## Chapter 14 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, May 16, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Harmonic Oscillator Equations Learning Goal: To derive the formulas for the major characteristics of motion as functions of time for a horizontal spring oscillator and to practice using the obtained formulas by answering some basic questions. A block of mass is attached to a spring whose spring constant is . The other end of the spring is fixed so that when the spring is unstretched, the mass is located at . . Assume that the +x direction is to the right. The mass is now pulled to the right a distance beyond the equilibrium position and released, at time , with zero initial velocity. Assume that the vertical forces acting on the block balance each other and that the tension of the spring is, in effect, the only force affecting the motion of the block. Therefore, the system will undergo simple harmonic motion. For such a system, the equation of motion is , and its solution, which provides the equation for , is . Part A At what time does the block come back to its original equilibrium position ( ) for the first time? Express your answer in terms of some or all of the variables: , , and . You did not open hints for this part. ANSWER: m k x = 0 A t = 0 a(t) = − x(t) km x(t) x(t) = Acos( t) km −− t1 x = 0 A k m Part B Find the velocity of the block as a function of time. Express your answer in terms of some or all of the variables: , , , and . You did not open hints for this part. ANSWER: Part C This question will be shown after you complete previous question(s). t1 = v k m A t v(t) = Typesetting math: 100% Part D Find the acceleration of the block as a function of time. Express your answer in terms of some of all of the variables: , , , and . ANSWER: Part E Specify when the magnitude of the acceleration of the block reaches its maximum value. Consider the following options: only once during one a. period of motion, b. when the block’s speed is zero, c. when the block is in the equilibrium position, d. when the block’s displacement equals either or , e. when the block’s speed is at a maximum. Choose the most complete answer. You did not open hints for this part. ANSWER: a k m A t a(t) = A −A Typesetting math: 100% Part F Find the kinetic energy of the block as a function of time. Express your answer in terms of some or all of the variables: , , , and . You did not open hints for this part. ANSWER: Part G Find , the maximum kinetic energy of the block. Express your answer in terms of some or all of the variables: , , and . ANSWER: a only b only c only d only e only b and d c and e b and c a and e d and e K k m A t K(t) = Kmax k m A Typesetting math: 100% Part H The kinetic energy of the block reaches its maximum when which of the following occurs? You did not open hints for this part. ANSWER: Mass and Simple Harmonic Motion Conceptual Question The shaker cart, shown in the figure, is the latest extreme sport craze. You stand inside of a small cart attached to a heavy-duty spring, the spring is compressed and released, and you shake back and forth, attempting to maintain your balance. Note that there is also a sandbag in the cart with you. Kmax = The displacement of the block is zero. The displacement of the block is . The acceleration of the block is at a maximum. The velocity of the block is zero. A Typesetting math: 100% At the instant you pass through the equilibrium position of the spring, you drop the sandbag out of the cart onto the ground. Part A What effect does dropping the sandbag out of the cart at the equilibrium position have on the amplitude of your oscillation? You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Instead of dropping the sandbag as you pass through equilibrium, you decide to drop the sandbag when the cart is at its maximum distance from equilibrium. Part C This question will be shown after you complete previous question(s). Part D It increases the amplitude. It decreases the amplitude. It has no effect on the amplitude. Typesetting math: 100% This question will be shown after you complete previous question(s). Simple Harmonic Motion Conceptual Question An object of mass is attached to a vertically oriented spring. The object is pulled a short distance below its equilibrium position and released from rest. Set the origin of the coordinate system at the equilibrium position of the object and choose upward as the positive direction. Assume air resistance is so small that it can be ignored. Refer to these graphs when answering the following questions. Part A Beginning the instant the object is released, select the graph that best matches the position vs. time graph for the object. You did not open hints for this part. ANSWER: m Typesetting math: 100% Part B Beginning the instant the object is released, select the graph that best matches the velocity vs. time graph for the object. You did not open hints for this part. ANSWER: Part C Beginning the instant the object is released, select the graph that best matches the acceleration vs. time graph for the object. A B C D E F G H A B C D E F G H Typesetting math: 100% You did not open hints for this part. ANSWER: Harmonic Oscillator Acceleration Learning Goal: To understand the application of the general harmonic equation to finding the acceleration of a spring oscillator as a function of time. One end of a spring with spring constant is attached to the wall. The other end is attached to a block of mass . The block rests on a frictionless horizontal surface. The equilibrium position of the left side of the block is defined to be . The length of the relaxed spring is . The block is slowly pulled from its equilibrium position to some position along the x axis. At time , the block is released with zero initial velocity. The goal of this problem is to determine the acceleration of the block as a function of time in terms of , , and . It is known that a general solution for the position of a harmonic oscillator is , where , , and are constants. Your task, therefore, is to determine the values of , , and in terms of , ,and and then use the connection between and to find the acceleration. A B C D E F G H k m x = 0 L xinit > 0 t = 0 a(t) k m xinit x(t) = C cos (t) + S sin (t) C S C S k m xinit x(t) a(t) Typesetting math: 100% Part A Combine Newton’s 2nd law and Hooke’s law for a spring to find the acceleration of the block as a function of time. Express your answer in terms of , , and the coordinate of the block . You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C a(t) k m x(t) a(t) = Typesetting math: 100% This question will be shown after you complete previous question(s). ± Introduction to Simple Harmonic Motion Consider the system shown in the figure. It consists of a block of mass attached to a spring of negligible mass and force constant . The block is free to move on a frictionless horizontal surface, while the left end of the spring is held fixed. When the spring is neither compressed nor stretched, the block is in equilibrium. If the spring is stretched, the block is displaced to the right and when it is released, a force acts on it to pull it back toward equilibrium. By the time the block has returned to the equilibrium position, it has picked up some kinetic energy, so it overshoots, stopping somewhere on the other side, where it is again pulled back toward equilibrium. As a result, the block moves back and forth from one side of the equilibrium position to the other, undergoing oscillations. Since we are ignoring friction (a good approximation to many cases), the mechanical energy of the system is conserved and the oscillations repeat themselves over and over. The motion that we have just described is typical of most systems when they are displaced from equilibrium and experience a restoring force that tends to bring them back to their equilibrium position. The resulting oscillations take the name of periodic motion. An important example of periodic motion is simple harmonic motion (SHM) and we will use the mass-spring system described here to introduce some of its properties. Part A Which of the following statements best describes the characteristic of the restoring force in the spring-mass system described in the introduction? You did not open hints for this part. ANSWER: m k The restoring force is constant. The restoring force is directly proportional to the displacement of the block. The restoring force is proportional to the mass of the block. The restoring force is maximum when the block is in the equilibrium position. Typesetting math: 100% Part B As shown in the figure, a coordinate system with the origin at the equilibrium position is chosen so that the x coordinate represents the