HW09 - Work and Kinetic Energy

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12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 1/22 HW09 - Work and Kinetic Energy Due: 11:59pm on Monday, November 21, 2022 To understand how points are awarded, read the Grading Policy for this assignment. When Push Comes to Shove Two forces, of magnitudes = 100 and = 45.0 , 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 = -2.00 . At some later time, the block has moved to the right, and its center is at a new position, = 6.00 . Part A Find the work done on the block by the force of magnitude = 100 as the block moves from = -2.00 to = 6.00 . Express your answer numerically, in joules. Hint 1. Formula for the work done by a force The work done by a force in producing a displacement is given by , where is the angle between the two vectors. ANSWER: Correct Part B Find the work done by the force of magnitude = 45.0 as the block moves from = -2.00 to = 6.00 . Express your answer numerically, in joules. = 8.00

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 2/22 Hint 1. Is the work positive or negative? The force of magnitude acts in the opposite direction to that of the motion of the block. Therefore, the work done by that force must be negative. ANSWER: Correct Part C What is the net work done on the block by the two forces? Express your answer numerically, in joules. ANSWER: Correct Part D Determine the change in the kinetic energy of the block as it moves from = -2.00 to = 6.00 . Express your answer numerically, in joules. Hint 1. Conservation of energy The work done on the block goes into changing its kinetic energy. Thus the net work done by the two forces is equal to the change in the kinetic energy. ANSWER: Correct Problem 9.5 - Enhanced - with Hints and Feedback A 25 box sliding to the left across a horizontal surface is brought to a halt in a distance of 15 by a horizontal rope pulling to the right with 15 tension. Part A How much work is done by tension? = -3.60 = 4.40 = 4.40

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 3/22 Express your answer with the appropriate units. Hint 1. How to approach the problem Recall the expression for work in terms of force and displacement. To determine the sign, remember that work is an energy transfer due to mechanical means. ANSWER: Correct Part B How much work is done by gravity? Express your answer with the appropriate units. Hint 1. How to approach the problem Recall the expression for work in terms of force and displacement. Pay attention to the fact that work also depends on the angle the force vector makes with the displacement vector. ANSWER: Correct Problem 9.9 - Enhanced - with Hints and Feedback The cable of a crane is lifting a 950 girder. The girder increases its speed from 0.25 to 1.25 in a distance of 2.3 . Part A How much work is done by gravity? Express your answer with the appropriate units. Hint 1. How to approach the problem Recall the expression for work in terms of force and displacement. Remember that its sign depends on the force and displacement directions. ANSWER: = -2.3 = 0

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12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 4/22 Correct Part B How much work is done by tension? Express your answer with the appropriate units. Hint 1. How to approach the problem The tension and gravity do work. Kinetic energy of the system changes. Use the energy principle to find the work done by tension. ANSWER: Correct The Vector Dot Product The dot product is one way of multiplying two vectors. Part A shows two vectors and . What is their dot product? Hint 1. How to approach the problem = −2.1×10 4 = 2.2×10 4

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 5/22 The dot product of vectors and is defined as , where and are the magnitudes of the vectors and is the angle between them. ANSWER: Correct Part B Vector , and vector . What is their dot product? Hint 1. How to approach the problem When written in terms of components and unit vectors, the dot product of vectors and is . ANSWER: Correct Part C shows two vectors and . What is the angle between them? Express your answer in degrees. Hint 1. How to approach the problem = 6 = 2

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 6/22 You could use geometry and trigonometry to find the angle, but it's easier to use the dot product. In Part A you learned an expression for the dot product involving the angle between the vectors. In Part B you learned how to use components to calculate the dot product. Combine these to find an expression for , then take the inverse cosine. ANSWER: Correct Problem 9.18 - Enhanced - with Expanded Hints The three ropes shown in the bird's-eye view of are used to drag a crate 3.2 across the floor. Part A How much work is done by each of the three forces? Express your answers in kilojoules separated by commas. Hint 1. How to approach the problem Recall the expression for the work performed by a constant force for any specific coordinate system. You may need to review The Vector Dot Product . Hint 2. Simplify: Work done by force What expression should be used to calculate the work of force ? is the distance traveled by the crate. ANSWER: = 83

