In order to derive a general expression for change in potential energy, we refer to the vertical motion of a ball, projected upward. As the ball moves up, initial kinetic energy of the ball decreases by the amount of negative work done by the gravity. In the meantime, however, separation between Earth and ball also increases. Consequently, the potential energy of "Earth - ball" system increases by the same amount. In effect, gravity transfers energy "from" the kinetic energy of the ball "to" the potential energy of the "Earth - ball" system.

Ball thrown vertically |
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In the reverse motion i.e. when ball starts moving downward, gravity transfers energy "from" potential energy of the "Earth - ball" system "to" the kinetic energy of the ball. The transfer of energy in "up" and "down" motion is same in quantity. This means that exchange of energy between "ball" and "Earth - ball" system is replicated in reverse motion.

A positive work on the object means that energy is being transferred from the system to the object. Consequently, the potential energy of the system decrease. As such, final potential energy of the system is lesser than initial potential energy. On the other hand, a negative work on the object means that energy is being transferred from the object to the system. Consequently, the potential energy of the system increases. As such, final potential energy of the system is greater than initial potential energy.

Between any two instants, the change in potential energy of the "Earth - ball" system is negative of the work done by the gravity on the "ball".

The result so obtained for gravitational force can be extended to any conservative force. The change in potential energy of a system, therefore, is defined "as the negative of the work done by conservative force". Note the wordings here. We have defined potential energy of a system - not to an individual entity. In practice, however, we refer potential energy to a particle with an implicit understanding that we mean a system - not an individual entity.

We must note following important aspects of potential energy as defined above :

1: We have derived this expression in the context of gravitational force, but is applicable to a class of forces (called conservative forces), which transfer energy in above manner. It is important to understand that there are non-conservative forces, which transfer energy to the system in other energy forms like heat etc. - not in the form of potential energy.

2: This is the expression for change in potential energy, which assumes different forms in different contexts (for different conservative forces).

3: We should realize that we deal here with the "change" in potential energy - not with "unique" value of potential energy corresponding to a specific configuration. This can be handled with the help of a known reference potential energy. The measurement of potential energy with respect to a chosen reference value enables us to assign a unique value to potential energy corresponding to a specific system configuration. We shall further elaborate this aspect while discussing specific potential energies.

4: The change in potential energy differs to kinetic energy in one aspect that it can assume negative value as well. We must remember that kinetic energy is always positive. A positive change in potential energy means that final potential energy is greater than initial potential energy. On the other hand, a negative value of change in potential energy means that final potential energy is lesser than initial potential energy.

5: Potential energy is defined only in the context of conservative force. For the sake of clarity, we should also realize that the conservative force doing work is internal to the system, whose potential energy is determined. We can rewrite the defining relation of potential energy that work refers to work by conservative as :

** Change in potential energy for constant and variable forces **

The relationship as given above forms the basic relation that is used to develop specific relations for different force systems. For constant conservative force, work is defined as :

Substituting in the equation, the change in potential energy involving constant force is :

For variable force having displacement in the same direction, the work is obtained as :

The change in potential energy is equal to the integral evaluated over two appropriate limits corresponding to displacement :

In mechanical context, we deal with gravitational and elastic potential energies. Elastic potential energy involves any configuration, which can be deformed elastically like spring. In this module, we will drive specific expressions of potential energy for these two systems.