displacement from the equilibrium position. (The positive direction is to the right.) What is the initial acceleration of the block, , when the block is released at a distance from its equilibrium position? Express your answer in terms of some or all of the variables , , and . You did not open hints for this part. ANSWER: Part C What is the acceleration of the block when it passes through its equilibrium position? Express your answer in terms of some or all of the variables , , and . You did not open hints for this part. ANSWER: a0 A A m k a0 = a1 A m k Typesetting math: 100% Part D This question will be shown after you complete previous question(s). Using the information found so far, select the correct phrases to complete the following statements. Part E You did not open hints for this part. ANSWER: Part F You did not open hints for this part. ANSWER: a1 = The magnitude of the block’s acceleration reaches its maximum value when the block is in the equilibrium position. at either its rightmost or leftmost position. between its rightmost position and the equilibrium position. between its leftmost position and the equilibrium position. Typesetting math: 100% Part G You did not open hints for this part. ANSWER: Part H Because of the periodic properties of SHM, the mathematical equations that describe this motion involve sine and cosine functions. For example, if the block is released at a distance from its equilibrium position, its displacement varies with time according to the equation , where is a constant characteristic of the system. If time is measured is seconds, must be expressed in radians per second so that the quantity is expressed in radians. Use this equation and the information you now have on the acceleration and speed of the block as it moves back and forth from one side of its equilibrium position to the other to determine the correct set of equations for the block’s x components of velocity and acceleration, and , respectively. In the expressions below, and are nonzero positive constants. You did not open hints for this part. The speed of the block is zero when it is in the equilibrium position. at either its rightmost or leftmost position. between its rightmost position and the equilibrium position. between its leftmost position and the equilibrium position. The speed of the block reaches its maximum value when the block is in the equilibrium position. at either its rightmost or leftmost position. between the rightmost position and the equilibrium position. between the leftmost position and the equilibrium position. A x t x = Acost t vx ax B C Typesetting math: 100% ANSWER: Period of a Pendulum Ranking Task Part A Six pendulums of mass and length as shown are released from rest at the same angle from vertical. Rank the pendulums according to the number of complete cycles of motion each pendulum goes through per minute. Rank from most to least complete cycles of motion per minute. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: , , , , vx = −Bsint ax = C cost vx = Bcost ax = C sint vx = −Bcost ax = −C cost vx = −Bsint ax = −C cost m L Typesetting math: 100% ± Gravity on Another Planet After landing on an unfamiliar planet, a space explorer constructs a simple pendulum of length 47.0 . The explorer finds that the pendulum completes 108 full swing cycles in a time of 138 . Part A What is the magnitude of the gravitational acceleration on this planet? Express your answer in meters per second per second. You did not open hints for this part. cm s Typesetting math: 100% ANSWER: ± Tactics Box 14.1 Identifying and Analyzing Simple Harmonic Motion Learning Goal: To practice Tactics Box 14.1 Identifying and analyzing simple harmonic motion. A complete description of simple harmonic motion must take into account several physical quantities and various mathematical relations among them. This Tactics Box summarizes the essential information needed to solve oscillation problems of this type. TACTICS BOX 14.1 Identifying and analyzing simple harmonic motion If the net force acting on a particle is a linear restoring force, the motion will be simple harmonic motion around the equilibrium 1. position. The position as a function of time is . The velocity as a function of time is . The maximum speed is . The equations are given here in terms of , but they can be written in terms of , or some other parameter if the situation calls for it. 2. 3. The amplitude and the phase constant are determined by the initial conditions through and . 4. The angular frequency (and hence the period ) depends on the physical properties of the situtaion. But does not depend on or . Mechanical energy is conserved. Thus .Energy conservation provides a relationship between position and velocity that is independent of time. 5. Part A The position of a 60 oscillating mass is given by , where is in seconds. Determine the velocity at . Express your answer in meters per second to two significant figures. You did not open hints for this part. ANSWER: gplanet = m/s2 x(t) = Acos(t + 0 ) vx(t) = −Asin(t + 0 ) vmax = A x y A 0 x0 = Acos 0 v0x = −Asin 0 T = 2/ A 0 1 m + k = k = m( 2 v2 x 1 2 x2 1 2 A2 1 2 vmax)2 g x(t) = (2.0 cm) cos(10t) t t = 0.40 s Typesetting math: 100% Part B Assume that the oscillating mass described in Part A is attached to a spring. What would the spring constant of this spring be? Express your answer in newtons per meter to two significant figures. You did not open hints for this part. ANSWER: Part C What is the total energy of the mass described in the previous parts? Express your answer in joules to two significant figures. You did not open hints for this part. ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. vx = m/s k k = N/m E E = J Typesetting math: 100%

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## EME 4003 Projects (15% of the Final Grade) There are two design projects in this class which are presented as Projects 1 & 2 below. Grades for the projects will be assigned on a group basis. Peer evaluations of each project will be required. These peer evaluations in addition to evaluation by the instructor are used to evaluate the individual participation on the group projects and will affect each individual’s grade. No late projects will be accepted. A quality design project could earn up to 5 extra points to the final grade for all the group members. The design projects should be geared towards real-world machine element design. Attention should be given to functional analysis, materials selection, critical point analysis, failure mode analysis, and application of knowledge gained during your mechanical engineering courses. Each group should consist of 4 students. Please talk to your classmates and form your group by Thursday, September 10. I will form and assign the teammates for the students with no group. Project 1 (3%): Due October 15 Each member of the group should report on one of his/her favorite machines or mechanisms and provide a one-page description (approximately 500 words) along with a picture of the machine/mechanism(it is helpful if each group member brings the actual component to the group meeting). The paper should include the innovative part of the mechanisms, explain the forces acting on one or more of the components and the maximum stress locations. Each group should review the reports submitted by its members and provide feedback. The final reports with the comments from each teammate should be submitted to the instructor by October 15.. Project 2(12%): Due December 10 In this project, your group will be responsible for analyzing an existing engineering product (a piece of equipment with multiple mechanical components at work). Your group is responsible to evaluate the product and determine the safety factors in the design (if any) and suggest the minimum values for the design parameters (geometry, dimension, material, etc.) to reduce the cost. Project Guideline Step 1: Select an engineering product (consumer product or fixture) that is subjected to constant and variable (fatigue) loads. For examples: tools in the garage, the pedal crank on your bike, automotive components (i.e. shock tower on your vehicle), circular saw, etc. Brainstorm ideas with your teammates and select the product. Step 2: Here are the tasks a) Analyze it completely for failure modes b) Investigate the loads acting on the components, their variations, and the location of the maximum stresses (several locations) c) Estimate the stresses at these locations and draw Mohr’s Circle for these locations. You may need to create a spreadsheet in Excel/MATLAB to preform your calculations. All values in the spreadsheet or program must be clearly labeled. Show all intermediate steps, results, as well as the final output. d) Are there any Safety Factors required at these locations? e) If a Safety Factor of 1 was used, estimate the new geometry, dimensions, etc. Step 3: Develop conclusions and recommendations about the product (discuss and comments about the design, can it be improved?) State all assumptions you made in your analysis that might have accounted for possible inaccurate answers. The final report should be at least 1500 words including introduction, discussion, and conclusion with any pictures, figures, equations, tables, and spreadsheets. Reference: INCORPORATING OPEN-ENDED PROJECTS INTO A MACHINE ELEMENTS COURSE, Kathy Schmidt, the University of Texas at Austin, Matthew Campbell, University of Texas at Austin, Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition

