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Assignment MasteringPhysics: Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentI... [ Print View ] PHCC 141: Physics for Scientists and Engineers I - Fall 2007 6b. Work, Energy, and Power Due at 11:59pm on Monday, October 1, 2007 Hide Grading Details Number of answer attempts per question is: 5 You gain credit for: correctly answering a question in a Part, or correctly answering a question in a Hint. You lose credit for: exhausting all attempts or requesting the answer to a question in a Part or Hint, or incorrectly answering a question in a Part. Late submissions: reduce your score by 100% over each day late. Hints are helpful clues or simpler questions that guide you to the answer. Hints are not available for all questions. There is no penalty for leaving questions in Hints unanswered. Grading of Incorrect Answers For Multiple-Choice or True/False questions, you lose 100%/(# of options - 1) credit per incorrect answer. For any other question, you lose 3% credit per incorrect answer. Integrate Force to Find Work Work Done by a Spring Consider a spring, with spring constant , one end of which is attached to a wall. The spring is initially unstretched, with the . unconstrained end of the spring at position Part A The spring is now compressed so that the unconstrained end moves from , find the work done by the spring as it is compressed. Hint A.1 Spring force as a function of position The spring force vector to . Using the work integral as a function of displacement from the spring's equilibrium position, is given by where is the spring constant and is a unit vector in the direction of the displacement of the spring (in this case, towards the right). Part A.2 Integrand of the work integral The work done by the spring is given by the integral of the dot product of the spring force and an infinitesimal displacement of the end of the spring: , where the infinitesmal displacement vector Express your answer in terms of , , and has been written as . . Write .) in terms of given quantities, and then compute the dot product to find an expression for the integrand. (Note, of 11 10/4/2007 4:06 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentI... Work Energy Problems Work-Energy Scaling A particle of mass moves along a straight line with initial speed . A force of magnitude along the direction of its motion. Part A Find , the final speed of the particle after it has traveled a distance . Part A.1 Find Find the final kinetic energy pushes the particle a distance , the final kinetic energy of the particle after it has been pushed a distance . Part A.1.a Find the final kinetic energy in terms of the work done Assume that the work done by the pushing force is . Find the final kinetic energy, a distance . Express your answer in terms of the particle's initial kinetic energy, , and ANSWER: = . , of the particle after it has been pushed Part A.1.b Find the work done Find , the work done by the force . and . Express your answer in terms of ANSWER: = , , and . Express the final kinetic energy in terms of , ANSWER: = Express the final speed in terms of , ANSWER: = , , and . Increase in mass For the next two parts, assume that the particle's mass is increased to remain the same. Part B By what multiplicative factor does the initial kinetic energy increase, and by what multiplicative factor done by the force increase (with respect to the case when the particle had a mass )? does the work , while all other parameters in the problem introduction If one of the quantities doubles, for instance, it would increase by a factor of 2. If a quantity stays the same, then the multiplicative factor would be 1. Part B.1 Find Express Find the work done in terms of some or all of the variables , , , and . = , the work done by the force . ANSWER: of 11 10/4/2007 4:06 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentI... You should enter the two factors separated by a comma. ANSWER: Part C The particle's change in speed over the distance ANSWER: less than , = 3,1 will be ______ the change in speed when it had a mass equal to . Increase in initial velocity For the final two parts, assume that the initial speed of the particle is increased to to . Part D By what factor does the initial kinetic energy increase (with respect to the first situation, with mass what factor does the work done by the force increase? Again, enter the two factors, separated by a comma. ANSWER: Part E The particle's change in speed over the distance Hint E.1 Some math help Hint not displayed ANSWER: less than , with the particle's mass once again equal and speed ), and by , = 9,1 will be ______ the change in speed when it had an initial velocity equal to . Make sure you understand this result: The amount of energy needed to increase an object's velocity by a certain fixed amount increases the faster the object is already moving. An elegant way to see this is to take the derivative of kinetic energy with respect to velocity: , or . This says that when the velocity of an object increases by an amount , its kinetic energy increases by , so if you have a very fast moving object (large ), it takes a large amount of work or energy to increase its velocity even just a little bit! Holding Force of a Nail A hammer of mass is moving at speed when it strikes a nail of negligible mass that is stuck in a wooden block. The hammer is observed to drive the nail a distance deeper into the block. Part A Find the magnitude of the force that the wooden block exerts on the nail, assuming that this force is independent of the depth of penetration of the nail into the wood. You may also assume that , so that the change in the hammer's gravitational potential energy, as it drives the nail into the block, is insignificant. of 11 10/4/2007 4:06 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentI... Hint A.1 How to approach the problem One way to solve this problem is to use the work-energy theorem. To stop the hammer from moving, the wooden block-nail system must do a certain amount of work on the hammer. One expression for this amount of work involves and the displacement of the hammer. In addition, the work-energy theorem implies that the initial kinetic energy of the hammer plus the work done on the hammer must equal the final kinetic energy