Assignment 5 Due: 11:59pm on Wednesday, March 5, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 6.13 A hand presses down on the book in the figure. Part A Is the normal force of the table on the book larger than, smaller than, or equal to ? ANSWER: Correct mg Equal to Larger than Smaller than mg mg mg Problem 6.2 The three ropes in the figure are tied to a small, very light ring. Two of these ropes are anchored to walls at right angles with the tensions shown in the figure. Part A What is the magnitude of the tension in the third rope? Express your answer using two significant figures. ANSWER: Correct Part B What is the direction of the tension in the third rope? Express your answer using two significant figures. T  3 T3 = 94 N T  3 Typesetting math: 100% ANSWER: Correct The Normal Force When an object rests on a surface, there is always a force perpendicular to the surface; we call this the normal force, denoted by . The two questions to the right will explore the normal force. Part A A man attempts to pick up his suitcase of weight by pulling straight up on the handle. However, he is unable to lift the suitcase from the floor. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man pulls upward on the suitcase? Hint 1. How to approach this problem First, identify the forces that act on the suitcase and draw a free-body diagram. Then use the fact that the suitcase is in equilibrium, , to examine how the forces acting on the suitcase relate to each other. Hint 2. Identify the correct free-body diagram Which of the figures represents the free-body diagram of the suitcase while the man is pulling on the handle with a force of magnitude ? = 58   below horizontal n ws n F = 0 fpull Typesetting math: 100% ANSWER: ANSWER: Correct Part B A B C D The magnitude of the normal force is equal to the magnitude of the weight of the suitcase. The magnitude of the normal force is equal to the magnitude of the weight of the suitcase minus the magnitude of the force of the pull. The magnitude of the normal force is equal to the sum of the magnitude of the force of the pull and the magnitude of the suitcase’s weight. The magnitude of the normal force is greater than the magnitude of the weight of the suitcase. Typesetting math: 100% Now assume that the man of weight is tired and decides to sit on his suitcase. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man is sitting on the suitcase? Hint 1. Identify the correct free-body diagram. Which of the figures represents the free-body diagram while the man is sitting atop the suitcase? Here the vector labeled is a force that has the same magnitude as the man’s weight. ANSWER: wm n wm Typesetting math: 100% ANSWER: Correct Recognize that the normal force acting on an object is not always equal to the weight of that object. This is an important point to understand. Problem 6.5 A construction worker with a weight of 880 stands on a roof that is sloped at 18 . Part A What is the magnitude of the normal force of the roof on the worker? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct A B C D The magnitude of the normal force is equal to the magnitude of the suitcase’s weight. The magnitude of the normal force is equal to the magnitude of the suitcase’s weight minus the magnitude of the man’s weight. The magnitude of the normal force is equal to the sum of the magnitude of the man’s weight and the magnitude of the suitcase’s weight. The magnitude of the normal force is less than the magnitude of the suitcase’s weight. N  n = 840 N Typesetting math: 100% Problem 6.6 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B For diagram the part A, find the value of the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: kg ax x ax = -0.67 m s2 ay, y Typesetting math: 100% Correct Part C For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D For diagram the part C, find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: ay = 0 m s2 ax x ax = 0.67 m s2 ay y Typesetting math: 100% Correct Problem 6.7 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay = 0 m s2 kg ax x ax = 0.99 m s2 Typesetting math: 100% Part B Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay y ay = 0 m s2 ax x ax = -0.18 m s2 ay y ay = 0 m s2 Typesetting math: 100% Problem 6.10 A horizontal rope is tied to a 53.0 box on frictionless ice. What is the tension in the rope if: Part A The box is at rest? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part B The box moves at a steady = 4.80 ? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part C The box = 4.80 and = 4.60 ? Express your answer to three significant figures and include the appropriate units. ANSWER: kg T = 0 N vx m/s T = 0 N vx m/s ax m/s2 Typesetting math: 100% Correct Problem 6.14 It takes the elevator in a skyscraper 4.5 to reach its cruising speed of 11 . A 60 passenger gets aboard on the ground floor. Part A What is the passenger’s weight before the elevator starts moving? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the passenger’s weight while the elevator is speeding up? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the passenger’s weight after the elevator reaches its cruising speed? T = 244 N s m/s kg w = 590 N w = 730 N Typesetting math: 100% Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Block on an Incline A block lies on a plane raised an angle from the horizontal. Three forces act upon the block: , the force of gravity; , the normal force; and , the force of friction. The coefficient of friction is large enough to prevent the block from sliding . Part A Consider coordinate system a, with the x axis along the plane. Which forces lie along the axes? ANSWER: w = 590 N  F  w F n F  f Typesetting math: 100% Correct Part B Which forces lie along the axes of the coordinate system b, in which the y axis is vertical? ANSWER: Correct only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w Typesetting math: 100% Usually the best advice is to choose coordinate system so that the acceleration of the system is directly along one of the coordinate axes. If the system isn’t accelerating, then you are better off choosing the coordinate system with the most vectors along the coordinate axes. But now you are going to ignore that advice. You will find the normal force, , using vertical coordinate system b. In these coordinates you will find the magnitude appearing in both the x and y equations, each multiplied by a trigonometric function. Part C Because the block is not moving, the sum of the y components of the forces acting on the block must be zero. Find an expression for the sum of the y components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the y-axis is also , as shown in the figure. ANSWER: F  n Fn Fn Ff Fw  F n Fny F  n Fn  F  n  Typesetting math: 100% Hint 2. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the x-axis is also , as shown in the figure. ANSWER: ANSWER: Fny = Fncos() F f Ffy F f Ff  F  f  Ffy = Ffsin() Fy = 0 = Fncos() + Ffsin() − Fw Typesetting math: 100% Correct Part D Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find an expression for the sum of the x components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the x component of Write an expression for , the x component of the force , using coordinate system b. Express your answer in terms of and . ANSWER: ANSWER: Correct Part E To find the magnitude of the normal force, you must express in terms of since is an unknown. Using the equations you found in the two previous parts, find an expression for involving and but not . Hint 1. How to approach the problem From your answers to the previous two parts you should have two force equations ( and ). Combine these equations to eliminate . The key is to multiply the Fn Ff Fw  F n Fnx F  n Fn  Fnx = −Fnsin() Fx = 0 = −Fnsin() + Ffcos() Fn Fw Ff Fn Fw  Ff Typesetting math: 100% Fy = 0 Fx = 0 Ff equation for the y components by and the equation for the x components by , then add or subtract the two equations to eliminate the term . An alternative motivation for the algebra is to eliminate the trig functions in front of by using the trig identity . At the very least this would result in an equation that is simple to solve for . ANSWER: Correct Congratulations on working this through. Now realize that in coordinate system a, which is aligned with the plane, the y-coordinate equation is , which leads immediately to the result obtained here for . CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra. Contact Forces Introduced Learning Goal: To introduce contact forces (normal and friction forces) and to understand that, except for friction forces under certain circumstances, these forces must be determined from: net Force = ma. Two solid objects cannot occupy the same space at the same time. Indeed, when the objects touch, they exert repulsive normal forces on each other, as well as frictional forces that resist their slipping relative to each other. These contact forces arise from a complex interplay between the electrostatic forces between the electrons and ions in the objects and the laws of quantum mechanics. As two surfaces are pushed together these forces increase exponentially over an atomic distance scale, easily becoming strong enough to distort the bulk material in the objects if