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12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 7/22 Hint 3. Simplify: Work done by force What expression should be used to calculate the work of force ? is the distance traveled by the crate. ANSWER: Hint 4. Simplify: Work done by force What expression should be used to calculate the work of force ? is the distance traveled by the crate. ANSWER: ANSWER: , , = 1.8,1.1,-2.1

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 8/22 Correct SOLVE: Model the crate as a particle and use , where is the work done by a force on a particle and is the particle’s displacement. For the tension : For the tension : For the tension : Problem 9.21 - Enhanced - with Expanded Hints A 400 particle moving along the -axis experiences the force shown in . The particle's velocity is 2.0 at = 0 . Part A What is its velocity at = 3 ? Express your answer with the appropriate units. Hint 1. How to approach the problem The work of the force goes to change the kinetic energy of the particle. By finding the work of the force , you can find the velocity of the particle at the point 3 . Hint 2. Simplify: Area under the graph You can find the work of the force using the integral: . You can use the geometric meaning of a definite integral the area under the graph of a function to quickly calculate this integral. Find the total area

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 9/22 under the force graph . Express your answer using two significant figures. ANSWER: ANSWER: Correct SOLVE: The energy principle is At 3 : Work and Kinetic Energy Two blocks of ice, one four times as heavy as the other, are at rest on a frozen lake. A person pushes each block the same distance . Ignore friction and assume that an equal force is exerted on each block. Part A Which of the following statements is true about the kinetic energy of the heavier block after the push? Hint 1. How to approach the problem The work-energy theorem states that the change in kinetic energy of an object equals the net work done on that object: . The work done on an object can also be related to the distance that the object moves while being acted on by a force : , where is the component of parallel to the direction of displacement. Hint 2. Find the work done on each block What can be said about the net work done on the heavier block? ANSWER: Area = 30 = 12.4

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12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 10/22 ANSWER: Correct The work-energy theorem states that the change in kinetic energy of an object equals the net work done on that object. The only force doing work on the blocks is the force from the person, which is the same in both cases. Since the initial kinetic energy of each block is zero, both blocks have the same final kinetic energy. Part B Compared to the speed of the heavier block, what is the speed of the light block after both blocks move the same distance ? Hint 1. How to approach the problem In Part A, you determined that the kinetic energy of the heavier block was the same as that of the lighter block. Relate this to the speed of the blocks. Hint 2. Proportional reasoning Proportional reasoning becomes easier with practice. First relate the kinetic energies of the blocks to each other. To accomplish this, let the subscript refer to the heavier block and the subscript to the lighter block. Now can be written as . The problem states that the heavier block is four time as massive as the lighter block. This can be represented by the expression . Substituting this expression into the expression for kinetic energy yields . How many times larger than is ? ANSWER: It is greater than the work done on the lighter block. It is equal to the work done on the lighter block. It is less than the work done on the lighter block. It is smaller than the kinetic energy of the lighter block. It is equal to the kinetic energy of the lighter block. It is larger than the kinetic energy of the lighter block. It cannot be determined without knowing the force and the mass of each block.

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 11/22 ANSWER: Correct Since the kinetic energy of the lighter block is equal to the kinetic energy of the heavier block, the lighter block must be moving faster than the heavier block. Part C Now assume that both blocks have the same speed after being pushed with the same force . What can be said about the distances the two blocks are pushed? Hint 1. How to approach the problem The work-energy theorem states that the change in kinetic energy of an object equals the net work done on that object: . The work done on an object can also be related to the distance that the object moves while being acted on by a force : , where is the component of parallel to the direction of displacement. Hint 2. Relate the kinetic energies of the blocks Let the subscript refer to the heavier block and the subscript to the lighter block. What is the ratio ? Hint 1. The kinetic energies To relate the kinetic energies of the blocks to each other, recall that and . ANSWER: = 4 one quarter as fast half as fast the same speed twice as fast four times as fast

