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A particle of mass m is moving around the origin with a constant force F pulling it towards the origin. If Bohr model is…

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A particle of mass m is moving around the origin with a constant force F pulling it towards the origin. If Bohr model is used to describe its motion, the radius of the \({\mathrm{n}}^{\text{th }}\) orbit and the particle's speed v in the orbit depend on n as

[NEET 2025]

a

\(\mathrm{r}\propto {\mathrm{n}}^{2/3};\mathrm{v}\propto {\mathrm{n}}^{1/3}\)

b

\(\mathrm{r}\propto {\mathrm{n}}^{4/3};\mathrm{v}\propto {\mathrm{n}}^{-1/3}\)

c

\(\mathrm{r}\propto {\mathrm{n}}^{1/3};\mathrm{v}\propto {\mathrm{n}}^{1/3}\)

d

\(\mathrm{r}\propto {\mathrm{n}}^{1/3};\mathrm{v}\propto {\mathrm{n}}^{2/3}\)

✓ Correct answer: a)

\(\mathrm{r}\propto {\mathrm{n}}^{2/3};\mathrm{v}\propto {\mathrm{n}}^{1/3}\)

Explanation

Given, force is constant

\(F=\frac{m{v}^{2}}{r}\)

\(\Rightarrow \frac{{v}^{2}}{r}=\text{constant}\)

\(\Rightarrow r\propto {v}^{2}\ldots \left(1\right)\)

\(L=mvr=\frac{nh}{2\pi }\) ...(2)

⇒ on solving equation (1) and equation (2)
\(v\propto {n}^{1/3}\) and \(r\propto {n}^{2/3}\)

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