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

Explanation

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