13.11 Conservation of energy  (Page 4/7)

3: We need to quickly visualize system type in accordance with our requirement. To understand this, let us get back to the example of “Earth-incline-block” system. We can interpret boundary and hence system in accordance with the situation and objective of analysis.

In an all inclusive scenario, we can consider “Earth-incline-block” system plus the “agent applying external force” as part of an “isolated” system.

Isolated system

We can relax the boundary condition a bit and define a closed system, which allows transfer of energy via work only. In that case, “Earth-incline-block” system is “closed” system, which allows transfer of energy via “work” – not via any other form of energy. In this case, we exclude the agent applying external force from the system definition.

Work on isolated system

In the next step, we can define a proper “closed” system, which allows exchange of energy via both “work” and “energy”. In that case, “Earth – block” system constitutes the closed system. External force transfers energy by doing work, whereas friction, between block and incline, produces heat, which is distributed between the defined system of “Earth-block” and the “incline”, which is ,now, part of the surrounding. This system is shown below :

Closed system

4: We see that energy is transferred between system and surrounding, if the system is either open or closed. Isolated system does not allow energy transfer. Clearly, system definition regulates transfer of energy between system and surrounding – not the transfer that takes place within the system from one form of energy to another. Energy transfers from one form to another can take place within the system irrespective of system types.

Work – kinetic energy theorem for a system

The mathematical statement of work – kinetic energy theorem for a particle is concise and straight forward :

$$W=\Delta K$$

The forces on the particle are external forces as we are dealing with a single particle. There can not be anything internal to a particle. For calculation of work, we consider all external forces acting on the particle. The forces include both conservative (subscripted with C) and non-conservative (subscripted with NC) forces. The change in the kinetic energy of the particle is written explicitly to be equal to work by external force (subscripted with E) :

$$W_E=\Delta K$$

When the context changes from single particle to many particles system, we need to redefine the context of the theorem. The forces on the particle are both internal (subscripted with I)and external (subscripted with E) forces. In this case, we need to calculate work and kinetic energy for each of the particles. The theorem takes the following form :

$$\Rightarrow \sum W_E+\sum W_I=\sum \Delta K$$

We may be tempted here to say that internal forces sum to zero. Hence, net work by internal forces is zero. But work is not force. Work also involves displacement. The displacement of particles within the system may be different. As such, we need to keep the work by internal forces as well. Now, internal work can be divided into two groups : (i) work by conservative force (example : gravity) and work by non-conservative force (example : friction). We, therefore, expand “work-kinetic energy” theorem as :