If ∮ F·dr
= 0 for a given force, that force is called conservative,
and it is possible to define a potential energy
corresponding to the force.
The gravitational potential
energy Ug increases by the amount of work we
would have to do against gravity to lift an object a given
distance. In general we could define Ug = mgy,
where y is the vertical distance lifted, with the potential
energy arbitrarily taken to be zero at the arbitrary point y =
0. Note that the concept of potential energy has no
meaning at all unless we specify a zero point; this has to be
done every time the concept is used!!
In the same way we could
define a spring potential energy, or elastic potential energy,
as Us = (1/2)kx2, where x is the
distance stretched from equilibrium.
CONSERVATION OF ENERGY: E =
Σi Ki + ΣjUcj is a
constant if all conservative forces that can do work on any
body in the system are included in the potential energy terms.
It is vital to
remember that the zero of PE is arbitrary and must be
specified in order to set up any conservation of
energy example.
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Including non-conservative
forces, like friction: ΔE = Wnc.
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Potential
energy plots contain a tremendous amount of information.
For example a particle with an energy of 2 J starting at
x = 2 m can never reach 3 m or 1 m. A particle
starting at x = 4 m with an energy of 3 J can
never reach 5.2 m or 1 m. A particle thrown toward the
origin of coordinates at 5 J, starting at 2 m, will
bounce back at 1 m, never reaching the origin.
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Most physics textbooks are
written by people who aren't professionally physicists, and
many meaningless terms are used. One of the most irritating is
“mechanical energy.” There is no such thing. There are only
three forms of energy known in our universe: (1) kinetic
energy, (2) potential energy, and (3) the energy related to
inertial mass, as given by the famous equation E = mc2.
All “forms” of energy are combinations of these basic three,
involving one or more potential energies associated with the
four fundamental forces of nature.
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