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%

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You work for a healthcare insurance company as an industrial engineer. As part of the company’s customer service department, there is a call center that responds to customer questions by phone. Your manager has asked you to perform an analysis of some data that the call center has collected over the last two years in an effort to determine how many workers are needed. In particular your manager wants to find out if there is a relationship between how long it takes to answer a call (independent variable) and whether or not customers will hang up (dependent variable). Is there a linear relationship between these two variables? Include a graph and analysis to support your opinion. If the goal of the call center is to have fewer than 15% of calls abandoned, how quickly must the call center respond? Learning Outcome #2. Data for Question 3 Average Answer Speed = Average number of seconds to answer an incoming call during the week % abandoned = % of calls that are abandoned (hang up) before being answered during the week.

You work for a healthcare insurance company as an industrial engineer. As part of the company’s customer service department, there is a call center that responds to customer questions by phone. Your manager has asked you to perform an analysis of some data that the call center has collected over the last two years in an effort to determine how many workers are needed. In particular your manager wants to find out if there is a relationship between how long it takes to answer a call (independent variable) and whether or not customers will hang up (dependent variable). Is there a linear relationship between these two variables? Include a graph and analysis to support your opinion. If the goal of the call center is to have fewer than 15% of calls abandoned, how quickly must the call center respond? Learning Outcome #2. Data for Question 3 Average Answer Speed = Average number of seconds to answer an incoming call during the week % abandoned = % of calls that are abandoned (hang up) before being answered during the week.

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For Day 12 and 13 Homework Cover Sheet Name:_________________________________________________ 1. Read Pages from 184-226, or watch the videos listed below  Properties of Subtraction http://www.youtube.com/watch?v=W9PEgpFyAYg (15 min)  Subtraction Algorithm http://www.youtube.com/watch?v=azaR-4ySSwQ (9 min) Visualizing Subtraction http://www.youtube.com/watch?v=PwQGc_1p0jQ (8 min)  Subtraction http://www.youtube.com/watch?v=E7Cj8QnEmNo (12 min)  Subtraction of Rational Expressions http://www.youtube.com/watch?v=Vuvmrq54b4w (8 min)  Prime factors and multiples of expressions http://www.youtube.com/watch?v=wy7pm8wjm_8 (8 min)  Multiples http://www.youtube.com/watch?v=f3ZdozzChjQ (9 min)  Least Common Multiples http://www.youtube.com/watch?v=wJCWNcytyXE (15 min)  Adding Rational Expressions Using LCM http://www.youtube.com/watch?v=O0V6hbTE-2s (12 min) 2. Attempt problems from workbook pages 51-66 Summary of the lectures you watched. List any parts of the video lecture (if there are any) that were unclear or you had trouble understanding. Please be specific and do not just say “All of it”. Questions you had difficulty with or felt stuck on- ALEKS Topics Mastered Word problem with powers of ten Combining like terms: Advanced Combining like terms: Integer coefficients Distributive property: Integer coefficients Elapsed time Estimating a decimal sum or difference Integer subtraction: Problem type 1 Integer subtraction: Problem type 2 Integer subtraction: Problem type 3 Multiplication involving binomials and trinomials in two variables Multiplying a univariate polynomial by a monomial with a negative coefficient Multiplying binomials with negative coefficients Signed decimal addition and subtraction Signed decimal addition and subtraction with 3 numbers Signed fraction addition or subtraction: Basic Simplifying a sum or difference of multivariate polynomials Simplifying a sum or difference of three univariate polynomials Simplifying a sum or difference of two univariate polynomials Subtracting a 1-digit number from a 2-digit number Subtraction and regrouping with zeros Subtraction with borrowing Subtraction with multiple regrouping steps Subtraction without borrowing Word problem with addition or subtraction of whole numbers Adding or subtracting complex numbers

