur is in the atmosphere where the flux is low and the residence time is long. The main pool of sulfur is in rocks. The flux of sulfur through the atmosphere is low and the residence is short. nucleic acids glucose phosphates some amino acids nitrogen and phosphorus are added to the lake nitrogen and phosphorus decrease near the surface of the lake nitrogen and phosphorus increase near the surface of the lake None of the choices is correct. Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 7 of 8 5/21/2014 8:01 PM ANSWER: Score Summary: Your score on this assignment is 0.0%. You received 0 out of a possible total of 21 points. and the soil decreases increases and the pool of nutrients in the soil decreases and the soil increases decreases and the pool of nutrients in the soil increases Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 8 of 8 5/21/2014 8:01 PM

Chapter 07 Reading Questions Due: 11:59pm on Friday, May 23, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Chapter 7 Reading Quiz Question 17 Part A A lake is currently at high pool, with the same amount of water flowing into the lake as is flowing over the spillway. Which of the following temporary changes would increase the resident time of water in this lake? ANSWER: Chapter 7 Reading Quiz Question 16 Part A A large reservoir behind a dam is rapidly rising, as rain and melting snow add more water than is being released out of the dam’s spillway. In this situation, _____. ANSWER: Chapter 7 Reading Quiz Question 1 Part A Which one of the following statements is correct? ANSWER: Double the rate of water flow into the lake and double the rate of water flow out of the lake, while keeping the lake at the same level. Keep the inflow into the lake the same, but release twice as much water from the lake, resulting in a lowering of the lake level. Decrease the inflow into the lake by half, and decrease the outflow of the lake by half. None of the choices would increase the resident time in the lake. the net flux is positive and the capital of water within the reservoir is decreasing the net flux is positive and the capital of water within the reservoir is increasing the net flux is negative and the capital of water within the reservoir is increasing the net flux is negative and the capital of water within the reservoir is decreasing Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 1 of 8 5/21/2014 8:01 PM Chapter 7 Reading Quiz Question 18 Part A A raging river cascades down a granite mountain and eventually reaches the ocean. At the mouth of the river is a beautiful sandy beach composed of fine grains of granite particles from the river. The entire process of producing this sand is a result of _____. ANSWER: Chapter 7 Reading Quiz Question 4 Part A The physical and chemical properties of soils are primarily determined by _____. ANSWER: Chapter 7 Reading Quiz Question 19 Part A Several inches of rain fall over a field of tall corn, soaking into the soil and draining into ditches. Within an hour, there is no standing water and the humidity over the field rises quickly. At a nearby shopping mall, the rainwater fell onto blacktop and drained to sewer pipes, which carried the water directly into a stream. Which of the following occurred in The cycling time of an element or molecule in an ecosystem is equal to the sum of all the flux times. The cycling time is how long it takes an element or molecule to pass through a biogeochemical cycle. The cycling time of water moving through an ecosystem is typically shorter than the resident time in any pool in this system. The amount of time that water spends in an ocean is the cycling time. mineral evaporation erosion, weathering, transport, and then deposition erosion, dissolution, and precipitation organisms consuming and eroding granite the properties of rock from which the soils develop the amount of precipitation that the soil experiences the range of temperatures that the soil experiences the types of animals that live and move through the soils Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 2 of 8 5/21/2014 8:01 PM the cornfield but not in the parking lot? ANSWER: Chapter 7 Reading Quiz Question 6 Part A Most of the water on Earth is found in _____. ANSWER: Chapter 7 Reading Quiz Question 5 Part A Which one of the following primarily results from the effects of solar energy? ANSWER: Chapter 7 Reading Quiz Question 20 Part A A rural Minnesota farmer grows a variety of vegetables to feed her family. In addition, she cuts down some of her dead trees for firewood to heat her home in the winter. This farmer is adding to the flux of the carbon cycle in her region by _____. precipitation evaporation runoff transpiration the polar ice caps lakes and streams aquifers the oceans evaporation of water from a lake the formation of ice on the top of a pond movement of ocean tides the movement of water over a waterfall Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 3 of 8 5/21/2014 8:01 PM ANSWER: Chapter 7 Reading Quiz Question 8 Part A In a terrestrial ecosystem, most carbon is stored in the biomass of _____. ANSWER: Chapter 7 Reading Quiz Question 7 Part A In which of the following countries would we expect that the terrestrial ecosystems have the highest net primary production and biomass? ANSWER: Chapter 7 Reading Quiz Question 22 Part A Some farmers in the Midwest of the United States rotate their crops from year to year, switching from soybeans to corn on the same fields. What is one of the advantages of doing this? encouraging photosynthesis as she raises crops burning carbon-based fuels by consuming vegetables grown on her farm All of the choices are correct. the animals living there air the top layers of soil containing dead organisms living plants China Australia Brazil United States Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 4 of 8 5/21/2014 8:01 PM ANSWER: Chapter 7 Reading Quiz Question 10 Part A Most nitrogen enters the biosphere through the process of _____ ANSWER: Chapter 7 Reading Quiz Question 9 Part A Where do we expect to find the least amount of nitrogen? ANSWER: Chapter 7 Reading Quiz Question 12 Part A Along the west coast of the United States, upwellings bring deep ocean waters to the surface, carrying with them _____, which greatly increases NPP. ANSWER: The corn crop benefits from reactive nitrogen added to the soil by the soybean crop. Both crops require the same fertilizing supplies, so farmers save by buying fertilizer in bulk. Soybeans add large amounts of carbon dioxide to the soil, which helps the corn crop. Corn adds large amounts of phosphorus to the soil, which helps the soybean crop. nitrogen fixation in which bacteria convert N2 to NH3 cellular respiration, in which animals convert N2 to NH4 fermentation in which bacteria convert N2 to HNO3 photosynthesis, in which plants convert N2 to NO2 in Earth’s crust in plants in animals in the atmosphere Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 5 of 8 5/21/2014 8:01 PM Chapter 7 Reading Quiz Question 11 Part A Which one of the following statements about the carbon, phosphorus, and nitrogen cycles is true? ANSWER: Chapter 7 Reading Quiz Question 24 Part A A large coal-burning power plant is about 50 miles upwind from a lake that used to be popular for fishing. But now, just five years after the plant was constructed, the fish populations are decreasing dramatically. Which one of the following impacts of this coal-burning power plant is most likely hurting the fish populations in this downwind lake? ANSWER: Chapter 7 Reading Quiz Question 14 Part A Which one of the following statements about sulfur is correct? ANSWER: oxygen phosphate carbon sulfur Phosphorus is virtually absent in the atmosphere. The major source of carbon used by plants is the soil. Bacteria drive the phosphorus cycle. The major source of nitrogen used by plants is the air. insufficient sunlight reaching the lake low oxygen levels from burning fossil fuels eutrophication of the lake acidification of the lake Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 6 of 8 5/21/2014 8:01 PM Chapter 7 Reading Quiz Question 13 Part A Nitrogen and sulfur are important to all organisms because they are important constituents of _____. ANSWER: Chapter 7 Reading Quiz Question 25 Part A In Iowa, a small, deep lake in the summer becomes stratified with warmer, less-dense water at the surface and colder, denser water near the bottom. As fall air temperatures decrease, the surface water cools and then drops toward the bottom, mixing the lake levels together. As a result of this mixing, _____. ANSWER: Chapter 7 Reading Quiz Question 15 Part A A fire spreads across hundreds of acres of prairie, burning most of the plant parts above the ground. Compared to before the fire, right after this fire the pool of nutrients in the prairie plants _____. The main pool of sulfur is in the atmosphere where the flux is high and the residence time is long. The main pool of sulfur is in rocks. The flux of sulfur through the atmosphere is high and the residence is short. The main pool of sulfur is in the atmosphere where the flux is low and the residence time is long. The main pool of sulfur is in rocks. The flux of sulfur through the atmosphere is low and the residence is short. nucleic acids glucose phosphates some amino acids nitrogen and phosphorus are added to the lake nitrogen and phosphorus decrease near the surface of the lake nitrogen and phosphorus increase near the surface of the lake None of the choices is correct. Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 7 of 8 5/21/2014 8:01 PM ANSWER: Score Summary: Your score on this assignment is 0.0%. You received 0 out of a possible total of 21 points. and the soil decreases increases and the pool of nutrients in the soil decreases and the soil increases decreases and the pool of nutrients in the soil increases Chapter 07 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 8 of 8 5/21/2014 8:01 PM

