Question

A particle with mass m is held in circular orbit around the origin by an attractive force F(r) =-Dr ; where D is a positive constant. Assume that the Bohr model idea that ,the angular momentum is quantized i.e. it takes only integral multiples of $\frac{h}{2\pi }$ holds.
The value of $n^{th}$ orbital radius is

• A. ${\left(\frac{n^2{h}^2}{{\pi }^{2}mD}\right)}^{\frac{1}{2}}$
• B. ${\left(\frac{n^2{h}^2}{{\pi }^{2}mD}\right)}^{\frac{1}{4}}$
• C. ${\left(\frac{n^2{h}^2}{2{\pi }^{2}mD}\right)}^{\frac{1}{4}}$
• D. ${\left(\frac{n^2{h}^2}{4{\pi }^{2}mD}\right)}^{\frac{1}{4}}$

Answer 1

The correct option is D ${\left(\frac{n^2{h}^2}{4{\pi }^{2}mD}\right)}^{\frac{1}{4}}$

$\frac{m{v}_n^2}{r_n}=D{r}_{n}m{v}_{n}rn=\frac{nh}{2\pi }\Rightarrow v_{n}rn\times \frac{rn}{Vn}=\frac{nh}{2\pi m}\times \sqrt{\frac{m}{D}}\Rightarrow r_n={\left(\frac{n^2{h}^2}{4{\pi }^{2}mD}\right)}^{\frac{1}{4}}$