For Day 12 and 13 Homework Cover Sheet Name:_________________________________________________ 1. Read Pages from 184-226, or watch the videos listed below  Properties of Subtraction http://www.youtube.com/watch?v=W9PEgpFyAYg (15 min)  Subtraction Algorithm http://www.youtube.com/watch?v=azaR-4ySSwQ (9 min) Visualizing Subtraction http://www.youtube.com/watch?v=PwQGc_1p0jQ (8 min)  Subtraction http://www.youtube.com/watch?v=E7Cj8QnEmNo (12 min)  Subtraction of Rational Expressions http://www.youtube.com/watch?v=Vuvmrq54b4w (8 min)  Prime factors and multiples of expressions http://www.youtube.com/watch?v=wy7pm8wjm_8 (8 min)  Multiples http://www.youtube.com/watch?v=f3ZdozzChjQ (9 min)  Least Common Multiples http://www.youtube.com/watch?v=wJCWNcytyXE (15 min)  Adding Rational Expressions Using LCM http://www.youtube.com/watch?v=O0V6hbTE-2s (12 min) 2. Attempt problems from workbook pages 51-66 Summary of the lectures you watched. List any parts of the video lecture (if there are any) that were unclear or you had trouble understanding. Please be specific and do not just say “All of it”. Questions you had difficulty with or felt stuck on- ALEKS Topics Mastered Word problem with powers of ten Combining like terms: Advanced Combining like terms: Integer coefficients Distributive property: Integer coefficients Elapsed time Estimating a decimal sum or difference Integer subtraction: Problem type 1 Integer subtraction: Problem type 2 Integer subtraction: Problem type 3 Multiplication involving binomials and trinomials in two variables Multiplying a univariate polynomial by a monomial with a negative coefficient Multiplying binomials with negative coefficients Signed decimal addition and subtraction Signed decimal addition and subtraction with 3 numbers Signed fraction addition or subtraction: Basic Simplifying a sum or difference of multivariate polynomials Simplifying a sum or difference of three univariate polynomials Simplifying a sum or difference of two univariate polynomials Subtracting a 1-digit number from a 2-digit number Subtraction and regrouping with zeros Subtraction with borrowing Subtraction with multiple regrouping steps Subtraction without borrowing Word problem with addition or subtraction of whole numbers Adding or subtracting complex numbers

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– 1 – Fall 2015 EECS 338 Assignment 2 Due: Oct. 1st, 2015 G. Ozsoyoglu Concurrent Programming with Semaphores; 140 points (100 pts) 1. Priority-based Searchers/Inserters/Deleters Problem without starvation. Three types of processes, namely, searchers, inserters, and deleters share access to a singly linked list L, and perform search, insert, or delete operations, respectively. The list L does not have duplicate values. a) Searchers merely search the list L, and report success (i.e., item searched is in L) or no-success (i.e., item searched is not in L) to a log file. Hence they can execute concurrently with each other. b) Inserters add new items to the end of the list L, and report success (i.e., item is not in L, and successfully inserted into L) or no-success (i.e., item is already in L, and no insertion takes place) to a log file. Insertions must be mutually exclusive to preclude two inserters from inserting new items at about the same time. However, one insert can proceed in parallel with any number of searches. c) Deleters remove items from anywhere in the list, and report success (i.e., the item is found in L and deleted) or no-success (i.e., item is not in L, and could not be deleted) to a log file. At most one deleter can access the list L at a time, and the deletion must be mutually exclusive with searches and insertions. d) Initial start. Searcher, inserter, and deleter processes are initially launched as follows. A user process that needs a search/insertion/deletion operation to the list L first forks a process, and then, in the forked process, performs an execv into a searcher/ inserter/deleter process. e) Log maintenance. Upon