of the hammer. This gives another expression for the work done that involves only the change in kinetic energy of the hammer. Equate the two expressions for the work done and solve for . Part A.2 Find the work done in terms of The work-energy theorem connects the work needed to stop the hammer with the change in the hammer's kinetic energy. Find the work done on the hammer by the nail. Don't forget to consider the sign of your answer. Express your answer in terms of ANSWER: Part A.3 What is = Find the change in kinetic energy of the hammer , the change in kinetic energy of the hammer? and . and . Express your answer in terms of ANSWER: = Express the magnitude of the force in terms of , , and . ANSWER: = Part B Now evaluate the magnitude of the holding force of the wooden block on the nail by assuming that the force necessary to pull the nail out is the same as that needed to drive it in, which we just derived. Assume a relatively heavy hammer (about 18 ounces), moving with speed . (If such a hammer were swung this hard upward and released, it would rise 5 m). Take the penetration depth to be 2 cm, which is appropriate for one hit on a relatively heavy construction nail. Express your answer to the nearest pound. (Note: ANSWER: = 281 lb .) Pulling a Block on an Incline with Friction A block of weight sits on an inclined plane as shown. A force of magnitude is applied to pull the block up the incline at constant speed. The coefficient of kinetic friction between the plane and the block is . Part A What is the total work Hint A.1 How to start done on the block by the force of friction as the block moves a distance up the incline? of 11 10/4/2007 4:06 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentI... Part B What is the total work done on the block by the applied force as the block moves a distance up the incline? Express your answer in terms of any or all of the variables , , , , , and . ANSWER: = Now the applied force is changed so that instead of pulling the block up the incline, the force pulls the block down the incline at a constant speed. Part C What is the total work done on the block by the force of friction as the block moves a distance down the incline? Express your answer in terms of any or all of the variables , , , , , and . ANSWER: = Part D What is the total work done on the box by the appled force in this case? Express your answer in terms of any or all of the variables , , , , , and . ANSWER: = Power Power Dissipation Puts a Drag on Racing The dominant form of drag experienced by vehicles (bikes, cars, planes, etc.) at operating speeds is called form drag. It increases quadratically with velocity (essentially because the amount of air you run into increases with and so does the amount of force you must exert on each small volume of air). Thus , where is the cross-sectional area of the vehicle and is called the coefficient of drag. Part A Consider a vehicle moving with constant velocity . Find the power dissipated by form drag. Hint A.1 How to approach the problem Because the velocity of the car is constant, the drag force is also constant. Therefore, you can use the result that the power provided by a constant force to an object moving with constant velocity is . Be careful to consider the relative direction of the force drag and the velocity. Express your answer in terms of , , and speed . of 11 10/4/2007 4:06 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentI... ANSWER: Part B = A certain car has an engine that provides a maximum power . Suppose that the maximum speed of the car, , is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power is 10 percent greater than the original power ( . Assume the following: The top speed is limited by air drag. The magnitude of the force of air drag at these speeds is proportional to the square of the speed. By what percentage, Part B.1 , is the top speed of the car increased? Find the relationship between speed and power If the magnitude of the air-drag force is proportional to the square of the car's speed, how is the power delivered, , related to the speed ? ANSWER: Part B.2 How is the algebra done? The relationship between the new power and the old power is . The relationship between the new top speed and the old top speed can be written as , where is the percent change in top speed. Finally, power is related to maximum speed by the formula . What is in terms of ? Hint B.2.a Help with some math Starting with the relationship , substitute in the expressions for and in terms of and : . Then, divide this last expression by the relationship . This is a general approach to scaling problems. The advantage is that the unknown constant of proportionality (in this case ) divides out. Express in terms of , the power relationship between ANSWER: = and . Express the percent increase in top speed numerically to two significant figures. ANSWER: = 3.2 % You'll note that your answer is very close to one-third of the percentage by which the power was increased. This dependence of small changes on each other, when the quantities are related by proportionalities of exponents, is common in physics and often makes a useful shortcut for estimations. of 11 10/4/2007 4:06 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentI... Problem 6.28 To stretch a spring a distance 2.97 Part A How much work must be done to compress this spring a distance 4.06 ANSWER: 21.9 J from its unstretched length? from its unstretched length, an amount of work of 11.7 must be done. Problem 6.34 Leg Presses. As part of your daily workout, you lie on your back and push with your feet against a platform attached to two stiff springs arranged side by side so that they are parallel to each other. When you push the platform, you compress the springs. You do an amount of work of 82.0 when you compress the springs a distance of 0.190 from their uncompressed length. Part A What magnitude of force must you apply to hold the platform in this position? ANSWER: Part B How much additional work must you do to move the platform a distance 0.190 ANSWER: Part C What maximum force must you apply to move the platform a distance 0.190 ANSWER: 1730 N farther? 246 J farther? 863 N Problem 6.43 Part A How many joules of energy does a 100-watt light bulb use per hour? ANSWER: Part B How fast would a 75.0 ANSWER: 98.0 m/s person have to run to have that amount of energy? 