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 nth orbit and the particle’s speed ν in the orbit depend on n as

The correct option is:
r ∝ n^2/3; ν ∝ n^1/3

Let us derive the dependence of the radius $r$ of the orbit and the particle's speed $v$ in the orbit on the principal quantum number $n$ using Bohr's quantization condition and the equation of motion for a particle in circular orbit.

First, the particle of mass $m$ is moving in a circle of radius $r$ under the action of a constant force $F$ directed towards the origin. This force provides the necessary centripetal force for circular motion:
$F=\frac{m{v}^2}{r}$
From this, we can express the square of the velocity as:
$v^2=\frac{Fr}{m}$
which gives:
$v=\sqrt{\frac{Fr}{m}}$
Thus, the speed is related to the radius by:
$v\propto r^{1/2}$

Second, we apply Bohr's angular momentum quantization rule:
$mvr=\frac{nh}{2\pi }$
where $n$ is the principal quantum number and $h$ is Planck's constant.
This quantization condition implies:
$vr\propto n$

Now, substitute the relation $v\propto r^{1/2}$ into the quantization relation:
$\left(r^{1/2}\right)·r\propto n$
$r^{3/2}\propto n$
By raising both sides to the power of 2/3, we obtain:
$r\propto n^{2/3}$

To find the dependence of the speed $v$ on $n$, we substitute the radius dependence back into the relation $v\propto r^{1/2}$:
$v\propto {\left(n^{2/3}\right)}^{1/2}$
$v\propto n^{1/3}$

Therefore, the relationships are:
$r\propto n^{2/3}$ and $v\propto n^{1/3}$