Question

The work done on a particle of mass $m$ by a force $K \left[\frac{ x }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}} \hat{ i }+\frac{ y }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}} \hat{ j }\right]$ ( $K$ being a constant of appropriate dimensions, when the particle is taken from the point $( a , 0)$ to the point $(0, a )$ along a circular path of radius a about the origin in the $x$-y plane is

• A. $\frac{ 2K \pi}{a}$
• B. $\frac{ K \pi}{a}$
• C. $\frac{ K \pi}{2a}$
• D. 0

Answer 1

Answer: (D)

0

Answer 2

$d w =\vec{ F } \cdot d \vec{ r }=\vec{ F } \cdot( d x \hat{ i }+ d \hat{ j })$
$= K \int \frac{ x dx }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}}+\frac{ ydy }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}}$
$x ^{2}+ y ^{2}= a ^{2}$
$w =\frac{ K }{ a ^{3}} \int\limits_{ a }^{0} x d x +\int\limits_{0}^{ a } ydy$
$=\frac{ K }{ a ^{3}}\left(\frac{- a ^{2}}{2}+\frac{ a ^{2}}{2}\right)=0$