Question

Consider a force $\overrightarrow{F}=(y\hat{i}+x^2\hat{j})N$ (where $x$ and $y$ are in metre) acting on a particle, displaces it from the origin $O$ to a point $P(10\text{m},10\text{m})$ along three different paths $\text{OLP, OMP}$ and $\text{OP}$. Then,

• A. Maximum work is done along the path OLP
• B. No work is done along any path.
• C. If the force is conservative, work done along three different paths $\text{OLP, OMP}$ and $\text{OP}$ is same .
• D. If the force is conservative, work done along three different paths $\text{OLP, OMP}$ and $\text{OP}$ is different.

Answer 1

Answer: (A), (C)

The correct options are
A Maximum work is done along the path OLP
C If the force is conservative, work done along three different paths $\text{OLP, OMP}$ and $\text{OP}$ is same .

$W=\int \overrightarrow{F}.\overrightarrow{ds}=\int (y\hat{i}+x^2\hat{j}).(dx\hat{i}+dy\hat{j})$
$=\int ydx+\int x^{2}dy$

Along path $\text{OLP}$
$W_{OLP}=W_{OL}+W_{LP}$
Here
$W_{OL}=\int 0dx+{\int }_0^0{x}^{2}dy=0$
$W_{LP}={\int }_{10}^{10}ydx+{\int }_0^{10}{10}^{2}dy=1000\text{J}$
$W_{OLP}=0+1000=1000\text{J}$

Along path $\text{OMP}$
$W_{OMP}=W_{OM}+W_{MP}$
$W_{OM}={\int }_0^{0}ydx+\int 0^{2}dy=0$
$W_{MP}={\int }_0^{10}10dx+{\int }_{10}^{10}{x}^{2}dy=100$

$W_{OMP}=0+100=100\text{J}$

For the line $OP,y=x$

$\therefore W_{OP}={\int }_0^{10}ydx+{\int }_0^{10}{x}^{2}dx=\frac{{10}^2}{2}+\frac{{10}^3}{3}=\frac{1150}{3}\text{J}$

If force is conservative, work is path independent.