A particle of mass m moves in a circular orbit in a central potential field

Question:

A particle of mass $\mathrm{m}$ moves in a circular orbit in a central potential field

$\mathrm{U}(\mathrm{r})=\mathrm{U}_{0} r^{4}$. If Bohr's quantization conditions are applied, radii of

possible orbitals $\mathrm{r}_{\mathrm{n}}$ vary with $\mathrm{n}^{1 / \alpha}$, where $\alpha$ is

Solution:

(3)

$\mathrm{F}=\frac{-\mathrm{dU}}{\mathrm{dr}}=-4 \mathrm{U}_{0} \mathrm{r}^{3}=\frac{\mathrm{mv}^{2}}{\mathrm{r}}$

$\mathrm{mv}^{2}=4 \mathrm{U}_{0} \mathrm{r}^{4}$

$\mathrm{V} \propto \mathrm{r}^{2}$

$\mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi}$

$\mathrm{r}^{3} \propto \mathrm{n}$

$\mathrm{r} \propto \mathrm{n} 1 / 3$

$=3$

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