start, each searcher/inserter/deleter writes to a log file, recording the time of insertion, process id, process type (i.e., searcher, inserter, or deleter), and the item that is being searched/inserted/deleted. f) Termination. Upon successful or unsuccessful completion, each searcher/inserter/deleter writes to the same log file, recording the time and the result of its execution. g) Priority-based service between three types. Searchers, inserters, and deleters perform their search, insert, delete operations, respectively, on a priority basis (not on a first-come-first-serve (FCFS) basis) between separate process types (i.e., searchers, inserters, deleters) as follows. Searchers search with the highest priority; inserters insert with the second highest priority (except that one inserter can proceed in parallel with any number of searchers), and deleters delete with the lowest priority. h) FCFS service within a single type. Processes of the same type are serviced FCFS. As an example, among multiple inserters, the order of insertions into L is FCFS. Similarly, among multiple deleters, the order of deletions into L is FCFS. Note that, among searchers, while the start of search among searchers is FCFS, due to concurrent searcher execution, the completions of multiple searchers may not be FCFS. i) Starvation avoidance. In addition to the above priority-based search/insert/delete operations, the following starvation-avoidance rule is enforced. o After 10 consecutive searchers search the list L, if there is at least one waiting inserter or deleter then newly arriving searchers are blocked until (a) all waiting inserters are first serviced FCFS, and, then (b) all waiting deleters are serviced FCFS. Then, both the standard priority-based service between process types and the FCFS service within a process type resume. You are to specify a semaphore-based algorithm to synchronize searcher, inserter and deleter processes. Note:  Explain your algorithm.  Make sure to state any assumptions you make in your solution.  Specify the initial states of your variables and semaphores.  Specify whether your semaphores are binary or nonbinary.  Do not bother specifying algorithms for sequential tasks: simply specify a well-defined function/procedure (i.e., one with well-defined input/output/functional specification). – 2 – (40 pts) 2. Four-of-a-Kind Problem is defined as follows.  There is a deck of 24 cards, split into 6 different kinds, 4 cards of each kind.  There are 4 players (i.e., processes) ??,0≤?≤3; each player can hold 4 cards.  Between each pair of adjacent (i.e., seated next to each other) players, there is a pile of cards.  The game begins by o someone dealing four cards to each player, and putting two cards on the pile between each pair of adjacent players, and o ?0 starting the game. If ?0 has four-of-a-kind, ?0 wins. Whoever gets four-of-a-kind first wins.  Players take turns to play clockwise. That is, ?0 plays, ?1 plays, ?2 plays, ?3 plays, ?0 plays, etc.  Each player behaves as follows. o So long as no one has won, keep playing. o If it is my turn and no one has won:  Check for Four-of-a-Kind. If yes, claim victory. Otherwise discard a card into the pile on the right; pick up a card from the pile on the left; and, check again: If Four-of-a-Kind, claim victory; otherwise revise turn so that the next player plays and wait for your turn.  There are no ties; when a player has claimed victory, all other players stop (when their turns to play come up). You are to specify a semaphore-based algorithm to the Four-of-a-Kind problem. Note:  Explain your algorithm.  Make sure to state any assumptions you make in your solution.  Specify the initial states of your variables and semaphores.  Specify whether your semaphores are binary or nonbinary.  Do not bother specifying algorithms for sequential tasks: simply specify a well-defined function/procedure (i.e., one with well-defined input/output/functional specification). P1 P0 P2 P3 pile 1 pile 2 pile 3 pile 0

