Question

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

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

Answer 1

Answer: (B)

$W=\displaystyle \int _{\left(\right. 0,0 \left.\right)}^{\left(\right. 1,1 \left.\right)} \overset{ \rightarrow }{F} . \overset{ \rightarrow }{d} x$
Here $\overset{ \rightarrow }{d}s=dx\hat{i}+ \, dy\hat{j} \, + \, dz\hat{k}$
$\therefore W=\displaystyle \int _{\left(\right. 0,0 \left.\right)}^{\left(\right. 1,1 \left.\right)} \left(\right. x^{2} d y + y d x \left.\right)$
$= \, \displaystyle \int _{\left(\right. 0,0 \left.\right)}^{\left(\right. 1,1 \left.\right)} \left(\right. x^{2} d y + x . d x \left.\right)$
(As $x=y$ )
$\therefore W=\left(\left[\frac{y^{3}}{3} + \frac{x^{2}}{2}\right]\right)_{\left(\right. 0,0 \left.\right)}^{\left(\right. 1,1 \left.\right)}=\frac{5}{6}J$