HW7-Interactions_and_Potential_Energy

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12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 1/18 HW7-Interactions and Potential Energy Due: 11:59pm on Wednesday, December 13, 2023 You will receive no credit for items you complete after the assignment is due. Grading Policy Spring and Projectile A child's toy consists of a block that attaches to a table with a suction cup, a spring connected to that block, a ball, and a launching ramp. The spring has a spring constant , the ball has a mass , and the ramp rises a height above the table, the surface of which is a height above the floor. Initially, the spring rests at its equilibrium length. The spring then is compressed a distance , where the ball is held at rest. The ball is then released, launching it up the ramp. When the ball leaves the launching ramp its velocity vector makes an angle with respect to the horizontal. Throughout this problem, ignore friction and air resistance. Part A Relative to the initial configuration (with the spring relaxed), when the spring has been compressed, the ball-spring system has ANSWER: Correct Part B As the spring expands (after the ball is released) the ball-spring system ANSWER: Correct Part C As the ball goes up the ramp, it ANSWER: gained kinetic energy gained potential energy lost kinetic energy lost potential energy gains kinetic energy and loses potential energy gains kinetic energy and gains potential energy loses kinetic energy and gains potential energy loses kinetic energy and loses potential energy

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 2/18 Correct Part D As the ball falls to the floor (after having reached its maximum height), it ANSWER: Correct Part E Which of the graphs shown best represents the potential energy of the ball-spring system as a function of the ball's horizontal displacement? Take the "zero" on the distance axis to represent the point at which the spring is fully compressed. Keep in mind that the ball is not attached to the spring, and neglect any recoil of the spring after the ball loses contact with it. ANSWER: Correct Part F Calculate , the speed of the ball when it leaves the launching ramp. Express the speed of the ball in terms of , , , , , and/or . gains kinetic energy and loses potential energy gains kinetic energy and gains potential energy loses kinetic energy and gains potential energy loses kinetic energy and loses potential energy gains kinetic energy and loses potential energy gains kinetic energy and gains potential energy loses kinetic energy and gains potential energy loses kinetic energy and loses potential energy A B C D

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 3/18 Hint 1. General approach Find an expression for the mechanical energy (kinetic plus potential) of the spring and ball when the spring is compressed. Then find an expression for the mechanical energy of the ball when it leaves the launching ramp. ( will be an unknown in this expression.) Since energy is conserved, you can set these two expressions equal to each other, and solve for . Hint 2. Find the initial mechanical energy Find the total mechanical energy of the ball-spring system when the spring is fully compressed. Take the gravitational potential energy to be zero at the floor. Hint 1. What contributes to the mechanical energy? The total initial mechanical energy is the sum of the potential energy of the spring, the gravitational potential energy, plus any initial kinetic energy of the ball. ANSWER: Hint 3. Find the mechanical energy at the end of the ramp Find the total mechanical energy of the ball when it leaves the launching ramp. (At this point, assume that the spring is relaxed and has no stored potential energy.) Again, take the gravitational potential energy to be zero at the floor. Express your answer in terms of and other given quantities. ANSWER: Hint 4. Is energy conserved? Because no nonconservative forces act on the system, energy is conserved: ANSWER: Correct Part G With what speed will the ball hit the floor? Express the speed in terms of , , , , , and/or . Hint 1. General approach Find an expression for the initial mechanical energy (kinetic plus potential) of the spring and ball. Then find an expression for the mechanical energy of the ball when it hits the floor. ( will be an unknown in this expression.) Since energy is conserved, you can set these two expressions equal to each other, and solve for . Hint 2. Initial mechanical energy For the initial mechanical energy, you can use either the expression you found for the mechanical energy of the ball at the top of the ramp or that for the total mechanical energy of the ball plus spring just before the ball was launched. These two expressions are equal. Hint 3. Find the final mechanical energy Find the total mechanincal energy of the ball when it hits the floor. Express your answer in terms of and other given quantities. = = =

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12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 4/18 ANSWER: Hint 4. Is energy conserved? Only conservative forces (gravity, spring) are acting on the ball, so energy is conserved: . ANSWER: Correct PhET Tutorial: Energy Skate Park: Basics Learning Goal: To understand conservation of mechanical energy involving gravitational potential energy and kinetic energy. For this problem, use the PhET simulation Energy Skate Park: Basics . This simulation allows you to explore the motion and energetics of a skater riding along a track. Start the simulation. When you click the simulation link, a new window will load. Click on Intro to get started. You can click and drag the skater to any location and release the skater from rest.Watch the skater skate up and down the track. Click on Bar Graph to see the relative magnitudes of the kinetic, potential, thermal, and total energies as a function of the skater's position. Try Pie Chart to see the same information as a pie chart. Play around with the simulation. When you are done, click on the Reset button before answering the questions. Part A Click on Bar Graph , and observe the kinetic energy bar as the skater goes back and forth. You can select Slow Motion below the track for a more accurate observation. Where on the track is the skater's kinetic energy the greatest? Hint 1. How to approach the problem The kinetic energy of an object is given by , where is the mass of the object and is the speed of the object. Thus, the skater is moving the fastest where the skater's kinetic energy is the greatest and vice versa. Look at the Bar Graph, and identify the location(s) where the skater's kinetic energy reaches its maximum value. = =