– 1 – Fall 2015 EECS 338 Assignment 2 Due: Oct. 1st, 2015 G. Ozsoyoglu Concurrent Programming with Semaphores; 140 points (100 pts) 1. Priority-based Searchers/Inserters/Deleters Problem without starvation. Three types of processes, namely, searchers, inserters, and deleters share access to a singly linked list L, and perform search, insert, or delete operations, respectively. The list L does not have duplicate values. a) Searchers merely search the list L, and report success (i.e., item searched is in L) or no-success (i.e., item searched is not in L) to a log file. Hence they can execute concurrently with each other. b) Inserters add new items to the end of the list L, and report success (i.e., item is not in L, and successfully inserted into L) or no-success (i.e., item is already in L, and no insertion takes place) to a log file. Insertions must be mutually exclusive to preclude two inserters from inserting new items at about the same time. However, one insert can proceed in parallel with any number of searches. c) Deleters remove items from anywhere in the list, and report success (i.e., the item is found in L and deleted) or no-success (i.e., item is not in L, and could not be deleted) to a log file. At most one deleter can access the list L at a time, and the deletion must be mutually exclusive with searches and insertions. d) Initial start. Searcher, inserter, and deleter processes are initially launched as follows. A user process that needs a search/insertion/deletion operation to the list L first forks a process, and then, in the forked process, performs an execv into a searcher/ inserter/deleter process. e) Log maintenance. Upon start, each searcher/inserter/deleter writes to a log file, recording the time of insertion, process id, process type (i.e., searcher, inserter, or deleter), and the item that is being searched/inserted/deleted. f) Termination. Upon successful or unsuccessful completion, each searcher/inserter/deleter writes to the same log file, recording the time and the result of its execution. g) Priority-based service between three types. Searchers, inserters, and deleters perform their search, insert, delete operations, respectively, on a priority basis (not on a first-come-first-serve (FCFS) basis) between separate process types (i.e., searchers, inserters, deleters) as follows. Searchers search with the highest priority; inserters insert with the second highest priority (except that one inserter can proceed in parallel with any number of searchers), and deleters delete with the lowest priority. h) FCFS service within a single type. Processes of the same type are serviced FCFS. As an example, among multiple inserters, the order of insertions into L is FCFS. Similarly, among multiple deleters, the order of deletions into L is FCFS. Note that, among searchers, while the start of search among searchers is FCFS, due to concurrent searcher execution, the completions of multiple searchers may not be FCFS. i) Starvation avoidance. In addition to the above priority-based search/insert/delete operations, the following starvation-avoidance rule is enforced. o After 10 consecutive searchers search the list L, if there is at least one waiting inserter or deleter then newly arriving searchers are blocked until (a) all waiting inserters are first serviced FCFS, and, then (b) all waiting deleters are serviced FCFS. Then, both the standard priority-based service between process types and the FCFS service within a process type resume. You are to specify a semaphore-based algorithm to synchronize searcher, inserter and deleter processes. Note:  Explain your algorithm.  Make sure to state any assumptions you make in your solution.  Specify the initial states of your variables and semaphores.  Specify whether your semaphores are binary or nonbinary.  Do not bother specifying algorithms for sequential tasks: simply specify a well-defined function/procedure (i.e., one with well-defined input/output/functional specification). – 2 – (40 pts) 2. Four-of-a-Kind Problem is defined as follows.  There is a deck of 24 cards, split into 6 different kinds, 4 cards of each kind.  There are 4 players (i.e., processes) ??,0≤?≤3; each player can hold 4 cards.  Between each pair of adjacent (i.e., seated next to each other) players, there is a pile of cards.  The game begins by o someone dealing four cards to each player, and putting two cards on the pile between each pair of adjacent players, and o ?0 starting the game. If ?0 has four-of-a-kind, ?0 wins. Whoever gets four-of-a-kind first wins.  Players take turns to play clockwise. That is, ?0 plays, ?1 plays, ?2 plays, ?3 plays, ?0 plays, etc.  Each player behaves as follows. o So long as no one has won, keep playing. o If it is my turn and no one has won:  Check for Four-of-a-Kind. If yes, claim victory. Otherwise discard a card into the pile on the right; pick up a card from the pile on the left; and, check again: If Four-of-a-Kind, claim victory; otherwise revise turn so that the next player plays and wait for your turn.  There are no ties; when a player has claimed victory, all other players stop (when their turns to play come up). You are to specify a semaphore-based algorithm to the Four-of-a-Kind problem. Note:  Explain your algorithm.  Make sure to state any assumptions you make in your solution.  Specify the initial states of your variables and semaphores.  Specify whether your semaphores are binary or nonbinary.  Do not bother specifying algorithms for sequential tasks: simply specify a well-defined function/procedure (i.e., one with well-defined input/output/functional specification). P1 P0 P2 P3 pile 1 pile 2 pile 3 pile 0

