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Chapter 7

Chapter 7. Potential Energy. 7.1 Potential Energy. Potential energy is the energy associated with the configuration of a system of objects that exert forces on each other The forces are internal to the system. Types of Potential Energy. There are many forms of potential energy, including:

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Chapter 7

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  1. Chapter 7 Potential Energy

  2. 7.1 Potential Energy • Potential energy is the energy associated with the configuration of a system of objects that exert forces on each other • The forces are internal to the system

  3. Types of Potential Energy • There are many forms of potential energy, including: • Gravitational • Electromagnetic • Chemical • Nuclear • One form of energy in a system can be converted into another

  4. System Example • This system consists of Earth and a book • Do work on the system by lifting the book through Dy • The work done is mgyb - mgya Fig 7.1

  5. Potential Energy • The energy storage mechanism is called potential energy • A potential energy can only be associated with specific types of forces • Potential energy is always associated with a system of two or more interacting objects

  6. Gravitational Potential Energy • Gravitational Potential Energy is associated with an object at a given distance above Earth’s surface • Assume the object is in equilibrium and moving at constant velocity • The work done on the object is done by and the upward displacement is

  7. Gravitational Potential Energy, cont • The quantity mgy is identified as the gravitational potential energy, Ug • Ug = mgy • Units are joules (J)

  8. Gravitational Potential Energy, final • The gravitational potential energy depends only on the vertical height of the object above Earth’s surface • In solving problems, you must choose a reference configuration for which the gravitational potential energy is set equal to some reference value, normally zero • The choice is arbitrary because you normally need the difference in potential energy, which is independent of the choice of reference configuration

  9. 7.2 Conservation of Mechanical Energy • The mechanical energy of a system is the algebraic sum of the kinetic and potential energies in the system • Emech = K + Ug • The statement of Conservation of Mechanical Energy for an isolated system is Kf + Ugf= Ki+ Ugi • An isolated system is one for which there are no energy transfers across the boundary

  10. Conservation of Mechanical Energy, example • Look at the work done by the book as it falls from some height to a lower height • Won book = DKbook • Also, W = mgyb – mgya • So, DK = -DUg Fig 7.2

  11. Active Figure 7.3 NEXT

  12. A ball of mass m is dropped from rest at a height h above the ground as in Figure 7.4. Ignore air resistance. Fig 7.4

  13. You are designing apparatus to support an actor ofmass 65 kg who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle as the angle that the actor’s cable makes with the vertical when he begins his motion from rest. What is the maximum value  can have such that the sandbag does not lift off the floor during the actor’s swing?

  14. Fig 7.5

  15. Since T = mbagg

  16. 7.3 Conservative Forces • A conservative force is a force between members of a system that causes no transformation of mechanical energy within the system • The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle • The work done by a conservative force on a particle moving through any closed path is zero

  17. Nonconservative Forces • A nonconservative force does not satisfy the conditions of conservative forces • Nonconservative forces acting in a system cause a change in the mechanical energy of the system

  18. Nonconservative Force, Example • Friction is an example of a nonconservative force • The work done depends on the path • The red path will take more work than the blue path Fig 7.7

  19. Elastic Potential Energy • Elastic Potential Energy is associated with a spring, Us = 1/2 k x2 • The work done by an external applied force on a spring-block system is • W = 1/2 kxf2 – 1/2 kxi2 • The work is equal to the difference between the initial and final values of an expression related to the configuration of the system

  20. Elastic Potential Energy, cont • This expression is the elastic potential energy: Us = 1/2 kx2 • The elastic potential energy can be thought of as the energy stored in the deformed spring • The stored potential energy can be converted into kinetic energy Fig 7.6

  21. Elastic Potential Energy, final • The elastic potential energy stored in a spring is zero whenever the spring is not deformed (U = 0 when x = 0) • The energy is stored in the spring only when the spring is stretched or compressed • The elastic potential energy is a maximum when the spring has reached its maximum extension or compression • The elastic potential energy is always positive • x2 will always be positive

  22. Active Figure 7.6 NEXT

  23. Conservation of Energy, Extended • Including all the types of energy discussed so far, Conservation of Energy can be expressed as DK + DU + DEint = DE system = 0 or K + U + E int = constant • K would include all objects • U would be all types of potential energy

  24. Problem Solving Strategy – Conservation of Mechanical Energy • Conceptualize: Define the isolated system and the initial and final configuration of the system • The system may include two or more interacting particles • The system may also include springs or other structures in which elastic potential energy can be stored • Also include all components of the system that exert forces on each other

  25. Problem-Solving Strategy, 2 • Categorize: Determine if any energy transfers across the boundary • If it does, use the nonisolated system model, DE system = ST • If not, use the isolated system model, DEsystem = 0 • Determine if any nonconservative forces are involved

  26. Problem-Solving Strategy, 3 • Analyze: Identify the configuration for zero potential energy • Include both gravitational and elastic potential energies • If more than one force is acting within the system, write an expression for the potential energy associated with each force

  27. Problem-Solving Strategy, 4 • If friction or air resistance is present, mechanical energy of the system is not conserved • Use energy with non-conservative forces instead • The difference between initial and final energies equals the energy transformed to or from internal energy by the nonconservative forces

  28. Problem-Solving Strategy, 5 • If the mechanical energy of the system is conserved, write the total energy as • Ei = Ki + Ui for the initial configuration • Ef = Kf + Uf for the final configuration • Since mechanical energy is conserved, Ei = Ef and you can solve for the unknown quantity

  29. Mechanical Energy and Nonconservative Forces • In general, if friction is acting in a system: • DEmech = DK + DU = -ƒkd • DU is the change in all forms of potential energy • If friction is zero, this equation becomes the same as Conservation of Mechanical Energy

  30. A 3.00-kg crate slides down a ramp at a loading dock. The ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp. Fig 7.8

  31. Nonconservative Forces, Example 1 (Slide) DEmech = DK + DU DEmech =(Kf – Ki) + (Uf – Ui) DEmech = (Kf + Uf) – (Ki + Ui) DEmech = 1/2 mvf2 – mgh = -ƒkd

  32. Nonconservative Forces, Example 2 (Spring-Mass) • Without friction, the energy continues to be transformed between kinetic and elastic potential energies and the total energy remains the same • If friction is present, the energy decreases • DEmech = -ƒkd

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