When a particle moves from f to i and from j to i along the paths shown in Fig. 8-28, and in the indicated directions, a conservative force $\overrightarrow{\mathbf{F}}$does the indicated amounts of work on it. How much work is done on the particle by$\overrightarrow{\mathbf{F}}$when the particle moves directly from f to j?

The figure shows a particle moving from f toi and from j toi along the path

The problem deals with the work done. This is the work is donewhenever a force moves something over a distance.The resultant work doneon the particle by$\overrightarrow{\mathbf{F}}$can be found when the particle moves directly from fto jby adding the work done along fto iand ito j.

Formula:

$W_{f\to j}=W_{f\to i}+W_{i\to j}$

As force $\overrightarrow{F}$is conservative, the net work done by force $\overrightarrow{F}$ in displacing the particle from f to j will be the sum of work done in moving the particle from f to i and then from i to j.

So, we have,

$W_{f\to j}=W_{f\to i}+W_{i\to j}$

From diagram,

role="math" localid="1657182443809" $$\begin{gathered} W_{i\to j}=W_{j\to i} \\ {W}_{i\to j}=-20\mathrm{J} \\ {W}_{f\mathit{\to }i}\mathit{=}(-20\mathrm{J})+(-20\mathrm{J}) \\ {W}_{f\mathit{\to }j}\mathit{=}-40\mathrm{J} \end{gathered}$$

Therefore, the work done on the particle by $\overrightarrow{F}$when the particle moves directly from f to j is$-40\mathrm{J}$