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 5/18 ANSWER: Correct The kinetic energy of an object is given by , where is the speed of the object and is the mass of the object. Thus, the skater's kinetic energy is greatest at the lowest point of the track, where the skater is moving the fastest. Part B Now observe the potential energy bar on the Bar Graph. As the skater is skating back and forth, where does the skater have the most potential energy? Hint 1. How to approach the problem The gravitational potential energy of an object is given by , where is the object's height above the potential energy reference, which is currently the ground. Thus, the skater's potential energy is greatest where the skater is the highest above the reference line. Look at the Bar Graph, and identify the location(s) where the skater's potential energy reaches its maximum value. ANSWER: Correct The gravitational potential energy of an object is given by , where is the object's height above the potential energy reference, which is currently the ground. Thus, the skater's potential energy is greatest at the locations where the skater turns to go back in the opposite direction, where the skater is the highest above the reference line. Notice that the skater's potential energy is greatest where the kinetic energy is the lowest, and vice versa. Part C Because we are ignoring friction, no thermal energy is generated and the total energy is the mechanical energy, the kinetic energy plus the potential energy: . Observe the total energy bar on the Bar Graph. As the skater is skating back and forth, which statement best describes the total energy? ANSWER: The skater's kinetic energy is at its maximum value at the lowest point of the track. the same everywhere. at its maximum value at the locations where the skater turns and goes back in the opposite direction. The skater's potential energy is the same everywhere. at its maximum value at the locations where the skater turns and goes back in the opposite direction. at its maximum value at the lowest point of the track.

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 6/18 Correct The mechanical energy (kinetic plus potential) is conserved. (Since there is no friction, the mechanical energy is equal to the total energy.) When the kinetic energy is relatively small, the potential energy is relatively large, and vice versa. Part D Ignoring friction, the total energy of the skater is conserved. This means that the kinetic plus potential energy at one location, say , must be equal to the kinetic plus potential energy at a different location, say . This is the principle of conservation of energy and can be expressed as . Since the energy is conserved, the change in the kinetic energy is equal to the negative of the change in the potential energy: . At the bottom of the simulation window, click on Playground . For this part of the tutorial, you should have the Friction slider (in the upper right part of the window) set to "None," which means no thermal energy is generated. Select the Show Grid option. Then, add a track by clicking and dragging on a new track (the shape with three circles in the bottom left of the window) and placing it near the skater. You can then click and drag on individual circles to stretch and /or bend the track and make it look as shown below. The bottom of the track should be 1 above the ground, and both ends of the track should be at a height of 7 Place the skater on the track 7 above the ground, and look at the resulting motion and the Bar Graph showing the energetics. Match the approximate numerical values on the left with the energy type categories on the right to complete the equations. Assume that the mass of the skater is 75.0 and that the acceleration of gravity is 9.8 . The total energy is the same at all locations of the track. greatest at the locations where the skater turns and goes back in the opposite direction and smallest at the lowest point of the track. smallest at the locations where the skater turns to go back in the opposite direction and greatest at the lowest point of the track.

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12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 7/18 Match the values in the left column to the appropriate blanks in the sentences on the right. Values can be used once, more than once, or not at all. ANSWER: Correct Part E Based on the previous question, which statement is true? ANSWER: Correct Because the total energy is conserved, the kinetic energy at the bottom of the hill plus the potential energy at the bottom of the hill must equal the initial potential energy (since the initial kinetic energy is zero): . Solving for the kinetic energy, we get , or . More generally, the change in the kinetic energy is equal to the negative of the change in the potential energy. Part F If the skater started from rest 4 above the ground (instead of 7 ), what would be the kinetic energy at the bottom of the ramp (which is still 1 above the ground)? Hint 1. How to approach the problem Use the Bar Graph to verify that the relationship between the kinetic, potential, and total energies still holds. ANSWER: Reset Help 5145 0 4410 735 1. Total energy at initial position = 5145 . 2. Potential energy at initial position = 5145 . 3. Kinetic energy at initial position = 0 . 4. Total energy at bottom of track = 5145 . 5. Potential energy at bottom of track = 735 . 6. Kinetic energy at bottom of track = 4410 . The kinetic energy at the bottom of the ramp is equal to the total energy. equal to the amount of potential energy loss in going from the initial location to the bottom. equal to the initial potential energy.

