When an external force performs work against a conservative force by for example lifting an object up or compressing a spring we are priming the object to fall or the spring to expand and in the process gain kinetic energy. We have stored mechanical energy in the system and created the potential for the later appearance of kinetic energy. To make an analogy lifting an object far above the ground in preparation for dropping it and releasing lots of kinetic energy is like saving money in the bank in preparation for a shopping spree.
To account for such configurational energy associated with a
conservative force we define potential energy: As a matter of fact we only
really define the difference in potential energy between two configurations
of a conservative system. The potential energy difference between
state A and B of a system is
the amount of work done by the conservative force when the system
goes from
condition B to condition A:
Be careful to bay attention to the signs in these expressions.
The second expression can be interpreted as the work which an external
force must supply to bring the system from state A to state B.
It is important to notice that potential energy is only defined down to an arbitrary additive constant. Another way of stating this is to say that the choice of the configuration which is defined to have zero potential energy is arbitrary though important to know as we use the concept of potential energy.
One example of potential energy is gravitational potential energy
associated with the motion of an object with mass, m, close
to the surface of the earth. According to
Eq. 41 and 43 It takes the form
Many times we choose the zero for gravitational potential energy
at the surface of the earth and then
where h is the height of the object over the surface of the earth.
Another important example is the potential energy associated with compression or
extension of a spring from its equilibrium length which takes the form
Potential energy functions can take more complicated forms as well.
For example the potential energy associated with the location of
two atoms in a molecule with respect to one another could
look something like Fig. 7-24 in the book. According to Eq. 43
we can actually derive the conservative force from knowledge of
the potential energy function:
We see that the force between atoms vanishes at the minimum
of the potential energy function so this is the stable
equilibrium separation of the atoms.
With this definition of potential energy the work kinetic
energy theorem becomes an energy conservation theorem stating that in the
absence of non-conservative forces the total mechanical energy
is conserved:
where we define the total mechanical energy as the sum of the potential
and the kinetic energy
