DEFINITION OF POTENTIAL ENERGY AND CONSERVATION OF MECHANICAL ENERGY :

 DEFINITION OF POTENTIAL ENERGY AND CONSERVATION OF MECHANICAL ENERGY :

We define the change in potential energy of a system corresponding to a conservative internal force as

 Uf − Ui = − W = − ∫ i f F → ⋅ dr → 

where W is the work done by the internal force on the system as the system passes from the initial configuration i to the final configuration f.

 We don’t (or can’t) define potential energy corresponding to a nonconservative internal force.

Suppose only conservative internal forces operate between the parts of the system and the potential energy U is defined corresponding to these forces. There are either no external forces or the work done by them is zero. 

We have Uf − Ui = − W = − (Kf − Ki) or, Uf + Kf = Ui + Ki . … (8.6)

 The sum of the kinetic energy and the potential energy is called the total mechanical energy. We see from equation (8.6) that the total mechanical energy of a system remains constant if the internal forces are conservative and the external forces do no work. This is called the principle of conservation of energy. 

The total mechanical energy K + U is not constant if nonconservative forces, such as friction, act between the parts of the system. We can’t apply the principle of conservation of energy in presence of nonconservative forces.

 The work-energy theorem is still valid even in the presence of nonconservative forces. Note that only a change in potential energy is defined above. We are free to choose the zero potential energy in any configuration just as we are free to choose the origin in space anywhere we like. If nonconservative internal forces operate within the system, or external forces do work on the system, the mechanical energy changes as the configuration changes

. According to the work-energy theorem, the work done by all the forces equals the change in the kinetic energy. Thus,

Wc + Wnc + Wext = Kf − Ki

where the three terms on the left denote the work done by the conservative internal forces, nonconservative internal forces and the external forces. As

 Wc = − (Uf − Ui) , 

we get

 Wnc + Wext = (Kf + Uf) − (Ki + Ui) = Ef − Ei … 

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