they approach too close. In everyday experience, contact forces are limited by the deformation or acceleration of the objects, rather than by the fundamental interatomic forces. Hence, we can conclude the following: The magnitude of contact forces is determined by , that is, by the other forces on, and acceleration of, the contacting bodies. The only exception is that the frictional forces cannot exceed (although they can be smaller than this or even zero). Normal and friction forces Two types of contact forces operate in typical mechanics problems, the normal and frictional forces, usually designated by and (or , or something similar) respectively. These are the components of the overall contact force: perpendicular to and parallel to the plane of contact. Kinetic friction when surfaces slide cos  sin  Ff cos() sin() Fn sin2() + cos2 () = 1 Fn Fn = Fwcos() Fy = Fn − FW cos() = 0 Fn F = ma μn n f Ffric n f Typesetting math: 100% When one surface is sliding past the other, experiments show three things about the friction force (denoted ): The frictional force opposes the relative motion at the 1. point of contact, 2. is proportional to the normal force, and 3. the ratio of the magnitude of the frictional force to that of the normal force is fairly constant over a wide range of speeds. The constant of proportionality is called the coefficient of kinetic friction, often designated . As long as the sliding continues, the frictional force is then (valid when the surfaces slide by each other). Static friction when surfaces don’t slide When there is no relative motion of the surfaces, the frictional force can assume any value from zero up to a maximum , where is the coefficient of static friction. Invariably, is larger than , in agreement with the observation that when a force is large enough that something breaks loose and starts to slide, it often accelerates. The frictional force for surfaces with no relative motion is therefore (valid when the contacting surfaces have no relative motion). The actual magnitude and direction of the static friction force are such that it (together with other forces on the object) causes the object to remain motionless with respect to the contacting surface as long as the static friction force required does not exceed . The equation is valid only when the surfaces are on the verge of sliding. Part A When two objects slide by one another, which of the following statements about the force of friction between them, is true? ANSWER: Correct Part B fk fk μk fk = μkn μsn μs μs μk fs ! μsn μsn fs = μsn The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μkn μkn μkn Typesetting math: 100% When two objects are in contact with no relative motion, which of the following statements about the frictional force between them, is true? ANSWER: Correct For static friction, the actual magnitude and direction of the friction force are such that it, together with any other forces present, will cause the object to have the observed acceleration. The magnitude of the force cannot exceed . If the magnitude of static friction needed to keep acceleration equal to zero exceeds , then the object will slide subject to the resistance of kinetic friction. Do not automatically assume that unless you are considering a situation in which the magnitude of the static friction force is as large as possible (i.e., when determining at what point an object will just begin to slip). Whether the actual magnitude of the friction force is 0, less than , or equal to depends on the magnitude of the other forces (if any) as well as the acceleration of the object through . Part C When a board with a box on it is slowly tilted to larger and larger angle, common experience shows that the box will at some point “break loose” and start to accelerate down the board. The box begins to slide once the component of gravity acting parallel to the board just begins to exceeds the maximum force of static friction. Which of the following is the most general explanation for why the box accelerates down the board? ANSWER: The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μsn μsn μsn μsn μsn fs = μsn μsn μsn F = ma Fg The force of kinetic friction is smaller than that of maximum static friction, but remains the same. Once the box is moving, is smaller than the force of maximum static friction but larger than the force of kinetic friction. Once the box is moving, is larger than the force of maximum static friction. When the box is stationary, equals the force of static friction, but once the box starts moving, the sliding reduces the normal force, which in turn reduces the friction. Fg Fg Fg Fg Typesetting math: 100% Correct At the point when the box finally does “break loose,” you know that the component of the box’s weight that is parallel to the board just exceeds (i.e., this component of gravitational force on the box has just reached a magnitude such that the force of static friction, which has a maximum value of , can no longer oppose it.) For the box to then accelerate, there must be a net force on the box along the board. Thus, the component of the box’s weight parallel to the board must be greater than the force of kinetic friction. Therefore the force of kinetic friction must be less than the force of static friction which implies , as expected. Part D Consider a problem in which a car of mass is on a road tilted at an angle . The normal force Select the best answer. ANSWER: Correct The key point is that contact forces must be determined from Newton’s equation. In the problem described above, there is not enough information given to determine the normal force (e.g., the acceleration is unknown). Each of the answer options is valid under some conditions ( , the car is sliding down an icy incline, or the car is going around a banked turn), but in fact none is likely to be correct if there are other forces on the car or if the car is accelerating. Do not memorize values for the normal force valid in different problems–you must determine from . Problem 6.17 Bonnie and Clyde are sliding a 323 bank safe across the floor to their getaway car. The safe slides with a constant speed if Clyde pushes from behind with 375 of force while Bonnie pulls forward on a rope with 335 of force. μsn μsn μkn μsn μk < μs M  is found using n = Mg n = Mg cos() n = Mg cos() F  = Ma  = 0 n F = ma kg N N Typesetting math: 100% Part A What is the safe's coefficient of kinetic friction on the bank floor? ANSWER: Correct Problem 6.19 A crate is placed on a horizontal conveyor belt. The materials are such that and . Part A Draw a free-body diagram showing all the forces on the crate if the conveyer belt runs at constant speed. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: 0.224 10 kg μs = 0.5 μk = 0.3 Typesetting math: 100% Correct Part B Draw a free-body diagram showing all the forces on the crate if the conveyer belt is speeding up. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Typesetting math: 100% Correct Part C What is the maximum acceleration the belt can have without the crate slipping? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct amax = 4.9 m s2 Typesetting math: 100% Problem 6.28 A 1100 steel beam is supported by two ropes. Part A What is the tension in rope 1? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the tension in rope 2? Express your answer to two significant figures and include the appropriate units. ANSWER: kg T1 = 7000 N Typesetting math: 100% Correct Problem 6.35 The position of a 1.4 mass is given by , where is in seconds. Part A What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.39 T2 = 4800 N kg x = (2t3 − 3t2 )m t t = 0 s F = -8.4 N t = 1 s F = 8.4 N Typesetting math: 100% A rifle with a barrel length of 61 fires a 8 bullet with a horizontal speed of 400 . The bullet strikes a block of wood and penetrates to a depth of 11 . Part A What resistive force (assumed to be constant) does the wood exert on the bullet? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How long does it take the bullet to come to rest after entering the wood? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.45 You and your friend Peter are putting new shingles on a roof pitched at 21 . You're sitting on the very top of the roof when Peter, who is at the edge of the roof directly below you, 5.0 away, asks you for the box of nails. Rather than carry the 2.0 box of nails down to Peter, you decide to give the box a push and have it slide down to him. Part A If the coefficient of kinetic friction between the box and the roof is 0.55, with what speed should you push the box to have it gently come to rest right at the edge of the roof? Express your answer to two significant figures and include the appropriate units. cm g m/s cm fk = 5800 N = 5.5×10−4 t s  m kg Typesetting math: 100% ANSWER: Correct Problem 6.54 The 2.0 wood box in the figure slides down a vertical wood wall while you push on it at a 45 angle. Part A What magnitude of force should you apply to cause the box to slide down at a constant speed? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct v = 3.9 ms kg  F = 23 N Typesetting math: 100% Score Summary: Your score on this assignment is 98.8%. You received 114.57 out of a possible total of 116 points. Typesetting math: 100%

