## Chapter 11 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Understanding Work and Kinetic Energy Learning Goal: To learn about the Work-Energy Theorem and its basic applications. In this problem, you will learn about the relationship between the work done on an object and the kinetic energy of that object. The kinetic energy of an object of mass moving at a speed is defined as . It seems reasonable to say that the speed of an object–and, therefore, its kinetic energy–can be changed by performing work on the object. In this problem, we will explore the mathematical relationship between the work done on an object and the change in the kinetic energy of that object. First, let us consider a sled of mass being pulled by a constant, horizontal force of magnitude along a rough, horizontal surface. The sled is speeding up. Part A How many forces are acting on the sled? ANSWER: Part B This question will be shown after you complete previous question(s). Part C K m v K = (1/2)mv2 m F one two three four This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). 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 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 Typesetting math: 91% This question will be shown after you complete previous question(s). Part J This question will be shown after you complete previous question(s). Part K This question will be shown after you complete previous question(s). Work-Energy Theorem Reviewed Learning Goal: Review the work-energy theorem and apply it to a simple problem. If you push a particle of mass in the direction in which it is already moving, you expect the particle’s speed to increase. If you push with a constant force , then the particle will accelerate with acceleration (from Newton’s 2nd law). Part A Enter a one- or two-word answer that correctly completes the following statement. If the constant force is applied for a fixed interval of time , then the _____ of the particle will increase by an amount . You did not open hints for this part. ANSWER: M F a = F/M t at Typesetting math: 91% Part B Enter a one- or two-word answer that correctly completes the following statement. If the constant force is applied over a given distance , along the path of the particle, then the _____ of the particle will increase by . ANSWER: Part C If the initial kinetic energy of the particle is , and its final kinetic energy is , express in terms of and the work done on the particle. ANSWER: Part D In general, the work done by a force is written as . Now, consider whether the following statements are true or false: The dot product assures that the integrand is always nonnegative. The dot product indicates that only the component of the force perpendicular to the path contributes to the integral. The dot product indicates that only the component of the force parallel to the path contributes to the integral. Enter t for true or f for false for each statement. Separate your responses with commas (e.g., t,f,t). ANSWER: D FD Ki Kf Kf Ki W Kf = F W = ( ) d f i F r r Typesetting math: 91% Part E Assume that the particle has initial speed . Find its final kinetic energy in terms of , , , and . You did not open hints for this part. ANSWER: Part F What is the final speed of the particle? Express your answer in terms of and . ANSWER: ± The Work Done in Pulling a Supertanker Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 2.20×106 , one at an angle 10.0 west of north, and the other at an angle 10.0 east of north, as they pull the tanker a distance 0.660 toward the north. Part A What is the total work done by the two tugboats on the supertanker? Express your answer in joules, to three significant figures. vi Kf vi M F D Kf = Kf M vf = N km Typesetting math: 91% You did not open hints for this part. ANSWER: Energy Required to Lift a Heavy Box As you are trying to move a heavy box of mass , you realize that it is too heavy for you to lift by yourself. There is no one around to help, so you attach an ideal pulley to the box and a massless rope to the ceiling, which you wrap around the pulley. You pull up on the rope to lift the box. Use for the magnitude of the acceleration due to gravity and neglect friction forces. Part A Once you have pulled hard enough to start the box moving upward, what is the magnitude of the upward force you must apply to the rope to start raising the box with constant velocity? Express the magnitude of the force in terms of , the mass of the box. J m g F m Typesetting math: 91% You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). 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 F = mg F μ Typesetting math: 91% What is the total work done on the block by the force of friction as the block moves a distance up the incline? Express the work done by friction in terms of any or all of the variables , , , , , and . You did not open hints for this part. ANSWER: 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. Wfric L μ m g L F Wfric = WF F L μ m g L F WF = Typesetting math: 91% 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: When Push Comes to Shove Two forces, of magnitudes = 75.0 and = 25.0 , act in opposite directions on a block, which sits atop a frictionless surface, as shown in the figure. Initially, the center of the block is at position = -1.00 . At some later time, the block has moved to the right, and its center is at a new position, = 1.00 . Wfric L μ m g L F Wfric = WF μ m g L F WF = F1 N F2 N xi cm xf cm Typesetting