A particle which is constant to move along the `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)` . Hero `K` and `a` are positive constant . For `x ge 0`, the fanctional from of the patential every `U(x) of the particle is

Options

A.
B.
C.
D.

Answer

D

Explanation

`F = (-dU)/(dx) rArr d U = -F dx`
`rArr U = -int_(0)^(x)(-Kx + a x^(3))dx = (k x^(2))/(2) - (a x^(4))/(4)`
`therefore` We get U = 0 at x = 0 and `x = sqrt(2K//a)`
and also U = negative for `x gt sqrt(2k//a)`.
So F = 0 at x = 0
i.e. slope of U-x graph is zero at x = 0.