Assignment 5 Due: 11:59pm on Wednesday, March 5, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 6.13 A hand presses down on the book in the figure. Part A Is the normal force of the table on the book larger than, smaller than, or equal to ? ANSWER: Correct mg Equal to Larger than Smaller than mg mg mg Problem 6.2 The three ropes in the figure are tied to a small, very light ring. Two of these ropes are anchored to walls at right angles with the tensions shown in the figure. Part A What is the magnitude of the tension in the third rope? Express your answer using two significant figures. ANSWER: Correct Part B What is the direction of the tension in the third rope? Express your answer using two significant figures. T  3 T3 = 94 N T  3 Typesetting math: 100% ANSWER: Correct The Normal Force When an object rests on a surface, there is always a force perpendicular to the surface; we call this the normal force, denoted by . The two questions to the right will explore the normal force. Part A A man attempts to pick up his suitcase of weight by pulling straight up on the handle. However, he is unable to lift the suitcase from the floor. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man pulls upward on the suitcase? Hint 1. How to approach this problem First, identify the forces that act on the suitcase and draw a free-body diagram. Then use the fact that the suitcase is in equilibrium, , to examine how the forces acting on the suitcase relate to each other. Hint 2. Identify the correct free-body diagram Which of the figures represents the free-body diagram of the suitcase while the man is pulling on the handle with a force of magnitude ? = 58   below horizontal n ws n F = 0 fpull Typesetting math: 100% ANSWER: ANSWER: Correct Part B A B C D The magnitude of the normal force is equal to the magnitude of the weight of the suitcase. The magnitude of the normal force is equal to the magnitude of the weight of the suitcase minus the magnitude of the force of the pull. The magnitude of the normal force is equal to the sum of the magnitude of the force of the pull and the magnitude of the suitcase’s weight. The magnitude of the normal force is greater than the magnitude of the weight of the suitcase. Typesetting math: 100% Now assume that the man of weight is tired and decides to sit on his suitcase. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man is sitting on the suitcase? Hint 1. Identify the correct free-body diagram. Which of the figures represents the free-body diagram while the man is sitting atop the suitcase? Here the vector labeled is a force that has the same magnitude as the man’s weight. ANSWER: wm n wm Typesetting math: 100% ANSWER: Correct Recognize that the normal force acting on an object is not always equal to the weight of that object. This is an important point to understand. Problem 6.5 A construction worker with a weight of 880 stands on a roof that is sloped at 18 . Part A What is the magnitude of the normal force of the roof on the worker? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct A B C D The magnitude of the normal force is equal to the magnitude of the suitcase’s weight. The magnitude of the normal force is equal to the magnitude of the suitcase’s weight minus the magnitude of the man’s weight. The magnitude of the normal force is equal to the sum of the magnitude of the man’s weight and the magnitude of the suitcase’s weight. The magnitude of the normal force is less than the magnitude of the suitcase’s weight. N  n = 840 N Typesetting math: 100% Problem 6.6 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B For diagram the part A, find the value of the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: kg ax x ax = -0.67 m s2 ay, y Typesetting math: 100% Correct Part C For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D For diagram the part C, find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: ay = 0 m s2 ax x ax = 0.67 m s2 ay y Typesetting math: 100% Correct Problem 6.7 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay = 0 m s2 kg ax x ax = 0.99 m s2 Typesetting math: 100% Part B Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay y ay = 0 m s2 ax x ax = -0.18 m s2 ay y ay = 0 m s2 Typesetting math: 100% Problem 6.10 A horizontal rope is tied to a 53.0 box on frictionless ice. What is the tension in the rope if: Part A The box is at rest? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part B The box moves at a steady = 4.80 ? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part C The box = 4.80 and = 4.60 ? Express your answer to three significant figures and include the appropriate units. ANSWER: kg T = 0 N vx m/s T = 0 N vx m/s ax m/s2 Typesetting math: 100% Correct Problem 6.14 It takes the elevator in a skyscraper 4.5 to reach its cruising speed of 11 . A 60 passenger gets aboard on the ground floor. Part A What is the passenger’s weight before the elevator starts moving? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the passenger’s weight while the elevator is speeding up? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the passenger’s weight after the elevator reaches its cruising speed? T = 244 N s m/s kg w = 590 N w = 730 N Typesetting math: 100% Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Block on an Incline A block lies on a plane raised an angle from the horizontal. Three forces act upon the block: , the force of gravity; , the normal force; and , the force of friction. The coefficient of friction is large enough to prevent the block from sliding . Part A Consider coordinate system a, with the x axis along the plane. Which forces lie along the axes? ANSWER: w = 590 N  F  w F n F  f Typesetting math: 100% Correct Part B Which forces lie along the axes of the coordinate system b, in which the y axis is vertical? ANSWER: Correct only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w Typesetting math: 100% Usually the best advice is to choose coordinate system so that the acceleration of the system is directly along one of the coordinate axes. If the system isn’t accelerating, then you are better off choosing the coordinate system with the most vectors along the coordinate axes. But now you are going to ignore that advice. You will find the normal force, , using vertical coordinate system b. In these coordinates you will find the magnitude appearing in both the x and y equations, each multiplied by a trigonometric function. Part C Because the block is not moving, the sum of the y components of the forces acting on the block must be zero. Find an expression for the sum of the y components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the y-axis is also , as shown in the figure. ANSWER: F  n Fn Fn Ff Fw  F n Fny F  n Fn  F  n  Typesetting math: 100% Hint 2. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the x-axis is also , as shown in the figure. ANSWER: ANSWER: Fny = Fncos() F f Ffy F f Ff  F  f  Ffy = Ffsin() Fy = 0 = Fncos() + Ffsin() − Fw Typesetting math: 100% Correct Part D Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find an expression for the sum of the x components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the x component of Write an expression for , the x component of the force , using coordinate system b. Express your answer in terms of and . ANSWER: ANSWER: Correct Part E To find the magnitude of the normal force, you must express in terms of since is an unknown. Using the equations you found in the two previous parts, find an expression for involving and but not . Hint 1. How to approach the problem From your answers to the previous two parts you should have two force equations ( and ). Combine these equations to eliminate . The key is to multiply the Fn Ff Fw  F n Fnx F  n Fn  Fnx = −Fnsin() Fx = 0 = −Fnsin() + Ffcos() Fn Fw Ff Fn Fw  Ff Typesetting math: 100% Fy = 0 Fx = 0 Ff equation for the y components by and the equation for the x components by , then add or subtract the two equations to eliminate the term . An alternative motivation for the algebra is to eliminate the trig functions in front of by using the trig identity . At the very least this would result in an equation that is simple to solve for . ANSWER: Correct Congratulations on working this through. Now realize that in coordinate system a, which is aligned with the plane, the y-coordinate equation is , which leads immediately to the result obtained here for . CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra. Contact Forces Introduced Learning Goal: To introduce contact forces (normal and friction forces) and to understand that, except for friction forces under certain circumstances, these forces must be determined from: net Force = ma. Two solid objects cannot occupy the same space at the same time. Indeed, when the objects touch, they exert repulsive normal forces on each other, as well as frictional forces that resist their slipping relative to each other. These contact forces arise from a complex interplay between the electrostatic forces between the electrons and ions in the objects and the laws of quantum mechanics. As two surfaces are pushed together these forces increase exponentially over an atomic distance scale, easily becoming strong enough to distort the bulk material in the objects if