13.13 Conservation of mechanical energy

Conservation of mechanical energy is description of an ideal mechanical process. It is characterized by absence of non-conservative force like friction. There is no external force either on the system. Therefore, the elements of this ideal process are :

• System : isolated
• Energy : kinetic and potential
• Forces : gravitational and elastic
• Transfer of energy : No transfer of energy across the system.

Characteristics of an ideal mechanical process

Conservation of mechanical energy applies to a mechanical process in which external force and non-conservative internal forces are absent.

There is no external force on the system. Hence, work by external force is zero. There is no exchange or transfer of energy across the system. Therefore, we use an isolated system to apply conservation of energy. We should, however, note that transfer of energy from one form to another takes place within the system, resulting from work done by internal force.

There is no “non-conservative” force like friction in the system. It means that there is no change in thermal energy of the system. The internal forces are only conservative force. This ensures that transfer of energy takes place only between kinetic and potential energy of the isolated system. Since potential energy is regained during the process, there is no dissipation of energy.

As there is no dissipation of energy involved, the system represents the most energy efficient reference for the particular process. One of the most striking feature of this system is that only force working in the system is conservative force. This has great simplifying effect on the analysis. The work by conservative force is independent of path and hence calculation of potential energy of the system is path independent as well. The independence of path, in turn, allows analysis of motion along paths, which are not straight.

Conservation of mechanical energy

The statement of conservation of energy for the ideal mechanical process is known as “conservation of mechanical energy". The equation for the mechanical process is :

$$W_E+W_F=\Delta K+\Delta U$$

Here,

$$W_E=0$$

$$W_F=0$$

Hence, for the isolated system,

$$\Rightarrow \Delta K+\Delta U=0$$

$$\Rightarrow \Delta E_{\mathrm{mech}}=0$$

$$\Rightarrow E_{\mathrm{mech}}=0$$

This is what is known as conservation of mechanical energy. We can interpret this equation in many ways and in different words :

1: When only conservative forces interact within an isolated system, sum of the change in kinetic and potential energy between two states is equal to zero.

$$\Delta K+\Delta U=0$$

2: When only conservative forces interact within an isolated system, sum of the kinetic and potential energy of an isolated system can not change.

$$\Delta K=-\Delta U$$

$$\Rightarrow K_f-K_i=-\left(U_f-U_i\right)$$

$$\Rightarrow K_i+U_i=K_f+U_f$$

3: When only conservative forces interact within an isolated system, the change in mechanical energy of an isolated system is zero.

$$\Delta E_{\mathrm{mech}}=0$$

4: When only conservative forces interact within an isolated system, the mechanical energy of an isolated system can not change.

$$E_{\mathrm{mech}}=0$$

We should be aware that there are two ways to apply conservation law. We can apply it in terms of energy for initial (subscripted with “i”) and final (subscripted with “f”) states or in terms of “change” in energy. This point will be clear as we work with examples.