2017.02.02 Homework 7_ Potential Energy and Energy Conservation.pdf - Homework 7 Potential Energy and Energy Conservation Due 11:59pm on Sunday March 5

2017.02.02 Homework 7_ Potential Energy and Energy Conservation.pdf

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Unformatted text preview: Homework 7: Potential Energy and Energy Conservation Due: 11:59pm on Sunday, March 5, 2017 To understand how points are awarded, read the Grading Policy for this assignment. Introduction to Potential Energy Learning Goal: Understand that conservative forces can be removed from the work integral by incorporating them into a new form of energy called potential energy that must be added to the kinetic energy to get the total mechanical energy. The first part of this problem contains short­answer questions that review the work­energy theorem. In the second part we introduce the concept of potential energy. But for now, please answer in terms of the work­energy theorem. Work­Energy Theorem The work­energy theorem states Kf = Ki + Wall , where Wall is the work done by all forces that act on the object, and Ki and Kf are the initial and final kinetic energies, respectively. Part A The work­energy theorem states that a force acting on a particle as it moves over a ______ changes the ______ energy of the particle if the force has a component parallel to the motion. Choose the best answer to fill in the blanks above: ANSWER: distance / potential distance / kinetic vertical displacement / potential none of the above Correct It is important that the force have a component acting in the direction of motion. For example, if a ball is attached to a string and whirled in uniform circular motion, the string does apply a force to the ball, but since the string's force is always perpendicular to the motion it does no work and cannot change the kinetic energy of the ball. Part B To calculate the change in kinetic energy, you must know the force as a function of _______. The work done by the force causes the kinetic energy change. Choose the best answer to fill in the blank above: ANSWER: acceleration work position potential energy Correct Part C To illustrate the work­energy concept, consider the case of a stone falling from xi to xf under the influence of gravity. Using the work­energy concept, we say that work is done by the gravitational _____, resulting in an increase of the ______ energy of the stone. Choose the best answer to fill in the blanks above: ANSWER: force / kinetic potential energy / potential force / potential potential energy / kinetic Correct Potential Energy You should read about potential energy in your text before answering the following questions. Potential energy is a concept that builds on the work­energy theorem, enlarging the concept of energy in the most physically useful way. The key aspect that allows for potential energy is the existence of conservative forces, forces for which the work done on an object does not depend on the path of the object, only the initial and final positions of the object. The gravitational force is conservative; the frictional force is not. The change in potential energy is the negative of the work done by conservative forces. Hence considering the initial and final potential energies is equivalent to calculating the work done by the conservative forces. When potential energy is used, it replaces the work done by the associated conservative force. Then only the work due to nonconservative forces needs to be calculated. In summary, when using the concept of potential energy, only nonconservative forces contribute to the work, which now changes the total energy: Kf + Uf = Ef = Wnc + Ei = Wnc + Ki + Ui ,where Uf and Ui are the final and initial potential energies, and Wnc is the work due only to nonconservative forces. Now, we will revisit the falling stone example using the concept of potential energy. Part D Rather than ascribing the increased kinetic energy of the stone to the work of gravity, we now (when using potential energy rather than work­energy) say that the increased kinetic energy comes from the ______ of the _______ energy. Choose the best answer to fill in the blanks above: ANSWER: work / potential force / kinetic change / potential Correct Part E This process happens in such a way that total mechanical energy, equal to the ______ of the kinetic and potential energies, is _______. Choose the best answer to fill in the blanks above: ANSWER: sum / conserved sum / zero sum / not conserved difference / conserved Correct Understanding Non­Conservative Forces You and three friends stand at the corners of a square whose sides are 8.0 m long in the middle of the gym floor, as shown in the figure . You take your physics book and push it from one person to the other. The book has a mass of 2.0 kg , and the coefficient of kinetic friction between the book and the floor is μk =0.20. Part A The book slides from you to Beth and then from Beth to Carlos, along the lines