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Conservative Forces and Potentials Which forces are conservative? § 7.4

# Conservative Forces and Potentials Which forces are conservative? § 7.4

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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?

2. x2.

4. x3.

8. x4.

Source: Young and Freedman, Figure 7.24a.

Poll Question

Which positions are accessible if E = E2?

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.