QUESTION 1 1. Convert 30 degrees 2 minutes to decimal degrees. Give this answer to 6 decimal places. Do not provide units. You know those are decimal degrees. 5 points QUESTION 2 1. Convert 342 degrees 6 minutes and 41 seconds to decimal degrees. Show your answers to only 6 decimal places. Do not give units. 5 points QUESTION 3 1. COMPUTE the sin of 52 degrees. Give the answer to 6 decimal places. 5 points QUESTION 4 1. What is the sine of 277 degrees and 16 minutes? Give your answer to 6 decimal places. Pay attention to rounding. 5 points QUESTION 5 1. This is a right triangle problem. Angle A is 90 degrees. Draw the triangle and label it as we did in lecture. If angle B is 24 degrees 43 minutes and side c is 395.82 feet, what is the distance in feet of side b? Give your answer to two decimal places. Do not provide units. Those are in feet – right? 10 points QUESTION 6 1. This is a right triangle problem with angle A being the 90 degree angle. It should look like the one from lecture. If angle B is 25 degrees 18 minutes and side c is 206.1 feet, what is the distance to two decimal places of side a? Give your answer to two decimal places. Do not provide units – those are in feet. 10 points QUESTION 7 1. You are given a right triangle with angle A being the 90 degree angle – just like in lecture. If angle C is 42 degrees 9 minutes and side a is 401.73 feet, what is the length of side c? Give your answer to two decimal places. The units are feet – don’t list those. 10 points QUESTION 8 Ad by Browse Safe | Close 1. It is desired to determine the height of a flagpole. Assuming that the ground is level, an instrument is set up 227.59 feet from the flagpole with its telescope centered 5.31 feet above the ground. The telescope is sighted horizontally to a point 5.31 feet from the bottom of the flagpole and then the angle at the instrument looking to the top of the pole is measured. That angle is 26 degrees 51 minutes. How tall is the flagpole from its base? Give your answer to two decimal places with NO units. 10 points QUESTION 9 1. You are hiking in the mountains. For every 100.00 feet you would be walking horizontally, you have increased your elevation by 4 feet. At what grade are you climbing? Give your answer to three decimal places. Hint: Your units will be in ft/ft. 5 points QUESTION 10 1. A grade of -0.9 percent is being considered for a mountain roadway. The elevation at the initial point is 2,848.25 feet and a horizontal distance of 4,377.51 needs to be covered. What is the elevation at the end of the grade? 10 points QUESTION 11 1. A slope distance was measured between two points (A and T) and determined to be 4,788.68 feet. At point A the elevation is 857.23 feet and at point T the elevation is 877.96 feet. What is the horizontal distance between A and T?

QUESTION 1 1. Convert 30 degrees 2 minutes to decimal degrees. Give this answer to 6 decimal places. Do not provide units. You know those are decimal degrees. 5 points QUESTION 2 1. Convert 342 degrees 6 minutes and 41 seconds to decimal degrees. Show your answers to only 6 decimal places. Do not give units. 5 points QUESTION 3 1. COMPUTE the sin of 52 degrees. Give the answer to 6 decimal places. 5 points QUESTION 4 1. What is the sine of 277 degrees and 16 minutes? Give your answer to 6 decimal places. Pay attention to rounding. 5 points QUESTION 5 1. This is a right triangle problem. Angle A is 90 degrees. Draw the triangle and label it as we did in lecture. If angle B is 24 degrees 43 minutes and side c is 395.82 feet, what is the distance in feet of side b? Give your answer to two decimal places. Do not provide units. Those are in feet – right? 10 points QUESTION 6 1. This is a right triangle problem with angle A being the 90 degree angle. It should look like the one from lecture. If angle B is 25 degrees 18 minutes and side c is 206.1 feet, what is the distance to two decimal places of side a? Give your answer to two decimal places. Do not provide units – those are in feet. 10 points QUESTION 7 1. You are given a right triangle with angle A being the 90 degree angle – just like in lecture. If angle C is 42 degrees 9 minutes and side a is 401.73 feet, what is the length of side c? Give your answer to two decimal places. The units are feet – don’t list those. 10 points QUESTION 8 Ad by Browse Safe | Close 1. It is desired to determine the height of a flagpole. Assuming that the ground is level, an instrument is set up 227.59 feet from the flagpole with its telescope centered 5.31 feet above the ground. The telescope is sighted horizontally to a point 5.31 feet from the bottom of the flagpole and then the angle at the instrument looking to the top of the pole is measured. That angle is 26 degrees 51 minutes. How tall is the flagpole from its base? Give your answer to two decimal places with NO units. 10 points QUESTION 9 1. You are hiking in the mountains. For every 100.00 feet you would be walking horizontally, you have increased your elevation by 4 feet. At what grade are you climbing? Give your answer to three decimal places. Hint: Your units will be in ft/ft. 5 points QUESTION 10 1. A grade of -0.9 percent is being considered for a mountain roadway. The elevation at the initial point is 2,848.25 feet and a horizontal distance of 4,377.51 needs to be covered. What is the elevation at the end of the grade? 10 points QUESTION 11 1. A slope distance was measured between two points (A and T) and determined to be 4,788.68 feet. At point A the elevation is 857.23 feet and at point T the elevation is 877.96 feet. What is the horizontal distance between A and T?

Question no Assignment 2 1 30.0333333 degrees 2 342.111389 degrees … 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

A shipment of 6 refrigerators to a restaurant contains 2 defective ones. The restaurant manager begins to randomly test the 6 refrigerators one at a time. a)Find the probability that the last defective refrigerator is found on the fourth test b)Find the probability that no more than four refrigerators need to be tested to find both of the defective refrigerators.

A shipment of 6 refrigerators to a restaurant contains 2 defective ones. The restaurant manager begins to randomly test the 6 refrigerators one at a time. a)Find the probability that the last defective refrigerator is found on the fourth test b)Find the probability that no more than four refrigerators need to be tested to find both of the defective refrigerators.

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unit 6 only Part 1: Analysis of a unit of work (1000-1500) Part 1 requires you to critically evaluate a unit of work given in terms of: • the range of approaches and methodologies to language learning and teaching this unit of work encompasses. Discuss whether there is a focus on a particular approach, eg, are the students asked to memorise / rote learn/ repeat (audio-lingual); are students required to complete a task (task based learning) or an information-gap type activity (communicative language learning); is there a focus on a specific genre? 300 – 400 • the clarity of the objectives and target language/ exponents being taught 200-300 • the selection and sequencing of the activities 200 – 300 • to what extent language exponents and skills are integrated in the activities 200 -300 • the learner group, their needs and their language level for which the unit of work would be most appropriate 100 Describe the learner group this unit is designed for: ESL students, students of English as an international language etc; what language level the unit assumes and; the students language learning needs. Part 2: Extension, addition, omission and substitution (1500 – 2000) This section of the assignment requires you to focus on the unit of work: • Comment on any extensions, additions, omissions or substitutions you would make if you were teaching this unit to the learner group you identified in Part 1, above. 500 • Give reasons for your decisions. 500 • Describe how you will assess student learning. 300 • Describe how you will evaluate the success of the unit of work. 200 • Identify any problems you anticipate in carrying out the unit of work and suggest how you would go about overcoming these. 300 • For added or substituted activities, list the resources you will need for these, and reference the materials you have used or drawn on. 200

unit 6 only Part 1: Analysis of a unit of work (1000-1500) Part 1 requires you to critically evaluate a unit of work given in terms of: • the range of approaches and methodologies to language learning and teaching this unit of work encompasses. Discuss whether there is a focus on a particular approach, eg, are the students asked to memorise / rote learn/ repeat (audio-lingual); are students required to complete a task (task based learning) or an information-gap type activity (communicative language learning); is there a focus on a specific genre? 300 – 400 • the clarity of the objectives and target language/ exponents being taught 200-300 • the selection and sequencing of the activities 200 – 300 • to what extent language exponents and skills are integrated in the activities 200 -300 • the learner group, their needs and their language level for which the unit of work would be most appropriate 100 Describe the learner group this unit is designed for: ESL students, students of English as an international language etc; what language level the unit assumes and; the students language learning needs. Part 2: Extension, addition, omission and substitution (1500 – 2000) This section of the assignment requires you to focus on the unit of work: • Comment on any extensions, additions, omissions or substitutions you would make if you were teaching this unit to the learner group you identified in Part 1, above. 500 • Give reasons for your decisions. 500 • Describe how you will assess student learning. 300 • Describe how you will evaluate the success of the unit of work. 200 • Identify any problems you anticipate in carrying out the unit of work and suggest how you would go about overcoming these. 300 • For added or substituted activities, list the resources you will need for these, and reference the materials you have used or drawn on. 200

