A particle of mass m, and angular momentum ℓ is moving in a circular orbit of radius r₀ under the influence of an attractive force $\overrightarrow{F}\left(r\right)=-\frac{k}{r^2}\hat{r}$. Keeping its angular momentum unchanged, the particle is displaced radially by a small distance δr ≪ r₀, due to which its radial distance varies periodically. The corresponding time period is:

The correct answer is $\frac{2\pi {\ell }^3}{m{k}^2}$.

Step 1: Effective Potential and Orbital Equilibrium
For a particle of mass m moving under a central force field $\overrightarrow{F}\left(r\right)=-\frac{k}{r^2}\hat{r}$, the corresponding potential energy is given by:

$V\left(r\right)=-\frac{k}{r}$

The effective potential energy $V_{\text{eff}}\left(r\right)$, incorporating the centrifugal potential term for a given conserved angular momentum ℓ, is:

$V_{\text{eff}}\left(r\right)=\frac{{\ell }^2}{2m{r}^2}-\frac{k}{r}$

For a stable circular orbit of radius r₀, the effective force must be zero at r = r₀:

${\left(\frac{d{V}_{\text{eff}}}{dr}\right)}_{r=r_0}=-\frac{{\ell }^2}{m{r}_0^3}+\frac{k}{r_0^2}=0$

Solving for r₀ gives the radius of the circular orbit:

$r_0=\frac{{\ell }^2}{mk}$

Step 2: Radial Oscillations for Small Perturbations
When the particle is given a small radial displacement δr ≪ r₀ while keeping its angular momentum ℓ fixed, it undergoes small oscillations about the minimum of the effective potential well.
The effective spring constant k_eff of these radial oscillations is determined by the second derivative of the effective potential evaluated at r = r₀:

$k_{\text{eff}}={\left(\frac{d^2{V}_{\text{eff}}}{d{r}^2}\right)}_{r=r_0}$

Differentiating $V_{\text{eff}}\left(r\right)$ a second time with respect to r:

$\frac{d^2{V}_{\text{eff}}}{d{r}^2}=\frac{3{\ell }^2}{m{r}^4}-\frac{2k}{r^3}$

Substituting $r_0=\frac{{\ell }^2}{mk}$ into the expression:

$k_{\text{eff}}=\frac{3{\ell }^2}{m{\left(\frac{{\ell }^2}{mk}\right)}^4}-\frac{2k}{{\left(\frac{{\ell }^2}{mk}\right)}^3}=\frac{3m^3{k}^4}{{\ell }^6}-\frac{2m^3{k}^4}{{\ell }^6}=\frac{m^3{k}^4}{{\ell }^6}$

Step 3: Calculating the Time Period
The angular frequency ω of the radial oscillations is:

$\omega =\sqrt{\frac{k_{\text{eff}}}{m}}=\sqrt{\frac{m^2{k}^4}{{\ell }^6}}=\frac{m{k}^2}{{\ell }^3}$

The time period T of the periodic variation in radial distance is given by:

$T=\frac{2\pi }{\omega }=\frac{2\pi {\ell }^3}{m{k}^2}$