13.3 Work - kinetic energy theorem

Work - kinetic energy theorem with multiple forces

Extension of work - kinetic energy theorem to multiple forces is simple. We can either determine net force of all external forces acting on the particle, compute work by the net force and then apply work - kinetic energy theorem. This approach requires that we consider free body diagram of the particle in the context of a coordinate system to find the net force on it.

$$\begin{array}{l}{\mathbf{F}}_N={\mathbf{F}}_1+{\mathbf{F}}_2+{\mathbf{F}}_3+\mathrm{........}+{\mathbf{F}}_n\end{array}$$

Work - kinetic energy theorem is written for the net force as :

$$\begin{array}{l}\Rightarrow K_f-K_i={\mathbf{F}}_N\mathbf{.}\mathbf{r}\end{array}$$

where ${\mathbf{F}}_N$ is the net force on the particle. Alternatively, we can determine work done by individual forces for the displacement involved and then sum them to equate with the change in kinetic energy. Most favour this second approach as it does not involve vector consideration with a coordinate system.

$$\begin{array}{l}\Rightarrow K_f-K_i=\sum W_i\end{array}$$

Application of work - kinetic energy theorem

Work - kinetic energy theorem is not an alternative to other techniques available for analyzing motion. What we want to mean here is that it provides a specific technique to analyze motion, including situations where details of motion are not available. The analysis typically does not involve intermediate details in certain circumstances. One such instance is illustrated here.

In order to illustrate the application of "work - kinetic energy" theory, we shall work with an example of a block being raised along an incline. We do not have information about the nature of motion - whether it is raised along the incline slowly or with constant speed or with varying speed. We also do not know - whether the applied external force was constant or varying. But, we know the end conditions that the block was stationary at the beginning of the motion and at the end of motion. So,

$$\begin{array}{l}\Rightarrow K_f-K_i=0\end{array}$$

It means that work done by the forces on the block should sum up to zero (according to "work - kinetic energy" theorem). If we know all other forces but one "unknown" force and also work done by the known forces, then we are in position to know the work done by the "unknown" force.

We should know that application of work-kinetic energy theorem is not limited to cases where initial and final velocities are zero or equal, but can be applied also to situations where velocities are not equal. We shall discuss these applications with references to specific forces like gravity and spring force in separate modules.

Example

Problem : A block of 2 kg is pulled up along a smooth incline of length 10 m and height 5 m by applying an external force. At the end of incline, the block is released to slide down to the bottom. Find (i) work done by the external force and (ii) kinetic energy of the particle at the end of round trip. (consider, g = 10 $m/s^2$ ).

Solution : This question is structured to bring out finer points about the "work - kinetic energy" theorem. There are three forces on the block while going up : (i) weight of the block, mg, and (ii) normal force, N, applied by the block and (iii) external force, F. On the other hand, there are only two forces while going down. The external force is absent in downward journey. The force diagram of the forces is shown here for upward motion of the block.