JEE Main & Advanced Physics Work, Energy, Power & Collision / कार्य, ऊर्जा और शक्ति Work Done by a Variable Force

When the magnitude and direction of a force varies with position, the work done by such a force for an infinitesimal displacement is given by  \[dW=\overrightarrow{F}.\,d\overrightarrow{s\,}\]

The total work done in going from A to B as shown in the figure is

\[W=\int_{A}^{B}{\,\overrightarrow{F}.\,d\overrightarrow{s\,}=\int_{A}^{B}{\,(F\cos \theta )ds}}\]

In terms of rectangular component \[\overrightarrow{F}={{F}_{x}}\hat{i}+{{F}_{y}}\hat{j}+{{F}_{z}}\hat{k}\]

\[d\overrightarrow{s\,}=dx\hat{i}+dy\hat{j}+dz\hat{k}\]

\[\therefore \,\,\,W=\int_{A}^{B}{({{F}_{x}}\hat{i}+{{F}_{y}}\hat{j}+{{F}_{z}}\hat{k})}.(dx\hat{i}+dy\hat{j}+dz\hat{k})\]

or \[W=\int_{{{x}_{A}}}^{{{x}_{B}}}{{{F}_{x}}dx+\int_{{{y}_{A}}}^{{{y}_{B}}}{{{F}_{y}}dy}}+\int_{{{z}_{A}}}^{{{z}_{B}}}{{{F}_{z}}dz}\]