Question

A force $F=x^2{y}^{2}i-x^2{y}^{2}j\left(N\right)$ acts on a particle which moves in the $XY$ plane. Find the work done by $F$ as it moves the particle from $A$ to $C$ (fig.) along each of the paths $ABC,ADC$ and $AC$.

Answer 1

Work done along ABC

$=\underset{0,0}{\overset{1,0}{\int }}\left(x^2{y}^2\hat{i}-x^2{y}^2\hat{j}\right)dx+\underset{1,0}{\overset{1,1}{\int }}\left(x^2{y}^2\hat{i}-x^2{y}^2\hat{j}\right)dy$

$=0+0$

Work done along ADC

$=\underset{0,0}{\overset{0,1}{\int }}\left(x^2{y}^2\hat{i}-x^2{y}^2\hat{j}\right)dy+\underset{0,1}{\overset{1,1}{\int }}\left(x^2{y}^2\hat{i}-x^2{y}^2\hat{j}\right)dx$

$=0+0$

Work done along AC

$=\underset{0,0}{\overset{1,1}{\int }}\left(x^2{y}^2\hat{i}-x^2{y}^2\hat{j}\right)dxdy$

$=0$