3.60105 J Problem 6.47 When its engine of power 75.0 rate of 2.60 . Part A is generating full power, a small single-engine airplane with mass 690 gains altitude at a of 11 10/4/2007 4:06 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentI... What fraction of the engine power is being used to make the airplane climb? (The remainder is used to overcome the effects of air resistance and of inefficiencies in the propeller and engine.) Take the free fall acceleration to be = 9.80 ANSWER: 23.4 % . Express your answer as a percentage Problem 6.52 A ski tow operates on a slope of angle 14.9 of length 280 . The rope moves at a speed of 13.0 54.0 riders at one time, with an average mass per rider of 65.0 . Part A Estimate the power required to operate the tow. ANSWER: 3.19104 W and provides power for Problem 6.67 An object that can move along the x-axis is attracted toward the origin with a force of magnitude Part A What is the force F when the object is at ANSWER: Part B What is the force F when the object is at ANSWER: Part C How much work is done by the force F when the object moves from ANSWER: -17.2 J = 1.00 to = 2.01 ? 36.5 N = 2.01 ? 4.50 N = 1.00 ? 4.50 . The next STP and EOC problems are OPTIONAL (no credit) "Challenge Problems" The 1st EOC problem may take longer to do, the second is more complicated (harder). Dragging a Board A uniform board of length and mass lies near a boundary that separates two regions. In region 1, the coefficient of kinetic friction between the board and the surface is , and in region 2, the coefficient is . The positive direction is shown in the figure. of 11 10/4/2007 4:06 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentI... Part A Find the net work constant velocity. Part A.1 done by friction in pulling the board directly from region 1 to region 2. Assume that the board moves at The net force of friction Suppose that the right edge of the board is a distance from the boundary, as shown. When the board is at this position, what is the magnitude of the force of friction, , acting on the board (assuming that it's moving)? Part A.1.a Fraction of board in region 2 Consider the part of the board in region 2 when the right edge of the board is a distance from the boundary. The magnitude of the force of friction acting on the board (only considering the friction from region 2) will be the coefficient of friction, multiplied by the magnitude of the normal force that acts on the board. Since the ground is horizontal, and the board is not accelerating in the vertical direction, the normal force should equal the board's weight. But, only a fraction of the board's total mass is in region 2. Find the fraction of the board in region 2 in terms of the given lengths; . ANSWER: Fraction of board in region 2 = Part A.1.b Force of friction in region 1 Now consider that part of the board in region 1. Again, only a fraction of the board's mass is in region 1. Using this fact, find the magnitude of the force of friction acting on the board, just due to friction in region 1. Part A.1.b.i Fraction of the board in region 1 When the right edge of the board is a distance from the boundary, what fraction of the board lies in region 1? . ANSWER: Fraction of board in region 1 = , , , , and . Express your answer in terms of ANSWER: = Express the force acting on the board in terms of , , , , , and . ANSWER: = Part A.2 Work as integral of force After you find the net force of friction that acts on the board, as a function of , to find the net work done by this force, you will need to perform the appropriate work integral, The lower limit of this integral will be at . What will be the upper limit? of 11 10/4/2007 4:06 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentI... Part B What is the total work done by the external force in pulling the board from region 1 to region 2? (Again, assume that the board moves at constant velocity.) Hint B.1 No acceleration Since the board is not accelerating, the sum of the external forces on it must be zero. Therefore the external force must be oppositely directed to that of friction. Express your answer in terms of ANSWER: = , , , , and . Problem 6.70 Proton Bombardment. A proton with mass is propelled at an initial speed of nucleus 5.00 m away. The proton is repelled by the uranium nucleus with a force of magnitude between the two objects and . Assume that the uranium nucleus remains at rest. Part A What is the speed of the proton when it is ANSWER: Part B As the proton approaches the uranium nucleus, the repulsive force slows down the proton until it comes momentarily to rest, after which the proton moves away from the uranium nucleus. How close to the uranium nucleus does the proton get? ANSWER: Part C What is the speed of the proton when it is again 5.00 m away from the uranium nucleus? ANSWER: 3.00105 m/s 2.821010 m 2.41105 m/s from the uranium nucleus? directly toward a uranium , where is the separation Problem 6.102 An airplane in flight is subject to an air resistance force proportional to the square of its speed . But there is an additional resistive force because the airplane has wings. Air flowing over the wings is pushed down and slightly forward, so from Newton's third law the air exerts a force on the wings and airplane that is up and slightly backward . The upward force is the lift force that keeps the airplane aloft, and the backward force is called induced drag. At flying speeds, induced drag is inversely proportional to , so that the total air resistance force can be expressed by , where and are positive constants that depend on the shape and size of the airplane and the density of the air. To simulate a Cessna 150, a small single-engine airplane, use and . In steady flight, the engine must provide a forward force that exactly balances the air resistance force. Part A Calculate the speed at which this airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel. 0 of 11 10/4/2007 4:06 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentI... Part B Calculate the speed for which the airplane will have the maximum endurance (that is, will remain in the air the longest time). ANSWER: Summary 14 of 14 items complete (11.44% avg. score) 16.02 of 110 points 1 of 11 10/4/2007 4:06 PM ... View Full Document