## College Physics (4th Edition)

If $r$ is given, then $a_c$ is fixed by the gravitational force, and then $v = \sqrt{a_c~r}$. Therefore, the speed $v$ is not independent of $r$.
At a given radius, the gravitational force on the object is fixed, which means that the centripetal acceleration $a_c$ toward the center of the circle is also fixed. $a_c = \frac{v^2}{r}$ If $r$ is given, then $a_c$ is fixed by the gravitational force, and then $v = \sqrt{a_c~r}$. Therefore, the speed $v$ is not independent of $r$.