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

Options

A. `vec F=2r^(3) hatr`
B. `vec F = (5)/(r) hat r`
C. `vec F=(3( x i + y j))/((x^(2)+y^(2))^(3//2))`
D. `vec F = (3(x^(2) i+ y j))/((x^(2)+y^(2))^(3//2))`

Answer

A, B, C

Explanation

Since , `W = int vecF . vecdr`
Clearly for forces `(A)` and `(B)` the integration do not require any information of the path taken.
For `( C) : W_(c) =`
`int (3(x i+yj))/((x^(2)+y^(2))^(3//2)). (dx i+ dy j) = 3 int(x dx + y dy)/((x^(2)+y^(2))^(3//2))`
Taking : `x^(2) + y^(2) = t`
`2x dx + 2y dy = dt`
`rArr xdx + ydy = (dt)/(2) rArr W_(c) = 3 int (dt//2)/(t^(3//2)) = (3)/(2) int (dt)/(t^(3//2))`
Which is solvable.
Hence `(A),(B)` and `( c)` are conservative forces. But `(D)` requires some more information on path. Hence non-conservative.