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## Potential energy Discussion

W J M Rankine coined the term potential energy 150 years ago.

gravitational potential energy

Δ*U _{g}* =

*mg*Δ

*h*

- acceleration due to gravity is nearly constant
- height change is small compared to the separation between centers

the more general form will be dealt with later

U = −_{g} | Gm_{1}m_{2} |

r |

## work-energy theorem, two possibilities

- conservative forces
- work done is independent of path
*W*= ∮**F**·*d***r**= 0- can be associated with a potential energy function

- nonconservative forces
- work done depends on path
*W*= ∮**F**·*d***r**> 0- cannot be associated with a potential energy function

## force and potential energy

- one dimensional
*F*(*r*) = −*dU**dr* - three dimensional, expanded notation
**F**(**r**) = −∂ *U***î**−∂ *U***ĵ**−∂ *U***k̂**∂ *x*∂ *y*∂ *z* - three dimensional, compact notation
**F**(**r**) = − ∇*U*

constant total energy, horizontal line above curve, kinetic energy is difference between line and curve

bounded and unbounded states, binding energy

stability of equilibrium

stable equilibrium | unstable equilibrium | neutral equilibrium |

[diagram] | [diagram] | [diagram] |

local maximum | local minimum | constant potential energy |