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The Rocket Equation The Tsiolovsky Rocket Equation describes the velocity that results from pushing matter (exploding rocket fuel) in the opposite direction to the direction you want to travel. This assignment requires you to do basic calculation using the Tsiolovsky Rocket Equation : v[t] = eV Log M M – bR t  – g t The parameters used are : ◼ eV exhaust velocity (m/s) ◼ pL payload (kg) ◼ fL fuel load (kg) ◼ M is the mass of the rocket (pL+fL, kg) ◼ bR the burn rate of fuel (kg/s) ◼ g the force due to gravity ms2 The variables calculated are : h(t) the height of the rocket at time t (m) v(t) the velocity of the rocket at time t (m/s) m(t) the mass of the rocket at time t (kg) Questions Question 1 (1 mark) Write an expression corresponding to the Tsiolovsky rocket equation and use integrate to find a function to describe the height of the rocket during fuel burn. Question 2 (2 marks) The fuel burns at a constant rate. Find the time (t0), velocity (vmax), and height (h0) of the rocket when the fuel runs out (calculate the time when the fuel runs out, and substitute this into the height Printed by Wolfram Mathematica Student Edition and velocity equations). Question 3 (2 marks) The second phase is when the only accelaration acting on the rocket is from gravity. This phase starts from the height and velocity of the previous question, and the velocity is given by the projectile motion equation, v(t) = vmax – g (t – t0). Use Solve to find the time when this equation equals 0. This will be the highest point the rocket reaches before returning to earth. Question 4 (1 marks) Integerate the projectile motion equation and add h0 to find the maximum height the rocket reaches. Question 5 (1 marks) Use Solve over the projectile motion equation to find the time when the height is 0. 2 assignment4.nb Printed by Wolfram Mathematica Student Edition

