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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/The following graphs depict the motion of an object starting from rest and moving without friction. Describe how you would calculate the object’s acceleration, instantaneous speed, and distance at time “p” from each graph (slope, area-under-curve, etc.). 15. An object is launched at an angle of 450 from the ground of a mystery planet. The object hits the ground 20m away after a total flight time of 4.0s. Assume no air resistance. a. What are the initial vertical and horizontal velocities? b. Calculate the acceleration due to gravity. c. Draw graphs to quantitatively represent the vertical and horizontal velocities for the entire 4.0s of flight. Linear Dynamics 1. A block of mass 3 kg, initially at rest, is pulled along a frictionless, horizontal surface with a force shown as a function of time by the graph above. Calculate the acceleration and speed after 2s. Questions 2-4: Two blocks of masses M and m, with M > m, are connected by a light string. The string passes over a frictionless pulley of negligible mass so that the blocks hang vertically like Atwood’s machine. The blocks are then released from rest as shown above. 2. Draw a free-body diagram for each mass. Compare and contrast the tension on each. 3. Compare and contrast the net-force acting on each block. 4. Draw a free-body diagram for the string holding the pulley. Explain whether the force increases, decreases, or remains the same as the blocks accelerate. Questions 5-6: A ball is released from the top of a curved hill as shown above; the hill has sufficient friction so that the ball rolls as it moves down the hill. 5. What can be inferred about the ball’s linear acceleration and speed as the ball goes from the top to the bottom? (Increase, decrease, or remain the same) 6. Draw a free-body diagram for each location in the diagram to compare the weight, normal, and friction forces as it rolls down hill. Questions 7-8: Consider the above block sitting on a smooth tabletop. It is connected by a light string that passes over a frictionless and massless pulley to a pulling force of 30N downward. 7. Use Newton’s 2nd Law to determine what will happen to the net force, mass, and acceleration of the entire system if the pulling force of 30N is replaced with another block weighing 30N. 8. What will happen to the tension on each body?

/The following graphs depict the motion of an object starting from rest and moving without friction. Describe how you would calculate the object’s acceleration, instantaneous speed, and distance at time “p” from each graph (slope, area-under-curve, etc.). 15. An object is launched at an angle of 450 from the ground of a mystery planet. The object hits the ground 20m away after a total flight time of 4.0s. Assume no air resistance. a. What are the initial vertical and horizontal velocities? b. Calculate the acceleration due to gravity. c. Draw graphs to quantitatively represent the vertical and horizontal velocities for the entire 4.0s of flight. Linear Dynamics 1. A block of mass 3 kg, initially at rest, is pulled along a frictionless, horizontal surface with a force shown as a function of time by the graph above. Calculate the acceleration and speed after 2s. Questions 2-4: Two blocks of masses M and m, with M > m, are connected by a light string. The string passes over a frictionless pulley of negligible mass so that the blocks hang vertically like Atwood’s machine. The blocks are then released from rest as shown above. 2. Draw a free-body diagram for each mass. Compare and contrast the tension on each. 3. Compare and contrast the net-force acting on each block. 4. Draw a free-body diagram for the string holding the pulley. Explain whether the force increases, decreases, or remains the same as the blocks accelerate. Questions 5-6: A ball is released from the top of a curved hill as shown above; the hill has sufficient friction so that the ball rolls as it moves down the hill. 5. What can be inferred about the ball’s linear acceleration and speed as the ball goes from the top to the bottom? (Increase, decrease, or remain the same) 6. Draw a free-body diagram for each location in the diagram to compare the weight, normal, and friction forces as it rolls down hill. Questions 7-8: Consider the above block sitting on a smooth tabletop. It is connected by a light string that passes over a frictionless and massless pulley to a pulling force of 30N downward. 7. Use Newton’s 2nd Law to determine what will happen to the net force, mass, and acceleration of the entire system if the pulling force of 30N is replaced with another block weighing 30N. 8. What will happen to the tension on each body?

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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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Learning Goals: Students will be able to determine the gravitational acceleration of “Planet X” 1. Research to find equations that would help you find g using a pendulum. Design an experiment and test your design using Moon and Jupiter. Write your procedure in a paragraph that another student could use to verify your results. Show your data, graphs, and calculations that support your strategy. 2. Use your procedure to find g on Planet X. Show your data, graphs, and calculations that support your conclusion. 3. Give your conclusion and write an error analysis.

Learning Goals: Students will be able to determine the gravitational acceleration of “Planet X” 1. Research to find equations that would help you find g using a pendulum. Design an experiment and test your design using Moon and Jupiter. Write your procedure in a paragraph that another student could use to verify your results. Show your data, graphs, and calculations that support your strategy. 2. Use your procedure to find g on Planet X. Show your data, graphs, and calculations that support your conclusion. 3. Give your conclusion and write an error analysis.

