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.
- The force on depends on the objects position (F = F(r)). Not on time nor velocity
- 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{ E_{tot} = T + U(r) = }[/math] constant
where
- [math]\displaystyle{ U(r) \equiv -\int_{r_o}^r \vec{F}(r) \cdot d\vec{}r }[/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
- 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]\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 = \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
Aside on non-conservative Lorentz force
Given the experience of air frictions 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