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Lab 5 Math 551 Fall 2015 Goal: In this assignment we will look at two fractals, namely the Sierpinski fractal and the Barnsley Fern. During the lab session, your lab instructor will teach you the necessary MATLAB code to complete the assignment, which will be discussed in the lab on Thursday October 8th or Friday October 9th in the lab (CW 144 or CW 145). What you have to submit: An m- le containing all of the commands necessary to perform all the tasks described below. Submit this le on Canvas. Click: \Assignments”, click \MATLAB Project 5″, click \Submit Assignment”, then upload your .m le and click \Submit Assignment” again. Due date: Friday October 16, 5pm. No late submission will be accepted. TASKS A fractal can be de ned as a self-similar detailed pattern repeating itself. Some of the most well know fractals (the Mandelbrot set and Julia set) can be viewed here: http://classes.yale.edu/fractals/ The Sierpinski Fractal The program srnpnski(m,dist,n) gets its name from the mathematician W. Sierpinski. The only parameter that must be speci ed is m which determines the number of vertices that will be part of a regular polygon. For larger m it produces a graph which is similar to a snow ake. The program starts with a randomly chosen seed position given by the internal variable s. At each stage one of the vertices is chosen at random and a new point is produced which is dist away from the old point to the vertex. The value of dist should be between 0 and 1. The default value is 0.5. This process is repeated n times. The default value of n is 1500. 1. Create a new Matlab function: func t i on s rpns k i (m, di s t , n) %This c r e a t e s a snowf lake from m v e r t i c e s us ing n i t e r a t i o n s . i f nargin <3, n=1500; end i f nargin <2, d i s t =0.5; end c l f a x i s ( [ ?1 ,1 , ?1 ,1] ) p=exp (2 pi  i  ( 1 :m)/m) ; pl o t (p , '  ' ) hold s=rand+i  rand ; f o r j =1:n r=c e i l (m rand ) ; s=d i s t  s+(1?d i s t )p( r ) ; pl o t ( s , ' . ' ) end 2. Try out the following commands s rpns k i ( 3 ) s rpns k i ( 3 , 0 . 5 , 2 5 0 0 ) s rpns k i ( 3 , 0 . 5 , 5 0 0 0 ) s rpns k i ( 3 , 0 . 4 ) 1 s rpns k i ( 3 , 0 . 2 ) s rpns k i ( 5 ) s rpns k i ( 5 , 0 . 4 ) s rpns k i ( 5 , 0 . 3 ) s rpns k i ( 6 , 0 . 3 ) s rpns k i ( 8 , 0 . 3 , 5 0 0 0 ) The Barnsley Fern The following program is the famous Barnsley Fern. The only external parameter is n, the number of iterations. 3. Create a new Matlab function: func t i on f e r n (n) A1=[ 0 . 8 5 , 0 . 0 4 ; ?0 . 0 4 , 0 . 8 5 ] ; A2=[ ?0 . 1 5 , 0 . 2 8 ; 0 . 2 6 , 0 . 2 4 ] ; A3=[ 0 . 2 , ?0 . 2 6 ; 0 . 2 3 , 0 . 2 2 ] ; A4=[ 0 , 0 ; 0 , 0 . 1 6 ] ; T1=[ 0 ; 1 . 6 ] ; T2=[ 0 ; 0 . 4 4 ] ; T3=[ 0 ; 1 . 6 ] ; T4=[ 0 , 0 ] ; P1=0.85; P2=0.07; P3=0.07; P4=0.01; c l f ; s=rand ( 2 , 1 ) ; pl o t ( s ( 1 ) , s ( 2 ) , ' . ' ) hold f o r j =1:n r=rand ; i f r<=P1 , s=A1 s+T1 ; e l s e i f r<=P1+P2 , s=A2 s+T2 ; e l s e i f r<=P1+P2+P3 , s=A3 s+T3 ; e l s e s=A4 s ; end pl o t ( s ( 1 ) , s ( 2 ) , ' . ' ) end 4. Try the following commands: f e r n (100) f e r n (500) f e r n (1000) f e r n (3000) f e r n (5000) f e r n (10000) 2 5. Change the parameters in the fern program: A1=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; A2=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; A3=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; T1=[ 1 ; 1 ] ; T2=[ 1 ; 5 0 ] ; T3=[ 5 0 ; 5 0 ] ; P1=0.33; P2=0.33; P3=0.34; Call the new program srptri.m. Try the command s r p t r i (5000) You should see a familiar looking result. 6. Change the parameters in the fern program: A1=[ 0 , 0 ; 0 , 0 . 5 ] ; A2=[ 0 . 4 2 , ?0 . 4 2 ; 0 . 4 2 , 0 . 4 2 ] ; A3=[ 0 . 4 2 , 0 . 4 2 ; ?0 . 4 2 , 0 . 4 2 ] ; A4=[ 0 . 1 , 0 ; 0 , 0 . 1 ] ; T1=[ 0 ; 0 ] ; T2=[ 0 ; 0 . 2 ] ; T3=[ 0 ; 0 . 2 ] ; T4=[ 0 , 0 . 2 ] ; P1=0.05; P2=0.4; P3=0.4; P4=0.15; Call the new program srptree.m. Try the command s r p t r e e (5000) This is an example of a fractal tree. Some nice animations of fractal trees can be seen here: http://classes.yale.edu/fractals/ MATLAB commands to learn: Cl f , c e i l , imaginary uni t I , i f . . e l s e i f . . e l s e . . end 3

