HW10 - Energy

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12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 1/19 HW10 - Energy Due: 11:59pm on Friday, December 2, 2022 To understand how points are awarded, read the Grading Policy for this assignment. Problem 10.3 The lowest point in Death Valley is 85 below sea level. The summit of nearby Mt. Whitney has an elevation of 4420 above sea level. Part A What is the change in potential energy of an energetic 76 hiker who makes it from the floor of Death Valley to the top of Mt. Whitney? Express your answer with the appropriate units. ANSWER: Correct Problem 10.9 - Enhanced - with Video Solution A 20 child is on a swing that hangs from 4.0- -long chains. For general problem-solving tips and strategies for this topic, you may want to view a Video Tutor Solution of Car rolling down a hill . Part A What is her maximum speed if she swings out to a 49 angle? Express your answer with the appropriate units. ANSWER: Correct Here we learn how to calculate the maximum speed of a swinging mass if we know the maximum angle of the swing. Problem 10.11 - Enhanced - with Video Solution = 3.4×10 6 = 5.2

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 2/19 A 1500 car traveling at 16 suddenly runs out of gas while approaching the valley shown in . The alert driver immediately puts the car in neutral so that it will roll. For general problem-solving tips and strategies for this topic, you may want to view a Video Tutor Solution of Car rolling down a hill . Part A What will be the car’s speed as it coasts into the gas station on the other side of the valley? Express your answer with the appropriate units. ANSWER: Correct Here we learn how to find the speed in a certain moment knowing its height, initial speed and initial height using the law of conservation of energy. Problem 10.13 - Enhanced - with Hints and Feedback A cannon tilted up at a 30 angle fires a cannon ball at 78 from atop a 16- -high fortress wall. Part A What is the ball's impact speed on the ground below? Express your answer with the appropriate units. Hint 1. How to approach the problem Use the law of conservation of mechanical energy to find how the ball’s potential energy transforms into kinetic energy. ANSWER: = 13 = 80

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 3/19 Correct Fun with a Spring Gun A spring-loaded toy gun is used to shoot a ball of mass straight up in the air, as shown in . The spring has spring constant . If the spring is compressed a distance of 25.0 centimeters from its equilibrium position and then released, the ball reaches a maximum height (measured from the equilibrium position of the spring). There is no air resistance, and the ball never touches the inside of the gun. Assume that all movement occurs in a straight line up and down along the y axis. Part A Which of the following statements are true? Check all that apply. Hint 1. Nonconservative forces Dissipative , or nonconservative , forces are those that always oppose the motion of the object on which they act. Forces such as friction and drag are dissipative forces. Hint 2. Forces acting on the ball The ball is acted on by the spring force only when the two are in contact. The force of tension in the spring is a conservative force. Also, the ball is always acted on by gravity, which is also a conservative, or nondissipative, force. ANSWER: Correct Part B Mechanical energy is conserved because no dissipative forces perform work on the ball. The forces of gravity and the spring have potential energies associated with them. No conservative forces act in this problem after the ball is released from the spring gun.

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12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 4/19 Find the muzzle velocity of the ball (i.e., the velocity of the ball at the spring's equilibrium position ). Hint 1. Determine how to approach the problem What physical relationship can you use to solve this problem? Choose the best answer. ANSWER: Hint 2. Energy equations Recall that kinetic energy is given by the equation , where is the speed of the object and is the object's mass. Gravitational potential energy is given by , where is the object's height measured from . The elastic potential energy of a spring is given by , where is the spring constant and is the spring's displacement from equilibrium. Hint 3. Determine which two locations you should examine Pick the two points along the ball's path that would be most useful to compare in order to find the solution to this problem. Choose from among the following three points: Check all that apply. ANSWER: Hint 4. Find the initial energy of the system A useful statement of mechanical energy conservation relating the initial and final kinetic ( ) and potential ( ) energies is . In this situation, the initial position is and the final position is , which is the equilibrium position of the spring. What kind(s) of energy does the system "spring-ball" have at the initial position? ANSWER: kinematics equations Newton's second law law of conservation of energy conservation of momentum , the location of the ball when the spring is compressed. , the equilibrium position of the spring. , the maximum height that the ball reaches above the point .

