NEET 2025 — Physics solution

NEET2025Physics

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

Solution

Constant centripetal force: $\frac{m v ^{2}}{r} = F$ , so $v^{2} = \frac{F r}{m}$ . ...(1)

Bohr quantisation: $m v r = \frac{nh}{2 π}$ , so $v = \frac{nh}{2 π m r}$ . ...(2)

Square (2) and substitute in (1): $\frac{n ^{2} h ^{2}}{4 π ^{2} m ^{2} r ^{2}} = \frac{F r}{m} \Rightarrow r^{3} \propto n^{2} \Rightarrow r \propto n^{2/3}$ .

From (2): $v \propto \frac{n}{r} \propto \frac{n}{n ^{2/3}} = n^{1/3}$ .

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