13.1 Work and energy  (Page 3/5)

Work

Work is the energy transferred by the force "to" or "from" the particle on which force is applied.

It is also clear that a positive work means transfer of energy "to" the particle and negative work means transfer of energy "from" the particle. Further, the term "work done" represents the process of transferring energy to the particle by the force.

Work done by a system of forces

The connection of work with kinetic energy is true for a special condition. The change in kinetic energy resulting from doing work on the body refers to work by net force - not work by any of the forces operating on the body. This fact should always be kept in mind, while attempting to explain motion in terms of work and energy.

Above observation is quite easy to comprehend. Consider motion of a block on a rough incline. As the block comes down the incline, component of gravity does the positive work i.e. transfers energy from gravitational system to the block. As a consequence, speed and therefore kinetic energy of the block increases. On the other hand, friction acts opposite to displacement and hence does negative work. It draws some of the kinetic energy of the block and transfers the same to surrounding as heat. We can see that gravity increases speed, whereas friction decreases speed. Since work by gravity is more than work by fraction in this case, the block comes down with increasing speed. The point is that net change in speed and hence kinetic energy is determined by both the forces, operating on the block. Hence, we should think of relation between work by "net" force and kinetic energy.

As mentioned earlier, a body under consideration may be subjected to more than one force. In that case, if we have to find the work by the net force, then we can adopt either of two approaches:

(i) Determine the net (resultant) force. Then, compute the work by net force.

$$\begin{array}{l}\mathbf{F}=\sum {\mathbf{F}}_i\\ W=\mathbf{F}\mathbf{.}\mathbf{r}=Fr\cos \theta \end{array}$$

(ii) Compute work by individual forces. Then, sum the works to compute work by net force.

$$\begin{array}{l}{W}_i={\mathbf{F}}_i\mathbf{.}{\mathbf{r}}_i\\ W=\sum W_i\end{array}$$

Either of two methods yields the same result. But, there is an important aspect about the procedures involved. If we determine the net force first, then we shall require to use free body diagram and a coordinate system to analyze forces to determine net force. Finally, we use the formula to compute work by the net force. On the other hand, if we follow the second approach, then we need to find the product of the components of individual forces along the displacement and displacement. Finally, we carry out the algebraic sum to find the work by the net force.

This is a very significant point. We can appreciate this point fully, when the complete framework of the analysis of motion, using concepts of work and energy, is presented. For the time being, we should understand that work together with energy provides an alternative to analyze motion. And, then think if we can completely do away with "free body diagram" and "coordinate system". Indeed, it is a great improvisation and simplification, if we can say so. But then this simplification comes at the cost of more involved understanding of physical process. Meaning of this paragraph will be more clear as we go through the modules on the related topics. For the time being, let us work out a simple example illustrating the point.