Conservative forces and potentials
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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

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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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