13.11 Conservation of energy

Work - kinetic energy theorem is used to analyze motion of a particle. This theorem, as pointed out, is a consideration of energy for describing motion of a particle in general, which may involve both conservative and non-conservative force. However, “work-kinetic energy” theorem is limited in certain important aspects. First, it is difficult to apply this theorem to many particle systems and second it is limited in application to mechanical process – involving motion.

Law of conservation of energy is an extension of this theorem that changes the context of analysis in two important ways. First, it changes the context of energy from a single particle situation to a system of particles. Second, law of energy conservation is extremely general that can be applied to situation or processes other than that of motion. It can be applied to thermal, chemical, electrical and all possible processes that we can think about. Motion is just one of the processes.

Clearly, we are embarking on a new analysis system. The changes in analysis framework require us to understand certain key concepts, which have not been used before. In this module, we shall develop these concepts and subsequently conservation law itself in general and, then, see how this law can be applied in mechanical context to analyze motion and processes, which are otherwise difficult to deal with. Along the way, we shall highlight advantages and disadvantages of the energy analysis with the various other analysis techniques, which have so far been used.

In this module, the “detailed” treatment of energy consideration will be restricted to process related to motion only (mechanical process).

Mechanical energy

Mechanical energy of a system comprises of kinetic and potential energies. Significantly, it excludes thermal energy. Idea of mechanical energy is that it represents a base line (ideal) case, in which a required task is completed with minimum energy. Consider the case of a ball, which is thrown upward with certain initial kinetic energy. We analyze motion assuming that there is no air resistance i.e. drag on the ball. For a given height, this assumption represents the baseline case, where requirement of initial kinetic energy for the given height is least. Mechanical energy is expressed mathematically as :

$$E_M=K+U$$

We can remind ourselves that potential energy arises due to “position” of the particle/ system, whereas kinetic energy arises due to “movement” of the particle/ system.

Other forms of energy

Different forms of energy are subject of individual detailed studies. They are topics of great deliberation in themselves. Here, we shall only briefly describe characteristics of other forms of energy. One important aspect of other forms of energy is that they are simply a macroscopic reflection of the same mechanical energy that we talk about in mechanics.

On a microscopic level or still smaller level, other forms of energy like thermal, chemical, electrical and nuclear energies are actually the same potential and kinetic energy. It is seen that scale of dimension involved with energy changes the ultimate or microscopic nature of mechanical energy as different forms of energy.