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 8/18 Correct Notice that the skater only drops down 3 , as compared to 6 for the case when the skater started 7 above the ground. In this case, the skater loses half as much potential energy as in the previous case, so the kinetic energy at the bottom is only half as much as the previous case. Part G One common application of conservation of energy in mechanics is to determine the speed of an object. Although the simulation doesn’t give the skater's speed, you can calculate it because the skater's kinetic energy is known at any location on the track. Consider again the case where the skater starts 7 above the ground and skates down the track. What is the skater's speed when the skater is at the bottom of the track? Express your answer numerically in meters per second to two significant figures. Hint 1. Find the kinetic energy at the bottom of the track What is the kinetic energy of the skater at the bottom of the track? ANSWER: Hint 2. Find the speed at the bottom of the track Because , the speed is given by . What is the speed at the bottom of the track? Express your answer numerically in meters per second to two significant figures. ANSWER: ANSWER: Correct For a 75- object having approximately 4410 of energy, the speed must be roughly 11 . Part H When the skater starts 7 above the ground, how does the speed of the skater at the bottom of the track compare to the speed of the skater at the bottom when the skater starts 4 above the ground? Hint 1. Find the kinetic energy at the bottom of the track How much more kinetic energy does the skater have at the bottom of the track? ANSWER: 4410 J 735 J 2940 J 2205 J 4410 J 5145 J 735 J 2205 J 11 11

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 9/18 Hint 2. Find the ratio of the speeds Because kinetic energy is proportional to the square of the speed, the ratio of the kinetic energy is equal to the ratio of the speeds squared: . What is the ratio of the speeds? Express your answer numerically with two significant figures. ANSWER: ANSWER: Correct The person will have twice as much kinetic energy. Because kinetic energy is proportional to the speed squared, the ratio of the speeds is equal to the square root of the ratio of the kinetic energies. In this case, since the ratio of the kinetic energies is 2, the ratio of the speeds is equal to the square root of 2, or roughly 1.4. Part I In the current window, click and drag a new track (the shape with three circles in the bottom left of the window), and place it near the upper left end of the existing track until the two connect. Then, click and drag on the circles to stretch and/or bend the track to make it look as shown below. If the skater starts from rest at position 1, rank, in increasing order from least to greatest, the kinetic energy of the skater at the five positions shown. Rank from smallest to largest. To rank items as equivalent, overlap them. ANSWER: four times as much the same amount twice as much = 1.4 The speed is the same. twice as fast. four times as fast. higher, but less than twice as fast.

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12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 10/18 Correct PhET Interactive Simulations University of Colorado http://phet.colorado.edu Conceptual Question 11.8 Sandy and Chris stand on the edge of a cliff and throw identical mass rocks at the same speed. Sandy throws her rock horizontally while Chris throws his upward at an angle of 45 to the horizontal. Part A Are the rocks moving at the same speed when they hit the ground, or is one moving faster than the other? If one is moving faster, which one? ANSWER: Correct Part B Explain. Drag the terms on the left to the appropriate blanks on the right to complete the sentences. Terms can be used more than once. ANSWER: Reset Help Chris's rock is faster when it hits the ground. Sandy's rock is faster when it hits the ground. The rocks hit the ground with the same speed. 1 3 2 5 4 largest KE smallest KE

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 11/18 Correct Conceptual Question 11.10 A spring gun shoots out a plastic ball at speed . The spring is then compressed twice the distance it was on the first shot. Part A By what factor is the ball's speed increased? ANSWER: Correct Conceptual Question 11.12 is the energy bar chart for a firefighter sliding down a fire pole from the second floor to the ground. Let the system consist of the firefighter, the pole, and the earth. Reset Help the same as smaller than greater than Both rocks are thrown from the same height. Therefore, the initial potential energy of Sandy's rock is the same as the initial potential energy of Chris's rock. The rocks have the same initial speed. Therefore, the initial kinetic energy of Sandy's rock is the same as the initial kinetic energy of Chris's rock. So, the total energy of Sandy's rock is the same as the total energy of Chris's rock. Both rocks reach the ground at the same height. Therefore, the final potential energy of Sandy's rock is the same as that of Chris's rock. Therefore, the kinetic energy (and the speed) of Sandy's rock is the same as that of Chris's rock. = 2

