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

We model the particle’s motion using the Bohr‑type quantisation while the particle experiences a constant attractive force F toward the origin.

1. Centripetal force condition
The inward force supplies the required centripetal force for uniform circular motion:

$F=\frac{m}{v}$

or equivalently

$m{v}^2=Fr$

2. Bohr angular‑momentum quantisation
The angular momentum of the particle is quantised in integer multiples of ħ:

$mvr=nħ$

From this we can express the speed:

$v=\frac{n}{ħ}\frac{1}{m}$

3. Combine the two relations
Insert the expression for v into the centripetal condition:

$m{v}^2=Fr$

Substituting v = nħ/(m r):

$m{\frac{n}{ħ}}^2\frac{}{}\{m{r}^2\}=Fr$

Simplify:

$\frac{n^2}{ħ}=Fr$

Multiply both sides by r²:

$n^2ħ=Fm{r}^3$

Thus

$r^3\propto n^2$

Taking the cube root gives the radius dependence:

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

4. Speed dependence
Use the quantisation relation v = nħ/(m r) and substitute the derived r ∝ n^(2/3):

$v\propto \frac{n}{n^{\frac{2}{3}}}$

Since n / n^(2/3) = n^(1/3), we obtain:

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

Result
The Bohr‑type r ∝ n^(2/3); v ∝ n^(1/3)

We model the particle’s motion using the Bohr‑type quantisation while the particle experiences a constant attractive force F toward the origin.

1. Centripetal force condition
The inward force supplies the required centripetal force for uniform circular motion:

$F=\frac{m}{v}$

or equivalently

$m{v}^2=Fr$

2. Bohr angular‑momentum quantisation
The angular momentum of the particle is quantised in integer multiples of ħ:

$mvr=nħ$

From this we can express the speed:

$v=\frac{n}{ħ}\frac{1}{m}$

3. Combine the two relations
Insert the expression for v into the centripetal condition:

$m{v}^2=Fr$

Substituting v = nħ/(m r):

$m{\frac{n}{ħ}}^2\frac{}{}\{m{r}^2\}=Fr$

Simplify:

$\frac{n^2}{ħ}=Fr$

Multiply both sides by r²:

$n^2ħ=Fm{r}^3$

Thus

$r^3\propto n^2$

Taking the cube root gives the radius dependence:

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

4. Speed dependence
Use the quantisation relation v = nħ/(m r) and substitute the derived r ∝ n^(2/3):

$v\propto \frac{n}{n^{\frac{2}{3}}}$

Since n / n^(2/3) = n^(1/3), we obtain:

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

Result
The Bohr‑type analysis for a particle under a constant central force yields the scaling laws

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

which matches the given correct option.