University Physics with Modern Physics Chapter 7 PDF

University Physics with Modern Physics Fifteenth Edition Chapter 7 Potential Energy and Energy Conservation Copyright © 2020 Pearson Education, Inc. All...

University Physics with Modern Physics Fifteenth Edition Chapter 7 Potential Energy and Energy Conservation Copyright © 2020 Pearson Education, Inc. All Rights Reserved Learning Outcomes In this chapter, you’ll learn… how to use gravitational potential energy in problems that involve vertical motion. how to use elastic potential energy in problems that involve a moving object attached to a stretched or compressed spring. the distinction between conservative and nonconservative forces. (Conservative forces always have a corresponding potential-energy function.) how to use energy diagrams to understand how an object moves under the influence of a conservative force. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Introduction How do energy concepts apply to the descending sandhill crane? We will see that we can think of energy as being stored and transformed from one form to another. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Gravitational Potential Energy (1 of 3) When a particle is in the gravitational field of the earth, there is a gravitational potential energy associated with the particle: As the basketball descends, gravitational potential energy is converted to kinetic energy and the basketball’s speed increases. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Gravitational Potential Energy (2 of 3) The change in gravitational potential energy is related to the work done by gravity. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Gravitational Potential Energy (3 of 3) When the object moves up,y increases, the work done by the gravitational force is negative, and the gravitational potential energy increases. Copyright © 2020 Pearson Education, Inc. All Rights Reserved The Conservation of Mechanical Energy (1 of 2) The total mechanical energy of a system is the sum of its kinetic energy and potential energy. A quantity that always has the same value is called a conserved quantity. When only the force of gravity does work on a system, the total mechanical energy of that system is conserved. This is an example of the conservation of mechanical energy. Video Tutor Demonstration: Chin Basher? Copyright © 2020 Pearson Education, Inc. All Rights Reserved The Conservation of Mechanical Energy (2 of 2) When only the force of gravity does work on a system, the total mechanical energy of that system is conserved. Video Tutor Solution: Example 7.1 Copyright © 2020 Pearson Education, Inc. All Rights Reserved When Forces Other Than Gravity Do Work Copyright © 2020 Pearson Education, Inc. All Rights Reserved Work and Energy Along a Curved Path We can use the same expression for gravitational potential energy whether the object’s path is curved or straight. W grav =mgy 1 −mgy 2 Copyright © 2020 Pearson Education, Inc. All Rights Reserved Conceptual Example 7.3 Two identical balls leave from the same height with the same speed but at different angles. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Elastic Potential Energy (1 of 2) A object is elastic if it returns to its original shape after being deformed. Elastic potential energy is the energy stored in an elastic object, such as a spring: Copyright © 2020 Pearson Education, Inc. All Rights Reserved Elastic Potential Energy (2 of 2) The Achilles tendon acts like a natural spring. When it stretches and then relaxes, this tendon stores and then releases elastic potential energy. This spring action reduces the amount of work your leg muscles must do as you run. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Work Done by a Spring Figure 7.13 below shows how a spring does work on a block as it is stretched and compressed. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Elastic Potential Energy The graph of elastic potential energy for an ideal spring is a parabola. x is the extension or compression of the spring. Elastic potential energy is never negative. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Situations with Both Gravitational and Elastic Forces When a situation involves both gravitational and elastic forces, the total potential energy is the sum of the gravitational potential energy and the elastic potential energy: U =U grav +U el Copyright © 2020 Pearson Education, Inc. All Rights Reserved Conservative and Nonconservative Forces A conservative force allows conversion between kinetic and potential energy. Gravity and the spring force are conservative. The work done between two points by any conservative force a) can be expressed in terms of a potential energy function. b) is reversible. c) is independent of the path between the two points. d) is zero if the starting and ending points are the same. A force (such as friction) that is not conservative is called a nonconservative force, or a dissipative force. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Conservative Forces The work done by a conservative force such as gravity depends on only the endpoints of a path, not the specific path taken between those points. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Nonconservative Forces As an automobile tire flexes as it rolls, nonconservative internal friction forces act within the rubber. Mechanical energy is lost and converted to internal energy of the tire. This causes the temperature and pressure of a tire to increase as it rolls. That’s why tire pressure is best checked before the car is driven, when the tire is cold. Video Tutor Solution: Example 7.11 Copyright © 2020 Pearson Education, Inc. All Rights Reserved Conservation of Energy Nonconservative forces do not store potential energy, but they do change the internal energy of a system. The law of conservation of energy means that energy is never created or destroyed; it only changes form. This law can be expressed as K  U  U in t  0. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Example Copyright © 2020 Pearson Education, Inc. All Rights Reserved Example Copyright © 2020 Pearson Education, Inc. All Rights Reserved Example Copyright © 2020 Pearson Education, Inc. All Rights Reserved Force and Potential Energy in One Dimension (1 of 3) In one dimension, a conservative force can be obtained from its potential energy function using: In regions whereU (x ) changes most rapidly withx , this corresponds to a large force magnitude. Also, whenFx (x ) is in the positivex -direction,U (x ) decreases with increasingx. A conservative force always acts to push the system toward lower potential energy. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Force and Potential Energy in One Dimension (2 of 3) Elastic potential energy and force as functions ofx for an ideal spring. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Force and Potential Energy in One Dimension (3 of 3) Gravitational potential energy and the gravitational force as functions ofy. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Force and Potential Energy in Three Dimensions (1 of 2) In three dimensions, the components of a conservative force can be obtained from its potential energy function using partial derivatives: Copyright © 2020 Pearson Education, Inc. All Rights Reserved Force and Potential Energy in Three Dimensions (2 of 2) When we take the partial derivative ofU with respect to each coordinate, multiply by the corresponding unit vector, and then take the vector sum, this is called the U : gradient of Video Tutor Solution: Example 7.14 Copyright © 2020 Pearson Education, Inc. All Rights Reserved Force and Potential Energy The greater the elevation of a hiker in Canada’s Banff National Park, the greater the gravitational potential U grav. energy Where the mountains have steep slopes,U grav has a large gradient and there’s a strong force pushing you along the mountain’s surface toward a region of lower elevation (and U hence lower grav). There’s zero force along the surface of the lake, which is all at the same elevation. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Energy Diagrams An energy diagram is a graph that shows both the potential-energy function U x ( ) and the total mechanical energyE. The figure illustrates the energy diagram for a glider attached to a spring on an air track. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Force and a Graph of Its Potential- Energy Function For any graph of potential energy versusx , the dU corresponding force is Fx  dx. Whenever the slope ofU is zero, the force there is zero, and this is a point of equilibrium. WhenU is at a minimum, the force near the minimum draws the object closer to the minimum, so it is a restoring force. This is called stable equilibrium. WhenU is at a maximum, the force near the maximum draws the object away from the maximum. This is called unstable equilibrium. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Unstable Equilibrium Each of these acrobats is in unstable equilibrium. The gravitational potential energy is lower no matter which way an acrobat tips, so if she begins to fall she will keep on falling. Staying balanced requires the acrobats’ constant attention. Copyright © 2020 Pearson Education, Inc. All Rights Reserved