The work done an a particle of mass `m` by a force
`K[(x)/((x^(2) + y^(2))^(3//2)) hati +(y)/((x^(2) + y^(2^(3//2))) hatj)]
(K being a constant of appropriate dimensions), when the partical is taken from the point `(a,0)` to the point `(0,a)` along a circular path of radius a about the origin in x - y plane is

Options

A. `(2 k pi)/(a)`
B. `(k pi)/(a)`
C. `(k pi)/(2a)`
D. zero

Answer

D

Explanation

`W = int vec F. d vec r`
=`k underset(r_(A)) overset(r_(B))int[(x hat i)/((x^(2) +y^(2))^(3//2))+(y hat j)/((x^(2) +y^(2))^(3//2))](d xhat i+ dy hatj)`
=`k underset(r_(A)) overset(r_(B))int (xdx)/((x^(2)+y^(2)))+(ydy)/((x^(2) +y^(2))^(3//2))`
=`k underset(r_(A)) overset(r_(B)) int (1)/((x^(2)+y^(2))^(3//2))[d((x^(2))/(2))+d((y^(2))/(2))]`
=`k underset(r_(A)) overset(r_(B)) int (1)/((x^(2)+y^(2))^(3//2)) (x^(2)+y^(2))`
=`k underset(r_(A)) overset(r_(B)) int (1)/(2r^(3))d(r^(2)) = k underset(r_(A)) overset(r_(B)) int (2rdr)/(2r^(3)) = k underset(r_(A)) overset(r_(B)) int (dr)/(r^(2))`
=`k[-(1)/(r)]_(r_(A))^(r_(B)) =k[(1)/(r_(A))-(1)/(r_(B))]`
But `r_(A) =a `and `r_(B) =a` , `:. W = 0`.