Q. 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 $n^{th}$ orbit and the particle's speed $v$ in the orbit depend on $n$ as
A) $r \propto n^{1/3}; v \propto n^{1/3}$
B) $r \propto n^{1/3}; v \propto n^{2/3}$
C) $r \propto n^{2/3}; v \propto n^{1/3}$
D) $r \propto n^{4/3}; v \propto n^{-1/3}$
Explanation
[Solution by Claude - not from source] $mvr=\frac{nh}{2\pi}$ and $F=\frac{mv^2}{r}$. Then $v=\frac{nh}{2\pi mr}$ gives $F=\frac{n^2h^2}{4\pi^2mr^3}\Rightarrow r\propto n^{2/3}$, and $v\propto n/r\propto n^{1/3}$. Answer (3).
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.