Radial Acceleration Proportionality

Question

Radial acceleration $a_c$ is given by $$\frac{v^2}{r}$$, which clearly demonstrates the inverse proportionality of acceleration and radius. It seems to me, however, that the equation: $$a_c=4\pi^2rf^2$$ demonstrates the exact opposite. Am I missing something?

Answer 1

The first equation says that the acceleration is inversely proportional to the radius, if the velocity is constant. In the second equation, the frequency is inversely proportional to the radius, if the velocity is constant.$$f=\frac{v}{2\pi r}$$ So both equations say the same thing, if the velocity is independent of radius. If the frequency is constant, you will see that in both equations the acceleration is proportional to the radius.