math: 91% Part A Find the work done on the block by the force of magnitude = 75.0 as the block moves from = -1.00 to = 1.00 . Express your answer numerically, in joules. You did not open hints for this part. ANSWER: Part B Find the work done by the force of magnitude = 25.0 as the block moves from = -1.00 to = 1.00 . Express your answer numerically, in joules. You did not open hints for this part. ANSWER: W1 F1 N xi cm xf cm W1 = J W2 F2 N xi cm xf cm Typesetting math: 91% Part C What is the net work done on the block by the two forces? Express your answer numerically, in joules. ANSWER: Part D Determine the change in the kinetic energy of the block as it moves from = -1.00 to = 1.00 . Express your answer numerically, in joules. You did not open hints for this part. ANSWER: Work from a Constant Force Learning Goal: W2 = J Wnet Wnet = J Kf − Ki xi cm xf cm Kf − Ki = J Typesetting math: 91% To understand how to compute the work done by a constant force acting on a particle that moves in a straight line. In this problem, you will calculate the work done by a constant force. A force is considered constant if is independent of . This is the most frequently encountered situation in elementary Newtonian mechanics. Part A Consider a particle moving in a straight line from initial point B to final point A, acted upon by a constant force . The force (think of it as a field, having a magnitude and direction at every position ) is indicated by a series of identical vectors pointing to the left, parallel to the horizontal axis. The vectors are all identical only because the force is constant along the path. The magnitude of the force is , and the displacement vector from point B to point A is (of magnitude , making and angle (radians) with the positive x axis). Find , the work that the force performs on the particle as it moves from point B to point A. Express the work in terms of , , and . Remember to use radians, not degrees, for any angles that appear in your answer. You did not open hints for this part. ANSWER: Part B Now consider the same force acting on a particle that travels from point A to point B. The displacement vector now points in the opposite direction as it did in Part A. Find the work done by in this case. Express your answer in terms of , , and . F( r) r F r F L L WBA F L F WBA = F L WAB F Typesetting math: 91% L F You did not open hints for this part. ANSWER: ± Vector Dot Product Let vectors , , and . Calculate the following: Part A You did not open hints for this part. ANSWER: WAB = A = (2, 1,−4) B = (−3, 0, 1) C = (−1,−1, 2) Typesetting math: 91% Part B What is the angle between and ? Express your answer using one significant figure. You did not open hints for this part. ANSWER: Part C ANSWER: Part D ANSWER: A B = AB A B AB = radians 2B 3C = Typesetting math: 91% Part E Which of the following can be computed? You did not open hints for this part. ANSWER: and are different vectors with lengths and respectively. Find the following: Part F Express your answer in terms of You did not open hints for this part. ANSWER: 2(B 3C) = A B C A (B C) A (B + C) 3 A V 1 V 2 V1 V2 V1 Typesetting math: 91% Part G If and are perpendicular, You did not open hints for this part. ANSWER: Part H If and are parallel, Express your answer in terms of and . You did not open hints for this part. ANSWER: ± Tactics Box 11.1 Calculating the Work Done by a Constant Force V = 1 V 1 V 1 V 2 V = 1 V 2 V 1 V 2 V1 V2 V = 1 V 2 Typesetting math: 91% Learning Goal: To practice Tactics Box 11.1 Calculating the Work Done by a Constant Force. Recall that the work done by a constant force at an angle to the displacement is . The vector magnitudes and are always positive, so the sign of is determined entirely by the angle between the force and the displacement. W F d W = Fd cos F d W Typesetting math: 91% TACTICS BOX 11.1 Calculating the work done by a constant force Force and displacement Work Sign of Energy transfer Energy is transferred into the system. The particle speeds up. increases. No energy is transferred. Speed and are constant. Energy is transferred out of the system. The particle slows down. decreases. A box has weight of magnitude = 2.00 accelerates down a rough plane that is inclined at an angle = 30.0 above the horizontal, as shown at left. The normal force acting on the box has a magnitude = 1.732 , the coefficient of kinetic friction between the box and the plane is = 0.300, and the displacement of the box is 1.80 down the inclined plane. W W 0 F(“r) + K < 90 F("r) cos + 90 0 0 K > 90 F(“r) cos − K 180 −F(“r) − FG N n N μk d m Typesetting math: 91% Part A What is the work done on the box by gravity? Express your answers in joules to two significant figures. You did not open hints for this part. ANSWER: 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). Wgrav Wgrav = J Typesetting math: 91% Work and Potential Energy on a Sliding Block with Friction A block of weight sits on a plane inclined at an angle as shown. The coefficient of kinetic friction between the plane and the block is . A force is applied to push the block up the incline at constant speed. Part A What is the work done on the block by the force of friction as the block moves a distance up the incline? Express your answer in terms of some or all of the following: , , , . You did not open hints for this part. ANSWER: w μ F Wf L μ w L Wf = Typesetting math: 91% Part B What is the work done by the applied force of magnitude ? Express your