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(C) The centripetal force is provided by the constant force $F$ directed towards the origin:$\frac{m v^2}{r} = F \quad ....(1)$According to Bohr's qua

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$r \propto n^{2/3} ; v \propto n^{1/3}$

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(C) The centripetal force is provided by the constant force $F$ directed towards the origin:$\frac{m v^2}{r} = F \quad ....(1)$According to Bohr's quantization condition for angular momentum:$mvr = \frac{nh}{2\pi} \Rightarrow v = \frac{nh}{2\pi mr} \quad ....(2)$Substitute $v$ from equation $(2)$ into equation $(1)$:$\frac{m}{r} \left( \frac{nh}{2\pi mr} \right)^2 = F$$\frac{m}{r} \cdot \frac{n^2 h^2}{4\pi^2 m^2 r^2} = F$$\frac{n^2 h^2}{4\pi^2 m r^3} = F$Rearranging for $r^3$:$r^3 = \frac{n^2 h^2}{4\pi^2 m F}$Since $h, m, F$ are constants,$r^3 \propto n^2$,which implies $r \propto n^{2/3}$.Now,substitute $r \propto n^{2/3}$ into the expression for $v$ from equation $(2)$:$v \propto \frac{n}{r} \propto \frac{n}{n^{2/3}} \propto n^{1 - 2/3} \propto n^{1/3}$.Thus,$r \propto n^{2/3}$ and $v \propto n^{1/3}$.

$A$ 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

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