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Conservative Forces and Potentials

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Conservative Forces and Potentials

Which forces are conservative?

§ 7.4

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)

Every conservative force is a spatial derivative of a potential energy function.

- Near-surface gravity:

Source: Young and Freedman, Figure 7.22b.

Every conservative force is a spatial derivative of a potential energy function.

- Hooke’s law spring:

Source: Young and Freedman, Figure 7.22a.

- 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

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.

Energy Diagrams

Keeping track—and more!

§ 7.5

Energy

K

0

K

r

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.

Energy

0

r

Where is the particle?

How does it behave?

E

U

Energy

E

0

r

If E is lower:

Where is the particle?

How does it behave?

U

Which points are stable equilibria?

Add correct answers together.

1.x1.

2.x2.

4.x3.

8.x4.

Source: Young and Freedman, Figure 7.24a.

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.

Particles can become trapped.

Source: Young and Freedman, Figure 7.24a.