Lab 5 Math 551 Fall 2015 Goal: In this assignment we will look at two fractals, namely the Sierpinski fractal and the Barnsley Fern. During the lab session, your lab instructor will teach you the necessary MATLAB code to complete the assignment, which will be discussed in the lab on Thursday October 8th or Friday October 9th in the lab (CW 144 or CW 145). What you have to submit: An m- le containing all of the commands necessary to perform all the tasks described below. Submit this le on Canvas. Click: \Assignments”, click \MATLAB Project 5″, click \Submit Assignment”, then upload your .m le and click \Submit Assignment” again. Due date: Friday October 16, 5pm. No late submission will be accepted. TASKS A fractal can be de ned as a self-similar detailed pattern repeating itself. Some of the most well know fractals (the Mandelbrot set and Julia set) can be viewed here: http://classes.yale.edu/fractals/ The Sierpinski Fractal The program srnpnski(m,dist,n) gets its name from the mathematician W. Sierpinski. The only parameter that must be speci ed is m which determines the number of vertices that will be part of a regular polygon. For larger m it produces a graph which is similar to a snow ake. The program starts with a randomly chosen seed position given by the internal variable s. At each stage one of the vertices is chosen at random and a new point is produced which is dist away from the old point to the vertex. The value of dist should be between 0 and 1. The default value is 0.5. This process is repeated n times. The default value of n is 1500. 1. Create a new Matlab function: func t i on s rpns k i (m, di s t , n) %This c r e a t e s a snowf lake from m v e r t i c e s us ing n i t e r a t i o n s . i f nargin <3, n=1500; end i f nargin <2, d i s t =0.5; end c l f a x i s ( [ ?1 ,1 , ?1 ,1] ) p=exp (2 pi  i  ( 1 :m)/m) ; pl o t (p , '  ' ) hold s=rand+i  rand ; f o r j =1:n r=c e i l (m rand ) ; s=d i s t  s+(1?d i s t )p( r ) ; pl o t ( s , ' . ' ) end 2. Try out the following commands s rpns k i ( 3 ) s rpns k i ( 3 , 0 . 5 , 2 5 0 0 ) s rpns k i ( 3 , 0 . 5 , 5 0 0 0 ) s rpns k i ( 3 , 0 . 4 ) 1 s rpns k i ( 3 , 0 . 2 ) s rpns k i ( 5 ) s rpns k i ( 5 , 0 . 4 ) s rpns k i ( 5 , 0 . 3 ) s rpns k i ( 6 , 0 . 3 ) s rpns k i ( 8 , 0 . 3 , 5 0 0 0 ) The Barnsley Fern The following program is the famous Barnsley Fern. The only external parameter is n, the number of iterations. 3. Create a new Matlab function: func t i on f e r n (n) A1=[ 0 . 8 5 , 0 . 0 4 ; ?0 . 0 4 , 0 . 8 5 ] ; A2=[ ?0 . 1 5 , 0 . 2 8 ; 0 . 2 6 , 0 . 2 4 ] ; A3=[ 0 . 2 , ?0 . 2 6 ; 0 . 2 3 , 0 . 2 2 ] ; A4=[ 0 , 0 ; 0 , 0 . 1 6 ] ; T1=[ 0 ; 1 . 6 ] ; T2=[ 0 ; 0 . 4 4 ] ; T3=[ 0 ; 1 . 6 ] ; T4=[ 0 , 0 ] ; P1=0.85; P2=0.07; P3=0.07; P4=0.01; c l f ; s=rand ( 2 , 1 ) ; pl o t ( s ( 1 ) , s ( 2 ) , ' . ' ) hold f o r j =1:n r=rand ; i f r<=P1 , s=A1 s+T1 ; e l s e i f r<=P1+P2 , s=A2 s+T2 ; e l s e i f r<=P1+P2+P3 , s=A3 s+T3 ; e l s e s=A4 s ; end pl o t ( s ( 1 ) , s ( 2 ) , ' . ' ) end 4. Try the following commands: f e r n (100) f e r n (500) f e r n (1000) f e r n (3000) f e r n (5000) f e r n (10000) 2 5. Change the parameters in the fern program: A1=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; A2=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; A3=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; T1=[ 1 ; 1 ] ; T2=[ 1 ; 5 0 ] ; T3=[ 5 0 ; 5 0 ] ; P1=0.33; P2=0.33; P3=0.34; Call the new program srptri.m. Try the command s r p t r i (5000) You should see a familiar looking result. 6. Change the parameters in the fern program: A1=[ 0 , 0 ; 0 , 0 . 5 ] ; A2=[ 0 . 4 2 , ?0 . 4 2 ; 0 . 4 2 , 0 . 4 2 ] ; A3=[ 0 . 4 2 , 0 . 4 2 ; ?0 . 4 2 , 0 . 4 2 ] ; A4=[ 0 . 1 , 0 ; 0 , 0 . 1 ] ; T1=[ 0 ; 0 ] ; T2=[ 0 ; 0 . 2 ] ; T3=[ 0 ; 0 . 2 ] ; T4=[ 0 , 0 . 2 ] ; P1=0.05; P2=0.4; P3=0.4; P4=0.15; Call the new program srptree.m. Try the command s r p t r e e (5000) This is an example of a fractal tree. Some nice animations of fractal trees can be seen here: http://classes.yale.edu/fractals/ MATLAB commands to learn: Cl f , c e i l , imaginary uni t I , i f . . e l s e i f . . e l s e . . end 3

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1 CE 321 PRINCIPLES ENVIRONMENTAL ENGINEERING LAB WORKSHEET No. 1 Due: One (1) Week After Each Lab Section, respectively MICROBIOLOGY Environmental engineers employ microbiology in a variety of applications. Testing for coliform bacteria is used to assess whether pathogens may be present in a water or wastewater sample. Coliforms are a type of bacteria that live in the intestines of warm blooded mammals, such as humans and cattle. They are not pathogens, but if they are present in a sample, it is taken as an indication that fecal material from humans or cattle has contacted the water. If fecal material is present, pathogens may be present, too. In water treatment, coliform counts must average less than one colony per 100 milliliters of sample tested. In wastewater treatment, typical acceptable levels might be 200-colonies/100 mL. There are two standard ways to test for coliforms, the Most Probable Number test, MPN (also called the multiple tube fermentation technique, MTF) and the membrane filter test, MF. Several companies market testing systems that are somewhat simpler, but these cannot be used by treatment plants until they receive EPA approval. Two recently accepted methods are the Minimal Media Test (Colilert system), and the Presence-Absence coliform test (P-A test). Wastewater treatment plant operators study the microorganism composition of the activated sludge units in order to assess and predict the performance of the biological floc. A sample of mixed liquor from the aeration basin is examined under the microscope, and based on the relative predominance of a variety of organisms that might be present; the operator can tell if the BOD application rates and wasting rates are as they should be. For your worksheet, please submit the items requested below (10 pts. each): 1. Examine a sample of activated sludge under the microscope (To be done together in class). Use the Atlas, Standard Methods, or other references to identify at least 5 different organisms you observed. List them and sketch them neatly on unlined paper. Describe their motility and any other distinctive characteristics as you observed it. 2. Explain what types of organisms you might expect to find in sludge with a high mean cell residence time (MCRT), and explain why these would predominate over the other types. 3. How can the predominance of a certain kind of microorganism in activated sludge affect the settling characteristics of the sludge? Give several examples. 2 4. Explain why coliforms are used as “indicator organisms” for water and wastewater testing. Name two pathogenic bacteria, two pathogenic viruses, and one pathogenic protozoan sometimes found in water supplies. 5. There is also a test for fecal coliforms. Use your class notes and outside references to explain the distinctions between the tests for total and fecal coliforms. Explain why one would use the fecal coliform test instead of the test for total coliforms. 6. Using outside references, indicate typical coliform limits for surface waters used for swimming and fishing; potable water; and wastewater treatment plant effluent. 7). In the recent past, EPA instituted regulations designed to insure that Giardia are removed from the water. Using your text or other references, explain what kind of organism this is, and explain the way in which EPA has set standards to insure they are removed during water treatment. 8. What is meant by “population dynamics”? What two factors usually control the population dynamics of a mixed culture? 9. Use the MPN test data from the samples prepared for class prior to determine the number of coliforms present in the wastewater samples. Please show your work and explain your reasoning. Total Coliforms Raw Intermediate Effluent Sample Volume No. Positive No. Positive No. Positive 10 5 5 4 1 5 5 2 0.1 5 3 1 0.01 5 1 0 0.001 2 1 —- 0.0001 1 —- —- FecalColiforms Raw Intermediate Effluent Sample Volume No. Positive No. Positive No. Positive 10 5 5 2 1 5 4 0 0.1 5 2 1 0.01 1 0 0 0.001 0 0 —– 0.0001 2 —– —– 3 10. Use the membrane filter test data given in class to determine the number of total coliforms and fecal coliforms present in the sample. Please show your work and explain your reasoning. Total Coliforms Fecal Coliforms Dilution Colonies Dilution Colonies Raw Influent 0.1 mL/100 mL 58 1 mL/100 mL 47 Intermediate 1 mL/100 mL 13 10 mL/100 mL 28 Wetland Effluent 10 mL/100 mL 10 100 mL/100 mL 15

