When a body is accelerated, either the magnitude or the direction of its velocity changes. If you tie a ball to a string and then spin it around, a constraining force must act on the ball, forcing it into a circular path. A formula for calculating this so-called centripetal force will be derived below.
The position of the sphere can be expressed by the position vector $\vec{r}$.
\begin{equation}
\vec{r} =
\begin{pmatrix}
\cos\alpha\\
\sin\alpha\\
\end{pmatrix} =
\begin{pmatrix}
\cos\omega t\\
\sin\omega t\\
\end{pmatrix}
\end{equation}
Here, the angle $\alpha$ has been replaced by the product $\omega t$.
If one differentiates $\vec{r}$ component-wise with respect to time, then, using the chain rule for the orbital velocity, one obtains
\begin{equation}
\vec{v} = \omega
\begin{pmatrix}
-\sin\omega t\\
\cos\omega t\\
\end{pmatrix}
\end{equation}
The scalar product of $\vec{v}$ and $\vec{r}$ results in the value 0 at any time $t$, i.e., the velocity vector is always perpendicular to the distance vector in a circular motion. Differentiating again with respect to time then yields the acceleration:
\begin{equation}
\vec{a} = -\omega^2
\begin{pmatrix}
\cos\omega t\\
\sin\omega t\\
\end{pmatrix}
= -\omega^2\vec{r}
\end{equation}
As can be seen, this is directed antiparallel to the position vector and always points toward the center of the circle. This is referred to as a radial acceleration. If one considers only the magnitude of $\vec{a}$ and multiplies it by the mass $m$, one obtains the following formula for the centripetal force:
\begin{equation}
{F_\mathrm{Z} = m\omega^2r}
\end{equation}
By replacing $\omega$ with $v/r$, one obtains an alternative representation that is also frequently found in the literature. In this case, the following relationship between the centripetal force and the orbital velocity results:
\begin{equation}
{F_\mathrm{Z} = \frac{mv^2}{r}}
\end{equation}
Which of the two formulas is used depends on the specific case at hand.