13.10 Absolute potential energy  (Page 2/3)

Definition of potential energy

Having decided the universal zero reference, we are now in position to define potential energy, using the expression obtained for the change in potential energy :

$$\Delta U=U_2-U_1=-W_C$$

If we set initial position at infinity, then $U_1=0$ . Let us denote potential energy of a system to be “U” for a given configuration. Then,

$$\Rightarrow U-0=-W_C$$

$$\Rightarrow U=-W_C=-\underset{\infty }{\overset{0}{\int }}{F}_{C}dr$$

Hence, we can now define potential energy as given here :

Potential energy

The potential energy of a system of particles is equal to “negative” of the work by the conservative force as a particle is brought from infinity to its position in the presence of other particles of the system.

We should understand that the work by conservative force is independent of path and hence no reference is made about path in the definition. This work has a unique value. Hence, it gives a unique value of potential to the system of particles.

Distribution of potential energy

There is a peculiar aspect of the definition of potential energy, presented above. It defines potential energy of the system of particles in terms of work on a “single” particle. This peculiarity can be explained as it defines work in the presence of other particles and as such accounts for the forces operating on the particle due to their presence.

This definition, however, is not clear about how potential energy is distributed among the particles in the system. The value of potential energy does not throw any light on this aspect. As a matter of fact, it is not possible to segregate potential energy for the individual constituents of the system. Potential energy, therefore, belongs to all of them – not to any one of them.

Potential energy and external force

The potential energy is defined in terms of work by conservative force and zero reference potential at infinity. It is equal to the “negative” of work by conservative force :

$$\Rightarrow U=-W_C=-\underset{\infty }{\overset{0}{\int }}{F}_{C}dr$$

Can we think to express this definition of potential energy in terms of external force? In earlier module, we have analyzed the motion of a body, which is raised by hand slowly to a certain vertical height. The significant point of this illustration was the manner in which body was raised. It was, if we can recall, raised slowly without imparting kinetic energy to the body being raised. It was described so with a purpose. The idea was to ensure that work by the external force (in this case, external force is equal to the normal force applied by the hand) is equal to the work by gravity.

Since speed of the body is zero at the end points, “work-kinetic energy” theorem reduces to :

$$W=K_f-K_i=0$$

$$W=0$$

This means that work by “net” force is zero. It follows, then, that works by gravity (conservative force) and external force are equal in magnitude, but opposite in sign.

$$\Rightarrow W=W_C+W_F=0$$

$$W_F=-W_C$$

Under this condition, the work by external force is equal to negative of work by conservative force :

$$\Rightarrow W_F=-\int F_{C}dr$$

where “ $F_C$ ” is conservative force. It means that if we work on the particle slowly without imparting it kinetic energy, then work by the external force is equal to negative of the work by conservative force. In other words, work by external force without a change in kinetic energy of the particle is equal to change in potential energy only. Equipped with this knowledge, we can define potential energy in terms of external force as :