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 12/22 Hint 3. Compare the amount of work done on each block In the previous hint, you found that . What is the ratio of the work done on the heavy block to the work done on the lighter block, ? ANSWER: ANSWER: Correct Because the heavier block has four times the mass of the lighter block, when the two blocks travel with the same speed, the heavier block will have four times as much kinetic energy. The work-energy theorem implies that four times more work must be done on the heavier block than on the lighter block. Since the same force is applied to both blocks, the heavier block must be pushed through four times the distance as the lighter block. Problem 9.44 - Enhanced - with Hints and Feedback Part A Starting from rest, a crate of mass is pushed up a frictionless slope of angle by a horizontal force of magnitude . Use work and energy to find an expression for the crate's speed when it is at height above the bottom of the slope. Express your answer in terms of the variables , , , , and free fall acceleration . Hint 1. How to approach the problem To find the crate's speed, you first need to find its kinetic energy. Recall how kinetic energy of an object depends on the amount of work done on this object by external forces. ANSWER: = 4 = 4 The heavy block must be pushed 16 times farther than the light block. The heavy block must be pushed 4 times farther than the light block. The heavy block must be pushed 2 times farther than the light block. The heavy block must be pushed the same distance as the light block. The heavy block must be pushed half as far as the light block.

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12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 13/22 Correct Part B Doug uses a 29 horizontal force to push a 3.0 crate up a 1.4 -high, 20 frictionless slope. What is the speed of the crate at the top of the slope? Express your answer with the appropriate units. Hint 1. How to approach the problem Simply use the expression you derived in the previous part. ANSWER: Correct ± Hooke's Law Learning Goal: To understand the use of Hooke's law for a spring. Hooke's law states that the restoring force on a spring when it has been stretched or compressed is proportional to the displacement of the spring from its equilibrium position. The equilibrium position is the position at which the spring is neither stretched nor compressed. Recall that means that is equal to a constant times . For a spring, the proportionality constant is called the spring constant and denoted by . The spring constant is a property of the spring and must be measured experimentally. The larger the value of , the stiffer the spring. In equation form, Hooke's law can be written . The minus sign indicates that the force is in the opposite direction to that of the spring's displacement from its equilibrium length and is "trying" to restore the spring to its equilibrium position. The magnitude of the force is given by , where is the magnitude of the displacement. In Haiti, public transportation is often by taptaps, small pickup trucks with seats along the sides of the pickup bed and railings to which passengers can hang on. Typically they carry two dozen or more passengers plus an assortment of chickens, goats, luggage, etc. Putting this much into the back of a pickup truck puts quite a large load on the truck springs. A truck has springs for each wheel, but for simplicity assume that the individual springs can be treated as one spring with a spring constant that includes the effect of all the springs. Also for simplicity, assume that all four springs compress equally when weight is added to the truck and that the equilibrium length of the springs is the length they have when they support the load of an empty truck. Part A = = 6.9

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 14/22 A 68 driver gets into an empty taptap to start the day's work. The springs compress 1.9×10 −2 . What is the effective spring constant of the spring system in the taptap? Enter the spring constant numerically in newtons per meter using two significant figures. Hint 1. How to approach the problem The compression of the springs is governed by Hooke's law. The amount the springs are compressed when the driver climbs into the truck is given in the problem statement. The force that acts to compress the springs is the force caused by the driver getting into the truck. ANSWER: Correct If you need to use the spring constant in subsequent parts, use the full precision value you calculated, only rounding as a final step before submitting your answer. Part B After driving a portion of the route, the taptap is fully loaded with a total of 24 people including the driver, with an average mass of 68 per person. In addition, there are three 15- goats, five 3- chickens, and a total of 25 of bananas on their way to the market. Assume that the springs have somehow not yet compressed to their maximum amount. How much are the springs compressed? Enter the compression numerically in meters using two significant figures. Hint 1. How to find the compression of the spring The spring compression is governed by Hooke's law. Use the spring constant you calculated to full precision in Part A prior to rounding your answer. To find the force add the total weight of the load on the truck. Only round as a final step before submitting your answer. ANSWER: Correct Part C Whenever you work a physics problem you should get into the habit of thinking about whether the answer is physically realistic. Think about how far off the ground a typical small truck is. Is the answer to Part B physically realistic? Select the best choice below. ANSWER: = 3.5×10 4 = 0.48 No, typical small pickup truck springs are not large enough to compress 0.48 . Yes, typical small pickup truck springs can easily compress 0.48 .

