Which of the following is//are conservative force (s) ?

Options

A. (a) `vecF=2r^3vecr`
B. (b) `vecF=-5/rhatr`
C. (c) `vecF=(3(xhati+yhatj))/((x^2+y^2)^(3//2))`
D. (d) `vecF=(3(yhati+xhatj))/((x^2+y^2)^(3//2)`

Answer

A, C

Explanation

Since, `W=intvecF*dvecr`
Clearly for forces (a) and (b), the integration does not require For c.
`W_(C)=int (3(xhati+yhatj))/((x^2+y^2)^(3//2))*(dxhati+dyhatj)=3int(xdx+ydy)/((x^2+y^2)^(3//2))`
Taking `x^2+y^2=t` , we get `2xdx+2ydy=dt`
`xdx=ydy=(dt)/(2)`
`W_c=3int(dt//2)/(t^3//2)=3/2int(dt)/(t^(3//2))`
which will be independent of path after solving.
Hence, a., b., and c., are conservative forces.
But d. requires some more information on path to solve it.
Hence, it is non-conservative.