A car at a constant 20 m/s through a curve is accelerating, and the speedometer has nothing to say about it. Acceleration is a change in velocity, and velocity carries a direction. Bend the direction and you have accelerated, whatever the needle reads. That acceleration points at the centre of the curve, which is why it is called centripetal — Latin for centre-seeking — and why its symbol carries the subscript c: $a_c$.

Two forms, one fact. From a path speed: $a_c = \dfrac{v^{2}}{r}$, where $v$ is the speed along the path in $\mathrm{m/s}$ and $r$ is the radius of the curve in metres; the answer is in $\mathrm{m/s^2}$. From a spin rate: $a_c = \omega^{2} r$, with $\omega$ in $\mathrm{rad/s}$. They are the same statement — substitute $v = \omega r$ into the first and the second falls out. Notice the radius changes sides: with a path speed, a wider curve is gentler; at a fixed rpm, a wider rotor is fiercer.

Give the moving thing a mass $m$ in kilograms and Newton prices the turn: $F_c = \dfrac{m v^{2}}{r}$, in newtons. That force is not a new kind of push — it is the NAME for whatever is already supplying it. On a level road it is tyre friction; on a rotor it is the steel of the arm; on a string it is tension. And the $v^{2}$ is the warning in the whole lesson: take a bend 40% faster and you have doubled the demand on grip you do not control.