answer in terms of some or all of the following: , , , . ANSWER: Part C What is the change in the potential energy of the block, , after it has been pushed a distance up the incline? Express your answer in terms of some or all of the following: , , , . ANSWER: Part D This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). W F μ w L W = “U L μ w L “U = Typesetting math: 91% Part F This question will be shown after you complete previous question(s). Where’s the Energy? Learning Goal: To understand how to apply the law of conservation of energy to situations with and without nonconservative forces acting. The law of conservation of energy states the following: In an isolated system the total energy remains constant. If the objects within the system interact through gravitational and elastic forces only, then the total mechanical energy is conserved. The mechanical energy of a system is defined as the sum of kinetic energy and potential energy . For such systems where no forces other than the gravitational and elastic forces do work, the law of conservation of energy can be written as , where the quantities with subscript “i” refer to the “initial” moment and those with subscript “f” refer to the final moment. A wise choice of initial and final moments, which is not always obvious, may significantly simplify the solution. The kinetic energy of an object that has mass \texttip{m}{m} and velocity \texttip{v}{v} is given by \large{K=\frac{1}{2}mv^2}. Potential energy, instead, has many forms. The two forms that you will be dealing with most often in this chapter are the gravitational and elastic potential energy. Gravitational potential energy is the energy possessed by elevated objects. For small heights, it can be found as U_{\rm g}=mgh, where \texttip{m}{m} is the mass of the object, \texttip{g}{g} is the acceleration due to gravity, and \texttip{h}{h} is the elevation of the object above the zero level. The zero level is the elevation at which the gravitational potential energy is assumed to be (you guessed it) zero. The choice of the zero level is dictated by convenience; typically (but not necessarily), it is selected to coincide with the lowest position of the object during the motion explored in the problem. Elastic potential energy is associated with stretched or compressed elastic objects such as springs. For a spring with a force constant \texttip{k}{k}, stretched or compressed a distance \texttip{x}{x}, the associated elastic potential energy is \large{U_{\rm e}=\frac{1}{2}kx^2}. When all three types of energy change, the law of conservation of energy for an object of mass \texttip{m}{m} can be written as K U Ki + Ui = Kf + Uf Typesetting math: 91% \large{\frac{1}{2}mv_{\rm i}^2+mgh_{\rm i}+\frac{1}{2}kx_{\rm i}^2=\frac{1}{2}mv_{\rm f \hspace{1 pt}}^2+mgh_{\rm f \hspace{1 pt}}+\frac{1}{2}kx_{\rm f \hspace{1 pt}}^2}. The gravitational force and the elastic force are two examples of conservative forces. What if nonconservative forces, such as friction, also act within the system? In that case, the total mechanical energy would change. The law of conservation of energy is then written as \large{\frac{1}{2}mv_{\rm i}^2+mgh_{\rm i}+\frac{1}{2}kx_{\rm i}^2+W_{\rm nc}=\frac{1}{2}mv_{\rm f \hspace{1 pt}}^2+mgh_{\rm f \hspace{1 pt}}+\frac{1}{2}kx_{\rm f \hspace{1 pt}}^2}, where \texttip{W_{\rm nc}}{W_nc} represents the work done by the nonconservative forces acting on the object between the initial and the final moments. The work \texttip{W_{\rm nc}}{W_nc} is usually negative; that is, the nonconservative forces tend to decrease, or dissipate, the mechanical energy of the system. In this problem, we will consider the following situation as depicted in the diagram : A block of mass \texttip{m}{m} slides at a speed \texttip{v}{v} along a horizontal, smooth table. It next slides down a smooth ramp, descending a height \texttip{h}{h}, and then slides along a horizontal rough floor, stopping eventually. Assume that the block slides slowly enough so that it does not lose contact with the supporting surfaces (table, ramp, or floor). You will analyze the motion of the block at different moments using the law of conservation of energy. Part A Which word in the statement of this problem allows you to assume that the table is frictionless? ANSWER: Part B straight smooth horizontal Typesetting math: 91% 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). 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 This question will be shown after you complete previous question(s). Part H Typesetting math: 91% This question will be shown after you complete previous question(s). Part I This question will be shown after you complete previous question(s). Part J This question will be shown after you complete previous question(s). Part K This question will be shown after you complete previous question(s). Sliding In Socks Suppose that the coefficient of kinetic friction between Zak’s feet and the floor, while wearing socks, is 0.250. Knowing this, Zak decides to get a running start and then slide across the floor. Part A If Zak’s speed is 3.00 \rm m/s when he starts to slide, what distance \texttip{d}{d} will he slide before stopping? Express your answer in meters. ANSWER: Typesetting math: 91% Part B This question will be shown after you complete previous question(s). Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. \rm m Typesetting math: 91%

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