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 poetential 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)

Forest_UCM_Energy#PE_.26_Conservative_Force