connecting these people. What is the work done by friction during this displacement? Express your answer using two significant figures. ANSWER: ­63 J Correct Part B You slide the book from you to Carlos along the diagonal of the square. What is the work done by friction during this displacement? Express your answer using two significant figures. ANSWER: ­44 J Correct Part C You slide the book to Kim who then slides it back to you. What is the total work done by friction during this motion of the book? Express your answer using two significant figures. ANSWER: ­63 J Correct Part D A conservative force is a force for which the work done on an object by that force depends only upon the initial and final positions of the object. Equivalently, a conservative force is a force for which the work done along a path from an initial position back to that initial position is always zero. Based upon your answers to this problem, is the friction force on the book conservative or nonconservative? ANSWER: conservative nonconservative Correct Part E For any conservative force, a potential energy function can be defined. This function is a function of position and allows one to calculate the work done by that force on an object based solely on the change in the potential energy from an initial position to a final position (without having to worry about the details of the specific path that the object took). Based on your answers in this problem, can a potential energy function be defined for the friction force? ANSWER: no yes Correct Projectile Motion and Conservation of Energy Ranking Task Part A Six baseball throws are shown below. In each case the baseball is thrown at the same initial speed and from the same height H above the ground. Assume that the effects of air resistance are negligible. Rank these throws according to the speed of the baseball the instant before it hits the ground. Rank from largest to smallest. To rank items as equivalent, overlap them. Hint 1. How to approach the problem Although this situation can be investigated using the concepts of projectile motion, the conservation of mechanical energy is a better approach. Consider the initial total energy (kinetic plus gravitational potential). By conservation of energy, the final total energy must be equal to the initial total energy. You can use the final energy to determine the final speed when it reaches the ground. Note that the launch angle does not affect the initial kinetic or initial gravitational potential energy of the ball. ANSWER: Reset largest Help smallest The correct ranking cannot be determined. Correct This answer is best understood in terms of conservation of energy. The initial energy of the ball is independent of the direction in which it is thrown. The initial and final potential energies of the ball are the same regardless of the trajectory. Therefore, the final kinetic energy, and therefore the final speed, of the ball must be the same no matter in what direction it is thrown. Energy Conservation I A baseball is thrown from the roof of a 23.7 m tall building with an initial velocity of magnitude 12.6 m/s directed at an angle of 53.1° above the horizontal. At a later time the ball hits the ground. Assume there is no air resistance. Part A Choose all the energy types which change from the initial state (thrown from the roof) to the final state (just about to hit the ground). ANSWER: Internal Energy Kinetic Energy Gravitational Potential Energy Elastic Potential Energy Correct Part B Use energy conservation to solve for the speed of the baseball just before it hits the ground. ANSWER: v ground = 25.0 m/s Correct Part C If the angle of the throw was greater than 53.1°, how would your answer to part B change? ANSWER: The final speed of the ball would be smaller than the answer in part B. The final speed of the ball would be equal to the answer in part B. The final speed of the ball would be greater than the answer in part B. Correct Energy Conservation II An elastic slingshot will shoot a 8.8 g pebble 23.9 m straight up before it reaches maximum height. Ignore air resistance. Part A Choose all energy types which change from the initial state (pebble pulled all the way back in the slingshot before being shot) to the final state (pebble at maximum height). ANSWER: Uel Uint K Ug Correct Part B Use energy conservation to solve for the elastic potential energy of the slingshot when it was pulled all the way back (i.e. in the initial state). ANSWER: Uel = 2.06 J Correct Hooke's Law and Elastic Potential Energy A force of 500 N stretches a certain spring a distance of 0.200 m . Part A What is the spring constant k of the spring? Hint 1. Force and Displacement Remember Hooke's Law: Fel ANSWER: = 2500 N/m k = −kx . Correct Part B What is the potential energy of the spring when it is stretched a distance of 0.200 m ? ANSWER: U1 = 50.0 J Correct Part C What is its potential energy when it is compressed a distance of 7.00 cm ? ANSWER: U2 = 