Question: Consider an arbitrary power-law central force (mathbf{F}(mathbf{r})=-k r^{n} hat{mathbf{r}}), where (k) and (n) are constants and (r) is the radius in spherical coordinates. Prove that

Consider an arbitrary power-law central force $\mathbf{F}(\mathbf{r})=-k r^{n} \hat{\mathbf{r}}$, where $k$ and $n$ are constants and $r$ is the radius in spherical coordinates. Prove that such a force is conservative, and find the associated potential energy of a particle subject to this force.

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