Q: A small particle of mass m moves in such a way that the potential energy U = a r^{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.

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Sol: The force at a distance r is,

$\large F = -\frac{dU}{dr} = -2 a r $

Suppose r be the radius of n^th orbit.then the necessary centripetal force is provided by the above force.

Thus , $\large \frac{m v^2}{r} = 2 a r $ …(i)

Further, the quantization of angular momentum gives,

$\large m v r = \frac{n h}{2 \pi} $ ….(ii)

Solving Eqs. (i) and (ii) for r , we get

$\large r = (\frac{n^2 h^2}{8 a m \pi^2})^{1/4} $