6.13 J Correct Energy Conservation III A 0.400 kg mass is attached to a horizontal spring with a spring constant of 50.0 N/m. The mass is initially stretched a distance of 48.0 cm from its equilibrium position. The mass is then released from rest and slides along a horizontal table. Part A Assume the table is frictionless and there is no air resistance. Select the energy types which change from the time the mass is released to the time it is compressed a distance of 39.0 cm on the other side of equilibrium. ANSWER: Uel Ug Uint K Correct Part B What will be the speed of the mass when it is compressed to the distance of 39.0 cm from equilibrium? ANSWER: v = 3.13 m/s Correct Now we will imagine that the table does exert a kinetic friction force on the mass (we will still ignore air resistance, however). We would like to analyze this scenario in two ways: 1. We will treat the mass­spring as an open system, where the table is exerting an external force on the mass. This system will be analyzed in parts C, D and E. 2. We will treat the mass­spring and the table as a closed system, with no external forces acting. This system will be analyzed in parts F and G. Part C Due to the kinetic friction from the table, after being released as before (from rest, stretched a distance of 48.0 cm from equilibrium), the mass reaches its maximum compression 40.0 cm on the other side of equilibrium. Find the change in the mechanical energy of the mass­spring from when it is released to when it reaches its maximum compression. (Remember that mechanical energy is defined as the sum of the kinetic energy and total potential energy of the system ). Hint 1. Kinetic Energy Think carefully about the speed in the initial state and the speed in the final state. This should make finding the change in kinetic energy very simple! ANSWER: ΔEmech = ­1.76 J Correct Part D Using your answer to part C, find the work done by the external force of the table on the mass. Hint 1. Conservation of Energy in an Open System Recall that for an open system, the change in the mechanical energy is equal to the amount of work done by external forces. In other words, however much mechanical energy was lost or gained by the system was caused by the work done by the external force. ANSWER: Wf ric,table = ­1.76 J Correct Part E Use your answer to part D to solve for the coefficient of kinetic friction between the table and the mass. Hint 1. Definition of Work Since the force of kinetic friction is constant, we remember that to calculate the amount of work done we use the formula W What is the angle ϕ between the force vector and displacement vector in this case? ⃗ = F ⋅ s ⃗ = F scosϕ . ANSWER: = 180 degrees ϕ Hint 2. Displacement In the formula to calculate work, we need to find the magnitude of the displacement vector, s, from the initial time to the final time. In this problem, how many centimeters does the mass slide in going from the initial state to the final state? ANSWER: = 88.0 cm s Hint 3. Normal Force To find the coefficient of friction, we first have to find the normal force that the table exerts on the mass. Draw a force diagram of the mass and analyze just the vertical direction to determine the magnitude of the normal force. ANSWER: = 3.92 N n ANSWER: μk = 0.510 Correct Now we will treat the mass­spring and table as our system, so there will be no external forces acting. The system is therefore a closed system. Part F In this system, select which types of energy must change from the initial state (released from rest stretched 48 cm from equilibrium) to the final state (maximum compression 40 cm from equilibrium)? Hint 1. Closed System Conservation of Energy Recall that for a closed system, the total energy of the system cannot change! Therefore if one type of energy changes at least one other type must change to offset that change. ANSWER: Uel Ug Uint K Correct Part G Use conservation of energy to solve for the change in the internal energy of the system. ANSWER: ΔUint = 1.76 J Correct Potential Energy Graphs and Motion Learning Goal: To be able to interpret potential energy diagrams and predict the corresponding motion of a particle. Potential energy diagrams for a particle are useful in predicting the motion of that particle. These diagrams allow one to determine the direction of the force acting on the particle at any point, the points of stable and unstable equilibrium, the particle's kinetic energy, etc. Consider the potential energy diagram shown. The curve represents the value of potential energy U as a function of the particle's coordinate x. The horizontal line above the curve represents the constant value of the total energy of the particle E . The total energy E is the sum of kinetic ( K ) and potential ( U ) energies of the particle. The key idea in interpreting the graph can be expressed in the equation Fx (x) = − dU (x) dx , where Fx (x) is the x component of the net force as function of the particle's coordinate x. Note the negative