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Determine the , , components of reaction at the fixed wall . The 150- force is parallel to the axis and the 200- force is parallel to the axis. Enter the , , and components of the reaction using three significant figures separated by commas.

Determine the , , components of reaction at the fixed wall . The 150- force is parallel to the axis and the 200- force is parallel to the axis. Enter the , , and components of the reaction using three significant figures separated by commas.

Instructions 1. The next sheet is an example problem from the text. Select the cells at the top of the columns to see the formulas and format. 2. Columns A,B, and C are the input data. Note that the upper boundary is used. Columns D and E are used to calculate the average and sample standard deviation. Column F calculates the z value using the upper boundary (Column B), the average, and the sample standard deviation. Column G is the area (cumulative probability) under the normal curve to the left of the z value in the same manner as Table A. Column H is the area [probability] for each cell. Note that the formula for Row 3 is different than the rest of the rows. Column I is the expected frequency for each cell. It equals the value in Column H times the total number of observed values (110). Column J is the chi-squared value which can be compared to a chi-squared table to determine the observed values (110). This column is really not necessary because the program calculates the observed values (110) and performs the chi-squared test at I21 and K21. Column K is an adjustment to bring the total number of observed values to 110. The chi-squared test for the adjustment gives a probability of 0.971that the distribution is normal. 3. The following sheet, called template, should be copied before using the program. This activity is accomplished by selecting EDIT, selecting MOVE OR COPY SHEET, selecting CREATE A COPY, and locating the new sheet, called template (2), in the dialog box. 4. The template is designed for 9 cells. If more or less cells are required, the appropriate changes must be made.

Instructions 1. The next sheet is an example problem from the text. Select the cells at the top of the columns to see the formulas and format. 2. Columns A,B, and C are the input data. Note that the upper boundary is used. Columns D and E are used to calculate the average and sample standard deviation. Column F calculates the z value using the upper boundary (Column B), the average, and the sample standard deviation. Column G is the area (cumulative probability) under the normal curve to the left of the z value in the same manner as Table A. Column H is the area [probability] for each cell. Note that the formula for Row 3 is different than the rest of the rows. Column I is the expected frequency for each cell. It equals the value in Column H times the total number of observed values (110). Column J is the chi-squared value which can be compared to a chi-squared table to determine the observed values (110). This column is really not necessary because the program calculates the observed values (110) and performs the chi-squared test at I21 and K21. Column K is an adjustment to bring the total number of observed values to 110. The chi-squared test for the adjustment gives a probability of 0.971that the distribution is normal. 3. The following sheet, called template, should be copied before using the program. This activity is accomplished by selecting EDIT, selecting MOVE OR COPY SHEET, selecting CREATE A COPY, and locating the new sheet, called template (2), in the dialog box. 4. The template is designed for 9 cells. If more or less cells are required, the appropriate changes must be made.

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1 Before-class Preparation: Project Management Use the following questions to guide your reading and preparation for the iRAT/tRAT quiz. 1. Pre-lecture reading: Read the Textbook Chapter 7 from page 59 to page 63 “Step 3: Plan and manage the project”. 1) What is the three-part basic sequence of project management activities? 2) Fill in the blanks. During the “Plan” activity, a schedule shows will work on issues and materials will be applied to task? 3) Fill in the blank. During the “Assess” activity, periodic help keep team and customers informed of the state of the project. 2. Pre-lecture reading: Read the handout “Project Management – Creating a Project Schedule”. 1) What are the three aspects that need to be addressed when planning a project? 2) What are the two ways of presenting a WBS? Are task dependencies considered when creating a WBS? 3) In a network diagram, why do some tasks have to be completed in a specific order? What is the predecessor(s) of a task? 4) In a Gantt chart, what information can be derived from the “bar” for each task? What about the arrows? 5) What is a critical path? How to determine the total duration of a project? In order to reduce the project duration, should tasks on the critical path get additional resources or tasks not on the critical path? 3. Pre-lecture homework: answer all questions above. First copy each question then write down the answer. Submit the homework to Blackboard and also bring in the completed homework to the RAT quiz to use as a cheat sheet.

1 Before-class Preparation: Project Management Use the following questions to guide your reading and preparation for the iRAT/tRAT quiz. 1. Pre-lecture reading: Read the Textbook Chapter 7 from page 59 to page 63 “Step 3: Plan and manage the project”. 1) What is the three-part basic sequence of project management activities? 2) Fill in the blanks. During the “Plan” activity, a schedule shows will work on issues and materials will be applied to task? 3) Fill in the blank. During the “Assess” activity, periodic help keep team and customers informed of the state of the project. 2. Pre-lecture reading: Read the handout “Project Management – Creating a Project Schedule”. 1) What are the three aspects that need to be addressed when planning a project? 2) What are the two ways of presenting a WBS? Are task dependencies considered when creating a WBS? 3) In a network diagram, why do some tasks have to be completed in a specific order? What is the predecessor(s) of a task? 4) In a Gantt chart, what information can be derived from the “bar” for each task? What about the arrows? 5) What is a critical path? How to determine the total duration of a project? In order to reduce the project duration, should tasks on the critical path get additional resources or tasks not on the critical path? 3. Pre-lecture homework: answer all questions above. First copy each question then write down the answer. Submit the homework to Blackboard and also bring in the completed homework to the RAT quiz to use as a cheat sheet.

