A particle of mass m moves in circular orbits with potential energy 𝑉(𝑟) = 𝐹𝑟, where F is a positive constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle’s orbit is denoted by R and its speed and energy are denoted by v and E, respectively, then for the $$n^{th}$$ orbit (here h is the Planck’s constant)

The constant attractive force acting on the particle is obtained from the given potential energy $$V(r)=Fr$$ as
$$F_{\text{radial}}=-\frac{dV}{dr}=-F$$ (directed toward the origin, magnitude $$F$$).

For a stable circular orbit of radius $$R$$ and speed $$v$$, the necessary centripetal force is provided by this constant force:

$$\frac{mv^{2}}{R}=F \qquad -(1)$$

The Bohr quantisation condition for angular momentum is

$$m v R=\frac{n h}{2\pi} \qquad -(2)$$

Step 1 ‑ Express $$v$$ from the Bohr condition:
$$v=\frac{n h}{2\pi m R} \qquad -(3)$$

Step 2 ‑ Substitute $$v$$ from $$(3)$$ into the force balance $$(1)$$ :

$$\frac{m}{R}\left(\frac{n h}{2\pi m R}\right)^{2}=F$$
$$\Longrightarrow\; \frac{n^{2}h^{2}}{4\pi^{2}m R^{3}}=F$$

Solving for the orbital radius $$R$$ gives

$$R^{3}=\frac{n^{2}h^{2}}{4\pi^{2}mF}$$
$$\boxed{\,R=\left(\frac{n^{2}h^{2}}{4\pi^{2}mF}\right)^{1/3}\,}$$

Hence $$R\propto n^{2/3}$$.

Step 3 ‑ Find the speed using $$(1)$$ or $$(3)$$. Using $$(1)$$:

$$v^{2}=\frac{F R}{m}$$
$$\Rightarrow v=\sqrt{\frac{F R}{m}}$$

Insert the expression for $$R$$:

$$v=\sqrt{\frac{F}{m}}\;\left(\frac{n^{2}h^{2}}{4\pi^{2}mF}\right)^{1/6}$$
$$\boxed{\,v\propto n^{1/3}\,}$$

Step 4 ‑ Total mechanical energy $$E$$ is the sum of kinetic and potential energies.

• Kinetic energy: $$K=\tfrac12 m v^{2}=\tfrac12 F R$$ (using $$(1)$$).
• Potential energy: $$U=F R$$ (given).

Therefore

$$E=K+U=\tfrac12 FR+FR=\tfrac32 FR$$

Substitute $$R=\left(\dfrac{n^{2}h^{2}}{4\pi^{2}mF}\right)^{1/3}$$:

$$E=\frac{3}{2}\,F\left(\frac{n^{2}h^{2}}{4\pi^{2}mF}\right)^{1/3}$$
$$\boxed{\,E=\frac{3}{2}\left(\frac{n^{2}h^{2}F^{2}}{4\pi^{2}m}\right)^{1/3}\,}$$

Verification of the options:

Option A: predicts $$R\propto n^{1/3}$$ — incorrect.
Option B: $$R\propto n^{2/3}$$ and $$V\propto n^{1/3}$$ (the symbol $$V$$ in the statement represents speed) — correct.
Option C: gives the exact energy expression we derived — correct.
Option D: differs in the numerical coefficient (2 instead of $$\tfrac32$$) — incorrect.

Hence the correct statements are:
Option B (radius and speed dependence) and Option C (energy expression).