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)/(2 pi)" " `.... (ii)
Solving Eqs. (i) and (ii) for r,
we get r = `((n^(2)h^(2))/(8am pi^(2)))^(1//4)`