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Q.18) 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
A) $r \propto n^{4 / 3} ; \quad v \propto n^{-1 / 3}$
B) $r \propto n^{1 / 3} ; \quad v \propto n^{1 / 3}$
C) $r \propto n^{1 / 3} ; \quad v \propto n^{2 / 3}$
D) $r \propto n^{2 / 3} ; \quad v \propto n^{1 / 3}$
Answers (1)
Solution:
From the centripetal force $F=\frac{m v^2}{r}$, we get $v^2 \propto r \Rightarrow v \propto \sqrt{r}$.
Using Bohr's quantization $m v r=n \hbar \Rightarrow v \propto \frac{n}{r}$.
Equating both gives $\frac{n}{r} \propto \sqrt{r} \Rightarrow n \propto r^{3 / 2} \Rightarrow r \propto n^{2 / 3}$.
Substituting back, $v \propto \frac{n}{r}=\frac{n}{n^{2 / 3}}=n^{1 / 3}$.
Hence, the answer is option (3).
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