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1 1 of 6 10/21/2009 6:33 PM Chapter 10 Homework Due: 9:00am on Thursday, October 22, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View] Conceptual Question 10.2 Upon what basic quantity does kinetic energy depend? Position Acceleration Speed Upon what basic quantity does potential energy depend? Speed Acceleration Position A spring is compressed 1.1. Conceptual Question How far must you compress a spring with twice the spring constant to store the same amount of energy? Express your answer using two significant figures. = 0.78 Conservation of Energy Ranking Task Six pendulums of various masses are released from various heights above a tabletop, as shown in the figures below. All the pendulums have the same length, are mounted the same distance above the table and just do not strike the tabletop when released. Rank each pendulum on the basis of its initial gravitational potential energy (before being released). Gravitational potential energy Rank from largest to smallest To rank items as equivalent, overlap them. View Rank each pendulum on the basis of the maximum kinetic energy it attains after release. Kinetic energy Rank from largest to smallest To rank items as equivalent, overlap them. View

2 2 of 6 10/21/2009 6:33 PM Rank each pendulum on the basis of its maximum speed. The role of mass Rank from largest to smallest To rank items as equivalent, overlap them. View Stretching a Spring As illustrated in the figure, a spring with spring constant is stretched from to, where is the equilibrium position of the spring. During which interval is the largest amount of energy required to stretch the spring? From to From to From to The energy required is the same in all three intervals. A graph of the force exerted on the spring versus the displacement of the spring is shown in the figure. Recall that on a graph of force as a function of position, the work done by the force is represented by the area under the curve. The work done by the hand in the first segment to pull the spring from to is represented by a single triangle. The area under the second segment from to is three times larger than the first segment, and the area under the third segment from to is five times larger than in the first segment. So more energy is required to pull the spring through the third segment. A spring is stretched from to, where is the equilibrium position of the spring. It is then compressed from to. What can be said about the energy required to stretch or compress the spring? More energy is required to stretch the spring than to compress it. The same amount of energy is required to either stretch or compress the spring. Less energy is required to stretch the spring than to compress it. The work done to stretch or compress a spring from equilibrium is given by,

3 3 of 6 10/21/2009 6:33 PM where is the distance away from equilibrium that the spring moves. Since is squared in the equation for work, stretching ( ) or compressing ( ) a spring by the same distance requires the same positive amount of work. Now consider two springs A and B that are attached to a wall. Spring A has a spring constant that is four times that of the spring constant of spring B. If the same amount of energy is required to stretch both springs, what can be said about the distance each spring is stretched? How to approach this problem The work done to stretch or compress a spring is given by, where is the distance away from equilibrium that the spring is displaced. Use this expression to relate the information provided about the work done on each spring and the spring constants to the distance each spring stretches. Hint C.2 Use proportional reasoning to find a relationship between the springs From the problem statement you know that, where and. Use this information to find an expression for. = Answer not displayed Spring A must stretch 4 times as far as spring B Spring A must stretch 2 times as far as spring B. Spring A must stretch the same distance as spring B. Spring A must stretch half the distance spring B stretches. Spring A must stretch one-quarter of the distance spring B stretches. The energy required to stretch a spring is proportional to and to. If is four times, must be half that of, so the energy required is the same for both springs. Part D Two identical springs are attached to two different masses, and, where is greater than. The masses lie on a frictionless surface. Both springs are compressed the same distance,, as shown in the figure. Which of the following statements descibes the energy required to compress spring A and spring B? Spring A requires more energy than spring B. Spring A requires the same amount of energy as spring B. Spring A requires less energy than spring B. Not enough information is provided to answer the question. Good job; you have realized an important fact. The work done on a spring to compress it a distance is given by. The amount of mass attached to the spring does not affect the work required to stretch or compress the spring. Spring Gun A spring-loaded toy gun is used to shoot a ball straight up in the air. The ball reaches a maximum height, measured from the equilibrium position of the spring.

4 4 of 6 10/21/2009 6:33 PM The same ball is shot straight up a second time from the same gun, but this time the spring is compressed only half as far before firing. How far up does the ball go this time? Neglect friction. Assume that the spring is ideal and that the distance by which the spring is compressed is negligible compared to. Potential energy of the spring The potential energy of a spring is proportional to the square of the distance the spring is compressed. The spring was compressed half the distance, so the mass, when launched, has one quarter of the energy as in the first trial. Hint A.2 Potential energy of the ball At the highest point in the ball's trajectory, all of the spring's potential energy has been converted into gravitational potential energy of the ball. height = Problem 10.9 A 59.0 skateboarder wants to just make it to the upper edge of a "quarter pipe," a track that is one-quarter of a circle with a radius of What speed does he need at the bottom? 8.52 m/s Problem A student places her 460 physics book on a frictionless table. She pushes the book against a spring, compressing the spring by 8.90, then releases the book. What is the book's speed as it slides away? The spring constant is m/s Problem A 44.0 marble moving at 1.90 strikes a 30.0 marble at rest. What is the speed of each marble immediately after the collision? = m/s = 2.26 m/s A Bullet Is Fired into a Wooden Block A bullet of mass is fired horizontally with speed at a wooden block of mass resting on a frictionless table. The bullet hits the block and becomes completely embedded within it. After the bullet has come to rest within the block, the block, with the bullet in it, is traveling at speed.

5 5 of 6 10/21/2009 6:33 PM Which of the following best describes this collision? Types of collisions perfectly elastic partially inelastic perfectly inelastic Which of the following quantities, if any, are conserved during this collision? When is kinetic energy conserved? kinetic energy only momentum only kinetic energy and momentum neither momentum nor kinetic energy What is the speed of the block/bullet system after the collision? Hint C.2 Find the momentum after the collision Use conservation of momentum Express your answer in terms of,, and. = A Girl on a Trampoline A girl of mass kilograms springs from a trampoline with an initial upward velocity of meters per second. At height meters above the trampoline, the girl grabs a box of mass kilograms. For this problem, use meters per second per second for the magnitude of the acceleration due to gravity. What is the speed of the girl immediately before she grabs the box? Hint A.2 Hint A.3 Initial kinetic energy Potential energy at height Express your answer numerically in meters per second. = 4.98 What is the speed of the girl immediately after she grabs the box?

6 6 of 6 10/21/2009 6:33 PM Hint B.2 Total initial momentum Express your answer numerically in meters per second. = 3.98 Is this "collision" elastic or inelastic? Definition of an inelastic collision elastic inelastic In inelastic collisions, some of the system's kinetic energy is lost. In this case the kinetic energy lost is converted to heat energy in the girl's muscles as she grabs the box, and sound energy. Part D What is the maximum height that the girl (with box) reaches? Measure with respect to the top of the trampoline. Hint D.1 Hint D.2 Finding Hint D.3 Finding Express your answer numerically in meters. = 2.81 Score Summary: Your score on this assignment is 99.6%. You received out of a possible total of 60 points.

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