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