A force $$F=-k(yi^+xj^)$$ where k is a positive constant, acts on a particle moving in the $$XY-plane$$. Starting from the origin, the particle is taken along the positive $$X-axis$$ to the point (a$$, $$0) and then parallel to the $$Y-axis$$ to the point (a$$, $$a). The total work done by the force on the particle is [AIIMS $$2017$$]

Question

A force $$F=-k(yi^+xj^)$$ where k is a positive  constant, acts on a particle moving in the $$XY-plane$$. Starting from the origin, the particle is taken along the positive $$X-axis$$ to the point (a$$, $$0) and then parallel to the $$Y-axis$$ to the point (a$$, $$a). The total work done by the force on the particle is [AIIMS $$2017$$]

Infinity Learn · Accepted Answer

Given,   $$F=-k(yi^+xj^)As$$, work done,   $$W=$$∫F·drSo,     W'$$=-k$$∫(yi^+xj^)·(dxi^+dyj^)⇒ $$W=-k$$∫(ydx+xdy)⇒$$W=-k$$∫(0$$,$$0)(a$$,$$a)d(xy)  ∵$$ydx+xdy=$$∫$$d(xy)Hence$$,    $$W=-k(xy)(0$$,$$0)(a$$,$$a)=-ka2$$

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