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## Assignment 12 Due: 11:59pm on Friday, May 9, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Problem 15.6 A 2.00 -diameter vat of liquid is 2.90 deep. The pressure at the bottom of the vat is 1.20 . Part A What is the mass of the liquid in the vat? Express your answer with the appropriate units. ANSWER: Correct Problem 15.8 A 120-cm-thick layer of oil floats on a 130-cm-thick layer of water. Part A What is the pressure at the bottom of the water layer? Express your answer with the appropriate units. ANSWER: Correct m m atm 6490 kg p = 1.25×105 Pa Problem 15.9 A research submarine has a 40.0 -diameter window 8.00 thick. The manufacturer says the window can withstand forces up to 1.20×106 . What is the submarine’s maximum safe depth? Part A The pressure inside the submarine is maintained at 1.0 atm. Express your answer with the appropriate units. ANSWER: Correct Problem 15.13 Part A What is the minimum hose diameter of an ideal vacuum cleaner that could lift a 13 dog off the floor? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Enhanced EOC: Problem 15.40 The 80 student in the figure balances a 1300 elephant on a hydraulic lift. cm cm N 947 m kg d = 4.0 cm kg kg Typesetting math: 100% You may want to review ( pages 415 – 419) . For help with math skills, you may want to review: Rearrangement of Equations Involving Multiplication and Division Part A What is the diameter of the piston the student is standing on? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Given that the height of the fluid on the two sides is the same in the figure, how is the pressure of the fluid on the two sides related? What is the definition of pressure? What is the area of the right cylinder? What is the force exerted by the elephant on the right cylinder? What is the additional pressure above atmospheric pressure in the fluid under the elephant? What is the additional pressure above atmospheric pressure under the student in the left cylinder? What is the force exerted by the student on the left cylinder? What is the area of the left cylinder? ANSWER: Correct Part B d = 0.50 m Typesetting math: 100% When a second student joins the first, the piston sinks 30 . What is the second student’s mass? Express your answer to two significant figures and include the appropriate units. You did not open hints for this part. ANSWER: Enhanced EOC: Problem 15.17 A 4.70 rock whose density is 4300 is suspended by a string such that half of the rock’s volume is under water. You may want to review ( pages 419 – 423) . For help with math skills, you may want to review: Conversion Factors Part A What is the tension in the string? Express your answer with the appropriate units. Hint 1. How to approach the problem What are the three forces acting on the rock? Draw a picture indicating the direction of the forces on the rock and an appropriate coordinate system indicating the positive direction. How is volume related to mass and density? What is the volume of the rock? What is the buoyant force on the rock given that half of the rock is underwater? What is the gravitational force on the rock? Given that the rock is suspended, what is the net force on the rock? Now, determine the tension in the string. cm m = kg kg/m3 Typesetting math: 100% ANSWER: Correct Problem 15.15 A block floats in water with its long axis vertical. The length of the block above water is 1.0 . Part A What is the block’s mass density? Express your answer with the appropriate units. ANSWER: Correct Crown of Gold? According to legend, the following challenge led Archimedes to the discovery of his famous principle: Hieron, king of Syracuse, was suspicious that a new crown that he had received from the royal goldsmith was not pure gold, as claimed. Archimedes was ordered to determine whether the crown was in fact made of pure gold, with the condition that only a nondestructive test would be allowed. Rather than answer the problem in the politically most expedient way (or perhaps extract a bribe from the goldsmith), Archimedes thought about the problem scientifically. The legend relates that when Archimedes stepped into his bath and caused it to overflow, he realized that he could answer the challenge by comparing the volume of water displaced by the crown with the volume of water displaced by an amount of pure gold equal in weight to the crown. If the crown was made of pure gold, the two volumes would be equal. If some other (less dense) metal had been substituted for some of the gold, then the crown would displace more water than the pure gold. A similar method of answering the challenge, based on the same physical principle, is to compute the ratio of the actual weight of the crown, , and the apparent weight of the crown when it is submerged in water, . See whether you can follow in Archimedes’ footsteps. The figure shows what is meant by weighing the crown while it is submerged in water. 40.7 N 2.0 cm × 2.0 cm × 8.0 cm cm 875 kg m3 Wactual Wapparent Typesetting math: 100% Part A Take the density of the crown to be . What is the ratio of the crown’s apparent weight (in water) to its actual weight ? Express your answer in terms of the density of the crown and the density of water . Hint 1. Find an expression for the actual weight of the crown Assume that the crown has volume . Find the actual weight of the crown. Express in terms of , (the magnitude of the acceleration due to gravity), and . ANSWER: Hint 2. Find an expression for the apparent weight of the crown Assume that the crown has volume , and take the density of water to be . Find the apparent weight of the crown submerged in water. Express your answer in terms of , (the magnitude of the acceleration due to gravity), , and . Hint 1. How to approach the problem c Wapparent Wactual c w V Wactual Wactual V g c Wactual = cV g V w Wapparent V g w c Typesetting math: 100% The apparent weight of the crown when it is submerged in water will be less than its actual weight (weight in air) due to the buoyant force, which opposes gravity. Hint 2. Find an algebraic expression for the buoyant force. Find the magnitude of the buoyant force on the crown when it is completely submerged in water. Express your answer in terms of , , and the gravitational acceleration . ANSWER: ANSWER: ANSWER: Correct Part B Imagine that the apparent weight of the crown in water is , and the actual weight is . Is the crown made of pure (100%) gold? The density of water is grams per cubic centimeter. The density of gold is grams per cubic centimeter. Hint 1. Find the ratio of weights for a crown of pure gold Given the expression you obtained for , what should the ratio of weights be if the crown is made of pure gold? Express your answer numerically, to two decimal places. Fbuoyant w V g Fbuoyant = wV g Wapparent = (c − w)gV = Wapparent Wactual 1 − w c Wapparent = 4.50 N Wactual = 5.00 N w = 1.00 g = 19.32 Wapparent Wactual Typesetting math: 100% ANSWER: ANSWER: Correct For the values given, , whereas for pure gold, . Thus, you can conclude that the the crown is not pure gold but contains some less-dense metal. The goldsmith made sure that the crown’s (true) weight was the same as that of the amount of gold he was allocated, but in so doing was forced to make the volume of the crown slightly larger than it would otherwise have been. Problem 15.23 A 1.0-cm-diameter pipe widens to 2.0 cm, then narrows to 0.5 cm. Liquid flows through the first segment at a speed of 2.0 . Part A What is the speed in the second segment? Express your answer with the appropriate units. ANSWER: Correct = 0.95 Wapparent Wactual Yes No = 4.50/5.00 = 0.90 Wapparent Wactual = 1 − = 0.95 Wapparent Wactual w g m/s 0.500 ms Typesetting math: 100% Part B What is the speed in the third segment? Express your answer with the appropriate units. ANSWER: Correct Part C What is the volume flow rate through the pipe? Express your answer with the appropriate units. ANSWER: Correct Understanding Bernoulli’s Equation Bernoulli’s equation is a simple relation that can give useful insight into the balance among fluid pressure, flow speed, and elevation. It applies exclusively to ideal fluids with steady flow, that is, fluids with a constant density and no internal friction forces, whose flow patterns do not change with time. Despite its limitations, however, Bernoulli’s equation is an essential tool in understanding the behavior of fluids in many practical applications, from plumbing systems to the flight of airplanes. For a fluid element of density that flows along a streamline, Bernoulli’s equation states that , where is the pressure, is the flow speed, is the height, is the acceleration due to gravity, and subscripts 1 and 2 refer to any two points along the streamline. The physical interpretation of Bernoulli’s equation becomes clearer if we rearrange the terms of the equation as follows: . 