question_answer

The work done by the force \[\vec{F}=A\,({{y}^{2}}\hat{i}+2{{x}^{2}}\hat{j}),\] where A is a constant and x & y are in meters around the path shown is:

A) zero      

B) A d

C) \[A\,{{d}^{2}}\]

D) \[A\,{{d}^{3}}\]

Correct Answer: D

Solution :

\[W=\int{\vec{F}\cdot d\bar{x}}=\int{A\,({{y}^{2}}\,}\hat{i}+2{{x}^{2}}\hat{j})\cdot (dx\,\hat{i}+dy\cdot \hat{j})\]

\[=A\int{({{y}^{2}}\,dx+}2{{x}^{2}}dy)\]

\[{{W}_{OA}}=0+0,\]\[{{W}_{AB}}=A\,[0+2{{d}^{2}}\,d]\]

\[{{W}_{BC}}=A\,[{{d}^{2}}\,(-\,d)+0],\]\[{{W}_{CD}}=A\,[0+0]\]

\[W=0+2A{{d}^{3}}-A{{d}^{3}}+0=A{{d}^{3}}\]