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 15/22 Correct The answer to Part B is not physically realistic because the springs of a typical light truck will compress their maximum amount (typically about 10 ) before the total weight of all the passengers and other cargo given in Part B is added to the truck. When this maximum compression is reached, the springs will bottom out, and the ride will be very rough. Part D Now imagine that you are a Haitian taptap driver and want a more comfortable ride. You decide to replace the springs with new springs that can handle the typical heavy load on your vehicle. What spring constant do you want your new spring system to have? ANSWER: Correct A spring constant with a large value is a stiff spring. It will take more force to compress (or stretch) a stiff spring. On a taptap, stiffer springs are less likely to bottom out under a heavy load. However, with a lighter load, for most vehicles, very stiff springs will not compress as much for a bump in the road. Hence very stiff springs will give a better ride with a very heavy load, but less-stiff springs (lower spring constant) will give a smoother ride with a light load. This is why larger vehicles need stiffer springs than smaller vehicles. Problem 9.25 - Enhanced - with Hints and Feedback A 39- -long vertical spring has one end fixed on the floor. Placing a 2.2 physics textbook on the spring compresses it to a length of 29 . Part A What is the spring constant? Express your answer with the appropriate units. Hint 1. How to approach the problem Use Hooke's law and the equilibrium equation for the book to find the spring constant. ANSWER: Correct substantially larger than the spring constant of the old springs slightly larger than the spring constant of the old springs slightly smaller than the spring constant of the old springs substantially smaller than the spring constant of the old springs = 220

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12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 16/22 Problem 9.27 - Enhanced - with Expanded Hints A 12- -long spring is attached to the ceiling. When a 2.3 mass is hung from it, the spring stretches to a length of 18 . Part A What is the spring constant ? Express your answer with the appropriate units. Hint 1. How to approach the problem Start by drawing a free-body diagram indicating forces acting on the weight. Choose a coordinate system. Be sure to use your coordinate system when determining the signs of all the forces. Then use Newton's second law and Hooke's law to find . Hint 2. Simplify: Newton's second law Choose the correct expression of Newton's second law for the situation described in the problem. ANSWER: ANSWER: Correct Assume an ideal spring that obeys Hooke’s law. Weight is directed vertically downward and restoring force is directed vertically upward. The spring force on the mass is . Notice that is negative, so is positive. This force is equal to , because the mass is at rest. We have . Solving for : Part B How long is the spring when a 3.0 mass is suspended from it? Express your answer with the appropriate units. Hint 1. How to approach the problem = 380

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 17/22 Use Newton's second law and Hooke's law to find how far the spring is stretched. Consider the initial length of the spring. Hint 2. Simplify: Hooke's law Use the spring constant found in the previous part and determine the spring displacement when a 3.0 mass is suspended from it. Express your answer using four significant figures with the appropriate units. ANSWER: ANSWER: Correct Again using : The length of the spring is when a mass of 3.0 is attached to the spring. The position of the end of the spring is negative because it is below the origin, but length must be a positive number. Thermal Energy of a Laptop shows your 1.5 laptop computer being dragged up a 25 slope at steady speed by a rope with tension 12 . Both the laptop and the slope are getting warmer because of friction. We want to find the increase in thermal energy if the laptop is pulled 2.0 . Part A How should you define the system for a problem involving thermal energy? Select everything that should be included in the system. Select all that apply. = 7.826×10 −2 = 20