they approach too close. In everyday experience, contact forces are limited by the deformation or acceleration of the objects, rather than by the fundamental interatomic forces. Hence, we can conclude the following: The magnitude of contact forces is determined by , that is, by the other forces on, and acceleration of, the contacting bodies. The only exception is that the frictional forces cannot exceed (although they can be smaller than this or even zero). Normal and friction forces Two types of contact forces operate in typical mechanics problems, the normal and frictional forces, usually designated by and (or , or something similar) respectively. These are the components of the overall contact force: perpendicular to and parallel to the plane of contact. Kinetic friction when surfaces slide cos  sin  Ff cos() sin() Fn sin2() + cos2 () = 1 Fn Fn = Fwcos() Fy = Fn − FW cos() = 0 Fn F = ma μn n f Ffric n f Typesetting math: 100% When one surface is sliding past the other, experiments show three things about the friction force (denoted ): The frictional force opposes the relative motion at the 1. point of contact, 2. is proportional to the normal force, and 3. the ratio of the magnitude of the frictional force to that of the normal force is fairly constant over a wide range of speeds. The constant of proportionality is called the coefficient of kinetic friction, often designated . As long as the sliding continues, the frictional force is then (valid when the surfaces slide by each other). Static friction when surfaces don’t slide When there is no relative motion of the surfaces, the frictional force can assume any value from zero up to a maximum , where is the coefficient of static friction. Invariably, is larger than , in agreement with the observation that when a force is large enough that something breaks loose and starts to slide, it often accelerates. The frictional force for surfaces with no relative motion is therefore (valid when the contacting surfaces have no relative motion). The actual magnitude and direction of the static friction force are such that it (together with other forces on the object) causes the object to remain motionless with respect to the contacting surface as long as the static friction force required does not exceed . The equation is valid only when the surfaces are on the verge of sliding. Part A When two objects slide by one another, which of the following statements about the force of friction between them, is true? ANSWER: Correct Part B fk fk μk fk = μkn μsn μs μs μk fs ! μsn μsn fs = μsn The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μkn μkn μkn Typesetting math: 100% When two objects are in contact with no relative motion, which of the following statements about the frictional force between them, is true? ANSWER: Correct For static friction, the actual magnitude and direction of the friction force are such that it, together with any other forces present, will cause the object to have the observed acceleration. The magnitude of the force cannot exceed . If the magnitude of static friction needed to keep acceleration equal to zero exceeds , then the object will slide subject to the resistance of kinetic friction. Do not automatically assume that unless you are considering a situation in which the magnitude of the static friction force is as large as possible (i.e., when determining at what point an object will just begin to slip). Whether the actual magnitude of the friction force is 0, less than , or equal to depends on the magnitude of the other forces (if any) as well as the acceleration of the object through . Part C When a board with a box on it is slowly tilted to larger and larger angle, common experience shows that the box will at some point “break loose” and start to accelerate down the board. The box begins to slide once the component of gravity acting parallel to the board just begins to exceeds the maximum force of static friction. Which of the following is the most general explanation for why the box accelerates down the board? ANSWER: The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μsn μsn μsn μsn μsn fs = μsn μsn μsn F = ma Fg The force of kinetic friction is smaller than that of maximum static friction, but remains the same. Once the box is moving, is smaller than the force of maximum static friction but larger than the force of kinetic friction. Once the box is moving, is larger than the force of maximum static friction. When the box is stationary, equals the force of static friction, but once the box starts moving, the sliding reduces the normal force, which in turn reduces the friction. Fg Fg Fg Fg Typesetting math: 100% Correct At the point when the box finally does “break loose,” you know that the component of the box’s weight that is parallel to the board just exceeds (i.e., this component of gravitational force on the box has just reached a magnitude such that the force of static friction, which has a maximum value of , can no longer oppose it.) For the box to then accelerate, there must be a net force on the box along the board. Thus, the component of the box’s weight parallel to the board must be greater than the force of kinetic friction. Therefore the force of kinetic friction must be less than the force of static friction which implies , as expected. Part D Consider a problem in which a car of mass is on a road tilted at an angle . The normal force Select the best answer. ANSWER: Correct The key point is that contact forces must be determined from Newton’s equation. In the problem described above, there is not enough information given to determine the normal force (e.g., the acceleration is unknown). Each of the answer options is valid under some conditions ( , the car is sliding down an icy incline, or the car is going around a banked turn), but in fact none is likely to be correct if there are other forces on the car or if the car is accelerating. Do not memorize values for the normal force valid in different problems–you must determine from . Problem 6.17 Bonnie and Clyde are sliding a 323 bank safe across the floor to their getaway car. The safe slides with a constant speed if Clyde pushes from behind with 375 of force while Bonnie pulls forward on a rope with 335 of force. μsn μsn μkn μsn μk < μs M  is found using n = Mg n = Mg cos() n = Mg cos() F  = Ma  = 0 n F = ma kg N N Typesetting math: 100% Part A What is the safe's coefficient of kinetic friction on the bank floor? ANSWER: Correct Problem 6.19 A crate is placed on a horizontal conveyor belt. The materials are such that and . Part A Draw a free-body diagram showing all the forces on the crate if the conveyer belt runs at constant speed. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: 0.224 10 kg μs = 0.5 μk = 0.3 Typesetting math: 100% Correct Part B Draw a free-body diagram showing all the forces on the crate if the conveyer belt is speeding up. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Typesetting math: 100% Correct Part C What is the maximum acceleration the belt can have without the crate slipping? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct amax = 4.9 m s2 Typesetting math: 100% Problem 6.28 A 1100 steel beam is supported by two ropes. Part A What is the tension in rope 1? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the tension in rope 2? Express your answer to two significant figures and include the appropriate units. ANSWER: kg T1 = 7000 N Typesetting math: 100% Correct Problem 6.35 The position of a 1.4 mass is given by , where is in seconds. Part A What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.39 T2 = 4800 N kg x = (2t3 − 3t2 )m t t = 0 s F = -8.4 N t = 1 s F = 8.4 N Typesetting math: 100% A rifle with a barrel length of 61 fires a 8 bullet with a horizontal speed of 400 . The bullet strikes a block of wood and penetrates to a depth of 11 . Part A What resistive force (assumed to be constant) does the wood exert on the bullet? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How long does it take the bullet to come to rest after entering the wood? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.45 You and your friend Peter are putting new shingles on a roof pitched at 21 . You're sitting on the very top of the roof when Peter, who is at the edge of the roof directly below you, 5.0 away, asks you for the box of nails. Rather than carry the 2.0 box of nails down to Peter, you decide to give the box a push and have it slide down to him. Part A If the coefficient of kinetic friction between the box and the roof is 0.55, with what speed should you push the box to have it gently come to rest right at the edge of the roof? Express your answer to two significant figures and include the appropriate units. cm g m/s cm fk = 5800 N = 5.5×10−4 t s  m kg Typesetting math: 100% ANSWER: Correct Problem 6.54 The 2.0 wood box in the figure slides down a vertical wood wall while you push on it at a 45 angle. Part A What magnitude of force should you apply to cause the box to slide down at a constant speed? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct v = 3.9 ms kg  F = 23 N Typesetting math: 100% Score Summary: Your score on this assignment is 98.8%. You received 114.57 out of a possible total of 116 points. Typesetting math: 100%