The Rocket Equation The Tsiolovsky Rocket Equation describes the velocity that results from pushing matter (exploding rocket fuel) in the opposite direction to the direction you want to travel. This assignment requires you to do basic calculation using the Tsiolovsky Rocket Equation : v[t] = eV Log M M – bR t  – g t The parameters used are : ◼ eV exhaust velocity (m/s) ◼ pL payload (kg) ◼ fL fuel load (kg) ◼ M is the mass of the rocket (pL+fL, kg) ◼ bR the burn rate of fuel (kg/s) ◼ g the force due to gravity ms2 The variables calculated are : h(t) the height of the rocket at time t (m) v(t) the velocity of the rocket at time t (m/s) m(t) the mass of the rocket at time t (kg) Questions Question 1 (1 mark) Write an expression corresponding to the Tsiolovsky rocket equation and use integrate to find a function to describe the height of the rocket during fuel burn. Question 2 (2 marks) The fuel burns at a constant rate. Find the time (t0), velocity (vmax), and height (h0) of the rocket when the fuel runs out (calculate the time when the fuel runs out, and substitute this into the height Printed by Wolfram Mathematica Student Edition and velocity equations). Question 3 (2 marks) The second phase is when the only accelaration acting on the rocket is from gravity. This phase starts from the height and velocity of the previous question, and the velocity is given by the projectile motion equation, v(t) = vmax – g (t – t0). Use Solve to find the time when this equation equals 0. This will be the highest point the rocket reaches before returning to earth. Question 4 (1 marks) Integerate the projectile motion equation and add h0 to find the maximum height the rocket reaches. Question 5 (1 marks) Use Solve over the projectile motion equation to find the time when the height is 0. 2 assignment4.nb Printed by Wolfram Mathematica Student Edition

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Extra Credit Due: 11:59pm on Thursday, May 15, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Man Running to Catch a Bus A man is running at speed (much less than the speed of light) to catch a bus already at a stop. At , when he is a distance from the door to the bus, the bus starts moving with the positive acceleration . Use a coordinate system with at the door of the stopped bus. Part A What is , the position of the man as a function of time? Answer symbolically in terms of the variables , , and . Hint 1. Which equation should you use for the man’s speed? Because the man’s speed is constant, you may use . ANSWER: c t = 0 b a x = 0 xman(t) b c t x(t) = x(0) + vt xman(t) = −b + ct

Extra Credit Due: 11:59pm on Thursday, May 15, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Man Running to Catch a Bus A man is running at speed (much less than the speed of light) to catch a bus already at a stop. At , when he is a distance from the door to the bus, the bus starts moving with the positive acceleration . Use a coordinate system with at the door of the stopped bus. Part A What is , the position of the man as a function of time? Answer symbolically in terms of the variables , , and . Hint 1. Which equation should you use for the man’s speed? Because the man’s speed is constant, you may use . ANSWER: c t = 0 b a x = 0 xman(t) b c t x(t) = x(0) + vt xman(t) = −b + ct

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Q1 A machining center in a local manufacturing company has five jobs to complete. The jobs are labeled 1, 2, 3, 4, and 5 based on the order they entered the shop. Your manager has asked you to compare two different job sequences to determine the best order for processing the jobs – First-In, First-Out (FIFO) and Shortest Processing Time (SPT). Compare the two sequences. Which sequence would you recommend and why? Be sure to include a variety of measures (average completion time, lateness, etc.) to compare the two sequencing rules. Include all calculations in your response q2 You work for a healthcare insurance company as an industrial engineer. As part of the company’s customer service department, there is a call center that responds to customer questions by phone. Your manager has asked you to perform an analysis of some data that the call center has collected over the last two years in an effort to determine how many workers are needed. In particular your manager wants to find out if there is a relationship between how long it takes to answer a call (independent variable) and whether or not customers will hang up (dependent variable). Is there a linear relationship between these two variables? Include a graph and analysis to support your opinion. If the goal of the call center is to have fewer than 15% of calls abandoned, how quickly must the call center respond? Learning Outcome #2. Data for Question 3 Average Answer Speed = Average number of seconds to answer an incoming call during the week % abandoned = % of calls that are abandoned (hang up) before being answered during the week. let me know if you can do excel work so I can send you the rest of the information and date

Q1 A machining center in a local manufacturing company has five jobs to complete. The jobs are labeled 1, 2, 3, 4, and 5 based on the order they entered the shop. Your manager has asked you to compare two different job sequences to determine the best order for processing the jobs – First-In, First-Out (FIFO) and Shortest Processing Time (SPT). Compare the two sequences. Which sequence would you recommend and why? Be sure to include a variety of measures (average completion time, lateness, etc.) to compare the two sequencing rules. Include all calculations in your response q2 You work for a healthcare insurance company as an industrial engineer. As part of the company’s customer service department, there is a call center that responds to customer questions by phone. Your manager has asked you to perform an analysis of some data that the call center has collected over the last two years in an effort to determine how many workers are needed. In particular your manager wants to find out if there is a relationship between how long it takes to answer a call (independent variable) and whether or not customers will hang up (dependent variable). Is there a linear relationship between these two variables? Include a graph and analysis to support your opinion. If the goal of the call center is to have fewer than 15% of calls abandoned, how quickly must the call center respond? Learning Outcome #2. Data for Question 3 Average Answer Speed = Average number of seconds to answer an incoming call during the week % abandoned = % of calls that are abandoned (hang up) before being answered during the week. let me know if you can do excel work so I can send you the rest of the information and date

