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} = \text{constant} $

From this, it follows that:

$ r \propto v^2 \quad \text{...(1)} $

In the Bohr model, the angular momentum $L$ is quantized and given by:

$ L = m v r = \frac{n h}{2 \pi} \quad \text{...(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 $n^{\text{th}}$ orbit and the speed of the particle depend on $n$ as follows:

$r \propto n^{2/3}$

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