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 5/19 Hint 5. Determine the final energy A useful statement of mechanical energy conservation relating the initial and final kinetic ( ) and potential ( ) energies is . In this situation, the initial position is and the final position is , which is the equilibrium position of the spring. What kind(s) of energy does the system "spring-ball" have at the final position? ANSWER: Hint 6. Creating an equation From the hints you now know what kinds of energy are present at the initial and final positions chosen for the ball in this part of the problem. You also know that . It has been determined that is zero and consists of two terms: gravitational potential energy and elastic potential energy. In addition, is zero. ANSWER: Correct Part C Find the maximum height of the ball. Express your answer numerically, in meters. kinetic only elastic potential only gravitational potential only kinetic and gravitational potential kinetic and elastic potential elastic and gravitational potentials kinetic only elastic potential only gravitational potential only kinetic and gravitational potential kinetic and elastic potential elastic and gravitational potentials = 4.78

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 6/19 Hint 1. Choose two locations to examine Pick the two points along the ball's movement that would be most useful to compare in order to find a solution to this problem. Choose from among the following three points: Check all that apply. ANSWER: Hint 2. Find the initial energy A useful statement of mechanical energy conservation is . Recall that in the problem statement, is set to correspond to the equilibrium position of the spring. Therefore, in this situation, the initial location is at and the final position should be taken as . What kind(s) of energy does the ball have at the initial location? ANSWER: Hint 3. Determine the final energy A useful statement of mechanical energy conservation is . In this situation, the initial location is at , and the final position should be taken as . What kind(s) of energy does the ball have at ? Hint 1. Find the speed of the ball at the top of its trajectory What is the speed of the ball at the top of its trajectory? Express your answer numerically, in meters per second. Hint 1. Motion in the vertical direction Recall from kinematics that a ball travels upward until its speed decreases to zero, at which point it starts falling back to Earth. , the location of the ball when the spring is compressed. , the equilibrium position of the spring. , the maximum height that the ball reaches measured from . kinetic only elastic potential only gravitational potential only kinetic and gravitational potential kinetic and elastic potential elastic and gravitational potentials

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12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 7/19 ANSWER: ANSWER: Hint 4. Creating an equation From the above hints, you now know what kind of energy is present at the inital and final positions chosen for the ball in this part of the problem. You know that . It was determined that is zero and that consists of two terms: gravitational potential energy and elastic potential energy. In addition, is zero. ANSWER: Correct In this problem you practiced applying the law of conservation of mechanical energy to a physical situation to find the muzzle velocity and the maximum height reached by the ball. Part D Which of the following actions, if done independently, would increase the maximum height reached by the ball? Check all that apply. ANSWER: = 0 kinetic only elastic potential only gravitational potential only kinetic and gravitational potential kinetic and elastic potential elastic and gravitational potentials = 1.17

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 8/19 Correct Problem 10.22 - Enhanced - with Hints and Feedback The spring in is compressed by 10 . It launches a block across a frictionless surface at 0.50 . The two springs in are identical to the spring of . They are compressed the same 10 and launch the same block. reducing the spring constant increasing the spring constant decreasing the distance the spring is compressed increasing the distance the spring is compressed decreasing the mass of the ball increasing the mass of the ball tilting the spring gun so that it is at an angle degrees from the horizontal

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 9/19 Part A What is the block's speed now? Express your answer with the appropriate units. Hint 1. How to approach the problem Recall the expression of potential energy of a spring. Use the law of conservation of mechanical energy to find the launching speed in case of one spring and that of two springs. Find the compression of each of the two combined springs. ANSWER: Correct Problem 10.31 - Enhanced - with Expanded Hints A system in which only one particle can move has the potential energy shown in . Suppose = 15 . Part A What is the -component of the force exerted on the particle at = 5 ? Express your answer with the appropriate units. Hint 1. How to approach the problem Use the graph in the problem statement. Then recall the definition for the slope of the potential energy curve to find the force. Hint 2. Simplify: the sign of force Determine the sign of the force at = 5 . = 0.35

