A particle which is constrained to move along x - axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as `F(x)=kx + ax^(3)`. Here, k and a are positive constants. For `x ge 0`, the functional form of the potential energy U (x) of the particle is

Options

A.
B.
C.
D.

Answer

D

Explanation

`du=-f.dx`
`u=-underset(0)overset(x)int(kx+ax^(3))dx`
`=(kx^(2))/(2)-(ax^(4))/(4)`
When `x=0, u=0`
`therefore (kx^(2))/(2)=(ax^(4))/(4)`
`k=(ax^(2))/(2)`
`x=sqrt((2k)/(a))`