8.00 ms 1.57×10−4 m3 s p1 +gh1 + = +g + 1 2 v21 p2 h2 1 2 v22 p v h g p1 − p2 = g(h2 −h1)+ ( − ) 1 2 v22 v21 Typesetting math: 100% The term on the left-hand side represents the total work done on a unit volume of fluid by the pressure forces of the surrounding fluid to move that volume of fluid from point 1 to point 2. The two terms on the right-hand side represent, respectively, the change in potential energy, , and the change in kinetic energy, , of the unit volume during its flow from point 1 to point 2. In other words, Bernoulli’s equation states that the work done on a unit volume of fluid by the surrounding fluid is equal to the sum of the change in potential and kinetic energy per unit volume that occurs during the flow. This is nothing more than the statement of conservation of mechanical energy for an ideal fluid flowing along a streamline. Part A Consider the portion of a flow tube shown in the figure. Point 1 and point 2 are at the same height. An ideal fluid enters the flow tube at point 1 and moves steadily toward point 2. If the cross section of the flow tube at point 1 is greater than that at point 2, what can you say about the pressure at point 2? Hint 1. How to approach the problem Apply Bernoulli’s equation to point 1 and to point 2. Since the points are both at the same height, their elevations cancel out in the equation and you are left with a relation between pressure and flow speeds. Even though the problem does not give direct information on the flow speed along the flow tube, it does tell you that the cross section of the flow tube decreases as the fluid flows toward point 2. Apply the continuity equation to points 1 and 2 and determine whether the flow speed at point 2 is greater than or smaller than the flow speed at point 1. With that information and Bernoulli’s equation, you will be able to determine the pressure at point 2 with respect to the pressure at point 1. Hint 2. Apply Bernoulli’s equation Apply Bernoulli’s equation to point 1 and to point 2 to complete the expression below. Here and are the pressure and flow speed, respectively, and subscripts 1 and 2 refer to point 1 and point 2. Also, use for elevation with the appropriate subscript, and use for the density of the fluid. Express your answer in terms of some or all of the variables , , , , , , and . Hint 1. Flow along a horizontal streamline p1 − p2 g(h2 − h1) 1 ( − ) 2 v22 v21 p v h p1 v1 h1 p2 v2 h2 Typesetting math: 100% Along a horizontal streamline, the change in potential energy of the flowing fluid is zero. In other words, when applying Bernoulli’s equation to any two points of the streamline, and they cancel out. ANSWER: Hint 3. Determine with respect to By applying the continuity equation, determine which of the following is true. Hint 1. The continuity equation The continuity equation expresses conservation of mass for incompressible fluids flowing in a tube. It says that the amount of fluid flowing through a cross section of the tube in a time interval must be the same for all cross sections, or . Therefore, the flow speed must increase when the cross section of the flow tube decreases, and vice versa. ANSWER: ANSWER: h1 = h2 p1 + = 1 2 v21 p2 + v2 2 2 v2 v1 $V A $t $V = = $t A1v1 A2v2 v2 > v1 v2 = v1 v2 < v1 Typesetting math: 100% Correct Thus, by combining the continuity equation and Bernoulli's equation, one can characterize the flow of an ideal fluid.When the cross section of the flow tube decreases, the flow speed increases, and therefore the pressure decreases. In other words, if , then and . Part B As you found out in the previous part, Bernoulli's equation tells us that a fluid element that flows through a flow tube with decreasing cross section moves toward a region of lower pressure. Physically, the pressure drop experienced by the fluid element between points 1 and 2 acts on the fluid element as a net force that causes the fluid to __________. Hint 1. Effects from conservation of mass Recall that, if the cross section of the flow tube varies, the flow speed must change to conserve mass. This means that there is a nonzero net force acting on the fluid that causes the fluid to increase or decrease speed depending on whether the fluid is flowing through a portion of the tube with a smaller or larger cross section. ANSWER: Correct Part C Now assume that point 2 is at height with respect to point 1, as shown in the figure. The ends of the flow tube have the same areas as the ends of the horizontal flow tube shown in Part A. Since the cross section of the flow tube is decreasing, Bernoulli's equation tells us that a fluid element flowing toward point 2 from point 1 moves toward a region of lower pressure. In this case, what is the pressure drop The pressure at point 2 is lower than the pressure at point 1. equal to the pressure at point 1. higher than the pressure at point 1. A2 < A1 v2 > v1 p2 < p1 A v decrease in speed increase in speed remain in equilibrium h Typesetting math: 100% experienced by the fluid element? Hint 1. How to approach the problem Apply Bernoulli's equation to point 1 and to point 2, as you did in Part A. Note that this time you must take into account the difference in elevation between points 1 and 2. Do you need to add this additional term to the other term representing the pressure drop between the two ends of the flow tube or do you subtract it? ANSWER: Correct Part D From a physical point of view, how do you explain the fact that the pressure drop at the ends of the elevated flow tube from Part C is larger than the pressure drop occurring in the similar but purely horizontal flow from Part A? The pressure drop is smaller than the pressure drop occurring in a purely horizontal flow. equal to the pressure drop occurring in a purely horizontal flow. larger than the pressure drop occurring in a purely horizontal flow. Typesetting math: 100% Hint 1. Physical meaning of the pressure drop in a tube As explained in the introduction, the difference in pressure between the ends of a flow tube represents the total work done on a unit volume of fluid by the pressure forces of the surrounding fluid to move that volume of fluid from one end to the other end of the flow tube. ANSWER: Correct In the case of purely horizontal flow, the difference in pressure between the two ends of the flow tube had to balance only the increase in kinetic energy resulting from the acceleration of the fluid. In an elevated flow tube, the difference in pressure must also balance the increase in potential energy of the fluid; therefore a higher pressure is needed for the flow to occur. Venturi Meter with Two Tubes A pair of vertical, open-ended glass tubes inserted into a horizontal pipe are often used together to measure flow velocity in the pipe, a configuration called a Venturi meter. Consider such an arrangement with a horizontal pipe carrying fluid of density . The fluid rises to heights and in the two open-ended tubes (see figure). The cross-sectional area of the pipe is at the position of tube 1, and at the position of tube 2. p1 − p2 A greater amount of work is needed to balance the increase in potential energy from the elevation change. decrease in potential energy from the elevation change. larger increase in kinetic energy. larger decrease in kinetic energy. h1 h2 A1 A2 Typesetting math: 100% Part A Find , the gauge pressure at the bottom of tube 1. (Gauge pressure is the pressure in excess of outside atmospheric pressure.) Express your answer in terms of quantities given in the problem introduction and , the magnitude of the acceleration due to gravity. Hint 1. How to approach the problem Use Bernoulli's law to compute the difference in pressure between the top and bottom of tube 1. The pressure at the top of the tube is defined to be atmospheric pressure. Note: