Bonus problem 1

This bonus question is worth up to 0.5 bonus points at the end of the term.

Suppose that we have a central attracting force in $\mathbb{R}^3$ of the form

$$\vec{F} = -\frac{C}{r^k}\hat{r}$$

where $C$ is a positive constant, $k$ is a positive integer and as usual,  $\vec{r}=\langle x,y,z\rangle$ , $r = |\vec{r}|$ and $\hat{r}=\frac{\vec{r}}{r}$.

Find the value of $k$ so that there exists a trajectory of a particle in motion acted on by the force given by a circle of some radius $R$ whose center is a distance  $R$  from the origin. In other words, this circle passes through the origin.

To get full credit for this bonus question, the write up must include both the correct answer as well as clear and coherent reasoning as to how you obtained the answer.