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7.4 Conservative Forces and Potential Energy

7.4 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 D U is negative when F and x are in the same direction.

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7.4 Conservative Forces and Potential Energy

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  1. 7.4 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

  2. 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

  3. 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

  4. 7.6 Potential Energy for Gravitational Forces • Generalizing gravitational potential energy uses Newton’s Law of Universal Gravitation: • The potential energy then is Fig 7.12

  5. Potential Energy for Gravitational Forces, Final • The result for the earth-object system can be extended to any two objects:

  6. Gravitational potential energy for three particles

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

  8. 7.7 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 Fig 7.15

  9. 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 Fig 7.16

  10. A particle is attached between two identical springs on a horizontal frictionless table. Both springs have spring constant k and are initially unstressed. (a) The particle is pulled a distance x along a direction perpendicular to the initial configuration of the springs as shown in Figure. Show that the force exerted by the springs on the particle is • (b) Determine the amount of work done by this force in moving the particle from x = A to x = 0.

  11. (a) Show that the potential energy of the system is • (b) Make a plot of U(x) versus x and identify all equilibrium points. • (c) If the particle of mass m is pulled in a distance d to the right and then released, what is its speed when it reaches the equilibrium point x = 0?

  12. Exercises of chapter 7 • 3, 5, 9, 14, 17, 26, 33, 39, 42, 54, 62

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