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Q7 using the formula below calculate the using the Hata okumura propagation model using the following variables there is an extra page after this page so you can show your work if don’t show your work you will losr your points

Q7 using the formula below calculate the using the Hata okumura propagation model using the following variables there is an extra page after this page so you can show your work if don’t show your work you will losr your points

Q7 using the formula below calculate the using the Hata … Read More...
EE214 Fall 2015 Problem Set1 I am submitting my own work in this exercise, and I am aware of the penalties for cheating that will be assessed if I submit work for credit that is not my own. Print Name Sign Name Date Contains material © Digilent, Inc. 7 pages 1. (15 points) Below are some circuit elements from a simple digital system. 3.3V 20mA VB 1Kohm VA 1.3V RB 1K RC RD SW1 SW2 RA VC When the pushbutton SW1 is not pressed, what is the voltage at VA? (1pt) When the SW1 is pressed, what is the voltage at VA? (1pt) When the SW1 is pressed, what current flows in the 1K resistor RA? (1pt) When SW1 is pressed, what power is dissipated in RA? (2pt) In the LED circuit, 1.3V is required at VB to forward-bias the LED and cause current to flow. Given there is a 1.3V drop across the LED, what resistance RB is required for 20mA to flow through the LED? (2pt) What power is dissipated in the LED? (1pt) In the circuit on the far right, if RC dissipates 25mW, what is VC? (2pt) Using the VC voltage you calculated, if RC is changed to 100Ohms, how much power would it dissipate? (2pt) Using the VC voltage you calculated and a 1K RC, if pressing SW2 causes the total circuit power to increase to 75mW, what value must RD be? (3pt) EE214 Problem Set 1 2. (20 points) Complete the truth tables below. Provide SOP equations for the bottom three tables. F <= Σ ( ) F <= Σ ( ) F <= Σ ( ) 3. (12 points) Write the number of transistors required for each logic gate below inside the gate symbol, and then write the logic gate name below the symbol. 4. (12 points) Complete truth tables for the circuits shown below A B F AND A B F OR A B F XOR A F INV A B C F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 ? = ? ̅ ∙ ? + ? A B C F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 ? = ? ∙ ? ∙? ̅ + ? ∙ ? A B C F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 ? = ? ∙? ̅+? ̅ ∙ ? A B C F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A B C F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A F B C A B C Y EE214 Problem Set 1 5. (18 points) Show the total transistor count and gate/input number for the circuits below. Then sketch equivalent circuits using NAND gates that use fewer transistors (do not minimize the circuits). 6. (12 points) Sketch circuits for the following logic equations F = A̅ ∙ B ∙ C + A ∙B̅ ∙C̅ +A̅ ∙ C F = A̅ ∙ B ∙C̅ ̅̅̅̅̅̅̅̅̅̅ + ̅A̅̅+̅̅̅B̅ F = (? +? ̅ ) ∙ ̅̅?̅̅̅̅̅+̅̅̅̅̅̅̅?̅̅̅∙̅̅?̅̅ G AB C D AB C D H G F F AB C EE214 Problem Set 1 7. (22 points) Sketch a circuit similar to the figure below that asserts logic 1 only when both switches are closed. Label the switches 1 and 2, and complete the truth table below. Then circle the correct term (high or low, and open or closed) to complete the following sentences describing the AND and OR relationships: AND Relationship: The output F is [high / low] when SW1 is [open / closed], and SW2 is [open / closed]. OR Relationship: The output F is [high / low] when SW1 is [open / closed], or SW2 is [open / closed]. Sketch a circuit similar to the figure below that asserts logic 0 whenever one or both switches are closed. Label the switches 1 and 2, and complete the truth table below. Circle the correct term (high or low, and open or closed) to complete the following sentences describing the AND and OR relationships: AND Relationship: The output F is [high / low] when SW1 is [open / closed], and SW2 is [open / closed]. OR Relationship: The output F is [high / low] when SW1 is [open / closed], or SW2 is [open / closed]. 8. (4 points) Complete the following. A pFET turns [ ON / OFF ] with LLV and conducts [ LHV / LLV ] well (circle one in each bracket). An nFET turns [ ON / OFF ] with LLV and conducts [ LHV / LLV ] well (circle one in each bracket). Vdd GND F SW1 SW2 Vdd GND F SW1 SW2 SW1 SW2 F SW1 SW2 F EE214 Problem Set 1 9. (8 points) Sketch circuits and write Verilog assignment statements for the following equations. F = m(1, 2, 6) F = M(0, 7) 10. (21 points) Complete the truth tables below (enter “on” or “off” under each transistor entry, and “1” or “0” for output F), and enter the gate name and schematic shapes in the tables. You get 1/2 point for each correct column, and 1/2 point each for correct names and shapes. Q1 Q2 Q3 Q4 A B F Vdd Q2 Q1 Q3 Q4 A B F Vdd A B Q1 Q2 Q3 Q4 F 0 0 0 1 1 0 1 1 Gate Name AND shape OR shape A B Q1 Q2 Q3 Q4 F 0 0 0 1 1 0 1 1 Gate Name AND shape OR shape EE214 Problem Set 1 Q2 Q1 Q3 Q4 A B F Q5 Q6 Vdd Q1 Q2 Q3 Q4 A B F Q5 Q6 Vdd (2 points) Enter the logic equation for the 3-input circuit above: A B Q1 Q2 Q3 Q4 F 0 0 0 1 1 0 1 1 Gate Name AND shape OR shape A B Q1 Q2 Q3 Q4 F 0 0 0 1 1 0 1 1 Gate Name AND shape OR shape A B C Q1 Q2 Q3 Q4 Q5 Q6 F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 F = Q1 Q2 Q4 Q5 A B F Q6 Vdd C Q3 EE214 Problem Set 1 11. (20 points) In a logic function with n inputs, there are 2? unique combinations of inputs and 22? possible logic functions. The table below has four rows that show the four possible combinations of two inputs (22 = 4), and 16 output columns that show all possible two-input logic function (222 = 16). Six of these output columns are associated with common logic functions of two variables. Circle the six columns, and label them with the appropriate logic gate name. Draw the circuit symbols for the functions represented. INPUTS ALL POSSIBLE FUNCTIONS A B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 A table like the one above for 3 inputs would need _________ rows and _________ columns. A table like the one above for 4 inputs would need _________ rows and _________ columns. A table like the one above for 5 inputs would need _________ rows and _________ columns. 12. (15 points) Find global minimum circuits for the following three logic signal outputs that are all functions of the same three inputs. Show all work. F1 =  m (0, 3, 4) F2 =  m (1, 6, 7) F3 =  m (0, 1, 3, 4)