1 CE 321 PRINCIPLES ENVIRONMENTAL ENGINEERING LAB WORKSHEET No. 1 Due: One (1) Week After Each Lab Section, respectively MICROBIOLOGY Environmental engineers employ microbiology in a variety of applications. Testing for coliform bacteria is used to assess whether pathogens may be present in a water or wastewater sample. Coliforms are a type of bacteria that live in the intestines of warm blooded mammals, such as humans and cattle. They are not pathogens, but if they are present in a sample, it is taken as an indication that fecal material from humans or cattle has contacted the water. If fecal material is present, pathogens may be present, too. In water treatment, coliform counts must average less than one colony per 100 milliliters of sample tested. In wastewater treatment, typical acceptable levels might be 200-colonies/100 mL. There are two standard ways to test for coliforms, the Most Probable Number test, MPN (also called the multiple tube fermentation technique, MTF) and the membrane filter test, MF. Several companies market testing systems that are somewhat simpler, but these cannot be used by treatment plants until they receive EPA approval. Two recently accepted methods are the Minimal Media Test (Colilert system), and the Presence-Absence coliform test (P-A test). Wastewater treatment plant operators study the microorganism composition of the activated sludge units in order to assess and predict the performance of the biological floc. A sample of mixed liquor from the aeration basin is examined under the microscope, and based on the relative predominance of a variety of organisms that might be present; the operator can tell if the BOD application rates and wasting rates are as they should be. For your worksheet, please submit the items requested below (10 pts. each): 1. Examine a sample of activated sludge under the microscope (To be done together in class). Use the Atlas, Standard Methods, or other references to identify at least 5 different organisms you observed. List them and sketch them neatly on unlined paper. Describe their motility and any other distinctive characteristics as you observed it. 2. Explain what types of organisms you might expect to find in sludge with a high mean cell residence time (MCRT), and explain why these would predominate over the other types. 3. How can the predominance of a certain kind of microorganism in activated sludge affect the settling characteristics of the sludge? Give several examples. 2 4. Explain why coliforms are used as “indicator organisms” for water and wastewater testing. Name two pathogenic bacteria, two pathogenic viruses, and one pathogenic protozoan sometimes found in water supplies. 5. There is also a test for fecal coliforms. Use your class notes and outside references to explain the distinctions between the tests for total and fecal coliforms. Explain why one would use the fecal coliform test instead of the test for total coliforms. 6. Using outside references, indicate typical coliform limits for surface waters used for swimming and fishing; potable water; and wastewater treatment plant effluent. 7). In the recent past, EPA instituted regulations designed to insure that Giardia are removed from the water. Using your text or other references, explain what kind of organism this is, and explain the way in which EPA has set standards to insure they are removed during water treatment. 8. What is meant by “population dynamics”? What two factors usually control the population dynamics of a mixed culture? 9. Use the MPN test data from the samples prepared for class prior to determine the number of coliforms present in the wastewater samples. Please show your work and explain your reasoning. Total Coliforms Raw Intermediate Effluent Sample Volume No. Positive No. Positive No. Positive 10 5 5 4 1 5 5 2 0.1 5 3 1 0.01 5 1 0 0.001 2 1 —- 0.0001 1 —- —- FecalColiforms Raw Intermediate Effluent Sample Volume No. Positive No. Positive No. Positive 10 5 5 2 1 5 4 0 0.1 5 2 1 0.01 1 0 0 0.001 0 0 —– 0.0001 2 —– —– 3 10. Use the membrane filter test data given in class to determine the number of total coliforms and fecal coliforms present in the sample. Please show your work and explain your reasoning. Total Coliforms Fecal Coliforms Dilution Colonies Dilution Colonies Raw Influent 0.1 mL/100 mL 58 1 mL/100 mL 47 Intermediate 1 mL/100 mL 13 10 mL/100 mL 28 Wetland Effluent 10 mL/100 mL 10 100 mL/100 mL 15

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6.[ Book Section 7.6] Using Table 7.2, determine which of these fields can be either an electric or a magnetic field (state whether they satisfy the necessary conditions): (a) A˙ (b) B˙ (c) C˙ = 2y3z˙ax + 3(x + 2)yz˙ay − (x − 1)z3˙az = ((z + 1)/ρ)cosφ˙aρ − 2sinφ/ρ˙az = 3/r2(3sinθ˙aφ + cosθ˙aθ )

6.[ Book Section 7.6] Using Table 7.2, determine which of these fields can be either an electric or a magnetic field (state whether they satisfy the necessary conditions): (a) A˙ (b) B˙ (c) C˙ = 2y3z˙ax + 3(x + 2)yz˙ay − (x − 1)z3˙az = ((z + 1)/ρ)cosφ˙aρ − 2sinφ/ρ˙az = 3/r2(3sinθ˙aφ + cosθ˙aθ )

Attached Files: File Operational Definitions for 670.doc (25.5 KB) Amply armed with all the information you have learned throughout these last 7 weeks (paying special attention to Chapters 11-14), complete a “mini public relations proposal.” Following is a checklist of what is expected in this proposal: 1. Name of the organization and a brief explanation/description (Example: it is a boutique that specializes in selling high-end bridal gowns; it is a nonprofit organization that raises money for children whose parents are wounded veterans, etc.) PLEASE NOTE: No fictitious organizations, please! 2. ONE Overaching Goal (to persuade, inform, educate, etc.) 3. ONE suggestion for the research you plan to conduct. Explain the method (survey, phone interviews, etc.), who you are researching, and why you think this method is most conducive for this communication campaign. 4. ONE behavioral objective (see handouts a) RECALL PLOT: public, level of obtainment, timeframe) b). RECALL that the objective is what you want your target public to do 5. ONE action strategy (RECALL that the strategy is what you are planning to do meet your objective – your gameplan) 6. ONE message strategy (what your message will say) 7. TWO communication tactics 8. ONE technique for measuring whether the objective was met IMPORTANT NOTES: > USE the prsa operational definitions (SEE ATTACHED HANDOUT) > USE subheads for each part of the proposal OR you can just number the components (1-8) > The rubric for this last report is very simple: points will be deducted for each component you do not include or if it is written incorrectly or does not meet all the critiera mapped out in the attached handout.

Attached Files: File Operational Definitions for 670.doc (25.5 KB) Amply armed with all the information you have learned throughout these last 7 weeks (paying special attention to Chapters 11-14), complete a “mini public relations proposal.” Following is a checklist of what is expected in this proposal: 1. Name of the organization and a brief explanation/description (Example: it is a boutique that specializes in selling high-end bridal gowns; it is a nonprofit organization that raises money for children whose parents are wounded veterans, etc.) PLEASE NOTE: No fictitious organizations, please! 2. ONE Overaching Goal (to persuade, inform, educate, etc.) 3. ONE suggestion for the research you plan to conduct. Explain the method (survey, phone interviews, etc.), who you are researching, and why you think this method is most conducive for this communication campaign. 4. ONE behavioral objective (see handouts a) RECALL PLOT: public, level of obtainment, timeframe) b). RECALL that the objective is what you want your target public to do 5. ONE action strategy (RECALL that the strategy is what you are planning to do meet your objective – your gameplan) 6. ONE message strategy (what your message will say) 7. TWO communication tactics 8. ONE technique for measuring whether the objective was met IMPORTANT NOTES: > USE the prsa operational definitions (SEE ATTACHED HANDOUT) > USE subheads for each part of the proposal OR you can just number the components (1-8) > The rubric for this last report is very simple: points will be deducted for each component you do not include or if it is written incorrectly or does not meet all the critiera mapped out in the attached handout.

