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 pay 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.
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. 11 and 18 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
Where we have chosen the zero for potential energy to correspond to the
relaxed state of the spring.