Question 530: Atoms & Nuclei

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 of the $n^{	ext {th }}$ orbit and the particle's speed $v$ in the orbit depend on $n$ as

Accepted Answer

When a particle of mass $m$ is subject to a constant force $F$ pulling it toward the origin, we can apply the Bohr model to describe its motion.
Starting with the relationship for centripetal force:
$ F = \frac{m v^2}{r} $
We can rearrange this to show:
$ \frac{v^2}{r} = 	ext{constant} $
From this, it follows that:
$ r \propto v^2 \quad 	ext{...(1)} $
In the Bohr model, the angular momentum $L$ is quantized and given by:
$ L = m v r = \frac{n h}{2 \pi} \quad 	ext{...(2)} $
Solving equations (1) and (2) together, we find:
$ v \propto n^{1/3} $
Substituting this into equation (1), we get:
$ r \propto n^{2/3} $
Thus, the radius of the $n^{	ext{th}}$ orbit and the speed of the particle depend on $n$ as follows:
$r \propto n^{2/3}$
$v \propto n^{1/3}$

Answer

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

Answer

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

Answer

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

Answer

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

Atoms & Nuclei

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 of the n^{\text {th }} orbit and the particle's speed v in the orbit depend on n as

Choose the correct answer: