Question

A particle is moved along the different paths OAC, OBC & ODC as shown in the fig. Path ODC is a parabola, $y=4x^2$. Find the work done by a force $\overrightarrow{F}=xy\hat{i}+x^{2}y\hat{j}$ on the particle along these paths. Is this force a conservative force?

• A. $4J,10J,\frac{9}{3}J$
• B. $6J,\frac{19}{3}J,5J$
• C. $8J,2J,\frac{19}{3}J$
• D. None of these

Answer 1

The correct option is C $8J,2J,\frac{19}{3}J$

$Letthesmalldisplacementdsbeds=dx\hat{i}+dy\hat{j}WorkdoneW=\int dw=\int F.dsWorkdonealongthepathOBCisgivenby{W}_{OB}+W_{BC}={\int }_{(0,0)}^{(0,4)}(xy\hat{i}+x^{2}y\hat{j}).(dx\hat{i}+dy\hat{j})+{\int }_{(0,4)}^{(1,4)}(xy\hat{i}+x^{2}y\hat{j}).(dx\hat{i}+dy\hat{j})={\int }_{(0,0)}^{(0,4)}(xydx+x^{2}ydy)+{\int }_{(0,4)}^{(1,4)}(xydx+x^{2}ydy)=0+2=2JWorkdonealongthepathOACisgivenby{W}_{OA}+W_{AC}={\int }_{(0,0)}^{(1,0)}(xy\hat{i}+x^{2}y\hat{j}).(dx\hat{i}+dy\hat{j})+{\int }_{(1,0)}^{(1,4)}(xy\hat{i}+x^{2}y\hat{j}).(dx\hat{i}+dy\hat{j})={\int }_{(0,0)}^{(0,4)}(xydx+x^{2}ydy)+{\int }_{(0,4)}^{(1,4)}(xydx+x^{2}ydy)=0+8=8JWorkdonealongthepathODCisgivenby{W}_{OC}={\int }_{(0,0)}^{(1,4)}(xydx+x^{2}ydy)={\int }_0^{1}4x^{3}dx+{\int }_0^4{y}^2/4dy=1+16/3=19/3JSooptionCissatisfiedandForceisnotconservativeinnature$\quad