Inside the tube, since the fluid is not flowing, the terms involving velocity in Bernoulli's equation can be ignored. Thus, Bernoulli's equation reduces to the formula for pressure as a function of depth in a fluid of uniform density. Hint 2. Simplified Bernoulli's equation For a fluid of uniform density that is not flowing, the pressure at a depth below the surface is given by , where is the pressure at the surface and is the magnitude of the acceleration due to gravity. ANSWER: Correct The fluid is pushed up tube 1 by the pressure of the fluid at the base of the tube, and not by its kinetic energy, since there is no streamline around the sharp edge of the tube. Thus energy is not conserved (there is turbulence at the edge of the tube) at the entrance of the tube. Since Bernoulli's law is essentially a statement of energy conservation, it must be applied separately to the fluid in the tube and the fluid flowing in the main pipe. However, the pressure in the fluid is the same just inside and just outside the tube. Part B Find , the speed of the fluid in the left end of the main pipe. Express your answer in terms of , , , and either and or , which is equal to . p1 g p h p = p0 + gh p0 g p1 = gh1 v1 h1 h2 g A1 A2 A1 A2 Typesetting math: 100% Hint 1. How to approach the problem Energy is conserved along the streamlines in the main flow. This means that Bernoulli's law can be applied to obtain a relationship between the fluid pressure and velocity at the bottom of tube 1, and the fluid pressure and velocity at the bottom of tube 2. Hint 2. Find in terms of What is , the pressure at the bottom of tube 2? Express your answer in terms of , , and any other given quantities. Hint 1. Recall Part A Obtain the solution for this part in the same way that you found an expression for in terms of in Part A of this problem. ANSWER: Hint 3. Find in terms of given quantities Find , the fluid pressure at the bottom of tube 2. Express your answer in terms of , , , , and . Hint 1. Find the pressure at the bottom of tube 2 Find , the fluid pressure at the bottom of tube 2. Express your answer in terms of , , and . ANSWER: p2 h2 p2 h2 g p1 h1 p2 = gh2 p2 p2 p1 v1 A1 A2 p2 p1 v1 v2 p2 = p1 + ( − ) 1 2 v1 2 v2 2 Typesetting math: 100% Hint 2. Find in terms of The fluid is incompressible, so you can use the continuity equation to relate the fluid velocities and in terms of the geometry of the pipe. Find , the fluid velocity at the bottom of tube 2, in terms of . Your answer may include and , the cross-sectional areas of the pipe. ANSWER: ANSWER: ANSWER: Correct Note that this result depends on the difference between the heights of the fluid in the tubes, a quantity that is more easily measured than the heights themselves. Problem 15.39 The container shown in the figure is filled with oil. It is open to the atmosphere on the left. v2 v1 v1 v2 v2 v1 A1 A2 v2 = A1 A2 v1 p2 = p1 + ( )(1 − ) 1 2 v1 2 ( ) A1 A2 2 v1 = 2g h1−h2 ( ) −1 A1 A2 2 −−−−−−−−−−−−−− Typesetting math: 100% Part A What is the pressure at point A? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part B What is the pressure difference between points A and B? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct PA = 106 kPa PB − PA = 4.4 kPa Typesetting math: 100% Part C What is the pressure difference between points A and C? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 15.48 You need to determine the density of a ceramic statue. If you suspend it from a spring scale, the scale reads 32.4 . If you then lower the statue into a tub of water, so that it is completely submerged, the scale reads 17 . Part A What is the density? Express your answer with the appropriate units. ANSWER: Correct Problem 15.60 Water flows from the pipe shown in the figure with a speed of 2.0 . PC − PA = 4.4 kPa N N statue = 2100 kg m3 m/s Typesetting math: 100% Part A What is the water pressure as it exits into the air? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the height of the standing column of water? Express your answer to two significant figures and include the appropriate units. ANSWER: Incorrect; Try Again P = 1.0×105 Pa h h = Typesetting math: 100% Score Summary: Your score on this assignment is 83.9%. You received 78.84 out of a possible total of 94 points. Typesetting math: 100%

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## What is the prime purpose of selecting a composite material over material from the other family groups? MODULE 3 – STRUCTURE OF SOLID MATERIALS The ability of a material to exist in different space lattices is called a. Allotropic b. Crystalline c. Solvent d. Amorphous Amorphous metals develop their microstructure as a result of ___________. a. Dendrites b. Directional solidification c. Slip d. Extremely rapid cooling In an alloy, the material that dissolves the alloying element is the ___________. a. Solute b. Solvent c. Matrix d. Allotrope What is the coordination number (CN) for the fcc structure formed by ions of sodium and chlorine that is in the chemical compound NaCl (salt) ? a. 6 b. 8 c. 14 d. 16 What pressure is normally used in constructing a phase diagram? a. 100 psi b. Depends on material c. Ambient d. Normal atmospheric pressure What line on a binary diagram indicates the upper limit of the solid solution phase? a. Liquidus b. Eutectic c. Eutectoid d. Solidus What holds the atoms (ions) together in a compound such as NaCl are electrostatic forces between ___________. a. Atom and ion b. Covalent bonds c. Electrons and nuclei d. Neutrons Diffusion of atoms through a solid takes place by two main mechanisms. One is diffusion through vacancies in the atomic structure. Another method of diffusion is ___________. a. Cold b. APF c. Substitutional d. Interstitial Give a brief explanation of the Lever rule (P117) Grain boundaries ___________ movement of dislocations through a solid. a. Improve b. Inhibit c. Do not affect Iron can be alloyed with carbon because it is ___________. a. Crystalline b. Amorphous c. A mixture d. Allotropic Metals can be cooled only to crystalline solids. a. T (true) b. F (false) Sketch an fcc unit cell. Metals are classified as crystalline materials. Name one metal that is an amorphous solid and name at least one recent application in which its use is saving energy or providing greater strength and/or corrosion resistance. MODULE 4 – MECHANICAL PROPERTIES Give two examples of a mechanical property. a. Thermal resistance b. Wear resistance c. Hardness d. Strength Scissors used in the home cut material by concentrating forces that ultimately produce a certain type of stress within the material. Identify this stress. a. Bearing stress b. Shearing stress c. Compressive stress An aluminum rod 1 in. in diameter (E =10.4 x 106psi) experiences an elastic tensile strain of 0.0048 in./in. Calculate the stress in the rod. a. 49,920 ksi b. 49,920 psi c. 49,920 msi A 1-in.-diameter steel circular rod is subject to a tensile load that reduces its cross-sectional area to 0.64 in2. Express the rod’s ductility using a standard unit of measure. a. 18.5% b. 1.85% c. 18.5 d. (a) and (c) What term is used to describe the low-temperature creep of polymerics? a. Springback b. Creep rupture c. Cold flow d. Creep forming MODULE 7 – TESTING, FAILURE ANALYSIS, STANDARDS, & INSPECTION Factors of safety are defined either in terms of the ultimate strength of a material or its yield strength. In other words, by the use of a suitable factor, the ultimate or yield strength is reduced in size to what is known as the design stress or safe working stress. Which factor of safety would be more appropriate for a material that will be subjected to repetitious, suddenly applied loads? Product liability court cases have risen sharply in recent years because of poor procedures in selecting materials for particular applications. Assuming that a knowledge of a material’s properties is a valid step in the selection process, cite two examples where such lack of knowledge could or did lead to failure or unsatisfactory performance. Make a sketch and fully dimension an Izod impact test specimen. Which agency publishes the Annual Book of standard test methods used worldwide for evaluation of materials? a. NASA b. NIST c. ASTM d. SPE

