Forest UCM Energy PE n ConsForces

Potential Energy and Conservative Forces

Conservative Forces

What is a conservative Force?

Two conditions are neccessary for a Force to be conservative.

1. The force on depends on the objects position (F = F(r)). Not on time nor velocity
2. The work done by the force in displacing the object between two points is independent of the path taken.

Examples of conservative forces

[math]\vec F = k\vec r[/math]

[math]\vec F = G \frac{m_1 m_2}{r^3} \vec r[/math]

[math]\vec F = q \vec E[/math]

Examples of Non Conservaive Forces

[math]\vec F = -bv -cv^2[/math]

[math]\vec{F} = q \vec v \times \vec B[/math]

Potential Energy

If ALL External forces are conservative

Then a potential energy U(r) may be defined such that the total energy of the system is constant (conserved)

[math]E_{tot} = T + U(r) =[/math] constant

where

[math]U(r) \equiv -\int_{r_o}^r \vec{F}(r) \cdot d\vec{}r[/math]

and

[math]r_0 =[/math] an arbitrary reference point where the potential is often chosen to be zero

It is not necessary to define the potential as zero at [math]r_0[/math]

remember

Positive Work INcreases the kinetic energy (T) but DEcreases the Potential energy (U)

Negative Work DEcreases the kinetic energy (T) but INcreases the Potential energy (U)

conservation of mechanical enegy

Let

[math]r_1[/math] and [math]r_2[/math]

be any two points used to locate and object.

the work done to move an object from an arbitrary reference point [math]r_0[/math] to [math]r_2[/math] maybe be written as

[math]\int_{r_0}^{r_2} \vec{F}(\vec r) \cdot dr = \int_{r_0}^{r_1} \vec{F}(\vec r) \cdot dr + \int_{r_1}^{r_2} \vec{F}(\vec r) \cdot dr[/math]

[math]E_{tot} = T + U(r) =[/math] constant

Aside on non-conservative Lorentz force

While the magnetic term of the lorentz force from a classical mechanics point of view is not conservative, one can consider some systems influence by this force where the energy is constant in time. The result is that an additional term is added to the kinetic energy part of the Hamiltonian that come from the ability to define a magnetic vector potential[math] (\vec{A})[/math].

ie

[math]\vec{E} = - \vec{\nabla} \phi - \frac{\part}{\part t} \vec{A}[/math]

but

[math]\vec{B} = \vec{\nabla} \times \vec A[/math]

http://lamp.tu-graz.ac.at/~hadley/ss1/IQHE/cpimf.php

http://insti.physics.sunysb.edu/itp/lectures/01-Fall/PHY505/09c/notes09c.pdf

http://www.tcm.phy.cam.ac.uk/~bds10/aqp/lec5.pdf

You can have systems where the magnetic field causes polarizable atoms to realign and as a result heat can be produced converting energy in the system to a form that would need to be tractable in order to conserve energy,

Forest_UCM_Energy#PE_.26_Conservative_Force