Please write clearly, show all work in an organized fashion, and circle answers. 1) Using the data shown in Figures 6.14 (at 25oC) and 6.21, combine both curves onto one plot, being careful to correctly plot the modulus, yield strength, tensile (ultimate) strength, and ductility. Discuss how the modulus, yield strength, and ductility compare for pure iron (figure 6.14) vs. the alloy steel. 2) The equation for the effect of grain size on yield strength is given by: y = I +kD-0.5 where y is the yield stress, I is the intrinsic resistance of the lattice to dislocation motion, k is the “blocking parameter” which measures the effectiveness of grain boundaries in blocking dislocation motion, and D is the grain diameter. Use this equation to determine the change in yield strength of a typical steel when the grain size is increased from 10micron to 50 micron (1 micron = 10-6 m), due to grain growth. . I = 150 MN/m2 and k = 0.70 MN/m1.5 . 3) Using the data shown in Callister Figure 7.19, draw an approximate stress-strain curve for the 1040 steel at 0% cold work and at 30% cold work, clearly indicating the yield strength, ductility, and tensile strength of the steel before and after cold-working (Young’s modulus of steel E = 250 MPa). 4) A fatigue test is carried out on a steel having an ultimate strength of 289 MPa. The number of cycles required to break the specimen at different stresses are given below: Stress Amplitude Fatigue Life (MPa) (cycles) 223 4.5 x 104 209 2.4 x 105 192 8.0 x 105 178 1.5 x 106 175 2.7 x 106 168 7.8 x 106 168 >1.0 x 107 (did not break) 165 >2.6 x 107 162 >2.2 x 107 a) Plot the data on linear-log scale, preferably with a computerized figure-plotting program. b) Determine the average fatigue strength at 106 cycles (hint: use curve-fitting software to fit the line). c) What is the ratio of the fatigue strength at 106 cycles to the ultimate strength? e) If you plan to use this material for 108 cycles, what is the maximum fatigue strength you would recommend (assuming 20% fluctuations in stress amplitude). Callister Homework Problems: 7.22, 8.4, 8.12 (see next page)

Please write clearly, show all work in an organized fashion, and circle answers. 1) Using the data shown in Figures 6.14 (at 25oC) and 6.21, combine both curves onto one plot, being careful to correctly plot the modulus, yield strength, tensile (ultimate) strength, and ductility. Discuss how the modulus, yield strength, and ductility compare for pure iron (figure 6.14) vs. the alloy steel. 2) The equation for the effect of grain size on yield strength is given by: y = I +kD-0.5 where y is the yield stress, I is the intrinsic resistance of the lattice to dislocation motion, k is the “blocking parameter” which measures the effectiveness of grain boundaries in blocking dislocation motion, and D is the grain diameter. Use this equation to determine the change in yield strength of a typical steel when the grain size is increased from 10micron to 50 micron (1 micron = 10-6 m), due to grain growth. . I = 150 MN/m2 and k = 0.70 MN/m1.5 . 3) Using the data shown in Callister Figure 7.19, draw an approximate stress-strain curve for the 1040 steel at 0% cold work and at 30% cold work, clearly indicating the yield strength, ductility, and tensile strength of the steel before and after cold-working (Young’s modulus of steel E = 250 MPa). 4) A fatigue test is carried out on a steel having an ultimate strength of 289 MPa. The number of cycles required to break the specimen at different stresses are given below: Stress Amplitude Fatigue Life (MPa) (cycles) 223 4.5 x 104 209 2.4 x 105 192 8.0 x 105 178 1.5 x 106 175 2.7 x 106 168 7.8 x 106 168 >1.0 x 107 (did not break) 165 >2.6 x 107 162 >2.2 x 107 a) Plot the data on linear-log scale, preferably with a computerized figure-plotting program. b) Determine the average fatigue strength at 106 cycles (hint: use curve-fitting software to fit the line). c) What is the ratio of the fatigue strength at 106 cycles to the ultimate strength? e) If you plan to use this material for 108 cycles, what is the maximum fatigue strength you would recommend (assuming 20% fluctuations in stress amplitude). Callister Homework Problems: 7.22, 8.4, 8.12 (see next page)

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