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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,. F = –( i dU / dx + j dU / dy + k dU / dz ). ( This is Calculus 3 stuff).

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conservative forces and potentials

Conservative Forces and Potentials

Which forces are conservative?

§ 7.4

forces and potentials
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)

forces and potentials1
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 potentials2
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
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
whiteboard work
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.
energy diagrams

Energy Diagrams

Keeping track—and more!

§ 7.5

energy diagram

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.

energy diagram1

Energy

0

r

Energy diagram

Where is the particle?

How does it behave?

E

U

energy diagram2

Energy

E

0

r

Energy diagram

If E is lower:

Where is the particle?

How does it behave?

U

poll question
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 question1
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
Potential Well

Particles can become trapped.

Source: Young and Freedman, Figure 7.24a.

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