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$$\Rightarrow \sum {W}_{E}+\sum {W}_{C}+\sum {W}_{NC}=\sum \Delta K$$
Generally, we drop the summation sign with the understanding that we mean “summation”. We simply say that both work and kinetic energy refers to the system – not to a particle :
$$\Rightarrow {W}_{E}+{W}_{C}+{W}_{NC}=\Delta K$$
However, work by conservative force is equal to negative of change in potential energy of the system.
$${W}_{C}=-\Delta U$$
Substituting in the equation above,
$$\Rightarrow {W}_{E}-\Delta U+{W}_{NC}=\Delta K$$
$$\Rightarrow {W}_{E}+{W}_{NC}=\Delta K+\Delta U=\Delta {E}_{\mathrm{mech}}$$
One of the familiar non-conservative force is friction. Work by friction is not path independent. One important consequence is that it does not transfer kinetic energy as potential energy as is the case of conservative force. Further, it only transfers kinetic energy of the particle into heat energy, but not in the opposite direction. What it means that friction is incapable to transfer heat energy into kinetic energy.
Nonetheless, friction converts kinetic energy of the particle into heat energy. A very sophisticated and precise set up measures this energy equal to the work done by the friction. We have seen that gravitational potential energy remains in the system. What about heat energy? Where does it go? If we consider a “Earth-incline-block” system, in which the block is released from the top, then we can visualize that heat so produced is distributed between “block” and “incline” (consider that there is no radiation loss). Next thing that we need to answer is “what does this heat energy do to the system?” We can infer that heat so produced raises the thermal energy of the system.
$${W}_{NC}=-\Delta {E}_{\mathrm{thermal}}$$
Note that work by friction is negative. Hence, we should put a negative sign to a positive change in thermal energy in order to equate it to work by friction.
Putting this in the equation of “Work-kinetic energy” expression, we have :
$$\Rightarrow {W}_{E}=\Delta K+\Delta U+\Delta {E}_{\mathrm{thermal}}=\Delta {E}_{\mathrm{mech}}+\Delta {E}_{\mathrm{thermal}}$$
We now turn to something which we have not studied so far, but we shall employ those concepts to complete the picture of conservation of energy in the most general case.
Without going into detail, we shall refer to a consideration of thermodynamics. Work on the system, besides bringing change in the kinetic energy, also brings about change in the “internal” energy of the system. Similarly, combination of internal and external forces can bring about change in other forms of energy as well. Hence, we can rewrite “Work-kinetic energy” expression as :
$$\Rightarrow {W}_{E}=\Delta {E}_{\mathrm{mech}}+\Delta {E}_{\mathrm{thermal}}+\Delta {E}_{\mathrm{others}}$$
This equation brings us close to the formulation of “conservation of energy” in general. We need to interpret this equation in the suitable context of system type. We can easily see here that we have developed this equation for a system, which allows energy transfers through work by external force. Hence, context here is that of special “closed” system, which allows transfer of energy only through work by external force. What if we choose a system boundary such that there is no external force. In that case, closed system becomes isolated system and
$${W}_{E}=0$$
Putting this in “work – kinetic energy” expression, we have :
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