When a particle of mass m moves on the x-axis in a potential of the form `v(x)=kx^(2)` it performs simple harmonic motion. The corresponding time period is proportional to m, k as can be seen easily using dimensional analysis. However, the motion of particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from `kx^(2)` and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the x-axis. Its potential energy is `v(x)=ax^(4)(a>0)`for |x| near the origin and becomes a constant equal to `v_(0)` for`|x|geX_(0)`.
Q If the total energy of the particle is E, it will perform periodic motion only if :

Options

A. propotional to `V_(0)`
B. propotional to `V_(0)/(mX_(0))`
C. propotional to `sqrt((V_(0))/(mX_(0)))`
D. zero

Answer

D

Explanation

`F = -(dU)/(dx)`
as for `|x| gt x_(0) V = V_(0) =` constant
`rArr (dU)/(dx) = 0 rArr F = 0`.