Conservative Forces and Potentials

1. Conservative Forces and Potentials Which forces are conservative? § 7.4

2. Forces and potentials Every conservative force is a spatial derivative of a potential energy function. Specifically, F = –(idU/dx + jdU/dy + kdU/dz) (This is Calculus 3 stuff)

3. Forces and potentials Every conservative force is a spatial derivative of a potential energy function. • Near-surface gravity: Source: Young and Freedman, Figure 7.22b.

4. Forces and potentials Every conservative force is a spatial derivative of a potential energy function. • Hooke’s law spring: Source: Young and Freedman, Figure 7.22a.

5. Equilibrium Potentials • Stable equilibrium: small excursions damped by a restoring force • Unstable equilibrium: small excursions amplified by non-restoring force • Force is zero at an equilibrium point • Potential is locally unchanging

6. Whiteboard Work A particle is in neutral equilibrium if the net force on it is zero and remains zero if the particle is displaced slightly in any direction. • Sketch a one-dimensional potential energy function near a point of neutral equilibrium. • Give an example of a neutral equilibrium potential.

7. Energy Diagrams Keeping track—and more! § 7.5

8. Energy K 0 K r Energy diagram Plot U as a function of position Mark total E as a horizontal line K = E – U (function of position) E U Diagram shows the partition of energy everywhere.

9. Energy 0 r Energy diagram Where is the particle? How does it behave? E U

10. Energy E 0 r Energy diagram If E is lower: Where is the particle? How does it behave? U

11. Poll Question Which points are stable equilibria? Add correct answers together. 1. x1. 2. x2. 4. x3. 8. x4. Source: Young and Freedman, Figure 7.24a.

12. Poll Question Which positions are accessible if E = E2? Add correct answers together. 1. x1. 2. x2. 4. x3. 8. x4. Source: Young and Freedman, Figure 7.24a.

13. Potential Well Particles can become trapped. Source: Young and Freedman, Figure 7.24a.