Shogin that F=-grad(U) in a centrally cymmetric force field

[Uku]

Homework Statement

I need to show that
$\vec{F}=(\vec{r}-\vec{r_{0}})f(\left\|\vec{r}-\vec{r_{0}}\right\|)$
in a centrally symmetric force field.

where $\vec{r_{0}}$ is the force field center and $f$ some sort of function.

Homework Equations

Newt. II

$m\vec{\ddot{r}}=\vec{F}$ and perhaps that in a central force field
$\vec{U(\vec{r})}=\vec{U(r)}$

The Attempt at a Solution

I can write:

$m\vec{\ddot{r}}=(\vec{r}-\vec{r_{0}})f(\left\|\vec{r}-\vec{r_{0}}\right\|)$

Is this the right approach?

 

[NateD]

I assume I have to use the fact that in a central force field \vec{U(\vec{r})}=\vec{U(r)}But I am not sure how to show that this is true. Is there something else I should be doing?