A particle of mass m moves around the origin under constant force F towards origin. Using Bohr model, how do radius r of nth orbit and speed v depend on n?

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NEET Physics QUESTION #1031

Question 1

A particle of mass \(m\) moves around the origin under constant force \(F\) towards origin. Using Bohr model, how do radius \(r\) of \(n\)th orbit and speed \(v\) depend on \(n\)?

\(r \propto n^{2/3};\ v \propto n^{1/3}\)✔️
\(r \propto n^{4/3};\ v \propto n^{-1/3}\)
\(r \propto n^{1/3};\ v \propto n^{1/3}\)
\(r \propto n^{1/3};\ v \propto n^{2/3}\)

Correct Answer Explanation

For circular orbit under constant force: \(F = \frac{mv^2}{r}\Rightarrow v^2 = \frac{Fr}{m}\). Bohr quantization: \(mvr = \frac{nh}{2\pi}\Rightarrow v = \frac{nh}{2\pi mr}\). Substituting: \(\frac{F r}{m} = \frac{n^2h^2}{4\pi^2m^2r^2}\Rightarrow r^3 \propto n^2\Rightarrow r \propto n^{2/3}\). Then \(v \propto \frac{n}{r} \propto \frac{n}{n^{2/3}} = n^{1/3}\).

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