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During deployment processing, soldiers undergo medical screening.After a wait, each soldier see same dicaltechnician who reviews his or her records.If the records review shows no problems—the soldier is physically qualified to deploy—then the soldier departs to the next step in deployment processing.It he records do reveal a potential medical issue,the soldier instead sees a doctor,who assesses the soldier’s condition and determines both deployability and for those who are medically disqualified, treatment needs.In rare cases ,the doctor in itiate a medical board to evaluate the soldier for retention in the military. Currently, 80 soldiers per hour arrive for deployment screening,and 80% of them pass there cords review.On average,20 people are waiting for the medical records review,which takes 6 minutes. When the records review indicates a soldier must see a doctor,the soldier reports to a waiting room,where an average of 8 soldiers are waiting. After a wait,the soldier sees a doctor, who reviews the soldier’s condition and either approves the soldier for deployment(75%of the time) or disapproves deployment and conducts a morein-deptxeam to determine treatment(20%of the time)or the need for a medical board(5%ofthetime).Each doctor’s exam takes,on average,6minutes if the solider is medically able to deploy(the doctor pretty much replicates the records review),15 minutes if the soldier requires some kind of treatment,and 30 minutes in those rare cases that require the doctor to initiate a medical review board. Assume the process is stable;that is,average inflow rate equals average outflow rate.[Finally,thisisNOTaqueuingproblem.] a. On average,how long does a soldier spend in the deployment process?

During deployment processing, soldiers undergo medical screening.After a wait, each soldier see same dicaltechnician who reviews his or her records.If the records review shows no problems—the soldier is physically qualified to deploy—then the soldier departs to the next step in deployment processing.It he records do reveal a potential medical issue,the soldier instead sees a doctor,who assesses the soldier’s condition and determines both deployability and for those who are medically disqualified, treatment needs.In rare cases ,the doctor in itiate a medical board to evaluate the soldier for retention in the military. Currently, 80 soldiers per hour arrive for deployment screening,and 80% of them pass there cords review.On average,20 people are waiting for the medical records review,which takes 6 minutes. When the records review indicates a soldier must see a doctor,the soldier reports to a waiting room,where an average of 8 soldiers are waiting. After a wait,the soldier sees a doctor, who reviews the soldier’s condition and either approves the soldier for deployment(75%of the time) or disapproves deployment and conducts a morein-deptxeam to determine treatment(20%of the time)or the need for a medical board(5%ofthetime).Each doctor’s exam takes,on average,6minutes if the solider is medically able to deploy(the doctor pretty much replicates the records review),15 minutes if the soldier requires some kind of treatment,and 30 minutes in those rare cases that require the doctor to initiate a medical review board. Assume the process is stable;that is,average inflow rate equals average outflow rate.[Finally,thisisNOTaqueuingproblem.] a. On average,how long does a soldier spend in the deployment process?

