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 $\frac{h}{2 π}$ holds.

The value of $n^{t h}$ orbital radius is

${(\frac{n^{2} h^{2}}{π^{2} m D})}^{\frac{1}{2}}$

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${(\frac{n^{2} h^{2}}{π^{2} m D})}^{\frac{1}{4}}$

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${(\frac{n^{2} h^{2}}{2 π^{2} m D})}^{\frac{1}{4}}$

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${(\frac{n^{2} h^{2}}{4 π^{2} m D})}^{\frac{1}{4}}$

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Solution

The correct option is D
${(\frac{n^{2} h^{2}}{4 π^{2} m D})}^{\frac{1}{4}}$

$\frac{m {v_{n}}^{2}}{r_{n}} = D r_{n} m v_{n} r n = \frac{n h}{2 π} \Rightarrow v_{n} r n \times \frac{r n}{v_{n}} = \frac{n h}{2 π m} \times \sqrt{\frac{m}{D}} \Rightarrow r_{n} = {(\frac{n^{2} h^{2}}{4 π^{2} m D})}^{\frac{1}{4}}$

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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 $\frac{h}{2 π}$ holds.

The value of energy in the $n^{t h}$ orbit (Let the potential energy be zero when r = 0)

List - I	List - II
A) Energy	1) $\frac{2 π k z e^{2}}{n h}$
B) Velocity	2) $\frac{- 2 π^{2} m k^{2} z^{2} e^{4}}{n^{2} h^{2}}$
C) Rydberg constant	3) $\frac{2 π^{2} m k^{2} e^{4}}{h^{2}}$
D) Radius	4) $\frac{n^{2} h^{2}}{4 π^{2} m k z e^{2}}$
	5) $\frac{- 4 π^{2} m k^{2} z^{2} e^{4}}{n^{2} h^{2}}$