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]\displaystyle{ \vec F = k\vec r }[/math]

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

[math]\displaystyle{ \vec F = q \vec E }[/math]

Examples of Non Conservaive Forces

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

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

where

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

and

[math]\displaystyle{ 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]\displaystyle{ r_0 }[/math]

remember

The potential is equal to the negative of the work done by conservative forces

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)

Example: Show that a the force on a charge by an electric field is conservative

Consider a charge q in a uniform electric field given by

[math]\displaystyle{ \vec{E} = E_0 \hat i }[/math]

[math]\displaystyle{ \Rightarrow \vec{F} = q E_0 \hat i }[/math]

The work done moving the charge between two points, point 1 and point 2, using the above force is

[math]\displaystyle{ W = \int_1^2 \vec{F} \cdot d \vec r = q E_0 \int_1^2 \hat i \cdot d \vec r }[/math]

[math]\displaystyle{ = \int_1^2 dx = qE_0 (x_2 - x_1) }[/math]

Since the work done by the above force only depends on the endpoints ([math]\displaystyle{ x_1 }[/math] and [math]\displaystyle{ x_2 }[/math]) the force is conservative.

The change in the potential energy is then

[math]\displaystyle{ \Delta U = -W = -qE_0 (x_2 - x_1) }[/math]

You can define a potential energy U with respect to an arbitrary reference point [math]\displaystyle{ (x_1=0) }[/math] such that

[math]\displaystyle{ U = -qE_0x }[/math]

conservation of mechanical energy

Let

[math]\displaystyle{ r_1 }[/math] and [math]\displaystyle{ 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]\displaystyle{ r_0 }[/math] to [math]\displaystyle{ r_2 }[/math] maybe be written as

[math]\displaystyle{ \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]

Re-arranging terms

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

[math]\displaystyle{ = -U(\vec{r}_2) - \left (-U(\vec{r}_1) \right) }[/math]

[math]\displaystyle{ = -\left [U(\vec{r}_2) - U(\vec{r}_1 \right] }[/math]

[math]\displaystyle{ = -\Delta U }[/math]

since

[math]\displaystyle{ \Delta T = W }[/math]

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

[math]\displaystyle{ = -\Delta U }[/math]

or

[math]\displaystyle{ \Delta T + \Delta U = 0 }[/math]

[math]\displaystyle{ \Delta (T + U) = 0 }[/math]

[math]\displaystyle{ T+U = }[/math] constant[math]\displaystyle{ \equiv E }[/math]

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

This conservation of Mechanical energy equation holds when the Force is a function of [math]\displaystyle{ r }[/math]

It does not mean that other conservation of energy equations may exist for forces that are not just functions of [math]\displaystyle{ r }[/math]

Frictional force example

Consider the inclined plan problem with friction

[math]\displaystyle{ \vec{N} + \vec{F}_f + m\vec g = m \vec a }[/math]

The work energy theorem states

[math]\displaystyle{ T_2-T_1 = \int_1^2 \sum_{i=1}^n \vec {F}_i \cdot d\vec r = W_2 - W_1 }[/math]

or

[math]\displaystyle{ \Delta T = W_{tot} = W_{cons} + W_{non-cons} }[/math]

or

[math]\displaystyle{ \Delta T - W_{cons} = W_{non-cons} }[/math]

[math]\displaystyle{ \Delta T + \Delta U = W_{non-cons} }[/math]

Here friction is proportional to the gravitational force, it does not depend on position though.

Aside on non-conservative Lorentz force

Given the experience of air friction's dependence on velocity, classical mechanic may argue that the velocity dependence of the Lorentz Force's magnetic term is also non-conservative

for example:

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,

But there is a special class of velocity dependent forces ( in particular forces that act perpendicular to an objects velocity) where a Hamiltonian can be formed and as a result the Energy is constant. The result of forming such a Hamiltonia is that an additional term is added to the kinetic energy part of the Hamiltonian that comes from the ability to define a magnetic vector potential[math]\displaystyle{ (\vec{A}) }[/math].

ie

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

but

[math]\displaystyle{ \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,

http://www.researchgate.net/post/Is_magnetic_force_a_conservative_or_non-conservative_force

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CDIQFjAE&url=http%3A%2F%2Fwww.phys.ttu.edu%2F~cmyles%2FPhys5306%2FLectures%2FLecture05.ppt&ei=JV8gVNn9Ns-2ogS-rIKADQ&usg=AFQjCNEa_Beegx-6zIvcKzdy_hhh6ngQgg&bvm=bv.75775273,d.cGU

Forest_UCM_Energy#PE_.26_Conservative_Force