A particle of mass $m$ is moving around the origin with a constant force $F$ pulling it towards the

Question

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 $n^{	ext {th }}$ orbit and the particle's speed $v$ in the orbit depend on $n$ as

Professor PC Thomas Classes · Answer

(a) : As force $(F)$ is constant, so $F=\frac{m v^2}{r}$ \ We can write, $F=\frac{m^2 v^2 r^2}{m r^3}=\frac{L^2}{m r^3} \quad$ (As, $L=m v r$ ) \ As force is constant, so $L^2 \propto m r^3$ \ sing, $L=\frac{n h}{2 \pi}$, we get $\frac{n^2 h^2}{4 \pi^2}=m r^3 \Rightarrow n^2 \propto r^3 \quad$ or $\quad r \propto n^{2 / 3}$ \ Also, $m v r=\frac{n h}{2 \pi}$ As $r \propto n^{2 / 3}$, so $m v\left(n^{2 / 3}ight) \propto \frac{n h}{2 \pi}$ or $\quad v \propto n^{1 / 3}$