6+0.75*6+.2*15+.05*30 = 15 miutes     Timeindeploymentsystem:15 (minutes)
Assignment 9 Due: 11:59pm on Friday, April 11, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Problem 11.2 Part A Evaluate the dot product if and . Express your answer using two significant figures. ANSWER: Correct Part B Evaluate the dot product if and . Express your answer using two significant figures. ANSWER: Correct Problem 11.4  A B = 5 − 6 A i ^ j ^ = −9 − 5 B i ^ j ^ A  B  = -15  A B = −5 + 9 A i ^ j ^ = 5 + 6 B i ^ j ^ A  B  = 29 Part A What is the angle between vectors and if and ? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part B What is the angle between vectors and if and ? Express your answer as an integer and include the appropriate units. ANSWER: Correct ± All Work and No Play Learning Goal: To be able to calculate work done by a constant force directed at different angles relative to displacement If an object undergoes displacement while being acted upon by a force (or several forces), it is said that work is being done on the object. If the object is moving in a straight line and the displacement and the force are known, the work done by the force can be calculated as , where is the work done by force on the object that undergoes displacement directed at angle relative to .  A B A = 2 + 5 ı ^  ^ B = −2 − 4 ı ^  ^  = 175  A B A = −6 + 2 ı ^  ^ B = − − 3 ı ^  ^  = 90 W =  = cos  F  s  F   s  W F  s  F  Note that depending on the value of , the work done can be positive, negative, or zero. In this problem, you will practice calculating work done on an object moving in a straight line. The first series of questions is related to the accompanying figure. Part A What can be said about the sign of the work done by the force ? ANSWER: Correct When , the cosine of is zero, and therefore the work done is zero. Part B cos  F  1 It is positive. It is negative. It is zero. There is not enough information to answer the question.  = 90  What can be said about the work done by force ? ANSWER: Correct When , is positive, and so the work done is positive. Part C The work done by force is ANSWER: Correct When , is negative, and so the work done is negative. Part D The work done by force is ANSWER: F  2 It is positive. It is negative. It is zero. 0 <  < 90 cos  F  3 positive negative zero 90 <  < 180 cos  F  4 Correct Part E The work done by force is ANSWER: Correct positive negative zero F  5 positive negative zero Part F The work done by force is ANSWER: Correct Part G The work done by force is ANSWER: Correct In the next series of questions, you will use the formula to calculate the work done by various forces on an object that moves 160 meters to the right. F  6 positive negative zero F  7 positive negative zero W =  = cos  F  s  F   s  Part H Find the work done by the 18-newton force. Use two significant figures in your answer. Express your answer in joules. ANSWER: Correct Part I Find the work done by the 30-newton force. Use two significant figures in your answer. Express your answer in joules. ANSWER: Correct Part J Find the work done by the 12-newton force. Use two significant figures in your answer. Express your answer in joules. W W = 2900 J W W = 4200 J W ANSWER: Correct Part K Find the work done by the 15-newton force. Use two significant figures in your answer. Express your answer in joules. ANSWER: Correct Introduction to Potential Energy Learning Goal: Understand that conservative forces can be removed from the work integral by incorporating them into a new form of energy called potential energy that must be added to the kinetic energy to get the total mechanical energy. The first part of this problem contains short-answer questions that review the work-energy theorem. In the second part we introduce the concept of potential energy. But for now, please answer in terms of the work-energy theorem. Work-Energy Theorem The work-energy theorem states , where is the work done by all forces that act on the object, and and are the initial and final kinetic energies, respectively. Part A The work-energy theorem states that a force acting on a particle as it moves over a ______ changes the ______ energy of the particle if the force has a component parallel to the motion. W = -1900 J W W = -1800 J Kf = Ki + Wall Wall Ki Kf Choose the best answer to fill in the blanks above: ANSWER: Correct It is important that the force have a component acting in the direction of motion. For example, if a ball is attached to a string and whirled in uniform circular motion, the string does apply a force to the ball, but since the string's force is always perpendicular to the motion it does no work and cannot change the kinetic energy of the ball. Part B To calculate the change in energy, you must know the force as a function of _______. The work done by the force causes the energy change. Choose the best answer to fill in the blank above: ANSWER: Correct Part C To illustrate the work-energy concept, consider the case of a stone falling from to under the influence of gravity. Using the work-energy concept, we say that work is done by the gravitational _____, resulting in an increase of the ______ energy of the stone. Choose the best answer to fill in the blanks above: distance / potential distance / kinetic vertical displacement / potential none of the above acceleration work distance potential energy xi xf ANSWER: Correct Potential Energy You should read about potential energy in your text before answering the following questions. Potential energy is a concept that builds on the work-energy theorem, enlarging the concept of energy in the most physically useful way. The key aspect that allows for potential energy is the existence of conservative forces, forces for which the work done on an object does not depend on the path of the object, only the initial and final positions of the object. The gravitational force is conservative; the frictional force is not. The change in potential energy is the negative of the work done by conservative forces. Hence considering the initial and final potential energies is equivalent to calculating the work done by the conservative forces. When potential energy is used, it replaces the work done by the associated conservative force. Then only the work due to nonconservative forces needs to be calculated. In summary, when using the concept of potential energy, only nonconservative forces contribute to the work, which now changes the total energy: , where and are the final and initial potential energies, and is the work due only to nonconservative forces. Now, we will revisit the falling stone example using the concept of potential energy. Part D Rather than ascribing the increased kinetic energy of the stone to the work of gravity, we now (when using potential energy rather than work-energy) say that the increased kinetic energy comes from the ______ of the _______ energy. Choose the best answer to fill in the blanks above: ANSWER: force / kinetic potential energy / potential force / potential potential energy / kinetic Kf + Uf = Ef = Wnc + Ei = Wnc + Ki + Ui Uf Ui Wnc Correct Part E This process happens in such a way that total mechanical energy, equal to the ______ of the kinetic and potential energies, is _______. Choose the best answer to fill in the blanks above: ANSWER: Correct Problem 11.7 Part A How much work is done by the force 2.2 6.6 on a particle that moves through displacement 3.9 Express your answer to two significant figures and include the appropriate units. ANSWER: work / potential force / kinetic change / potential sum / conserved sum / zero sum / not conserved difference / conserved F  = (− + i ^ ) N j ^ ! = r m i ^ Correct Part B How much work is done by the force 2.2 6.6 on a particle that moves through displacement 3.9 Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 11.10 A 1.8 book is lying on a 0.80- -high table. You pick it up and place it on a bookshelf 2.27 above the floor. Part A How much work does gravity do on the book? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B W = -8.6 J F  = (− + i ^ ) N j ^ ! = r m? j ^ W = 26 J kg m m Wg = -26 J How much work does your hand do on the book? