A particle, which is constrained to move along x-axis, is subjected to a force in the some direction which varies with the distance x of the particle from the origin an `F (x) =-kx + ax^(3)`. Here, k and a are positive constants. For `x(ge0,` the functional form of the potential energy (u) U of the U (x) the particle is.
(a) , (b) , (c) , (d) .

Options

A.
B.
C.
D.

Answer

D

Explanation

`dU (x) = -Fdx " " :. " " U(x) = int_(0)^(x) -F dx = (kx^2)/(2) - (ax^4)/(4)`
`U = 0 "at" x = 0 "and at" x = sqrt((2k)/(a)) , " "U(x) ` is negative for `x > sqrt((2k)/(a))`
`F = 0` at x = 0. It means slope of U–x graph is zero at x = 0.