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## Assignment 11 Due: 11:59pm on Wednesday, April 30, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 13.2 The gravitational force of a star on orbiting planet 1 is . Planet 2, which is twice as massive as planet 1 and orbits at twice the distance from the star, experiences gravitational force . Part A What is the ratio ? ANSWER: Correct Conceptual Question 13.3 A 1500 satellite and a 2200 satellite follow exactly the same orbit around the earth. Part A What is the ratio of the force on the first satellite to that on the second satellite? ANSWER: Correct F1 F2 F1 F2 = 2 F1 F2 kg kg F1 F2 = 0.682 F1 F2 Part B What is the ratio of the acceleration of the first satellite to that of the second satellite? ANSWER: Correct Problem 13.2 The centers of a 15.0 lead ball and a 90.0 lead ball are separated by 9.00 . Part A What gravitational force does each exert on the other? Express your answer with the appropriate units. ANSWER: Correct Part B What is the ratio of this gravitational force to the weight of the 90.0 ball? ANSWER: a1 a2 = 1 a1 a2 kg g cm 1.11×10−8 N g 1.26×10−8 Typesetting math: 100% Correct Problem 13.6 The space shuttle orbits 310 above the surface of the earth. Part A What is the gravitational force on a 7.5 sphere inside the space shuttle? Express your answer with the appropriate units. ANSWER: Correct ± A Satellite in Orbit A satellite used in a cellular telephone network has a mass of 2310 and is in a circular orbit at a height of 650 above the surface of the earth. Part A What is the gravitational force on the satellite? Take the gravitational constant to be = 6.67×10−11 , the mass of the earth to be = 5.97×1024 , and the radius of the Earth to be = 6.38×106 . Express your answer in newtons. Hint 1. How to approach the problem Use the equation for the law of gravitation to calculate the force on the satellite. Be careful about the units when performing the calculations. km kg Fe on s = 67.0 N kg km Fgrav G N m2/kg2 me kg re m Typesetting math: 100% Hint 2. Law of gravitation According to Newton’s law of gravitation, , where is the gravitational constant, and are the masses of the two objects, and is the distance between the centers of mass of the two objects. Hint 3. Calculate the distance between the centers of mass What is the distance from the center of mass of the satellite to the center of mass of the earth? Express your answer in meters. ANSWER: ANSWER: Correct Part B What fraction is this of the satellite’s weight at the surface of the earth? Take the free-fall acceleration at the surface of the earth to be = 9.80 . Hint 1. How to approach the problem All you need to do is to take the ratio of the gravitational force on the satellite to the weight of the satellite at ground level. There are two ways to do this, depending on how you define the force of gravity at the surface of the earth. ANSWER: F = Gm1m2/r2 G m1 m2 r r = 7.03×10r 6 m = 1.86×10Fgrav 4 N g m/s2 0.824 Typesetting math: 100% Correct Although it is easy to find the weight of the satellite using the constant acceleration due to gravity, it is instructional to consider the weight calculated using the law of gravitation: . Dividing the gravitational force on the satellite by , we find that the ratio of the forces due to the earth’s gravity is simply the square of the ratio of the earth’s radius to the sum of the earth’s radius and the height of the orbit of the satellite above the earth, . This will also be the fraction of the weight of, say, an astronaut in an orbit at the same altitude. Notice that an astronaut’s weight is never zero. When people speak of “weightlessness” in space, what they really mean is “free fall.” Problem 13.8 Part A What is the free-fall acceleration at the surface of the moon? Express your answer with the appropriate units. ANSWER: Correct Part B What is the free-fall acceleration at the surface of the Jupiter? Express your answer with the appropriate units. ANSWER: Correct w = G m/ me r2e Fgrav = Gmem/(re + h)2 w [re/(re + h)]2 gmoon = 1.62 m s2 gJupiter = 25.9 m s2 Typesetting math: 100% Enhanced EOC: Problem 13.14 A rocket is launched straight up from the earth’s surface at a speed of 1.90×104 . You may want to review ( pages 362 – 365) . For help with math skills, you may want to review: Mathematical Expressions Involving Squares Part A What is its speed when it is very far away from the earth? Express your answer with the appropriate units. Hint 1. How to approach the problem What is conserved in this problem? What is the rocket’s initial kinetic energy in terms of its unknown mass, ? What is the rocket’s initial gravitational potential energy in terms of its unknown mass, ? When the rocket is very far away from the Earth, what is its gravitational potential energy? Using conservation of energy, what is the rocket’s kinetic energy when it is very far away from the Earth? Therefore, what is the rocket’s velocity when it is very far away from the Earth? ANSWER: Correct Problem 13.13 Part A m/s m m 1.54×104 ms Typesetting math: 100% What is the escape speed from Venus? Express your answer with the appropriate units. ANSWER: Correct Problem 13.17 The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 4.2 earth years. Part A What is the asteroid’s orbital radius? Express your answer with the appropriate units. ANSWER: Correct Part B What is the asteroid’s orbital speed? Express your answer with the appropriate units. ANSWER: vescape = 10.4 km s = 3.89×1011 R m = 1.85×104 v ms Typesetting math: 100% Correct Problem 13.32 Part A At what height above the earth is the acceleration due to gravity 15.0% of its value at the surface? Express your answer with the appropriate units. ANSWER: Correct Part B What is the speed of a satellite orbiting at that height? Express your answer with the appropriate units. ANSWER: Correct Problem 13.36 Two meteoroids are heading for earth. Their speeds as they cross the moon’s orbit are 2 . 1.01×107 m 4920 ms km/s Typesetting math: 100% Part A The first meteoroid is heading straight for earth. What is its speed of impact? Express your answer with the appropriate units. ANSWER: Correct Part B The second misses the earth by 5500 . What is its speed at its closest point? Express your answer with the appropriate units. ANSWER: Incorrect; Try Again Problem 14.2 An air-track glider attached to a spring oscillates between the 11.0 mark and the 67.0 mark on the track. The glider completes 11.0 oscillations in 32.0 . Part A What is the period of the oscillations? Express your answer with the appropriate units. v1 = 11.3 km s km v2 = cm cm s Typesetting math: 100% ANSWER: Correct Part B What is the frequency of the oscillations? Express your answer with the appropriate units. ANSWER: Correct Part C What is the angular frequency of the oscillations? Express your answer with the appropriate units. ANSWER: Correct Part D What is the amplitude? Express your answer with the appropriate units. 2.91 s 0.344 Hz 2.16 rad s Typesetting math: 100% ANSWER: Correct Part E What is the maximum speed of the glider? Express your answer with the appropriate units. ANSWER: Correct Good Vibes: Introduction to Oscillations Learning Goal: To learn the basic terminology and relationships among the main characteristics of simple harmonic motion. Motion that repeats itself over and over is called periodic motion. There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth. The last example differs from the first two, in that it represents a special kind of periodic motion called simple harmonic motion. The conditions that lead to simple harmonic motion are as follows: There must be a position of stable equilibrium. There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object’s displacement from its equilibrium position. Mathematically, the restoring force is given by , where is the displacement from equilibrium and is a constant that depends on the properties of the oscillating system. The resistive forces in the system must be reasonably small. In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them. Consider a block of mass attached to a spring with force constant , as shown in the figure. The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at . If the 28.0 cm 60.5 cms F F = −kx x k m k x = 0 Typesetting math: 100% block is pulled to the right a distance and then released, will be the amplitude of the resulting oscillations. Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block. Part A After the block is released from , it will ANSWER: Correct As the block begins its motion to the left, it accelerates. Although the restoring force decreases as the block approaches equilibrium, it still pulls the block to the left, so by the time the equilibrium position is reached, the block has gained some speed. It will, therefore, pass the equilibrium position and keep moving, compressing the spring. The spring will now be pushing the block to the right, and the block will slow down, temporarily coming to rest at . After is reached, the block will begin its motion to the right, pushed by the spring. The block will pass the equilibrium position and continue until it reaches , completing one cycle of motion. The motion will then repeat; if, as we’ve assumed, there is no friction, the motion will repeat indefinitely. The time it takes the block to complete one cycle is called the period. Usually, the period is denoted and is measured in seconds. The frequency, denoted , is the number of cycles that are completed per unit of time: . In SI units, is measured in inverse seconds, or hertz ( ). A A x = A remain at rest. move to the left until it reaches equilibrium and stop there. move to the left until it reaches and stop there. move to the left until it reaches and then begin to move to the right. x = −A x = −A x = −A x = −A x = A T f f = 1/T f Hz Typesetting math: 100% Part B If the period is doubled, the frequency is ANSWER: Correct Part C An oscillating object takes 0.10 to complete one cycle; that is, its period is 0.10 . What is its frequency ? Express your answer in hertz. ANSWER: Correct unchanged. doubled. halved. s s f f = 10 Hz Typesetting math: 100% Part D If the frequency is 40 , what is the period ? Express your answer in seconds. ANSWER: Correct The following questions refer to the figure that graphically depicts the oscillations of the block on the spring. Note that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents time. Part E Which points on the x axis are located a distance from the equilibrium position? ANSWER: Hz T T = 0.025 s A Typesetting math: 100% Correct Part F Suppose that the period is . Which of the following points on the t axis are separated by the time interval ? ANSWER: Correct Now assume for the remaining Parts G – J, that the x coordinate of point R is 0.12 and the t coordinate of point K is 0.0050 . Part G What is the period ? Express your answer in seconds. Hint 1. How to approach the problem In moving from the point to the point K, what fraction of a full wavelength is covered? Call that fraction . Then you can set . Dividing by the fraction will give the R only Q only both R and Q T T K and L K and M K and P L and N M and P m s T t = 0 a aT = 0.005 s a Typesetting math: 100% period . ANSWER: Correct Part H How much time does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? Express your answer in seconds. ANSWER: Correct Part I What distance does the object cover during one period of oscillation? Express your answer in meters. ANSWER: Correct Part J What distance does the object cover between the moments labeled K and N on the graph? T T = 0.02 s t t = 0.01 s d d = 0.48 m d Typesetting math: 100% Express your answer in meters. ANSWER: Correct Problem 14.4 Part A What is the amplitude of the oscillation shown in the figure? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct d = 0.36 m A = 20.0 cm Typesetting math: 100% Part B What is the frequency of this oscillation? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the phase constant? Express your answer to two significant figures and include the appropriate units. ANSWER: Incorrect; Try Again Problem 14.10 An air-track glider attached to a spring oscillates with a period of 1.50 . At the glider is 4.60 left of the equilibrium position and moving to the right at 33.4 . Part A What is the phase constant? Express your answer to three significant figures and include the appropriate units. ANSWER: f = 0.25 Hz 0 = s t = 0 s cm cm/s Typesetting math: 100% Incorrect; Try Again Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Problem 14.12 A 140 air-track glider is attached to a spring. The glider is pushed in 12.2 and released. A student with a stopwatch finds that 14.0 oscillations take 19.0 . Part A What is the spring constant? Express your answer with the appropriate units. ANSWER: 0 = g cm s Typesetting math: 100% Correct Problem 14.14 The position of a 50 g oscillating mass is given by , where is in s. If necessary, round your answers to three significant figures. Determine: Part A The amplitude. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part B The period. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part C 3.00 Nm x(t) = (2.0 cm)cos(10t − /4) t 2.00 cm 0.628 s Typesetting math: 100% The spring constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Part D The phase constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Incorrect; Try Again Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). Part G Typesetting math: 100% This question will be shown after you complete previous question(s). Part H This question will be shown after you complete previous question(s). Part I This question will be shown after you complete previous question(s). Enhanced EOC: Problem 14.17 A spring with spring constant 16 hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down 4.0 and released. The ball makes 35 oscillations in 18 seconds. You may want to review ( pages 389 – 391) . For help with math skills, you may want to review: Differentiation of Trigonometric Functions Part A What is its the mass of the ball? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the period of oscillation? What is the angular frequency of the oscillations? How is the angular frequency related to the mass and spring constant? What is the mass? N/m cm s Typesetting math: 100% ANSWER: Correct Part B What is its maximum speed? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the amplitude of the oscillations? How is the maximum speed related to the amplitude of the oscillations and the angular frequency? ANSWER: Correct Changing the Period of a Pendulum A simple pendulum consisting of a bob of mass attached to a string of length swings with a period . Part A If the bob’s mass is doubled, approximately what will the pendulum’s new period be? Hint 1. Period of a simple pendulum The period of a simple pendulum of length is given by m = 110 g vmax = 49 cms m L T Typesetting math: 10T0% L , where is the acceleration due to gravity. ANSWER: Correct Part B If the pendulum is brought on the moon where the gravitational acceleration is about , approximately what will its period now be? Hint 1. How to approach the problem Recall the formula of the period of a simple pendulum. Since the gravitational acceleration appears in the denominator, the period must increase when the gravitational acceleration decreases. ANSWER: T = 2 Lg −− g T/2 T &2T 2T g/6 T/6 T/&6 &6T 6T Typesetting math: 100% Correct Part C If the pendulum is taken into the orbiting space station what will happen to the bob? Hint 1. How to approach the problem Recall that the oscillations of a simple pendulum occur when a pendulum bob is raised above its equilibrium position and let go, causing the pendulum bob to fall. The gravitational force acts to bring the bob back to its equilibrium position. In the space station, the earth’s gravity acts on both the station and everything inside it, giving them the same acceleration. These objects are said to be in free fall. ANSWER: Correct In the space station, where all objects undergo the same acceleration due to the earth’s gravity, the tension in the string is zero and the bob does not fall relative to the point to which the string is attached. Problem 14.20 A 175 ball is tied to a string. It is pulled to an angle of 8.0 and released to swing as a pendulum. A student with a stopwatch finds that 15 oscillations take 13 . Part A How long is the string? Express your answer to two significant figures and include the appropriate units. It will continue to oscillate in a vertical plane with the same period. It will no longer oscillate because there is no gravity in space. It will no longer oscillate because both the pendulum and the point to which it is attached are in free fall. It will oscillate much faster with a period that approaches zero. g ( s Typesetting math: 100% ANSWER: Correct Problem 14.22 Part A What is the length of a pendulum whose period on the moon matches the period of a 2.1- -long pendulum on the earth? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 14.42 An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk ( = 0.17 ) driven back and forth in SHM at by an electromagnetic coil. Part A The maximum restoring force that can be applied to the disk without breaking it is 4.4×104 . What is the maximum oscillation amplitude that won’t rupture the disk? Express your answer to two significant figures and include the appropriate units. ANSWER: L = 19 cm m lmoon = 0.35 m m g 1.0 MHz N amax = 6.6 μm Typesetting math: 100% Correct Part B What is the disk’s maximum speed at this amplitude? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 81.4%. You received 117.25 out of a possible total of 144 points. vmax = 41 ms

