The work done on a particle of mass m by a force K [( x /( x 2+ y 2)3 / 2) hat i +( y /( x 2+ y 2)3 / 2) hat j ] ( 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

Question

The work done on a particle of mass m by a force K [( x /( x 2+ y 2)3 / 2) hat i +( y /( x 2+ y 2)3 / 2) hat j ] ( 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) ( 2K π/a) (B) ( K π/a) (C) ( K π/2a) (D) 0

Tardigrade Admin · Accepted Answer

d w = vec F ⋅ d vec r = vec F ⋅( d x hat i + d hat j ) = K ∫ ( x dx /( x 2+ y 2)3 / 2)+( ydy /( x 2+ y 2)3 / 2) x 2+ y 2= a 2 w =( K / a 3) ∫ limits a 0 x d x +∫ limits0 a ydy =( K / a 3)((- a 2/2)+( a 2/2))=0

$\frac{2 K π}{a}$

$\frac{K π}{a}$

$\frac{K π}{2 a}$

$0$

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

Q. The work done on a particle of mass m by a force K[(x2+y2)3/2x​i^+(x2+y2)3/2y​j^​] ( 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

a2Kπ​

aKπ​

2aKπ​

0

dw=F⋅dr=F⋅(dxi^+dj^​) =K∫(x2+y2)3/2xdx​+(x2+y2)3/2ydy​ x2+y2=a2 w=a3K​a∫0​xdx+0∫a​ydy =a3K​(2−a2​+2a2​)=0

$\frac{2 K π}{a}$

$\frac{K π}{a}$

$\frac{K π}{2 a}$

$0$

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