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Potential energy
Consider a body moving in a conservative forcefield
. Let us
arbitrarily pick some point in this field. We can define a
function which possesses a unique value at every point in
the field. The value of this function associated with some
general point is simply

(155) 
In other words, is just the energy transferred to the field (i.e., minus
the work done by the field) when the
body moves from point to point . Of course, the value of at point
is zero: i.e., . Note that the above definition uniquely specifies , since
the work done when a body moves between two points in a conservative
forcefield is independent of the path taken between these points. Furthermore,
the above definition would make no sense in a nonconservative field, since
the work done when a body moves between two points in such a
field is dependent on the chosen path: hence, would have an infinite number
of different values corresponding to the infinite number of different paths the body could take
between points and .
According to the workenergy theorem,

(156) 
In other words, the net change in the kinetic energy of the body, as it
moves from point to point , is equal to the work done
on the body by the forcefield during this process.
However, comparing with Eq. (155), we can see that

(157) 
In other words, the increase in the kinetic energy of the body, as it moves
from point to point , is equal to the decrease in the
function evaluated between these same two points. Another way
of putting this is

(158) 
i.e., the sum of the kinetic energy and the function remains
constant as the body moves around in the forcefield. It should be clear, by now, that the
function represents some form of potential energy.
The above discussion leads to the following important conclusions. Firstly,
it should be possible to associate a potential energy (i.e., an energy
a body possesses by virtue of its position) with any conservative
forcefield. Secondly, any forcefield for which we can define
a potential energy must necessarily be conservative. For instance, the existence
of gravitational potential energy is proof that gravitational fields
are conservative.
Thirdly, the concept of potential energy is meaningless
in a nonconservative forcefield (since the potential energy at a
given point cannot be uniquely defined). Fourthly, potential energy is
only defined to within an arbitrary additive constant. In other words,
the point in space at which we set the potential energy to zero
can be chosen at will. This implies that only differences in potential
energies between different points in space have any physical
significance. For instance, we have seen that the definition of
gravitational potential energy is , where represents height above
the ground. However, we could just as well write
, where
is the height of some arbitrarily chosen reference point
(e.g., the top of Mount Everest, or the bottom of the
Dead Sea). Fifthly, the difference in potential energy between two points
represents the net energy transferred to the associated forcefield when a body moves
between these two points. In other words, potential energy is not, strictly speaking, a
property of the bodyinstead, it is a property of the forcefield within which
the body moves.
Next: Hooke's law
Up: Conservation of energy
Previous: Conservative and nonconservative forcefields
Richard Fitzpatrick
20060202