Assignment 5 Due: 11:59pm on Wednesday, March 5, 2014 You … Read More...
(b) Based on the lessons learned, best practices and any additional steps you came up with in part (a), what if project manager X then got a job at Bank of America. Would it be possible for him/her to implement lean in the banking industry based on experience from the previous positions held at the automotive plant and the pharmaceutical company? Please state yes or no and explain the logic clearly for the same. Also, explain the steps that project manager X could take to implement lean at Bank of America (in the service industry) [10 points] You can refer to your class notes and will also have to do research online for both parts (a) and (b). Please state all the references used for each question.

(b) Based on the lessons learned, best practices and any additional steps you came up with in part (a), what if project manager X then got a job at Bank of America. Would it be possible for him/her to implement lean in the banking industry based on experience from the previous positions held at the automotive plant and the pharmaceutical company? Please state yes or no and explain the logic clearly for the same. Also, explain the steps that project manager X could take to implement lean at Bank of America (in the service industry) [10 points] You can refer to your class notes and will also have to do research online for both parts (a) and (b). Please state all the references used for each question.

Yes, lean can be applied to the banking industry.   … Read More...
1. (2 marks total) a. Multiply 109 x 309 b. Divide 1988 by 16 exactly 2. (4 marks total) a. Write 2/11 as a decimal to 2 decimal places b. Calculate 35% of 62 c. Add 103/4 to 92/3 d. Subtract 79.04 from 115.225 giving your answer correct to 2 decimal places 3. Circle the fractions in the list which are equivalent to 0.80 (2 marks) 2/7 32/40 8/10 8/20 8/25 9/24 36/45 40/50 4. Write the numerical value of: 3-3 (2 marks total) 5. Simplify z + 67 = 3z + 33 (1 mark total) 6. Solve to 1 decimal place 3y – 34 = 2y + 89 (1 mark total) 7. Solve the following equations to 2 decimal places (3 marks total) a. 37x + 1 = 35 b. 27 – a = 7.45 c. 3(y + 2) = 14 8. A 7-sided polygon is called a Heptagon. (3 marks total) a. What is the total of a Heptagon’s interior angles? b. If the Heptagon is regular (all angles the same), calculate the size of each interior angle to 2 decimal places. 9. Calculate the size of angle a and angle b. (2 mark total) 10. How many centilitres are there in 1.25 litres? (1 mark total) 11. The diagram below shows a stone carving with a hole on it; determine its volume (not including hole), if its thickness is 8 cm. Give your answer in cm3 to 2 decimal points. Assume π = 3.14 (6 marks total) 12. The diagram below shows a piece of alloy plate with a hole in it made from aluminium, copper and magnesium with a mass ratio of 35:3:2. Calculate the following to 2 decimal places. All measurements are in cm. (7 marks total) a. Using the formula A = 1/2(a+b)h calculate the height of the shape below. b. The volume of the solid part (not including the hole) of the shape below to 3 decimal places if it was 0.25cm thick. c. The mass of each material if the total mass of the plate is 62 kg. 10 cm Hole dia = 3 cm Cross sectional area of solid (not including hole) = 28.935 cm2 8 cm 13. A 66kg boy is running at 3 m/s. Calculate his Kinetic Energy using the formula KE = 1/2mv2 (2 marks total) 14. A rocket has a mass of 2,000 kg. What is its acceleration if the forces of its engines are 50kN? Show working out to receive full marks. (1 marks total) 250,000,000 m/s² 25 m/s² 25,000 m/s² 15. In the diagram below a force of 125N (F1) is applied to a lever 20cm (D1) away from the fulcrum, (4 marks total) Fulcrum (a) How far away in metres would a force of 5N (F2) need to be to balance the load? (b) How much force (F2) would need to be applied 0.7m away to balance the same load (F1)? 16. For the circuit shown in the diagram below, calculate: (3 mark total) a. The total circuit resistance. b. The value of the current I. c. Calculate the voltage of the battery cell if the current was 3amp and the resistors stayed the same. 17. In the diagram of a hydraulic system, the area of piston A is 8cm2 and the area of piston B is 48cm2. (2 mark total) If the Force IN is 16 N, calculate the force OUT. 18. Plot the graph 2y = x3 – 4 using a value range for x from 0 to 3 (3 marks total) 14 12 10 8 6 4 2 0 -2 Choosing appropriate scale (1 mark) Accurately plotting y values (1 mark) X 0 1 2 3 Y Accurately plotting line of best fit. (1 mark) SPARE PAPER

