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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 diagramPlot U as a function of position
Energy
0r
Mark total E as a horizontal lineK = E – U
E
UK
K
Diagram shows the partition of energy everywhere.
(function of position)
Energy diagramWhere is the particle?
How does it behave?
Energy
0r
E
U
Energy diagramIf E is lower:
Where is the particle?
How does it behave?
Energy
0r EU
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.
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.
Potential Well
Particles can become trapped.
Source: Young and Freedman, Figure 7.24a.