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 12/18 Part A What is the bar height of ? Express your answer in joules as an integer. ANSWER: Correct Part B What is the bar height of ? Express your answer in joules as an integer. ANSWER: Correct Conceptual Question 11.13 A particle with the potential energy shown in is moving to the right at with total energy . Part A At what value or values of is this particle's speed a maximum? Express your answer in meters. ANSWER: Correct Part B Does this particle have a turning point or points in the range of covered by the graph? ANSWER: = 0 = 2 = 6

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12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 13/18 Correct Part C Where? Express your answers in meters. Enter your answers in ascending order separated by commas. ANSWER: Correct Part D If is changed appropriately, could the particle remain at rest at any point or points in the range of covered by the graph? ANSWER: Correct Part E Where? Express your answers in meters. Enter your answers in ascending order separated by commas. ANSWER: Correct Problem 11.6 The maximum energy a bone can absorb without breaking is surprisingly small. Experimental data show that both leg bones, together, of a healthy human adult can absorb about 210 before breaking. Part A From what maximum height could a 67 person jump and land rigidly upright on both feet without breaking their legs? Assume that all energy is absorbed by the leg bones in a rigid landing. Express your answer with the appropriate units. ANSWER: No Yes = 2,8 No Yes = 1,3,4,6

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 14/18 Correct Problem 11.13 - Enhanced - with Video Solution A 20 child is on a swing that hangs from 2.4- -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 50 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 11.22 - Enhanced - with Hints and Feedback The elastic energy stored in your tendons can contribute up to 35 of your energy needs when running. Sports scientists find that (on average) the knee extensor tendons in sprinters stretch 42 while those of nonathletes stretch only 35 . The spring constant of the tendon is the same for both groups, 33 . Part A What is the difference in maximum stored energy between the sprinters and the nonathletes? Express your answer with the appropriate units. Hint 1. How to approach the problem Recall how to find the potential energy of an elastic system and use it to find the difference in energy. ANSWER: Correct Problem 11.27 is the potential-energy diagram for a 500 particle that is released from rest at . = 0.32 = 4.1 = 8.9

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 15/18 Part A What is the particle's speed at ? Express your answer with the appropriate units. ANSWER: Correct Part B What is the particle's speed at ? Express your answer with the appropriate units. ANSWER: Correct Part C What is the particle's speed at ? Express your answer with the appropriate units. ANSWER: Correct Problem 11.33 Hydrogen is a diatomic molecule composed of two hydrogen atoms. is an energy diagram for hydrogen as a function of the atomic separation . = 3.5 = 2.8 = 4.5

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12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 16/18 Part A What is the minimum light-photon energy that can dissociate a hydrogen molecule? Express your answer to one significant figure and include the appropriate units. ANSWER: Correct Problem 11.39 A particle moving along the -axis is in a system with potential energy , where is in . Part A What is the -component of the force on the particle at = 2 ? Express your answer with the appropriate units. ANSWER: Correct Part B What is the -component of the force on the particle at = 5 ? Express your answer with the appropriate units. ANSWER: Correct Part C What is the -component of the force on the particle at = 8 ? = 7×10 −19 = 4.5 = 0.72

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 17/18 Express your answer with the appropriate units. ANSWER: Correct Problem 11.42 Part A How much work is done by the environment in the process shown in ? Express your answer with the appropriate units. ANSWER: Correct Part B Is energy transferred from the environment to the system or from the system to the environment? ANSWER: Correct Problem 11.44 In an average human, basic life processes require energy to be supplied at a steady rate of 100 . Part A What daily energy intake, in calories, is required to maintain these basic processes? Express your answer in calories. ANSWER: = 0.28 = -1.00 Energy is transferred from the environment to the system. Energy is transferred from the system to the environment.

12/6/23, 12:10 AM HW7-Interactions and Potential Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10892095 18/18 Correct Problem 11.49 The basis of muscle action is the power stroke of the myosin protein pulling on an actin filament. It takes the energy of one molecule of ATP, , to produce a displacement of 28 against a force of 1.0 . Part A What is the efficiency? Express your answer as a percentage. ANSWER: Correct Problem 11.53 Fleas have remarkable jumping ability. A 0.50 flea, jumping straight up, would reach a height of 40 if there were no air resistance. In reality, air resistance limits the height to 20 . Part A What is the flea's kinetic energy as it leaves the ground? Express your answer with the appropriate units. ANSWER: Correct Part B At its highest point, what fraction of the initial kinetic energy has been converted to potential energy? ANSWER: Correct Score Summary: Your score on this assignment is 100%. You received 29 out of a possible total of 29 points. = 2100 = 55 = 2.0×10 −3 = 0.50

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