13.11 Conservation of energy  (Page 6/7)

$$\Rightarrow \Delta E_{\mathrm{mech}}+\Delta E_{\mathrm{thermal}}+\Delta E_{\mathrm{others}}=0$$

If we denote “E” to represent all types of energy, then :

$$\Rightarrow \Delta E=0$$

$$\Rightarrow E=\mathrm{constant}$$

Above two equations are the mathematical expressions of conservation of energy in the most general case. We read this law in words in two ways corresponding to above two equations.

Definition 1: The change in the total energy of an isolated system is zero.

Definition 2: The total energy of an isolated system can not change.

From above two interpretations, it emerges that “energy can neither be created nor destroyed”.

Relativity and conservation of energy

Einstien’s special theory of relativity establishes equivalence of mass and energy. This equivalence is stated in following mathematical form.

$$E=m{c}^2$$

Where “c” is the speed of light in the vacuum. An amount of mass, “m” can be converted into energy “E” and vice – versa. In a process known as mass annihilation of a positron and electron, an amount of energy is released as,

$$e^{+}+e^{-}=hv$$

The energy released as a result is given by :

$$E=2X9.11X{10}^{-31}X3X{\left({10}^8\right)}^2=163.98X{10}^{-15}=1.64X{10}^{-13}J$$

Similarly, in a process known as “pair production”, energy is converted into a pair of positron and electron as :

$$hv=e^{+}+e^{-}$$

This revelation of equivalence of “mass” and “energy” needs to be appropriately interpreted in the context of conservation of energy. The mass – energy equivalence appears to contradict the statement that “energy can neither be created nor destroyed”.

There is, however, another perspective. We may consider “mass” and “energy” completely exchangeable with each other in accordance with the relation given by Einstein. When we say that “energy can neither be created nor destroyed”, we mean to include “mass” also as “energy”. However, if we want to be completely explicit, then we can avoid the statement. Additionally, we need to modify the statements of conservation law as :

Definition 1: The change in the total equivalent mass - energy of an isolated system is zero.

Definition 2: The total equivalent mass - energy of an isolated system can not change.

Alternatively, we have yet another option to exclude nuclear reaction or any other process involving mass-energy conversion within the system. In that case, we can retain the earlier statements of conservation law with the qualification that process excludes “mass-energy” conversion.

In order to maintain semblance to pre-relativistic revelation, we generally retain the form of conservation law, which is stated in terms of “energy” with an implicit understanding that we mean to include “mass” as just another form of “energy”.

Nature of motion and conservation law

So far we have studied motion in translation. We need to clarify the context of conservation of energy with respect to other motion type i.e. rotational motion. In subsequent modules, we shall learn that there is a corresponding “work-energy theorem”, “potential energy concept” and actually a parallel system for analysis for rotational motion. It is, therefore, logical to think that constituents of the isolated system that we have considered for development of conservation of energy can possess rotational kinetic and potential energy as well. Thus, conservation law is all inclusive of motion types and associated energy. When we consider potential energy of the system, we mean to incorporate both translational and rotational potential energy. Similar is the case with other forms of energy – wherever applicable.