A small particle of mass m moves in such a way that the
potential energy `U = ar^2`, where a is constant and r is the distance of the
particle from the origin. Assuming Bhor model of quantization of angular
momentum and circular orbits, find the radius of nth allowed orbit.

Explanation

The force at a distance `r` is,
`F= -(dU)/(dr)= -2 ar`
Suppose `r` be the radius of `nth` orbit. The necessary centipetal force is provided by the above force. Thus,
`(mv^(2))/(r )=2ar`
Further, the equantization of angular momentum gives,
`mvr=(nh)/(2 pi)`
Solving Eqs. (i) and (ii) for `r`, we get
`r=((n^(2)h^(2))/(8ampi^(2)))^(1//4)`