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 18/22 ANSWER: Correct Friction occurs between the laptop and the surface, and both objects will experience a temperature increase. Both need to be inside the system. Gravity and tension are external forces acting on the system. Part B How much work does tension do on the laptop as it moves 2.0 ? Express your answer with the appropriate units. Hint 1. How to approach the problem Work done by force acting on an object as it moves through displacement is , where is the angle between the two vectors. What is the angle between the tension and the laptop's displacement ? Express your answer in degrees. ANSWER: ANSWER: Correct Part C How much work does the normal force do on the laptop as it moves 2.0 ? Express your answer with the appropriate units. Hint 1. How to approach the problem Work done by force acting on an object as it moves through displacement is , where is the angle between the two vectors. What is the angle between the normal force and the laptop's displacement ? Express your answer in degrees. The air The entire earth The laptop The rope The surface = 0 = 24

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12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 19/22 ANSWER: ANSWER: Correct Part D How much work does the gravity do on the laptop as it moves 2.0 ? Express your answer with the appropriate units. Hint 1. How to approach the problem Work done by force acting on an object as it moves through displacement is , where is the angle between the two vectors. What is the angle between the gravitational force and the laptop's displacement ? Express your answer in degrees. ANSWER: ANSWER: Correct Part E What is the laptop's change in kinetic energy? Express your answer with the appropriate units. Hint 1. How to approach the problem The laptop is moving at a constant speed, so . ANSWER: = 90 = 0 = 115 = -12 = 0

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 20/22 Correct Part F What is the increase in thermal energy of the laptop and the surface? Express your answer with the appropriate units. Hint 1. How to approach the problem The energy principle states that . Parts B, C, and D give you the information you need to calculate . You found in Part E. ANSWER: Correct Problem 9.36 - Enhanced - with Hints and Feedback Part A How much work must you do to push a 13.0 block of steel across a steel table ( = 0.60) at a steady speed of 1.00 for 4.20 ? Express your answer with the appropriate units. Hint 1. How to approach the problem Recall the expression for work in terms of force and displacement. Write the equilibrium equation for the block to find the force you have to apply to keep the block moving at a steady speed. ANSWER: Correct Part B What is your power output while doing so? Express your answer with the appropriate units. Hint 1. How to approach the problem = 12 = 321

12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 21/22 Use your answer from the previous part to calculate power using its definition. Alternatively, you can use the expression for power in terms of force and velocity. ANSWER: Correct Problem 9.37 - Enhanced - with Expanded Hints Part A How much work does an elevator motor do to lift a 1600 elevator a height of 170 ? Express your answer with the appropriate units. Hint 1. How to approach the problem Treat the elevator as a particle, and determine what forces do work on it. Recall how the work done on a particle by a force is related to the magnitude of the force, the particle's displacement, and the angle between the force and displacement. To determine the work done by the motor, apply the energy principle, given that the elevator is at rest at the beginning and end of its movement, and no work is dissipated as thermal energy. Hint 2. Simplify: work done by gravity What is the work done by gravity on an elevator of mass moving upward by a distance of ? ANSWER: ANSWER: = 76.4 = 2.7×10 6

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12/19/22, 12:15 PM HW09 - Work and Kinetic Energy https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072427 22/22 Correct MODEL: Model the elevator as a particle, and apply the energy principle. SOLVE: The tension in the cable does work on the elevator to lift it. Because the cable is pulled by the motor, we say that the motor does the work of lifting the elevator. The energy principle tells us . Using , , and gives Part B How much power must the motor supply to do this in 40 at constant speed? Express your answer with the appropriate units. Hint 1. How to approach the problem By definition, power is the rate of transfer of energy. Focusing on work as the source of energy transfer, power is simply the rate of doing work. Hint 2. Simplify: work and power Choose the correct relationship between the work done by the motor, power supplied by the motor, and time of doing work. ANSWER: ANSWER: Correct The power required by the motor is REVIEW: Since 1 horsepower ( ) is 746 , the power of the motor is 89 . This is a reasonable amount of power to lift a mass of 1600 to a height of 170 in 40 . Score Summary: Your score on this assignment is 99.8%. You received 18.96 out of a possible total of 19 points. = 6.7×10 4

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HW09 - Work and Kinetic Energy

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