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(A) Given potential energy $U = \frac{1}{2}kr^2$.The force is $F = -\frac{dU}{dr} = -kr$.For circular motion,the centripetal force is provided by the

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$r_n \propto \sqrt{n}, E_n \propto n$

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(A) Given potential energy $U = \frac{1}{2}kr^2$.The force is $F = -\frac{dU}{dr} = -kr$.For circular motion,the centripetal force is provided by the potential field:$\frac{mv^2}{r} = kr \implies mv^2 = kr^2$ .... $(i)$According to Bohr's quantization condition:$mvr = \frac{nh}{2\pi} \implies v = \frac{nh}{2\pi mr}$ .... $(ii)$Substituting $(ii)$ into $(i)$:$m(\frac{nh}{2\pi mr})^2 = kr^2$$\frac{n^2h^2}{4\pi^2mr^2} = kr^2 \implies r^4 = \frac{n^2h^2}{4\pi^2mk} \implies r^2 \propto n \implies r_n \propto \sqrt{n}$.Total energy $E = K + U = \frac{1}{2}mv^2 + \frac{1}{2}kr^2$.From $(i)$,$\frac{1}{2}mv^2 = \frac{1}{2}kr^2$,so $E = \frac{1}{2}kr^2 + \frac{1}{2}kr^2 = kr^2$.Since $r^2 \propto n$,we have $E_n \propto n$.

$A$ particle of mass $m$ moves in a circular orbit in a central potential field $U(r) = \frac{1}{2}kr^2$. If Bohr's quantization conditions are applied,radii of possible orbits and energy levels vary with quantum number $n$ as

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