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Chapter 7

Chapter 7. Potential Energy. 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. 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

  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. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

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

  25. 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

  26. Conservative Forces and Potential Energy • Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system • The work done by such a force, F, is • DU is negative when F and x are in the same direction

  27. Conservative Forces and Potential Energy • The conservative force is related to the potential energy function through • The conservative force acting between parts of a system equals the negative of the derivative of the potential energy associated with that system • This can be extended to three dimensions

  28. Conservative Forces and Potential Energy – Check • Look at the case of an object located some distance y above some reference point: • This is the expression for the vertical component of the gravitational force

  29. Nonisolated System in Steady State • A system can be nonisolated with 0 = DT • This occurs if the rate at which energy is entering the system is equal to the rate at which it is leaving • There can be multiple competing transfers

  30. Nonisolated System in Steady State, House Example

  31. Potential Energy for Gravitational Forces • Generalizing gravitational potential energy uses Newton’s Law of Universal Gravitation: • The potential energy then is

  32. Potential Energy for Gravitational Forces, Final • The result for the earth-object system can be extended to any two objects: • For three or more particles, this becomes

  33. Electric Potential Energy • Coulomb’s Law gives the electrostatic force between two particles • This gives an electric potential energy function of

  34. Energy Diagrams and Stable Equilibrium • The x = 0 position is one of stable equilibrium • Configurations of stable equilibrium correspond to those for which U(x) is a minimum • x=xmax and x=-xmax are called the turning points

  35. Energy Diagrams and Unstable Equilibrium • Fx = 0 at x = 0, so the particle is in equilibrium • For any other value of x, the particle moves away from the equilibrium position • This is an example of unstable equilibrium • Configurations of unstable equilibrium correspond to those for which U(x) is a maximum

  36. Neutral Equilibrium • Neutral equilibrium occurs in a configuration when U is constant over some region • A small displacement from a position in this region will produce neither restoring nor disrupting forces

  37. Potential Energy in Fuels • Fuel represents a storage mechanism for potential energy • Chemical reactions release energy that is used to operate the automobile

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