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Que 5: the temperature T at a location in the Norther Hemisphere depends on the longitude x, the latitude y, and the time t. what are the meaning of the partial derivatives ∂T/∂t, ∂T/∂x,∂T/∂y ? Moscow lies at 46.73 0 N, 117 0 W. suppose that at 9 am on January 1st the wind is blowing hot air to the northeast so the air to the west and south is warm and the air to the north and east is cooler. Would you expect fx (46.73 0 , 117 0, 9), fy (46.73 0 , 117 0, 9) to be positive negative or positive?

Que 5: the temperature T at a location in the Norther Hemisphere depends on the longitude x, the latitude y, and the time t. what are the meaning of the partial derivatives ∂T/∂t, ∂T/∂x,∂T/∂y ? Moscow lies at 46.73 0 N, 117 0 W. suppose that at 9 am on January 1st the wind is blowing hot air to the northeast so the air to the west and south is warm and the air to the north and east is cooler. Would you expect fx (46.73 0 , 117 0, 9), fy (46.73 0 , 117 0, 9) to be positive negative or positive?

Sometimes during embryonic development, neural tube defects occur, and they can result in which of the following? Question 9 options: Spina bifida Down syndrome Klinefelter’s syndrome YY syndrome

Sometimes during embryonic development, neural tube defects occur, and they can result in which of the following? Question 9 options: Spina bifida Down syndrome Klinefelter’s syndrome YY syndrome

Sometimes during embryonic development, neural tube defects occur, and they … Read More...
The current recommendation for most women, after becoming sexually active or starting at age 21, is to have a Pap smear done Question 9 options: once every year once every 6 months once every 3 years once every 5 years

The current recommendation for most women, after becoming sexually active or starting at age 21, is to have a Pap smear done Question 9 options: once every year once every 6 months once every 3 years once every 5 years