A particle of mass m is moving around the… - NEET UG 2025 Physics PYQ

NEET UG 2025 — Physics Modern Physics

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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

Official previous-year question

Held on 30 Apr 2025 · Verified 9 Jul 2026.

Options

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
$r \propto n^{4 / 3} ; v \propto n^{-1 / 3}$

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Solution

$\begin{aligned} & \frac{m v^2}{\mathrm{r}}=\mathrm{F}....(1) \\ & \mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi} \Rightarrow v=\frac{\mathrm{nh}}{2 \pi \mathrm{mr}}....(2) \end{aligned}$ From equation (1) $\begin{aligned} & \frac{\mathrm{m}}{\mathrm{r}}\left(\frac{\mathrm{nh}}{2 \pi \mathrm{mr}}\right)^2=\mathrm{F} \\ & \Rightarrow \frac{\mathrm{n}^2 \mathrm{~h}^2}{4 \pi^2 \mathrm{mF}}=\mathrm{r}^3 \Rightarrow \mathrm{r}^3 \propto \mathrm{n}^2 \\ & \Rightarrow \mathrm{r} \propto \mathrm{n}^{2 / 3} \end{aligned}$ From (2) $v \propto \frac{n}{r} \propto \frac{n}{n^{2 / 3}} \Rightarrow v \propto n^{1 / 3}$

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