Question

A body is displaced from $\left(0m,0m\right)$ to $\left(1m,1m\right)$ along the path $x=y$ by a force $\overrightarrow{F}=\left(x^2\hat{j}+y\hat{i}\right)N$ . The work done by this force will be

• A. $\frac{4}{3}J$
• B. $\frac{5}{6}J$
• C. $\frac{3}{2}J$
• D. $\frac{7}{5}J$

Answer 1

Answer: (B)

As force is a function of $r$ , therefore
$W={\int }_{\left(0,0\right)}^{\left(1,1\right)}\overrightarrow{F}.\overrightarrow{d}x$
Here $d\overrightarrow{s}=dx\hat{i}+dy\hat{j}+dz\hat{k}$
$\therefore$ $W={\int }_{\left(0,0\right)}^{\left(1,1\right)}\left(x^{2}dy+ydx\right)={\int }_{\left(0,0\right)}^{\left(1,1\right)}\left(y^{2}dy+xdx\right)$ (as $x=y$ )
$\therefore$ $W={\left(\left[\frac{y^3}{3}+\frac{x^2}{2}\right]\right)}_{\left(0,0\right)}^{\left(1,1\right)}=\frac{5}{6}J$