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A non-relativistic particle of mass m is held in a circular orbit around the origin by an attractive force. The potential energy of the force is given by $U (r) = \frac{k r^{2}}{2}$ Assuming the Bohr quantization of the angular momentum of the particle select correct alternatives.

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$v^{2} = (\frac{n ħ}{m}) {(\frac{k}{m})}^{1 / 2}$

$r = (\frac{n ħ}{k}) {(\frac{k}{m})}^{1 / 2}$

$E_{n} = n ħ {(\frac{k}{m})}^{1 / 2}$

$E_{n} = n ħ (\frac{k}{m})$

answer is A, C.

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Detailed Solution

$Centripetal force F = - \frac{d U}{d r} = kr = \frac{{mv}^{2}}{r}$

Angular momentum mvr = n $ħ$

solve above equations for v and r
$E_{n} = K E_{n} + P E_{n} = \frac{1}{2} m v^{2} + \frac{k r^{2}}{2}$

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