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12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 10/19 ANSWER: ANSWER: Correct Use the definition . is the negative of the slope of the potential energy graph at position . Between and the slope is Thus, at . Part B What is the -component of the force exerted on the particle at = 15 ? Express your answer with the appropriate units. Hint 1. How to approach the problem Use the graph in the problem statement. Then recall the definition for the slope of the potential energy curve to find the force. Hint 2. Simplify: the sign of force Determine the sign of the force at = 15 . ANSWER: ANSWER: Correct Use the definition . The slope between and is zero, at . positive negative The force is zero. = 150 negative positive The force is zero. = 0

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 11/19 Part C What is the -component of the force exerted on the particle at = 25 ? Express your answer with the appropriate units. Hint 1. How to approach the problem Use the graph in the problem statement. Then recall the definition for the slope of the potential energy curve to find the force. Hint 2. Simplify: the sign of force Determine the sign of the force at = 25 . ANSWER: ANSWER: Correct Use the definition . Between and Thus, at . Part D What is the -component of the force exerted on the particle at = 35 ? Express your answer with the appropriate units. Hint 1. How to approach the problem Use the graph in the problem statement. Then recall the definition for the slope of the potential energy curve to find the force. Hint 2. Simplify: the sign of force Determine the sign of the force at = 35 . ANSWER: positive negative The force is zero. = -75

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 12/19 ANSWER: Correct Use the definition . Between and Thus, at . Problem 10.36 - Enhanced - with Hints and Feedback A particle moves from A to D in while experiencing force . Part A How much work does the force do if the particle follows path ABD? Express your answer with the appropriate units. Hint 1. Calculating the work done by a constant force Recall that the work done by a force on a particle can be found as a dot product of the displacement vector and the force vector. Use the components of the vectors in your expressions. ANSWER: negative positive The force is zero. = -75

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12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 13/19 Correct Part B How much work does the force do if the particle follows path ACD? Express your answer with the appropriate units. Hint 1. Calculating the work done by a constant force Recall that the work done by a force on a particle can be found as a dot product of the displacement vector and the force vector. Use the components of the vectors in your expressions. ANSWER: Correct Part C Is this a conservative force? Hint 1. Conservative force A force for which the work done on a particle as it moves from the initial to the final position is independent of the path followed is called a conservative force. ANSWER: Correct Problem 10.40 - Enhanced - with Expanded Hints A cable with 25.0 of tension pulls straight up on a 1.50 block that is initially at rest. Part A What is the block's speed after being lifted 2.00 ? Solve this problem using work and energy. = 56 = 56 Yes No

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 14/19 Express your answer with the appropriate units. Hint 1. How to approach the problem Use the law of conservation of energy to find the speed. Take into account the change in potential and kinetic energy and the work of the tension force during the uplifting of the block. Hint 2. Simplify: law of conservation of energy Choose the correct expression for speed from the law of conservation of energy. ANSWER: ANSWER: Correct The tension of in the cable is an external force that does work on the block increasing the gravitational potential energy of the block. We placed the origin of our coordinate system at the initial resting position of the block, so we have and . Also, , and . The conservation of energy equation is Problem 10.60 - Enhanced - with Hints and Feedback A 100 particle experiences the one-dimensional, conservative force shown in . Let the zero of the potential energy be at = 0 . = 5.24

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 15/19 Part A What is the potential energy at = 1.0 ? Hint: Use the definition of potential energy and the geometric interpretation of work. Express your answer with the appropriate units. Hint 1. Geometric interpretation of work Work done by a variable force equals the area under the force-versus-position graph. ANSWER: Correct Part B What is the potential energy at = 2.0 ? Hint: Use the definition of potential energy and the geometric interpretation of work. Express your answer with the appropriate units. Hint 1. Geometric interpretation of work Work done by a variable force equals the area under the force-versus-position graph. ANSWER: Correct Part C = 20.0 = 40.0

