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  • A small particle of mass m moves in such a way that the potential energy of particle is given as <img alt="\mathrm{U}=-\frac{1}{2} \mathrm{~m} \alpha^2 \mathrm{r}^2" src="/latex-image/?%5Cmathrm%7B

A small particle of mass m moves in such a way that the potential energy of particle is given as \mathrm{U}=-\frac{1}{2} \mathrm{~m} \alpha^2 \mathrm{r}^2 where \alpha is constant and \mathrm{r} is the distance of particle from centre. If Bohr's model of quantization of angular momentum and circular orbit is valid for the particle. ( \mathrm{h}= Planck's constant). The radius of \mathrm{n}^{\text {th }} orbit of the particle is

Option: 1

\left(\frac{\mathrm{nh}}{4 \pi \mathrm{m} \alpha}\right)^{1 / 2}


Option: 2

\left(\frac{\mathrm{nh}}{2 \pi \mathrm{m} \alpha}\right)^{1 / 2}


Option: 3

\left(\frac{\mathrm{n}^2 \mathrm{~h}}{4 \pi \mathrm{m} \alpha}\right)^{1 / 2}


Option: 4

\left(\frac{\mathrm{nh}}{2 \pi \mathrm{m} \alpha}\right)^{1 / 3}


Answers (1)

best_answer

\mathrm{F}=-\frac{\mathrm{dU}}{\mathrm{dr}}=\frac{\mathrm{mV}_{\mathrm{n}}^2}{\mathrm{r}_{\mathrm{n}}}

                 \mathrm{m \alpha}^2 \mathrm{r}_{\mathrm{n}}=\frac{\mathrm{mV} \mathrm{V}_{\mathrm{n}}^2}{\mathrm{r}_{\mathrm{n}}}                     (1)

 and         \mathrm{mv}_{\mathrm{n}} \mathrm{r}_{\mathrm{n}}=\frac{\mathrm{nh}}{2 \pi}                                (2)

Solving equations (1) and (2)

\mathrm{r}_{\mathrm{n}}=\left(\frac{\mathrm{nh}}{2 \pi \mathrm{m} \alpha}\right)^{1 / 2}

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Shailly goel

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