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 11.12 The three ropes shown in the bird's-eye view of the figure are used to drag a crate 3.3 across the floor. Part A How much work is done by each of the three forces? Express your answers using two significant figures. Enter your answers numerically separated by commas. ANSWER: WH = 26 J m W1 , W2 , W3 = 1.9,1.2,-2.1 kJ Correct Enhanced EOC: Problem 11.16 A 1.2 particle moving along the x-axis experiences the force shown in the figure. The particle's velocity is 4.6 at . You may want to review ( pages 286 - 287) . For help with math skills, you may want to review: The Definite Integral Part A What is its velocity at ? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the work–kinetic energy theorem? What is the kinetic energy at ? How is the work done in going from to related to force shown in the graph? Using the work–kinetic energy theorem, what is the kinetic energy at ? What is the velocity at ? ANSWER: kg m/s x = 0m x = 2m x = 0 m x = 0 m x = 2 m x = 2 m x = 2 m Correct Part B What is its velocity at ? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the work–kinetic energy theorem? What is the kinetic energy at ? How is the work done in going from to related to force shown in the graph? Can the work be negative? Using the work–kinetic energy theorem, what is the kinetic energy at ? What is the velocity at ? ANSWER: Correct Work on a Sliding Block A block of weight sits on a frictionless inclined plane, which makes an angle with respect to the horizontal, as shown. A force of magnitude , applied parallel to the incline, pulls the block up the plane at constant speed. v = 6.2 ms x = 4m x = 0 m x = 0 m x = 4 m x = 4 m x = 4 m v = 4.6 ms w  F Part A The block moves a distance up the incline. The block does not stop after moving this distance but continues to move with constant speed. What is the total work done on the block by all forces? (Include only the work done after the block has started moving, not the work needed to start the block moving from rest.) Express your answer in terms of given quantities. Hint 1. What physical principle to use To find the total work done on the block, use the work-energy theorem: . Hint 2. Find the change in kinetic energy What is the change in the kinetic energy of the block, from the moment it starts moving until it has been pulled a distance ? Remember that the block is pulled at constant speed. Hint 1. Consider kinetic energy If the block's speed does not change, its kinetic energy cannot change. ANSWER: ANSWER: L Wtot Wtot = Kf − Ki L Kf − Ki = 0 Wtot = 0 Correct Part B What is , the work done on the block by the force of gravity as the block moves a distance up the incline? Express the work done by gravity in terms of the weight and any other quantities given in the problem introduction. Hint 1. Force diagram Hint 2. Force of gravity component What is the component of the force of gravity in the direction of the block's displacement (along the inclined plane)? Express your answer in terms of and . Hint 1. Relative direction of the force and the motion Remember that the force of gravity acts down the plane, whereas the block's displacement is directed up the plane. ANSWER: Wg L w w  ANSWER: Correct Part C What is , the work done on the block by the applied force as the block moves a distance up the incline? Express your answer in terms of and other given quantities. Hint 1. How to find the work done by a constant force Remember that the work done on an object by a particular force is the integral of the dot product of the force and the instantaneous displacement of the object, over the path followed by the object. In this case, since the force is constant and the path is a straight segment of length up the inclined plane, the dot product becomes simple multiplication. ANSWER: Correct Part D What is , the work done on the block by the normal force as the block moves a distance up the inclined plane? Express your answer in terms of given quantities. Hint 1. First step in computing the work Fg|| = −wsin() Wg = −wLsin() WF F L F L WF = FL Wnormal L The work done by the normal force is equal to the dot product of the force vector and the block's displacement vector. The normal force and the block's displacement vector are perpendicular. Therefore, what is their dot product? ANSWER: ANSWER: Correct Problem 11.20 A particle moving along the -axis has the potential energy , where is in . Part A What is the -component of the force on the particle at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the -component of the force on the particle at ? Express your answer to two significant figures and include the appropriate units. N  L = 0 Wnormal = 0 y U = 3.2y3 J y m y y = 0 m Fy = 0 N y y = 1 m ANSWER: Correct Part C What is the -component of the force on the particle at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 11.28 A cable with 25.0 of tension pulls straight up on a 1.08 block that is initially at rest. Part A What is the block's speed after being lifted 2.40 ? Solve this problem using work and energy. Express your answer with the appropriate units. ANSWER: Correct Fy = -9.6 N y y = 2 m Fy = -38 N N kg m vf = 8.00 ms Problem 11.29 Part A How much work does an elevator motor do to lift a 1500 elevator a height of 110 ? Express your answer with the appropriate units. ANSWER: Correct Part B How much power must the motor supply to do this in 50 at constant speed? Express your answer with the appropriate units. ANSWER: Correct Problem 11.32 How many energy is consumed by a 1.20 hair dryer used for 10.0 and a 11.0 night light left on for 16.0 ? Part A Hair dryer: Express your answer with the appropriate units. kg m Wext = 1.62×106 J s = 3.23×104 P W kW min W hr ANSWER: Correct Part B Night light: Express your answer with the appropriate units. ANSWER: Correct Problem 11.42 A 2500 elevator accelerates upward at 1.20 for 10.0 , starting from rest. Part A How much work does gravity do on the elevator? Express your answer with the appropriate units. ANSWER: Correct W = 7.20×105 J = 6.34×105 W J kg m/s2 m −2.45×105 J Part B How much work does the tension in the elevator cable do on the elevator? Express your answer with the appropriate units. ANSWER: Correct Part C Use the work-kinetic energy theorem to find the kinetic energy of the elevator as it reaches 10.0 . Express your answer with the appropriate units. ANSWER: Correct Part D What is the speed of the elevator as it reaches 10.0 ? Express your answer with the appropriate units. ANSWER: Correct 2.75×105 J m 3.00×104 J m 4.90 ms Problem 11.47 A horizontal spring with spring constant 130 is compressed 17 and used to launch a 2.4 box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.15. Part A Use work and energy to find how far the box slides across the rough surface before stopping. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 11.49 Truck brakes can fail if they get too hot. In some mountainous areas, ramps of loose gravel are constructed to stop runaway trucks that have lost their brakes. The combination of a slight upward slope and a large coefficient of rolling friction as the truck tires sink into the gravel brings the truck safely to a halt. Suppose a gravel ramp slopes upward at 6.0 and the coefficient of rolling friction is 0.45. Part A Use work and energy to find the length of a ramp that will stop a 15,000 truck that enters the ramp at 30 . Express your answer to two significant figures and include the appropriate units. ANSWER: Correct N/m cm kg l = 53 cm kg m/s l = 83 m Problem 11.51 Use work and energy to find an expression for the speed of the block in the following figure just before it hits the floor. Part A Find an expression for the speed of the block if the coefficient of kinetic friction for the block on the table is . Express your answer in terms of the variables , , , , and free fall acceleration . ANSWER: Part B Find an expression for the speed of the block if the table is frictionless. Express your answer in terms of the variables , , , and free fall acceleration . ANSWER: μk M m h μk g v = M m h g Problem 11.57 The spring shown in the figure is compressed 60 and used to launch a 100 physics student. The track is frictionless until it starts up the incline. The student's coefficient of kinetic friction on the incline is 0.12 . Part A What is the student's speed just after losing contact with the spring? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How far up the incline does the student go? Express your answer to two significant figures and include the appropriate units. ANSWER: v = cm kg 30 v = 17 ms Correct Score Summary: Your score on this assignment is 93.6%. You received 112.37 out of a possible total of 120 points. !s = 41 m