EE214 Fall 2015 Problem Set1 I am submitting my own work in this exercise, and I am aware of the penalties for cheating that will be assessed if I submit work for credit that is not my own. Print Name Sign Name Date Contains material © Digilent, Inc. 7 pages 1. (15 points) Below are some circuit elements from a simple digital system. 3.3V 20mA VB 1Kohm VA 1.3V RB 1K RC RD SW1 SW2 RA VC When the pushbutton SW1 is not pressed, what is the voltage at VA? (1pt) When the SW1 is pressed, what is the voltage at VA? (1pt) When the SW1 is pressed, what current flows in the 1K resistor RA? (1pt) When SW1 is pressed, what power is dissipated in RA? (2pt) In the LED circuit, 1.3V is required at VB to forward-bias the LED and cause current to flow. Given there is a 1.3V drop across the LED, what resistance RB is required for 20mA to flow through the LED? (2pt) What power is dissipated in the LED? (1pt) In the circuit on the far right, if RC dissipates 25mW, what is VC? (2pt) Using the VC voltage you calculated, if RC is changed to 100Ohms, how much power would it dissipate? (2pt) Using the VC voltage you calculated and a 1K RC, if pressing SW2 causes the total circuit power to increase to 75mW, what value must RD be? (3pt) EE214 Problem Set 1 2. (20 points) Complete the truth tables below. Provide SOP equations for the bottom three tables. F <= Σ ( ) F <= Σ ( ) F <= Σ ( ) 3. (12 points) Write the number of transistors required for each logic gate below inside the gate symbol, and then write the logic gate name below the symbol. 4. (12 points) Complete truth tables for the circuits shown below A B F AND A B F OR A B F XOR A F INV A B C F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 ? = ? ̅ ∙ ? + ? A B C F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 ? = ? ∙ ? ∙? ̅ + ? ∙ ? A B C F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 ? = ? ∙? ̅+? ̅ ∙ ? A B C F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A B C F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A F B C A B C Y EE214 Problem Set 1 5. (18 points) Show the total transistor count and gate/input number for the circuits below. Then sketch equivalent circuits using NAND gates that use fewer transistors (do not minimize the circuits). 6. (12 points) Sketch circuits for the following logic equations F = A̅ ∙ B ∙ C + A ∙B̅ ∙C̅ +A̅ ∙ C F = A̅ ∙ B ∙C̅ ̅̅̅̅̅̅̅̅̅̅ + ̅A̅̅+̅̅̅B̅ F = (? +? ̅ ) ∙ ̅̅?̅̅̅̅̅+̅̅̅̅̅̅̅?̅̅̅∙̅̅?̅̅ G AB C D AB C D H G F F AB C EE214 Problem Set 1 7. (22 points) Sketch a circuit similar to the figure below that asserts logic 1 only when both switches are closed. Label the switches 1 and 2, and complete the truth table below. Then circle the correct term (high or low, and open or closed) to complete the following sentences describing the AND and OR relationships: AND Relationship: The output F is [high / low] when SW1 is [open / closed], and SW2 is [open / closed]. OR Relationship: The output F is [high / low] when SW1 is [open / closed], or SW2 is [open / closed]. Sketch a circuit similar to the figure below that asserts logic 0 whenever one or both switches are closed. Label the switches 1 and 2, and complete the truth table below. Circle the correct term (high or low, and open or closed) to complete the following sentences describing the AND and OR relationships: AND Relationship: The output F is [high / low] when SW1 is [open / closed], and SW2 is [open / closed]. OR Relationship: The output F is [high / low] when SW1 is [open / closed], or SW2 is [open / closed]. 8. (4 points) Complete the following. A pFET turns [ ON / OFF ] with LLV and conducts [ LHV / LLV ] well (circle one in each bracket). An nFET turns [ ON / OFF ] with LLV and conducts [ LHV / LLV ] well (circle one in each bracket). Vdd GND F SW1 SW2 Vdd GND F SW1 SW2 SW1 SW2 F SW1 SW2 F EE214 Problem Set 1 9. (8 points) Sketch circuits and write Verilog assignment statements for the following equations. F = m(1, 2, 6) F = M(0, 7) 10. (21 points) Complete the truth tables below (enter “on” or “off” under each transistor entry, and “1” or “0” for output F), and enter the gate name and schematic shapes in the tables. You get 1/2 point for each correct column, and 1/2 point each for correct names and shapes. Q1 Q2 Q3 Q4 A B F Vdd Q2 Q1 Q3 Q4 A B F Vdd A B Q1 Q2 Q3 Q4 F 0 0 0 1 1 0 1 1 Gate Name AND shape OR shape A B Q1 Q2 Q3 Q4 F 0 0 0 1 1 0 1 1 Gate Name AND shape OR shape EE214 Problem Set 1 Q2 Q1 Q3 Q4 A B F Q5 Q6 Vdd Q1 Q2 Q3 Q4 A B F Q5 Q6 Vdd (2 points) Enter the logic equation for the 3-input circuit above: A B Q1 Q2 Q3 Q4 F 0 0 0 1 1 0 1 1 Gate Name AND shape OR shape A B Q1 Q2 Q3 Q4 F 0 0 0 1 1 0 1 1 Gate Name AND shape OR shape A B C Q1 Q2 Q3 Q4 Q5 Q6 F 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 F = Q1 Q2 Q4 Q5 A B F Q6 Vdd C Q3 EE214 Problem Set 1 11. (20 points) In a logic function with n inputs, there are 2? unique combinations of inputs and 22? possible logic functions. The table below has four rows that show the four possible combinations of two inputs (22 = 4), and 16 output columns that show all possible two-input logic function (222 = 16). Six of these output columns are associated with common logic functions of two variables. Circle the six columns, and label them with the appropriate logic gate name. Draw the circuit symbols for the functions represented. INPUTS ALL POSSIBLE FUNCTIONS A B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 A table like the one above for 3 inputs would need _________ rows and _________ columns. A table like the one above for 4 inputs would need _________ rows and _________ columns. A table like the one above for 5 inputs would need _________ rows and _________ columns. 12. (15 points) Find global minimum circuits for the following three logic signal outputs that are all functions of the same three inputs. Show all work. F1 =  m (0, 3, 4) F2 =  m (1, 6, 7) F3 =  m (0, 1, 3, 4)

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