Q 12-12-003NEETNEET 2025Top questionMedium
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 $r$ of the $n^{\text{th}}$ orbit and the particle's speed $v$ in the orbit depend on $n$ as
Answer: (C) $r \propto n^{2/3};\ v \propto n^{1/3}$
Centripetal force: $\dfrac{mv^2}{r} = F$, so $v^2 \propto r$.
Bohr quantization: $mvr = \dfrac{nh}{2\pi}$, so $v \propto \dfrac{n}{r}$.
Combining: $\dfrac{n^2}{r^2} \propto r \;\Rightarrow\; r^3 \propto n^2 \;\Rightarrow\; r \propto n^{2/3}$.
Then $v \propto \dfrac{n}{r} \propto n^{1/3}$.
Solution by Sreeraj P, M.Sc Physics