A particle of mass is moving around the origin with a constant force pulling it towards the origin. If Bohr model is used to describe its motion, the radius of the orbit and the particle’s speed in the orbit depend on as [2025]

Question

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 $n^{th}$ orbit and the particle’s speed $v$ in the orbit depend on $n$ as [2025]

Smart Achievers · Accepted Answer

(1)

As force (F) is constant, so

$F = \frac{m v^{2}}{r}$

We can write, $F = \frac{m^{2} v^{2} r^{2}}{m r^{3}} = \frac{L^{2}}{m r^{3}} (As, L = m v r)$

As force is constant, so $L^{2} \propto m r^{3}$

Using, $L = \frac{n h}{2 π},$ we get $\frac{n^{2} h^{2}}{4 π^{2}} = m r^{3} \Rightarrow n^{2} \propto r^{3} or r \propto n^{2 / 3}$

Also, $m v r = \frac{n h}{2 π}$

As $r \propto n^{2 / 3},$ so $m v (n^{2 / 3}) \propto \frac{n h}{2 π}$

or $v \propto n^{1 / 3}$

Solution

Q.
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 $n^{th}$ orbit and the particle’s speed $v$ in the orbit depend on $n$ as [2025]

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

2 $r \propto n^{4 / 3}; v \propto n^{- 1 / 3}$

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

4 $r \propto n^{1 / 3}; v \propto n^{2 / 3}$

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Solution

Q. 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 nth orbit and the particle’s speed v in the orbit depend on n as [2025]

1 r∝n2/3; v∝n1/3

2 r∝n4/3; v∝n-1/3

3 r∝n1/3; v∝n1/3

4 r∝n1/3; v∝n2/3