Assignment 9 Due: 11:59pm on Friday, April 11, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Problem 11.2 Part A Evaluate the dot product if and . Express your answer using two significant figures. ANSWER: Correct Part B Evaluate the dot product if and . Express your answer using two significant figures. ANSWER: Correct Problem 11.4  A B = 5 − 6 A i ^ j ^ = −9 − 5 B i ^ j ^ A  B  = -15  A B = −5 + 9 A i ^ j ^ = 5 + 6 B i ^ j ^ A  B  = 29 Part A What is the angle between vectors and if and ? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part B What is the angle between vectors and if and ? Express your answer as an integer and include the appropriate units. ANSWER: Correct ± All Work and No Play Learning Goal: To be able to calculate work done by a constant force directed at different angles relative to displacement If an object undergoes displacement while being acted upon by a force (or several forces), it is said that work is being done on the object. If the object is moving in a straight line and the displacement and the force are known, the work done by the force can be calculated as , where is the work done by force on the object that undergoes displacement directed at angle relative to .  A B A = 2 + 5 ı ^  ^ B = −2 − 4 ı ^  ^  = 175  A B A = −6 + 2 ı ^  ^ B = − − 3 ı ^  ^  = 90 W =  = cos  F  s  F   s  W F  s  F  Note that depending on the value of , the work done can be positive, negative, or zero. In this problem, you will practice calculating work done on an object moving in a straight line. The first series of questions is related to the accompanying figure. Part A What can be said about the sign of the work done by the force ? ANSWER: Correct When , the cosine of is zero, and therefore the work done is zero. Part B cos  F  1 It is positive. It is negative. It is zero. There is not enough information to answer the question.  = 90  What can be said about the work done by force ? ANSWER: Correct When , is positive, and so the work done is positive. Part C The work done by force is ANSWER: Correct When , is negative, and so the work done is negative. Part D The work done by force is ANSWER: F  2 It is positive. It is negative. It is zero. 0 <  < 90 cos  F  3 positive negative zero 90 <  < 180 cos  F  4 Correct Part E The work done by force is ANSWER: Correct positive negative zero F  5 positive negative zero Part F The work done by force is ANSWER: Correct Part G The work done by force is ANSWER: Correct In the next series of questions, you will use the formula to calculate the work done by various forces on an object that moves 160 meters to the right. F  6 positive negative zero F  7 positive negative zero W =  = cos  F  s  F   s  Part H Find the work done by the 18-newton force. Use two significant figures in your answer. Express your answer in joules. ANSWER: Correct Part I Find the work done by the 30-newton force. Use two significant figures in your answer. Express your answer in joules. ANSWER: Correct Part J Find the work done by the 12-newton force. Use two significant figures in your answer. Express your answer in joules. W W = 2900 J W W = 4200 J W ANSWER: Correct Part K Find the work done by the 15-newton force. Use two significant figures in your answer. Express your answer in joules. ANSWER: Correct Introduction to Potential Energy Learning Goal: Understand that conservative forces can be removed from the work integral by incorporating them into a new form of energy called potential energy that must be added to the kinetic energy to get the total mechanical energy. The first part of this problem contains short-answer questions that review the work-energy theorem. In the second part we introduce the concept of potential energy. But for now, please answer in terms of the work-energy theorem. Work-Energy Theorem The work-energy theorem states , where is the work done by all forces that act on the object, and and are the initial and final kinetic energies, respectively. Part A The work-energy theorem states that a force acting on a particle as it moves over a ______ changes the ______ energy of the particle if the force has a component parallel to the motion. W = -1900 J W W = -1800 J Kf = Ki + Wall Wall Ki Kf Choose the best answer to fill in the blanks above: ANSWER: Correct It is important that the force have a component acting in the direction of motion. For example, if a ball is attached to a string and whirled in uniform circular motion, the string does apply a force to the ball, but since the string's force is always perpendicular to the motion it does no work and cannot change the kinetic energy of the ball. Part B To calculate the change in energy, you must know the force as a function of _______. The work done by the force causes the energy change. Choose the best answer to fill in the blank above: ANSWER: Correct Part C To illustrate the work-energy concept, consider the case of a stone falling from to under the influence of gravity. Using the work-energy concept, we say that work is done by the gravitational _____, resulting in an increase of the ______ energy of the stone. Choose the best answer to fill in the blanks above: distance / potential distance / kinetic vertical displacement / potential none of the above acceleration work distance potential energy xi xf ANSWER: Correct Potential Energy You should read about potential energy in your text before answering the following questions. Potential energy is a concept that builds on the work-energy theorem, enlarging the concept of energy in the most physically useful way. The key aspect that allows for potential energy is the existence of conservative forces, forces for which the work done on an object does not depend on the path of the object, only the initial and final positions of the object. The gravitational force is conservative; the frictional force is not. The change in potential energy is the negative of the work done by conservative forces. Hence considering the initial and final potential energies is equivalent to calculating the work done by the conservative forces. When potential energy is used, it replaces the work done by the associated conservative force. Then only the work due to nonconservative forces needs to be calculated. In summary, when using the concept of potential energy, only nonconservative forces contribute to the work, which now changes the total energy: , where and are the final and initial potential energies, and is the work due only to nonconservative forces. Now, we will revisit the falling stone example using the concept of potential energy. Part D Rather than ascribing the increased kinetic energy of the stone to the work of gravity, we now (when using potential energy rather than work-energy) say that the increased kinetic energy comes from the ______ of the _______ energy. Choose the best answer to fill in the blanks above: ANSWER: force / kinetic potential energy / potential force / potential potential energy / kinetic Kf + Uf = Ef = Wnc + Ei = Wnc + Ki + Ui Uf Ui Wnc Correct Part E This process happens in such a way that total mechanical energy, equal to the ______ of the kinetic and potential energies, is _______. Choose the best answer to fill in the blanks above: ANSWER: Correct Problem 11.7 Part A How much work is done by the force 2.2 6.6 on a particle that moves through displacement 3.9 Express your answer to two significant figures and include the appropriate units. ANSWER: work / potential force / kinetic change / potential sum / conserved sum / zero sum / not conserved difference / conserved F  = (− + i ^ ) N j ^ ! = r m i ^ Correct Part B How much work is done by the force 2.2 6.6 on a particle that moves through displacement 3.9 Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 11.10 A 1.8 book is lying on a 0.80- -high table. You pick it up and place it on a bookshelf 2.27 above the floor. Part A How much work does gravity do on the book? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B W = -8.6 J F  = (− + i ^ ) N j ^ ! = r m? j ^ W = 26 J kg m m Wg = -26 J How much work does your hand do on the book? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 11.12 The three ropes shown in the bird's-eye view of the figure are used to drag a crate 3.3 across the floor. Part A How much work is done by each of the three forces? Express your answers using two significant figures. Enter your answers numerically separated by commas. ANSWER: WH = 26 J m W1 , W2 , W3 = 1.9,1.2,-2.1 kJ Correct Enhanced EOC: Problem 11.16 A 1.2 particle moving along the x-axis experiences the force shown in the figure. The particle's velocity is 4.6 at . You may want to review ( pages 286 - 287) . For help with math skills, you may want to review: The Definite Integral Part A What is its velocity at ? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the work–kinetic energy theorem? What is the kinetic energy at ? How is the work done in going from to related to force shown in the graph? Using the work–kinetic energy theorem, what is the kinetic energy at ? What is the velocity at ? ANSWER: kg m/s x = 0m x = 2m x = 0 m x = 0 m x = 2 m x = 2 m x = 2 m Correct Part B What is its velocity at ? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the work–kinetic energy theorem? What is the kinetic energy at ? How is the work done in going from to related to force shown in the graph? Can the work be negative? Using the work–kinetic energy theorem, what is the kinetic energy at ? What is the velocity at ? ANSWER: Correct Work on a Sliding Block A block of weight sits on a frictionless inclined plane, which makes an angle with respect to the horizontal, as shown. A force of magnitude , applied parallel to the incline, pulls the block up the plane at constant speed. v = 6.2 ms x = 4m x = 0 m x = 0 m x = 4 m x = 4 m x = 4 m v = 4.6 ms w  F Part A The block moves a distance up the incline. The block does not stop after moving this distance but continues to move with constant speed. What is the total work done on the block by all forces? (Include only the work done after the block has started moving, not the work needed to start the block moving from rest.) Express your answer in terms of given quantities. Hint 1. What physical principle to use To find the total work done on the block, use the work-energy theorem: . Hint 2. Find the change in kinetic energy What is the change in the kinetic energy of the block, from the moment it starts moving until it has been pulled a distance ? Remember that the block is pulled at constant speed. Hint 1. Consider kinetic energy If the block's speed does not change, its kinetic energy cannot change. ANSWER: ANSWER: L Wtot Wtot = Kf − Ki L Kf − Ki = 0 Wtot = 0 Correct Part B What is , the work done on the block by the force of gravity as the block moves a distance up the incline? Express the work done by gravity in terms of the weight and any other quantities given in the problem introduction. Hint 1. Force diagram Hint 2. Force of gravity component What is the component of the force of gravity in the direction of the block's displacement (along the inclined plane)? Express your answer in terms of and . Hint 1. Relative direction of the force and the motion Remember that the force of gravity acts down the plane, whereas the block's displacement is directed up the plane. ANSWER: Wg L w w  ANSWER: Correct Part C What is , the work done on the block by the applied force as the block moves a distance up the incline? Express your answer in terms of and other given quantities. Hint 1. How to find the work done by a constant force Remember that the work done on an object by a particular force is the integral of the dot product of the force and the instantaneous displacement of the object, over the path followed by the object. In this case, since the force is constant and the path is a straight segment of length up the inclined plane, the dot product becomes simple multiplication. ANSWER: Correct Part D What is , the work done on the block by the normal force as the block moves a distance up the inclined plane? Express your answer in terms of given quantities. Hint 1. First step in computing the work Fg|| = −wsin() Wg = −wLsin() WF F L F L WF = FL Wnormal L The work done by the normal force is equal to the dot product of the force vector and the block's displacement vector. The normal force and the block's displacement vector are perpendicular. Therefore, what is their dot product? ANSWER: ANSWER: Correct Problem 11.20 A particle moving along the -axis has the potential energy , where is in . Part A What is the -component of the force on the particle at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the -component of the force on the particle at ? Express your answer to two significant figures and include the appropriate units. N  L = 0 Wnormal = 0 y U = 3.2y3 J y m y y = 0 m Fy = 0 N y y = 1 m ANSWER: Correct Part C What is the -component of the force on the particle at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 11.28 A cable with 25.0 of tension pulls straight up on a 1.08 block that is initially at rest. Part A What is the block's speed after being lifted 2.40 ? Solve this problem using work and energy. Express your answer with the appropriate units. ANSWER: Correct Fy = -9.6 N y y = 2 m Fy = -38 N N kg m vf = 8.00 ms Problem 11.29 Part A How much work does an elevator motor do to lift a 1500 elevator a height of 110 ? Express your answer with the appropriate units. ANSWER: Correct Part B How much power must the motor supply to do this in 50 at constant speed? Express your answer with the appropriate units. ANSWER: Correct Problem 11.32 How many energy is consumed by a 1.20 hair dryer used for 10.0 and a 11.0 night light left on for 16.0 ? Part A Hair dryer: Express your answer with the appropriate units. kg m Wext = 1.62×106 J s = 3.23×104 P W kW min W hr ANSWER: Correct Part B Night light: Express your answer with the appropriate units. ANSWER: Correct Problem 11.42 A 2500 elevator accelerates upward at 1.20 for 10.0 , starting from rest. Part A How much work does gravity do on the elevator? Express your answer with the appropriate units. ANSWER: Correct W = 7.20×105 J = 6.34×105 W J kg m/s2 m −2.45×105 J Part B How much work does the tension in the elevator cable do on the elevator? Express your answer with the appropriate units. ANSWER: Correct Part C Use the work-kinetic energy theorem to find the kinetic energy of the elevator as it reaches 10.0 . Express your answer with the appropriate units. ANSWER: Correct Part D What is the speed of the elevator as it reaches 10.0 ? Express your answer with the appropriate units. ANSWER: Correct 2.75×105 J m 3.00×104 J m 4.90 ms Problem 11.47 A horizontal spring with spring constant 130 is compressed 17 and used to launch a 2.4 box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.15. Part A Use work and energy to find how far the box slides across the rough surface before stopping. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 11.49 Truck brakes can fail if they get too hot. In some mountainous areas, ramps of loose gravel are constructed to stop runaway trucks that have lost their brakes. The combination of a slight upward slope and a large coefficient of rolling friction as the truck tires sink into the gravel brings the truck safely to a halt. Suppose a gravel ramp slopes upward at 6.0 and the coefficient of rolling friction is 0.45. Part A Use work and energy to find the length of a ramp that will stop a 15,000 truck that enters the ramp at 30 . Express your answer to two significant figures and include the appropriate units. ANSWER: Correct N/m cm kg l = 53 cm kg m/s l = 83 m Problem 11.51 Use work and energy to find an expression for the speed of the block in the following figure just before it hits the floor. Part A Find an expression for the speed of the block if the coefficient of kinetic friction for the block on the table is . Express your answer in terms of the variables , , , , and free fall acceleration . ANSWER: Part B Find an expression for the speed of the block if the table is frictionless. Express your answer in terms of the variables , , , and free fall acceleration . ANSWER: μk M m h μk g v = M m h g Problem 11.57 The spring shown in the figure is compressed 60 and used to launch a 100 physics student. The track is frictionless until it starts up the incline. The student's coefficient of kinetic friction on the incline is 0.12 . Part A What is the student's speed just after losing contact with the spring? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How far up the incline does the student go? Express your answer to two significant figures and include the appropriate units. ANSWER: v = cm kg 30 v = 17 ms Correct Score Summary: Your score on this assignment is 93.6%. You received 112.37 out of a possible total of 120 points. !s = 41 m

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