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12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 16/19 What is the potential energy at = 3.0 ? Hint: Use the definition of potential energy and the geometric interpretation of work. Express your answer with the appropriate units. Hint 1. Geometric interpretation of work Work done by a variable force equals the area under the force-versus-position graph. ANSWER: Correct Part D What is the potential energy at = 4.0 ? Hint: Use the definition of potential energy and the geometric interpretation of work. Express your answer with the appropriate units. Hint 1. Geometric interpretation of work Work done by a variable force equals the area under the force-versus-position graph. ANSWER: Correct Part E Suppose the particle is shot toward the right from = 1.0 with a speed of 21 . Where is the particle's turning point? Express your answer with the appropriate units. Hint 1. Turning point A turning point occurs where velocity is zero, hence, the total energy equals the potential energy. ANSWER: Correct Problem 10.70 = 60.0 = 70.0 = 2.1

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 17/19 A pendulum is formed from a small ball of mass on a string of length . As the figure shows, a peg is height above the pendulum's lowest point. Part A From what minimum angle must the pendulum be released in order for the ball to go over the top of the peg without the string going slack? Express your answer with the appropriate units. ANSWER: Correct Loop the Loop A roller coaster car may be approximated by a block of mass . The car, which starts from rest, is released at a height above the ground and slides along a frictionless track. The car encounters a loop of radius , as shown. Assume that the initial height is great enough so that the car never loses contact with the track. Part A Find an expression for the kinetic energy of the car at the top of the loop. Express the kinetic energy in terms of , , , and . = 80.4

12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 18/19 Hint 1. Find the potential energy at the top of the loop What is the potential energy of the car when it is at the top of the loop? Define the gravitational potential energy to be zero at . Express your answer in terms of and other given quantities. ANSWER: ANSWER: Correct Part B Find the minimum initial height at which the car can be released that still allows the car to stay in contact with the track at the top of the loop. Express the minimum height in terms of . Hint 1. How to approach this part Meaning of "stay in contact" For the car to just stay in contact through the loop, without falling, the normal force that acts on the car when it's at the top of the loop must be zero (i.e., ). Find the velocity at the top such that the remaining force on the car i.e. its weight provides the necessary centripetal acceleration. If the velocity were any greater, you would additionally require some force from the track to provide the necessary centripetal acceleration. If the velocity were any less, the car would fall off the track. Use the above described condition to find the velocity and then the result from the above part to find the required height. Hint 2. Acceleration at the top of the loop Assuming that the speed of the car at the top of the loop is , and that the car stays on the track, find the acceleration of the car. Take the positive y direction to be upward. Express your answer in terms of and any other quantities given in the problem introduction. ANSWER: Hint 3. Normal force at the top of the loop Suppose the car stays on the track and has speed at the top of the loop. Use Newton's 2nd law to find an expression for , the magnitude of the normal force that the loop exerts on the car when the car is at the top of the loop. Express your answer in terms of , , , and . = = =

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12/19/22, 12:16 PM HW10 - Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072428 19/19 Hint 1. Find the sum of forces at the top of the loop Find the sum of the forces acting on the car at the top of the loop. Remember that the positive y direction is upward . Express your answer in terms of , , and . ANSWER: ANSWER: Hint 4. Solving for The requirement to stay in contact results in an expression for in terms of and . Substitute this into your expression for kinetic energy, found in Part A, to determine a relation between and . ANSWER: Correct For the car will still complete the loop, though it will require some normal reaction even at the very top. For the car will just oscillate. Do you see this? For , the cart will lose contact with the track at some earlier point. That is why roller coasters must have a lot of safety features. If you like, you can check that the angle at which the cart loses contact with the track is given by . Where is the angle measured counterclockwise from the horizontal positive x -axis, where the origin of the x -axis is at the center of the loop. Score Summary: Your score on this assignment is 99.3%. You received 14.9 out of a possible total of 15 points. = = =

HW10 - Energy

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