1. (2 marks total) a. Multiply 109 x 309 b. Divide 1988 by 16 exactly 2. (4 marks total) a. Write 2/11 as a decimal to 2 decimal places b. Calculate 35% of 62 c. Add 103/4 to 92/3 d. Subtract 79.04 from 115.225 giving your answer correct to 2 decimal places 3. Circle the fractions in the list which are equivalent to 0.80 (2 marks) 2/7 32/40 8/10 8/20 8/25 9/24 36/45 40/50 4. Write the numerical value of: 3-3 (2 marks total) 5. Simplify z + 67 = 3z + 33 (1 mark total) 6. Solve to 1 decimal place 3y – 34 = 2y + 89 (1 mark total) 7. Solve the following equations to 2 decimal places (3 marks total) a. 37x + 1 = 35 b. 27 – a = 7.45 c. 3(y + 2) = 14 8. A 7-sided polygon is called a Heptagon. (3 marks total) a. What is the total of a Heptagon’s interior angles? b. If the Heptagon is regular (all angles the same), calculate the size of each interior angle to 2 decimal places. 9. Calculate the size of angle a and angle b. (2 mark total) 10. How many centilitres are there in 1.25 litres? (1 mark total) 11. The diagram below shows a stone carving with a hole on it; determine its volume (not including hole), if its thickness is 8 cm. Give your answer in cm3 to 2 decimal points. Assume π = 3.14 (6 marks total) 12. The diagram below shows a piece of alloy plate with a hole in it made from aluminium, copper and magnesium with a mass ratio of 35:3:2. Calculate the following to 2 decimal places. All measurements are in cm. (7 marks total) a. Using the formula A = 1/2(a+b)h calculate the height of the shape below. b. The volume of the solid part (not including the hole) of the shape below to 3 decimal places if it was 0.25cm thick. c. The mass of each material if the total mass of the plate is 62 kg. 10 cm Hole dia = 3 cm Cross sectional area of solid (not including hole) = 28.935 cm2 8 cm 13. A 66kg boy is running at 3 m/s. Calculate his Kinetic Energy using the formula KE = 1/2mv2 (2 marks total) 14. A rocket has a mass of 2,000 kg. What is its acceleration if the forces of its engines are 50kN? Show working out to receive full marks. (1 marks total) 250,000,000 m/s² 25 m/s² 25,000 m/s² 15. In the diagram below a force of 125N (F1) is applied to a lever 20cm (D1) away from the fulcrum, (4 marks total) Fulcrum (a) How far away in metres would a force of 5N (F2) need to be to balance the load? (b) How much force (F2) would need to be applied 0.7m away to balance the same load (F1)? 16. For the circuit shown in the diagram below, calculate: (3 mark total) a. The total circuit resistance. b. The value of the current I. c. Calculate the voltage of the battery cell if the current was 3amp and the resistors stayed the same. 17. In the diagram of a hydraulic system, the area of piston A is 8cm2 and the area of piston B is 48cm2. (2 mark total) If the Force IN is 16 N, calculate the force OUT. 18. Plot the graph 2y = x3 – 4 using a value range for x from 0 to 3 (3 marks total) 14 12 10 8 6 4 2 0 -2 Choosing appropriate scale (1 mark) Accurately plotting y values (1 mark) X 0 1 2 3 Y Accurately plotting line of best fit. (1 mark) SPARE PAPER

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2. When Protagoras said “Man is the measure of all things,” why was this a different and new way of seeing the world? To what degree does contemporary US culture agree with Protagoras?

2. When Protagoras said “Man is the measure of all things,” why was this a different and new way of seeing the world? To what degree does contemporary US culture agree with Protagoras?

  2.    When Protagoras said “Man is the measure of … Read More...
Read: http://xnet.kp.org/permanentejournal/winter03/leader.html This article talks about physicians as leaders. It is written by a physician for physicians, so it provides insight into how doctors think of themselves in leadership. How can you use this understanding of doctors and leadership in managing your own healthcare facility? After all, the organizational chart shows the board of directors and CEO at the top, but physicians are just as important in leading any hospital or clinic. How will you integrate physicians as leaders in your own organization?

Read: http://xnet.kp.org/permanentejournal/winter03/leader.html This article talks about physicians as leaders. It is written by a physician for physicians, so it provides insight into how doctors think of themselves in leadership. How can you use this understanding of doctors and leadership in managing your own healthcare facility? After all, the organizational chart shows the board of directors and CEO at the top, but physicians are just as important in leading any hospital or clinic. How will you integrate physicians as leaders in your own organization?

The physicians always take a lead in creating patient-cantered care. … Read More...
10.2 California Imaging Center, a not-for-profit business, is evaluating the purchase of new diagnostic equipment. The equipment, which costs $600,000 has an expected life of five years and an estimated salvage value of $200,000 at that time. The equipment is expected to be used 15 times a day for 250 days a year for each year of the project’s life. On average, each procedure is expected to generate $80 in cash collections during the first year of use. Thus, net revenues for Year 1 are estimated at 15 X 250 X $80 =$300,000. Labor and maintenance costs are expected to be $100,000 during the first year of operation, while utilities will cost another $10,000 and cash overhead will increase by $5,000 in Year 1. The cost for expendable supplies is expected to average $5 per procedure during the first year. All costs and revenues are expected to increase at 5 percent inflation rate after the first year. The center’s corporate cost of capital is 10 percent. a. Estimate the project’s net cash flows over its five-year estimated life. (hint: use the following format as a guide.) Year 0 1 2 3 4 5 Equipment Cost Net revenues Less: labor/maintenance costs Utilities cost Supplies Incremental overhead Operating income Equipment salvage value Net cash flow b. What are the project’s NPV and IRR? (Assume for now that the project has average risk.) c. Assume the project is assessed to have high risk and California Imaging Center adds or subtracts 3 percentage points to adjust for project risk. Now, what is the project’s NPV? Does the risk assessment change how the project’s IRR is interpreted?

