Conservative Forces and Potentials Which forces are conservative? § 7.4

Forces and potentials Every conservative force is a spatial derivative of a potential energy function. Specifically, ( This is Calculus 3 stuff) F = –(i dU/dx + j dU/dy + k dU/dz)

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.

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.

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

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. a.Sketch a one-dimensional potential energy function near a point of neutral equilibrium. b.Give an example of a neutral equilibrium potential.

Energy Diagrams Keeping track—and more! § 7.5

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

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

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

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

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

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