Two particles of the same mass m are moving in circular orbits because of force, given by F   r = - 16…

Centripetal force is given as $F\left(r\right)=\frac{m{v}^2}{r}$.

Given in the problem, $\frac{m{V}^2}{r}=\frac{16}{r}+r^3 ...\left(i\right)$

Kinetic energy is stated as $K{E}_0=\frac{1}{2}m{v}^2$.

$K{E}_0=\frac{1}{2}\left[16+r^4\right]$ (From equation$\left(i\right)$)

The ratio of kinetic energies of first and second particle is $\frac{K_1}{K_2}=\frac{\frac{16+1}{2}}{\frac{16+256}{2}}=\frac{17}{272}$

$\frac{K_1}{K_2} \simeq 6 \times {10}^{-2}$.