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## 3. The probability density function for mechanical component is given by: fT(t) = 1/(b-a) when t <=a<=b = 0; elsewhere Determine: • Cumulative distribution of the failures (5 points) • Reliability of the components (5 points) • Hazard rate for the components (5 points) • Mean, standard deviation of the failure distribution and reliability of components at the end of 2 years, when c=0.0025 (5 points) • Plot the probability density function, probability time distribution function, Reliability function and Hard Rate function for the given distribution when a=6000 and b=12000 (5 points)

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## Research and Study Skills Module Assessment 1 Information. Bibliography This is designed to help set you up for the coming academic year/s. The Bibliography will provide you with vital references for every module undertaken during this postgraduate course. Clearly, a thorough Bibliography will help with this ICA, but also with successive coursework and practice. Your Bibliography will be split into 2 sections, you must pick 2 subjects from the 3 listed for your subject area: Civil Engineering: 1. Coastal protection and sea defence methods; 2. Bearing capacity, settlement and ground modification 3. Project planning methods utilised in civil engineering projects. Energy and Environmental Management: 1 Energy generation by geothermal means 2 The physical science base for climate change 3 Life Cycle Assessment (LCA) Project Management 1. IT in Construction Industry 2. Development of Project manager Competency 3. Project Success: Project management Biotechnology 1. Biofuel production using Algae 2. Use of Genetically Modified Organisms and Health and Safety measures 3. Genetic disease and gene-therapy treatment Electronics and Communications 1. Transistors, amplifiers and analogue circuits design 2. Wireless telecommunications 3. Industrial communications protocols Control and Electronics 1. Transistors, amplifiers and analogue circuits design 2. Robust control system design 3. Model predictive control Mechanical Engineering 1. Finite Element Methods 2. Machine Design 3. Applied Continuum Mechanics or Automotive Engineering & Vehicle Design Advanced Manufacturing Systems 1. Finite Element Methods 2. Manufacturing Process Technology or Manufacturing Systems 3. Production Management Petroleum Engineering 1. Enhanced Oil Recovery Methods (screening criteria) 2. Horizontal drilling technology (why and how) 3. Types of oil & gas separators (performance affecting parameters) You must choose two areas for your subject area to seek out suitable reference material for. In the first section, you must critically select between 10 and 15 key references from one of your subject areas and provide a 300 word summary of the content of one of these references. In the second section, you must again select between 10 and 15 references from the second of your subject areas and then produce a 600 word critical comparison of two of these references. In this critical comparison you should be comparing the research conducted in each of your two chosen sources to one another. The Bibliography should be in the Harvard format. Marks for the assessed bibliography will be distributed for the following criteria: 1. Appropriateness of references (25%) 2. Range of sources (25%) 3. Appropriate content of summaries (40%) 4. Format (10%)

Section 1: Transistors, amplifiers and analogue circuits design References: … Read More...

## SUMMARY ASSIGNMENT SHEET Upload to Turnitin.com by due date. Create an account using Class ID: 10423941 and Password: English. Due Dates: • Peer Response Workshop with Rough Draft: Tuesday, September 8th • First Draft Due (submit for feedback): Thursday, September 10th • Final Draft Due (highlighting, labeling & reflection done in class): Thursday, September 17th What is Summary? Summary is a comprehensive and objective restatement of the main ideas of a text (an article, book, movie, event, etc.) and while the act of summarizing might seem like an easy and obvious undertaking there is a noticeable difference between summarizing and summarizing well. For example, many students, without even realizing it, leave out key information when they summarize because they forget to consider how much their reader already knows about the topic or reading. Be careful to explain fully so readers do not have to guess what you mean. In his textbook, A Brief Guide to Writing from Readings, Stephen Wilhoit suggests that to avoid the pitfalls of unclear or disjointed summaries, they should keep in mind the qualities of a good summary. These qualities include: Neutrality – The writer avoids inserting his or her opinion or interpreting the original text’s content in any way. This requires the writer avoid language that is evaluative, such as: good, bad, effective, ineffective, interesting, boring, etc. Also, keep 1st person (I, we, our, us) out of the summary; instead, summary should be written in grammatical 3rd person (For example, he she, the author, they, etc). Brevity – The summary should not be longer than the original text, but rather highlight the most important information from that text while leaving out unnecessary details and still maintaining accuracy. Independence – The summary should make sense to someone who has not read the text. There should be no confusion about the main content and organization of the original source. This also requires that the summary be accurate—that it not misinterpret any part of the original text. Mastering the craft of summarizing puts students in the position to do well on many assignments in college, not just English essays. In most fields, from the humanities to the sciences, summary is a required task. Being able to summarize lab results accurately and briefly, for example, is critical in a chemistry or engineering class. Summarizing the various theories of sociology or education helps a person apply them to his or her fieldwork. In business, summarizing ideas effectively can be useful in numerous scenarios. The assignment: • Your summary will follow guidelines that integrate both informative and explanatory summary components: Informative summary “simply conveys the author’s main ideas, data, [and] arguments” (Wilhoit 62). Explanatory summary maintains the order of the author’s main points, and also includes the author, title and publication information. • The text: “Who are You and What are You Doing Here?” by Mark Edmundson, pgs. 115-27 The basic structure is as follows: • An Introductory Paragraph including: o The title of the essay, the author’s name, and a brief bio o The topic of the essay—what the text is about o The author’s main idea (essentially your thesis statement) • Body Paragraph(s) including: o Topic Sentences with transitions for each main idea the author addresses o Supporting points following the same order as the article o A concluding sentence that expounds upon, or echoes, the main idea • A Work Cited page that o Includes only the source text Evaluation: A successful summary will include all of the following: • Briefly summarizes the main ideas of the text • Comprehensive, accurate and independent summarization of the text • Objective and neutral restatement of the main ideas • Contain no directly quoted sentences • Clearly introduces the author and title • Body paragraphs that reflect the text’s main ideas • Clear and effective transitions • Mostly free of mechanical and grammatical errors • A correct Work Cited page • Formatted with correct MLA standards • Follows the basic structure of a summary Possible Points: • Peer Response Workshop w/ Rough Draft: 10 points • Final Draft: 80 points • Highlighted revisions and labeling: 5 points • Reflection: 5 points _______________________________________________________________ • Total possible points for the Summary Assignment: 100 Points

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## 11. Define mechanical work and provide both an equation and the proper units for this quantity. 12. If the gauge pressure of a tire is 5 atm, what is the total pressure inside the tire? (hint: not 5 atm!) 13. Express the number of seconds in 1 year in scientific notation using units of kilo- and Megaseconds. 14. Express the average diameter of a human hair (Google!) in feet and meters (again, Sci. Notation!). 15. Convert your answers from #14 above into deci-, centi-, milli-, and micrometers. 16. For what functions, y(x), is the relationship dy/dx = Δy/Δx always true? 17. Seperate log(xn/y) into simple log form with no exponents. 18. Differentiate the functions y(x) = 4×3 + 3×2 + 2x + 1, f(x) = ln (x3), and P(r) = 14 e2r + 3. 19. Differentiate the functions y(x) = 3 sin 2x, f(x) = –2 cos x2, and Pr(x) = A sin2 kx. 20. What is the inverse of frequency? What are SI units of frequency and inverse frequency?

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