10.2 California Imaging Center, a not-for-profit business, is evaluating the purchase of new diagnostic equipment. The equipment, which costs $600,000 has an expected life of five years and an estimated salvage value of $200,000 at that time. The equipment is expected to be used 15 times a day for 250 days a year for each year of the project’s life. On average, each procedure is expected to generate $80 in cash collections during the first year of use. Thus, net revenues for Year 1 are estimated at 15 X 250 X $80 =$300,000. Labor and maintenance costs are expected to be $100,000 during the first year of operation, while utilities will cost another $10,000 and cash overhead will increase by $5,000 in Year 1. The cost for expendable supplies is expected to average $5 per procedure during the first year. All costs and revenues are expected to increase at 5 percent inflation rate after the first year. The center’s corporate cost of capital is 10 percent. a. Estimate the project’s net cash flows over its five-year estimated life. (hint: use the following format as a guide.) Year 0 1 2 3 4 5 Equipment Cost Net revenues Less: labor/maintenance costs Utilities cost Supplies Incremental overhead Operating income Equipment salvage value Net cash flow b. What are the project’s NPV and IRR? (Assume for now that the project has average risk.) c. Assume the project is assessed to have high risk and California Imaging Center adds or subtracts 3 percentage points to adjust for project risk. Now, what is the project’s NPV? Does the risk assessment change how the project’s IRR is interpreted?

info@checkyourstudy.cominfo@checkyourstudy.com 10.2 California Imaging Center, a not-for-profit business, is evaluating … Read More...
Problems Marking scheme 1. Let A be a nonzero square matrix. Is it possible that a positive integer k exists such that ?? = 0 ? For example, find ?3 for the matrix [ 0 1 2 0 0 1 0 0 0 ] A square matrix A is nilpotent of index k when ? ≠ 0 , ?2 ≠ 0 , … . . , ??−1 ≠ 0, ??? ?? = 0. In this task you will explore nilpotent matrices. 1. The matrix in the example given above is nilpotent. What is its index? ( 2 marks ) 2. Use a software program to determine which of the following matrices are nilpotent and find their indices ( 12 marks ) A. [ 0 1 0 0 ] B. [ 0 1 1 0 ] C. [ 0 0 1 0 ] D. [ 1 0 1 0 ] E. [ 0 0 1 0 0 0 0 0 0 ] F. [ 0 0 0 1 0 0 1 1 0 ] 3. Find 3×3 nilpotent matrices of indices 2 and 3 ( 2 marks ) 4. Find 4×4 nilpotent matrices of indices 2, 3, and 4 ( 2 marks ) 5. Find nilpotent matrix of index 5 ( 2 marks ) 6. Are nilpotent matrices invertible? prove your answer ( 3 marks ) 7. When A is nilpotent, what can you say about ?? ? prove your answer ( 3 marks ) 8. Show that if ? is nilpotent , then ? − ? is invertible ( 4 marks ) 30% 2. A radio transmitter circuit contains a resisitance of 2.0 Ω, a variable inductor of 100 − ? ℎ?????? and a voltage source of 4.0 ? . find the current ? in the circuit as a function of the time t for 0 ≤ ? ≤ 100? if the intial curent is zero. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 3. An object falling under the influence of gravity has a variable accelertaion given by 32 − ? , where ? represents the velocity. If the object starts from rest, find an expression for the velocity in terms of the time. Also, find the limiting value of the velocity. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 4. When the angular displacement ? of a pendulum is small ( less than 60), the pendulum moves with simple harmonic motion closely approximated by ?′′ + ? ? ? = 0 . Here , ?′ = ?? ?? and ? is the accelertaion due to gravity , and ? is the length of the pendulum. Find ? as a function of time ( in s ) if ? = 9.8 ?/?2, ? = 1.0 ? ? = 0.1 and ?? ?? = 0 when ? = 0 . sketch the cuve using any graphical tool. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 5. Find the equation relating the charge and the time in a electric circuit with the following elements: ? = 0.200 ? , ? = 8.00 Ω , ? = 1.00 ?? , ? = 0. In this circuit , ? = 0 and ? = 0.500 ? when ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 6. A spring is stretched 1 m by ? 20 − ? Weight. The spring is stretched 0.5 m below the equilibrium position with the weight attached and the then released. If it is a medium that resists the motion with a force equal to 12?, where v is the velocity, sketch and find the displacement y of the weight as a function of the time. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 7. A 20?? inductor, a 40.0 Ω resistor, a 50.0 ?? capacitor, and voltage source of 100 ?−100?are connected in series in an electric circuit. Find the charge on the capacitor as a function of time t , if ? = 0 and ? = 0 ?ℎ?? ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 10% quality and neatness and using Math equations in MS word. –

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

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10. Thinking back upon this course’s content, please discuss the authors’ assertion: “The logistics manager of the future will be much more of a change leader and much less of a technician.”

10. Thinking back upon this course’s content, please discuss the authors’ assertion: “The logistics manager of the future will be much more of a change leader and much less of a technician.”