sign: It means that the x component of the net force is negative when the derivative is positive and vice versa. For instance, if the particle is moving to the right, and its potential energy is increasing, the net force would be pulling the particle to the left. If you are still having trouble visualizing this, consider the following: If a massive particle is increasing its gravitational potential energy (that is, moving upward), the force of gravity is pulling in the opposite direction (that is, downward). If the x component of the net force is zero, the particle is said to be in equilibrium. There are two kinds of equilibrium: Stable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle back toward the equilibrium point (think of a ball rolling between two hills). Unstable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle further away from the equilibrium point (think of a ball on top of a hill). In answering the following questions, we will assume that there is a single varying force F acting on the particle along the x axis. Therefore, we will use the term force instead of the cumbersome x component of the net force. Part A The force acting on the particle at point A is __________. Hint 1. Sign of the derivative If a function increases (as x increases) in a certain region, then the derivative of the function in that region is positive. Hint 2. Sign of the component If x increases to the right, as in the graph shown, then a (one­dimensional) vector with a positive x component points to the right, and vice versa. ANSWER: directed to the right directed to the left equal to zero Correct Consider the graph in the region of point A. If the particle is moving to the right, it would be "climbing the hill," and the force would "pull it down," that is, pull the particle back to the left. Another, more abstract way of thinking about this is to say that the slope of the graph at point A is positive; therefore, ⃗ the direction of F is negative. Part B The force acting on the particle at point C is __________. Hint 1. Sign of the derivative If a function increases (as x increases) in a certain region, then the derivative of the function in that region is positive, and vice versa. Hint 2. Sign of the component If x increases to the right, as in the graph shown, then a (one­dimensional) vector with a positive x component points to the right, and vice versa. ANSWER: directed to the right directed to the left equal to zero Correct Part C The force acting on the particle at point B is __________. Hint 1. Derivative of a function at a local maximum At a local maximum, the derivative of a function is equal to zero. ANSWER: directed to the right directed to the left equal to zero Correct The slope of the graph is zero; therefore, the derivative dU /dx = 0 , and |F |⃗ = 0 . Part D The acceleration of the particle at point B is __________. Hint 1. Relation between acceleration and force The relation between acceleration and force is given by Newton's 2nd law, F = ma . ANSWER: directed to the right directed to the left equal to zero Correct If the net force is zero, so is the acceleration. The particle is said to be in a state of equilibrium. Part E If the particle is located slightly to the left of point B, its acceleration is __________. Hint 1. The force on such a particle To the left of B, U (x) is an increasing function and so its derivative is positive. This implies that the x component of the force on a particle at this location is negative, or that the force is directed to the left, just like at A. What can you say now about the acceleration? ANSWER: directed to the right directed to the left equal to zero Correct Part F If the particle is located slightly to the right of point B, its acceleration is __________. Hint 1. The force on such a particle To the right of B, U (x) is a decreasing function and so its derivative is negative. This implies that the x component of the force on a particle at this location is positive, or that the force is directed to the right, just like at C. What can you now say about the acceleration? ANSWER: directed to the right directed to the left equal to zero Correct As you can see, small deviations from equilibrium at point B cause a force that accelerates the particle further away; hence the particle is in unstable equilibrium. Part G Name all labeled points on the graph corresponding to unstable equilibrium. List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE. Hint 1. Definition of unstable equilibrium Unstable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle further away from the equilibrium point (think of a ball on top of a hill). ANSWER: BF Correct Part H Name all labeled points on the graph corresponding to stable equilibrium. List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE. Hint 1. Definition of stable equilibrium Stable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle back towa...
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