How to calculate the centripetal force along a cartesian axis?

Question

If a centrifuge without a ballast is centered on the origin of an $x$-$y$ coordinate plane, and starts at the $x$-axis rotating with increasing velocity counter-clockwise around the origin, how can the centripetal force of this accelerating centrifuge be calculated in the $x$-direction, and in the $y$-direction separately? This is to mathematically understand the "knocking" that is happening in an unbalanced centrifuge.

My attempt: Look up a polar function for a spiral as the centripital force is radial, then convert to cartesian coordinates and integrate dy or dx. Attempted this but ran into the problem that for a spiral there are multiple arms of the spiral for each x or y coordinate.

Answer 1

If a centrifuge is knocking it means the center of mass is not aligned with the center of rotation. We can assume the COM is rotating in a perfect circle around the rotation center. In order to rotate in a circle, the COM needs to accelerate with $\vec a_c=-\frac{v^2}r\hat r$, where $\vec r$ is the displacement of the COM with respect to the rotation center and $\hat r$ is the unit vector of $\vec r$. Since the centrifuge ring is accelerating, it must be applying a force equal to $\vec F=-m\vec a_c$ to the apparatus, which is quickly rotating because $\hat r$ is also rotating.

Answer 2

the centrifugal force is:

$$m\,\omega^2\,r\,\hat{\mathbf{r}}$$

using polar coordinate you obtain the components in $x~,y~$ coordinate system

$$F_x= m\,\omega^2\,r\,\cos(\phi)\\ F_y=m\,\omega^2\,r\,\sin(\phi)$$

with $~\phi=-\omega\,t~$

$$F_x= m\,\omega^2\,r\,\cos(\omega\,t)\\ F_y=-m\,\omega^2\,r\,\sin(\omega\,t)$$