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) $r \propto n^{2/3};\ v \propto n^{1/3}$
Concept. Combine the force condition for a circular orbit with Bohr's quantisation $mvr = \dfrac{nh}{2\pi}$.
Step 1. Force condition. The constant force $F$ is the centripetal force: $\dfrac{mv^2}{r} = F$, so $v^2 \propto r$.
Step 2. Quantisation. $mvr = \dfrac{nh}{2\pi}$, so $v \propto \dfrac{n}{r}$.
Step 3. Eliminate $v$.
$$\frac{n^2}{r^2} \propto r \;\Rightarrow\; r^3 \propto n^2 \;\Rightarrow\; r \propto n^{2/3}$$
Step 4. Speed. $v \propto \dfrac{n}{r} \propto \dfrac{n}{n^{2/3}} = n^{1/3}$.
Answer. $r \propto n^{2/3}$, $v \propto n^{1/3}$ (Option C).
Solution by Sreeraj P, M.Sc Physics