Work Done by a Variable Force
Category : JEE Main & Advanced
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 \[W=\int_{{{x}_{A}}}^{{{x}_{B}}}{{{F}_{x}}dx+\int_{{{y}_{A}}}^{{{y}_{B}}}{{{F}_{y}}dy}}+\int_{{{z}_{A}}}^{{{z}_{B}}}{{{F}_{z}}dz}\]
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