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EXPERIMENT 6 FET CHARACTERISTIC CURVES ________________________________________ Bring a diskette to save your data. ________________________________________ OBJECT: The objective of this lab is to investigate the DC characteristics and operation of a field effect transistor (FET). The FET recommended to be used in this lab is 2N5486 n-channel FET. • Gathering data for the DC characteristics ________________________________________ APPARATUS: Dual DC Power Supply, Voltmeter, and 1k resistors, 2N5486 N-Channel FET. ________________________________________ THEORY: A JFET (Junction Field Effect Transistor) is a three terminal device (drain, source, and gate) similar to the BJT. The difference between them is that the JFET is a voltage controlled constant current device, whereas BJT is a current controlled current source device. Whereas for BJT the relationship between an output parameter, iC, and an input parameter, iB, is given by a constant , the relationship in JFET between an output parameter, iD, and an input parameter, vGS, is more complex. PROCEDURE: Measuring ID versus VDS (Output Characteristics) 1. Build the circuit shown below. 2. Obtain the output characteristics i.e. ID versus VDS. a. Set VGS = 0. Vary the voltage across drain (VDS) from 0 to 8 V with steps of 1 V and measure the corresponding drain current (ID). b. Repeat the procedure for different values of VGS. (0V, -0.5V, -1V, -1.5V, -2V, -2.5V, -3.0V, -3.5V, -4.0V). 3. Record the values in Table 1 and plot the graph ID vs. VGS. VGS 0 -0.5 -1.0 -1.5` -2.0 -2.5 -3.0 -3.5 -4.0 VDS ID ID ID ID ID ID ID ID ID 0 0 0.002mA 0.002mA 0.002mA 0.002mA 0.002mA 0.002mA 0.002mA 0mA 1 0 0.7 mA 0.7 mA 0.66 mA 0.6 mA 0.6 mA 0.5 0.1mA 0mA 2 0 1.5 mA 1.3 mA 1.3mA 1.2 mA 1.1 mA 0.7 0.1mA 0mA 3 0 2.1 mA 2.6 mA 1.9 mA 1.8 mA 1.5 mA 0.8 mA 0.1mA 0mA 4 0 2.7 mA 2.6 mA 2.5 mA 2.4 mA 1.7 mA 0.8 mA 0.1mA 0mA 5 0 3.4 mA 3.3 mA 3.1 mA 2.8 mA 1.8 mA 0.9 mA 0.1mA 0mA 6 0 4.1 mA 3.4 mA 3.7 mA 3.2 mA 1.9 mA 0.9 mA 0.1mA 0mA 7 0 4.7 mA 4.5 mA 4.2 mA 3.4 mA 1.9 mA 0.9 mA 0.1mA 0mA 8 0 5.3 mA 5.1 mA 6.6 mA 3.5 mA 2.0 mA 0.9 mA 0.1mA 0mA Table 1. vds=0:8; id=[0 6.2e-3 9.7e-3 11.3e-3 11.9e-3 12.2e-3 12.3e-3 12.3e-3 12.32e-3]; plot(vds,id);grid on;hold on id2=[0 5.23e-3 8.05e-3 9.15e-3 9.57e-3 9.77e-3 9.88e-3 9.9e-3 9.92e-3]; plot(vds,id2);grid on;hold on id3=[0 4.29e-3 6.41e-3 7.17e-3 7.46e-3 7.60e-3 7.67e-3 7.73e-3 7.76e-3]; plot(vds,id3);grid on;hold on ________________________________________ Measuring ID versus VGS (Transconductance Characteristics) 1. For the same circuit, obtain the transconductance characteristics. i.e. ID versus VGS. a. Set a particular value of voltage for VDS, i.e. 5V. Start with a gate voltage VGS of 0 V, and measure the corresponding drain current (ID). b. Then decrease VGS in steps of 0.5 V until VGS is -4V. c. At each step record the drain current. VDS = 5 V VGS ID 0 3.42 mA -0.5 3.36 mA -1.00 3.27 mA -1.50 3.12 mA -2.00 2.79 mA -2.50 1.84 mA -3.00 0.71 mA -3.50 0.11 mA -4.00 0 mA Table 2. 2. Plot the graph with ID versus VGS using Excel, MATLAB, or some other program. Discussion Questions—Make sure you answer the following questions in your discussion. Use all of the data obtained to answer the following questions: 1. Discuss the output and transconductance curves obtained in lab? Are they what you expected? 2. Are the output characteristics spaced evenly? Should they be? 3. What are the applications of a JFET?

EXPERIMENT 6 FET CHARACTERISTIC CURVES ________________________________________ Bring a diskette to save your data. ________________________________________ OBJECT: The objective of this lab is to investigate the DC characteristics and operation of a field effect transistor (FET). The FET recommended to be used in this lab is 2N5486 n-channel FET. • Gathering data for the DC characteristics ________________________________________ APPARATUS: Dual DC Power Supply, Voltmeter, and 1k resistors, 2N5486 N-Channel FET. ________________________________________ THEORY: A JFET (Junction Field Effect Transistor) is a three terminal device (drain, source, and gate) similar to the BJT. The difference between them is that the JFET is a voltage controlled constant current device, whereas BJT is a current controlled current source device. Whereas for BJT the relationship between an output parameter, iC, and an input parameter, iB, is given by a constant , the relationship in JFET between an output parameter, iD, and an input parameter, vGS, is more complex. PROCEDURE: Measuring ID versus VDS (Output Characteristics) 1. Build the circuit shown below. 2. Obtain the output characteristics i.e. ID versus VDS. a. Set VGS = 0. Vary the voltage across drain (VDS) from 0 to 8 V with steps of 1 V and measure the corresponding drain current (ID). b. Repeat the procedure for different values of VGS. (0V, -0.5V, -1V, -1.5V, -2V, -2.5V, -3.0V, -3.5V, -4.0V). 3. Record the values in Table 1 and plot the graph ID vs. VGS. VGS 0 -0.5 -1.0 -1.5` -2.0 -2.5 -3.0 -3.5 -4.0 VDS ID ID ID ID ID ID ID ID ID 0 0 0.002mA 0.002mA 0.002mA 0.002mA 0.002mA 0.002mA 0.002mA 0mA 1 0 0.7 mA 0.7 mA 0.66 mA 0.6 mA 0.6 mA 0.5 0.1mA 0mA 2 0 1.5 mA 1.3 mA 1.3mA 1.2 mA 1.1 mA 0.7 0.1mA 0mA 3 0 2.1 mA 2.6 mA 1.9 mA 1.8 mA 1.5 mA 0.8 mA 0.1mA 0mA 4 0 2.7 mA 2.6 mA 2.5 mA 2.4 mA 1.7 mA 0.8 mA 0.1mA 0mA 5 0 3.4 mA 3.3 mA 3.1 mA 2.8 mA 1.8 mA 0.9 mA 0.1mA 0mA 6 0 4.1 mA 3.4 mA 3.7 mA 3.2 mA 1.9 mA 0.9 mA 0.1mA 0mA 7 0 4.7 mA 4.5 mA 4.2 mA 3.4 mA 1.9 mA 0.9 mA 0.1mA 0mA 8 0 5.3 mA 5.1 mA 6.6 mA 3.5 mA 2.0 mA 0.9 mA 0.1mA 0mA Table 1. vds=0:8; id=[0 6.2e-3 9.7e-3 11.3e-3 11.9e-3 12.2e-3 12.3e-3 12.3e-3 12.32e-3]; plot(vds,id);grid on;hold on id2=[0 5.23e-3 8.05e-3 9.15e-3 9.57e-3 9.77e-3 9.88e-3 9.9e-3 9.92e-3]; plot(vds,id2);grid on;hold on id3=[0 4.29e-3 6.41e-3 7.17e-3 7.46e-3 7.60e-3 7.67e-3 7.73e-3 7.76e-3]; plot(vds,id3);grid on;hold on ________________________________________ Measuring ID versus VGS (Transconductance Characteristics) 1. For the same circuit, obtain the transconductance characteristics. i.e. ID versus VGS. a. Set a particular value of voltage for VDS, i.e. 5V. Start with a gate voltage VGS of 0 V, and measure the corresponding drain current (ID). b. Then decrease VGS in steps of 0.5 V until VGS is -4V. c. At each step record the drain current. VDS = 5 V VGS ID 0 3.42 mA -0.5 3.36 mA -1.00 3.27 mA -1.50 3.12 mA -2.00 2.79 mA -2.50 1.84 mA -3.00 0.71 mA -3.50 0.11 mA -4.00 0 mA Table 2. 2. Plot the graph with ID versus VGS using Excel, MATLAB, or some other program. Discussion Questions—Make sure you answer the following questions in your discussion. Use all of the data obtained to answer the following questions: 1. Discuss the output and transconductance curves obtained in lab? Are they what you expected? 2. Are the output characteristics spaced evenly? Should they be? 3. What are the applications of a JFET?

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